Issue 
A&A
Volume 596, December 2016



Article Number  A108  
Number of page(s)  19  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201628897  
Published online  12 December 2016 
Planck intermediate results
XLVII. Planck constraints on reionization history
^{1} APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris Cité, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France
^{2} Aalto University Metsähovi Radio Observatory and Dept of Radio Science and Engineering, PO Box 13000, 00076 Aalto, Finland
^{3} African Institute for Mathematical Sciences, 6−8 Melrose Road, 7945 Muizenberg, Cape Town, South Africa
^{4} Agenzia Spaziale Italiana Science Data Center, via del Politecnico snc, 00133 Roma, Italy
^{5} AixMarseille Université, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, 13388 Marseille, France
^{6} AixMarseille Université, Centre de Physique Théorique, 163 avenue de Luminy, 13288 Marseille, France
^{7} Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, UK
^{8} Astrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZuluNatal, Westville Campus, Private Bag X54001, 4000 Durban, South Africa
^{9} CITA, University of Toronto, 60 St. George St., Toronto, ON M5S 3H8, Canada
^{10} CNRS, IRAP, 9 Av. colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France
^{11} California Institute of Technology, Pasadena, CA 91125, USA
^{12} Computational Cosmology Center, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
^{13} DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark
^{14} Département de Physique Théorique, Université de Genève, 24 quai E. Ansermet, 1211 Genève 4, Switzerland
^{15} Departamento de Astrofísica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain
^{16} Departamento de Física, Universidad de Oviedo, Avda. Calvo Sotelo s/n, 33003 Oviedo, Spain
^{17} Department of Astrophysics/IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands
^{18} Department of Physics & Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada
^{19} Department of Physics and Astronomy, Dana and David Dornsife College of Letter, Arts and Sciences, University of Southern California, Los Angeles, CA 90089, USA
^{20} Department of Physics and Astronomy, University College London, London WC1E 6BT, UK
^{21} Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, UK
^{22} Department of Physics, Gustaf Hällströmin katu 2a, University of Helsinki, 00560 Helsinki, Finland
^{23} Department of Physics, Princeton University, Princeton, NJ 08544, USA
^{24} Department of Physics, University of California, Berkeley, CA 94720, USA
^{25} Department of Physics, University of California, One Shields Avenue, Davis, CA 95064, USA
^{26} Department of Physics, University of California, Santa Barbara, CA 93106, USA
^{27} Department of Physics, University of Illinois at UrbanaChampaign, 1110 West Green Street, Urbana, IL 61801 USA
^{28} Dipartimento di Fisica e Astronomia G. Galilei, Università degli Studi di Padova, via Marzolo 8, 35131 Padova, Italy
^{29} Dipartimento di Fisica e Astronomia, Alma Mater Studiorum, Università degli Studi di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy
^{30} Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, via Saragat 1, 44122 Ferrara, Italy
^{31} Dipartimento di Fisica, Università La Sapienza, P.le A. Moro 2, 00133 Roma, Italy
^{32} Dipartimento di Fisica, Università degli Studi di Milano, via Celoria, 16, 20133 Milano, Italy
^{33} Dipartimento di Fisica, Università di Roma Tor Vergata, via della Ricerca Scientifica, 1, 00133 Roma, Italy
^{34} Dipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica, 1, 00133 Roma, Italy
^{35} Discovery Center, Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 2100 Copenhagen, Denmark
^{36} European Space Agency, ESAC, Planck Science Office, Camino bajo del Castillo, s/n, Urbanización Villafranca del Castillo, 28691 Villanueva de la Cañada, Madrid, Spain
^{37} European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
^{38} Gran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’ Aquila, Italy
^{39} HGSFP and University of Heidelberg, Theoretical Physics Department, Philosophenweg 16, 69120 Heidelberg, Germany
^{40} Haverford College Astronomy Department, 370 Lancaster Avenue, Haverford, PA 19041, USA
^{41} Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, 00560 Helsinki, Finland
^{42} INAF–Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35131 Padova, Italy
^{43} INAF–Osservatorio Astronomico di Roma, via di Frascati 33, 00040 Monte Porzio Catone, Italy
^{44} INAF–Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, 34127 Trieste, Italy
^{45} INAF/IASF Bologna, via Gobetti 101, 40127 Bologna, Italy
^{46} INAF/IASF Milano, via E. Bassini 15, 20133 Milano, Italy
^{47} INFNCNAF, viale Berti Pichat 6/2, 40127 Bologna, Italy
^{48} INFN, Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy
^{49} INFN, Sezione di Ferrara, via Saragat 1, 44122 Ferrara, Italy
^{50} INFN, Sezione di Roma 1, Università di Roma Sapienza, P.le Aldo Moro 2, 00185 Roma, Italy
^{51} INFN, Sezione di Roma 2, Università di Roma Tor Vergata, via della Ricerca Scientifica, 1, 00185 Roma, Italy
^{52} Imperial College London, Astrophysics group, Blackett Laboratory, Prince Consort Road, London, SW7 2AZ, UK
^{53} Institut d’Astrophysique Spatiale, CNRS, Univ. ParisSud, Université ParisSaclay, Bât. 121, 91405 Orsay Cedex, France
^{54} Institut d’Astrophysique de Paris, CNRS (UMR 7095), 98bis boulevard Arago, 75014 Paris, France
^{55} Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
^{56} Institute of Theoretical Astrophysics, University of Oslo, Blindern, 0371 Oslo, Norway
^{57} Instituto de Astrofísica de Canarias, C/Vía Láctea s/n, La Laguna, 38205 Tenerife, Spain
^{58} Instituto de Física de Cantabria (CSICUniversidad de Cantabria), Avda. de los Castros s/n, 39005 Santander, Spain
^{59} Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, 35131 Padova, Italy
^{60} Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
^{61} Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, TheUniversity of Manchester, Oxford Road, Manchester, M13 9PL, UK
^{62} Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
^{63} Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, UK
^{64} LAL, Université ParisSud, CNRS/IN2P3, 91405 Orsay, France
^{65} LERMA, CNRS, Observatoire de Paris, 61 avenue de l’Observatoire, 75000 Paris, France
^{66} Laboratoire Traitement et Communication de l’Information, CNRS (UMR 5141) and Télécom ParisTech, 46 rue Barrault, 75634 Paris Cedex 13, France
^{67} Laboratoire de Physique Subatomique et Cosmologie, Université GrenobleAlpes, CNRS/IN2P3, 53 rue des Martyrs, 38026 Grenoble Cedex, France
^{68} Laboratoire de Physique Théorique, Université ParisSud 11 & CNRS, Bâtiment 210, 91405 Orsay, France
^{69} Lawrence Berkeley National Laboratory, Berkeley, California, USA
^{70} MaxPlanckInstitut für Astrophysik, KarlSchwarzschildStr. 1, 85741 Garching, Germany
^{71} Mullard Space Science Laboratory, University College London, Surrey RH5 6NT, UK
^{72} Nicolaus Copernicus Astronomical Center, Bartycka 18, 00716 Warsaw, Poland
^{73} Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 1165 Copenhagen, Denmark
^{74} Nordita (Nordic Institute for Theoretical Physics), Roslagstullsbacken 23, 106 91 Stockholm, Sweden
^{75} SISSA, Astrophysics Sector, via Bonomea 265, 34136 Trieste, Italy
^{76} School of Chemistry and Physics, University of KwaZuluNatal, Westville Campus, Private Bag X54001, 4000 Durban, South Africa
^{77} School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK
^{78} Simon Fraser University, Department of Physics, 8888 University Drive, Burnaby BC, Canada
^{79} Sorbonne UniversitéUPMC, UMR 7095, Institut d’Astrophysique de Paris, 98bis boulevard Arago, 75014 Paris, France
^{80} Space Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, 117997 Moscow, Russia
^{81} Space Sciences Laboratory, University of California, Berkeley, CA 94720, USA
^{82} SubDepartment of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
^{83} The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, 106 91 Stockholm, Sweden
^{84} UPMC Univ. Paris 06, UMR 7095, 98bis boulevard Arago, 75014 Paris, France
^{85} Université de Toulouse, UPSOMP, IRAP, 31028 Toulouse Cedex 4, France
^{86} University of Granada, Departamento de Física Teórica y del Cosmos, Facultad de Ciencias, 18010 Granada, Spain
^{87} Warsaw University Observatory, Aleje Ujazdowskie 4, 00478 Warszawa, Poland
Received: 11 May 2016
Accepted: 23 July 2016
We investigate constraints on cosmic reionization extracted from the Planck cosmic microwave background (CMB) data. We combine the Planck CMB anisotropy data in temperature with the lowmultipole polarization data to fit ΛCDM models with various parameterizations of the reionization history. We obtain a Thomson optical depth τ = 0.058 ± 0.012 for the commonly adopted instantaneous reionization model. This confirms, with data solely from CMB anisotropies, the low value suggested by combining Planck 2015 results with other data sets, and also reduces the uncertainties. We reconstruct the history of the ionization fraction using either a symmetric or an asymmetric model for the transition between the neutral and ionized phases. To determine better constraints on the duration of the reionization process, we also make use of measurements of the amplitude of the kinetic SunyaevZeldovich (kSZ) effect using additional information from the highresolution Atacama Cosmology Telescope and South Pole Telescope experiments. The average redshift at which reionization occurs is found to lie between z = 7.8 and 8.8, depending on the model of reionization adopted. Using kSZ constraints and a redshiftsymmetric reionization model, we find an upper limit to the width of the reionization period of Δz < 2.8. In all cases, we find that the Universe is ionized at less than the 10% level at redshifts above z ≃ 10. This suggests that an early onset of reionization is strongly disfavoured by the Planck data. We show that this result also reduces the tension between CMBbased analyses and constraints from other astrophysical sources.
Key words: cosmic background radiation / dark ages, reionization, first stars / polarization
© ESO, 2016
1. Introduction
The process of cosmological recombination happened around redshift z ≃ 1100, after which the ionized fraction fell precipitously (Peebles 1968; Zel’dovich et al. 1969; Seager et al. 2000) and the Universe became mostly neutral. However, observations of the GunnPeterson effect (Gunn & Peterson 1965) in quasar spectra (Becker et al. 2001, 2015; Fan et al. 2006b; Venemans et al. 2013) indicate that intergalactic gas had become almost fully reionized by redshift z ≃ 6. Reionization is thus the second major change in the ionization state of hydrogen in the Universe. Details of the transition from the neutral to ionized Universe are still the subject of intense investigations (for a recent review, see the book by Mesinger 2016). In the currently conventional picture, early galaxies reionize hydrogen progressively throughout the entire Universe between z ≃ 12 and z ≃ 6, while quasars take over to reionize helium from z ≃ 6 to ≃ 2. But many questions remain. When did the epoch of reionization (EoR) start, and how long did it last? Are early galaxies enough to reionize the entire Universe or is another source required? We try to shed light on these questions using the traces left by the EoR in the cosmic microwave background (CMB) anisotropies.
The CMB is affected by the total column density of free electrons along each line of sight, parameterized by its Thomson scattering optical depth τ. This is one of the six parameters of the baseline Λ cold dark matter (CDM) cosmological model and is the key measurement for constraining reionization. Largescale anisotropies in polarization are particularly sensitive to the value of τ. The WMAP mission was the first to extract a τ measurement through the correlation between the temperature field and the Emode polarization (i.e., the TE power spectrum) over a large fraction of the sky. This measurement is very demanding, since the expected level of the Emode polarization power spectrum at low multipoles (ℓ< 10) is only a few times 10^{2}μK^{2}, lower by more than two orders of magnitude than the level of the temperature anisotropy power spectrum. For such weak signals the difficulty is not only to have enough detector sensitivity, but also to reduce and control both instrumental systematic effects and foreground residuals to a very low level. This difficulty is illustrated by the improvements over time in the WMAPderived τ estimates. The 1yr results gave a value of τ = 0.17 ± 0.04, based on the temperaturepolarization TE crosspower spectrum (Kogut et al. 2003). In the 3yr release, this was revised down to 0.10 ± 0.03 using Emodes alone, whereas the combined TT, TE, and EE power spectra gave 0.09 ± 0.03 (Page et al. 2007). Error bars improved in further WMAP analyses, ending up with 0.089 ± 0.014 after the 9yr release (see Dunkley et al. 2009; Komatsu et al. 2011; Hinshaw et al. 2013). In 2013, the first Planck satellite^{1} cosmological results were based on Planck temperature power spectra combined with the polarized WMAP data and gave the same value τ = 0.089 ± 0.014 (Planck Collaboration XVI 2014). However, using a preliminary version of the Planck 353 GHz polarization maps to clean the dust emission (in place of the WMAP dust model), the optical depth was reduced by approximately 1σ to τ = 0.075 ± 0.013 (Planck Collaboration XV 2014).
