© ESO, 2014

1. Introduction

This paper, one of a set associated with the 2013 release of data from the Planck¹ mission (Planck Collaboration I 2014), describes the constraints on cosmic strings, semi-local strings and global textures. Such cosmic defects are a generic outcome of symmetry-breaking phase transitions in the early Universe (Kibble 1976) and further motivation came from a potential role in large-scale structure formation (Zeldovich 1980; Vilenkin 1981a). Cosmic strings appear in a variety of supersymmetric and other grand unified theories, forming at the end of inflation (see, for example, Jeannerot et al. 2003). However, further interest in cosmic (super-)strings has been motivated by their emergence in higher-dimensional theories for the origin of our Universe, such as brane inflation. These superstring variants come in a number of D − and F −string forms, creating hybrid networks with more complex dynamics (see, e.g., Polchinski 2005). Cosmic strings can have an enormous energy per unit length μ that can give rise to a number of observable effects, including gravitational lensing and a background of gravitational waves. Here, we shall concentrate on the impact of strings on the cosmic microwave background (CMB), which includes the generation of line-like discontinuities in temperature. Comparable effects can also be caused by other types of cosmic defects, notably semi-local strings and global textures. As well as influencing the CMB power spectrum, each type of topological defect should have a counterpart non-Gaussian signature giving us the ability distinguish between different defects, alternative scenarios, or systematic effects. The discovery of any of these objects would profoundly influence our understanding of fundamental physics, identifying GUT-scale symmetry breaking patterns, perhaps even providing direct evidence for extra dimensions. Conversely, the absence of these objects will tightly constrain symmetry breaking schemes, again providing guidance for high energy theory. For a general introduction to cosmic strings and other defects, refer to Vilenkin & Shellard (2000); Hindmarsh & Kibble (1995); Copeland & Kibble (2010).

High resolution numerical simulations of cosmic strings using the Nambu-Goto action indicate that cosmological networks tend towards a scale-invariant solution with typically tens of long strings stretching across each horizon volume. These strings continuously source gravitational perturbations on sub-horizon scales, the magnitude of that are determined by the dimensionless parameter: $\begin{array}{ccc}\frac{\mathit{G\mu }}{{\mathit{c}}^{\mathrm{2}}}\mathrm{=}{\left(\frac{\mathit{\eta }}{{\mathit{m}}_{\mathrm{Pl}}}\right)}^{\mathrm{2}}\mathit{,}& & \end{array}$(1)where η is the energy scale of the string-forming phase transition and ${\mathit{m}}_{\mathrm{Pl}}\mathrm{\equiv }\sqrt{ħ\mathit{c}\mathit{/}\mathit{G}}$ is the Planck mass. String effects on the CMB power spectrum have been estimated using a phenomenological string model and, with WMAP and SDSS data, these estimates yield a 2σ upper bound of Gμ/c² < 2.6 × 10^-7 (Battye & Moss 2010). A consequence is that strings can be responsible for no more than 4.4% of the CMB anisotropy signal at multipole ℓ = 10.

As we shall discuss, the evolution of Nambu-Goto string networks is computationally challenging and quantitative uncertainties remain, notably in characterising the string small-scale structure and loop production. An alternative approach has been to use field theory simulations of cosmological vortex-strings. These yield a significantly lower number of strings per horizon volume (less than half), reflecting the importance of radiative effects on the microphysical scales being probed numerically. The degree of convergence with Nambu-Goto string simulations is difficult to determine computationally at present, but there are also global strings for which radiative effects of comparable magnitude are expected to remain important on cosmological scales. It is prudent in this paper, therefore, to constrain both varieties of strings, labelling the field theory constraints as AH from the Abelian-Higgs (local U(1)) model used to describe them. Given these quantitative differences, such as the lower density, field theory strings produce a weaker constraint Gμ_AH/c² < 5.7 × 10^-7 using WMAP data alone (Bevis et al. 2008) and (Urrestilla et al. 2011). The shape of the string-induced power spectrum also has a different shape, which allows up to a 9.5% contribution at ℓ = 10. These WMAP constraints can be improved by adding small-scale CMB anisotropy in a joint analysis. The Nambu-Goto strings limit improves to become Gμ/c² < 1.7 × 10^-7 (using SPT data, Dvorkin et al. 2011) and field theory strings yield Gμ_AH/c² < 4.2 × 10^-7 (Urrestilla et al. 2011). Power-spectrum based constraints on global textures were studied in Bevis et al. (2004) and (Urrestilla et al. 2008), with the latter paper giving a 95% limit of Gμ/c² < 4.5 × 10^-6. Urrestilla et al. (2008) also provide constraints on semi-local strings, Gμ/c² < 5.3 × 10^-6.

