Issue 
A&A
Volume 571, November 2014
Planck 2013 results



Article Number  A16  
Number of page(s)  66  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201321591  
Published online  29 October 2014 
Planck 2013 results. XVI. Cosmological parameters
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^{67}
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Received: 28 March 2013
Accepted: 24 February 2014
This paper presents the first cosmological results based on Planck measurements of the cosmic microwave background (CMB) temperature and lensingpotential power spectra. We find that the Planck spectra at high multipoles (ℓ ≳ 40) are extremely well described by the standard spatiallyflat sixparameter ΛCDM cosmology with a powerlaw spectrum of adiabatic scalar perturbations. Within the context of this cosmology, the Planck data determine the cosmological parameters to high precision: the angular size of the sound horizon at recombination, the physical densities of baryons and cold dark matter, and the scalar spectral index are estimated to be θ_{∗} = (1.04147 ± 0.00062) × 10^{2}, Ω_{b}h^{2} = 0.02205 ± 0.00028, Ω_{c}h^{2} = 0.1199 ± 0.0027, and n_{s} = 0.9603 ± 0.0073, respectively(note that in this abstract we quote 68% errors on measured parameters and 95% upper limits on other parameters). For this cosmology, we find a low value of the Hubble constant, H_{0} = (67.3 ± 1.2) km s^{1} Mpc^{1}, and a high value of the matter density parameter, Ω_{m} = 0.315 ± 0.017. These values are in tension with recent direct measurements of H_{0} and the magnituderedshift relation for Type Ia supernovae, but are in excellent agreement with geometrical constraints from baryon acoustic oscillation (BAO) surveys. Including curvature, we find that the Universe is consistent with spatial flatness to percent level precision using Planck CMB data alone. We use highresolution CMB data together with Planck to provide greater control on extragalactic foreground components in an investigation of extensions to the sixparameter ΛCDM model. We present selected results from a large grid of cosmological models, using a range of additional astrophysical data sets in addition to Planck and highresolution CMB data. None of these models are favoured over the standard sixparameter ΛCDM cosmology. The deviation of the scalar spectral index from unity isinsensitive to the addition of tensor modes and to changes in the matter content of the Universe. We find an upper limit of r_{0.002}< 0.11 on the tensortoscalar ratio. There is no evidence for additional neutrinolike relativistic particles beyond the three families of neutrinos in the standard model. Using BAO and CMB data, we find N_{eff} = 3.30 ± 0.27 for the effective number of relativistic degrees of freedom, and an upper limit of 0.23 eV for the sum of neutrino masses. Our results are in excellent agreement with big bang nucleosynthesis and the standard value of N_{eff} = 3.046. We find no evidence for dynamical dark energy; using BAO and CMB data, the dark energy equation of state parameter is constrained to be w = 1.13_{0.10}^{+0.13}. We also use the Planck data to set limits on a possible variation of the finestructure constant, dark matter annihilation and primordial magnetic fields. Despite the success of the sixparameter ΛCDM model in describing the Planck data at high multipoles, we note that this cosmology does not provide a good fit to the temperature power spectrum at low multipoles. The unusual shape of the spectrum in the multipole range 20 ≲ ℓ ≲ 40 was seen previously in the WMAP data and is a real feature of the primordial CMB anisotropies. The poor fit to the spectrum at low multipoles is not of decisive significance, but is an “anomaly” in an otherwise selfconsistent analysis of the Planck temperature data.
Key words: cosmic background radiation / cosmological parameters / early Universe / inflation / primordial nucleosynthesis
© ESO, 2014
1. Introduction
The discovery of the cosmic microwave background (CMB) by Penzias & Wilson (1965) established the modern paradigm of the hot big bang cosmology. Almost immediately after this seminal discovery, searches began for anisotropies in the CMB – the primordial signatures of the fluctuations that grew to form the structure that we see today^{1}.After a number of earlier detections, convincing evidence for a dipole anisotropy was reported by Smoot et al. (1977), but despite many attempts, the detection of higherorder anisotropies proved elusive until the first results from the Cosmic Background Explorer (COBE; Smoot et al. 1992). The COBE results established the existence of a nearly scaleinvariant spectrum of primordial fluctuations on angular scales larger than 7°, consistent with the predictions of inflationary cosmology, and stimulated a new generation of precision measurements of the CMB of which this set of papers forms a part.
CMB anisotropies are widely recognized as one of the most powerful probes of cosmology and earlyUniverse physics. Given a set of initial conditions and assumptions concerning the background cosmology, the angular power spectrum of the CMB anisotropies can be computed numerically to high precision using linear perturbation theory (see Sect. 2). The combination of precise experimental measurements and accurate theoretical predictions can be used to set tight constraints on cosmological parameters. The influential results from the Wilkinson Microwave Anisotropy Probe (WMAP) satellite (Bennett et al. 2003; Spergel et al. 2003), following on from earlier groundbased and suborbital experiments^{2}, demonstrated the power of this approach, which has been followed by all subsequent CMB experiments.
Fig. 1 Planck foregroundsubtracted temperature power spectrum (with foreground and other “nuisance” parameters fixed to their bestfit values for the base ΛCDM model). The power spectrum at low multipoles (ℓ = 2–49, plotted on a logarithmic multipole scale) is determined by the Commander algorithm applied to the Planck maps in the frequency range 30–353 GHz over 91% of the sky. This is used to construct a lowmultipole temperature likelihood using a BlackwellRao estimator, as described in Planck Collaboration XV (2014). The asymmetric error bars show 68% confidence limits and include the contribution from uncertainties in foreground subtraction. At multipoles 50 ≤ ℓ ≤ 2500 (plotted on a linear multipole scale) we show the bestfit The CMB spectrum computed from the CamSpec likelihood (see Planck Collaboration XV 2014) after removal of unresolved foreground components.This spectrum is averaged over the frequency range 100–217 GHz using frequencydependent diffuse sky cuts (retaining 58% of the sky at 100 GHz and 37% of the sky at 143 and 217 GHz) and is samplevariance limited to ℓ ~ 1600. The light grey points show the power spectrum multipolebymultipole. The blue points show averages in bands of width Δℓ = 25 together with 1σ errors computed from the diagonal components of the bandaveraged covariance matrix (which includes contributions from beam and foreground uncertainties). The red line shows the temperature spectrum for the bestfit base ΛCDM cosmology. The lower panel shows the power spectrum residuals with respect to this theoretical model. The green lines show the ± 1σ errors on the individual power spectrum estimates at high multipoles computed from the CamSpec covariance matrix. Note the change in vertical scale in the lower panel at ℓ = 50. 
Planck^{3} is the thirdgeneration space mission, following COBE and WMAP, dedicated to measurements of the CMB anistropies. The primary aim of Planck (Planck Collaboration 2005) is to measure the temperature and polarization anisotropies with microKelvin sensitivity per resolution element over the entire sky. The wide frequency coverage of Planck (30–857 GHz) was chosen to provide accurate discrimination of Galactic emission from the primordial anisotropies and to enable a broad range of ancilliary science, such as detections of galaxy clusters, extragalactic point sources and the properties of Galactic dust emission. This paper, one of a set associated with the 2013 release of data from the Planck mission (Planck Collaboration I 2014), describes the first cosmological parameter results from the Planck temperature power spectrum.
The results from WMAP (see Bennett et al. 2013 and Hinshaw et al. 2012 for the final nineyear WMAP results) together with those from highresolution groundbased CMB experiments (e.g., Reichardt et al. 2012b; Story et al. 2013; Sievers et al. 2013) are remarkably consistent with the predictions of a “standard” cosmological model. This model is based upon a spatiallyflat, expanding Universe whose dynamics are governed by General Relativity and whose constituents are dominated by cold dark matter (CDM) and a cosmological constant (Λ) at late times. The primordial seeds of structure formation are Gaussiandistributed adiabatic fluctuations with an almost scaleinvariant spectrum. This model (which is referred to as the base ΛCDM model in this paper) is described by only six key parameters. Despite its simplicity, the base ΛCDM model has proved to be successful in describing a wide range of cosmological data in addition to the CMB, including the Type Ia supernovae magnitudedistance relation, baryon acoustic oscillation measurements, the largescale clustering of galaxies and cosmic shear (as reviewed in Sect. 5).
Nevertheless, there have been some suggestions of new physics beyond that assumed in the base ΛCDM model. Examples include various largeangle “anomalies” in the CMB (as reviewed by the WMAP team in Bennett et al. 2011) and hints of new physics, such as additional relativistic particles, that might steepen the high multipole “damping tail” of the CMB temperature power spectrum (Dunkley et al. 2011; Hou et al. 2014). Furthermore, developments in earlyUniverse cosmology over the past 20 years or so have led to a rich phenomenology (see e.g., Baumann 2009, for a review). It is easy to construct models that preserve the main features of simple singlefield inflationary models, but lead to distinctive observational signatures such as nonGaussianity, isocurvature modes or topological defects.
A major goal of the Planck experiment is to test the ΛCDM model to high precision and identify areas of tension. From previous CMB experiments and other cosmological probes, we know that any departures from the standard sixparameter ΛCDM cosmology are likely to be small and challenging to detect. Planck, with its combination of high sensitivity, wide frequency range and allsky coverage, is uniquely wellsuited to this challenge.
The focus of this paper is to investigate cosmological constraints from the temperature power spectrum measured by Planck. Figure 1 summarizes some important aspects of the Planck temperature power spectrum; we plot this as (a notation we use throughout this paper) versus multipole ℓ. The temperature likelihood used in this paper is a hybrid: over the multipole range ℓ = 2–49, the likelihood is based on a componentseparation algorithm applied to 91% of the sky (Planck Collaboration XII 2014; Planck Collaboration XV 2014). The likelihood at higher multipoles is constructed from crossspectra over the frequency range 100–217 GHz, as discussed in Planck Collaboration XV (2014). It is important to recognize that unresolved foregrounds (and other factors such as beams and calibration uncertainties) need to be modelled to high precision to achieve the science goals of this paper. There is therefore no unique “Planck primordial temperature spectrum”. Figure 1 is based on a full likelihood solution for foreground and other “nuisance” parameters assuming a cosmological model. A change in the cosmology will lead to small changes in the Planck primordial CMB power spectrum because of differences in the foreground solution. Neverthess, Fig. 1 provides a good illustration of the precision achieved by Planck. The precision is so high that conventional power spectrum plots (shown in the upper panel of Fig. 1) are usually uninformative. We therefore place high weight in this paper on plots of residuals with respect to the bestfit model (shown in the lower panel). Figure 1 also serves to illustrate the highly interconnected nature of this series of papers. The temperature likelihood used in this paper utilizes data from both the Planck Low Frequency Instrument (LFI) and High Frequency Instrument (HFI). The dataprocessing chains for these two instruments and beam calibrations are described in Planck Collaboration II (2014), Planck Collaboration VI (2014), and associated papers (Planck Collaboration III 2014; Planck Collaboration IV 2014; Planck Collaboration V 2014; Planck Collaboration VII 2014; Planck Collaboration VIII 2014; Planck Collaboration IX 2014; Planck Collaboration X 2014). Component separation is described in Planck Collaboration XII (2014) and the temperature power spectrum and likelihood, as used in this paper, are described in Planck Collaboration XV (2014). Planck Collaboration XV (2014) also presents a detailed analysis of the robustness of the likelihood to various choices, such as frequency ranges and sky masks (and also compares the likelihood to results from an independent likelihood code based on different assumptions, see also Appendix C). Consistency of the Planck maps across frequencies is demonstrated in Planck Collaboration XI (2014), and the level of consistency with WMAP is assessed.
This paper is closely linked to other papers reporting cosmological results in this series. We make heavy use of the gravitational lensing power spectrum and likelihood estimated from an analysis of the 4point function of the Planck maps (Planck Collaboration XVII 2014). The present paper concentrates on simple parameterizations of the spectrum of primordial fluctuations. Tests of specific models of inflation, isocurvature modes, broken scaleinvariance etc. are discussed in Planck Collaboration XXII (2014). Here, we assume throughout that the initial fluctuations are Gaussian and statistically isotropic. Precision tests of nonGaussianity, from Planck estimates of the 3 and 4point functions of the temperature anisotropies, are presented in Planck Collaboration XXIV (2014). Tests of isotropy and additional tests of nonGaussianity using Planck data are discussed in Planck Collaboration XXIII (2014) and Planck Collaboration XXVI (2014).
The outline of the paper is as follows. In Sect. 2 we define our notation and cosmological parameter choices. This section also summarizes aspects of the Markov chain Monte Carlo (MCMC) sampler used in this paper and of the CMB Boltzmann code used to predict theoretical temperature power spectra. Section 3 presents results on cosmological parameters using Planck data alone. For this data release we do not use Planck polarization data in the likelihood, and we therefore rely on WMAP polarization data at low multipoles to constrain the optical depth, τ, from reionization. An interesting aspect of Sect. 3 is to assess whether CMB gravitational lensing measurements from Planck can be used to constrain the optical depth without the use of WMAP polarization measurements.
Section 4 introduces additional CMB temperature data from highresolution experiments. This section presents a detailed description of how we have modified the Planck model for unresolved foreground and “nuisance” parameters introduced in Planck Collaboration XV (2014) to enable the Planck spectra to be used together with those from other CMB experiments. Combining highresolution CMB experiments with Planck mitigates the effects of unresolved foregrounds which, as we show, can affect cosmological parameters (particularly for extensions to the base ΛCDM model) if the foreground parameters are allowed too much freedom. Section 4 ends with a detailed analysis of whether the base ΛCDM model provides an acceptable fit to the CMB temperature power spectra from Planck and other experiments.
It is well known that certain cosmological parameter combinations are highly degenerate using CMB power spectrum measurements alone (Zaldarriaga et al. 1997; Efstathiou & Bond 1999; Howlett et al. 2012). These degeneracies can be broken by combining with other cosmological data (though the Planck lensing analysis does help to break the principal “geometrical” degeneracy, as discussed in Sect. 5.1). Section 5 discusses additional “astrophysical” data that are used in combination with Planck. Since the Planck temperature data are so precise, we have been selective in the additional data sets that we have chosen to use. Section 5 discusses our rationale for making these choices.
Having made a thorough investigation of the base ΛCDM model, Sect. 6 describes extended models, including models with nonpowerlaw spectral indices, tensor modes, curvature, additional relativistic species, neutrino masses and dynamical dark energy. This section also discusses constraints on models with annihilating dark matter, primordial magnetic fields and a timevariable finestructure constant.
Finally, we present our conclusions in Sect. 7. Appendix A compares the Planck and WMAP base ΛCDM cosmologies. Appendix B contrasts the Planck bestfit ΛCDM cosmology with that determined recently by combining data from the South Pole Telescope with WMAP (Story et al. 2013). Appendix C discusses the dependence of our results for extended models on foreground modelling and likelihood choices, building on the discussion in Planck Collaboration XV (2014) for the base ΛCDM model.
Since the appearance of the first draft of this paper, there have been a number of developments that affect both the Planck data and some of the constraints from supplementary astrophysical data used in this paper.
The primary developments are as follows. [1] After the submission of this paper, we discovered a minor error in the ordering of the beam transfer functions applied to each of the CamSpec 217 × 217 GHz crossspectra before their coaddition to form a single spectrum. Correcting for this error changes the mean 217 × 217 GHz spectrum by a smooth function with an amplitude of a few (μK)^{2}. An extensive analysis of a revised likelihood showed that this error has negligible impact on cosmological parameters and that it is absorbed by small shifts in the foreground parameters. Since the effect is so minor, we have decided not to change any of the numbers in this paper and not to revise the public version of the CamSpec likelihood. [2] The foregroundcorrected 217 × 217 GHz spectrum shows a small negative residual (or “dip”) with respect to the bestfit base ΛCDM theoretical model at multipoles ℓ ≈ 1800. This can be seen most clearly in Fig. 7 in this paper. After submission of this paper we found evidence that this feature is a residual systematic in the data associated with incomplete 4 K line removal (see Planck Collaboration VI 2014 for a discussion of the 4 K line removal algorithm). The 4 K lines, at specific frequencies in the detector timelines, are caused by an electromagneticinterference/electromagneticcompatibility (EMIEMC) problem between the ^{4}He JouleThomson (4 K) cooler drive electronics and the readout electronics. This interference is timevariable. Tests in which we have applied more stringent flagging of 4 K lines show that the ℓ = 1800 feature is reduced to negligible levels in all sky surveys, including Survey 1 in which the effect is strongest. The 2014 Planck data release will include improvements in the 4 K line removal. It is important to emphasise that this systematic is a small effect. Analysis of cosmological parameters, removing the multipole range around ℓ = 1800 (and also analysis of the full mission data, where the effect is diluted by the additional sky surveys) shows that the impact of this feature on cosmological parameters is small (i.e., less than half a standard deviation) even for extensions to the base ΛCDM cosmology. Some quantitiative tests of the impact of this systematic on cosmology are summarized in Appendix C. [3] An error was found in the dark energy model used for theoretical predictions with equation of state w ≠ − 1, leading to fewpercent C_{ℓ} errors at very low multipoles in extreme models with w ≳ − 0.5. We have checked, using the corrected October 2013 camb version, that this propagates to only a very small error on marginalized parameters and that the results presented in this paper are consistent to within the stated numerical accuracy. [4] After this paper was submitted, Humphreys et al. (2013) presented the final results of a longterm campaign to establish a new geometric maser distance to NGC 4258. Their revised distance of (7.60 ± 0.23) Mpc leads to a lowering of the Hubble constant, based on the Cepheid distance scale, to H_{0} = (72.0 ± 3.0) km s^{1} Mpc^{1}, partially alleviating the tension between the Riess et al. (2011) results and the Planck results on H_{0} discussed in Sect. 5.3 and subsequent sections. [5] In a recent paper, Betoule et al. (2013) present results from an extensive programme that improves the photometric calibrations of the SDSS and SNLS supernovae surveys. An analysis of the SDSSII and SNLS supernovae samples, including revisions to the photometric calibrations, favours a higher value of Ω_{m} = 0.295 ± 0.034 for the base ΛCDM model, consistent with the Planck results discussed in Sect. 5.4 (Betoule et al. 2014).
A detailed discussion of the impact of the changes discussed here on cosmology will be deferred until the Planck 2014 data release, which will include improvements to the lowlevel data processing and, by which time, improved complementary astrophysical data sets (such as a revised SNLS compilation) should be available to us. In revising this paper, we have taken the view that this, and other Planck papers in this 2013 release, should be regarded as a snapshot of the Planck analysis as it was in early 2013. We have therefore kept revisions to a minimum. Nevertheless, readers of this paper, and users of products from the Planck Legacy Archive^{4} (such as parameter tables and MCMC chains), should be aware of developments since the first submission of this paper.
2. Model, parameters, and methodology
2.1. Theoretical model
We shall treat anisotropies in the CMB as small fluctuations about a FriedmannRobertsonWalker metric whose evolution is described by General Relativity. We shall not consider modified gravity scenarios or “active” sources of fluctuations such as cosmic defects. The latter are discussed in Planck Collaboration XXV (2014). Under our assumptions, the evolution of the perturbations can be computed accurately using a CMB Boltzmann code once the initial conditions, ionization history and constituents of the Universe are specified. We discuss each of these in this section, establishing our notation. Our conventions are consistent with those most commonly adopted in the field and in particular with those used in the camb^{5} Boltzmann code (Lewis et al. 2000), which is the default code used in this paper.
2.1.1. Matter and radiation content
We adopt the usual convention of writing the Hubble constant at the present day as H_{0} = 100 h km s^{1} Mpc^{1}. For our baseline model, we assume that the cold dark matter is pressureless, stable and noninteracting, with a physical density ω_{c} ≡ Ω_{c}h^{2}. The baryons, with density ω_{b} ≡ Ω_{b}h^{2}, are assumed to consist almost entirely of hydrogen and helium; we parameterize the mass fraction in helium by Y_{P}. The process of standard big bang nucleosynthesis (BBN) can be accurately modelled, and gives a predicted relation between Y_{P}, the photonbaryon ratio, and the expansion rate (which depends on the number of relativistic degrees of freedom). By default we use interpolated results from the PArthENoPE BBN code (Pisanti et al. 2008) to set Y_{P}, following Hamann et al. (2011), which for the Planck bestfitting base model (assuming no additional relativistic components and negligible neutrino degeneracy) gives Y_{P} = 0.2477. We shall compare our results with the predictions of BBN in Sect. 6.4.
The photon temperature today is well measured to be T_{0} = 2.7255 ± 0.0006 K (Fixsen 2009); we adopt T_{0} = 2.7255 K as our fiducial value. We assume full thermal equilibrium prior to neutrino decoupling. The decoupling of the neutrinos is nearly, but not entirely, complete by the time of electronpositron annihilation. This leads to a slight heating of the neutrinos in addition to that expected for the photons and hence to a small departure from the thermal equilibrium prediction T_{γ} = (11/4)^{1/3}T_{ν} between the photon temperature T_{γ} and the neutrino temperature T_{ν}. We account for the additional energy density in neutrinos by assuming that they have a thermal distribution with an effective energy density (1)with N_{eff} = 3.046 in the baseline model (Mangano et al. 2002, 2005). This density is divided equally between three neutrino species while they remain relativistic.
In our baseline model we assume a minimalmass normal hierarchy for the neutrino masses, accurately approximated for current cosmological data as a single massive eigenstate with m_{ν} = 0.06 eV (Ω_{ν}h^{2} ≈ ∑ m_{ν}/ 93.04 eV ≈ 0.0006; corrections and uncertainties at the meV level are well below the accuracy required here). This is consistent with global fits to recent oscillation and other data (Forero et al. 2012), but is not the only possibility. We discuss more general neutrino mass constraints in Sect. 6.3.
We shall also consider the possibility of extra radiation, beyond that included in the Standard Model. We model this as additional massless neutrinos contributing to the total N_{eff} determining the radiation density as in Eq. (1). We keep the mass model and heating consistent with the baseline model at N_{eff} = 3.046, so there is one massive neutrino with , and massless neutrinos with . In the case where N_{eff}< 1.015 we use one massive eigenstate with reduced temperature.
2.1.2. Ionization history
To make accurate predictions for the CMB power spectra, the background ionization history has to be calculated to high accuracy. Although the main processes that lead to recombination at z ≈ 1090 are well understood, cosmological parameters from Planck can be sensitive to subpercent differences in the ionization fraction x_{e} (Hu et al. 1995; Lewis et al. 2006; RubinoMartin et al. 2010; Shaw & Chluba 2011). The process of recombination takes the Universe from a state of fully ionized hydrogen and helium in the early Universe, through to the completion of recombination with residual fraction x_{e} ~ 10^{4}. Sensitivity of the CMB power spectrum to x_{e} enters through changes to the sound horizon at recombination, from changes in the timing of recombination, and to the detailed shape of the recombination transition, which affects the thickness of the lastscattering surface and hence the amount of smallscale diffusion (Silk) damping, polarization, and lineofsight averaging of the perturbations.
Since the pioneering work of Peebles (1968) and Zeldovich et al. (1969), which identified the main physical processes involved in recombination, there has been significant progress in numerically modelling the many relevant atomic transitions and processes that can affect the details of the recombination process (Hu et al. 1995; Seager et al. 2000; Wong et al. 2008; Hirata & Switzer 2008; Switzer & Hirata 2008; RubinoMartin et al. 2010; Grin & Hirata 2010; Chluba & Thomas 2011; AliHaimoud et al. 2010; AliHaimoud & Hirata 2011). In recent years a consensus has emerged between the results of two multilevel atom codes HyRec^{6} (Switzer & Hirata 2008; Hirata 2008; AliHaimoud & Hirata 2011), and CosmoRec^{7} (Chluba et al. 2010; Chluba & Thomas 2011), demonstrating agreement at a level better than that required for Planck (differences less that 4 × 10^{4} in the predicted temperature power spectra on small scales).
These recombination codes are remarkably fast, given the complexity of the calculation. However, the recombination history can be computed even more rapidly by using the simple effective threelevel atom model developed by Seager et al. (2000) and implemented in the recfast code^{8}, with appropriately chosen small correction functions calibrated to the full numerical results (Wong et al. 2008; RubinoMartin et al. 2010; Shaw & Chluba 2011). We use recfast in our baseline parameter analysis, with correction functions adjusted so that the predicted power spectra C_{ℓ} agree with those from the latest versions of HyRec (January 2012) and CosmoRec (v2) to better than 0.05%^{9}. We have confirmed, using importance sampling, that cosmological parameter constraints using recfast are consistent with those using CosmoRec at the 0.05σ level. Since the results of the Planck parameter analysis are crucially dependent on the accuracy of the recombination history, we have also checked, following Lewis et al. (2006), that there is no strong evidence for simple deviations from the assumed history. However, we note that any deviation from the assumed history could significantly shift parameters compared to the results presented here and we have not performed a detailed sensitivity analysis.
The background recombination model should accurately capture the ionization history until the Universe is reionized at late times via ultraviolet photons from stars and/or active galactic nuclei. We approximate reionization as being relatively sharp, with the midpoint parameterized by a redshift z_{re} (where x_{e} = f/ 2) and width parameter Δz_{re} = 0.5. Hydrogen reionization and the first reionization of helium are assumed to occur simultaneously, so that when reionization is complete x_{e} = f ≡ 1 + f_{He} ≈ 1.08 (Lewis 2008), where f_{He} is the heliumtohydrogen ratio by number. In this parameterization, the optical depth is almost independent of Δz_{re} and the only impact of the specific functional form on cosmological parameters comes from very small changes to the shape of the polarization power spectrum on large angular scales. The second reionization of helium (i.e., He^{+} → He^{++}) produces very small changes to the power spectra (Δτ ~ 0.001, where τ is the optical depth to Thomson scattering) and does not need to be modelled in detail. We include the second reionization of helium at a fixed redshift of z = 3.5 (consistent with observations of Lymanα forest lines in quasar spectra, e.g., Becker et al. 2011), which is sufficiently accurate for the parameter analyses described in this paper.
2.1.3. Initial conditions
In our baseline model we assume purely adiabatic scalar perturbations at very early times, with a (dimensionless) curvature power spectrum parameterized by (2)with n_{s} and dn_{s}/ dlnk taken to be constant. For most of this paper we shall assume no “running”, i.e., a powerlaw spectrum with dn_{s}/ dlnk = 0. The pivot scale, k_{0}, is chosen to be k_{0} = 0.05 Mpc^{1}, roughly in the middle of the logarithmic range of scales probed by Planck. With this choice, n_{s} is not strongly degenerate with the amplitude parameter A_{s}.
The amplitude of the smallscale linear CMB power spectrum is proportional to e. Because Planck measures this amplitude very accurately there is a tight linear constraint between τ and lnA_{s} (see Sect. 3.4). For this reason we usually use lnA_{s} as a base parameter with a flat prior, which has a significantly more Gaussian posterior than A_{s}. A linear parameter redefinition then also allows the degeneracy between τ and A_{s} to be explored efficiently. (The degeneracy between τ and A_{s} is broken by the relative amplitudes of largescale temperature and polarization CMB anisotropies and by the nonlinear effect of CMB lensing.)
We shall also consider extended models with a significant amplitude of primordial gravitational waves (tensor modes). Throughout this paper, the (dimensionless) tensor mode spectrum is parameterized as a powerlaw with^{10}(3)We define r_{0.05} ≡ A_{t}/A_{s}, the primordial tensortoscalar ratio at k = k_{0}. Our constraints are only weakly sensitive to the tensor spectral index, n_{t} (which is assumed to be close to zero), and we adopt the theoretically motivated singlefield inflation consistency relation n_{t} = − r_{0.05}/ 8, rather than varying n_{t} independently. We put a flat prior on r_{0.05}, but also report the constraint at k = 0.002 Mpc^{1} (denoted r_{0.002}), which is closer to the scale at which there is some sensitivity to tensor modes in the largeangle temperature power spectrum. Most previous CMB experiments have reported constraints on r_{0.002}. For further discussion of the tensortoscalar ratio and its implications for inflationary models see Planck Collaboration XXII (2014).