In the 2015 Planck analysis (Planck Collaboration XIII 2016), the Low Frequency Instrument (LFI) lowresolution maps polarization at 70 GHz were used. Foreground cleaning was performed using the LFI 30 GHz and High Frequency Instrument (HFI) 353 GHz maps, operating effectively as polarized synchrotron and dust templates, respectively. The optical depth was found to be τ = 0.078 ± 0.019, and this decreased to 0.066 ± 0.016 when adding CMB lensing data. This value is also in agreement with the constraints from the combination “PlanckTT+lensing+BAO”, yielding τ = 0.067 ± 0.016, which uses no information from lowℓ polarization.
In this paper and its companion (Planck Collaboration Int. XLVI 2016), we derive the first estimate of τ from the PlanckHFI polarization data at large scales. For the astrophysical interpretation, the power spectra are estimated using a PCL estimate which is more conservative. Indeed, it gives a slightly larger distribution on τ than the QML estimator used in Planck Collaboration Int. XLVI (2016) but is less sensitive to the limited number of simulations available for the analysis. Using only Emode polarization, the Plancklollipop likelihood gives for a standard instantaneous reionization model, when all other ΛCDM parameters are fixed to their Planck2015 bestfit values. We show that in combination with the Planck temperature data the error bars are improved and we find τ = 0.058 ± 0.012.
In the ΛCDM model, improved accuracy on the reionization optical depth helps to reduce the degeneracies with other parameters. In particular, the measurement of τ reduces the correlation with the normalization of the initial power spectrum A_{s} and its spectral index n_{s}. In addition to this τ is a particularly important source of information for constraining the history of reionization, which is the main subject of this paper. When combined with direct probes at low redshift, a better knowledge of the value of the CMB optical depth parameter may help to characterize the duration of the EoR, and thus tell us when it started.
In addition to the effect of reionization on the polarized largescale CMB anisotropies, reionization generates CMB temperature anisotropies through the kinetic SunyaevZeldovich (kSZ) effect (Sunyaev & Zeldovich 1980), caused by the Doppler shift of photons scattering off electrons moving with bulk velocities. Simulations have shown that early homogeneous and patchy reionization scenarious differently affect the shape of the kSZ power spectrum, allowing us to place constraints on the reionization history (e.g., McQuinn et al. 2005; Aghanim et al. 2008). Zahn et al. (2012) derived the first constraints on the epoch of reionization from the combination of kSZ and lowℓ CMB polarization, specifically using the lowℓ polarization power spectrum from WMAP and the very high multipoles of the temperature angular power spectrum from the South Pole Telescope (SPT, Reichardt et al. 2012). However one should keep in mind that kSZ signal is complicated to predict and depends on detailed astrophysics which makes the constraints on reionization difficult to interpret (Mesinger et al. 2012).
In this paper, we investigate constraints on the epoch of reionization coming from Planck. Section 2 first briefly describes the pre2016 data and likelihood used in this paper, which are presented in detail in Planck Collaboration Int. XLVI (2016). In Sect. 3 we then present the parameterizations we adopt for the ionization fraction, describing the reionization history as a function of redshift. In Sect. 4, we show the results obtained from the CMB observables (i.e., the optical depth τ and the amplitude of the kSZ effect) in the case of “instantaneous” reionization. Section 5 presents results based on the CMB measurements by considering different models for the ionization history. In particular, we derive limits on the reionization redshift and duration. Finally, in Sect. 6, we derive the ionization fraction as a function of redshift and discuss how our results relate to other astrophysical constraints.
2. Data and likelihood
2.1. Data
The present analysis is based on the pre2016 full mission intensity and polarization PlanckHFI maps at 100 and 143 GHz. The data processing and the beam description are the same as in the Planck 2015 release and have been detailed in Planck Collaboration VII (2016). PlanckHFI polarization maps are constructed from the combination of polarized detectors that have fixed polarization direction. The Planck scanning strategy produces a relatively low level of polarization angle measurement redundancy on the sky, resulting in a high level of IQU mixing, as shown in Planck Collaboration VIII (2016). As a consequence, any instrumental mismatch between detectors from the same frequency channel produces leakage from intensity to polarization. This temperaturetopolarization leakage was at one point the main systematic effect present in the PlanckHFI data, and prevented robust lowℓ polarization measurements from being included in the previous Planck data releases.
The maps that we use here differ in some respects from those data released in 2015. The updated mapmaking procedure, presented in Planck Collaboration Int. XLVI (2016), now allows for a significant reduction of the systematic effects in the maps. In particular, the relative calibration within a channel is now accurate to better than 0.001%, which ensures a very low level of gainmismatch between detectors. The major systematic effect that remains in the pre2016 maps is due to imperfections of the correction for nonlinearity in the analoguetodigital converters (ADCs) but produces very low level of residuals in the maps. In addition to the 100 GHz and 143 GHz maps, we also make use of 30 GHz LFI data (Planck Collaboration II 2016) and 353 GHz HFI data to remove polarized foregrounds.
Using the pre2016 endtoend simulations, we show that the power spectrum bias induced by the remaining nonlinearities is very small and properly accounted for in the likelihood. Figure 1 shows the bias (in the quantity , where C_{ℓ} is the conventional power spectrum) computed as the mean of the EE crosspower spectra from simulated maps, including realistic noise and systematic effects without and with Galactic foregrounds. In the latter case, the foregrounds are removed for each simulation using the 30 GHz and 353 GHz maps as templates for synchrotron and dust, respectively. The resulting bias in the EE100 × 143 crosspower spectrum can be used to correct the measured crossspectrum, but in fact has very little impact on the likelihood.
Fig. 1 Bias in the 100 × 143 crosspower spectrum computed from simulations, including instrumental noise and systematic effects, with or without foregrounds (dark blue and light blue), compared to the cosmic variance level (in grey). 
Fig. 2 Variance of the 100 × 143EE crosspower spectrum for simulations, including instrumental noise and noise plus systematic effects, compared to cosmic variance. 
Furthermore, we use endtoend simulations to propagate the systematic uncertainties to the crosspower spectra and all the way to the cosmological parameters. Figure 2 shows the impact on the variance due to the inclusion of the main ADC nonlinearity systematic effect, compared to realistic noise and cosmic variance. The resulting C_{ℓ} covariance matrix is estimated from these Monte Carlos. In the presence of such systematic effects, the variance of the C_{ℓ} is shown to be higher by roughly a factor of 2 compared to the pure noise case.
Polarized foregrounds at PlanckHFI frequencies are essentially dominated by Galactic dust emission, but also include a small contribution from synchrotron emission. We use the 353 GHz and 30 GHz Planck maps as templates to subtract dust and synchrotron, respectively, using a single coefficient for each component over 95% of the sky (see Planck Collaboration IX 2016; Planck Collaboration X 2016). However, foreground residuals in the maps are still dominant over the CMB polarized signal near the Galactic plane. We therefore apply a very conservative mask, based on the amplitude of the polarized dust emission, which retains 50% of the sky for the cosmological analysis. Outside this mask, the foreground residuals are found to be lower than 0.3 and 0.4 μK in Q and U Stokes polarization maps at 100 and 143 GHz, respectively. We have checked that our results are very stable when using a larger sky fraction of 60%.
In this paper, we also make use of the constraints derived from the observation of the GunnPeterson effect on highredshift quasars. As suggested by Fan et al. (2006a), these measurements show that the Universe was almost fully reionized at redshift z ≃ 6. We later discuss the results obtained with and without imposing a prior on the redshift of the end of reionization.
Fig. 3 EE and TE power spectra for various τ values ranging from 0.04 to 0.08. The ionization fraction is modelled using a redshiftsymmetric tanh function with δz = 0.5. Grey bands represent the cosmic variance (fullsky) associated with the τ = 0.06 model. 
2.2. Likelihood
For temperature anisotropies, we use the combined Planck likelihood (hereafter “PlanckTT”), which includes the TT power spectrum likelihood at multipoles ℓ> 30 (using the Plik code) and the lowℓ temperatureonly likelihood based on the CMB map recovered from the componentseparation procedure (specifically Commander) described in detail in Planck Collaboration XI (2016).
For polarization, we use the Planck lowℓEE polarization likelihood (hereafter lollipop), a crossspectrabased likelihood approach described in detail in Mangilli et al. (2015) and applied to Planck data as discussed here in Appendix A. The multipole range used is ℓ = 4–20. Crossspectra are estimated using the pseudoC_{ℓ} estimator Xpol (a generalization to polarization of the algorithm presented in Tristram et al. 2005). For a fullsky analysis, the statistics of the reconstructed C_{ℓ} are given by a χ^{2} distribution that is uncorrelated between multipoles. For a cutsky analysis, the distribution is more complex and includes ℓtoℓ correlations. Hamimeche & Lewis (2008) proposed an approximation of the likelihood for cutsky autopower spectra that was adapted by Mangilli et al. (2015) to be suitable for crossspectra. Crossspectra between independent data sets show common sky signal, but are not biased by the noise because this should be uncorrelated. This approximation assumes that any systematic residuals are not correlated between the different data sets; We have shown using realistic simulations (including PlanckHFI noise characteristics and systematic effect residuals), that the bias in the crossspectra is very small and can be corrected for at the powerspectrum level. Nevertheless, we choose to remove the first two multipoles (ℓ = 2 and ℓ = 3), since they may still be partially contaminated by systematics. Using those simulations, we derive the C_{ℓ} covariance matrix used in the likelihood, which propagates both the noise and the systematic uncertainties. For the astrophysical interpretation, the powerspectra are estimated with a PCL estimate which is more conservative. Indeed, it gives a slightly larger distribution on τ than a QML estimator but is less sensitive to the limited number of simulations available for the analysis.
With Planck sensitivity in polarization, the results from the lowℓEE power spectrum dominate the constraints compared to the TE power spectrum, as can be seen in Fig. 3. This is because of the relatively larger cosmic variance for TE (arising from the temperature term) and the intrinsically weaker dependence on τ (∝τ compared with τ^{2} for EE), as well as the fact that there is only partial correlation between T and E. As a consequence, we do not consider the TE data in this analysis. Furthermore, we do not make use of the highℓ likelihoods in EE and TE from Planck, since they do not carry additional information on reionization parameters.