Constraints on cosmic strings from non-Gaussianity require high resolution realisations of string-induced CMB maps that are extremely challenging to produce. Low resolution small-angle and full-sky CMB maps calculated with the full recombination physics included, have indicated some evidence for a significant kurtosis from strings (Landriau & Shellard 2011). More progress has been made in creating high resolution maps from string lensing after recombination (see Ringeval & Bouchet (2012) and references therein) and identifying, in principle, the bispectrum and trispectrum, which can be predicted for strings analytically (Hindmarsh et al. 2009, 2010; Regan & Shellard 2010). The first WMAP constraint on cosmic strings using the analytic CMB trispectrum yielded Gμ/c² < 1.1 × 10^-6 at 95% confidence (Fergusson et al. 2010b). An alternative approach is to fit pixel-space templates to a map, this method was applied to global textures templates in Cruz et al. (2007) and Feeney et al. (2013, 2012).

The most stringent constraints that are claimed for the string tension arise from predicted backgrounds of gravitational waves that are created by decaying loops (Vilenkin 1981b). However, these constraints are strongly dependent on uncertain string physics, most notably the network loop production scale and the nature of string radiation from cusps, i.e., points on the strings approaching the speed of light c. The most optimistic constraint based on the European Pulsar Timing Array is Gμ/c² < 4.0 × 10^-9 (van Haasteren et al. 2011), but a much more conservative estimate of Gμ/c² < 5.3 × 10^-7 can be found in Sanidas et al. (2012), together with a string parameter constraint survey and an extensive discussion of these uncertainties. Such gravitational wave limits do not apply to global strings or to strings for which other radiative channels are available.

Alternative topological defects scenarios also have strong motivations and we report limits on textures and global monopoles in this paper as well. Of particular recent interest are hybrid networks of cosmic strings where the correlation length is reduced by having several interacting varieties (e.g., F- and D-strings) or a small reconnection probability, p < 1. We expect to investigate these models using the Planck full mission data.

The outline of this paper is as follows. In Sect. 2 we briefly describe the different types of topological defects that we consider, and their impact on the CMB anisotropies. We also discuss how the CMB power spectrum is computed and how we obtain CMB maps with a cosmic string contribution. In Sect. 3 we present the defect constraints from the CMB power spectrum (with numbers given in Table 2), while Sect. 4 discusses searches for topological defects with the help of their non-Gaussian signature. We finally present the overall conclusions in Sect. 5.