2.1.4. Dark energy
In our baseline model we assume that the dark energy is a cosmological constant with current density parameter Ω_{Λ}. When considering a dynamical dark energy component, we parameterize the equation of state either as a constant w or as a function of the cosmological scale factor, a, with (4)and assume that the dark energy does not interact with other constituents other than through gravity. Since this model allows the equation of state to cross below −1, a singlefluid model cannot be used selfconsistently. We therefore use the parameterized postFriedmann (PPF) model of Fang et al. (2008a). For models with w> − 1, the PPF model agrees with fluid models to significantly better accuracy than required for the results reported in this paper.
Cosmological parameters used in our analysis.
2.1.5. Power spectra
Over the past decades there has been significant progress in improving the accuracy, speed and generality of the numerical calculation of the CMB power spectra given an ionization history and set of cosmological parameters (see e.g., Bond & Efstathiou 1987; Sugiyama 1995; Ma & Bertschinger 1995; Hu et al. 1995; Seljak & Zaldarriaga 1996; Hu & White 1997b; Zaldarriaga et al. 1998; Lewis et al. 2000; Lesgourgues & Tram 2011). Our baseline numerical Boltzmann code is camb^{11} (Lewis et al. 2000), a parallelized lineofsight code developed from cmbfast (Seljak & Zaldarriaga 1996) and Cosmics (Bertschinger 1995; Ma & Bertschinger 1995), which calculates the lensed CMB temperature and polarization power spectra. The code has been publicly available for over a decade and has been very well tested (and improved) by the community. Numerical stability and accuracy of the calculation at the sensitivity of Planck has been explored in detail (Hamann et al. 2009; Lesgourgues 2011b; Howlett et al. 2012), demonstrating that the raw numerical precision is sufficient for numerical errors on parameter constraints from Planck to be less than 10% of the statistical error around the assumed cosmological model. (For the high multipole CMB data at ℓ > 2000 introduced in Sect. 4, the default camb settings are adequate because the power spectra of these experiments are dominated by unresolved foregrounds and have large errors at high multipoles.) To test the potential impact of camb errors, we importancesample a subset of samples from the posterior parameter space using higher accuracy settings. This confirms that differences purely due to numerical error in the theory prediction are less than 10% of the statistical error for all parameters, both with and without inclusion of CMB data at high multipoles. We also performed additional tests of the robustness and accuracy of our results by reproducing a fraction of them with the independent Boltzmann code class (Lesgourgues 2011a; Blas et al. 2011).
In the parameter analysis, information from CMB lensing enters in two ways. Firstly, all the CMB power spectra are modelled using the lensed spectra, which includes the approximately 5% smoothing effect on the acoustic peaks due to lensing. Secondly, for some results we include the Planck lensing likelihood, which encapsulates the lensing information in the (mostly squeezedshape) CMB trispectrum via a lensing potential power spectrum (Planck Collaboration XVII 2014). The theoretical predictions for the lensing potential power spectrum are calculated by camb, optionally with corrections for the nonlinear matter power spectrum, along with the (nonlinear) lensed CMB power spectra. For the Planck temperature power spectrum, corrections to the lensing effect due to nonlinear structure growth can be neglected, however the impact on the lensing potential reconstruction is important. We use the halofit model (Smith et al. 2003) as updated by Takahashi et al. (2012) to model the impact of nonlinear growth on the theoretical prediction for the lensing potential power.
2.2. Parameter choices
2.2.1. Base parameters
The first section of Table 1 lists our base parameters that have flat priors when they are varied, along with their default values in the baseline model. When parameters are varied, unless otherwise stated, prior ranges are chosen to be much larger than the posterior, and hence do not affect the results of parameter estimation. In addition to these priors, we impose a “hard” prior on the Hubble constant of [20,100] km s^{1} Mpc^{1}.
2.2.2. Derived parameters
Matterradiation equality z_{eq} is defined as the redshift at which ρ_{γ} + ρ_{ν} = ρ_{c} + ρ_{b} (where ρ_{ν} approximates massive neutrinos as massless).
The redshift of last scattering, z_{∗}, is defined so that the optical depth to Thomson scattering from z = 0 (conformal time η = η_{0}) to z = z_{∗} is unity, assuming no reionization. The optical depth is given by (5)where (and n_{e} is the density of free electrons and σ_{T} is the Thomson cross section). We define the angular scale of the sound horizon at last scattering, θ_{∗} = r_{s}(z_{∗}) /D_{A}(z_{∗}), where r_{s} is the sound horizon (6)with R ≡ 3ρ_{b}/ (4ρ_{γ}). The parameter θ_{MC} in Table 1 is an approximation to θ_{∗} that is used in CosmoMC and is based on fitting formulae given in Hu & Sugiyama (1996).
Baryon velocities decouple from the photon dipole when Compton drag balances the gravitational force, which happens at τ_{d} ~ 1, where (Hu & Sugiyama 1996) (7)Here, again, τ is from recombination only, without reionization contributions. We define a drag redshift z_{drag}, so that τ_{d}(η(z_{drag})) = 1. The sound horizon at the drag epoch is an important scale that is often used in studies of baryon acoustic oscillations; we denote this as r_{drag} = r_{s}(z_{drag}). We compute z_{drag} and r_{drag} numerically from camb (see Sect. 5.2 for details of application to BAO data).
The characteristic wavenumber for damping, k_{D}, is given by (8)We define the angular damping scale, θ_{D} = π/ (k_{D}D_{A}), where D_{A} is the comoving angular diameter distance to z_{∗}.
For our purposes, the normalization of the power spectrum is most conveniently given by A_{s}. However, the alternative measure σ_{8} is often used in the literature, particularly in studies of largescale structure. By definition, σ_{8} is the rms fluctuation in total matter (baryons + CDM + massive neutrinos) in 8 h^{1} Mpc spheres at z = 0, computed in linear theory. It is related to the dimensionless matter power spectrum, , by (9)where R = 8 h^{1} Mpc and j_{1} is the spherical Bessel function of order 1.
In addition, we compute Ω_{m}h^{3} (a welldetermined combination orthogonal to the acoustic scale degeneracy in flat models; see e.g., Percival et al. 2002 and Howlett et al. 2012), 10^{9}A_{s}e^{− 2τ} (which determines the smallscale linear CMB anisotropy power), r_{0.002} (the ratio of the tensor to primordial curvature power at k = 0.002 Mpc^{1}), Ω_{ν}h^{2} (the physical density in massive neutrinos), and the value of Y_{P} from the BBN consistency condition.
2.3. Likelihood
Planck Collaboration XV (2014) describes the Planck temperature likelihood in detail. Briefly, at high multipoles (ℓ ≥ 50) we use the 100, 143 and 217 GHz temperature maps (constructed using HEALPix Górski et al. 2005) to form a high multipole likelihood following the CamSpec methodology described in Planck Collaboration XV (2014). Apodized Galactic masks, including an apodized point source mask, are applied to individual detector/detectorset maps at each frequency. The masks are carefully chosen to limit contamination from diffuse Galactic emission to low levels (less than 20 μK^{2} at all multipolesused in the likelihood) before correction for Galactic dust emission^{12}. Thus we retain 57.8% of the sky at 100 GHz and 37.3% of the sky at 143 and 217 GHz. Maskdeconvolved and beamcorrected crossspectra (following Hivon et al. 2002) are computed for all detector/detectorset combinations and compressed to form averaged 100 × 100, 143 × 143, 143 × 217 and 217 × 217 pseudospectra (note that we do not retain the 100 × 143 and 100 × 217 crossspectra in the likelihood). Semianalytic covariance matrices for these pseudospectra (Efstathiou 2004) are used to form a highmultipole likelihood in a fiducial Gaussian likelihood approximation (Bond et al. 2000; Hamimeche & Lewis 2008).
At low multipoles (2 ≤ ℓ ≤ 49) the temperature likelihood is based on a BlackwellRao estimator applied to Gibbs samples computed by the Commander algorithm (Eriksen et al. 2008) from Planck maps in the frequency range 30–353 GHz over 91% of the sky. The likelihood at low multipoles therefore accounts for errors in foreground cleaning.
Detailed consistency tests of both the high and lowmultipole components of the temperature likelihood are presented in Planck Collaboration XV (2014). The highmultipole Planck likelihood requires a number of additional parameters to describe unresolved foreground components and other “nuisance” parameters (such as beam eigenmodes). The model adopted for Planck is described in Planck Collaboration XV (2014). A selfcontained account is given in Sect. 4 which generalizes the model to allow matching of the Planck likelihood to the likelihoods from highresolution CMB experiments. A complete list of the foreground and nuisance parameters is given in Table 4.
2.4. Sampling and confidence intervals
We sample from the space of possible cosmological parameters with MCMC exploration using CosmoMC (Lewis & Bridle 2002). This uses a MetropolisHastings algorithm to generate chains of samples for a set of cosmological parameters, and also allows for importance sampling of results to explore the impact of small changes in the analysis. The set of parameters is internally orthogonalized to allow efficient exploration of parameter degeneracies, and the baseline cosmological parameters are chosen following Kosowsky et al. (2002), so that the linear orthogonalisation allows efficient exploration of the main geometric degeneracy (Bond et al. 1997). The codehas been thoroughly tested by the community and has recently been extended to sample efficiently large numbers of “fast” parameters by use of a speedordered Cholesky parameter rotation and a fastparameter “dragging” scheme described by Neal (2005) and Lewis (2013).
Cosmological parameter values for the sixparameter base ΛCDM model.
For our main cosmological parameter runs we execute eight chains until they are converged, and the tails of the distribution are well enough explored for the confidence intervals for each parameter to be evaluated consistently in the last half of each chain. We check that the spread in the means between chains is small compared to the standard deviation, using the standard Gelman and Rubin (Gelman & Rubin 1992) criterion R − 1 < 0.01 in the leastconverged orthogonalized parameter. This is sufficient for reliable importance sampling in most cases. We perform separate runs when the posterior volumes differ enough that importance sampling is unreliable. Importancesampled and extended datacombination chains used for this paper satisfy R − 1 < 0.1, and in almost all cases are closer to 0.01. We discard the first 30% of each chain as burn in, where the chains may be still converging and the sampling may be significantly nonMarkovian. This is due to the way CosmoMC learns an accurate orthogonalisation and proposal distribution for the parameters from the sample covariance of previous samples.
From the samples, we generate estimates of the posterior mean of each parameter of interest, along with a confidence interval. We generally quote 68% limits in the case of twotail limits, so that 32% of samples are outside the limit range, and there are 16% of samples in each tail. For parameters where the tails are significantly different shapes, we instead quote the interval between extremal points with approximately equal marginalized probability density. For parameters with prior bounds we either quote onetail limits or no constraint, depending on whether the posterior is significantly nonzero at the prior boundary. Our onetail limits are always 95% limits. For parameters with nearly symmetric distribution we sometimes quote the mean and standard deviation (± 1σ). The samples can also be used to estimate one, two and threedimensional marginalized parameter posteriors. We use variablewidth Gaussian kernel density estimates in all cases.
We have also performed an alternative analysis to the one described above, using an independent statistical method based on frequentist profile likelihoods (Wilks 1938). This gives fits and error bars for the baseline cosmological parameters in excellent agreement for both Planck and Planck combined with highresolution CMB experiments, consistent with the Gaussian form of the posteriors found from full parameter space sampling.
In addition to posterior means, we also quote maximumlikelihood parameter values. These are generated using the BOBYQA bounded minimization routine^{13}. Precision is limited by stability of the convergence, and values quoted are typically reliable to within Δχ^{2} ~ 0.6, which is the same order as differences arising from numerical errors in the theory calculation. For poorly constrained parameters the actual value of the bestfit parameters is not very numerically stable and should not be overinterpreted; in particular, highly degenerate parameters in extended models and the foreground model can give many apparently different solutions within this level of accuracy. The bestfit values should be interpreted as giving typical theory and foreground power spectra that fit the data well, but are generally nonunique at the numerical precision used; they are however generally significantly better fits than any of the samples in the parameter chains. Bestfit values are useful for assessing residuals, and differences between the bestfit and posterior means also help to give an indication of the effect of asymmetries, parametervolume and priorrange effects on the posterior samples. We have crosschecked a small subset of the bestfits with the widely used MINUIT software (James 2004), which can give somewhat more stable results.
3. Constraints on the parameters of the base ΛCDM model from Planck
In this section we discuss parameter constraints from Planck alone in the ΛCDM model. Planck provides a precision measurement of seven acoustic peaks in the CMB temperature power spectrum. The range of scales probed by Planck is sufficiently large that many parameters can be determined accurately without using lowℓ polarization information to constrain the optical depth, or indeed without using any other astrophysical data.
However, because the data are reaching the limit of astrophysical confusion, interpretation of the peaks at higher multipoles requires a reliable model for unresolved foregrounds. We model these here parametrically, as described in Planck Collaboration XV (2014), and marginalize over the parameters with wide priors. We give a detailed discussion of consistency of the foreground model in Sect. 4, making use of other highℓ CMB observations, although as we shall see the parameters of the base ΛCDM model have a weak sensitivity to foregrounds.
As foreground modelling is not especially critical for the base ΛCDM model, we have decided to present the Planck constraints early in this paper, ahead of the detailed descriptions of the foreground model, supplementary highresolution CMB data sets, and additional astrophysical data sets. The reader can therefore gain a feel for some of the key Planck results before being exposed to the lengthier discussions of Sects. 4 and 5, which are essential for the analysis of extensions to the base ΛCDM cosmology presented in Sect. 6.
In addition to the temperature power spectrum measurement, the Planck lensing reconstruction (discussed in more detail in Sect. 5.1 and Planck Collaboration XVII 2014) provides a different probe of the perturbation amplitudes and geometry at late times. CMB lensing can break degeneracies inherent in the temperature data alone, especially the geometric degeneracy in nonflat models, providing a strong constraint on spatial curvature using only CMB data. The lensing reconstruction constrains the matter fluctuation amplitude, and hence the accurate measurement of the temperature anisotropy power can be used together with the lensing reconstruction to infer the relative suppression of the temperature anisotropies due to the finite optical depth to reionization. The largescale polarization from nine years of WMAP observations (Bennett et al. 2013) gives a constraint on the optical depth consistent with the Planck temperature and lensing spectra. Nevertheless, the WMAP polarization constraint is somewhat tighter, so by including it we can further improve constraints on some parameters.
Fig. 2 Comparison of the base ΛCDM model parameters for Planck+lensing only (colourcoded samples), and the 68% and 95% constraint contours adding WMAP lowℓ polarization (WP; red contours), compared to WMAP9 (Bennett et al. 2013; grey contours). 
We therefore also consider the combination of the Planck temperature power spectrum with a WMAP polarization lowmultipole likelihood (Bennett et al. 2013) at ℓ ≤ 23 (denoted WP), as discussed in Planck Collaboration XV (2014)^{14}. We refer to this CMB data combination as Planck+WP.
Table 2 summarizes our constraints on cosmological parameters from the Planck temperature power spectrum alone (labelled “Planck”), from Planck in combination with Planck lensing (Planck+lensing) and with WMAP lowℓ polarization (Planck+WP). Figure 2 shows a selection of corresponding constraints on pairs of parameters and fully marginalized oneparameter constraints compared to the final results from WMAP (Bennett et al. 2013).
3.1. Acoustic scale
The characteristic angular size of the fluctuations in the CMB is called the acoustic scale. It is determined by the comoving size of the sound horizon at the time of last scattering, r_{s}(z_{∗}), and the angular diameter distance at which we are observing the fluctuations, D_{A}(z_{∗}). With accurate measurement of seven acoustic peaks, Planck determines the observed angular size θ_{∗} = r_{s}/D_{A} to better than 0.1% precision at 1σ: (10)Since this parameter is constrained by the positions of the peaks but not their amplitudes, it is quite robust; the measurement is very stable to changes in data combinations and the assumed cosmology. Foregrounds, beam uncertainties, or any systematic effects which only contribute a smooth component to the observed spectrum will not substantially affect the frequency of the oscillations, and hence this determination is likely to be Planck’s most robust precision measurement. The situation is analogous to baryon acoustic oscillations measurements in largescale structure surveys (see Sect. 5.2), but the CMB acoustic measurement has the advantage that it is based on observations of the Universe when the fluctuations were very accurately linear, so second and higherorder effects are expected to be negligible^{15}.
The tight constraint on θ_{∗} also implies tight constraints on some combinations of the cosmological parameters that determine D_{A} and r_{s}. The sound horizon r_{s} depends on the physical matter density parameters, and D_{A} depends on the latetime evolution and geometry. Parameter combinations that fit the Planck data must be constrained to be close to a surface of constant θ_{∗}. This surface depends on the model that is assumed. For the base ΛCDM model, the main parameter dependence is approximately described by a 0.3% constraint in the threedimensional Ω_{m}–h–Ω_{b}h^{2} subspace: (11)Reducing further to a twodimensional subspace gives a 0.6% constraint on the combination (12)The principle component analysis direction is actually Ω_{m}h^{2.93} but this is conveniently close to Ω_{m}h^{3} and gives a similar constraint. The simple form is a coincidence of the ΛCDM cosmology, error model, and particular parameter values of the model (Percival et al. 2002; Howlett et al. 2012). The degeneracy between H_{0} and Ω_{m} is illustrated in Fig. 3: parameters are constrained to lie in a narrow strip where Ω_{m}h^{3} is nearly constant, but the orthogonal direction is much more poorly constrained. The degeneracy direction involves consistent changes in the H_{0}, Ω_{m}, and Ω_{b}h^{2} parameters, so that the ratio of the sound horizon and angular diameter distance remains nearly constant. Changes in the density parameters, however, also have other effects on the power spectrum and the spectral index n_{s} also changes to compensate. The degeneracy is not exact; its extent is much more sensitive to other details of the power spectrum shape. Additional data can help further to restrict the degeneracy. Figure 3 shows that adding WMAP polarization has almost no effect on the Ω_{m}h^{3} measurement, but shrinks the orthogonal direction slightly from Ω_{m}h^{3} = 1.03 ± 0.13 to Ω_{m}h^{3} = 1.04 ± 0.11.
Fig. 3 Constraints in the Ω_{m}–H_{0} plane. Points show samples from the Planckonly posterior, coloured by the corresponding value of the spectral index n_{s}. The contours (68% and 95%) show the improved constraint from Planck+lensing+WP. The degeneracy direction is significantly shortened by including WP, but the wellconstrained direction of constant Ω_{m}h^{3} (set by the acoustic scale), is determined almost equally accurately from Planck alone. 
3.2. Hubble parameter and dark energy density
The Hubble constant, H_{0}, and matter density parameter, Ω_{m}, are only tightly constrained in the combination Ω_{m}h^{3} discussed above, but the extent of the degeneracy is limited by the effect of Ω_{m}h^{2} on the relative heights of the acoustic peaks. The projection of the constraint ellipse shown in Fig. 3 onto the axes therefore yields useful marginalized constraints on H_{0} and Ω_{m} (or equivalently Ω_{Λ}) separately. We find the 2% constraint on H_{0}: (13)The corresponding constraint on the dark energy density parameter is (14)and for the physical matter density we find (15)Note that these indirect constraints are highly model dependent. The data only measure accurately the acoustic scale, and the relation to underlying expansion parameters (e.g., via the angulardiameter distance) depends on the assumed cosmology, including the shape of the primordial fluctuation spectrum. Even small changes in model assumptions can change H_{0} noticeably; for example, if we neglect the 0.06 eV neutrino mass expected in the minimal hierarchy, and instead take ∑ m_{ν} = 0, the Hubble parameter constraint shifts to (16)
3.3. Matter densities
Planck can measure the matter densities in baryons and dark matter from the relative heights of the acoustic peaks. However, as discussed above, there is a partial degeneracy with the spectral index and other parameters that limits the precision of the determination. With Planck there are now enough well measured peaks that the extent of the degeneracy is limited, giving Ω_{b}h^{2} to an accuracy of 1.5% without any additional data: (17)Adding WMAP polarization information shrinks the errors by only 10%.
The dark matter density is slightly less accurately measured at around 3%: (18)
Fig. 4 Marginalized constraints on parameters of the base ΛCDM model for various data combinations. 
3.4. Optical depth
Smallscale fluctuations in the CMB are damped by Thomson scattering from free electrons produced at reionization. This scattering suppresses the amplitude of the acoustic peaks by e^{−2τ} on scales that correspond to perturbation modes with wavelength smaller than the Hubble radius at reionization. Planck measures the smallscale power spectrum with high precision, and hence accurately constrains the damped amplitude e. With only unlensed temperature power spectrum data, there is a large degeneracy between τ and A_{s}, which is weakly broken only by the power in largescale modes that were still superHubble scale at reionization. However, lensing depends on the actual amplitude of the matter fluctuations along the line of sight. Planck accurately measures many acoustic peaks in the lensed temperature power spectrum, where the amount of lensing smoothing depends on the fluctuation amplitude. Furthermore Planck’s lensing potential reconstruction provides a more direct measurement of the amplitude, independently of the optical depth. The combination of the temperature data and Planck’s lensing reconstruction can therefore determine the optical depth τ relatively well. The combination gives (19)As shown in Fig. 4 this provides marginal confirmation (just under 2σ) that the total optical depth is significantly higher than would be obtained from sudden reionization at z ~ 6, and is consistent with the WMAP9 constraint, τ = 0.089 ± 0.014, from largescale polarization (Bennett et al. 2013). The largescale Emode polarization measurement is very challenging because it is a small signal relative to polarized Galactic emission on large scales, so this Planck polarizationfree result is a valuable crosscheck. The posterior for the Planck temperature power spectrum measurement alone also consistently peaks at τ ~ 0.1, where the constraint on the optical depth is coming from the amplitude of the lensing smoothing effect and (to a lesser extent) the relative power between small and large scales.
Since lensing constrains the underlying fluctuation amplitude, the matter density perturbation power is also well determined: (20)Much of the residual uncertainty is caused by the degeneracy with the optical depth. Since the smallscale temperature power spectrum more directly fixes σ_{8}e^{−τ}, this combination is tightly constrained: (21)The estimate of σ_{8} is significantly improved to σ_{8} = 0.829 ± 0.012 by using the WMAP polarization data to constrain the optical depth, and is not strongly degenerate with Ω_{m}. (We shall see in Sect. 5.5 that the Planck results are discrepant with recent estimates of combinations of σ_{8} and Ω_{m} from cosmic shear measurements and counts of rich clusters of galaxies.)
3.5. Spectral index
The scalar spectral index defined in Eq. (2) is measured by Planck data alone to 1% accuracy: (22)Since the optical depth τ affects the relative power between large scales (that are unaffected by scattering at reionization) and intermediate and small scales (that have their power suppressed by e^{−2τ}), there is a partial degeneracy with n_{s}. Breaking the degeneracy between τ and n_{s} using WMAP polarization leads to a small improvement in the constraint: (23)Comparing Eqs. (22) and (23), it is evident that the Planck temperature spectrum spans a wide enough range of multipoles to give a highly significant detection of a deviation of the scalar spectral index from exact scale invariance (at least in the base ΛCDM cosmology) independent of WMAP polarization information.
One might worry that the spectral index parameter is degenerate with foreground parameters, since these act to increase smoothly the amplitudes of the temperature power spectra at high multipoles. The spectral index is therefore liable to potential systematic errors if the foreground model is poorly constrained. Figure 4 shows the marginalized constraints on the ΛCDM parameters for various combinations of data, including adding highresolution CMB measurements. As discussed in Sect. 4, the use of highresolution CMB provides tighter constraints on the foreground parameters (particularly “minor” foreground components) than from Planck data alone. However, the small shifts in the means and widths of the distributions shown in Fig. 4 indicate that, for the base ΛCDM cosmology, the errors on the cosmological parameters are not limited by foreground uncertainties when considering Planck alone. The effects of foreground modelling assumptions and likelihood choices on constraints on n_{s} are discussed in Appendix C.
Summary of the CMB temperature data sets used in this analysis.
4. Planck combined with highresolution CMB experiments: the base ΛCDM model
The previous section adopted a foreground model with relatively loose priors on its parameters. As discussed there and in Planck Collaboration XV (2014), for the base ΛCDM model, the cosmological parameters are relatively weakly correlated with the parameters of the foreground model and so we expect that the cosmological results reported in Sect. 3 are robust. Fortunately, we can get an additional handle on unresolved foregrounds, particularly “minor” components such as the kinetic SZ effect, by combining the Planck data with data from highresolution CMB experiments. The consistency of results obtained with Planck data alone and Planck data combined with highresolution CMB data gives added confidence to our cosmological results, particularly when we come to investigate extensions to the base ΛCDM cosmology (Sect. 6). In this section, we review the highresolution CMB data (hereafter, usually denoted highL) that we combine with Planck and then discuss how the foreground model is adapted (with additional “nuisance” parameters) to handle multiple CMB data sets. We then discuss the results of an MCMC analysis of the base ΛCDM model combining Planck data with the highℓ data.
4.1. Overview of the highℓ CMB data sets
The Atacama Cosmology Telescope (ACT) mapped the sky from 2007 to 2010 in two distinct regions, the equatorial stripe (ACTe) along the celestial equator, and the southern stripe (ACTs) along declination −55°, observing in total about 600 deg^{2}. The ACT data sets at 148 and 218 GHz are presented in Das et al. (2014, hereafter D13) and cover the angular scales 540 < ℓ < 9440 at 148 GHz and 1540 < ℓ < 9440 at 218 GHz. Beam errors are included in the released covariance matrix. We include the ACT 148 × 148 spectra for ℓ ≥ 1000, and the ACT 148 × 218 and 218 × 218 spectra for ℓ ≥ 1500. The inclusion of ACT spectra to ℓ = 1000 improves the accuracy of the intercalibration parameters between the highℓ experiments and Planck.
The South Pole Telescope observed a region of sky over the period 2007–10. Spectra are reported in Keisler et al. (2011, hereafter K11) and Story et al. (2013, hereafter S12) for angular scales 650 < ℓ < 3000 at 150 GHz, and in Reichardt et al. (2012b, hereafter R12) for angular scales 2000 < ℓ < 10 000 at 95, 150 and 220 GHz. Beam errors are included in the released covariance matrices used to form the SPT likelihood. The parameters of the base ΛCDM cosmology derived from the WMAP7+S12 data and (to a lesser extent) from K11 are in tension with Planck. Since the S12 spectra have provided the strongest CMB constraints on cosmological parameters prior to Planck, this discrepancy merits a more detailed analysis, which is presented in Appendix B. The S12 and K11 data are not used in combination with Planck in this paper. Since the primary purpose of including highℓ CMB data is to provide stronger constraints on foregrounds, we use only the R12 SPT data at ℓ > 2000 in combination with Planck. We ignore any correlations between ACT/SPT and Planck spectra over the overlapping multipole ranges.
Table 3 summarizes some key features of the CMB data sets used in this paper.
Fig. 5 Top: Planck spectra at 100, 143 and 217 GHz without subtraction of foregrounds. Middle: SPT spectra from R12 at 95, 150 and 220 GHz, recalibrated to Planck using the bestfit calibration, as discussed in the text. The S12 SPT spectrum at 150 GHz is also shown, but without any calibration correction. This spectrum is discussed in detail in Appendix B, but is not used elsewhere in this paper. Bottom: ACT spectra (weighted averages of the equatorial and southern fields) from D13 at 148 and 220 GHz, and the 148 × 220 GHz crossspectrum, with no extragalactic foreground corrections, recalibrated to the Planck spectra as discussed in the text. The solid line in each panel shows the bestfit base ΛCDM model from the combined Planck+WP+highL fits listed in Table 5. 
4.2. Model of unresolved foregrounds and “nuisance” parameters
The model for unresolved foregrounds used in the Planck likelihood is described in detail in Planck Collaboration XV (2014). Briefly, the model includes power spectrum templates for clustered extragalactic point sources (the cosmic infrared background, hereafter CIB), thermal (tSZ) and kinetic (kSZ) SunyaevZeldovich contributions, and the crosscorrelation (tSZ×CIB) between infrared galaxies and the thermal SunyaevZeldovich effect. The model also includes amplitudes for the Poisson contributions from radio and infrared galaxies. The templates are described in Planck Collaboration XV (2014) and are kept fixed here. (Appendix C discusses briefly a few tests showing the impact of varying some aspects of the foreground model.) The model for unresolved foregrounds is similar to the models developed by the ACT and the SPT teams (e.g., R12; Dunkley et al. 2013). The main difference is in the treatment of the Poisson contribution from radio and infrared galaxies. In the ACT and SPT analyses, spectral models are assumed for radio and infrared galaxies. The Poisson point source contributions can then be described by an amplitude for each population, assuming either fixed spectral parameters or solving for them. In addition, one can add additional parameters to describe the decorrelation of the point source amplitudes with frequency (see e.g., Millea et al. 2012). The Planck model assumes free amplitudes for the point sources at each frequency, together with appropriate correlation coefficients between frequencies. The model is adapted to handle the ACT and SPT data as discussed later in this section.