Planck temperature observations are complemented at smaller angular scales by measurements from the groundbased Atacama Cosmology Telescope (ACT) and South Pole Telescope (SPT). As explained in Planck Collaboration XI (2016), the highℓ likelihood (hereafter VHL) includes ACT power spectra at 148 and 218 GHz (Das et al. 2014), with a revised binning (described in Calabrese et al. 2013) and final beam estimates (Hasselfield et al. 2013), together with SPT measurements in the range 2000 <ℓ< 13 000 from the 2540 deg^{2} SPTSZ survey at 95, 150, and 220 GHz (George et al. 2015). To assess the consistency between these data sets, we extend the Planck foreground models up to ℓ = 13 000, with additional nuisance parameters for ACT and SPT (as described in Planck Collaboration XIII 2016). We use the same models for cosmic infrared background (CIB) fluctuations, the thermal SZ (tSZ) effect, kSZ effect, and CIB × tSZ components. The kSZ template used in the Planck 2015 results assumed homogeneous reionization. In order to investigate inhomogeneous reionization, we have modified the kSZ template when necessary, as discussed in Sect. 4.2.
We use the CMB lensing likelihood (Planck Collaboration XV 2016) in addition to the CMB anisotropy likelihood. The lensing information can be used to break the degeneracy between the normalization of the initial power spectrum A_{s} and τ (as discussed in Planck Collaboration XIII 2016). Despite this potential for improvement, we show in Sect. 4.1 that Planck’s lowℓ polarization signaltonoise ratio is sufficiently high that the lensing does not bring much additional information for the reionization constraints.
The Planck reference cosmology used in this paper corresponds to the PlanckTT+lowP+lensing best fit, as described in Table 4, Col. 2 of Planck Collaboration XIII (2016), namely Ω_{b}h^{2} = 0.02226, Ω_{c}h^{2} = 0.1197, Ω_{m} = 0.308, n_{s} = 0.9677, H_{0} = 67.81 km s^{1} Mpc^{1}, for which Y_{P} = 0.2453. This bestfit model comes from the combination of three Planck likelihoods: the temperature power spectrum likelihood at high ℓ; the “lowP” temperature+polarization likelihood, based on the foregroundcleaned LFI 70 GHz polarization maps, together with the temperature map from the Commander componentseparation algorithm; and the power spectrum of the lensing potential as measured by Planck.
3. Parametrization of reionization history
The epoch of reionization (EoR) is the period during which the cosmic gas transformed from a neutral to ionized state at the onset of the first sources. Details of the transition are thus strongly connected to many fundamental questions in cosmology, such as what were the properties of the first galaxies and the first (mini)quasars, how did the formation of very metalpoor stars proceed, etc. We certainly know that, at some point, luminous sources started emitting ultraviolet radiation that reionized the neutral regions around them. After a sufficient number of ionizing sources had formed, the average ionized fraction of the gas in the Universe rapidly increased until hydrogen became fully ionized. Empirical, analytic, and numerical models of the reionization process have highlighted many pieces of the essential physics that led to the birth to the ionized intergalactic medium (IGM) at late times (Couchman & Rees 1986; MiraldaEscude & Ostriker 1990; Meiksin & Madau 1993; Aghanim et al. 1996; Gruzinov & Hu 1998; Madau et al. 1999; Gnedin 2000; Barkana & Loeb 2001; Ciardi et al. 2003; Furlanetto et al. 2004; Pritchard et al. 2010; Pandolfi et al. 2011; Mitra et al. 2011; Iliev et al. 2014). Such studies provide predictions on the various reionization observables, including those associated with the CMB.
The most common physical quantity used to characterize reionization is the Thomson scattering optical depth defined as (1)where n_{e} is the number density of free electrons at time t′, σ_{T} is the Thomson scattering crosssection, t_{0} is the time today, t(z) is the time at redshift z, and we can use the Friedmann equation to convert dt to dz. The reionization history is conveniently expressed in terms of the ionized fraction x_{e}(z) ≡ n_{e}(z) /n_{H}(z) where n_{H}(z) is the hydrogen number density. In practice, the CMB is sensitive to the average over all sky directions of x_{e}(1 + δ_{b}) (where δ_{b} denotes the baryon overdensity). The IGM is likely to be very inhomogeneous during reionization process, with ionized bubbles embedded in neutral surroundings, which would impact the relation between the optical depth and the reionisation parameters (see Liu et al. 2016) at a level which is neglected in this paper.
In this study, we define the redshift of reionization, z_{re} ≡ z_{50%}, as the redshift at which x_{e} = 0.5 × f. Here the normalization, f = 1 + f_{He} = 1 + n_{He}/n_{H}, takes into account electrons injected into the IGM by the first ionization of helium (corresponding to 25 eV), which is assumed to happen roughly at the same time as hydrogen reionization. We define the beginning and the end of the EoR by the redshifts z_{beg} ≡ z_{10%} and z_{end} ≡ z_{99%} at which x_{e} = 0.1 × f and 0.99 × f, respectively. The duration of the EoR is then defined as Δz = z_{10%}−z_{99%}^{2}. Moreover, to ensure that the Universe is fully reionized at low redshift, we impose the condition that the EoR is completed before the second helium reionization phase (corresponding to 54 eV), noting that it is commonly assumed that quasars are necessary to produce the hard photons needed to ionize helium. To be explicit about how we treat the lowest redshifts we assume that the full reionization of helium happens fairly sharply at z_{He} = 3.5 (Becker et al. 2011), following a transition of hyperbolic tangent shape with width δz = 0.5. While there is still some debate on whether helium reionization could be inhomogeneous and extended (and thus have an early start, Worseck et al. 2014), we have checked that varying the helium reionization redshift between 2.5 and 4.5 changes the total optical depth by less than 1%.
The simplest and most widelyused parameterizations describes the EoR as a steplike transition between an essentially vanishing ionized fraction^{3}x_{e} at early times, to a value of unity at low redshifts. When calculating the effect on anisotropies it is necessary to give a nonzero width to the transition, and it can be modelled using a tanh function (Lewis 2008): (2)where y = (1 + z)^{3 / 2} and . The key parameters are thus z_{re}, which measures the redshift at which the ionized fraction reaches half its maximum and a width δz. The tanh parameterization of the EoR transition allows us to compute the optical depth of Eq. (1) for a onestage almost redshiftsymmetric^{4} reionization transition, where the redshift interval between the onset of the reionization process and its half completion is (by construction) equal to the interval between half completion and full completion. In this parameterization, the optical depth is mainly determined by z_{re} and almost degenerate with the width δz. This is the model used in the Planck 2013 and 2015 cosmological papers, for which we have fixed δz = 0.5 (corresponding to Δz = 1.73). In this case, we usually talk about “instantaneous” reionization.
Fig. 4 Left: evolution of the ionization fraction for several functions, all having the same optical depth, τ = 0.06: green and blue are for redshiftsymmetric instantaneous (δz = 0.05) and extended reionization (δz = 0.7), respectively; red is an example of a redshiftasymmetric parameterization; and light blue and magenta are examples of an ionization fraction defined in redshift bins, with two bins inverted between these two examples. Right: corresponding EE power spectra with cosmic variance in grey. All models have the same optical depth τ = 0.06 and are essentially indistinguishable at the reionization bump scale. 
A redshiftasymmetric parameterization is a better, more flexible description of numerical simulations of the reionization process (e.g., Ahn et al. 2012; Park et al. 2013; Douspis et al. 2015). A function with this behaviour is also suggested by the constraints from ionizing background measurements of starforming galaxies and from lowredshift lineofsight probes such as quasars, Lymanα emitters, or γray bursts (Faisst et al. 2014; Chornock et al. 2014; Ishigaki et al. 2015; Robertson et al. 2015; Bouwens et al. 2015). The two simplest choices of redshiftasymmetric parameterizations are polynomial or exponential functions of redshift (Douspis et al. 2015). These two parameterizations are in fact very similar, and we adopt here a power law defined by two parameters: the redshift at which reionization ends (z_{end}); and the exponent α. Specifically we have (3)In the following, we fix z_{early} = 20, the redshift around which the first emitting sources form, and at which we smoothly match x_{e}(z) to the ionized fraction left over from recombination. We checked that our results are not sensitive to the precise value of z_{early}, as long as it is not dramatically different.
Nonparametric reconstructions of the ionization fraction have also been proposed to probe the reionization history. Such methods are based on exploring reionization parameters in bins of redshift (Lewis et al. 2006). They should be particularly useful for investigating exotic reionization histories, e.g., double reionization (Cen 2003). However, the CMB largescale (ℓ ≲ 10) polarization anisotropies are mainly sensitive to the overall value of the optical depth, which determines the amplitude of the reionization bump in the EE power spectrum (see Fig. 3). We have estimated the impact on for the two different models (tanh and power law) having the same τ = 0.06 and found differences of less than 4% for ℓ< 10. Even for a double reionization model, Fig. 4 shows that the impact on is quite weak, given the actual measured value of τ, and cannot be distinguished relative to the cosmic variance spread (i.e., even for a fullsky experiment). We also checked that Planck data do not allow for modelindependent reconstruction of x_{e} in redshift bins. Principal component analysis has been proposed as an explicit approach to try to capture the details of the reionization history in a small set of parameters (Hu & Holder 2003; Mortonson & Hu 2008). Although these methods are generally considered to be nonparametric, they are in fact based on a description of x_{e}(z) in bins of redshift, expanded around a given fiducial model for . Moreover, the potential bias on the τ measurement when analysing a more complex reionization history using a simple sharp transition model (Holder et al. 2003; Colombo & Pierpaoli 2009) is considerably reduced for the (lower) τ values as suggested by the Planck results. Consequently, we do not consider the nonparametric approach further.
4. Measuring reionization observables
Reionization leaves imprints in the CMB power spectra, both in polarization at very large scales and in intensity via the suppression of TT power at higher ℓ. Reionization also affects the kSZ effect, due to the rescattering of photons off newly liberated electrons. We sample from the space of possible parameters with MCMC exploration using CAMEL^{5}. This uses an adaptativeMetropolis algorithm to generate chains of samples for a set of parameters.
4.1. Largescale CMB polarization
Thomson scattering between the CMB photons and free electrons generates linear polarization from the quadrupole moment of the CMB radiation field at the scattering epoch. This occurs at recombination and also during the epoch of reionization. Rescattering of the CMB photons at reionization generates an additional polarization anisotropy at large angular scales, because the horizon size at this epoch subtends a much larger angular size. The multipole location of this additional anisotropy (essentially a bump) in the EE and TE angular power spectra relates to the horizon size at the new “lastrescattering surface” and thus depends on the redshift of reionization. The height of the bump is a function of the optical depth or, in other words, of the history of the reionization process. Such a signature (i.e., a polarization bump at large scales) was first observed by WMAP, initially in the TE angular power spectrum (Kogut et al. 2003), and later in combination with all power spectra (Hinshaw et al. 2013).