2. Theoretical modelling and forecasting

2.1. Cosmic strings and their cosmological consequences

2.1.1. String network evolution

A detailed quantitative understanding of the cosmological evolution of string networks is an essential pre-requisite for making accurate predictions about the cosmological consequences of strings. Fortunately, all string network simulations to date have demonstrated convincingly that the large-scale properties of strings approach a self-similar scale-invariant regime soon after formation. If we treat the string as a one-dimensional object, then it sweeps out a two-dimensional worldsheet in spacetime $\begin{array}{ccc}{\mathit{x}}^{\mathit{\mu }}\mathrm{=}{\mathit{x}}^{\mathit{\mu }}\mathrm{\left(}{\mathit{\zeta }}^{\mathit{a}}\mathrm{\right)}\mathit{,} \mathit{a}\mathrm{=}\mathrm{0}\mathit{,}\hspace{0.17em}\mathrm{1}\mathit{,}& & \end{array}$(2)where the worldsheet parameters ζ⁰ and ζ¹ are time-like and space-like respectively. The Nambu-Goto action that governs string motion then becomes $\begin{array}{ccc}\mathit{S}\mathrm{=}\hspace{0.17em}\mathrm{-}\hspace{0.17em}\mathit{\mu }\mathrm{\int }\sqrt{\mathrm{-}\mathit{\gamma }}\hspace{0.17em}\hspace{0.17em}{\mathrm{d}}^{\hspace{0.17em}\mathrm{2}}\mathit{\zeta ,}& & \end{array}$(3)where γ_ab = g_μν∂_ax^μ∂_bx^ν is the two-dimensional worldsheet metric (γ = det(γ_ab)) induced by the spacetime metric g_μν. The Nambu-Goto action Eq. (3) can be derived systematically from a field theory action, such as that for the Abelian-Higgs model describing U(1) vortex-strings: $\hspace{0.17em}\begin{array}{c}\begin{array}{c}\end{array}\mathit{S}\mathrm{=}\mathrm{\int }{\mathrm{d}}^{\mathrm{4}}\mathit{x}\sqrt{\mathrm{-}\mathit{g}}\hspace{0.17em}\left[\mathrm{\left(}{\mathit{D}}_{\mathit{\mu }}\mathit{\phi }{\mathrm{)}}^{\mathrm{\ast }}\mathrm{\right(}{\mathit{D}}^{\mathit{\mu }}\mathit{\phi }\mathrm{)}\mathrm{-}\frac{\mathrm{1}}{\mathrm{4}{\mathit{e}}^{\mathrm{2}}}{\mathit{F}}_{\mathit{\mu \nu }}{\mathit{F}}^{\mathit{\mu \nu }}\mathrm{-}\frac{\mathit{\lambda }}{\mathrm{4}}\mathrm{(}\mathrm{|}\mathit{\phi }{\mathrm{|}}^{\mathrm{2}}\mathrm{-}{\mathit{\eta }}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{2}}\right]\mathit{,}\end{array}$(4)where φ is a complex scalar field, F^μν is the U(1) field strength and D_μ = ∂_μ + iA_μ is the gauge-covariant derivative with e and λ dimensionless coupling constants. The transverse degrees of freedom in φ can be integrated out provided the string is not strongly curved, that is, the string width δ ≈ ħc/η ≪ L where L is the typical radius of curvature. For a cosmological string network today with Gμ/c² ~ 10^-7, these two lengthscales are separated by over 40 orders of magnitude, so this should be a valid approximation.