Figure 5 illustrates the importance of unresolved foregrounds in interpreting the power spectra of the three CMB data sets. The upper panel of Fig. 5 shows the Planck temperature spectra at 100, 143, and 217 GHz, without corrections for unresolved foregrounds (to avoid overcrowding, we have not plotted the 143 × 217 spectrum). The solid (red) lines show the bestfit base ΛCDM CMB spectrum corresponding to the combined Planck+ACT+SPT+WMAP polarization likelihood analysis, with parameters listed in Table 5. The middle panel shows the SPT spectra at 95, 150 and 220 GHz from S12 and R12. In this figure, we have recalibrated the R12 power spectra to match Planck using calibration parameters derived from a full likelihood analysis of the base ΛCDM model. The S12 spectrum plotted is exactly as tabulated in S12, i.e., we have not recalibrated this spectrum to Planck. (The consistency of the S12 spectrum with the theoretical model is discussed in further detail in Appendix B.) The lower panel of Fig. 5 shows the ACT spectra from D13, recalibrated to Planck with calibration coefficients determined from a joint likelihood analysis. The power spectra plotted are an average of the ACTe and ACTs spectra, and include the small Galactic dust corrections described in D13.
The smallscale SPT (R12) and ACT (D13) data are dominated by the extragalactic foregrounds and hence are highly effective in constraining the multiparameter foreground model. In contrast, Planck has limited angular resolution and therefore limited ability to constrain unresolved foregrounds. Planck is sensitive to the Poisson point source contribution at each frequency and to the CIB contribution at 217 GHz. Planck has some limited sensitivity to the tSZ amplitude from the 100 GHz channel (and almost no sensitivity at 143 GHz). The remaining foreground contributions are poorly constrained by Planck and highly degenerate with each other in a Planckalone analysis. The main gain in combining Planck with the highresolution ACT and SPT data is in breaking some of the degeneracies between foreground parameters which are poorly determined from Planck data alone.
An important extension of the foreground parameterization described here over that developed in Planck Collaboration XV (2014) concerns the use of effective frequencies. Different experiments (and different detectors within a frequency band) have nonidentical bandpasses (Planck Collaboration IX 2014) and this needs to be taken into account in the foreground modelling. Consider, for example, the amplitude of the CIB template at 217 GHz, , introduced in Planck Collaboration XV (2014). The effective frequency for a dustlike component for the averaged 217 GHz spectrum used in the Planck likelihood is 225.7 GHz. To avoid cumbersome notation, we solve for the CIB amplitude at the CMB effective frequency of217 GHz. The actual amplitude measured in the Planck217 GHz band is , reflecting the different effective frequencies of a dustlike component compared to the blackbody primordial CMB (see Eq. (30) below). With appropriate effective frequencies, the single amplitude can be used to parameterize the CIB contributions to the ACT and SPT power spectra in their respective 218 and 220 GHz bands. A similar methodology is applied to match the tSZ amplitudes for each experiment.
The relevant effective frequencies for the foreground parameterization discussed below are listed in Table 3. For the high resolution experiments, these are as quoted in R12 and Dunkley et al. (2013). For Planck these effective frequencies were computed from the individual HFI bandpass measurements (Planck Collaboration IX 2014), and vary by a few percent from detector to detector. The numbers quoted in Table 3 are based on an approximate average of the individual detector bandpasses using the weighting scheme for individual detectors/detectorsets applied in the CamSpec likelihood. (The resulting bandpass correction factors for the tSZ and CIB amplitudes should be accurate to better than 5%.)Note that all temperatures in this section are in thermodynamic units.
Astrophysical parameters used to model foregrounds in our analysis, plus instrumental calibration and beam parameters.
The ingredients of the foreground model and associated “nuisance” parameters are summarized in the following paragraphs.
Calibration factors:
to combine the Planck, ACT and SPT likelihoods it is important to incorporate relative calibration factors, since the absolute calibrations of ACT and SPT have large errors (e.g., around 3.5% in power for the SPT 150 GHz channel). We introduce three map calibration parameters , and to rescale the R12 SPT spectra. These factors rescale the crossspectra at frequencies ν_{i} and ν_{j} as (24)In the analysis of ACT, we solve for different map calibration factors for the ACTe and ACTs spectra, , , , and . In addition, we solve for the 100 × 100 and 217 × 217Planckpowerspectrum calibration factors c_{100} and c_{217}, with priors as described in Planck Collaboration XV (2014); see also Table 4. (The use of map calibration factors for ACT and SPT follows the conventions adopted by the ACT and SPT teams, while for the Planck power spectrum analysis we have consistently used powerspectrum calibration factors.)
In a joint parameter analysis of Planck+ACT+SPT, the inclusion of these calibration parameters leads to recalibrations that match the ACT, SPT and Planck100 GHz and 217 GHz channels to the calibration of the Planck143 × 143 spectrum (which, in turn, is linked to the calibration of the HFI 1435 detector, as described in Planck Collaboration XV 2014). It is worth mentioning here that the Planck143 × 143 GHz spectrum is 2.5% lower than the WMAP9 combined V+W power spectrum (Hinshaw et al. 2012). This calibration offset between Planck HFI channels and WMAP is discussed in more detail in Planck Collaboration XI (2014) and in Appendix A.
Bestfit values and 68% confidence limits for the base ΛCDM model.
Poisson point source amplitudes:
to avoid any possible biases in modelling a mixed population of sources (synchrotron+dusty galaxies) with differing spectra, we solve for each of the Poisson point source amplitudes as free parameters. Thus, for Planck we solve for , , and , giving the amplitude of the Poisson point source contributions to for the 100 × 100, 143 × 143, and 217 × 217 spectra. The units of are therefore μK^{2}. The Poisson point source contribution to the 143 × 217 spectrum is expressed as a correlation coefficient, : (25)Note that we do not use the Planck100 × 143 and 100 × 217 spectra in the likelihood, and so we do not include correlation coefficients or . (These spectra carry little additional information on the primordial CMB, but would require additional foreground parameters had we included them in the likelihood.)
In an analogous way, the point source amplitudes for ACT and SPT are characterized by the amplitudes , , , , and (all in units of μK^{2}) and three correlation coefficients , , and . The last of these correlation coefficients is common to ACT and SPT.
Kinetic SZ:
the kSZ template used here is from Trac et al. (2011). We solve for the amplitude A^{kSZ} (in units of μK^{2}): (26)
Thermal SZ:
we use the ϵ = 0.5 tSZ template from Efstathiou & Migliaccio (2012) normalized to a frequency of 143 GHz.
For crossspectra between frequencies ν_{i} and ν_{j}, the tSZ template is normalized as (27)where ν_{0} is the reference frequency of 143 GHz, is the template spectrum at 143 GHz, and (28)The tSZ contribution is therefore characterized by the amplitude in units of μK^{2}.
We neglect the tSZ contribution for any spectra involving the Planck217 GHz, ACT 218 GHz, and SPT 220 GHz channels, since the tSZ effect has a null point at ν = 217 GHz. (For Planck the bandpasses of the 217 GHz detectors see less than 0.1% of the 143 GHz tSZ power.)
Cosmic infrared background:
the CIB contributions are neglected in the Planck100 GHz and SPT 95 GHz bands and in any crossspectra involving these frequencies. The CIB power spectra at higher frequencies are characterized by three amplitude parameters and a spectral index, where and are expressed in μK^{2}. As explained above, we define these amplitudes at the Planck CMB frequencies of 143 and 217 GHz and compute scalings to adjust these amplitudes to the effective frequencies for a dustlike spectrum for each experiment. The scalings are (30)where B_{ν}(T_{d}) is the Planck function at a frequency ν, (31)converts antenna temperature to thermodynamic temperature, ν_{i} and ν_{j} refer to the Planck/ACT/SPT dust effective frequencies, and ν_{i0} and ν_{j0} refer to the corresponding reference CMB Planck frequencies. In the analysis presented here, the parameters of the CIB spectrum are fixed to β_{d} = 2.20 and T_{d} = 9.7 K, as discussed in Addison et al. (2012a). The model of Eq. (30) then relates the Planck reference amplitudes of Eqs. (29b), (29c) to the neighbouring Planck, ACT, and SPT effective frequencies, assuming that the CIB is perfectly correlated over these small frequency ranges.
It has been common practice in recent CMB parameter studies to fix the slope of the CIB spectrum to γ^{CIB} = 0.8 (e.g., Story et al. 2013; Dunkley et al. 2013). In fact, the shape of the CIB spectrum is poorly constrained at frequencies below 353 GHz and we have decided to reflect this uncertainty by allowing the slope γ^{CIB} to vary. We adopt a Gaussian prior on γ^{CIB} with a mean of 0.7 and a dispersion of 0.2. In reality, the CIB spectrum is likely to have some degree of curvature reflecting the transition between linear (twohalo) and nonlinear (onehalo) clustering (see e.g., Cooray & Sheth 2002; Planck Collaboration XVIII 2011; Amblard et al. 2011; Thacker et al. 2013). However, a single power law is an adequate approximation within the restricted multipole range (500 ≲ ℓ ≲ 3000) over which the CIB contributes significantly to the Planck/ACT/SPT highfrequency spectra (as judged by the foregroundcorrected power spectrum residuals shown in Figs. 7–9 below). The prior on γ^{CIB} is motivated, in part, by the mapbased Planck CIB analysis discussed in Planck Collaboration XXX (2014) (see also Planck Collaboration XVIII 2014). Appendix C explores different parameterizations of the CIB power spectrum.
ThermalSZ/CIB crosscorrelation:
the crosscorrelation between dust emission from CIB galaxies and SZ emission from clusters (tSZ×CIB) is expected to be nonzero. Because of uncertainties in the modelling of the CIB, it is difficult to compute this correlation with a high degree of precision. Addison et al. (2012b) present a halomodel approach to model this term and conclude that anticorrelations of around 10–20% are plausible between the clustered CIB components and the SZ at 150 GHz. The tSZ×CIB correlation is therefore expected to make a minor contribution to the unresolved foreground emission, but it is nevertheless worth including to determine how it might interact with other subdominant components, in particular the kSZ contribution. We use the Addison et al. (2012b) template spectrum in this paper and model the frequency dependence of the power spectrum as follows: (32)where is the Addison et al. (2012b) template spectrum normalized to unity at ℓ = 3000 and and are given by Eqs. (27) and (31). The tSZ×CIB contribution is therefore characterized by the dimensionless crosscorrelation coefficient ξ^{tSZ × CIB}. With the definition of Eq. (32), a positive value of ξ^{tSZ × CIB} corresponds to an anticorrelation between the CIB and the tSZ signals.
Galactic dust:
for the masks used in the Planck CamSpec likelihood, Galactic dust makes a small contribution to of around 5 μK^{2} to the 217 × 217 power spectrum, 1.5 μK^{2} to the 143 × 217 spectrum, and around 0.5 μK^{2} to the 143 × 143 spectrum. We subtract the Galactic dust contributions from these power spectra using a “universal” dust template spectrum (at high multipoles this is accurately represented by a power law ). The template spectrum is based on an analysis of the 857 GHz Planck maps described in Planck Collaboration XV (2014), which uses maskdifferenced power spectra to separate Galactic dust from an isotropic extragalactic CIB contribution. This Galactic dust correction is kept fixed with an amplitude determined by template fitting the 217 and 143 GHz Planck maps to the 857 GHz map, as described in Planck Collaboration XV (2014). Galactic dust contamination is ignored in the 100 × 100 spectrum^{16}. The Galactic dust template spectrum is actually a good fit to the dust contamination at low multipoles, ℓ ≪ 1000; however, we limit the effects of any inaccuracies in dust subtraction at low multipoles by truncating the 217 × 217 and 143 × 217 spectra at a minimum multipole of ℓ_{min} = 500. (At multipoles ℓ ≲ 1000, the Planck temperature power spectra are signal dominated, so the 100 × 100 and 143 × 143 spectra contain essentially all of the information on cosmology.)
Compared to the contribution of Poisson point sources and the CIB, Galactic dust is a minor foreground component at 217 GHz within our default mask, which retains 37% of the sky. However, the contribution of Galactic dust emission rises rapidly as more sky area is used. Extending the sky mask to 65% of the sky (using the sequence of masks described in Planck Collaboration XV 2014), Galactic dust contributes to around 50 μK^{2} at 217 GHz (rising to around 200 μK^{2} on the scale of the first acoustic peak) and becomes a major foreground component, with an amplitude close to the net contribution of Poisson point sources and the clustered CIB. There is therefore a tradeoff between limiting the signaltonoise at 143 and 217 GHz, by restricting the sky area, and potential systematic errors associated with modelling Galactic dust over a large area of sky (i.e., sensitivity to the assumption of a “universal” dust template spectrum). We have chosen to be conservative in this first cosmological analysis of Planck by limiting the sky area at 143 and 217 GHz so that dust contamination is a minor foreground at high multipoles. As a further test of the importance of Galactic dust, we have analysed a Planck likelihood that retains only 24.7% of the sky (see Planck Collaboration XV 2014) at 217 GHz. Within this mask the CIB dominates over Galactic dust at multipoles ℓ ≳ 500. There is a signaltonoise penalty in using such a small area of sky at 217 GHz, but otherwise the results from this likelihood are in good agreement with the results presented here. With the conservative choices adopted in this paper, Galactic dust has no significant impact on our cosmological results.
We follow R12 and subtract a smallscale dust contribution of from the R12 220 GHz spectrum. This correction was determined by crosscorrelating the SPT data with model 8 of Finkbeiner et al. (1999). For the ACT data we marginalize over a residual Galactic dust component , with different amplitudes for the southern and equatorial spectra, imposing Gaussian priors and frequency scaling as described in Dunkley et al. (2013).
Notice that the spectral index of the SPT dust correction is significantly steeper than the dust correction applied to the Planck spectra. In future analyses it would be useful to derive more accurate dust corrections for the highresolution CMB data by crosscorrelating the SPT and ACT maps with the Planck 545 and 857 GHz maps. Since the dust corrections are relatively small for the highresolution data used here, we adopt the correction described above in this paper.
In application of the likelihood to Planck data alone, the model for unresolved foregrounds and relative calibrations contains 13 parameters. In addition, we can solve for up to 20 beam eigenmode amplitudes (five amplitudes for each of the four spectra used in the Planck likelihood; see Planck Collaboration XV 2014). In practice, we find that (usually) only the first beam eigenmode for the 100 × 100 spectrum, , has a posterior distribution that differs perceptibly from the prior, and we obtain nearly identical results on both foreground and cosmological parameters if we treat only the amplitude of this eigenmode as a parameter and analytically marginalize over the rest. This is the default adopted in this paper. (The analytic marginalization improves stability of the minimisation for bestfit searches, and makes the Planck likelihood less cumbersome for the user.)
The addition of ACT and SPT data introduces 17 extra parameters. We provide a summary of the 50 foreground and nuisance parameters in Table 4, including the prior ranges adopted in our MCMC analysis^{17}. The choice of priors for many of these parameters is, to a large extent, subjective. They were chosen at an early stage in the Planck analysis to reflect “theoretically plausible” allowed ranges of the foreground parameters and to be broad compared to the results from highresolution CMB experiments (which evolved over the course of this analysis as results from more ACT and SPT data were published). The foreground parameters from ACT and SPT depend on the assumptions of the underlying cosmology, and hence it is possible to introduce biases in the solutions for extensions to the base ΛCDM cosmology if overly restrictive foreground priors are imposed on the Planck data. Using the priors summarized in Table 4, the consistency between the Planckalone results and the solutions for Planck combined with ACT and SPT provides a crude (but informative) measure of the sensitivity of cosmological results on the foreground model. Appendix C discusses the effects on extended ΛCDM models of varying the priors on minor foreground components.
Fig. 6 Comparison of the posterior distributions of the foreground parameters for Planck+WP (red) and Planck+WP+highL (black). 
4.3. The base ΛCDM model
Cosmological and foreground parameters for the base sixparameter ΛCDM model are listed in Table 5, which gives bestfit values and 68% confidence limits. The first two columns list the parameters derived from the Planck+WP analysis discussed in Sect. 3, and are repeated here for easy reference. The next two columns list the results of combining the Planck+WP likelihoods with the ACT and SPT likelihoods following the model described above. We refer to this combination as “Planck+WP+highL” in this paper. The remaining columns list the parameter constraints combining the Planck+WP+highL likelihood with the Planck lensing and BAO likelihoods (see Sect. 5). Table 5 lists the cosmological parameters for the base ΛCDM model and a selection of derived cosmological parameters. These parameters are remarkably stable for such data combinations. We also list the values of the parameters describing the Planck foregrounds. A full list of all parameter values, including nuisance parameters, is given in the Explanatory Supplement (Planck Collaboration 2013).
A comparison of the foreground parameter constraints from Planck+WP and Planck+WP+highL is shown in Fig. 6; the corresponding cosmological parameter constraints are shown in Fig. 4.
We can draw the following general conclusions.

The cosmological parameters for the baseΛCDM model are extremely insensitive to theforeground model described in the previous subsection. Theaddition of the ACT and SPT data causes the posteriordistributions of cosmological parameters to shift by much lessthan one standard deviation.
With Planck data alone, the CIB amplitude at 217 GHz is strongly degenerate with the 217 GHz Poisson point source amplitude. This degeneracy is broken by the addition of the highresolution CMB data. This degeneracy must be borne in mind when interpreting Planckonly solutions for CIB parameters; the sum of the Poisson point source and CIB contributions are well constrained by Planck at 217 GHz (and in good agreement with the mapbased CIB Planck analysis reported in Planck Collaboration XI 2014), whereas the individual contributions are not. Another feature of the CIB parameters is that we typically find smaller values of the CIB spectral index, γ^{CIB}, in Planckalone solutions compared to Planck+highL solutions (which can be seen in Fig. 6). This provided additional motivation to treat γ^{CIB} as a parameter in the Planck likelihood rather than fixing it to a particular value. There is evidence from the Planck spectra (most clearly seen by differencing the 217 × 217 and 143 × 143 spectra) that the CIB spectrum at 217 GHz flattens in slope over the multipole range 500 ≲ ℓ ≲ 1000. This will be explored in further detail in future papers (see also Appendix C).
The addition of the ACT and SPT data constrains the thermal SZ amplitude, which is poorly determined by Planck alone. In the Planckalone analysis, the tSZ amplitude is strongly degenerate with the Poisson point source amplitude at 100 GHz. This degeneracy is broken when the highresolution CMB data are added to Planck.
The last two points are demonstrated clearly in Fig. 7, which shows the residuals of the Planck spectra with respect to the bestfit cosmology for the Planck+WP analysis compared to the Planck+WP+highL fits. The addition of highresolution CMB data also strongly constrains the net contribution from the kSZ and tSZ×CIB components (dotted lines), though these components are degenerate with each other (and tend to cancel).
Fig. 7 Power spectrum residual plots illustrating the accuracy of the foreground modelling. For each crossspectrum, there are two subfigures. The upper subfigures show the residuals with respect to the Planck+WP bestfit solution (from Table 5). The lowers subfigure show the residuals with respect to the Planck+WP+highL solution The upper panel in each subfigure shows the residual between the measured power spectrum and the bestfit (lensed) CMB power spectrum. The lower panels show the residuals after further removing the bestfit foreground model. The lines in the upper panels show the various foreground components. Major foreground components are shown by the solid lines, colour coded as follows: total foreground spectrum (red); Poisson point sources (orange); clustered CIB (blue); thermal SZ (green); and Galactic dust (purple). Minor foreground components are shown by the dotted lines colour coded as follows: kinetic SZ (green); tSZ×CIB crosscorrelation (purple). We also show residuals for the two spectra 100 × 143 and 100 × 217 that are not used in the Planck likelihood. For these, we have assumed Poisson pointsource correlation coefficients of unity.The χ^{2} values of the residuals, and the number of bandpowers, are listed in the lower panels. 
Fig. 8 SPT power spectra at high multipoles using the foreground model developed in this paper. The SPT R12 power spectra for each frequency combination are shown by the blue points, together with 1σ error bars. The foreground components, determined from the Planck+WP+highL analysis of ΛCDM models, are shown in the upper panels using the same colour coding as in Fig. 7. Here, the spectrum of the bestfit CMB is shown in red and the total spectra are the upper green curves. The lower panel in each subfigure shows the residuals with respect to the bestfit base ΛCDM cosmology+foreground model. The χ^{2} values of the residuals, and the number of SPT bandpowers, are listed in the lower panels. 
Although the foreground parameters for the Planck+WP fits can differ substantially from those for Planck+WP+highL, the total foreground spectra are insensitive to the addition of the highresolution CMB data. For example, for the 217 × 217 spectrum, the differences in the total foreground solution are less than 10 μK^{2} at ℓ = 2500. The net residuals after subtracting both the foregrounds and CMB spectrum (shown in the lower panels of each subplot in Fig. 7) are similarly insensitive to the addition of the highresolution CMB data. The foreground model is sufficiently complex that it has a high “absorptive capacity” to any smoothlyvarying frequencydependent differences between spectra (including beam errors).
Goodnessoffit tests for the Planck spectra.
To quantify the consistency of the model fits shown in Fig. 7 for Planck we compute the χ^{2} statistic (33)for each of the spectra, where the sums extend over the multipole ranges ℓ_{min} and ℓ_{max} used in the likelihood, ℳ_{ℓℓ′} is the covariance matrix for the spectrum (including corrections for beam eigenmodes and calibrations), is the bestfit primordial CMB spectrum and is the bestfit foreground model appropriate to the data spectrum. We expect χ^{2} to be approximately Gaussian distributed with a mean of N_{ℓ} = ℓ_{max} − ℓ_{min} + 1 and dispersion . Results are summarized in Table 6 for the Planck+WP+highL bestfit parameters of Table 5. (The χ^{2} values for the Planck+WP fit are almost identical.) Each of the spectra gives an acceptable global fit to the model, quantifying the high degree of consistency of these spectra described in Planck Collaboration XV (2014). (Note that Planck Collaboration XV 2014 presents an alternative way of investigating consistency between these spectra via power spectrum differences.)
Figures 8 and 9 show the fits and residuals with respect to the bestfit Planck+WP+highL model of Table 5, for each of the SPT and ACT spectra. The SPT and ACT spectra are reported as bandpowers, with associated window functions and . The definitions of these window functions differ between the two experiments.
For SPT, the contribution of the CMB and foreground spectra in each band is (34)(Note that this differs from the equations given in R12 and S12.)
For ACT, the window functions operate on the power spectra: (35)In Fig. 9. we plot , where ℓ_{b} is the effective multipole for band b.
The upper panels of each of the subplots in Figs. 8 and 9 show the spectra of the bestfit CMB, and the total CMB+foreground, as well as the individual contributions of the foreground components using the same colour codings as in Fig. 7. The lower panel in each subplot shows the residuals with respect to the bestfit cosmology+foreground model. For each spectrum, we list the value of χ^{2}, neglecting correlations between the (broad) ACT and SPT bands, together with the number of data points. The quality of the fits is generally very good. For SPT, the residuals are very similar to those inferred from Fig. 3 of R12. The SPT 150 × 220 spectrum has the largest χ^{2} (approximately a 1.8σ excess). This spectrum shows systematic positive residuals of a few μK^{2} over the entire multipole range. For ACT, the residuals and χ^{2} values are close to those plotted in Fig. 4 of Dunkley et al. (2013). All of the ACT spectra plotted in Fig. 9 are well fit by the model (except for some residuals at multipoles ℓ ≲ 2000, which are also seen by Dunkley et al. 2013).
Fig. 10 PlanckTT power spectrum. The points in the upper panel show the maximumlikelihood estimates of the primary CMB spectrum computed as described in the text for the bestfit foreground and nuisance parameters of the Planck+WP+highL fit listed in Table 5. The red line shows the bestfit base ΛCDM spectrum. The lower panel shows the residuals with respect to the theoretical model. The error bars are computed from the full covariance matrix, appropriately weighted across each band (see Eqs. (36a) and (36b)) and include beam uncertainties and uncertainties in the foreground model parameters. 
Having determined a solution for the bestfit foreground and other “nuisance” parameters, we can correct the four spectra used in the Planck likelihood and combine them to reconstruct a “bestfit” primary CMB spectrum and covariance matrix as described in Planck Collaboration XV (2014). This bestfit Planck CMB spectrum is plotted in the upper panels of Figs. 1 and 10 for Planck+WP+highL foreground parameters. The spectrum in Fig. 10 has been bandaveraged in bins of width Δℓ ~ 31 using a window function W_{b}(l): (36a)(36b)Here, and denote the minimum and maximum multipole ranges of band b, and is the covariance matrix of the bestfit spectrum , computed as described in Planck Collaboration XV (2014), and to which we have added corrections for beam and foreground errors (using the curvature matrix of the foreground model parameters from the MCMC chains). The solid lines in the upper panels of Figs. 1 and 10 show the spectrum for the bestfit ΛCDM cosmology. The residuals with respect to this cosmology are plotted in the lower panel. To assess the goodnessoffit, we compute χ^{2}: (37)using the covariance matrix for the bestfit data spectrum (including foreground and beam errors^{18}). The results are given in the last line of Table 6 labelled “All.” The lower panel of Fig. 10 shows the residuals with respect to the bestfit cosmology (on an expanded scale compared to Fig. 1). There are some visually striking residuals in this plot, particularly in the regions ℓ ~ 800 and ℓ ~ 1300–1500 (where we see “oscillatory” behaviour). As discussed in detail in Planck Collaboration XV (2014), these residuals are reproducible to high accuracy across Planck detectors and across Planck frequencies; see also Fig. 7. There is therefore strong evidence that the residuals at these multipoles, which are in the largely signal dominated region of the spectrum, are real features of the primordial CMB sky. These features are compatible with statistical fluctuations of a Gaussian ΛCDM model, and are described accurately by the covariance matrix used in the Planck likelihood. As judged by the χ^{2} statistic listed in Table 6, the best fit reconstructed Planck spectrum is compatible with the base ΛCDM cosmology to within 1.6σ^{19}.
Fig. 11 PlanckTE (left) and EE spectra (right) computed as described in the text. The red lines show the polarization spectra from the base ΛCDM Planck+WP+highL model, which is fitted to the TT data only. 
To the extremely high accuracy afforded by the Planck data, the power spectrum at high multipoles is compatible with the predictions of the base six parameter ΛCDM cosmology. This is the main result of this paper. Figure 1 does, however, suggest that the power spectrum of the bestfit base ΛCDM cosmology has a higher amplitude than the observed power spectrum at multipoles ℓ ≲ 30. We return to this point in Sect. 7.
Finally, Fig. 11 shows examples of PlanckTE and EE spectra. These are computed by performing a straight average of the (scalar) beamcorrected 143 × 143, 143 × 217, and 217 × 217 crossspectra (ignoring autospectra). There are 32TE and ET crossspectra contributing to the mean TE spectrum plotted in Fig. 11, and six EE spectra contributing to the mean EE spectrum. Planck polarization data, including LFI and 353 GHz data not shown here, will be analysed in detail, and incorporated into a Planck likelihood, following this data release. The purpose of presenting these figures here is twofold: first, to demonstrate the potential of Planck to deliver high quality polarization maps and spectra, as described in the Planck “bluebook” (Planck Collaboration 2005); and, second, to show the consistency of these polarization spectra with the temperature spectrum shown in Fig. 10. As discussed in Planck Collaboration VI (2014) and Planck Collaboration XV (2014), at present, the HFI polarization spectra at low multipoles (ℓ ≲ 200) are affected by systematic errors that cause biases. For the HFI channels used in Fig. 11, there are two primary sources of systematic error arising from nonlinear gainlike variations, and residual bandpass mismatches between detectors. However, these systematics rapidly become unimportant at higher multipoles^{20}.
The errors on the mean TE and EE spectra shown in Fig. 11 are computed from the analytic formulae given in Efstathiou (2006), using an effective beamwidth adjusted to reproduce the observed scatter in the polarization spectra at high multipoles. The spectra are then bandaveraged as in Eq. (37). The error bars shown in Fig. 11 are computed from the diagonal components of the bandaveraged covariance matrices.