In Fig. 3 we show for the “instantaneous” reionization case (specifically the redshiftsymmetric parameterization with δz = 0.5) power spectra for the Emode polarization power spectrum and the temperaturepolarization crosspower spectrum . The curves are computed with the CLASS Boltzmann solver (Lesgourgues 2011) using τ values ranging from 0.04 to 0.08. For the range of optical depth considered here and given the amount of cosmic variance, the TE spectrum has only a marginal sensitivity to τ, while in EE the ability to distinguish different values of τ is considerably stronger.
In Fig. 4 (left panel), the evolution of the ionized fraction x_{e} during the EoR is shown for five different parameterizations of the reionization history, all yielding the same optical depth τ = 0.06. Despite the differences in the evolution of the ionization fraction, the associated curves (Fig. 4, right panel) are almost indistinguishable. This illustrates that while CMB largescale anisotropies in polarization are only weakly sensitive to the details of the reionization history, they can nevertheless be used to measure the reionization optical depth, which is directly related to the amplitude of the lowℓ bump in the Emode power spectrum.
We use the Planck data to provide constraints on the Thomson scattering optical depth for “instantaneous” reionization. Figure 5 shows the posterior distributions for τ obtained with the different data sets described in Sect. 2 and compared to the 2015 PlanckTT+lowP results (Planck Collaboration XIII 2016). We show the posterior distribution for the lowℓPlanck polarized likelihood (lollipop) and in combination with the highℓPlanck likelihood in temperature (PlanckTT). We also consider the effect of adding the SPT and ACT likelihoods (VHL) and the Planck lensing likelihood, as described in Planck Collaboration XV (2016).
Fig. 5 Posterior distribution for τ from the various combinations of Planck data. The grey band shows the lower limit on τ from the GunnPeterson effect. 
The different data sets show compatible constraints on the optical depth τ. The comparison between posteriors indicates that the optical depth measurement is driven by the lowℓ likelihood in polarization (i.e., lollipop). The Planck constraints on τ for a ΛCDM model when considering the standard “instantaneous” reionization assumption (symmetric model with fixed δz = 0.5), for the various data combinations are: We can see an improvement of the posterior width when adding temperature anisotropy data to the lollipop likelihood. This comes from the fact that the temperature anisotropies help to fix other ΛCDM parameters, in particular the normalization of the initial power spectrum A_{s}, and its spectral index, n_{s}. CMB lensing also helps to reduce the degeneracy with A_{s}, while getting rid of the tension with the phenomenological lensing parameter A_{L} when using PlanckTT only (see Planck Collaboration XIII 2016), even if the impact on the error bars is small. Comparing the posteriors in Fig. 6 with the constraints from PlanckTT alone (see Fig. 45 in Planck Collaboration XI 2016) shows that indeed, the polarization likelihood is sufficiently powerful that it breaks the degeneracy between n_{s} and τ. The impact on other ΛCDM parameters is small, typically below 0.3σ (as shown more explicitly in Appendix B). The largest changes are for τ and A_{s}, where the lollipop likelihood dominates the constraint. The parameter σ_{8} shifts towards slightly smaller values by about 1σ. This is in the right direction to help resolve some of the tension with cluster abundances and weak galaxy lensing measurements, discussed in Planck Collaboration XX (2014) and Planck Collaboration XIII (2016); however, some tension still remains.
Combining with VHL data gives compatible results, with consistent error bars. The slight shift toward lower τ value (by 0.3σ) is related to the fact that the PlanckTT likelihood alone pushes towards higher τ values (see Planck Collaboration XIII 2016), while the addition of VHL data helps to some extent in reducing the tension on τ between highℓ and lowℓ polarization.
Fig. 6 Constraints on τ, A_{s}, n_{s}, and σ_{8} for the ΛCDM cosmology from PlanckTT, showing the impact of replacing the lowP likelihood from Planck 2015 release with the new lollipop likelihood. The top panels show results without lensing, while the bottom panels are with lensing. 
As mentioned earlier, astrophysics constraints from measurements of the GunnPeterson effect provide strong evidence that the IGM was highly ionized by a redshift of z ≃ 6. This places a lower limit on the optical depth (using Eq. (1)), which in the case of instantaneous reionization in the standard ΛCDM cosmology corresponds to τ = 0.038.
4.2. Kinetic SunyaevZeldovich effect
The Thomson scattering of CMB photons off ionized electrons induces secondary anisotropies at different stages of the reionization process. In particular, we are interested here in the effect of photons scattering off electrons moving with bulk velocity, which is called the “kinetic Sunyaev Zeldovich” or kSZ effect. It is common to distinguish between the “homogeneous” kSZ effect, arising when the reionization is complete (e.g., Ostriker & Vishniac 1986), and “patchy” (or inhomogeneous) reionization (e.g., Aghanim et al. 1996), which arises during the process of reionization, from the proper motion of ionized bubbles around emitting sources. These two components can be described by their power spectra, which can be computed analytically or derived from numerical simulations. In Planck Collaboration XI (2016), we used a kSZ template based on homogeneous simulations, as described in Trac et al. (2011).
In the following, we assume that the kSZ power spectrum is given by (8)where and the superscripts “hkSZ” and “pkSZ” stand for “homogeneous” and “patchy” reionization, respectively. For the homogeneous reionization, we use the kSZ template power spectrum given by Shaw et al. (2012) calibrated with a simulation that includes the effects of cooling and starformation (which we label “CSF”). For the patchy reionization kSZ effect we use the fiducial model of Battaglia et al. (2013).
Fig. 7 Power spectrum templates for the kSZ effect. The different lines correspond to: homogeneous reionization as used in Planck Collaboration XI (2016; dark blue), based on Trac et al. (2011); “CSF” (light blue), which is a homogeneous reionization model from Shaw et al. (2012); Patchy (green dashed) based on patchy reionization model from Battaglia et al. (2013); and the sum of CSF and patchy (red). 
In the range ℓ = 1000–7000, the shape of the kSZ power spectrum is relatively flat and does not vary much with the detailed reionization history. The relative contributions (specifically “CSF” and “patchy”) to the kSZ power spectrum are shown in Fig. 7 and compared to the “homogeneous” template used in Planck Collaboration XI (2016), rescaled to unity at ℓ = 3000.
The kSZ power spectrum amplitude does depend on the cosmological parameters (Shaw et al. 2012; Zahn et al. 2012). To deal with this, we adopt the scalings from Shaw et al. (2012), which gives the amplitude at ℓ = 3000, : (9)The amplitude of the kSZ power spectrum at ℓ = 3000 for the fiducial cosmology, A_{kSZ} is another observable of the reionization history that can be probed by CMB data. Its scalings with the reionization redshift and the duration of the EoR can be extracted from simulations. We assume for the patchy and homogeneous kSZ effect, the scalings of Battaglia et al. (2013) and Shaw et al. (2012), respectively. For the Planck base ΛCDM cosmology given in Sect. 2.2, we find (in μ): For the measured value τ = 0.058 ± 0.012, Eqs. (10) and (11) give amplitudes for the homogeneous and patchy reionization contributions of and , respectively.
For the multipole range of Planck, the amplitude of the kSZ spectrum is dominated by other foregrounds, including Galactic dust, point sources, CIB fluctuations, and the tSZ effect. Moreover, the Planck signaltonoise ratio decreases rapidly above ℓ = 2000, where the kSZ signal is maximal. This is why we cannot constrain the kSZ amplitude using Planck data alone. Combining with additional data at higher multipoles helps to disentangle the different foregrounds. We explicitly use the band powers from SPT (George et al. 2015) and ACT (Das et al. 2014), covering the multipole range up to ℓ = 13 000.
Despite our best efforts to account for the details, the kSZ amplitude is weak and there are large uncertainties in the models (both homogeneous and patchy). Moreover, there are correlations between the different foreground components, coming both from the astrophysics (we use the same halo model to derive the power spectra for the CIB and for CIB×tSZ as the one used for the kSZ effect) and from the adjustments in the data. We carried out several tests to check the robustness of the constraints on A_{kSZ} with respect to the template used for the CIB, CIB×tSZ, and kSZ contributions. In particular, the CIB×tSZ power spectrum amplitude is strongly anticorrelated with the kSZ amplitude and poorly constrained by the CMB data. As a consequence, if we neglect the CIB×tSZ contribution, the kSZ amplitude measured in CMB data is substantially reduced, leading to an upper limit much lower than the one derived when including the CIB×tSZ correlation. In the following discussion we consider only the more realistic case (and thus more conservative in terms of constraints on A_{kSZ}) where the CIB×tSZ correlation contributes to the highℓ signal.
Fig. 8 68% and 95% confidence intervals on the reionization optical depth, τ, and the amplitude of the kinetic SZ effect, A_{kSZ}, from the CMB (lollipop+PlanckTT+VHL). 
We combine the Planck likelihoods in TT (PlanckTT) and from lowℓEE polarization (lollipop) with the very highℓ data from ACT and SPT (VHL), assuming a redshiftsymmetric parameterization of the reionization. Figure 8 shows the 2D posterior distribution for τ and A_{kSZ} after marginalization over the other cosmological and nuisance parameters.
Figure 9 compares the constraints on the kSZ power at ℓ = 3000, A_{kSZ}, obtained for three different kSZ templates: the “homogeneous” reionization template from Trac et al. (2011), which neglects contributions from inhomogeneous reionization; a more complex model “CSF & patchy,” including both homogeneous and patchy contributions; and a pure “patchy” template from Battaglia et al. (2013). We find very similar upper limits on A_{kSZ}, even in the case of pure patchy reionization.
Fig. 9 Constraints on the kSZ amplitude at ℓ = 3000 using lollipop+PlanckTT+VHL likelihoods. The three cases correspond to different kSZ templates. 
Using the “CSF & patchy” model, the upper limit is (12)Compared to Planck 2013 results, the maximum likelihood value (PlanckTT+WP+highL, Planck Collaboration XVI 2014) is reduced to an upper limit in this new analysis. The data presented here provide the best constraint to date on the kSZ power and is a factor of 2 lower than the limit reported in George et al. (2015). Our limit is certainly not in tension with the homogeneous kSZ template, which predicts A_{kSZ} = 1.79 μ. However, it does not leave much room for any additional kSZ power coming from patchy reionization.
Consistent with George et al. (2015), we find the total kSZ power to be stable against varying tSZ and CIB templates. We also find very little dependence on the choice of the kSZ template (Fig. 9). This confirms that there is only a modest amount of information in the angular shape of the kSZ signal with the current data.
5. Constraints on the reionization history
We now interpret our measurements of the reionization observables in terms of constraint on the reionization history. We mainly focus on the determination of the reionization redshift z_{re} and its duration Δz = z_{beg}−z_{end}. We show only the results for Δz greater than unity, which corresponds to approximatively 90 Myr at redshift z = 8. We first begin by looking at constraints on the EoR for symmetric and asymmetric models using Planck data only (lollipop+PlanckTT). Then we introduce the VHL data and discuss additional constraints from the kSZ amplitude. In each case, we also derive the constraints that follow from postulating that reionization should be completed at a redshift of 6 (see Sect. 2.1), i.e., when imposing the prior z_{end}> 6.
5.1. Redshiftsymmetric parameterization