In an expanding universe, the Nambu-Goto action Eq. (3) yields a Hubble-damped wave equation governing the string motion. These equations can be solved numerically, provided “kinks” or velocity discontinuities are treated carefully. However, they can also be averaged analytically to describe the scale-invariant evolution of the whole string network in terms of two quantities, the energy density ρ and the rms velocity v. Any string network divides fairly neatly into two distinct populations of long (or “infinite”) strings ρ_∞ stretching beyond the Hubble radius and the small loops ρ_l with length l ≪ H^-1 that the long strings create Kibble (1985). Assuming the long strings form a Brownian random walk characterised by a correlation length L, we have $\begin{array}{ccc}{\mathit{\rho }}_{\mathrm{\infty }}\mathrm{=}\frac{\mathit{\mu }}{{\mathit{L}}^{\mathrm{2}}}\mathrm{\equiv }\frac{\mathit{\zeta \mu }}{{\mathit{t}}^{\mathrm{2}}}\mathit{,}& & \end{array}$(5)and the averaged equations of motion become simply $\begin{array}{ccc}\mathrm{2}\frac{\mathrm{d}\mathit{L}}{\mathrm{d}\mathit{t}}& \mathrm{=}& \mathrm{2}\mathit{HL}\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{v}}^{\mathrm{2}}\mathrm{)}\mathrm{+}\begin{array}{c}\mathrm{˜}\\ \mathit{c}\end{array}\mathit{v,}\\ \frac{\mathrm{d}{\mathit{v}}_{\mathrm{\infty }}}{\mathrm{d}\mathit{t}}& \mathrm{=}& \mathrm{(}\mathrm{1}\mathrm{-}{{\mathit{v}}^{\mathrm{2}}}^{\mathrm{)}}\left[\frac{\mathit{k}\mathrm{\left(}\mathit{v}\mathrm{\right)}}{\mathit{L}}\mathrm{-}\mathrm{2}\mathit{Hv}\right]\mathit{,}\end{array}$(6) where $\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{c}\end{array}$ measures the network loop production rate and k(v) is a curvature parameter with $\mathit{k}\mathrm{\approx }\mathrm{2}\sqrt{\mathrm{2}}\mathrm{(}\mathrm{1}\mathrm{-}\sqrt{\mathrm{2}}\mathit{v}\mathrm{)}$. This is the velocity-dependent one-scale (VOS) model and, with a single parameter $\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{c}\end{array}$, it provides a good fit to both Nambu and field theory simulations, notably through the radiation-matter transition (Martins & Shellard 1996).

A general consensus has emerged from the three main simulation codes describing Nambu-Goto string networks (Martins & Shellard 2006; Ringeval et al. 2007; Blanco-Pillado et al. 2011). These independent codes essentially solve for left- and right-moving modes along the string using special techniques to handle contact discontinuities or kinks, including “shock fronting”, artificial compression methods and an exact solver for piecewise linear strings, respectively. The consistency between simulations is shown in Table 1 for the string density parameter ζ defined in Eq. (5). Averaging yields the radiation era density ζ = 10.7 and a matter era value ζ = 3.3. Note that these asymptotic values and the intervening matter-radiation transition can be well-described by the VOS model Eq. (6) with $\begin{array}{c}{}_{\mathrm{˜}}\\ \mathit{c}\end{array}\mathrm{=}\mathrm{0.23}$. The matter era VOS value appears somewhat anomalous from the other two simulations, but this is obtained from larger simulations in a regime where convergence is very slow, so it may more closely reflect the true asymptotic value. These simulations have also advanced the study of string small-scale structure and the loop distribution, about which there had been less consensus (see, e.g., Blanco-Pillado et al. 2011). However, note that CMB anisotropy is far less sensitive to this issue compared to constraints from gravitational waves.

Field theory simulations using lattice gauge techniques have also been employed to study the evolution of string networks in an expanding universe. Comparatively, these three-dimensional simulations are constrained to a lower dynamic range and the simulations require the solution of modified field equations to prevent the string core width shrinking below the lattice resolution. On the other hand, field theory simulations include field radiation and therefore provide a more complete account of the string physics. In Table 1 the lower string densities obtained from two sets of Abelian-Higgs simulations are given (Moore et al. 2002; Bevis et al. 2007b). The evolution can be fitted with a VOS model Eq. (6) with $\begin{array}{c}{}_{\mathrm{˜}}\\ \mathit{c}\end{array}\mathrm{=}\mathrm{0.57}$, which is 150% higher than for Nambu-Goto strings. Field theory simulations have further important applications, particularly for describing delocalised topological defects such as textures, for describing models that do not form stable defects like semilocal strings, and because they include radiative effects naturally. Radiative effects observed in current Abelian-Higgs simulations are comparable to the radiative damping anticipated for cosmological global strings and so the AH analysis below should offer some insight into this case.