The solid lines in the upper panels of Fig. 11 show the theoretical TE and EE spectra expected in the bestfit Planck+WP+highL ΛCDM model (i.e., the model used to compute the theory TT spectrum plotted in Fig. 10). These theoretical spectra are determined entirely from the TT analysis and make no use of the Planck polarization data. As with the TT spectra, the ΛCDM model provides an extremely good match to the polarization spectra. Furthermore, polarized foreground emission is expected to be unimportant at high multipoles (e.g., Tucci & Toffolatti 2012) and so no foreground corrections have been made to the spectra in Fig. 11. The agreement between the polarization spectra and the theoretical spectra therefore provides strong evidence that the bestfit cosmological parameters listed in Table 5 are not strongly affected by the modelling of unresolved foregrounds in the TT analysis.
5. Comparison of the Planck base ΛCDM model with other astrophysical data sets
Unlike CMB data, traditional astrophysical data sets – e.g., measurements of the Hubble parameter, type Ia supernovae (SNe Ia), and galaxy redshift surveys – involve complex physical systems that are not understood at a fundamental level. Astronomers are therefore reliant on internal consistency tests and empirical calibrations to limit the possible impact of systematic effects. Examples include calibrating the metallicity dependence of the Cepheid period luminosity relation, calibrating the colourdeclinerateluminosity relation of Type Ia supernovae, or quantifying the relationship between the spatial distributions of galaxies and dark matter. In addition, there are more mundane potential sources of error, which can affect certain types of astrophysical observations (e.g., establishing consistent photometric calibration systems). We must be open to the possibility that unknown, or poorly quantified, systematic errors may be present in the astrophysical data, especially when used in combination with the high precision data from Planck.
We have seen in the previous section that the base ΛCDM model provides an acceptable fit to the PlanckTT power spectra (and the PlanckTE and EE spectra) and also to the ACT and SPT temperature power spectra. The cosmological parameters of this model are determined to high precision. We therefore review whether these parameters provide acceptable fits to other astrophysical data. If they do not, then we need to assess whether the discrepancy is a pointer to new physics, or evidence of some type of poorly understood systematic effect. Unless stated otherwise, we use the Planck+WP+highL parameters listed in Table 5 as the default “Planck” parameters for the base ΛCDM model.
5.1. CMB lensing measured by Planck
Weak gravitational lensing by largescale structure subtly alters the statistics of the CMB anisotropies, encoding information about the latetime Universe which is otherwise degenerate in the primary anisotropies laid down at last scattering (see Lewis & Challinor 2006, for a review). The lensing deflections are given by the gradient of the lensing potential , which corresponds to an integrated measure of the matter distribution along the line of sight with peak sensitivity to structures around redshift 2. The rms deflection is expected to be around 2.5 arcmin and to be coherent over several degrees. We include the effect of lensing on the temperature power spectrum in all our parameter analysis, but for some results we also include the lensing information encoded in the nonGaussian trispectrum (connected 4point function) of the CMB. Lensing generates a nonzero trispectrum, which, at leading order, is proportional to the power spectrum of the lensing potential (Hu 2001).
In Planck Collaboration XVII (2014), we present a detailed analysis of CMB lensing with Planck data, including estimation of from the trispectrum computed from Planck’s maps. This paper also describes the construction of a lensing likelihood. Briefly, we first reconstruct an estimate of the lensing potential using nearoptimal quadratic estimators, following Okamoto & Hu (2003), with various Galactic and pointsource masks. The empirical power spectrum of this reconstruction, after subtraction of the Gaussian noise bias (i.e., the disconnected part of the 4point function), is then used to estimate in bandpowers. The associated bandpower errors are estimated from simulations. The lensing power spectrum is estimated from channelcoadded Planck maps at 100, 143 and 217 GHz in the multipole range ℓ = 10–1000, and also from a minimumvariance combination of the 143 and 217 GHz maps. An empirical correction for the shotnoise trispectrum of unresolved point sources is made to each spectrum, based on the measured amplitude of a generalized kurtosis of the appropriate maps. Additionally, the N^{(1)} bias of Kesden et al. (2003), computed for a fiducial ΛCDM spectrum determined from a prepublication analysis of the Planck data, is subtracted from each spectrum. This latter correction is proportional to and accounts for subdominant couplings of the trispectrum, which mix lensing power over a range of scales into the power spectrum estimates. Excellent internal consistency of the various estimates is found over the full multipole range.
The Planck lensing likelihood is based on reconstructions from the minimumvariance combination of the 143 and 217 GHz maps with 30% of the sky masked. Conservatively, only multipoles in the range ℓ = 40–400 are included, with a bandpower width Δℓ = 45. The range ℓ = 40–400 captures 90% of the signaltonoise on a measurement of the amplitude of a fiducial , while minimizing the impact of imperfections in modelling the effect of survey anisotropies on the largescale φ reconstruction (the “meanfield” of Planck Collaboration XVII 2014), and the large Gaussian noise bias on small scales. Note, however, that by restricting the range of angular scales we do lose some ability to distinguish between scaledependent modifications of , such as from massive neutrinos, and almost scaleindependent modifications, such as from changes in the equation of state of unclustered dark energy or spatial curvature.Correlated uncertainties in the beam transfer functions, pointsource corrections, and the cosmology dependence of the N^{(1)} bias give very broadband correlations between the bandpowers. These are modelled as a sum of rankone corrections to the covariance matrix and induce bandpower correlations that are small, less than 4%, but very broad. Bandpower correlations induced by masking are estimated to be less than 5% for neighbouring bins and are neglected. The likelihood is modelled as a Gaussian in the bandpowers with a fiducial (i.e., parameterindependent) covariance. For verification of this approximation, see Schmittfull et al. (2013).
The connected fourpoint function is related to the fullyreduced trispectrum by (38)(Hu 2001). In the context of lensing reconstruction, the CMB trispectrum due to lensing takes the form (39)where is the lensed temperature power spectrum and F_{ℓ1Lℓ2} is a geometric modecoupling function (Hu 2001; Hanson et al. 2011). Our estimates of derive from the measured trispectrum. They are normalized using the fiducial lensed power spectrum to account for the factors of in Eq. (39). In the likelihood, we renormalize the parameterdependent to account for the mismatch between the parameterdependent and that in the fiducial model. Since the bestfit ΛCDM model we consider in this section has a lensed temperature power spectrum that is very close to that of the fiducial model, the renormalisation factor differs from unity by less than 0.25%.
The estimated lensing power spectrum is not independent of the measured temperature power spectrum , but the dependence is very weak for Planck, and can be accurately ignored (Schmittfull et al. 2013; Planck Collaboration XVII 2014). As discussed in detail in Schmittfull et al. (2013), there are several effects to consider. First, the reconstruction noise in the estimated φ derives from chance correlations in the unlensed CMB. If, due to cosmic variance, the unlensed CMB fluctuates high at some scale, the noise in the reconstruction will generally increase over a broad range of scales. Over the scales relevant for Planck lensing reconstruction, the correlation between the measured and from this effect is less than 0.2% and, moreover, is removed by a datadependent Gaussian noise bias removal that we adopt following Hanson et al. (2011) and Namikawa et al. (2013). The second effect derives from cosmic variance of the lenses. If a lens on a given scale fluctuates high, the estimated will fluctuate high at that scale. In tandem, there will be more smoothing of the acoustic peaks in the measured , giving broadband correlations that are negative at acoustic peaks and positive at troughs. The maximum correlation is around 0.05%. If we consider estimating the amplitude of a fiducial lensing power spectrum independently from the smoothing effect of and the measured in the range ℓ = 40–400, the correlation between these estimates due to the cosmic variance of the lenses is only 4%. This amounts to a misestimation of the error on a lensing amplitude in a joint analysis of and , treated as independent, of only 2%. For physical parameters, the misestimation of the errors is even smaller: Schmittfull et al. (2013) estimate around 0.5% from a Fisher analysis. A third negligible effect is due to the T − φ correlation sourced by the late integrated SachsWolfe effect (see Planck Collaboration XIX 2014). This produces only local correlations between the measured and which are less than 0.5% by ℓ = 40 and fall rapidly on smaller scales. They produce a negligible correlation between lensing amplitude estimates for the multipole ranges considered here. The T − φ correlation is potentially a powerful probe of dark energy dynamics (e.g., Verde & Spergel 2002) and modified theories of gravity (e.g., Acquaviva et al. 2004). The power spectrum can be measured from the Planck data using the CMB 3point function (Planck Collaboration XXIV 2014) or, equivalently, by crosscorrelating the φ reconstruction with the largeangle temperature anisotropies (Planck Collaboration XIX 2014) although the detection significance is only around 3σ. The powerspectrum based analysis in this paper discards the small amount of information in the T − φ correlation from Planck. In summary, we can safely treat the measured temperature and lensing power spectra as independent and simply multiply their respective likelihoods in a joint analysis.
We note that ACT (Das et al. 2011, 2014) and SPT (van Engelen et al. 2012) have both measured the lensing power spectrum with significances of 4.6σ and 6.3σ, respectively, in the multipole ranges ℓ = 75–2050 and ℓ = 100–1500. The Planck measurements used here represent a 26σ detection. We therefore do not expect the published lensing measurements from these other experiments to carry much statistical weight in a joint analysis with Planck, despite the complementary range of angular scales probed, and we choose not to include them in the analyses in this paper.
In the lensing likelihood, we characterize the estimates of with a set of eight (dimensionless) amplitudes Â_{i}, where (40)Here, is a binning operation with (41)for ℓ within the band defined by a minimum multipole and a maximum . The inverse of the weighting function, V_{ℓ}, is an approximation to the variance of the measured and is the lensing power spectrum of the fiducial model, which is used throughout the analysis. The Â_{i} are therefore nearoptimal estimates of the amplitude of the fiducial power spectrum within the appropriate multipole range, normalized to unity in the fiducial model. Given some parameterdependent model , the expected values of the Â_{i} are (42)where the term involving , which depends on the parameterdependent , accounts for the renormalisation step described above. The lensing amplitudes Â_{i} are compared to the for the bestfitting ΛCDM model to the Planck+WP+highL data combination (i.e., not including the lensing likelihood) in Table 7. The differences between Â_{i} and are plotted in the bottom panel of Fig. 12 while in the top panel the bandpower estimates are compared to in the bestfitting model. The Planck measurements of are consistent with the prediction from the bestfit ΛCDM model to Planck+WP+highL. Using the full covariance matrix, we find χ^{2} = 10.9 with eight degrees of freedom, giving an acceptable probability to exceed of approximately 21%. It is worth recalling here that the parameters of the ΛCDM model are tightly constrained by the CMB 2point function (as probed by our Planck+WP+highL data combination) which derives from physics at z ≈ 1100 seen in angular projection. It is a significant further vindication of the ΛCDM model that its predictions for the evolution of structure and geometry at much lower redshifts (around z = 2) fit so well with Planck’s CMB lensing measurements.
Fig. 12 Planck measurements of the lensing power spectrum compared to the prediction for the bestfitting Planck+WP+highL ΛCDM model parameters. In the top panel, the data points are the measured bandpowers and ± 1σ error ranges from the diagonal of the covariance matrix. The measured bandpowers are compared to the in the bestfit model (black line). The grey region shows the 1σ range in due to ΛCDM parameter uncertainties. The lower panel shows the differences between the bandpower amplitudes Â_{i} and the predictions for their expectation values in the bestfit model, . 
Planck CMB lensing constraints.
The discussion above does not account for the small spread in the predictions across the Planck+WP+highL ΛCDM posterior distribution. To address this, we introduce a parameter which, at any point in parameter space, scales the lensing trispectrum. Note that does not alter the lensed temperature power spectrum, so it can be used to assess directly how well the ΛCDM predictions from agree with the lensing measurements; in ΛCDM we have . The marginalized posterior distribution for in a joint analysis of Planck+WP+highL and the Planck lensing likelihood is given in Fig. 13. The agreement with is excellent, with (43)The significance of the detection of lensing using in ΛCDM is a little less than the 26σ detection of lensing power reported in Planck Collaboration XVII (2014), due to the small spread in from ΛCDM parameter uncertainties.
Fig. 13 Marginalized posterior distributions for (dashed) and A_{L} (solid). For we use the data combination Planck+ lensing+ WP+ highL. For A_{L} we consider Planck+ lensing+ WP+ highL (red), Planck+ WP + highL (green), Planck+WP (blue) and Planck− lowL + highL+ τprior (cyan; see text). 
Lensing also affects the temperature power spectrum, primarily by smoothing the acoustic peaks and troughs on the scales relevant for Planck. The most significant detection of the lensing effect in the power spectrum to date is from SPT. Introducing a parameter A_{L} (Calabrese et al. 2008) which takes when computing the lensed temperature power spectrum (we shall shortly extend the action of this parameter to include the computation of the lensing trispectrum), Story et al. (2013) report (68%; SPT+WMAP7). Results for A_{L} from Planck in combination with WMAP lowℓ polarization and the highℓ power spectra from ACT and SPT are also shown in Fig. 13. Where we include the Planck lensing measurements, we define A_{L} to scale the explicit in Eq. (39), as well as modulating the lensing effect in the temperature power spectrum. Figure 13 reveals a preference for A_{L}> 1 from the Planck temperature power spectrum (plus WMAP polarization). This is most significant when combining with the highℓ experiments for which we find (44)i.e., a 2σ preference for A_{L}> 1. Including the lensing measurements, the posterior narrows but shifts to lower A_{L}, becoming consistent with A_{L} = 1 at the 1σ level as expected from the results.
Fig. 14 Effect of allowing A_{L} to vary on the degeneracies between Ω_{b}h^{2} and n_{s} (left) and Ω_{m}h^{2} and n_{s} (right). In both panels the data combination is Planck+WP+highL. The contours enclose the 68% and 95% confidence regions in the base ΛCDM model with A_{L} = 1. The samples are from models with variable A_{L} and are colourcoded by the value of A_{L}. 
We do not yet have a full understanding of what is driving the preference for high A_{L} in the temperature power spectrum. As discussed in Appendix C, the general preference is stable to assumptions about foreground modelling and cuts of the Planck data in the likelihood. To gain some insight, we consider the range of multipoles that drive the preference for A_{L}> 1. For our favoured data combination of Planck+WP+highL, Δχ^{2} = − 5.2 going from the bestfit A_{L} = 1 model to the bestfit model with variable A_{L}. The improvement in fit comes only from the lowℓ temperature power spectrum (Δχ^{2} = − 1.9) and the ACT+SPT data (Δχ^{2} = − 3.3); for this data combination, there is no preference for high A_{L} from the Planck temperature data at intermediate and high multipoles (Δχ^{2} = + 0.2). The situation at lowℓ is similar if we exclude the highℓ experiments, with Δχ^{2} = − 1.6 there, but there is then a preference for the high A_{L} bestfit from the Planck data on intermediate and small scales (Δχ^{2} = − 3.4). However, as discussed in Sect. 4, there is more freedom in the foreground model when we exclude the highℓ data, and this can offset smooth differences in the CMB power spectra such as the transfer of power from large to small scales by lensing that is enhanced for A_{L}> 1.
Since the lowℓ temperature data seem to be partly responsible for pulling A_{L} high, we consider the effect of removing the lowℓ likelihood from the analysis. In doing so, we also remove the WMAP largeangle polarization which we compensate by introducing a simple prior on the optical depth; we use a Gaussian with mean 0.09 and standard deviation 0.013, similar to the constraint from WMAP polarization (Hinshaw et al. 2012). We denote this data combination, including the highℓ experiments, by Planck−lowL+highL+τprior and show the posterior for A_{L} in Fig. 13. As anticipated, the peak of the posterior moves to lower A_{L} giving (68% CL). The Δχ^{2} = + 1.1 between the bestfit model (now at A_{L} = 1.18) and the A_{L} = 1 model for the Planck data (i.e. no preference for the higher A_{L}) while Δχ^{2} = − 3.6 for the highℓ experiments.
Since varying A_{L} alone does not alter the power spectrum on large scales, why should the lowℓ data prefer higher A_{L}? The reason is due to a chain of parameter degeneracies that are illustrated in Fig. 14, and the deficit of power in the measured C_{ℓ}s on large scales compared to the bestfit ΛCDM model (see Fig. 1 and Sect. 7). In models with a powerlaw primordial spectrum, the temperature power spectrum on large scales can be reduced by increasing n_{s}. The effect of an increase in n_{s} on the relative heights of the first few acoustic peaks can be compensated by increasing ω_{b} and reducing ω_{m}, as shown by the contours in Fig. 14. However, on smaller scales, corresponding to modes that entered the sound horizon well before matterradiation equality, the effects of baryons on the midpoint of the acoustic oscillations (which modulates the relative heights of even and odd peaks) is diminished since the gravitational potentials have pressuredamped away during the oscillations in the radiationdominated phase (e.g., Hu & White 1996, 1997a). Moreover, on such scales the radiationdriving at the onset of the oscillations that amplifies their amplitude happens early enough to be unaffected by small changes in the matter density. The net effect is that, in models with A_{L} = 1, the extent of the degeneracy involving n_{s}, ω_{b} and ω_{m} is limited by the higherorder acoustic peaks, and there is little freedom to lower the largescale temperature power spectrum by increasing n_{s} while preserving the good fit at intermediate and small scales. Allowing A_{L} to vary changes this picture, letting the degeneracy extend to higher n_{s}, as shown by the samples in Fig. 14. The additional smoothing of the acoustic peaks due to an increase in A_{L} can mitigate the effect of increasing n_{s} around the fifth peak, where the signaltonoise for Planck is still high^{21}. This allows one to decrease the spectrum at low ℓ, while leaving it essentially unchanged on those smaller scales where Planck still has good sensitivity. Above ℓ ~ 2000, the bestfit A_{L} model has a little more power than the base model (around 3 μK^{2} at ℓ = 2000), while the Planck, ACT, and SPT data have excess power over the bestfit A_{L} = 1ΛCDM+foreground model at the level of a few μK^{2} (see Sect. 4). It is plausible that this may drive the preference for high A_{L} in the χ^{2} of the highℓ experiments. We note that a similar 2σ preference for A_{L}> 1 is also found combining ACT and WMAP data (Sievers et al. 2013) and, as we find here, this tension is reduced when the lensing power spectrum is included in the fit.
To summarize, there is no preference in the Planck lensing power spectrum for A_{L}> 1. The general preference for high A_{L} from the CMB power spectra in our favoured data combination (Planck+WP+highL) is mostly driven by two effects: the difficulty that ΛCDM models have in fitting the lowℓ spectrum when calibrated from the smallerscale spectrum; and, plausibly, from excess residuals at the μK^{2} level in the highℓ spectra relative to the bestfit A_{L} = 1ΛCDM+foregrounds model on scales where extragalactic foreground modelling is critical.
5.2. Baryon acoustic oscillations
Baryon acoustic oscillations (BAO) in the matter power spectrum were first detected in analyses of the 2dF Galaxy Redshift Survey (Cole et al. 2005) and the SDSS redshift survey (Eisenstein et al. 2005). Since then, accurate BAO measurements have been made using a number of different galaxy redshift surveys, providing constraints on the distance luminosity relation spanning the redshift range 0.1 ≲ z ≲ 0.7^{22}. Here we use the results from four redshift surveys: the SDSS DR7 BAO measurements at effective redshifts z_{eff} = 0.2 and z_{eff} = 0.35, analysed by Percival et al. (2010); the z = 0.35 SDSS DR7 measurement at z_{eff} = 0.35 reanalysed by Padmanabhan et al. (2012); the WiggleZ measurements at z_{eff} = 0.44, 0.60 and 0.73 analysed by Blake et al. (2011); the BOSS DR9 measurement at z_{eff} = 0.57 analysed by Anderson et al. (2012); and the 6dF Galaxy Survey measurement at z = 0.1 discussed by Beutler et al. (2011).
BAO surveys measure the distance ratio (45)where r_{s}(z_{drag}) is the comoving sound horizon at the baryon drag epoch (when baryons became dynamically decoupled from the photons) and D_{V}(z) is a combination of the angulardiameter distance, D_{A}(z), and the Hubble parameter, H(z), appropriate for the analysis of sphericallyaveraged twopoint statistics: (46)In the ΛCDM cosmology (allowing for spatial curvature), the angular diameter distance to redshift z is (47)where (48)and sin_{K} = sinh for Ω_{K}> 0 and sin_{K} = sin for Ω_{K}< 0. (The small effects of the 0.06 eV massive neutrino in our base cosmology are ignored in Eq. (48).) Note that the luminosity distance, D_{L}, relevant for the analysis of Type Ia supernovae (see Sect. 5.4) is related to the angular diameter distance via .
Fig. 15 Acousticscale distance ratio r_{s}/D_{V}(z) divided by the distance ratio of the Planck base ΛCDM model. The points are colourcoded as follows: green star (6dF); purple squares (SDSS DR7 as analysed by Percival et al. 2010); black star (SDSS DR7 as analysed by Padmanabhan et al. 2012); blue cross (BOSS DR9); and blue circles (WiggleZ). The grey band shows the approximate ± 1σ range allowed by Planck (computed from the CosmoMC chains). 
Different groups fit and characterize BAO features in different ways. For example, the WiggleZ team encode some shape information on the power spectrum to measure the acoustic parameter A(z), introduced by Eisenstein et al. (2005), (49)which is almost independent of ω_{m}. To simplify the presentation, Fig. 15 shows estimates of r_{s}/D_{V}(z) and 1σ errors, as quoted by each of the experimental groups, divided by the expected relation for the Planck base ΛCDM parameters. Note that the experimental groups use the approximate formulae of Eisenstein & Hu (1998) to compute z_{drag} and r_{s}(z_{drag}), though they fit power spectra computed with Boltzmann codes, such as camb, generated for a set of fiducialmodel parameters. The measurements have now become so precise that the small difference between the Eisenstein & Hu (1998) approximations and the accurate values of z_{drag} and r_{drag} = r_{s}(z_{drag}) returned by camb need to be taken into account. In CosmoMC we multiply the accurate numerical value of r_{s}(z_{drag}) by a constant factor of 1.0275 to match the EisensteinHu approximation in the fiducial model. This correction is sufficiently accurate over the range of ω_{m} and ω_{b} allowed by the CMB in the base ΛCDM cosmology (see e.g. Mehta et al. 2012) and also for the extended ΛCDM models discussed in Sect. 6.
The Padmanabhan et al. (2012) result plotted in Fig. 15 is a reanalysis of the z_{eff} = 0.35 SDSS DR7 sample discussed by Percival et al. (2010). Padmanabhan et al. (2012) achieve a higher precision than Percival et al. (2010) by employing a reconstruction technique (Eisenstein et al. 2007) to correct (partially) the baryon oscillations for the smearing caused by galaxy peculiar velocities. The Padmanabhan et al. (2012) results are therefore strongly correlated with those of Percival et al. (2010). We refer to the Padmanabhan et al. (2012) “reconstructioncorrected” results as SDSS(R). A similar reconstruction technique was applied to the BOSS survey by Anderson et al. (2012) to achieve 1.6% precision in D_{V}(z = 0.57) /r_{s}, the most precise determination of the acoustic oscillation scale to date.
All of the BAO measurements are compatible with the base ΛCDM parameters from Planck. The grey band in Fig. 15 shows the ± 1σ range in the acousticscale distance ratio computed from the Planck+WP+highL CosmoMC chains for the base ΛCDM model. To get a qualitative feel for how the BAO measurements constrain parameters in the base ΛCDM model, we form χ^{2}, (50)where x is the data vector, x^{ΛCDM} denotes the theoretical prediction for the ΛCDM model and is the inverse covariance matrix for the data vector x. The data vector is as follows: D_{V}(0.106) = (457 ± 27) Mpc (6dF); r_{s}/D_{V}(0.20) = 0.1905 ± 0.0061, r_{s}/D_{V}(0.35) = 0.1097 ± 0.0036 (SDSS); A(0.44) = 0.474 ± 0.034, A(0.60) = 0.442 ± 0.020, A(0.73) = 0.424 ± 0.021 (WiggleZ); D_{V}(0.35) /r_{s} = 8.88 ± 0.17 (SDSS(R)); and D_{V}(0.57) /r_{s} = 13.67 ± 0.22, (BOSS). The offdiagonal components of for the SDSS and WiggleZ results are given in Percival et al. (2010) and Blake et al. (2011). We ignore any covariances between surveys. Since the SDSS and SDSS(R) results are based on the same survey, we include either one set of results or the other in the analysis described below, but not both together.
The EisensteinHu values of r_{s} for the Planck and WMAP9 base ΛCDM parameters differ by only 0.9%, significantly smaller than the errors in the BAO measurements. We can obtain an approximate idea of the complementary information provided by BAO measurements by minimizing Eq. (50) with respect to either Ω_{m} or H_{0}, fixing ω_{m} and ω_{b} to the CMB bestfit parameters. (We use the Planck+WP+highL parameters from Table 5.) The results are listed in Table 8^{23}.
Approximate constraints with 68% errors on Ω_{m} and H_{0} (in units of km s^{1} Mpc^{1}) from BAO, with ω_{m} and ω_{b} fixed to the bestfit Planck+WP+highL values for the base ΛCDM cosmology.
As can be seen, the results are very stable from survey to survey and are in excellent agreement with the base ΛCDM parameters listed in Tables 2 and 5. The values of are also reasonable. For example, for the six data points of the 6dF+SDSS(R)+BOSS+WiggleZ combination, we find , evaluated for the Planck+WP+highL bestfit ΛCDM parameters.
The high value of Ω_{m} is consistent with the parameter analysis described by Blake et al. (2011) and with the “tension” discussed by Anderson et al. (2012) between BAO distance measurements and direct determinations of H_{0} (Riess et al. 2011; Freedman et al. 2012). Furthermore, if the errors on the BAO measurements are accurate, the constraints on Ω_{m} and H_{0} (for fixed ω_{m} and ω_{b}) are of comparable accuracy to those from Planck.
The results of this section show that BAO measurements are an extremely valuable complementary data set to Planck. The measurements are basically geometrical and free from complex systematic effects that plague many other types of astrophysical measurements. The results are consistent from survey to survey and are of comparable precision to Planck. In addition, BAO measurements can be used to break parameter degeneracies that limit analyses based purely on CMB data. For example, from the excellent agreement with the base ΛCDM model evident in Fig. 15, we can infer that the combination of Planck and BAO measurements will lead to tight constraints favouring Ω_{K} = 0 (Sect. 6.2) and a dark energy equationofstate parameter, w = − 1 (Sect. 6.5).Since the BAO measurements are primarily geometrical, they are used in preference to more complex astrophysical data sets to break CMB parameter degeneracies in this paper.
Finally, we note that we choose to use the 6dF+SDSS(R)+ BOSS data combination in the likelihood analysis of Sect. 6. This choice includes the two most accurate BAO measurements and, since the effective redshifts of these samples are widely separated, it should be a very good approximation to neglect correlations between the surveys.
5.3. The Hubble constant
A striking result from the fits of the base ΛCDM model to Planck power spectra is the low value of the Hubble constant, which is tightly constrained by CMB data alone in this model. From the Planck+WP+highL analysis we find (51)A low value of H_{0} has been found in other CMB experiments, most notably from the recent WMAP9 analysis. Fitting the base ΛCDM model, Hinshaw et al. (2012) find^{24}(52)consistent with Eq. (51) to within 1σ. We emphasize here that the CMB estimates are highly model dependent. It is important therefore to compare with astrophysical measurements of H_{0}, since any discrepancies could be a pointer to new physics.
There have been remarkable improvements in the precision of the cosmic distance scale in the past decade or so. The final results of the Hubble Space Telescope (HST) key project (Freedman et al. 2001), which used Cepheid calibrations of secondary distance indicators, resulted in a Hubble constant of H_{0} = (72 ± 8) km s^{1} Mpc^{1} (where the error includes estimates of both 1σ random and systematic errors). This estimate has been used widely in combination with CMB observations and other cosmological data sets to constrain cosmological parameters (e.g., Spergel et al. 2003, 2007). It has also been recognized that an accurate measurement of H_{0} with around 1% precision, when combined with CMB and other cosmological data, has the potential to reveal exotic new physics, for example, a timevarying dark energy equation of state, additional relativistic particles, or neutrino masses (see e.g., Suyu et al. 2012, and references therein). Establishing a more accurate cosmic distance scale is, of course, an important problem in its own right. The possibility of uncovering new fundamental physics provides an additional incentive.