We use the Planck CMB likelihoods in temperature (PlanckTT) and polarization (lollipop) to derive constraints on ΛCDM parameters, including the reionization redshift z_{re} and width Δz for a redshiftsymmetric parameterization. Figure 10 shows (in blue) the posterior on z_{re} and Δz after marginalization over the other cosmological and nuisance parameters. As discussed in Sect. 3, the largescale polarized CMB anisotropies are almost insensitive to the width δz of the tanh function. We thus recover the degeneracy in the direction of Δz. Imposing an additional GunnPeterson constraint on the ionization fraction at very low redshift can break this degeneracy. This is illustrated in Fig. 10, where we show (in green) the results of the same analysis with an additional prior z_{end}> 6. In this case, we find δz< 1.3 at 95% CL, which corresponds to a reionization duration (z_{beg}−z_{end}) of (13)
Fig. 10 Posterior distributions (in blue) of z_{re} and Δz for a redshiftsymmetric parameterization using the CMB likelihoods in polarization and temperature (lollipop+PlanckTT). The green contours and lines show the distribution after imposing the additional prior z_{end}> 6. 
The posterior distribution of z_{re} is shown in Fig. 10 after marginalizing over Δz, with and without the additional constraint z_{end}> 6. This suggests that the reionization process occurred at redshift This redshift is lower than the values derived previously from WMAP9 data, in combination with ACT and SPT (Hinshaw et al. 2013), namely z_{re} = 10.3 ± 1.1. It is also lower than the value z_{re} = 11.1 ± 1.1 derived in Planck Collaboration XVI (2014), based on Planck 2013 data and the WMAP9 polarization likelihood.
Although the uncertainty is now smaller, this new reionization redshift value is entirely consistent with the Planck 2015 results (Planck Collaboration XIII 2016) for PlanckTT+lowP alone, or in combination with other data sets, (specifically for PlanckTT+lowP+lensing+ BAO) estimated with δz fixed to 0.5. The constraint from lollipop+ PlanckTT when fixing δz to 0.5 is . This slightly lower value (compared to the one obtained when letting the reionization width be free) is explained by the shape of the degeneracy surface. Allowing for larger duration when keeping the same value of τ pushes towards higher reionization redshifts; marginalizing over Δz thus shifts the posterior distribution to slightly larger z_{re} values.
Fig. 11 Posterior distributions on the end and beginning of reionization, i.e., z_{end} and z_{beg}, using the redshiftsymmetric parameterization without (blue) and with (green) the prior z_{end}> 6. 
In addition to the posteriors for z_{re} and δz using the redshiftsymmetric parameterization, the distributions of the end and beginning of reionization, z_{end} (i.e., z_{99%}) and z_{beg} (i.e., z_{10%}), are plotted in Fig. 11. In such a model, the end of reionization strongly depends on the constraint at low redshift. On the other hand, the constraints on z_{beg} depend only slightly on the lowredshift prior. These results show that the Universe is ionized at less than the 10% level above z = 9.4 ± 1.2.
5.2. Redshiftasymmetric parameterization
We now explore more complex reionization histories using the redshiftasymmetric parameterization of x_{e}(z) described in Sect. 3. In the same manner as in Sect. 5.1, also examine the effect of imposing the additional constraint from the GunnPeterson effect.
The distributions of the two parameters, z_{end} and z_{beg}, are plotted in Fig. 12. With the redshiftasymmetric parameterization, we obtain (imposing the prior on z_{end}), which disfavours any major contribution to the ionized fraction from sources that could form as early as z ≳ 15.
Fig. 12 Posterior distributions of z_{end} and z_{beg} using the redshiftasymmetric parameterization without (blue) and with (green) the prior z_{end}> 6. 
In Fig. 13, we interpret the results in terms of reionization redshift and duration of the EoR, finding These values are within 0.4σ of the results for the redshiftsymmetric model. For the duration of the EoR, the upper limits on Δz are
Fig. 13 Posterior distributions for z_{re} and Δz using the redshiftasymmetric parameterization without (blue) and with (green) the prior z_{end}> 6. 
5.3. Combination with the kSZ effect
In order to try to obtain better constraints on the reionization width, we now make use of the additional information coming from the amplitude of the kinetic SZ effect. Since Planck alone is not able to provide accurate limits on the kSZ amplitude, we combine the Planck likelihoods in temperature and polarization with the measurements of the CMB TT power spectrum at highresolution from the ACT and SPT experiments, “VHL.”
Using the redshiftsymmetric model, when adding the VHL data, we recover essentially the same results as in Sect. 5.1. The reionization redshift is slightly lower, as suggested by the results on τ (see Eq. (7) and the discussion in Sect. 4.1). We also see the same degeneracy along the Δz direction.
With the addition of kSZ information, we are able to break the degeneracy with Δz. This might allow us to determine how much kSZ power originated during reionization (i.e., patchy kSZ) and how much at later times, when the Universe became fully ionized (i.e., homogeneous kSZ). We use the templates from Shaw et al. (2012) and Battaglia et al. (2013) for the homogeneous and patchy kSZ contributions, respectively, with the dependency on ΛCDM cosmological parameters as described in Sect. 4.2. Those specific relations rely on a redshiftsymmetric model for the description of the EoR. Note, however, that the results presented here are derived from specific simulations of the reionization process, and so explicit scalings need to be assumed, as discussed by Zahn et al. (2012) and George et al. (2015).
As described in Sect. 4.2, the amplitude of the kSZ power primarily depends on the duration of reionization, while the epoch is essentially constrained by the optical depth. Using the 2D distribution for τ and A_{kSZ}, as measured by Planck in combination with very highℓ temperature data (Fig. 8), we derive a 2D likelihood function for z_{re} and Δz. We can then sample the reionization parameters (the epoch z_{re} and duration Δz of the EoR), compute the associated optical depth and kSZ power and derive constraints based on the 2D likelihood. The allowed models in terms of z_{re} and Δz are shown in Fig. 14 (in blue). We also plot (in green) the same constraints with the additional prior z_{end}> 6.
Fig. 14 Posterior distributions on the duration Δz and the redshift z_{re} of reionization from the combination of CMB polarization and kSZ effect constraints using the redshiftsymmetric parameterization without (blue) and with (green) the prior z_{end}> 6. 
As discussed in Sect. 4.2, the measurement of the total kSZ power constrains the amplitude of patchy reionization, resulting in an upper limit of This is compatible with the constraints from George et al. (2015), where an upper limit was quoted of z_{20%}−z_{99%}< 5.4 at 95% CL. Our 95% CL upper limits on this same quantity are 4.3 and 2.5 without and with the prior on z_{end}, respectively.
For the reionization redshift, we find which is compatible within 1σ with the results from CMB Planck data alone without the kSZ constraint (Sect. 5.1).
The distributions of z_{end} and z_{beg} are plotted in Fig. 15. Within the redshiftsymmetric parameterization, we obtain (with the prior on z_{end}).
Fig. 15 Posterior distributions of z_{end} and z_{beg} using the redshiftsymmetric parameterization, combining Planck and VHL data, and using information from the kSZ amplitude, without (blue) and with (green) the prior z_{end}> 6. 
Adding information from the kSZ amplitude allows for somewhat tighter constraints to be placed on the reionization duration Δz and the beginning of reionization (corresponding to the 10% ionization limit) z_{beg}. However, as discussed in Sect. 4.2, those results are very sensitive to details of the simulations used to predict both the shape and the parameter dependences of the kSZ template in the different reionization scenarios (patchy or homogeneous).
6. Discussion
The CMB has long held the promise of measuring the Thomson optical depth in order to derive constraints on the reionization history of the Universe. Despite its importance, this constraint is fundamentally limited by cosmic variance in polarization and is further challenged by foregrounds and systematic effects. The first results, from WMAP, gave τ = 0.17 ± 0.04, suggesting a reionization redshift between 11 and 30 (Kogut et al. 2003). This was revised in the final 9yr WMAP results to a central value of τ = 0.084 (Hinshaw et al. 2013), which, in the instantaneous reionization model, implies z_{re} = 10.4. However, with the context of the same model, the Planck 2015 results (Planck Collaboration XIII 2016), either alone () or in combination with other data sets (), showed that the reionization redshift was smaller. The main result we present here, , further confirms that reionization occurred rather late, leaving little room for any significant ionization at z ≳ 15. This is consistent with what is suggested by other reionization probes, which we now discuss (for reviews, see e.g., Becker et al. 2015; McQuinn 2016).
The transition from neutral to ionized gas is constrained by absorption spectra of very distant quasars and gamma ray bursts (GRBs), revealing neutral hydrogen in intergalactic clouds. They show, through the GunnPeterson effect, that the diffuse gas in the Universe is mostly ionized up to a redshift of about 6 (Fan et al. 2006a). Given the decline in their abundance beyond redshift z ≃ 6, quasars and other active galactic nuclei (AGN) cannot be major contributors to the early stages of reionization (e.g., Willott et al. 2010; Fontanot et al. 2012, but see Madau & Haardt2015; Khaire et al.2016, for alternative AGNonly models). A faint AGN population can produce significant photoionization rates at redshifts of 4–6.5, consistent with the observed highly ionized IGM in the Lyα forest of highz quasar spectra (Giallongo et al. 2015). Starforming galaxies at redshifts z ≳ 6 have therefore been postulated to be the most likely sources of early reionization, and their timedependent abundance and spectral properties are crucial ingredients for understanding how intergalactic hydrogen ceased to be neutral (for reviews, see Barkana & Loeb 2001; Fan et al. 2006a; Robertson et al. 2010; McQuinn 2016). The luminosity function of early starforming galaxies, in particular in the UV domain, is thus an additional and powerful probe of the reionization history (e.g., Kuhlen & FaucherGiguère 2012; Robertson et al. 2013, 2015; Bouwens et al. 2015). Based on comparison of the 9yr WMAP results to optical depth values inferred from the UV luminosity function of highz galaxies, it has been suggested that either the UV luminosity density flattens, or physical parameters such as the escape fraction and the clumping factor evolved significantly, or alternatively, additional, undetected sources (such as Xray binaries and faint AGN) must have existed at z ≳ 11 (e.g., Kuhlen & FaucherGiguère 2012; Ellis et al. 2013; Cai et al. 2014; Ishigaki et al. 2015).
The Planck results, both from the 2015 data release and those presented here, strongly reduce the need for a significant contribution of Lyman continuum emission at early times. Indeed, as shown in Fig. 16, the present CMB results on the Thomson optical depth, τ = 0.058 ± 0.012, are perfectly consistent with the best models of starformation rate densities derived from the UV and IR luminosity functions, as directly estimated from observations of highredshift galaxies (Ishigaki et al. 2015; Robertson et al. 2015; Bouwens et al. 2015). With the present value of τ, if we maintain a UVluminosity density at the maximum level allowed by the luminosity density constraints at redshifts z< 9, then the currently observed galaxy population at M_{UV}< −17 seems to be sufficient to comply with all the observational constraints without the need for highredshift (z = 10–15) galaxies.
Fig. 16 Evolution of the integrated optical depth for the tanh functional form (with δz = 0.5, blue shaded area). The two envelopes mark the 68% and 95% confidence intervals. The red, black, and orange dashed lines are the models from Bouwens et al. (2015), Robertson et al. (2015), and Ishigaki et al. (2015), respectively, using highredshift galaxy UV and IR fluxes and/or direct measurements. 