2.1.2. String gravity and the CMB

Despite the enormous energy per unit length μ, the spacetime around a straight cosmic string is locally flat. The string has an equation of state p_z = −ρ, p_x = p_y = 0 (for one lying along the z-direction), so there is no source term in the relativistic version of the Poisson equation ∇²Φ = 4πG(ρ + p_x + p_y + p_z). The straight string exhibits no analogue of the Newtonian pull of gravity on any surrounding matter. But this does not mean the string has no gravitational impact; on the contrary, a moving string has dramatic effects on nearby matter or propagating CMB photons.

The spacetime metric about a straight static string takes the simple form, $\begin{array}{ccc}\mathrm{d}{\mathit{s}}^{\mathrm{2}}\mathrm{=}\mathrm{d}{\mathit{t}}^{\mathrm{2}}\mathrm{-}\mathrm{d}{\mathit{z}}^{\mathrm{2}}\mathrm{-}\mathrm{d}{\mathit{r}}^{\mathrm{2}}\mathrm{-}{\mathit{r}}^{\mathrm{2}}\mathrm{d}{\mathit{\theta }}^{\mathrm{2}}\mathit{,}& & \end{array}$(7)that looks like Minkowski space in cylindrical coordinates, but for the fact that the azimuthal coordinate θ has a restricted range 0 ≤ θ ≤ 2π(1−4Gμ). The spacetime is actually conical with a global deficit angle Δ = 8πGμ, that is, an angular wedge of width Δ is removed from the space and the remaining edges identified (see Fig. 1). This means that distant galaxies on the opposite side of a cosmic string can be gravitationally lensed to produce characteristic double images.

Fig. 1 The spacetime around a cosmic string is conical, as if a narrow wedge were removed from a flat sheet and the edges identified. For this reason cosmic strings can create double images of distant objects. Strings moving across the line of sight will cause line-like discontinuities in the CMB radiation.

Fig. 2 Characteristic CMB temperature discontinuity created by a cosmic string. Here, the simulated Nambu-Goto string has produced a cusp, a small region on the string that approaches the speed of light, which has generated a localised CMB signal.

Cosmic strings create line-like discontinuities in the CMB signal. As the string moves across the line of sight, the CMB photons are boosted towards the observer, causing a relative CMB temperature shift across the string, given by (Gott III 1985; Kaiser & Stebbins 1984) $\begin{array}{ccc}\frac{\mathit{\delta T}}{\mathit{T}}\mathrm{=}\mathrm{8}\mathit{\pi G\mu }{\mathit{v}}_{\mathrm{s}}\hspace{0.17em}{\mathit{\gamma }}_{\mathrm{s}}\mathit{,}& & \end{array}$(8)where v_s is the transverse velocity of the string and ${\mathit{\gamma }}_{\mathrm{s}}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{v}}_{\mathrm{s}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{2}}$. This rather simple picture, however, is complicated in an expanding universe with a wiggly string network and relativistic matter and radiation components. The energy-momentum tensor T_μν(x,t) essentially acts as a source term for the metric fluctuations that perturb the CMB photons and create temperature anisotropies. Essentially, the problem can be recast using Green’s (or transfer) functions G^μν that project forward the contributions of strings from early times to today: $\begin{array}{ccc}\frac{\mathrm{\Delta }\mathit{T}}{\mathit{T}}\mathrm{\left(}\hat{{n}}\mathit{,}{{x}}_{\mathrm{obs}}\mathit{,}{\mathit{t}}_{\mathrm{0}}\mathrm{\right)}\mathrm{=}\mathrm{\int }{\mathrm{d}}^{\mathrm{4}}\mathit{x}\hspace{0.17em}{\mathit{G}}^{\mathit{\mu \nu }}\mathrm{\left(}\hat{{n}}\mathit{,}{x}\mathit{,}{{x}}_{\mathrm{obs}}\mathit{,t,}{\mathit{t}}_{\mathrm{0}}\mathrm{\right)}\hspace{0.17em}{\mathit{T}}_{\mathit{\mu \nu }}\mathrm{\left(}{x}\mathit{,t}\mathrm{\right)}\mathit{,}& & \end{array}$(9)where $\hat{{n}}$ is the line-of-sight direction for photon propagation and x_obs is the observer position. The actual quantitative solution of this problem entails a sophisticated formalism to solve the Boltzmann equation and then to follow photon propagation along the observer’s line-of-sight. An example of the line-like discontinuity signal created by a cosmic string in the CMB is shown in Fig. 2. In this case, a string cusp has formed on the string, causing a strongly localised signal and reflecting the Lorentz boost factor in Eq. (9).