Fig. 16 Comparison of H_{0} measurements, with estimates of ± 1σ errors, from a number of techniques (see text for details). These are compared with the spatiallyflat ΛCDM model constraints from Planck and WMAP9. 
Fig. 17 MCMC samples and contours in the r_{∗}Ω_{m}h^{2} plane (left) and the D_{A}(z_{∗})Ω_{m}h^{2} plane (right) for ΛCDM models analysed with Planck+WP+highL. The lines in these plots show the expected degeneracy directions in the base ΛCDM cosmology. Samples are colourcoded by the values of Ω_{b}h^{2} (left) and H_{0} (right). 
Two recent analyses have greatly improved the precision of the cosmic distance scale. Riess et al. (2011) use HST observations of Cepheid variables in the host galaxies of eight SNe Ia to calibrate the supernova magnituderedshift relation. Their “best estimate” of the Hubble constant, from fitting the calibrated SNe magnituderedshift relation, is (53)where the error is 1σ and includes known sources of systematic errors. At face value, this measurement is discrepant with the Planck estimate in Eq. (51) at about the 2.5σ level.
Freedman et al. (2012), as part of the Carnegie Hubble Program, use Spitzer Space Telescope midinfrared observations to recalibrate secondary distance methods used in the HST key project. These authors find (54)We have added the two sources of error in quadrature in the error range shown in Fig. 16. This estimate agrees well with Eq. (53) and is also discordant with the Planck value (Eq. 16) at about the 2.5σ level. The error analysis in Eq. (54) does not include a number of known sources of systematic error and is very likely an underestimate. For this reason, and because of the relatively good agreement between Eqs. (53) and (54), we do not use the estimate in Eq. (54) in the likelihood analyses described in Sect. 6.
The dominant source of error in the estimate in Eq. (53) comes from the first rung in the distance ladder. Using the megamaserbased distance to NGC4258, Riess et al. (2011) find (74.8 ± 3.1) km s^{1} Mpc^{1}^{25}. Using parallax measurements for 10 Milky Way Cepheids, they find (75.7 ± 2.6) km s^{1} Mpc^{1}, and using Cepheid observations and a revised distance to the Large Magellanic Cloud, they find (71.3 ± 3.8) km s^{1} Mpc^{1}. These estimates are consistent with each other, and the combined estimate in Eq. (53) uses all three calibrations. The fact that the error budget of measurement (53) is dominated by the “firstrung” calibrators is a point of concern. A mild underestimate of the distance errors to these calibrators could eliminate the tension with Planck.
Figure 16 includes three estimates of H_{0} based on “geometrical” methods.^{26}The estimate labelled “MCP” shows the result H_{0} = (68.0 ± 4.8) km s^{1} Mpc^{1} from the Megamaser Cosmology Project (Braatz et al. 2013) based on observations of megamasers in UGC 3789, NGC 6264 and Mrk 1419 (see also Reid et al. 2013, for a detailed analysis of UGC 3789). The point labelled “RXJ11311231” shows the estimate derived from gravitational lensing time delay measurements of the system RXJ11311231, observed as part of the “COSmological MOnitoring of GRAvitational Lenses” (COSMOGRAIL) project (Suyu et al. 2013,see also Courbin et al. 2011; Tewes et al. 2013). Finally, the point labelled SZ clusters shows the value (Bonamente et al. 2006), derived by combining tSZ and Xray measurements of rich clusters of galaxies (see Carlstrom et al. 2002, and references therein). These geometrical methods bypass the need for local distance calibrators, but each has its own sources of systematic error that need to be controlled. The geometrical methods are consistent with the Cepheidbased methods, but at present, the errors on these methods are quite large. The COSMOGRAIL measurement (which involved a “blind” analysis to prevent experimenter bias) is discrepant at about 2.5σ with the Planck value in Eq. (51). We note here a number of other direct measurements of H_{0} (Jones et al. 2005; Sandage et al. 2006; Oguri 2007; Tammann & Reindl 2013) that give lower values than the measurements summarized in Fig. 16.
The tension between the CMBbased estimates and the astrophysical measurements of H_{0} is intriguing and merits further discussion. In the base ΛCDM model, the sound horizon depends primarily on Ω_{m}h^{2} (with a weaker dependence on Ω_{b}h^{2}). This is illustrated by the lefthand panel of Fig. 17, which shows samples from the Planck+WP+highL MCMC chains in the r_{∗}Ω_{m}h^{2} plane colour coded according to Ω_{b}h^{2}. The acoustic scale parameter θ_{∗} is tightly constrained by the CMB power spectrum, and so a change in r_{∗} must be matched by a corresponding shift in the angular diameter distance to the last scattering surface D_{A}(z_{∗}). In the base ΛCDM model, D_{A} depends on H_{0} and Ω_{m}h^{2}, as shown in the righthand panel of Fig. 17. The 2.7 km s^{1} Mpc^{1} shift in H_{0} between Planck and WMAP9 is primarily a consequence of the slightly higher matter density determined by Planck (Ω_{m}h^{2} = 0.143 ± 0.003) compared to WMAP9 (Ω_{m}h^{2} = 0.136 ± 0.004). A shift of around 7 km s^{1} Mpc^{1}, necessary to match the astrophysical measurements of H_{0} would require an even larger change in Ω_{m}h^{2}, which is disfavoured by the Planck data. The tension between Planck and the direct measurements of H_{0} cannot be easily resolved by varying the parameters of the base ΛCDM model. Section 6 explore whether there are any extensions to the base ΛCDM model that can relieve this tension. In that section, results labelled “H_{0}” include a Gaussian prior on H_{0} based on the Riess et al. (2011) measurement given in Eq. (53).
Fig. 18 Magnitude residuals relative to the base ΛCDM model that best fits the SNLS combined sample (left) and the Union2.1 sample (right). The error bars show the 1σ (diagonal) errors on m_{B}. The filled grey regions show the residuals between the expected magnitudes and the bestfit to the SNe sample as Ω_{m} varies across the ± 2σ range allowed by Planck+WP+highL in the base ΛCDM cosmology. The colour coding of the SNLS samples are as follows: low redshift (blue points); SDSS (green points); SNLS threeyear sample (orange points); and HST high redshift (red points). 
Bestfit parameters for the SNLS compilations.
5.4. Type Ia supernovae
In this subsection, we analyse two SNe Ia samples: the sample of 473 SNe as reprocessed by Conley et al. (2011), which we refer to as the “SNLS” compilation; and the updated Union2.1 compilation of 580 SNe described by Suzuki et al. (2012).
5.4.1. The SNLS compilation
The SNLS “combined” compilation consists of 123 SNe Ia at low redshifts, 242 SNe Ia from the threeyear Supernova Legacy Survey (SNLS; see Regnault et al. 2009; Guy et al. 2010; Conley et al. 2011), 93 intermediate redshift SNe Ia from the Sloan Digital Sky Survey (SDSS; Holtzman et al. 2008; Kessler et al. 2009) and 14 objects at high redshift observed with (HST; Riess et al. 2007).
The “combined” sample of Conley et al. (2011) combines the results of two lightcurve fitting codes, SiFTO (Conley et al. 2008) and SALT2 (Guy et al. 2007), to produce a peak apparent Bband magnitude, m_{B}, stretch parameter s and colour for each supernova. To explore the impact of lightcurve fitting, we also analyse separately the SiFTO and SALT2 parameters. The SiFTO and SALT2 samples differ by a few SNe from the combined sample because of colour and stretch constraints imposed on the samples. We also use ancillary data, such as estimates of the stellar masses of the host galaxies and associated covariance matrices, as reported by Conley et al. (2011)^{27}.
In this section, we focus exclusively on the base ΛCDM model (i.e., w = − 1 and Ω_{K} = 0). For a flat Universe, the expected apparent magnitudes are then given by (55)where is the dimensionless luminosity distance^{28} and ℳ_{B} absorbs the Hubble constant. As in Sullivan et al. (2011), we express values of the parameter(s) ℳ_{B} in terms of an effective absolute magnitude (56)for a value of H_{0} = 70 km s^{1} Mpc^{1}.
The likelihood for this sample is then constructed as in Conley et al. (2011) and Sullivan et al. (2011): (57)where M_{B} is the vector of effective absolute magnitudes and C_{SNe} is the sum of the nonsparse covariance matrices of Conley et al. (2011) quantifying statistical and systematic errors. As in Sullivan et al. (2011), we divide the sample according to the estimated stellar mass of the host galaxy and solve for two parameters, for M_{host}< 10^{10}M_{⊙} and for M_{host} ≥ 10^{10}M_{⊙}. We adopt the estimates of the “intrinsic” scatter in m_{B} for each SNe sample given in Table 4 of Conley et al. (2011).
Fits to the SNLS combined sample are shown in the lefthand panel of Fig. 18. The bestfit parameters for the combined, SiFTO and SALT2 samples are given in Table 9. In the base ΛCDM model, the SNe data provide a constraint on Ω_{m}, independent of the CMB. As can be seen from Table 9 (and also in the analyses of Conley et al. 2011 and Sullivan et al. 2011), the SNLS combined compilation favours a lower value of Ω_{m} than we find from the CMB. The key question, of course, is whether the SNe data are statistically compatible with the Planck data. The last three rows of Table 9 give the bestfit SNe parameters constraining Ω_{m} to the Planck+WP+highL bestfit value Ω_{m} = 0.317. The grey bands in Fig. 18 show the magnitude residuals expected for a ± 2σ variation in the value of Ω_{m} allowed by the CMB data. The CMB band lies systematically low by about 0.1 mag over most of the redshift range shown in Fig. 18a.
Table 9 also lists the χ^{2} values for the Ω_{m} = 0.317 fits^{29}. The likelihood ratio for the SiFTO fits is (58)This is almost a 2σ discrepancy. (The discrepancy would appear to be much more significant if only the diagonal statistical errors were included in the covariance matrix in Eq. (57)). The likelihood ratio for the combined sample is slightly larger (0.095) and is larger still for the SALT2 sample (0.33). In summary, there is some tension between the SNLS compilations and the base ΛCDM value of Ω_{m} derived from Planck. The degree of tension depends on the lightcurve fitter and is stronger for the SiFTO and combined SNLS compilations^{30}.
5.4.2. The Union2.1 compilation
The Union2.1 compilation (Suzuki et al. 2012) is the latest application of a scheme for combining multiple SNe data sets described by Kowalski et al. (2008). The Union2.1 compilation contains 19 data sets and includes early highredshift SNe data (e.g., Riess et al. 1998; Perlmutter et al. 1999) as well as recent data from the HST Cluster Supernova Survey (Amanullah et al. 2010; Suzuki et al. 2012). The SNLS and Union2.1 compilations contain 256 SNe in common and are therefore not independent.
The SALT2 model (Guy et al. 2007) is used to fit the light curves returning a Bband magnitude at maximum light, a lightcurve shape parameter and a colour correction.(Note that the version of SALT2 used in the Union2.1 analysis is not exactly the same as that used in the SNLS analysis.) As in Eq. (55), the theoreticallypredicted magnitudes include nuisance parameters α and β multiplying the shape and colour corrections, and an additional nuisance parameter δ describing the variation of SNe luminosity with host galaxy mass (see Eq. 3 of Suzuki et al. 2012). The CosmoMC module associated with the Union2.1 sample^{31} holds the nuisance parameters fixed (α = 0.1218, β = 2.4657, and δ = − 0.03634) and computes a χ^{2} via Eq. (57) using a fixed covariance matrix that includes a model for systematic errors. An analysis of the base ΛCDM model then requires minimization with respect to only two parameters, Ω_{m} and ℳ_{B} (or equivalently, M_{B}).
Maximizing the Union2.1 likelihood, we find bestfit parameters of Ω_{m} = 0.296 and M_{B} = − 19.272 (defined as in Eq. (56) for a value of H_{0} = 70 km s^{1} Mpc^{1}) and (580 SNe). The magnitude residuals with respect to this fit are shown in the righthand panel of Fig. 18. Notice that the scatter in this plot is significantly larger than the scatter of the SNLS compilation (lefthand panel) reflecting the more diverse range of data and the lower precision of some of the earlier SNe data used in the Union2.1 compilation. Nevertheless, the Union2.1 bestfit is close to (and clearly compatible with) the Planck base ΛCDM value of Ω_{m}.
5.4.3. SNe: Summary
The results of this subsection are summarized in Fig. 19. This shows the posterior distributions for Ω_{m} in the base ΛCDM cosmology, marginalized over nuisance parameters, for each of the SNe samples. These distributions are broad (with the Union2.1 distribution somewhat broader than the SNLS distributions) and show substantial overlap. There is no obvious inconsistency between the SNe samples. The posterior distribution for Ω_{m} in the base ΛCDM model fit to Planck+WP+highL is shown by the narrow green curve. This is consistent with the Union2.1 and SNLS SALT2 results, but is in some tension with the distributions from the SNLS combined and SNLS SiFTO samples. As we see in Sect. 6, Planck combined with Planck lensing and BAO measurements overwhelm SNe data for most of the extensions of the ΛCDM model considered in this paper. However, the results presented here suggest that there could be residual systematic errors in the SNe data that are not properly accounted for in the covariance matrices. Hints of new physics based on combining CMB and SNe data should therefore be treated with caution.
5.5. Additional data
In this subsection we review a number of other astrophysical data sets that have sometimes been combined with CMB data. These data sets are not used with Planck in this paper, either because they are statistically less powerful than the data reviewed in previous subsections and/or they involve complex physics (such as the behaviour of intracluster gas in rich clusters of galaxies) which is not yet well understood.
Fig. 19 Posterior distributions for Ω_{m} (assuming a flat cosmology) for the SNe compilations described in the text. The posterior distribution for Ω_{m} from the Planck+WP+highL fits to the base ΛCDM model is shown by the solid green line. 
5.5.1. Shape information on the galaxy/matter power spectrum
Reid et al. (2010) present an estimate of the dark matter halo power spectrum, P_{halo}(k), derived from 110,756 luminous red galaxies (LRGs) from the SDSS 7th data release (Abazajian et al. 2009). The sample extends to redshifts z ≈ 0.5, and is processed to identify LRGs occupying the same dark matter halo, reducing the impact of redshiftspace distortions and recovering an approximation to the halo density field. The power spectrum P_{halo}(k) is reported in 45 bands, covering the wavenumber range 0.02 h Mpc^{1}<k< 0.2 h Mpc^{1}. The window functions, covariance matrix and CosmoMC likelihood module are available on the NASA LAMBDA web site^{32}.
The halo power spectrum is plotted in Fig. 20. The blue line shows the predicted halo power spectrum from our bestfit base ΛCDM parameters convolved with the Reid et al. (2010) window functions. Here we show the predicted halo power spectrum for the bestfit values of the “nuisance” parameters b_{0} (halo bias), a_{1}, and a_{2} (defined in Eq. 15 of Reid et al. 2010) which relate the halo power spectrum to the dark matter power spectrum (computed using camb).The Planck model gives , very close to the value of the bestfit model of Reid et al. (2010).
Fig. 20 Bandpower estimates of the halo power spectrum, P_{halo}(k), from Reid et al. (2010) together with 1σ errors. (Note that these data points are strongly correlated.) The line shows the predicted spectrum for the bestfit Planck+WP+highL base ΛCDM parameters. 
Figure 20 shows that the Planck parameters provide a good match to the shape of the halo power spectrum. However, we do not use these data (in this form) in conjunction with Planck. The BAO scale derived from these and other data is used with Planck, as summarized in Sect. 5.2. As discussed by Reid et al. (2010, see their Fig. 5) there is little additional information on cosmology once the BAO features are filtered from the spectrum, and hence little to be gained by adding this information to Planck. The corrections for nonlinear evolution, though small in the wavenumber range 0.1–0.2 h Mpc^{1}, add to the complexity of using shape information from the halo power spectrum.
5.5.2. Cosmic shear
Another key cosmological observable is the distortion of distant galaxy images by the gravitational lensing of largescale structure, often called cosmic shear. The shear probes the (nonlinear) matter density projected along the line of sight with a broad kernel. It is thus sensitive to the geometry of the Universe and the growth of largescale structure, with a strong sensitivity to the amplitude of the matter power spectrum.
The most recent, and largest, cosmic shear data sets are provided by the CFHTLenS survey (Heymans et al. 2012; Erben et al. 2013), which covers^{33}154 deg^{2} in five optical bands with accurate shear measurements and photometric redshifts. The CFHTLenS team has released several cosmic shear results that are relevant to this paper. Benjamin et al. (2013) present results from a twobin tomographic analysis and Heymans et al. (2013) from a finely binned tomographic analysis. Kilbinger et al. (2013) present constraints from a 2D analysis. The constraints from all of the analyses show a high degree of consistency.
Heymans et al. (2013) estimate shear correlation functions associated with six redshift bins. Assuming a flat, ΛCDM model, from the weak lensing data alone they find σ_{8}(Ω_{m}/ 0.27)^{0.46 ± 0.02} = 0.774 ± 0.04 (68% errors) which is consistent with the constraint found by Benjamin et al. (2013). For comparison, we find (59)which is discrepant at about the 2σ level. Combining the tomographic lensing data with CMB constraints from WMAP7, Heymans et al. (2013) are able to constrain the individual parameters of the flat, ΛCDM model to be Ω_{m} = 0.255 ± 0.014 and h = 0.717 ± 0.016. The bestfit Planck value of Ω_{m} is 4σ away from this value, while h is discrepant at nearly 3σ. As might be expected, given the good agreement between the Planck and BAO distance scales, the bestfit CFHTLenS ΛCDM cosmology is also discrepant with the BOSS data, predicting a distance ratio to z = 0.57 which is 5% lower than measured by BOSS (Anderson et al. 2012). This is discrepant at approximately the 3σ level, comparable to the discrepancy with the Planck values. The source of the discrepancies between Planck and the CFHTLenS tomographic analyses is at present unclear, and further work will be needed to resolve them.
Kilbinger et al. (2013) give a tight constraint in the σ_{8}–Ω_{m} plane for flat ΛCDM models from their 2D (i.e., nontomographic) analysis. They find σ_{8}(Ω_{m}/ 0.27)^{0.6} = 0.79 ± 0.03, which, when combined with WMAP7, gives Ω_{m} = 0.283 ± 0.010 and h = 0.69 ± 0.01. These results are still discrepant with the Planck bestfit, but with lower significance than the results reported by Heymans et al. (2013).
It is also worth noting that a recent analysis of galaxygalaxy lensing in the SDSS survey (Mandelbaum et al. 2013) leads to the constraint σ_{8}(Ω_{m}/ 0.25)^{0.57} = 0.80 ± 0.05 for the base ΛCDM cosmology. This is about 2.4σ lower than expected from Planck.
5.5.3. Counts of rich clusters
For the base ΛCDM model we find σ_{8} = 0.828 ± 0.012 from Planck+WP+highL. This value is in excellent agreement with the WMAP9 value of σ_{8} = 0.821 ± 0.023 (Hinshaw et al. 2012). There are other ways to probe the power spectrum normalization, in addition to the cosmic shear measurements discussed above. For example, the abundances of rich clusters of galaxies are particularly sensitive to the normalization (see e.g., Komatsu & Seljak 2002). Recently, a number of studies have used tSZcluster mass scaling relations to constrain combinations of σ_{8} and Ω_{m} (e.g., Benson et al. 2013; Reichardt et al. 2013; Hasselfield et al. 2013) including an analysis of a sample of Planck tSZ clusters (see Planck Collaboration XXVIII 2014; Planck Collaboration XXIX 2014) reported in this series of papers (Planck Collaboration XX 2014)^{34}.
Constraints on oneparameter extensions to the base ΛCDM model.
The Planck analysis uses a relation between cluster mass and tSZ signal based on comparisons with Xray mass measurements. To take departures from hydrostatic equilibrium into account, Xray temperature calibration, modelling of the selection function, uncertainties in scaling relations and analysis uncertainties, Planck Collaboration XX (2014) assume a “bias” between the Xray derived masses and the true cluster masses. If the mass bias, (1 − b), is allowed to vary uniformly between 0.7 and 1.0, Planck Collaboration XX (2014) find σ_{8}(Ω_{m}/ 0.27)^{0.3} = 0.76 ± 0.03 for the base ΛCDM model. In comparison, for the same model we find which is a significant (around 3σ) discrepancy that remains unexplained. Qualitatively similar results are found from analyses of SPT clusters [σ_{8}(Ω_{m}/ 0.27)^{0.3} = 0.77 ± 0.04]. Key difficulties with this type of measurement, as discussed in Planck Collaboration XX (2014), include adequately modelling selection biases and calibrating cluster masses. These effects are discussed in the analysis of ACT clusters by Hasselfield et al. (2013), who adopt a number of approaches, including folding in dynamical mass measurements, to calibrate biases in clusters mass estimates. Some of these approaches give joint σ_{8}–Ω_{m} constraints consistent with the base ΛCDM parameters reported here.
At this stage of our understanding of the biases and scatter in the cluster mass calibrations, we believe that for the purposes of this paper it is premature to use cluster counts together with CMB measurements to search for new physics. Planck Collaboration XX (2014) explore a number of possibilities for reducing the tension between Planck CMB measurements and tSZ cluster counts, including nonzero neutrino masses.
Fig. 21 68% and 95% confidence regions on oneparameter extensions of the base ΛCDM model for Planck+WP (red) and Planck+WP+BAO (blue). Horizontal dashed lines correspond to the fixed base model parameter value, and vertical dashed lines show the mean posterior value in the base model for Planck+WP. 
6. Extensions to the base ΛCDM model
6.1. Grid of models
To explore possible deviations from ΛCDM we have analysed an extensive grid of models that covers many wellmotivated extensions of ΛCDM. As in the exploration of the base ΛCDM cosmology, we have also considered a variety of data combinations for each model. For models involving more than one additional parameter we restrict ourselves to Planck+WP combinations in order to obtain tighter constraints by leveraging the relative amplitude of the power spectrum at very low ℓ and high ℓ. Most models are run with Planck, Planck+WP, and Planck+WP+highL; additionally all are importance sampled with Planck lensing (Sect. 5.1), BAO (Sect. 5.2), SNe (Sect. 5.4), and the Riess et al. (2011) direct H_{0} measurement (Sect. 5.3). For models where the nonCMB data give a large reduction in parameter volume (e.g. Ω_{K} models), we run separate chains instead of importance sampling.
These runs provide no compelling evidence for deviations from the base ΛCDM model, and indeed, as shown in Table 10 and Fig. 21, the posteriors for individual extra parameters generally overlap the fiducial model within one standard deviation. The inclusion of BAO data shrinks further the allowed scope for deviation. The parameters of the base ΛCDM model are relatively robust to inclusion of additional parameters, but the errors on some do broaden significantly when additional degeneracies open up, as can be seen in Fig. 21
The full grid results are available online^{35}. Here we summarize some of the key results, and also consider a few additional extensions.
6.2. EarlyUniverse physics
Inflationary cosmology offers elegant explanations of key features of our Universe, such as its large size and near spatially flat geometry. Within this scenario, the Universe underwent a brief period of accelerated expansion (Starobinsky 1979, 1982; Kazanas 1980; Guth 1981; Sato 1981; Linde 1982; Albrecht & Steinhardt 1982) during which quantum fluctuations were inflated in scale to become the classical fluctuations that we see today. In the simplest inflationary models, the primordial fluctuations are predicted to be adiabatic, nearly scaleinvariant and Gaussian (Mukhanov & Chibisov 1981; Hawking 1982; Starobinsky 1982; Guth & Pi 1982; Bardeen et al. 1983), in good agreement with CMB observations and other probes of largescale structure.
Despite this success, the fundamental physics behind inflation is not yet understood and there is no convincing evidence that rules out alternative scenarios for the early Universe. A large number of phenomenological models of inflation, some inspired by string theory, have been discussed in the literature (see Liddle & Lyth 2000; Bassett et al. 2006; Linde 2008, for reviews), as well as alternatives to inflation including prebig bang scenarios (e.g., Gasperini & Veneziano 1993; Khoury et al. 2001; Boyle et al. 2004; Creminelli & Senatore 2007; Brandenberger 2012). Many of these models lead to distinctive signatures, such as departures from Gaussianity, isocurvature perturbations, or oscillatory features in the power spectrum, that are potentially observable. The detection of such signatures would offer valuable information on the physics of the early Universe and is one of the main science goals of Planck.
In this section we discuss basic aspects of the primordial power spectrum, such as the spectral index, departures from a pure power law, limits on tensor modes etc., and discuss the implications for inflationary cosmology. Tests of more complex models, such as multifield inflation, are discussed in a separate paper (Planck Collaboration XXII 2014). In Planck Collaboration XXIV (2014), the Planck maps are used to constrain possible deviations from Gaussianity via measurements of the bispectrum and trispectrum. Planck Collaboration XXIII (2014) considers departures from statistical isotropy and additional tests of nonGaussianity.
6.2.1. Scale dependence of primordial fluctuations
The primordial fluctuations in the base ΛCDM model are parameterized as a pure power law with a spectral index n_{s} (Eq. (2)). Prior to Planck, CMB observations have favoured a power law index with slope n_{s}< 1, which is expected in simple singlefield slowroll inflationary models (see e.g., Mukhanov 2007 and Eq. (65a) below). The final WMAP nineyear data give n_{s} = 0.972 ± 0.013 at 68% confidence (Hinshaw et al. 2012). Combining this with dampingtail measurements from ACT and SPT data gives n_{s} = 0.968 ± 0.009, indicating a departure from scale invariance at the 3σ level. The addition of BAO data has resulted in a stronger preference for n_{s}< 1 (Anderson et al. 2012; Hinshaw et al. 2012; Story et al. 2013; Sievers et al. 2013). These constraints assume the basic sixparameter ΛCDM cosmological model. Any new physics that affects the damping tail of the CMB spectrum, such as additional relativistic particles, can alter these constraints substantially and still allow a precisely scaleinvariant spectrum.
With Planck, a robust detection of the deviation from scale invariance can now be made from a single set of CMB observations spanning three decades in scale from ℓ = 2 to ℓ = 2500. We find (60)for the base ΛCDM model, a roughly 6σ departure from scale invariance. This is consistent with the results from previous CMB experiments cited above. The statistical significance of this result is high enough that the difference between a purely scale invariant spectrum can be seen easily in a plot of the power spectrum. Figure 22 shows the Planck spectrum of Fig. 10 plotted as compared to the base ΛCDM fit with n_{s} = 0.96 (red dashed line) and to the bestfit base ΛCDM cosmology with n_{s} = 1. The n_{s} = 1 model has more power at small scales and is strongly excluded by the Planck data.
Fig. 22 Planck power spectrum of Fig. 10 plotted as against multipole, compared to the bestfit base ΛCDM model with n_{s} = 0.96 (red dashed line). The bestfit base ΛCDM model with n_{s} constrained to unity is shown by the blue line. 
Fig. 23 Upper: posterior distribution for n_{s} for the base ΛCDM model (black) compared to the posterior when a tensor component and running scalar spectral index are added to the model (red) Middle: constraints (68% and 95%) in the n_{s}–dn_{s}/ dlnk plane for ΛCDM models with running (blue) and additionally with tensors (red). Lower: constraints (68% and 95%) on n_{s} and the tensortoscalar ratio r_{0.002} for ΛCDM models with tensors (blue) and additionally with running of the spectral index (red). The dotted line show the expected relation between r and n_{s} for a V(φ) ∝ φ^{2} inflationary potential (Eqs. (65a) and (65b)); here N is the number of inflationary efoldings as defined in the text. The dotted line should be compared to the blue contours, since this model predicts negligible running. All of these results use the Planck+WP+highL data combination. 
Fig. 24 Constraints on n_{s} for ΛCDM models with nonstandard relativistic species, N_{eff}, (upper) and helium fraction, Y_{P}, (lower). We show 68% and 95% contours for various data combinations. Note the tightening of the constraints with the addition of BAO data. 