The Planck data are certainly consistent with a fully reionized Universe at z ≃ 6. Moreover, they seem to be in good agreement with recent observational constraints on reionization in the direction of particular objects. The H i absorption along the line of sight to a distant γray burst, GRB140515A (Chornock et al. 2014), suggests a Universe containing about a 10% fraction of neutral hydrogen at z = 6–6.3. At even higher redshifts z ≃ 7, observation of Lyα emitters suggests that at least 70% of the IGM is neutral (Tilvi et al. 2014; Schenker et al. 2014; Faisst et al. 2014). Similarly, quasar nearzone detection and analysis (including sizes, and Lyα and β transmission properties) have been used to place constraints on z_{end} from signatures of the ionization state of the IGM around individual sources (Wyithe & Loeb 2004; Mesinger & Haiman 2004, 2007; Wyithe et al. 2005; Carilli et al. 2010; Mortlock et al. 2011; Schroeder et al. 2013). However, interpretation of the observed evolution of the nearzone sizes may be complicated by the opacity caused by absorption systems within the ionized IGM (e.g., Bolton et al. 2011; Bolton & Haehnelt 2013; Becker et al. 2015). Similarly, it is difficult to completely exclude the possibility that damped Lyα systems contribute to the damping wings of quasar spectra blueward of the Lyα line (e.g., Mesinger & Furlanetto 2008; Schroeder et al. 2013). Nevertheless, most such studies, indicate that the IGM is significantly neutral at redshifts between 6 and 7 (see also Keating et al. 2015), in agreement with the current Planck results, as shown in Fig. 17.
Fig. 17 Reionization history for the redshiftsymmetric parameterization compared with other observational constraints coming from quasars, Lyα emitters, and the Lyα forest (compiled by Bouwens et al. 2015). The red points are measurements of ionized fraction, while black arrows mark upper and lower limits. The dark and light blue shaded areas show the 68% and 95% allowed intervals, respectively. 
Fig. 18 Constraints on ionization fraction during reionization. The allowed models, in terms of z_{re} and Δz, translate into an allowed region in x_{e}(z) (68% and 95% in dark blue and light blue, respectively), including the z_{end}> 6 prior here. Left: constraints from CMB data using a redshiftsymmetric function (x_{e}(z) as a hyperbolic tangent with δz = 0.5). Centre: constraints from CMB data using a redshiftasymmetric parameterization (x_{e}(z) as a power law). Right: constraints from CMB data using a redshiftsymmetric parameterization with additional constraints from the kSZ effect. 
Although there are already all the constraints described above, understanding the formation of the first luminous sources in the Universe is still very much a work in progress. Our new (and lower) value of the optical depth leads to better agreement between the CMB and other astrophysical probes of reionization; however, the fundamental questions remain regarding how reionization actually proceeded.
7. Conclusions
We have derived constraints on cosmic reionization using Planck data. The CMB Planck power spectra, combining the EE polarization at lowℓ with the temperature data, give, for a socalled “instantaneous” reionization history (a redshiftsymmetric tanh function x_{e}(z) with δz = 0.5), a measurement of the Thomson optical depth (24)which is significantly more accurate than previous measurements. Thanks to the relatively high signaltonoise ratio of the lowℓ polarization signal, the combination with lensing or data from highresolution CMB anisotropy experiments (ACT and SPT) does not bring much additional constraining power. The impact on other ΛCDM parameters is only significant for the amplitude of the initial scalar power spectrum A_{s} and (to a lesser extent) on its tilt n_{s}. Other parameters are very stable compared to the Planck 2015 results.
Using Planck data, we have derived constraints on two models for the reionization history x_{e}(z) that are commonly used in the literature: a redshiftsymmetric form using a hyperbolic tangent transition function; and a redshiftasymmetric form parameterized by a power law. We have also investigated the effect of imposing the condition that the reionization is completed by z = 6.
Allowing the ionization fraction shape and duration to vary, we have found very compatible bestfit estimates for the optical depth (0.059 and 0.060 for the symmetric and asymmetric model, respectively), showing that the CMB is indeed more sensitive to the value of the optical depth than to the exact shape of the reionization history. However, the value of the reionization redshift does slightly depend on the model considered. In the case of a symmetric parameterization, we have found slightly larger estimates of z_{re} than in the case of instantaneous reionization. This can be understood through the shape of the degeneracy surface between the reionization parameters. For an asymmetric parameterization, z_{re} is smaller, due to the fact that x_{e}(z) changes more rapidly at the end of reionization than the beginning. We specifically find: Assuming two different parameterizations of the reionization history shows how much results on effective parameters (like the redshift of reionization or its duration) are sensitive to the assumption of the reionization history shape. The best models of symmetric and asymmetric parameterization give similar values for τ, and provide reionization redshifts which differ by less than 0.4σ. Constraints on the limits of possible early reionization are similar, leading to 10% reionization levels at around z = 10.
To derive constraints on the duration of the reionization epoch, we combined CMB data with measurements of the amplitude of the kSZ effect. In the case of a redshiftsymmetric model, we found (27)using the additional constraint that the Universe is entirely reionized at redshift 6 (i.e., z_{end}> 6).
Constraints on reionization parameters for the different models presented in this paper when including the z_{end}> 6 prior.
Our final constraints on the reionization history are summarized on Table 1 and plotted in Fig. 18 for each of the aforementioned cases, i.e., the redshiftsymmetric and redshiftasymmetric models, using only the CMB, and the redshiftsymmetric case using CMB+kSZ (all with prior z_{end}> 6). Plotted this way, the constraints are not very tight and are still fairly model dependent. Given the low value of τ as measured now by Planck, the CMB is not able to give tight constraints on details of the reionization history. However, the Planck data suggest that an early onset of reionization is disfavoured. In particular, in all cases, we found that the Universe was less than 10% ionized for redshift z > 10. Furthermore, comparisons with other tracers of the ionization history show that our new result on the optical depth eliminates most of the tension between CMBbased analyses and constraints from other astrophysical data. Additional sources of reionization, nonstandard early galaxies, or significantly evolving escape fractions or clumping factors, are thus not needed.
Ongoing and future experiments like LOFAR, MWA, and SKA, aimed at measuring the redshifted 21cm signal from neutral hydrogen during the EoR, should be able to probe reionization directly and measure its redshift and duration to high accuracy. Moreover, since reionization appears to happen at redshifts below 10, experiments measuring the global emission of the 21m line over the sky (e.g., EDGES; Bowman & Rogers 2010, LEDA; Greenhill & Bernardi 2012, DARE; Burns et al. 2012, NenuFAR; Zarka et al. 2012, SARAS; Patra et al. 2013, SCIHI; Voytek et al. 2014, ZEBRA; Mahesh et al. 2014, and BIGHORNS, Sokolowski et al. 2015) will also be able to derive very competitive constraints on the models (e.g., Liu et al. 2016; Fialkov & Loeb 2016).
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 and led by Principal Investigators from France and Italy, telescope reflectors provided through a collaboration between ESA and a scientific consortium led and funded by Denmark, and additional contributions from NASA (USA).
The ionized fraction is actually matched to the relic free electron density from recombination, calculated using recfast Seager et al. (2000).
For convenience, we refer to this parameterization as “redshift symmetric” in the rest of the paper, even although it is actually symmetric in y rather than z. The asymmetry is maximum in the instantaneous case, but the difference in x_{e} values around, for example, z_{re} = 8 ± 1, is less than 1%.
Available at camel.in2p3.fr
Acknowledgments
The Planck Collaboration acknowledges the support of: ESA; CNES, and CNRS/INSUIN2P3INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MINECO, J.A., and RES (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); ERC and PRACE (EU). A description of the Planck Collaboration and a list of its members, indicating which technical or scientific activities they have been involved in, can be found at http://www.cosmos.esa.int/web/planck/planckcollaboration
References
 Aghanim, N., Desert, F. X., Puget, J. L., & Gispert, R. 1996, A&A, 311, 1 [NASA ADS] [Google Scholar]
 Aghanim, N., Majumdar, S., & Silk, J. 2008, Rep. Prog. Phys., 71, 066902 [NASA ADS] [CrossRef] [Google Scholar]
 Ahn, K., Iliev, I. T., Shapiro, P. R., et al. 2012, ApJ, 756, L16 [NASA ADS] [CrossRef] [Google Scholar]
 Barkana, R., & Loeb, A. 2001, Phys. Rep., 349, 125 [NASA ADS] [CrossRef] [Google Scholar]
 Battaglia, N., Natarajan, A., Trac, H., Cen, R., & Loeb, A. 2013, ApJ, 776, 83 [NASA ADS] [CrossRef] [Google Scholar]
 Becker, R. H., Fan, X., White, R. L., et al. 2001, AJ, 122, 2850 [NASA ADS] [CrossRef] [Google Scholar]
 Becker, G. D., Bolton, J. S., Haehnelt, M. G., & Sargent, W. L. W. 2011, MNRAS, 410, 1096 [NASA ADS] [CrossRef] [Google Scholar]
 Becker, G. D., Bolton, J. S., & Lidz, A. 2015, PASA, 32, e045 [NASA ADS] [CrossRef] [Google Scholar]
 Bolton, J. S., & Haehnelt, M. G. 2013, MNRAS, 429, 1695 [NASA ADS] [CrossRef] [Google Scholar]
 Bolton, J. S., Haehnelt, M. G., Warren, S. J., et al. 2011, MNRAS, 416, L70 [NASA ADS] [CrossRef] [Google Scholar]
 Bouwens, R. J., Illingworth, G. D., Oesch, P. A., et al. 2015, ApJ, 811, 140 [NASA ADS] [CrossRef] [Google Scholar]
 Bowman, J. D., & Rogers, A. E. E. 2010, Nature, 468, 796 [NASA ADS] [CrossRef] [Google Scholar]
 Burns, J. O., Lazio, J., Bale, S., et al. 2012, Adv. Space Res., 49, 433 [NASA ADS] [CrossRef] [Google Scholar]
 Cai, Z.Y., Lapi, A., Bressan, A., et al. 2014, ApJ, 785, 65 [NASA ADS] [CrossRef] [Google Scholar]
 Calabrese, E., Hlozek, R. A., Battaglia, N., et al. 2013, Phys. Rev. D, 87, 103012 [NASA ADS] [CrossRef] [Google Scholar]
 Carilli, C. L., Wang, R., Fan, X., et al. 2010, ApJ, 714, 834 [NASA ADS] [CrossRef] [Google Scholar]
 Cen, R. 2003, ApJ, 591, 12 [NASA ADS] [CrossRef] [Google Scholar]
 Chornock, R., Berger, E., Fox, D. B., et al. 2014, ArXiv eprints [arXiv:1405.7400] [Google Scholar]
 Ciardi, B., Ferrara, A., & White, S. D. M. 2003, MNRAS, 344, L7 [NASA ADS] [CrossRef] [Google Scholar]
 Colombo, L. P. L., & Pierpaoli, E. 2009, New Astron., 14, 269 [NASA ADS] [CrossRef] [Google Scholar]
 Couchman, H. M. P., & Rees, M. J. 1986, MNRAS, 221, 53 [NASA ADS] [CrossRef] [Google Scholar]
 Das, S., Louis, T., Nolta, M. R., et al. 2014, J. Cosmol. Astropart. Phys., 4, 14 [NASA ADS] [CrossRef] [Google Scholar]
 Douspis, M., Aghanim, N., Ilić, S., & Langer, M. 2015, A&A, 580, L4 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Dunkley, J., Komatsu, E., Nolta, M. R., et al. 2009, ApJS, 180, 306 [NASA ADS] [CrossRef] [Google Scholar]
 Ellis, R. S., McLure, R. J., Dunlop, J. S., et al. 2013, ApJ, 763, L7 [NASA ADS] [CrossRef] [Google Scholar]
 Faisst, A. L., Capak, P., Carollo, C. M., Scarlata, C., & Scoville, N. 2014, ApJ, 788, 87 [NASA ADS] [CrossRef] [Google Scholar]
 Fan, X., Strauss, M. A., Becker, R. H., et al. 2006a, AJ, 132, 117 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Fan, X., Strauss, M. A., Richards, G. T., et al. 2006b, AJ, 131, 1203 [NASA ADS] [CrossRef] [Google Scholar]
 Fialkov, A., & Loeb, A. 2016, ApJ, 821, 59 [NASA ADS] [CrossRef] [Google Scholar]
 Fontanot, F., Cristiani, S., & Vanzella, E. 2012, MNRAS, 425, 1413 [NASA ADS] [CrossRef] [Google Scholar]
 Furlanetto, S. R., Zaldarriaga, M., & Hernquist, L. 2004, ApJ, 613, 1 [NASA ADS] [CrossRef] [Google Scholar]
 George, E. M., Reichardt, C. L., Aird, K. A., et al. 2015, ApJ, 799, 177 [NASA ADS] [CrossRef] [Google Scholar]
 Giallongo, E., Grazian, A., Fiore, F., et al. 2015, A&A, 578, A83 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Gnedin, N. Y. 2000, ApJ, 535, 530 [NASA ADS] [CrossRef] [Google Scholar]
 Greenhill, L. J., & Bernardi, G. 2012, ArXiv eprints [arXiv:1201.1700] [Google Scholar]
 Gruzinov, A., & Hu, W. 1998, ApJ, 508, 435 [NASA ADS] [CrossRef] [Google Scholar]
 Gunn, J. E., & Peterson, B. A. 1965, ApJ, 142, 1633 [NASA ADS] [CrossRef] [Google Scholar]