2.2. Semi-local strings

The tight constraints on the presence of cosmic strings that we will discuss later in this paper start to put pressure on the wide class of inflation models that generate such defects (Hindmarsh 2011). The power of these constraints would be reduced if the strings could be made unstable. This is the basic motivation behind semilocal strings: a duplication of the complex scalar field φ in the Abelian-Higgs action (4), occurring naturally in a range of inflation models (Urrestilla et al. 2004; Dasgupta et al. 2004, 2007; Achucarro et al. 2006), transforms the stable cosmic strings into non-topological semilocal strings (Vachaspati & Achucarro 1991) as the vacuum manifold becomes S³, which is simply-connected. The existence and stability of the semilocal strings is thus a question of dynamics rather than due to the topology of the vacuum manifold. In general we do not expect to form long strings, but rather shorter string segments, as the semilocal strings can have ends. The evolution of these segments is very complicated and arises directly from the field evolution, so that it is only practicable to simulate these defects with the help of field theory (Urrestilla et al. 2008).

2.3. Global defects

A large alternative class of defects is due to the breaking of a global O(N) symmetry (rather than a gauge symmetry as in the case of cosmic strings) of a N-component scalar field φ. The energy density of global defects is significantly less localised than those that result from gauge symmetry breaking due to the absence of the screening by a gauge field, and there are thus long-range forces between the defects. The field self-ordering is therefore very efficient for all types of defects with N ≥ 2, leading to a generic scaling of the defect energy density with the background energy density (see e.g., Durrer et al. 2002). For this reason global monopoles (N = 3) do not overclose the Universe as their local counterparts would. In this paper we study specifically the case N = 4 called “texture”, which can arise naturally in many multi-field inflation models that involve a non-zero vacuum expectation value and symmetry breaking. In this case there are no stable topological defects present, but contrary to local texture, global texture can have a non-negligible impact on the perturbations in the cosmos, with the field self-ordering leading to “unwinding events”. In spite of their non-topological nature, the field evolution is closely related to the one of lower-dimensional stable global defects due to the long-range nature of the forces. This is similar to the case of the non-topological semilocal strings of the previous section, and indeed the semilocal example can be seen as an intermediate case between cosmic strings and global texture: Starting from the semilocal action, we can on the one hand revert to the cosmic string action by removing one of the complex scalar fields, and on the other hand we find the texture action if we remove the gauge field.

The normalisation of global defects is usually given in terms of the parameter ε = 8πGη²/c² when using an action like Eq. (4) (with a second complex scalar field but without the gauge fields). However, for a simpler comparison with the cosmic string results we can recast this in terms of Gμ/c² ≡ ε/ 4 and quote limits on Gμ/c² also for the texture model, as in Urrestilla et al. (2008).

2.4. CMB power spectra from cosmic defects

The CMB power spectrum from topological defects, like strings, is more difficult to compute than the equivalent for inflationary scenarios that predict a spectrum dominated by an adiabatic component with a possible, but highly constrained, isocurvature component. In defect-based scenarios the perturbations are sourced continuously throughout the history of the Universe, as opposed to adiabatic and isocurvature modes that are the result of initial conditions. In principle this requires knowledge of the source, quantified by the unequal-time correlator (UETC) of the defect stress-energy tensor, from the time of defect formation near the GUT scale to the present day – a dynamic range of about 10⁵² – something that will never be possible to simulate. Fortunately, we can use the scaling assumption to extrapolate the results of simulations with substantially smaller dynamic range. This has allowed a qualitative picture to emerge of the characteristics of the power spectra from defects, though quantitative predictions differ. Here, we will focus on spectra calculated in two different ways for cosmic strings, as well as spectra from semilocal strings and texture models.