The unique contribution of Planck, compared to previous experiments, is that we are able to show that the departure from scale invariance is robust to changes in the underlying theoretical model. For example, Figs. 21 and 23 show that the departure from scale invariance is not sensitive to the parameterization of the primordial fluctuations. Even if we allow a possible running of the spectral index (the parameter dn_{s}/ dlnk defined in Eq. (2)) and/or a component of tensor fluctuations, the Planck data favour a tilted spectrum at a high significance level.
Our extensive grid of models allows us to investigate correlations of the spectral index with a number of cosmological parameters beyond those of the base ΛCDM model (see Figs. 21 and 24). As expected, n_{s} is uncorrelated with parameters describing latetime physics, including the neutrino mass, geometry, and the equation of state of dark energy. The remaining correlations are with parameters that affect the evolution of the early Universe, including the number of relativistic species, or the helium fraction. This is illustrated in Fig. 24: modifying the standard model by increasing the number of neutrinos species, or the helium fraction, has the effect of damping the smallscale power spectrum. This can be partially compensated by an increase in the spectral index. However, an increase in the neutrino species must be accompanied by an increased matter density to maintain the peak positions. A measurement of the matter density from the BAO measurements helps to break this degeneracy. This is clearly seen in the upper panel of Fig. 24, which shows the improvement in the constraints when BAO measurements are added to the Planck+WP+highL likelihood. With the addition of BAO measurements we find more than a 3σ deviation from n_{s} = 1 even in this extended model, with a bestfit value of n_{s} = 0.969 ± 0.010 for varying relativistic species. As discussed in Sect. 6.3, we see no evidence from the Planck data for nonstandard neutrino physics.
The simplest singlefield inflationary models predict that the running of the spectral index should be of second order in inflationary slowroll parameters and therefore small [dn_{s}/ dlnk ~ (n_{s} − 1)^{2}], typically about an order of magnitude below the sensitivity limit of Planck (see e.g., Kosowsky & Turner 1995; Baumann et al. 2009). Nevertheless, it is easy to construct inflationary models that have a larger scale dependence (e.g., by adjusting the third derivative of the inflaton potential) and so it is instructive to use the Planck data to constrain dn_{s}/ dlnk. A test for dn_{s}/ dlnk is of particularly interest given the results from previous CMB experiments.
Early results from WMAP suggested a preference for a negative running at the 1–2σ level. In the final 9year WMAP analysis no significant running was seen using WMAP data alone, with dn_{s}/ dlnk = − 0.019 ± 0.025 (68% confidence; Hinshaw et al. 2012. Combining WMAP data with the first data releases from ACT and SPT, Hinshaw et al. (2012) found a negative running at nearly the 2σ level with dn_{s}/ dlnk = − 0.022 ± 0.012 (see also Dunkley et al. 2011 and Keisler et al. 2011 for analysis of ACT and SPT with earlier data from WMAP). The ACT 3year release, which incorporated a new region of sky, gave dn_{s}/ dlnk = − 0.003 ± 0.013 (Sievers et al. 2013) when combined with WMAP 7 year data. With the wide field SPT data at 150 GHz, a negative running was seen at just over the 2σ level, dn_{s}/ dlnk = − 0.024 ± 0.011 (Hou et al. 2014).
The picture from previous CMB experiments is therefore mixed. The latest WMAP data show a 1σ trend for a running, but when combined with the S12 SPT data, this trend is amplified to give a potentially interesting result. The latest ACT data go in the other direction, giving no support for a running spectral index when combined with WMAP^{36}.
The results from Planck data are as follows (see Figs. 21 and 23): The consistency between (61a) and (61b) shows that these results are insensitive to modelling of unresolved foregrounds. The preferred solutions have a small negative running, but not at a high level of statistical significance. Closer inspection of the bestfits shows that the change in χ^{2} when dn_{s}/ dlnk is included as a parameter comes almost entirely from the low multipole temperature likelihood. (The fits to the high multipole Planck likelihood have a Δχ^{2} = − 0.4 when dn_{s}/ dlnk is included.) The slight preference for a negative running is therefore driven by the spectrum at low multipoles ℓ ≲ 50. The tendency for negative running is partly mitigated by including the Planck lensing likelihood (Eq. (61c)).
The constraints on dn_{s}/ dlnk are broadly similar if tensor fluctuations are allowed in addition to a running of the spectrum (Fig. 23). Adding tensor fluctuations, the marginalized posterior distributions for dn_{s}/ dlnk give As with Eqs. (61a)–(61c) the tendency to favour negative running is driven by the low multipole component of the temperature likelihood not by the Planck spectrum at high multipoles.
This is one of several examples discussed in this section where marginal evidence for extensions to the base ΛCDM model are favoured by the TT spectrum at low multipoles. (The low multipole spectrum is also largely responsible for the pull of the lensing amplitude, A_{L}, to values greater than unity discussed in Sect. 5.1). The mismatch between the bestfit base ΛCDM model and the TT spectrum at multipoles ℓ ≲ 30 is clearly visible in Fig. 1. The implications of this mismatch are discussed further in Sect. 7.
Beyond a simple running, various extended parameterizations have been developed by e.g., Bridle et al. (2003), Shafieloo & Souradeep (2008), Verde & Peiris (2008), and Hlozek et al. (2012), to test for deviations from a powerlaw spectrum of fluctuations. Similar techniques are applied to the Planck data in Planck Collaboration XXII (2014).
6.2.2. Tensor fluctuations
In the base ΛCDM model, the fluctuations are assumed to be purely scalar modes. Primordial tensor fluctuations could also contribute to the temperature and polarization power spectra (e.g., Grishchuk 1975; Starobinsky 1979; Basko & Polnarev 1980; Crittenden et al. 1993, 1995). The most direct way of testing for a tensor contribution is to search for a magnetictype parity signature via a largescale Bmode pattern in CMB polarization (Seljak 1997; Zaldarriaga & Seljak 1997; Kamionkowski et al. 1997). Direct Bmode measurements are challenging as the expected signal is small; upper limits measured by BICEP and QUIET give 95% upper limits of r_{0.002}< 0.73 and r_{0.002}< 2.8 respectively (Chiang et al. 2010; QUIET Collaboration et al. 2012)^{37}.
Measurements of the temperature power spectrum can also be used to constrain the amplitude of tensor modes. Although such limits can appear to be much tighter than the limits from Bmode measurements, it should be borne in mind that they are indirect because they are derived within the context of a particular theoretical model. In the rest of this subsection, we review temperature based limits on tensor modes and then present the results from Planck.
Adding a tensor component to the base ΛCDM model, the WMAP 9year results constrain r_{0.002}< 0.38 at 95% confidence (Hinshaw et al. 2012). Including smallscale ACT and SPT data this improves to r_{0.002}< 0.17, and to r_{0.002}< 0.12 with the addition of BAO data. These limits are degraded substantially, however, in models which allow running of the scalar spectral index in addition to tensors. For such models, the WMAP data give r_{0.002}< 0.50, and this limit is not significantly improved by adding high resolution CMB and BAO data.
The precise determination of the fourth, fifth and sixth acoustic peaks by Planck now largely breaks the degeneracy between the primordial fluctuation parameters. For the Planck+WP+highL likelihood we find As shown in Figs. 21 and 23, the tensor amplitude is weakly correlated with the scalar spectral index; an increase in n_{s} that could match the first three peaks cannot fit the fourth and higher acoustic peak in the Planck spectrum. Likewise, the shape constraints from the fourth and higher acoustic peaks give a reduction in the correlations between a tensor mode and a running in the spectral index, leading to significantly tighter limits than from previous CMB experiments. These numbers in Eqs. (63a) and (63b) are driven by the temperature spectrum and change very little if we add nonCMB data such as BAO measurements. The Planck limits are largely decoupled from assumptions about the latetime evolution of the Universe and are close to the tightest possible limits achievable from the temperature power spectrum alone (Knox & Turner 1994; Knox 1995).
These limits on a tensor mode have profound implications for inflationary cosmology. The limits translate directly to an upper limit on the energy scale of inflation, (64)(Linde 1983; Lyth 1984), and to the parameters of “largefield” inflation models. Slowroll inflation driven by a power law potential V(φ) ∝ φ^{α} offers a simple example of largefield inflation. The field values in such a model must necessarily exceed the Planck scale m_{Pl} , and lead to a scalar spectral index and tensor amplitude of where N is the number of efoldings between the end of inflation and the time that our present day Hubble scale crossed the inflationary horizon (see e.g., Lyth & Riotto 1999). The 95% confidence limits from the Planck data are now close to the predictions of α = 2 models for N ≈ 50–60 efolds (see Fig. 23). Largefield models with quartic potentials (e.g., Linde 1982) are now firmly excluded by CMB data. Planck constraints on powerlaw and on broader classes of inflationary models are discussed in detail in Planck Collaboration XXIV (2014). Improved limits on Bmodes will be required to further constrain high field models of inflation.
Fig. 25 Planck+WP+highL data combination (samples; colourcoded by the value of H_{0}) partially breaks the geometric degeneracy between Ω_{m} and Ω_{Λ} due to the effect of lensing in the temperature power spectrum. These limits are significantly improved by the inclusion of the Planck lensing reconstruction (black contours). Combining also with BAO (right; solid blue contours) tightly constrains the geometry to be nearly flat. 
6.2.3. Curvature
An explanation of the near flatness of our observed Universe was one of the primary motivations for inflationary cosmology. Inflationary models that allow a large number of efoldings predict that our Universe should be very accurately spatially flat^{38}. Nevertheless, by introducing fine tunings it is possible to construct inflation models with observationally interesting open geometries (e.g., Gott 1982; Linde 1995; Bucher et al. 1995; Linde 1999) or closed geometries (Linde 2003). Even more speculatively, there has been interest in models with open geometries from considerations of tunnelling events between metastable vacua within a “string landscape” (Freivogel et al. 2006). Observational limits on spatial curvature therefore offer important additional constraints on inflationary models and fundamental physics.
CMB temperature power spectrum measurements suffer from a wellknown “geometrical degeneracy” (Bond et al. 1997; Zaldarriaga et al. 1997). Models with identical primordial spectra, physical matter densities and angular diameter distance to the last scattering surface, will have almost identical CMB temperature power spectra. This is a near perfect degeneracy (see Fig. 25) and is broken only via the integrated SachsWolfe (ISW) effect on large angular scales and gravitational lensing of the CMB spectrum (Stompor & Efstathiou 1999). The geometrical degeneracy can also be broken with the addition of probes of late time physics, including BAO, Type Ia supernova, and measurement of the Hubble constant (e.g., Spergel et al. 2007).
Recently, the detection of the gravitational lensing of the CMB by ACT and SPT has been used to break the geometrical degeneracy, by measuring the integrated matter potential distribution. ACT constrained Ω_{Λ} = 0.61 ± 0.29 (68% CL) in Sherwin et al. (2011), with the updated analysis in Das et al. (2014) giving Ω_{K} = − 0.031 ± 0.026 (68% CL) (Sievers et al. 2013). The SPT lensing measurements combined with seven year WMAP temperature spectrum improved this limit to Ω_{K} = − 0.0014 ± 0.017 (68 % CL) (van Engelen et al. 2012).
With Planck we detect gravitational lensing at about 26σ through the 4point function (Sect. 5.1 and Planck Collaboration XVII 2014). This strong detection of gravitational lensing allows us to constrain the curvature to percent level precision using observations of the CMB alone: These constraints are improved substantially by the addition of BAO data. We then find These limits are consistent with (and slightly tighter than) the results reported by Hinshaw et al. (2012) from combining the nineyear WMAP data with high resolution CMB measurements and BAO data. We find broadly similar results to Eqs. (67a) and (67b) if the Riess et al. (2011)H_{0} measurement, or either of the SNe compilations discussed in Sect. 5.4, are used in place of the BAO measurements.
In summary, there is no evidence from Planck for any departure from a spatially flat geometry. The results of Eqs. (67a) and (67b) suggest that our Universe is spatially flat to an accuracy of better than a percent.
6.3. Neutrino physics and constraints on relativistic components
A striking illustration of the interplay between cosmology and particle physics is the potential of CMB observations to constrain the properties of relic neutrinos, and possibly of additional light relic particles in the Universe (see e.g., Dodelson et al. 1996; Hu et al. 1995; Bashinsky & Seljak 2004; Ichikawa et al. 2005; Lesgourgues & Pastor 2006; Hannestad 2010). In the following subsections, we present Planck constraints on the mass of ordinary (active) neutrinos assuming no extra relics, on the density of light relics assuming they all have negligible masses, and finally on models with both light massive and massless relics.
6.3.1. Constraints on the total mass of active neutrinos
The detection of solar and atmospheric neutrino oscillations proves that neutrinos are massive, with at least two species being nonrelativistic today. The measurement of the absolute neutrino mass scale is a challenge for both experimental particle physics and observational cosmology. The combination of CMB, largescale structure and distance measurements already excludes a large range of masses compared to betadecay experiments. Current limits on the total neutrino mass ∑ m_{ν} (summed over the three neutrino families) from cosmology are rather model dependent and vary strongly with the data combination adopted. The tightest constraints for flat models with three families of neutrinos are typically around 0.3 eV (95% CL; e.g., de Putter et al. 2012). Since ∑ m_{ν} must be greater than approximately 0.06 eV in the normal hierarchy scenario and 0.1 eV in the degenerate hierarchy (GonzalezGarcia et al. 2012), the allowed neutrino mass window is already quite tight and could be closed further by current or forthcoming observations (Jimenez et al. 2010; Lesgourgues et al. 2013).
Cosmological models, with and without neutrino mass, have different primary CMB power spectra. For observationallyrelevant masses, neutrinos are still relativistic at recombination and the unique effects of masses in the primary power spectra are small. The main effect is around the first acoustic peak and is due to the early integrated SachsWolfe (ISW) effect; neutrino masses have an impact here even for a fixed redshift of matterradiation equality (Lesgourgues & Pastor 2012; Hall & Challinor 2012; Hou et al. 2014; Lesgourgues et al. 2013). To date, this effect has been the dominant one in constraining the neutrino mass from CMB data, as demonstrated in Hou et al. (2014). As we shall see here, the Planck data move us into a new regime where the dominant effect is from gravitational lensing. Increasing neutrino mass, while adjusting other parameters to remain in a highprobability region of parameter space, increases the expansion rate at z ≳ 1 and so suppresses clustering on scales smaller than the horizon size at the nonrelativistic transition (Kaplinghat et al. 2003; Lesgourgues et al. 2006). The net effect for lensing is a suppression of the CMB lensing potential and, for orientation, by ℓ = 1000 the suppression is around 10% in power for ∑ m_{ν} = 0.66 eV.
Here we report constraints assuming three species of degenerate massive neutrinos. At the level of sensitivity of Planck, the effect of mass splittings is negligible, and the degenerate model can be assumed without loss of generality.
Fig. 26 Marginalized posterior distributions for ∑ m_{ν} in flat models from CMB data. We show results for Planck+WP+highL without (solid black) and with (red) marginalization over A_{L}, showing how the posterior is significantly broadened by removing the lensing information from the temperature anisotropy power spectrum. The effect of replacing the lowℓ temperature and (WMAP) polarization data with a τ prior is shown in solid blue (Planck−lowL+highL+τprior) and of further removing the highℓ data in dotdashed blue (Planck−lowL+τprior). We also show the result of including the lensing likelihood with Planck+WP+highL (dashed black) and Planck−lowL+highL+τprior (dashed blue). 
Combining the Planck+WP+highL data, we obtain an upper limit on the summed neutrino mass of (68)The posterior distribution is shown by the solid black curve in Fig. 26. To demonstrate that the dominant effect leading to the constraint is gravitational lensing, we remove the lensing information by marginalizing over A_{L}^{39}. We see that the posterior broadens considerably (see the red curve in Fig. 26) to give (69)taking us back close to the value of 1.3 eV (for A_{L} = 1) from the nineyear WMAP data (Hinshaw et al. 2012), corresponding to the limit above which neutrinos become nonrelativistic before recombination. (The resolution of WMAP gives very little sensitivity to lensing effects.)
As discussed in Sect. 5.1, the Planck+WP+highL data combination has a preference for high A_{L}. Since massive neutrinos suppress the lensing power (like a low A_{L}) there is a concern that the same tensions which drive A_{L} high may give artificially tight constraints on ∑ m_{ν}. We can investigate this issue by replacing the lowℓ data with a prior on the optical depth (as in Sect. 5.1) and removing the highℓ data. Posterior distributions with the τ prior, and additionally without the highℓ data, are shown in Fig. 26 by the solid blue and dotdashed blue curves, respectively. The constraint on ∑ m_{ν} does not degrade much by replacing the lowℓ data with the τ prior only, but the degradation is more severe when the highℓ data are also removed: ∑ m_{ν}< 1.31 eV (95% CL).
Including the lensing likelihood (see Sect. 5.1) has a significant, but surprising, effect on our results. Adding the lensing likelihood to the Planck+WP+highL data combination weakens the limit on ∑ m_{ν}, (70)as shown by the dashed black curve in Fig. 26. This is representative of a general trend that the Planck lensing likelihood favours larger ∑ m_{ν} than the temperature power spectrum. Indeed, if we use the data combination Planck−lowL+highL+τprior, which gives a weaker constraint from the temperature power spectrum, adding lensing gives a bestfit away from zero (∑ m_{ν} = 0.46 eV; dashed blue curve in Fig. 26). However, the total χ^{2} at the bestfit is very close to that for the bestfitting base model (which, recall, has one massive neutrino of mass 0.06 eV), with the improved fit to the lensing data (Δχ^{2} = − 2.35) being cancelled by the poorer fit to highℓ CMB data (Δχ^{2} = − 2.15). There are rather large shifts in other cosmological parameters between these bestfit solutions corresponding to shifts along the acousticscale degeneracy direction for the temperature power spectrum. Note that, as well as the change in H_{0} (which falls to compensate the increase in ∑ m_{ν} at fixed acoustic scale), n_{s}, ω_{b} and ω_{c} change significantly keeping the lensed temperature spectrum almost constant. These latter shifts are similar to those discussed for A_{L} in Sect. 5.1, with nonzero ∑ m_{ν} acting like A_{L}< 1. The lensing power spectrum is lower by 5.4% for the highermass best fit at ℓ = 400 and larger below ℓ ≈ 45 (e.g. by 0.6% at ℓ = 40), which is a similar trend to the residuals from the bestfit minimalmass model shown in the bottom panel of Fig. 12. Planck Collaboration XVII (2014) explores the robustness of the estimates to various data cuts and foregroundcleaning methods. The first (ℓ = 40–85) bandpower is the least stable to these choices, although the variations are not statistically significant. We have checked that excluding this bandpower does not change the posterior for ∑ m_{ν} significantly, as expected since most of the constraining power on ∑ m_{ν} comes from the bandpowers on smaller scales. At this stage, it is unclear what to make of this mild preference for high masses from the 4point function compared to the 2point function. As noted in Planck Collaboration XVII (2014), the lensing measurements from ACT (Das et al. 2014) and SPT (van Engelen et al. 2012) show similar trends to those from Planck where they overlap in scale. With further Planck data (including polarization), and forthcoming measurements from the full 2500 deg^{2} SPT temperature survey, we can expect more definitive results on this issue in the near future.
Apart from its impact on the earlyISW effect and lensing potential, the total neutrino mass affects the angulardiameter distance to last scattering, and can be constrained through the angular scale of the first acoustic peak. However, this effect is degenerate with Ω_{Λ} (and so the derived H_{0}) in flat models and with other latetime parameters such as Ω_{K} and w in more general models (Howlett et al. 2012). Latetime geometric measurements help in reducing this “geometric” degeneracy. Increasing the neutrino masses at fixed θ_{∗} increases the angulardiameter distance for 0 ≤ z ≤ z_{∗} and reduces the expansion rate at low redshift (z ≲ 1) but increases it at higher redshift. The sphericallyaveraged BAO distance D_{V}(z) therefore increases with increasing neutrino mass at fixed θ_{∗}, and the Hubble constant falls; see Fig. 8 of Hou et al. (2014). With the BAO data of Sect. 5.2, we find a significantly lower bound on the neutrino mass: (71)Following the philosophy of this paper, namely to give higher weight to the BAO data compared to more complex astrophysical data, we quote the result of Eq. (71) in the abstract as our most reliable limit on the neutrino mass. The ΛCDM model with minimal neutrino masses was shown in Sect. 5.3 to be in tension with recent direct measurements of H_{0} which favour higher values. Increasing the neutrino mass will only make this tension worse and drive us to artificially tight constraints on ∑ m_{ν}. If we relax spatial flatness, the CMB geometric degeneracy becomes threedimensional in models with massive neutrinos and the constraints on ∑ m_{ν} weaken considerably to (72)
6.3.2. Constraints on N_{eff}
As discussed in Sect. 2, the density of radiation in the Universe (besides photons) is usually parameterized by the effective neutrino number N_{eff}. This parameter specifies the energy density when the species are relativistic in terms of the neutrino temperature assuming exactly three flavours and instantaneous decoupling. In the Standard Model, N_{eff} = 3.046, due to noninstantaneous decoupling corrections (Mangano et al. 2005).
However, there has been some mild preference for N_{eff}> 3.046 from recent CMB anisotropy measurements (Komatsu et al. 2011; Dunkley et al. 2011; Keisler et al. 2011; Archidiacono et al. 2011; Hinshaw et al. 2012; Hou et al. 2014). This is potentially interesting, since an excess could be caused by a neutrino/antineutrino asymmetry, sterile neutrinos, and/or any other light relics in the Universe. In this subsection we discuss the constraints on N_{eff} from Planck in scenarios where the extra relativistic degrees of freedom are effectively massless.
The physics of how N_{eff} is constrained by CMB anisotropies is explained in Bashinsky & Seljak (2004), Hou et al. (2013) and Lesgourgues et al. (2013). The main effect is that increasing the radiation density at fixed θ_{∗} (to preserve the angular scales of the acoustic peaks) and fixed z_{eq} (to preserve the earlyISW effect and so firstpeak height) increases the expansion rate before recombination and reduces the age of the Universe at recombination. Since the diffusion length scales approximately as the square root of the age, while the sound horizon varies proportionately with the age, the angular scale of the photon diffusion length, θ_{D}, increases, thereby reducing power in the damping tail at a given multipole. Combining Planck, WMAP polarization and the highℓ experiments gives (73)The marginalized posterior distribution is given in Fig. 27 (black curve).The result in Eq. (73) is consistent with the value of N_{eff} = 3.046 of the Standard Model, but it is important to aknowledge that it is difficult to constrain N_{eff} accurately using CMB temperature measurements alone. Evidently, the nominal mission data from Planck do not strongly rule out a value as high as N_{eff} = 4.
Fig. 27 Marginalized posterior distribution of N_{eff} for Planck+ WP+ highL (black) and additionally BAO (blue), the H_{0} measurement (red), and both BAO and H_{0} (green). 
Increasing N_{eff} at fixed θ_{∗} and z_{eq} necessarily raises the expansion rate at low redshifts too. Combining CMB with distance measurements can therefore improve constraints (see Fig. 27) although for the BAO observable r_{drag}/D_{V}(z) the reduction in both r_{drag} and D_{V}(z) with increasing N_{eff} partly cancel. With the BAO data of Sect. 5.2, the N_{eff} constraint is tightened to (74)Our constraints from CMB alone and CMB+BAO are compatible with the standard value N_{eff} = 3.046 at the 1σ level, giving no evidence for extra relativistic degrees of freedom.
Fig. 28 Left: 2D joint posterior distribution between N_{eff} and ∑ m_{ν} (the summed mass of the three active neutrinos) in models with extra massless neutrinolike species. Right: samples in the N_{eff} plane, colourcoded by Ω_{c}h^{2}, in models with one massive sterile neutrino family, with effective mass , and the three active neutrinos as in the base ΛCDM model. The physical mass of the sterile neutrino in the thermal scenario, , is constant along the grey dashed lines, with the indicated mass in eV. The physical mass in the DodelsonWidrow scenario, , is constant along the dotted lines (with the value indicated on the adjacent dashed lines).Note the pile up of points at low values of N_{eff}, caused because the sterile neutrino component behaves like cold dark matter there, introducing a strong degeneracy between the two components, as described in the text. 
Since N_{eff} is positively correlated with H_{0}, the tension between the Planck data and direct measurements of H_{0} in the base ΛCDM model (Sect. 5.3) can be reduced at the expense of high N_{eff}. The marginalized constraint is (75)For this data combination, the χ^{2} for the bestfitting model allowing N_{eff} to vary is lower by 5.3 than for the base N_{eff} = 3.046 model. The H_{0} fit is much better, with Δχ^{2} = − 4.4, but there is no strong preference either way from the CMB. The lowℓ temperature power spectrum does weakly favour the high N_{eff} model (Δχ^{2} = − 1.4) – since N_{eff} is positively correlated with n_{s} (see Fig. 24) and increasing n_{s} reduces power on large scales – as does the rest of the Planck power spectrum (Δχ^{2} = − 1.8). The highℓ experiments mildly disfavour high N_{eff} in our fits (Δχ^{2} = 1.9). Further including the BAO data pulls the central value downwards by around 0.5σ (see Fig. 27): (76)The χ^{2} at the bestfit for this data combination (N_{eff} = 3.48) is lower by 4.2 than the bestfitting N_{eff} = 3.046 model. While the high N_{eff} bestfit is preferred by Planck+WP (Δχ^{2} = − 3.1) and the H_{0} data (Δχ^{2} = − 3.3 giving an acceptable χ^{2} = 1.8 for this data point), it is disfavoured by the highℓ CMB data (Δχ^{2} = 2.0) and slightly by BAO (Δχ^{2} = 0.5). We conclude that the tension between direct H_{0} measurements and the CMB and BAO data in the base ΛCDM can be relieved at the cost of additional neutrinolike physics, but there is no strong preference for this extension from the CMB damping tail.
Throughout this subsection, we have assumed that all the relativistic components parameterized by N_{eff} consist of ordinary freestreaming relativistic particles. Extra radiation components with a different sound speed or viscosity parameter (Hu 1998) can provide a good fit to prePlanck CMB data (Archidiacono et al. 2013), but are not investigated in this paper.
6.3.3. Simultaneous constraints on N_{eff} and either ∑m_{ν} or m
It is interesting to investigate simultaneous contraints on N_{eff} and ∑ m_{ν}, since extra relics could coexist with neutrinos of sizeable mass, or could themselves have a mass in the eV range. Joint constraints on N_{eff} and ∑ m_{ν} have been explored several times in the literature. These two parameters are known to be partially degenerate when largescale structure data are used (Hannestad & Raffelt 2004; Crotty et al. 2004), but their impact in the CMB is different and does not lead to significant correlations.
Joint constraints on N_{eff} and ∑ m_{ν} are always modeldependent: they vary strongly with assumptions about how the total mass is split between different species (and they would also be different for models in which massive species have chemical potentials or a nonthermal phasespace distribution). We present here Planck constraints for two different models and describe the scenarios that motivate them.
First, as in the previous subsection we assume that the three active neutrinos share a mass of ∑ m_{ν}/ 3, and may coexist with extra massless species contributing to N_{eff}. In this model, when N_{eff} is greater than 3.046, ΔN_{eff} = N_{eff} − 3.046 gives the density of extra massless relics with arbitrary phasespace distribution. When N_{eff}< 3.046, the temperature of the three active neutrinos is reduced accordingly, and no additional relativistic species are assumed. In this case, the CMB constraint is (77)These bounds tighten somewhat with the inclusion of BAO data, as illustrated in Fig. 28; we find (78)We see that the joint constraints do not differ very much from the bounds obtained when introducing these parameters separately. The physical effects of neutrino masses and extra relativistic relics are sufficiently different to be resolved separately at the level of accuracy of Planck.
In the second model, we assume the existence of one massive sterile neutrino, in addition to the two massless and one massive active neutrino of the base model. The active neutrino mass is kept fixed at 0.06 eV. In particle physics, this assumption can be motivated in several ways. For example, there has recently been renewed interest in models with one light sterile neutrino in order to explain the MiniBoone anomaly reported in AguilarArevalo et al. (2013), as well as reactor and Gallium anomalies (Giunti et al. 2013). The statistical significance of these results is marginal and they should not be overinterpreted. However, they do motivate investigating a model with three active neutrinos and one heavier sterile neutrino with mass m_{sterile}. If the sterile neutrino were to thermalize with the same temperature as active neutrinos, this model would have N_{eff} ≈ 4.