 Hamimeche, S., & Lewis, A. 2008, Phys. Rev. D, 77, 103013 [NASA ADS] [CrossRef] [Google Scholar]
 Hasselfield, M., Moodley, K., Bond, J. R., et al. 2013, ApJS, 209, 17 [NASA ADS] [CrossRef] [Google Scholar]
 Hinshaw, G., Larson, D., Komatsu, E., et al. 2013, ApJS, 208, 19 [NASA ADS] [CrossRef] [Google Scholar]
 Holder, G. P., Haiman, Z., Kaplinghat, M., & Knox, L. 2003, ApJ, 595, 13 [NASA ADS] [CrossRef] [Google Scholar]
 Hu, W., & Holder, G. P. 2003, Phys. Rev. D, 68, 023001 [NASA ADS] [CrossRef] [Google Scholar]
 Iliev, I. T., Mellema, G., Ahn, K., et al. 2014, MNRAS, 439, 725 [NASA ADS] [CrossRef] [Google Scholar]
 Ishigaki, M., Kawamata, R., Ouchi, M., et al. 2015, ApJ, 799, 12 [NASA ADS] [CrossRef] [Google Scholar]
 Keating, L. C., Haehnelt, M. G., Cantalupo, S., & Puchwein, E. 2015, MNRAS, 454, 681 [NASA ADS] [CrossRef] [Google Scholar]
 Khaire, V., Srianand, R., Choudhury, T. R., & Gaikwad, P. 2016, MNRAS, 457, 4051 [NASA ADS] [CrossRef] [Google Scholar]
 Kogut, A., Spergel, D. N., Barnes, C., et al. 2003, ApJS, 148, 161 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, ApJS, 192, 18 [NASA ADS] [CrossRef] [Google Scholar]
 Kuhlen, M., & FaucherGiguère, C.A. 2012, MNRAS, 423, 862 [NASA ADS] [CrossRef] [Google Scholar]
 Lesgourgues, J. 2011, ArXiv eprints [arXiv:1104.2932] [Google Scholar]
 Lewis, A. 2008, Phys. Rev. D, 78, 023002 [NASA ADS] [CrossRef] [Google Scholar]
 Lewis, A., Weller, J., & Battye, R. 2006, MNRAS, 373, 561 [NASA ADS] [CrossRef] [Google Scholar]
 Liu, A., Pritchard, J. R., Allison, R., et al. 2016, Phys. Rev. D, 93, 043013 [NASA ADS] [CrossRef] [Google Scholar]
 Madau, P., & Haardt, F. 2015, ApJ, 813, L8 [NASA ADS] [CrossRef] [Google Scholar]
 Madau, P., Haardt, F., & Rees, M. J. 1999, ApJ, 514, 648 [NASA ADS] [CrossRef] [Google Scholar]
 Mahesh, N., Subrahmanyan, R., Udaya Shankar, N., & Raghunathan, A. 2014, ArXiv eprints [arXiv:1406.2585] [Google Scholar]
 Mangilli, A., Plaszczynski, S., & Tristram, M. 2015, MNRAS, 453, 3174 [NASA ADS] [CrossRef] [Google Scholar]
 McQuinn, M. 2016, ARA&A, 54, 313 [NASA ADS] [CrossRef] [Google Scholar]
 McQuinn, M., Furlanetto, S. R., Hernquist, L., Zahn, O., & Zaldarriaga, M. 2005, ApJ, 630, 643 [NASA ADS] [CrossRef] [Google Scholar]
 Meiksin, A., & Madau, P. 1993, ApJ, 412, 34 [NASA ADS] [CrossRef] [Google Scholar]
 Mesinger, A. 2016, Understanding the Epoch of Cosmic Reionization (Springer International Publishing), Astrophys. Space Sci. Libr., 423 [Google Scholar]
 Mesinger, A., & Furlanetto, S. R. 2008, MNRAS, 385, 1348 [NASA ADS] [CrossRef] [Google Scholar]
 Mesinger, A., & Haiman, Z. 2004, ApJ, 611, L69 [NASA ADS] [CrossRef] [Google Scholar]
 Mesinger, A., & Haiman, Z. 2007, ApJ, 660, 923 [NASA ADS] [CrossRef] [Google Scholar]
 Mesinger, A., McQuinn, M., & Spergel, D. N. 2012, MNRAS, 422, 1403 [NASA ADS] [CrossRef] [Google Scholar]
 MiraldaEscude, J., & Ostriker, J. P. 1990, ApJ, 350, 1 [NASA ADS] [CrossRef] [Google Scholar]
 Mitra, S., Choudhury, T. R., & Ferrara, A. 2011, MNRAS, 413, 1569 [NASA ADS] [CrossRef] [Google Scholar]
 Mortlock, D. J., Warren, S. J., Venemans, B. P., et al. 2011, Nature, 474, 616 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Mortonson, M. J., & Hu, W. 2008, ApJ, 672, 737 [NASA ADS] [CrossRef] [Google Scholar]
 Ostriker, J. P., & Vishniac, E. T. 1986, ApJ, 306, L51 [NASA ADS] [CrossRef] [Google Scholar]
 Page, L., Hinshaw, G., Komatsu, E., et al. 2007, ApJS, 170, 335 [NASA ADS] [CrossRef] [Google Scholar]
 Pandolfi, S., Ferrara, A., Choudhury, T. R., Melchiorri, A., & Mitra, S. 2011, Phys. Rev. D, 84, 123522 [NASA ADS] [CrossRef] [Google Scholar]
 Park, H., Shapiro, P. R., Komatsu, E., et al. 2013, ApJ, 769, 93 [NASA ADS] [CrossRef] [Google Scholar]
 Patra, N., Subrahmanyan, R., Raghunathan, A., & Udaya Shankar, N. 2013, Exp. Astron., 36, 319 [NASA ADS] [CrossRef] [Google Scholar]
 Peebles, P. J. E. 1968, ApJ, 153, 1 [NASA ADS] [CrossRef] [Google Scholar]
 Planck Collaboration XV. 2014, A&A, 571, A15 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planck Collaboration XVI. 2014, A&A, 571, A16 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planck Collaboration XX. 2014, A&A, 571, A20 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planck Collaboration II. 2016, A&A, 594, A2 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planck Collaboration VII. 2016, A&A, 594, A7 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planck Collaboration VIII. 2016, A&A, 594, A8 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planck Collaboration IX. 2016, A&A, 594, A9 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planck Collaboration X. 2016, A&A, 594, A10 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planck Collaboration XI. 2016, A&A, 594, A11 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planck Collaboration XIII. 2016, A&A, 594, A13 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planck Collaboration XV. 2016, A&A, 594, A15 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planck Collaboration Int. XLVI. 2016, A&A, 596, A107 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Pritchard, J. R., Loeb, A., & Wyithe, J. S. B. 2010, MNRAS, 408, 57 [NASA ADS] [CrossRef] [Google Scholar]
 Reichardt, C. L., Shaw, L., Zahn, O., et al. 2012, ApJ, 755, 70 [NASA ADS] [CrossRef] [Google Scholar]
 Robertson, B. E., Ellis, R. S., Dunlop, J. S., McLure, R. J., & Stark, D. P. 2010, Nature, 468, 49 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Robertson, B. E., Furlanetto, S. R., Schneider, E., et al. 2013, ApJ, 768, 71 [NASA ADS] [CrossRef] [Google Scholar]
 Robertson, B. E., Ellis, R. S., Furlanetto, S. R., & Dunlop, J. S. 2015, ApJ, 802, L19 [NASA ADS] [CrossRef] [Google Scholar]
 Schenker, M. A., Ellis, R. S., Konidaris, N. P., & Stark, D. P. 2014, ApJ, 795, 20 [NASA ADS] [CrossRef] [Google Scholar]
 Schroeder, J., Mesinger, A., & Haiman, Z. 2013, MNRAS, 428, 3058 [NASA ADS] [CrossRef] [Google Scholar]
 Seager, S., Sasselov, D. D., & Scott, D. 2000, ApJS, 128, 407 [NASA ADS] [CrossRef] [Google Scholar]
 Shaw, L. D., Rudd, D. H., & Nagai, D. 2012, ApJ, 756, 15 [NASA ADS] [CrossRef] [Google Scholar]
 Sokolowski, M., Tremblay, S. E., Wayth, R. B., et al. 2015, PASA, 32, 004 [NASA ADS] [CrossRef] [Google Scholar]
 Sunyaev, R. A., & Zeldovich, I. B. 1980, MNRAS, 190, 413 [NASA ADS] [CrossRef] [Google Scholar]
 Tilvi, V., Papovich, C., Finkelstein, S. L., et al. 2014, ApJ, 794, 5 [NASA ADS] [CrossRef] [Google Scholar]
 Trac, H., Bode, P., & Ostriker, J. P. 2011, ApJ, 727, 94 [NASA ADS] [CrossRef] [Google Scholar]
 Tristram, M., MacíasPérez, J. F., Renault, C., & Santos, D. 2005, MNRAS, 358, 833 [NASA ADS] [CrossRef] [Google Scholar]
 Venemans, B. P., Findlay, J. R., Sutherland, W. J., et al. 2013, ApJ, 779, 24 [NASA ADS] [CrossRef] [Google Scholar]
 Voytek, T. C., Natarajan, A., García, J. M. J., Peterson, J. B., & LópezCruz, O. 2014, ApJ, 782, L9 [NASA ADS] [CrossRef] [Google Scholar]
 Willott, C. J., Delorme, P., Reylé, C., et al. 2010, AJ, 139, 906 [NASA ADS] [CrossRef] [Google Scholar]
 Worseck, G., Prochaska, J. X., Hennawi, J. F., & McQuinn, M. 2014, ArXiv eprints [arXiv:1405.7405] [Google Scholar]
 Wyithe, J. S. B., & Loeb, A. 2004, Nature, 432, 194 [NASA ADS] [CrossRef] [Google Scholar]
 Wyithe, J. S. B., Loeb, A., & Carilli, C. 2005, ApJ, 628, 575 [NASA ADS] [CrossRef] [Google Scholar]
 Zahn, O., Reichardt, C. L., Shaw, L., et al. 2012, ApJ, 756, 65 [NASA ADS] [CrossRef] [Google Scholar]
 Zarka, P., Girard, J. N., Tagger, M., & Denis, L. 2012, in SF2A2012: Proc. Annual meeting of the French Society of Astronomy and Astrophysicss, eds. S. Boissier, P. de Laverny, N. Nardetto, et al., 687 [Google Scholar]
 Zel’dovich, Y. B., Kurt, V. G., & Syunyaev, R. A. 1969, Sov. J. Exp. Theoret. Phys., 28, 146 [Google Scholar]
Appendix A: the Lollipop likelihood
lollipop, the LOwℓ LIkelihood on POlarized Powerspectra, is a likelihood function based on crosspower spectra for the low multipoles. The idea behind this approach is that the noise can be considered as uncorrelated between maps and that systematics will be considerably reduced in crosscorrelation compared to autocorrelation.
At low multipoles and for incomplete sky coverage, the C_{ℓ}s are not Gaussian distributed and are correlated between multipoles. lollipop uses the approximation presented in Hamimeche & Lewis (2008), modified as described in Mangilli et al. (2015) to apply to crosspower spectra. The idea is to apply a change of variable C_{ℓ} → X_{ℓ} so that the new variable X_{ℓ} is Gaussian. Similarly to Hamimeche & Lewis (2008), we define (A.1)where , are the measured crosspower spectra, C_{ℓ} are the powerspectra of the model to evaluate, is a fiducial model, and O_{ℓ} are the offsets needed in the case of crossspectra. For multidimensional CMB modes (i.e., T, E, and B), C_{ℓ} is a 3 × 3 matrix of powerspectra: (A.2)and the g function is applied to the eigenvalues of .
In the case of autospectra, the offsets are replaced by the noise bias effectively present in the measured powerspectra. For crosspower spectra, the noise bias is null and here we use the effective offsets defined from the C_{ℓ} noise variance: (A.3)The distribution of the new variable X can be approximated as Gaussian, with a covariance given by the covariance of the C_{ℓ}s. The likelihood function of the C_{ℓ} given the data is then (A.4)where the C_{ℓ} covariance matrix M_{ℓℓ′} is estimated via Monte Carlo simulations.
In this paper, we restrict ourselves to the onefield approximation in order to derive a likelihood function based only on the EE power spectrum at very low multipoles. We use a conservative sky fraction including 50% of the sky, with a Galactic mask based on a threshold on the polarisation amplitude measured in the 353 GHz Planck channel, further apodized using a 4° Gaussian taper (see Fig. A.1). We use Xpol (a pseudoC_{ℓ} estimator described in Tristram et al. 2005 extended to polarisation) to derive crosspower spectra between the 100 and 143 GHz channel maps from Planck. We also reject
the first two multipoles (ℓ = 2 and 3), since they are more subject to contamination by residual instrumental effects (see Planck Collaboration Int. XLVI 2016).
Fig. A.1 Galactic mask used for the lollipop likelihood, covering 50% of the sky. 
This likelihood has been tested on Monte Carlo simulations including signal (CMB and foregrounds), realistic noise, and systematic effects. The simulated maps are then foregroundsubtracted, using the same procedure as for the data. We constructed the C_{ℓ} covariance matrix M_{ℓℓ′} using those simulations. Figure A.2 shows the distribution of the recovered τ values for an input model with τ = 0.06, fixing all other cosmological parameters to the Planck 2015 bestfit values (including A_{s}e^{− 2τ}).
Fig. A.2 Distribution of the peak value of the posterior distribution for optical depth from endtoend simulations including noise, systematic effects, Galactic dust signal, and CMB with the fiducial value of τ = 0.06. 
To validate the choice of multipole and the stability of the result on τ, we performed several consistency checks on the Planck data. Among them, we varied the minimum multipole used (from ℓ = 2 to ℓ = 4) and allowed for larger sky coverage (increasing to 60% of the sky). The results are summarized in Fig. A.3.
Fig. A.3 Posterior distributions for optical depth, showing the effect of changing two of the choices made in our analysis. Top: different choices of minimum multipole. Bottom: different choices of sky fraction used. 
Appendix B: Impact on ΛCDM parameters
Fig. B.1 ΛCDM parameters for PlanckTT combined with the lowℓ polarization likelihood from the Planck 2015 release (lowP, in grey) and from this work (lollipop, in red). 
In addition to the restricted parameter set shown in Fig. 6, we describe here the impact of the lollipop likelihood on ΛCDM parameters in general. Figure B.1 compares results from lollipop+PlanckTT with the lowP+PlanckTT 2015. The new lowℓ polarization results are sufficiently powerful that they break the degeneracy between n_{s} and τ. The contours for τ and A_{s}, where the lollipop likelihood dominates the constraint, are significantly reduced. The impact on other ΛCDM parameters are small, typically below 0.3σ.
All Tables
Constraints on reionization parameters for the different models presented in this paper when including the z_{end}> 6 prior.
All Figures
Fig. 1 Bias in the 100 × 143 crosspower spectrum computed from simulations, including instrumental noise and systematic effects, with or without foregrounds (dark blue and light blue), compared to the cosmic variance level (in grey). 