Defect-based power spectra are dominated by different physical effects across the range of angular scales. (i) On large angular scales the spectra are dominated by an integrated Sachs-Wolfe (ISW) component due to the strings along the line-of-sight between the time of last scattering and the present day. The scaling assumption implies that this component will be close to scale invariant, although in practice it typically has a mildly blue spectrum. (ii) At intermediate scale the dominant contribution comes from anisotropies created at the time of last scattering. In contrast to the strong series of acoustic peaks created in adiabatic and isocurvature models, defects produce only a broad peak because their contributions are not coherent. (iii) At very small angular scales, the spectra are again dominated by the ISW effect because, rather than decaying exponentially due to the effects of Silk damping, there is only power-law decay with the exponent being a characteristic of the specific type of defect.

The standard lore is to treat the defect stress energy tensor, θ_μν, as being covariantly conserved at first order, which is known as the “stiff approximation”. In principle, this means that it is necessary to measure two independent quantities from the simulations, or model them. The other two component are then computed from the conservation equations. In practice things are a little more complicated since it is necessary to provide the UETC $\begin{array}{ccc}{\mathit{U}}_{\mathit{\mu \nu \alpha \beta }}\mathrm{\left(}\mathit{k,\tau ,}{\mathit{\tau }}^{\mathrm{\prime }}\mathrm{\right)}\mathrm{=}\mathrm{⟨}{\mathit{\theta }}_{\mathit{\mu \nu }}\mathrm{\left(}\mathit{k,\tau }\mathrm{\right)}{\mathit{\theta }}_{\mathit{\alpha \beta }}\mathrm{\left(}\mathit{k,}{\mathit{\tau }}^{\mathrm{\prime }}\mathrm{\right)}\mathrm{⟩}\mathit{,}& & \end{array}$(10)where τ is the conformal time and k is the wavenumber. Once one has the UETC, then there two ways to proceed. The first involves creating realisations of the defect stress-energy whose power spectra are computed then averaged to give the total power spectrum. The other approach involves diagonalization of the UETC. During pure matter or radiation domination, the scaling property of defect evolution means that quantities are measured relative to the horizon scale, so that the UETC is only a function of x = kτ and x′ = kτ′. These functions U(x,x′) can be discretized and then are symmetric matrices that we can diagonalize. The resulting eigenvectors can be inserted as sources into a Boltzmann code, and the resulting C_ℓ are then summed up, weighted by the eigenvalues (Pen et al. 1997; Durrer et al. 2002). Even though the power spectrum resulting from each “eigen-source” exhibits a series of acoustic peaks, the summation over many such spectra smears them out, as they are not coherent (unlike inflationary perturbations). This smearing-out explains why defect power spectra generically are smooth, as mentioned above.

Fig. 3 Cosmic string power spectra used in this analysis: NAMBU (black dashed), AH-mimic (blue dotted) and AH (red solid). The spectra have been set to equal power at ℓ = 10. When normalised to 850 μK at ℓ = 10 they correspond to string tensions of Gμ/c2 = 1.17 × 10-6, 1.89 × 10-6 and 2.04 × 10-6 respectively. Note that the limits discussed in this paper mean that the CMB spectra presented here are less than 3% of the overall power spectrum amplitude and hence the differences observed at high ℓ do not have a large effect.