Since we wish to be more general, we assume that the extra eigenstate is either: (i) thermally distributed with an arbitrary temperature T_{s}; or (ii) distributed proportionally to active neutrinos with an arbitrary scaling factor χ_{s} in which the scaling factor is a function of the activesterile neutrino mixing angle. This second case corresponds the DodelsonWidrow scenario (Dodelson & Widrow 1994). The two cases are in fact equivalent for cosmological observables and do not require separate analyses (Colombi et al. 1996; Lesgourgues et al. 2013). Sampling the posterior with flat priors on N_{eff} and m_{sterile} would not be efficient, since in the limit of small temperature T_{s}, or small scaling factor χ_{s}, the mass would be unbounded. Hence we adopt a flat prior on the “effective sterile neutrino mass” defined as ^{40}. In the case of a thermallydistributed sterile neutrino, this parameter is related to the true mass via (79)Here, recall that T_{ν} = (4/11)^{1/3}T_{γ} is the active neutrino temperature in the instantaneousdecoupling limit and that the effective number is defined via the energy density, ΔN_{eff} = (T_{s}/T_{ν})^{4}. In the DodelsonWidrow case the relation is given by (80)with ΔN_{eff} = χ_{s}. For a thermalized sterile neutrino with temperature T_{ν} (i.e., the temperature the active neutrinos would have if there were no heating at electronpositron annihilation), corresponding to ΔN_{eff} = 1, the three masses are equal to each other.
Assuming flat priors on N_{eff} and with , we find the results shown in Fig. 28. The physical mass, in the thermal scenario is constant along the dashed lines in the figure and takes the indicated value in eV. The physical mass, , in the DodelsonWidrow scenario is constant on the dotted lines. For low N_{eff} the physical mass of the neutrinos becomes very large, so that they become nonrelativistic well before recombination. In the limit in which the neutrinos become nonrelativistic well before any relevant scales enter the horizon, they will behave exactly like cold dark matter, and hence are completely unconstrained within the overall total constraint on the dark matter density. For intermediate cases where the neutrinos become nonrelativistic well before recombination they behave like warm dark matter. The approach to the massive limit gives the tail of allowed models with large and low N_{eff} shown in Fig. 28, with increasing being compensated by decreased Ω_{c}h^{2} to maintain the total level required to give the correct shape to the CMB power spectrum.
For low and ΔN_{eff} away from zero the physical neutrino mass is very light, and the constraint becomes similar to the massless case. The different limits are continuously connected, and given the complicated shape seen in Fig. 28 it is clearly not appropriate to quote fully marginalized parameter constraints that would depend strongly on the assumed upper limit on . Instead we restrict attention to the case where the physical mass is , which roughly defines the region where (for the CMB) the particles are distinct from cold or warm dark matter. Using the Planck+WP+highL (abbreviated to CMB below) data combination, this gives the marginalized oneparameter constraints (81)Combining further with BAO these tighten to (82)These bounds are only marginally compatible with a fully thermalized sterile neutrino (N_{eff} ≈ 4) with subeV mass that could explain the oscillation anomalies. The above contraints are also appropriate for the DodelsonWidrow scenario, but for a physical mass cut of .
The thermal and DodelsonWidrow scenarios considered here are representative of a large number of possible models that have recently been investigated in the literature (Hamann et al. 2011; Diamanti et al. 2013; Archidiacono et al. 2012; Hannestad et al. 2012).
6.4. Big bang nucleosynthesis
Observations of light elements abundances created during big bang nucleosynthesis (BBN) provided one of the earliest precision tests of cosmology and were critical in establishing the existence of a hot big bang. Uptodate accounts of nucleosynthesis are given by Iocco et al. (2009) and Steigman (2012). In the standard BBN model, the abundance of light elements (parameterized by for helium4 and for deuterium, where n_{i} is the number density of species i)^{41} can be predicted as a function of the baryon density ω_{b}, the number of relativistic degrees of freedom parameterized by N_{eff}, and of the lepton asymmetry in the electron neutrino sector. Throughout this subsection, we assume for simplicity that lepton asymmetry is too small to play a role at BBN. This is a reasonable assumption, since Planck data cannot improve existing constraints on the asymmetry^{42}. We also assume that there is no significant entropy increase between BBN and the present day, so that our CMB constraints on the baryontophoton ratio can be used to compute primordial abundances.
To calculate the dependence of and on the parameters ω_{b} and N_{eff}, we use the accurate public code PArthENoPE (Pisanti et al. 2008), which incorporates values of nuclear reaction rates, particle masses and fundamental constants, and an updated estimate of the neutron lifetime (τ_{n} = 880.1 s; Beringer et al. 2012). Experimental uncertainties on each of these quantities lead to a theoretical error for and . For helium, the error is dominated by the uncertainty in the neutron lifetime, leading to^{43}. For deuterium, the error is dominated by uncertainties in several nuclear rates, and is estimated to be (Serpico et al. 2004).
These predictions for the light elements can be confronted with measurements of their abundances, and also with CMB data (which is sensitive to ω_{b}, N_{eff}, and Y_{P}). We shall see below that for the base cosmological model with N_{eff} = 3.046 (or even for an extended scenario with free N_{eff}) the CMB data predict the primordial abundances, under the assumption of standard BBN, with smaller uncertainties than those estimated for the measured abundances. Furthermore, the CMB predictions are consistent with direct abundance measurements.
6.4.1. Observational data on primordial abundances
The observational constraint on the primordial helium4 fraction used in this paper is (68% CL) from the recent data compilation of Aver et al. (2012), based on spectroscopic observations of the chemical abundances in metalpoor H ii regions. The error on this measurement is dominated by systematic effects that will be difficult to resolve in the near future. It is reassuring that the independent and conservative method presented in Mangano & Serpico (2011) leads to an upper bound for that is consistent with the above estimate. The recent measurement of the protoSolar helium abundance by Serenelli & Basu (2010) provides an even more conservative upper bound, at the 2σ level.
For the primordial abundance of deuterium, data points show excess scatter above the statistical errors, indicative of systematic errors. The compilation presented in Iocco et al. (2009), based on data accumulated over several years, gives (68% CL). Pettini & Cooke (2012) report an accurate deuterium abundance measurement in the z = 3.04984 lowmetallicity damped Lyα system in the spectrum of QSO SDSS J1419+0829, which they argue is particularly well suited to deuterium abundance measurements. These authors find (68% CL), a significantly tighter constraint than that from the Iocco et al. (2009) compilation. The PettiniCooke measurement is, however, a single data point, and it is important to acquire more observations of similar systems to assess whether their error estimate is consistent with possible sources of systematic error. We adopt a conservative position in this paper and compare both the Iocco et al. (2009) and the Pettini & Cooke (2012) measurements to the CMB predictions
We consider only the ^{4}He and D abundances in this paper. We do not discuss measurements of ^{3}He abundances since these provide only an upper bound on the true primordial ^{3}He fraction. Likewise, we do not discuss lithium. There has been a long standing discrepancy between the low lithium abundances measured in metalpoor stars in our Galaxy and the predictions of BBN. At present it is not clear whether this discrepancy is caused by systematic errors in the abundance measurements, or has an “astrophysical” solution (e.g., destruction of primordial lithium) or is caused by new physics (see Fields 2011, for a recent review).
Fig. 29 Predictions of standard BBN for the primordial abundance of ^{4}He (top) and deuterium (bottom), as a function of the baryon density. The width of the green stripes corresponds to 68% uncertainties on nuclear reaction rates. The horizontal bands show observational bounds on primordial element abundances compiled by various authors, and the red vertical band shows the Planck+WP+highL bounds on ω_{b} (all with 68% errors). BBN predictions and CMB results assume N_{eff} = 3.046 and no significant lepton asymmetry. 
6.4.2. Planck predictions of primordial abundances in standard BBN
We first restrict ourselves to the base cosmological model, with no extra relativistic degrees of freedom beyond ordinary neutrinos (and a negligible lepton asymmetry), leading to N_{eff} = 3.046 (Mangano et al. 2005). Assuming that standard BBN holds, and that there is no entropy release after BBN, we can compute the spectrum of CMB anisotropies using the relation Y_{P}(ω_{b}) given by PArthENoPE. This relation is used as the default in the grid of models discussed in this paper; we use the CosmoMC implementation developed by Hamann et al. (2008). The Planck+WP+highL fits to the base ΛCDM model gives the following estimate of the baryon density, (83)as listed in Table 5. In Fig. 29, we show this bound together with theoretical BBN predictions for and . The bound of Eq. (83) leads to the predictions where the errors here are 68% and include theoretical errors that are added in quadrature to those arising from uncertainties in ω_{b}. (The theoretical error dominates the total error in the case of Y_{P}.)^{44} For helium, this prediction is in very good agreement with the data compilation of Aver et al. (2012), with an error that is 26 times smaller. For deuterium, the CMB+BBN prediction lies midway between the bestfit values of Iocco et al. (2009) and Pettini & Cooke (2012), but agrees with both at approximately the 1σ level. These results strongly support standard BBN and show that within the framework of the base ΛCDM model, Planck observations lead to extremely precise predictions of primordial abundances.
6.4.3. Estimating the helium abundance directly from Planck data
In the CMB analysis, instead of fixing Y_{P} to the BBN prediction, , we can relax any BBN prior and let this parameter vary freely. The primordial helium fraction has an influence on the recombination history and affects CMB anisotropies mainly through the redshift of last scattering and the diffusion damping scale (Hu et al. 1995; Trotta & Hansen 2004; Ichikawa & Takahashi 2006; Hamann et al. 2008). Extending the base ΛCDM model by adding Y_{P} as a free parameter with a flat prior in the range [0.1,0.5], we find (85)Constraints in the Y_{P}ω_{b} plane are shown in Fig. 30. This figure shows that the CMB data have some sensitivity to the helium abundance. In fact, the error on the CMB estimate of Y_{P} is only 2.7 times larger than the direct measurements of the primordial helium abundance by Aver et al. (2012). The CMB estimate of Y_{P} is consistent with the observational measurements adding further support in favour of standard BBN.
Fig. 30 Constraints in the ω_{b}Y_{P} plane from CMB and abundance measurements. The CMB constraints are for Planck+WP+highL (red 68% and 95% contours) in ΛCDM models with Y_{P} allowed to vary freely. The horizontal band shows observational bounds on ^{4}He compiled by Aver et al. (2012) with 68% errors, while the grey region at the top of the figure delineates the conservative 95% upper bound inferred from Solar helium abundance by Serenelli & Basu (2010). The green stripe shows the predictions of standard BBN for the primordial abundance of ^{4}He as a function of the baryon density (with 68% errors on nuclear reaction rates). Both BBN predictions and CMB results assume N_{eff} = 3.046 and no significant lepton asymmetry. 
6.4.4. Extension to the case with extra relativistic relics
We now consider the effects of additional relativistic degrees of freedom on photons and ordinary neutrinos (obeying the standard model of neutrino decoupling) by adding N_{eff} as a free parameter. In the absence of lepton asymmetry, we can predict the BBN primordial abundances as a function of the two parameters ω_{b} and N_{eff}.
Fig. 31 Constraints in the ω_{b}N_{eff} plane from the CMB and abundance measurements. The blue stripes shows the 68% confidence regions from measurements of primordial element abundances assuming standard BBN: ^{4}He bounds compiled by Aver et al. (2012); and deuterium bounds complied by Iocco et al. (2009) or measured by Pettini & Cooke (2012). We show for comparison the 68% and 95% contours inferred from Planck+WP+highL, when N_{eff} is left as a free parameter in the CMB analysis (and Y_{P} is fixed as a function of ω_{b} and N_{eff} according to BBN predictions). These constraints assume no significant lepton asymmetry. 
Figure 31 shows the regions in the ω_{b}N_{eff} plane preferred by primordial abundance measurements, and by the CMB data if the standard BBN picture is correct. The regions allowed by the abundance measurements are defined by the χ^{2} statistic (86)where y(ω_{b},N_{eff}) is the BBN prediction for either or , the quantity y_{obs} is the observed abundance, and the two errors in the denominator are the observational and theoretical uncertainties. Figure 31 shows the edges of the 68% preferred regions in the ω_{b}N_{eff} plane, given by .
For the CMB data, we fit a cosmological model with seven free parameters (the six parameters of the base ΛCDM model, plus N_{eff}) to the Planck+WP+highL data, assuming that the primordial helium fraction is fixed by the standard BBN prediction . Figure 31 shows the joint 68% and 95% confidence contours in the ω_{b}N_{eff} plane. The preferred regions in this plane from abundance measurements and the CMB agree remarkably well. The CMB gives approximately three times smaller error bars than primordial abundance data on both parameters.
We can derive constraints on N_{eff} from primordial element abundances and CMB data together by combining their likelihoods. The CMBonly confidence interval for N_{eff} is (87)When combined with the data reported respectively by Aver et al. (2012), Iocco et al. (2009), and Pettini & Cooke (2012), the 68% confidence interval becomes (88)Since there is no significant tension between CMB and primordial element results, all these bounds are in agreement with the CMBonly analysis. The small error bar derived from combining the CMB with the Pettini & Cooke (2012) data point shows that further deuterium observations combined with Planck data have the potential to pin down the value of N_{eff} to high precision.
6.4.5. Simultaneous constraints on both N_{eff} and Y_{P}
In this subsection, we discuss simultaneous constraints on both N_{eff} and Y_{P} by adding them to the six parameters of the base ΛCDM model. Both N_{eff} and Y_{P} have an impact on the damping tail of the CMB power spectrum by altering the ratio , where is the photon diffusion length at last scattering and r_{∗} is the sound horizon there. There is thus an approximate degeneracy between these two parameters along which the ratio is nearly constant. The extent of the degeneracy is limited by the characteristic phase shift of the acoustic oscillations that arises due to the free streaming of the neutrinos (Bashinsky & Seljak 2004). As discussed by Hou et al. (2013), the early ISW effect also partly breaks the degeneracy, but this is less important than the effect of the phase shifts.
Fig. 32 2D joint posterior distribution for N_{eff} and Y_{P} with both parameters varying freely, determined from Planck+WP+highL data. Samples are colourcoded by the value of the angular ratio θ_{D}/θ_{∗}, which is constant along the degeneracy direction. The N_{eff}Y_{P} relation from BBN theory is shown by the dashed curve. The vertical line shows the standard value N_{eff} = 3.046. The region with Y_{P}> 0.294 is highlighted in grey, delineating the region that exceeds the 2σ upper limit of the recent measurement of initial Solar helium abundance (Serenelli & Basu 2010), and the blue horizontal region is the 68% confidence region from the Aver et al. (2012) compilation of ^{4}He measurements. 
The joint posterior distribution for N_{eff} and Y_{P} from the Planck+WP+highL likelihood is shown in Fig. 32, with each MCMC sample colourcoded by the value of the observationallyrelevant angular ratio θ_{D}/θ_{∗} ∝ (k_{D}r_{∗})^{1}. The main constraint on N_{eff} and Y_{P} comes from the precise measurement of this ratio by the CMB, leaving the degeneracy along the constant θ_{D}/θ_{∗} direction. The relation between N_{eff} and Y_{P} from BBN theory is shown in the figure by the dashed curve^{45}. The standard BBN prediction with N_{eff} = 3.046 is contained within the 68% confidence region. The grey region is for Y_{P}> 0.294 and is the 2σ conservative upper bound on the primordial helium abundance from Serenelli & Basu (2010). Most of the samples are consistent with this bound. The inferred estimates of N_{eff} and Y_{P} from the Planck+WP+highL data are With Y_{P} allowed to vary, N_{eff} is no longer tightly constrained by the value of θ_{D}/θ_{∗}. Instead, it is constrained, at least in part, by the impact that varying N_{eff} has on the phase shifts of the acoustic oscillations. As discussed in Hou et al. (2014), this effect explains the observed correlation between N_{eff} and θ_{∗}, which is shown in Fig. 33. The correlation in the ΛCDM+N_{eff} model is also plotted in the figure showing that the N_{eff}Y_{P} degeneracy combines with the phase shifts to generate a larger dispersion in θ_{∗} in such models.
Fig. 33 2D joint posterior distribution between N_{eff} and θ_{∗} for ΛCDM models with variable N_{eff} (blue) and variable N_{eff} and Y_{P} (red). Both cases are for Planck+WP+highL data. 
6.5. Dark energy
A major challenge for cosmology is to elucidate the nature of the dark energy driving the accelerated expansion of the Universe. Perhaps the most straightforward explanation is that dark energy is a cosmological constant. An alternative is dynamical dark energy (Wetterich 1988; Ratra & Peebles 1988; Caldwell et al. 1998b), usually based on a scalar field. In the simplest models, the field is very light, has a canonical kinetic energy term and is minimally coupled to gravity. In such models the dark energy sound speed equals the speed of light and it has zero anisotropic stress. It thus contributes very little to clustering. We shall only consider such models in this subsection.
A cosmological constant has an equation of state w ≡ p/ρ = − 1, while scalar field models typically have time varying w with w ≥ − 1. The analysis performed here is based on the “parameterized postFriedmann” (PPF) framework of Hu & Sawicki (2007) and Hu (2008) as implemented in camb (Fang et al. 2008b,a) and discussed earlier in Sect. 2. This allows us to investigate both regions of parameter space in which w< − 1 (sometimes referred to as the “phantom” domain) and models in which w changes with time.
Fig. 34 Marginalized posterior distributions for the dark energy equation of state parameter w (assumed constant), for Planck+WP alone (green) and in combination with SNe data (SNSL in blue and the Union2.1 compilation in red) or BAO data (black). A flat prior on w from −3 to −0.3 was assumed and, importantly for the CMBonly constraints, the prior [20,100] km s^{1} Mpc^{1} on H_{0}. The dashed grey line indicates the cosmological constant solution, w = − 1. 
Figure 34 shows the marginalized posterior distributions for w for an extension of the base ΛCDM cosmology to models with constant w. We present results for Planck+WP and in combination with SNe or BAO data. (Note that adding in the highℓ data from ACT and SPT results in little change to the posteriors shown in Fig. 34.) As expected, the CMB alone does not strongly constrain w, due to the twodimensional geometric degeneracy in these models. We can break this degeneracy by combining the CMB data with lower redshift distance measures. Adding in BAO data tightens the constraints substantially, giving (90)in good agreement with a cosmological constant (w = − 1). Using supernovae data leads to the constraints The combination with SNLS data favours the phantom domain (w< − 1) at 2σ, while the Union2.1 compilation is more consistent with a cosmological constant.
If instead we combine Planck+WP with the Riess et al. (2011) measurement of H_{0}, we find (92)which is in tension with w = − 1 at more than the 2σ level.
The results in Eqs. (90)–(92) reflect the tensions between the supplementary data sets and the Planck base ΛCDM cosmology discussed in Sect. 5. The BAO data are in excellent agreement with the Planck base ΛCDM model, so there is no significant preference for w ≠ − 1 when combining BAO with Planck. In contrast, the addition of the H_{0} measurement, or SNLS SNe data, to the CMB data favours models with exotic physics in the dark energy sector. These trends form a consistent theme throughout this section. The SNLS data favours a lower Ω_{m} in the ΛCDM model than Planck, and hence larger dark energy density today. The tension can be relieved by making the dark energy fall away faster in the past than for a cosmological constant, i.e., w< − 1.
The constant w models are of limited physical interest. If w ≠ − 1 then it is likely to change with time. To investigate this we consider the simple linear relation in Eq. (4), w(a) = w_{0} + w_{a}(1 − a), which has often been used in the literature (Chevallier & Polarski 2001; Linder 2003). This parameterization approximately captures the lowredshift behaviour of light, slowlyrolling minimallycoupled scalar fields (as long as they do not contribute significantly to the total energy density at early times) and avoids the complexity of scanning a large number of possible potential shapes and initial conditions. The dynamical evolution of w(a) can lead to distinctive imprints in the CMB (Caldwell et al. 1998a) which would show up in the Planck data.
Fig. 35 2D marginalized posterior distribution for w_{0} and w_{a} for Planck+WP+BAO data. The contours are 68% and 95%, and the samples are colourcoded according to the value of H_{0}. Independent flat priors of −3 <w_{0}< − 0.3 and −2 <w_{a}< 2 are assumed. Dashed grey lines show the cosmological constant solution w_{0} = − 1 and w_{a} = 0. 
Figure 35 shows contours of the joint posterior distribution in the w_{0}w_{a} plane using Planck+WP+BAO data (colourcoded according to the value of H_{0}). The points are coloured by the value of H_{0}, which shows a clear variation with w_{0} and w_{a} revealing the threedimensional nature of the geometric degeneracy in such models. The cosmological constant point (w_{0},w_{a}) = (−1,0) lies within the 68% contour and the marginalized posteriors for w_{0} and w_{a} are Including the H_{0} measurement in place of the BAO data moves (w_{0},w_{a}) away from the cosmological constant solution towards negative w_{a} at just under the 2σ level.
Figure 36 shows likelihood contours for (w_{0},w_{a}), now adding SNe data to Planck. As discussed in detail in Sect. 5, there is a dependence of the base ΛCDM parameters on the choice of SNe data set, and this is reflected in Fig. 36. The results from the Planck+WP+Union2.1 data combination are in better agreement with a cosmological constant than those from the Planck+WP+SNLS combination. For the latter data combination, the cosmological constant solution lies on the 2σ boundary of the (w_{0},w_{a}) distribution.
Fig. 36 2D marginalized posterior distributions for w_{0} and w_{a}, for the data combinations Planck+WP+BAO (grey), Planck+WP+Union2.1 (red) and Planck+WP+SNLS (blue). The contours are 68% and 95%, and dashed grey lines show the cosmological constant solution. 
Dynamical dark energy models might also give a nonnegligible contribution to the energy density of the Universe at early times. Such early dark energy (EDE; Wetterich 2004) models may be very close to ΛCDM recently, but have a nonzero dark energy density fraction, Ω_{e}, at early times. Such models complement the (w_{0},w_{a}) analysis by investigating how much dark energy can be present at high redshifts. EDE has two main effects: it reduces structure growth in the period after last scattering; and it changes the position and height of the peaks in the CMB spectrum.
The model we adopt here is that of Doran & Robbers (2006): (94)It requires two additional parameters to those of the base ΛCDM model: Ω_{e}, the dark energy density relative to the critical density at early times (assumed constant in this treatment); and the presentday dark energy equation of state parameter w_{0}. Here is the present matter density and is the present dark energy abundance (for a flat Universe). Note that the model of Eq. (94) has dark energy present over a large range of redshifts; the bounds on Ω_{e} can be substantially weaker if dark energy is only present over a limited range of redshifts (Pettorino et al. 2013). The presence or absence of dark energy at the epoch of last scattering is the dominant effect on the CMB anisotropies and hence the constraints are insensitive to the addition of low redshift supplementary data such as BAO.
The most precise bounds on EDE arise from the analysis of CMB anisotropies (Doran et al. 2001; Caldwell et al. 2003; Calabrese et al. 2011; Reichardt et al. 2012a; Sievers et al. 2013; Hou et al. 2014; Pettorino et al. 2013). Using Planck+WP+highL, we find (95)(The limit for Planck+WP is very similar: Ω_{e}< 0.010.) These bounds are consistent with and improve the recent ones of Hou et al. (2014), who give Ω_{e}< 0.013 at 95% CL, and Sievers et al. (2013), who find Ω_{e}< 0.025 at 95% CL.
In summary, the results on dynamical dark energy (except for those on early dark energy discussed above) are dependent on exactly what supplementary data are used in conjunction with the CMB data. (Planck lensing does not significantly improve the constraints on the models discussed here.) Using the direct measurement of H_{0}, or the SNLS SNe sample, together with Planck we see preferences for dynamical dark energy at about the 2σ level reflecting the tensions between these data sets and Planck in the ΛCDM model. In contrast, the BAO measurements together with Planck give tight constraints which are consistent with a cosmological constant. Our inclination is to give greater weight to the BAO measurements and to conclude that there is no strong evidence that the dark energy is anything other than a cosmological constant.
6.6. Dark matter annihilation
Energy injection from dark matter (DM) annihilation can change the recombination history and affect the shape of the angular CMB spectra (Chen & Kamionkowski 2004; Padmanabhan & Finkbeiner 2005; Zhang et al. 2006; Mapelli et al. 2006). As recently shown in several papers (see e.g., Galli et al. 2009, 2011; Giesen et al. 2012; Hutsi et al. 2011; Natarajan 2012; Evoli et al. 2013) CMB anisotropies offer an opportunity to constrain DM annihilation models.
Highenergy particles injected in the highredshift thermal gas by DM annihilation are typically cooled down to the keV scale by high energy processes; once the shower has reached this energy scale, the secondary particles produced can ionize, excite or heat the thermal gas (Shull & van Steenberg 1985; Valdes et al. 2010); the first two processes modify the evolution of the free electron fraction x_{e}, while the third affects the temperature of the baryons.
The rate of energy release, dE/dt, per unit volume by a relic annihilating DM particle is given by (96)where p_{ann} is, in principle, a function of redshift z, defined as (97)where ⟨ σv ⟩ is the thermally averaged annihilation crosssection, m_{χ} is the mass of the DM particle, ρ_{c} is the critical density of the Universe today, g is a degeneracy factor equal to 1/2 for Majorana particles and 1/4 for Dirac particles (in the following, constraints refer to Majorana particles), and the parameter f(z) indicates the fraction of energy which is absorbed overall by the gas at redshift z.
In Eq. (97), the factor f(z) depends on the details of the annihilation process, such as the mass of the DM particle and the annihilation channel (see e.g., Slatyer et al. 2009). The functional shape of f(z) can be taken into account using generalized parameterizations (Finkbeiner et al. 2012; Hutsi et al. 2011). However, as shown in Galli et al. (2011), Giesen et al. (2012), and Finkbeiner et al. (2012) it is possible to neglect the redshift dependence of f(z) to first approximation, since current data shows very little sensitivity to variations of this function. The effects of DM annihilation can therefore be well parameterized by a single constant parameter, p_{ann}, that encodes the dependence on the properties of the DM particles.
We compute here the theoretical angular power in the presence of DM annihilations, by modifying the RECFAST routine in the camb code as in Galli et al. (2011) and by making use of the package CosmoMC for Monte Carlo parameter estimation. We checked that we obtain the same results by using the CLASS Boltzmann code (Lesgourgues 2011a) and the Monte Python package (Audren et al. 2013), with DM annihilation effects calculated either by RECFAST or HyRec (AliHaimoud & Hirata 2011), as detailed in Giesen et al. (2012). Besides p_{ann}, we sample the parameters of the base ΛCDM model and the foreground/nuisance parameters described in Sect. 4.
From Planck+WP we find (98)This constraint is weaker than that found from the full WMAP9 temperature and polarization likelihood, p_{ann}< 1.2 × 10^{6} m^{3}s^{1}kg^{1} because the Planck likelihood does not yet include polarization information at intermediate and high multipoles. In fact, the damping effect of DM annihilation on the CMB temperature power spectrum is highly degenerate with other cosmological parameters, in particular with the scalar spectral index and the scalar amplitude, as first shown by Padmanabhan & Finkbeiner (2005). As a consequence, the constraint on the scalar spectral index is significantly weakened when p_{ann} is allowed to vary, , to be compared to the constraint listed in Table 2 for the base ΛCDM cosmology, n_{s} = 0.9603 ± 0.0073.
These degeneracies can be broken by polarization data. The effect of DM annihilation on polarization is in fact an overall enhancement of the amplitude at large and intermediate scales, and a damping at small scales (see e.g., Fig. 1 in Galli et al. 2009 or Fig. 3 in Giesen et al. 2012). We thus expect the constraint to improve significantly with the forthcoming Planck polarization data release. We verified that adding BAO, HST or highL data to Planck+WP improves the constraints only marginally, as these data sets are not able to break the degeneracy between p_{ann} and n_{s}.
On the other hand, we observe a substantial improvement in the constraints when we combine the Planck+WP data with the Planck lensing likelihood data. For this data combination we find an upper limit of (99)The improvement over Eq. (98) comes from the constraining power of the lensing likelihood on A_{s} and n_{s}, that partially breaks the degeneracy with p_{ann}.
Our results are consistent with previous work and show no evidence for DM annihilation. Future release of Planck polarization data will help to break the degeneracies which currently limit the accuracy of the constraints presented here.