In the text 
Fig. 2 Variance of the 100 × 143EE crosspower spectrum for simulations, including instrumental noise and noise plus systematic effects, compared to cosmic variance. 

In the text 
Fig. 3 EE and TE power spectra for various τ values ranging from 0.04 to 0.08. The ionization fraction is modelled using a redshiftsymmetric tanh function with δz = 0.5. Grey bands represent the cosmic variance (fullsky) associated with the τ = 0.06 model. 

In the text 
Fig. 4 Left: evolution of the ionization fraction for several functions, all having the same optical depth, τ = 0.06: green and blue are for redshiftsymmetric instantaneous (δz = 0.05) and extended reionization (δz = 0.7), respectively; red is an example of a redshiftasymmetric parameterization; and light blue and magenta are examples of an ionization fraction defined in redshift bins, with two bins inverted between these two examples. Right: corresponding EE power spectra with cosmic variance in grey. All models have the same optical depth τ = 0.06 and are essentially indistinguishable at the reionization bump scale. 

In the text 
Fig. 5 Posterior distribution for τ from the various combinations of Planck data. The grey band shows the lower limit on τ from the GunnPeterson effect. 

In the text 
Fig. 6 Constraints on τ, A_{s}, n_{s}, and σ_{8} for the ΛCDM cosmology from PlanckTT, showing the impact of replacing the lowP likelihood from Planck 2015 release with the new lollipop likelihood. The top panels show results without lensing, while the bottom panels are with lensing. 

In the text 
Fig. 7 Power spectrum templates for the kSZ effect. The different lines correspond to: homogeneous reionization as used in Planck Collaboration XI (2016; dark blue), based on Trac et al. (2011); “CSF” (light blue), which is a homogeneous reionization model from Shaw et al. (2012); Patchy (green dashed) based on patchy reionization model from Battaglia et al. (2013); and the sum of CSF and patchy (red). 

In the text 
Fig. 8 68% and 95% confidence intervals on the reionization optical depth, τ, and the amplitude of the kinetic SZ effect, A_{kSZ}, from the CMB (lollipop+PlanckTT+VHL). 

In the text 
Fig. 9 Constraints on the kSZ amplitude at ℓ = 3000 using lollipop+PlanckTT+VHL likelihoods. The three cases correspond to different kSZ templates. 

In the text 
Fig. 10 Posterior distributions (in blue) of z_{re} and Δz for a redshiftsymmetric parameterization using the CMB likelihoods in polarization and temperature (lollipop+PlanckTT). The green contours and lines show the distribution after imposing the additional prior z_{end}> 6. 

In the text 
Fig. 11 Posterior distributions on the end and beginning of reionization, i.e., z_{end} and z_{beg}, using the redshiftsymmetric parameterization without (blue) and with (green) the prior z_{end}> 6. 

In the text 
Fig. 12 Posterior distributions of z_{end} and z_{beg} using the redshiftasymmetric parameterization without (blue) and with (green) the prior z_{end}> 6. 

In the text 
Fig. 13 Posterior distributions for z_{re} and Δz using the redshiftasymmetric parameterization without (blue) and with (green) the prior z_{end}> 6. 

In the text 
Fig. 14 Posterior distributions on the duration Δz and the redshift z_{re} of reionization from the combination of CMB polarization and kSZ effect constraints using the redshiftsymmetric parameterization without (blue) and with (green) the prior z_{end}> 6. 

In the text 
Fig. 15 Posterior distributions of z_{end} and z_{beg} using the redshiftsymmetric parameterization, combining Planck and VHL data, and using information from the kSZ amplitude, without (blue) and with (green) the prior z_{end}> 6. 

In the text 
Fig. 16 Evolution of the integrated optical depth for the tanh functional form (with δz = 0.5, blue shaded area). The two envelopes mark the 68% and 95% confidence intervals. The red, black, and orange dashed lines are the models from Bouwens et al. (2015), Robertson et al. (2015), and Ishigaki et al. (2015), respectively, using highredshift galaxy UV and IR fluxes and/or direct measurements. 

In the text 
Fig. 17 Reionization history for the redshiftsymmetric parameterization compared with other observational constraints coming from quasars, Lyα emitters, and the Lyα forest (compiled by Bouwens et al. 2015). The red points are measurements of ionized fraction, while black arrows mark upper and lower limits. The dark and light blue shaded areas show the 68% and 95% allowed intervals, respectively. 

In the text 
Fig. 18 Constraints on ionization fraction during reionization. The allowed models, in terms of z_{re} and Δz, translate into an allowed region in x_{e}(z) (68% and 95% in dark blue and light blue, respectively), including the z_{end}> 6 prior here. Left: constraints from CMB data using a redshiftsymmetric function (x_{e}(z) as a hyperbolic tangent with δz = 0.5). Centre: constraints from CMB data using a redshiftasymmetric parameterization (x_{e}(z) as a power law). Right: constraints from CMB data using a redshiftsymmetric parameterization with additional constraints from the kSZ effect. 

In the text 
Fig. A.1 Galactic mask used for the lollipop likelihood, covering 50% of the sky. 

In the text 
Fig. A.2 Distribution of the peak value of the posterior distribution for optical depth from endtoend simulations including noise, systematic effects, Galactic dust signal, and CMB with the fiducial value of τ = 0.06. 

In the text 
Fig. A.3 Posterior distributions for optical depth, showing the effect of changing two of the choices made in our analysis. Top: different choices of minimum multipole. Bottom: different choices of sky fraction used. 

In the text 
Fig. B.1 ΛCDM parameters for PlanckTT combined with the lowℓ polarization likelihood from the Planck 2015 release (lowP, in grey) and from this work (lollipop, in red). 

In the text 
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