6.7. Constraints on a stochastic background of primordial magnetic fields
Largescale magnetic fields of the order of a few μG observed in galaxies and galaxy clusters may be the product of the amplification during structure formation, of primordial magnetic seeds (Ryu et al. 2012). Several models of the early Universe predict the generation of primordial magnetic fields (hereafter PMF), either during inflation or during later phase transitions (see Widrow 2002; and Widrow et al. 2012, for reviews).
PMF have an impact on cosmological perturbations and in particular on CMB anisotropy angular power spectra (Subramanian 2006), that can be used to constrain the PMF amplitude. In this section we derive the constraints from Planck data on a stochastic background of PMF. We are mainly interested in constraints from CMB temperature anisotropies. Therefore, we do not consider the effect of Faraday rotation on CMB polarization anisotropies (Kosowsky & Loeb 1996; Kosowsky et al. 2005) nor nonGaussianities associated with PMF (Brown & Crittenden 2005; Caprini et al. 2009; Seshadri & Subramanian 2009; Trivedi et al. 2010). We restrict the analysis reported here to the nonhelical case.
A stochastic background of PMF is modelled as a fully inhomogeneous component whose energymomentum tensor is quadratic in the fields. We assume the usual magnetohydrodynamics limit, in which PMF are frozen and the time evolution is simply given by the dilution with cosmological expansion, B(k,η) = B(k) /a(η)^{2}. We model the PMF with a simple powerlaw power spectrum: P_{B}(k) = Ak^{nB}, with a sharp cut off at the damping scale k_{D}, as computed in Jedamzik et al. (1998) and Subramanian & Barrow (1998), to model the suppression of PMF on small scales.
It is customary to specify the amplitude of the PMF power spectrum with B_{λ}, the rootmeansquare of the field smoothed over length scale λ, defined such that (100)Given our assumed model and conventions, PMF are fully described by two parameters: the smoothed amplitude B_{λ}; and the spectral index n_{B}. Here, we set λ = 1 Mpc and hence use B_{1 Mpc} as the parameter.
The components of the energy momentum tensor of PMF source all types of linear cosmological perturbations, i.e., scalar, vector, and tensor. In particular, the source terms are given by the magnetic energy density and anisotropic stress for scalar magnetized perturbations, whereas vector and tensor modes are sourced only by the magnetic anisotropic stress. In addition, both scalar and vector perturbations are affected by the Lorentz force; PMF induce a Lorentz force on baryons modifying their evolution and in particular their velocity, but during the tightcoupling regime between matter and radiation the Lorentz force also has an indirect effect on photons.
For the computation of magnetized angular power spectra, we use the analytic approximations for the PMF energymomentum tensor components given in Paoletti & Finelli (2011). We consider here the regular mode for magnetic scalar perturbations, with the initial conditions of Paoletti et al. (2009) (see Giovannini 2004 for earlier calculations) and Shaw & Lewis (2010) (which describes the singular passive mode, depending on the generation time of PMF).
Previous analyses show that the main impact of PMF on the CMB anisotropy angular power spectrum is at small angular scales, well into the Silk damping regime. The dominant mode is the magnetic vector mode which peaks at ℓ ~ 2000–3000 (Mack et al. 2002; Lewis 2004). The scalar magnetic mode is the dominant PMF contribution on large and intermediate angular scales (Giovannini 2007; Giovannini & Kunze 2008; Finelli et al. 2008). The tensor contribution is always subdominant with respect to the other two and it is negligible for the purposes of this analysis.
We include the scalar and vector magnetized contributions to the angular power spectrum within the MCMC analysis to derive the constraints on the PMF amplitude and spectral index using PlanckTT data. We vary the magnetic parameters B_{1 Mpc}/ nG and n_{B}, in addition to the other cosmological parameters of the base ΛCDM cosmology (this analysis assumes massless neutrinos, rather than the default value of a single eigenstate of mass 0.06 eV used in the rest of this paper). We adopt as prior ranges for the parameters [0 ,10] for B_{1 Mpc}/ nG and [− 2.99 ,3] for the spectral index n_{B}. The lower bound n_{B}> − 3 is necessary to avoid infrared divergences in the PMF energy momentum tensor correlators.
Fig. 37 Constraints on the rootmeansquare amplitude of the primordial magnetic field (for a smoothing scale of 1 Mpc) obtained with Planck+WP (black) and Planck+WP+highL (red). 
We perform analyses with Planck+WP and Planck+WP+ highL likelihood combinations. Results are shown in Fig. 37. We find that the cosmological parameters are in agreement with those estimated assuming no PMF, confirming that the magnetic parameters are not degenerate with the cosmological parameters of the base ΛCDM model. The constraints on PMF with the Planck+WP likelihood are B_{1 Mpc}< 4.1 nG, with a preference for negative spectral indices at the 95% confidence level. These limits are improved using Planck+WP+highL to B_{1 Mpc}< 3.4 nG with n_{B}< 0 preferred at the 95% confidence level. The new constraints are consistent with, and slightly tighter, than previous limits based on combining WMAP7 data with highresolution CMB data (see e.g. Paoletti & Finelli 2011, 2013; Shaw & Lewis 2012).
Fig. 38 Left: likelihood contours (68% and 95%) in the α/α_{0}–H_{0} plane for the WMAP9 (red), Planck+WP (blue), Planck+WP+H_{0} (purple), and Planck+WP+BAO (green) data combinations. Middle: as left, but in the α/α_{0}Ω_{b}h^{2} plane. Right: marginalized posterior distributions of α/α_{0} for these data combinations. 
Constraints on the cosmological parameters of the base ΛCDM model with the addition of a varying finestructure constant.
6.8. Constraints on variation of the finestructure constant
The ΛCDM model assumes the validity of General Relativity on cosmological scales, as well as the physics of the standard model of particle physics. One possible extension, which may have motivations in fundamental physics, is to consider variations of dimensionless constants. Such variations can be constrained through tests on astrophysical scales (Uzan 2003, 2011).
A number of physical systems have been used, spanning different time scales, to set constraints on variations of the fundamental constants. These range from atomic clocks in the laboratory at a redshift z = 0 to BBN at z ~ 10^{8}. However, apart from the claims of varying α based on high resolution quasar absorptionline spectra (Webb et al. 2001; Murphy et al. 2003)^{46}, there is no other evidence for timevariable fundamental constants.
CMB temperature anisotropies have been used extensively to constrain the variation of fundamental constants over cosmic time scales. The temperature power spectrum is sensitive to the variation of the finestructure constant α, the electrontoproton mass ratio μ, and the gravitational constant . A variation of G can affect the Friedmann equation, and also raises the issue of consistency in the overall theory of gravity. However, a variation of the nongravitational constants (α and m_{e}) is more straightforward to analyse, mostly inducing a modification of the interaction between light and atoms (shifts in the energy levels and binding energy of hydrogen and helium). This induces a modification of the ionization history of the Universe. In particular, a variation of α modifies the redshift of recombination through the shift in the energy levels and the Thomson scattering crosssection. An increase in α induces a shift of the position of the first acoustic peak, which is inversely proportional to the sound horizon at last scattering. The larger redshift of last scattering also produces a larger early ISW effect, and hence a higher amplitude of the first acoustic peak. Finally, an increase in α decreases diffusive damping at high multipoles. For earlier studies of varying constants using the CMB (see e.g., Kaplinghat et al. 1999; Avelino et al. 2000; Martins et al. 2004; Rocha et al. 2004; Nakashima et al. 2008, 2010; Menegoni et al. 2009; Landau & Scóccola 2010).
The analysis presented here focusses solely on the time variation of the finestructure constant α, in addition to the parameters of the base ΛCDM model, using a modified form of the RECFAST recombination code (Hannestad 1999; Martins et al. 2004; Rocha et al. 2004). Selected results are given in Table 11, which compares parameter constraints from Planck and from our own analysis of the full WMAP9 TT, TE and EE likelihood. From CMB data alone, Planck improves the constraints from a 2% variation in α to about 0.4%. Planck thus improves the limit by a factor of around five, while the constraints on the parameters of the base ΛCDM model change very little with the addition of a timevarying α. These results are in good agreement with earlier forecasts (Rocha et al. 2004).
Given the apparent tension between the base ΛCDM parameters from Planck and direct measurements of H_{0} discussed in Sect. 5.3), we include further information from the H_{0} prior and BAO data (see Sect. 5.2). Figure 38 compares the constraints in the (α/α_{0},H_{0}) and (α/α_{0},Ω_{b}h^{2}) planes and also shows the marginalized posterior distribution of α/α_{0} for the various data combinations.
The constraint on α is slightly improved by including the BAO data (via a tightening of the parameters of the base ΛCDM model). Note that the central value of the prior on H_{0} is outside the 95% confidence region, even for the Planck+WP+H_{0} combination. Adding a varying α does not resolve the tension between direct measurements of H_{0} and the value determined from the CMB.
In summary, Planck data improve the constraints on α/α_{0}, with respect to those from WMAP9 by a factor of about five. Our analysis of Planck data limits any variation in the finestructure constant from z ~ 10^{3} to the present day to be less than approximately 0.4%.
7. Discussion and conclusions^{47}
The most important conclusion from this paper is the excellent agreement between the Planck temperature power spectrum at high multipoles with the predictions of the base ΛCDM model. The base ΛCDM model also provides a good match to the Planck power spectrum of the lensing potential, , and to the TE and EE power spectra at high multipoles.
The high statistical significance of the Planck detection of gravitational lensing of the CMB leads to some interesting science conclusions using Planck data alone. For example, gravitational lensing breaks the “geometrical degeneracy” and we find that the geometry of the Universe is consistent with spatial flatness to percentlevel precision using CMB data alone. The Planck lensing power spectrum also leads to an interesting constraint on the reionization optical depth of τ = 0.089 ± 0.032, independent of CMB polarization measurements at low multipoles.
The parameters of the base ΛCDM model are determined to extremely high precision by the Planck data. For example, the scalar spectral index is determined as n_{s} = 0.9585 ± 0.0070, a 6σ deviation from exact scale invariance. Even in the base ΛCDM model, we find quite large changes in some parameters compared to previous CMB experiments^{48}. In particular, from Planck we find a low value of the Hubble constant, H_{0} = (67.3 ± 1.2) km s^{1} Mpc^{1}, and a high matter density, Ω_{m} = 0.315 ± 0.016. If we accept that the base ΛCDM model is the correct cosmology, then as discussed in Sect. 5Planck is in tension with direct measurements of the Hubble constant (at about the 2.5σ level) and in mild tension with the SNLS Type Ia supernova compilation (at about the 2σ level). For the base ΛCDM model, we also find a high amplitude for the presentday matter fluctuations, σ_{8} = 0.828 ± 0.012, in agreement with previous CMB experiments. This value is higher than that inferred from counts of rich clusters of galaxies, including our own analysis of Planck cluster counts (Planck Collaboration XX 2014),and in tension with the cosmic shear measurements discussed in Sect. 5.5.2.
One possible interpretation of these tensions is that systematic errors are not completely understood in some astrophysical measurements. The fact that the Planck results for the base ΛCDM model are in such good agreement with BAO data, which are based on a simple geometrical measurement, lends support to this view. An alternative explanation is that the base ΛCDM model is incorrect. In summary, at high multipoles, the base ΛCDM cosmology provides an excellent fit to the spectra from Planck, ACT and SPT (for all frequency combinations), as illustrated in Figs. 7–9, but the parameters derived from the CMB apparently conflict with some types of astrophysical measurement.
Fig. 39 Left: PlanckTT spectrum at low multipoles with 68% ranges on the posteriors. The “rainbow” band show the best fits to the entire Planck+WP+highL likelihood for the base ΛCDM cosmology, colourcoded according to the value of the scalar spectral index n_{s}. Right: limits (68% and 95%) on the relative amplitude of the base ΛCDM fits to the Planck+WP likelihood fitted only to the Planck TT likelihood over the multipole range 2 ≤ ℓ ≤ ℓ_{max}. 
Before summarizing our results on extensions to the base ΛCDM model, it is worth making some remarks on foreground modelling and the impact of this modelling on our error estimates. The addition of CMB data at high multipoles helps to constrain the model of unresolved foregrounds, in particular, the contribution from “minor” components, such as the kinetic SZ, which are poorly constrained from Planck alone. For the base ΛCDM model, the cosmological parameters are not limited by foreground modelling^{49}, as illustrated in Fig. 4. As discussed in Appendix C, foreground modelling becomes more important in analysing extended CDM models, particularly those that have strong parameter degeneracies that are broken only via precision measurements of the damping tail in the CMB spectrum. As a crude measure of the importance of foreground modelling, we can compare parameter values with and without inclusion of the ACT and SPT data at high multipoles. A large shift in parameter values indicates a possible sensitivity to foreground modelling, and so any such result should be treated with caution. We have thus normally adopted the Planck+WP+highL likelihood combination as offering the most reliable results for extensions to the base ΛCDM cosmology.
From an analysis of an extensive grid of models, we find no strong evidence to favour any extension to the base Λ CDM cosmology, either from the CMB temperature power spectrum alone, or in combination with the Planck lensing power spectrum and other astrophysical data sets.
We find the following notable results using CMB data alone:

The deviation of the scalar spectral index from unity is robust to theaddition of tensor modes and to changes in the matter content ofthe Universe. For example, adding a tensor component we find n_{s} = 0.9600 ± 0.0072, a 5.5σdeparture from n_{s} = 1.

A 95% upper limit on the tensortoscalar ratio of r_{0.002}< 0.11. The combined contraints on n_{s} and r_{0.002} are on the borderline of compatibility with singlefield inflation with a quadratic potential (Fig. 23).

A 95% upper limit on the summed neutrino mass of ∑ m_{ν}< 0.66 eV.

A determination of the effective number of neutrinolike relativistic degrees of freedom of N_{eff} = 3.36 ± 0.34, compatible with the standard value of 3.046.

The results from Planck are consistent with the results of standard big bang nucleosynthesis. In fact, combining the CMB data with the most recent results on the deuterium abundance, leads to the constraint N_{eff} = 3.02 ± 0.27, again compatible with the standard value of 3.046.

New limits on a possible variation of the finestructure constant, dark matter annihilation and primordial magnetic fields.
We also find a number of marginal (around 2σ) results, perhaps indicative of internal tension within the Planck data. Examples include the preference of the (phenomenological) lensing parameter for values greater than unity (A_{L} = 1.23 ± 0.11; Eq. (44)) and for negative running (dn_{s}/ dlnk = − 0.015 ± 0.09; Eq. (61b)). In Planck Collaboration XXII (2014), the Planck data indicate a preference for anticorrelated isocurvature modes and for models with a truncated power spectrum on large scales. None of these results have a decisive level of statistical significance, but they can all be traced to an unusual aspect of the temperature power spectrum at low multipoles. As can be seen in Fig. 1, and on an expanded scale in the lefthand panel of Fig. 39, the measured power spectrum shows a dip relative to the bestfit base ΛCDM cosmology in the multipole range 20 ≲ ℓ ≲ 30 and an excess at ℓ = 40. The existence of “glitches” in the power spectrum at low multipoles was noted by the WMAP team in the firstyear papers (Hinshaw et al. 2003; Spergel et al. 2003) and acted as motivation to fit an inflation model with a steplike feature in the potential (Peiris et al. 2003). Similar investigations have been carried out by a number of authors, (see e.g., Mortonson et al. 2009, and references therein). At these low multipoles, the Planck spectrum is in excellent agreement with the WMAP nineyear spectrum (Planck Collaboration XV 2014), so it is unlikely that any of the features such as the low quadrupole or “dip” in the multipole range 20–30 are caused by instrumental effects or Galactic foregrounds. These are real features of the CMB anisotropies.
The Planck data, however, constrain the parameters of the base ΛCDM model to such high precision that there is little remaining flexibility to fit the lowmultipole part of the spectrum. To illustrate this point, the righthand panel of Fig. 39 shows the 68% and 95% limits on the relative amplitude of the base ΛCDM model (sampling the chains constrained by the full likelihood) fitted only to the PlanckTT likelihood over the multipole range 2 ≤ ℓ ≤ ℓ_{max}. From multipoles ℓ_{max} ≈ 25 to multipoles ℓ_{max} ≈ 35, we see more than a 2σ departure from values of unity. (The maximum deviation from unity is 2.7σ at ℓ = 30.) It is difficult to know what to make of this result, and we present it here as a “curiosity” that needs further investigation. The Planck temperature data are remarkably consistent with the predictions of the base ΛCDM model at high multipoles, but it is also conceivable that the ΛCDM cosmology fails at low multipoles. There are other indications, from both WMAP and Planck data for “anomalies” at low multipoles (Planck Collaboration XXIII 2014), that may be indicative of new physics operating on the largest scales in our Universe. Interpretation of largescale anomalies (including the results shown in Fig. 39) is difficult in the absence of a theoretical framework. The problem here is assessing the role of a posteriori choices, i.e., that inconsistencies attract our attention and influence our choice of statistical test. Nevertheless, we know so little about the physics of the early Universe that we should be open to the possibility that there is new physics beyond that assumed in the base ΛCDM model. Irrespective of the interpretation, the unusual shape of the low multipole spectrum is at least partly responsible for some of the 2σ effects seen in the analysis of extensions to the ΛCDM model discussed in Sect. 6.
Supplementary information from astrophysical data sets has played an important role in the analysis of all previous CMB experiments. For Planck the interpretation of results combined with nonCMB data sets is not straightforward (as a consequence of the tensions discussed in Sect. 5). For the base ΛCDM model, the statistical power of the Planck data is so high that we find very similar cosmological parameters if we add the Riess et al. (2011) constraint on H_{0}, or either of the two SNe samples, to those derived from the CMB data alone. In these cases, the solutions simply reflect the tensions discussed in Sect. 5, for example, including the H_{0} measurement with the Planck+WP likelihood we find H_{0} = (68.6 ± 1.2) km s^{1} Mpc^{1}, discrepant with the direct measurement at the 2.2σ level.
The interpretation becomes more complex for extended models where astrophysical data is required to constrain parameters that cannot be determined accurately from CMB measurements alone. As an example, it is well known that CMB data alone provide weak constraints on the dark energy equation of state parameter w (see Fig. 34). The addition of BAO data to the CMB data gives a tight constraint of w = − 1.13 ± 0.12^{50}. However, adding the SNLS SNe data gives w = − 1.135 ± 0.069 and adding the H_{0} measurement gives w = − 1.244 ± 0.095. Adding either of the two data sets which show tension with the CMB measurements for the base ΛCDM model, draws the solutions into the phantom domain (w< − 1) at about the 2σ level. In contrast, if we use the BAO data in addition to the CMB, we find no evidence for dynamical dark energy; these data are compatible with a cosmological constant, as assumed in the base ΛCDM model.
The impact of additional astrophysical data is particularly complex in our investigation of neutrino physics (Sect. 6.3). We use the effective number of relativistic degrees of freedom, N_{eff} as an illustration. From the CMB data alone, we find N_{eff} = 3.36 ± 0.34. Adding BAO data gives N_{eff} = 3.30 ± 0.27. Both of these values are consistent with the standard value of 3.046. Adding the H_{0} measurement to the CMB data gives N_{eff} = 3.62 ± 0.25and relieves the tension between the CMB data and H_{0} at the expense of new neutrinolike physics (at around the 2.3σ level). It is possible to alleviate the tensions between the CMB, BAO, H_{0} and SNLS data by invoking new physics such as an increase in N_{eff}. However, none of these cases are favoured significantly over the base ΛCDM model by the Planck data (and they are often disfavoured). Any preference for new physics comes almost entirely from the astrophysical data sets. It is up to the reader to decide how to interpret such results, but it is simplistic to assume that all astrophysical data sets have accurately quantified estimates of systematic errors. We have therefore tended to place greater weight on the CMB and BAO measurements in this paper rather than on more complex astrophysical data.
Our overall conclusion is that the Planck data are remarkably consistent with the predictions of the base ΛCDM cosmology. However, the mismatch with the temperature spectrum at low multipoles, evident in Figs. 1 and 39, and the existence of other “anomalies” at low multipoles, is possibly indicative that the model is incomplete. The results presented here are based on a first, and relatively conservative, analysis of the Planck data. The 2014 data release will use data obtained over the full mission lifetime of Planck, including polarization data. It remains to be seen whether these data, together with new astrophysical data sets and CMB polarization measurements, will offer any convincing evidence for new physics.
For a good review of the early history of CMB studies see Peebles et al. (2009).
It is worth highlighting here the preWMAP constraints on the geometry of the Universe by the BOOMERang (Balloon Observations of Millimetric Extragalactic Radiation and Geomagnetics; de Bernardis et al. 2000) and MAXIMA (Millimeterwave Anisotropy Experiment Imaging Array; Balbi et al. 2000) experiments, for example.
Planck (http://www.esa.int/Planck) is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states (in particular the lead countries France and Italy), with contributions from NASA (USA) and telescope reflectors provided by a collaboration between ESA and a scientific consortium led and funded by Denmark.
As described in Planck Collaboration XV (2014), we use spectra calculated on different masks to isolate the contribution of Galactic dust at each frequency, which we subtract from the 143 × 143, 143 × 217 and 217 × 217 power spectra (i.e., the correction is applied to the power spectra, not in the map domain). The Galactic dust templates are shown in Fig. 7 and are less than 5 (μK)^{2} at high multipoles for the 217 × 217 spectrum and negligible at lower frequencies. The residual contribution from Galactic dust after correction in the 217 × 217 spectrum is smaller than 0.5 (μK)^{2} and smaller than the errors from other sources such as beam uncertainties.
The WP likelihood is based on the WMAP likelihood module as distributed at http://lambda.gsfc.nasa.gov
Note, however, that Planck’s measurement of θ_{∗} is now so accurate that O(10^{3}) effects from aberration due to the relative motion between our frame and the CMB restframe are becoming nonnegligible; see Planck Collaboration XXVII (2014). The statistical anisotropy induced would lead to dipolar variations at the 10^{3} level in θ_{∗} determined locally on small regions of the sky. For Planck, we average over many such regions and we expect that the residual effect (due to asymmetry in the Galactic mask) on the marginalised values of other parameters is negligible.
Planck Collaboration XXII (2014) describes a specific statistical test designed to find features in the primordial power spectrum. This test responds to the extended “dip” in the Planck power spectrum centred at about ℓ ~ 1800, tentatively suggesting 2.4–3.1σ evidence for a feature.As discussed in Sect. 1, after submission of the Planck 2013 papers, we found strong evidence that this feature is a small systematic in the 217 × 217 spectrum caused by incomplete removal of 4 K cooler lines. This feature can be seen in the residual plots in Fig. 7 and contributes to the high (almost 2σ) values of χ^{2} in the 217 × 217 residual plots.
Detections of a BAO feature have recently been reported in the threedimensional correlation function of the Lyα forest in large samples of quasars at a mean redshift of z ≈ 2.3 (Busca et al. 2013; Slosar et al. 2013). These remarkable results, probing cosmology well into the matterdominated regime, are based on new techniques that are less mature than galaxy BAO measurements. For this reason, we do not include Lyα BAO measurements as supplementary data to Planck. For the models considered here and in Sect. 6, the galaxy BAO results give significantly tighter constraints than the Lyα results.
As an indication of the accuracy of Table 8, the full likelihood results for the Planck+WP+6dF+SDSS(R)+BOSS BAO data sets give Ω_{m} = 0.308 ± 0.010 and H_{0} = 67.8 ± 0.8 km s^{1} Mpc^{1}, for the base ΛCDM model.
As noted in Sect. 1, after the submission of this paper Humphreys et al. (2013) reported a new geometric maser distance to NGC 4258 that leads to a reduction of the Riess et al. (2011) NGC 4258 value of H_{0} from (74.8 ± 3.1) km s^{1} Mpc^{1} to H_{0} = (72.0 ± 3.0) km s^{1} Mpc^{1}.
https://tspace.library.utoronto.ca/handle/1807/25390. We use the module supplied with CosmoMC.
We caution the reader that, generally, the obtained from Eq. (57) differ from that quoted in the online parameter tables in cases where the SNLS data is importance sampled. For importance sampling, we modified the SNLS likelihood to marginalize numerically over the α and β parameters.
As noted in Sect. 1, recent revisions to the photometric calibrations between the SDSS and SNLS observations relieve some of the tensions discussed in this paper between the SNe data and the Planck base ΛCDM cosmology.
There is additionally a study of the statistical properties of the Planckderived Comptony map (Planck Collaboration XXI 2014) from which other parameter estimates can be obtained.
The differences between the Planck results reported here and the WMAP7+SPT results (Hou et al. 2014) are discussed in Appendix B.
As discussed in Planck Collaboration II (2014) and Planck Collaboration VI (2014), residual lowlevel polarization systematics in both the LFI and HFI data preclude a PlanckBmode polarization analysis at this stage.
The power spectrum of the temperature anisotropies is predominantly sensitive to changes in only one mode of the lensing potential power spectrum (Smith et al. 2006). It follows that marginalizing over the single parameter A_{L} is nearly equivalent to marginalizing over the full amplitude and shape information in the lensing power spectrum as regards constraints from the temperature power spectrum.
Observations of the primordial abundances are usually reported in terms of these number ratios. For helium, differs from the mass fraction Y_{P}, used elsewhere in this paper, by 0.5% due to the binding energy of helium. Since the CMB is only sensitive to Y_{P} at the 10% level, the distinction between definitions based on the mass or number fraction is ignored when comparing helium constraints from the CMB with those from observational data on primordial abundances.
A primordial lepton asymmetry could modify the outcome of BBN only if it were very large (of the order of 10^{3} or bigger). Such a large asymmetry is not motivated by particle physics, and is strongly constrained by BBN. Indeed, by taking neutrino oscillations in the early Universe into account, which tend to equalize the distribution function of three neutrino species, Mangano et al. (2012) derived strong bounds on the lepton asymmetry. CMB data cannot improve these bounds, as shown by Castorina et al. (2012); an exquisite sensitivity to N_{eff} would be required. Note that the results of Mangano et al. (2012) assume that N_{eff} departs from the standard value only due to the lepton asymmetry. A model with both a large lepton asymmetry and extra relativistic relics could be constrained by CMB data. However, we do not consider such a contrived scenario in this paper.
Serpico et al. (2004) quotes , but since that work, the uncertainty on the neutron lifetime has been reevaluated, from σ(τ_{n}) = 0.8 s to σ(τ_{n}) = 1.1 s (Beringer et al. 2012).
See however Srianand et al. (2004, 2007).
The tension between the Planck and SPT S12 results is discussed in detail in Appendix B.
Even in the restricted case of the base ΛCDM model, parameters can shift as a result of small changes to the theoretical assumptions. An example is given in Sect. 3.2, where we show that changing from our default assumption of ∑ m_{ν} = 0.06 eV to ∑ m_{ν} = 0, causes an upward shift of 0.4σ in the value of H_{0}.
The spectrum is a combination of all of the crossspectra computed from the nineyear coadded maps per differencing assembly. Crossspectra are first combined by band into VV, VW and WW spectra and the beam corrected spectra are then corrected for unresolved point sources, i.e., a Poisson term is removed to minimise residuals with respect to the WMAP bestfit ΛCDM spectrum. The spectra are then coadded with inverse noise weighting to form a single V+W spectrum.
In Fig. B.2 we use the window functions provided by S12 to bandaverage the Planck and theory data points at high multipoles.
The constraint on A_{L} for Planck+WP is not given in Table C.1; the result is (95% CL).
It is worth noting that the results presented in this section are consistent with those derived from a Fisher matrix analysis as described in Appendix A, which includes a model for the 217 × 217 GHz systematic effect.
Acknowledgments
The development of Planck has been supported by: ESA; CNES and CNRS/INSUIN2P3INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN and JA (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). A description of the Planck Collaboration and a list of its members, including the technical or scientific activities in which they have been involved, can be found at http://www.sciops.esa.int/index.php?project=planck&page=Planck_Collaboration.We thank the referee for a comprehensive and helpful report. We also thank JeanPhilippe Uzan for his contributions to Sect. 6.8. We additionally acknowledge useful comments on the first version of this paper from a large number of scientists who have helped improve the clarity of the revised version. We mention specifically Jim Braatz, John Carlstrom, Alex Conley, Raphael Flauger, Liz Humphreys, Adam Riess, Beth Reid, Uros Seljak, David Spergel, Mark Sullivan, and Reynald Pain.
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