There are also several methods to obtain predictions for the UETCs of cosmic strings and other topological defects. The first approach we will consider for cosmic strings is to use what has become known as the Unconnected Segment Model (USM; Albrecht et al. 1997, 1999; Pogosian & Vachaspati 1999). In its simplest form this models the cosmic string energy momentum tensor as that of an ensemble of line segments of correlation length ξd_H(t), moving with an rms velocity ⟨ v² ⟩ ^1/2, where d_H(t) is the horizon distance. In addition one can take into account the effects of string “wiggles” due to small-scale structure via a coefficient, β = μ_eff/μ quantifying the ratio of the renormalised mass per unit length to the true value. The model parameters ξ, ⟨ v² ⟩ ^1/2 and β are computed from simulations. In our calculations we link the USM sources to the line-of-sight Boltzmann solver CMBACT (Pogosian & Vachaspati 1999) to create an ensemble of realisations from which we find an averaged angular power spectrum.

There are two USM-based models that we will use in this analysis which we believe span the realistic possibilities – we note a more general approach marginalising over three string parameters is proposed in Foreman et al. (2011) (see also recent work in Avgoustidis et al. 2012). The first USM model, which we will refer to as NAMBU, is designed to model the observational consequences of simulations of cosmic string simulations performed in the Nambu-Goto approximation. In these simulations the scaling regime is different in the radiation and matter eras, with (ξ, ⟨ v² ⟩ ^1/2/c,β)_rad = (0.13,0.65,1.9) and (ξ, ⟨ v² ⟩ ^1/2/c,β)_mat = (0.21,0.60,1.5) and the extrapolation between the two is modelled by using the velocity dependent one-scale model (Martins & Shellard 1996). In the second, which we will refer to as AH-mimic, we attempt to model the field theory simulations using the Abelian-Higgs model described below, with (ξ, ⟨ v² ⟩ ^1/2/c,β) = (0.3,0.5,1) independent of time.

The other approach that we will consider is to measure the UETC directly from a simulation of cosmic strings in the Abelian-Higgs model, which we will refer to as AH. The Abelian-Higgs model involves a complex scalar field φ and a gauge field A_μ described earlier Eq. (4), for which the dimensionless coupling constants e and λ are chosen with λ = 2e², so that the characteristic scales of the magnetic and scalar energies are equal, (see Bevis et al. (2007b,a) for further details about the model). We then simulate the evolution of the fields on a grid, starting from random initial conditions designed to mimic a phase transition, followed by a brief period of diffusive evolution, to rapidly reach a scaling solution expected to be typical of the configuration found long after the phase transition. As the simulation is performed in comoving coordinates, the string width is effectively decreasing as time passes. To enlarge the dynamical range available, we partially compensate this shrinking with an artificial string fattening. We perform runs for various values of the fattening parameter to ensure that the results are not affected by it.

Fig. 4 Comparison between global texture (black dashed) and semilocal (blue dotted) string power spectra and the AH field theory strings (red solid), normalised to unity at ℓ = 10. As expected, the SL spectrum lies in between the TX and the AH spectra. The AH spectrum was recomputed for the Planck cosmological model with sources from Bevis et al. (2010), and the SL and TX spectra were taken from Urrestilla et al. (2008).

During the simulations, we compute the energy-momentum tensor at regular intervals and decompose it into scalar, vector and tensor parts. We store these components once scaling is reached, and compute the UETCs by correlating them with later values of the energy-momentum tensor. UETCs from several runs are averaged, diagonalized and then fed into a modified version of the CMBEASY Boltzmann code (Doran 2005) to compute the CMB power spectra (both temperature and polarisation). The spectra used in this paper were derived from field-theory simulations on a 1024³ grid and used the extrapolation to sub-string scales described in Bevis et al. (2010), which are expected to be accurate at the 10% level to ℓ_max ≈ 4000.

In Figs. 3 and 4 we present the spectra we will use in subsequent analysis. The higher dashed black curve is the spectrum computed using the USM for the NAMBU model, and the smaller dashed blue and solid red curves the AH-mimic model and the AH model, respectively. We should note that when normalised to the amplitude of the observed CMB anisotropies on large-scales at ℓ = 10, the three models give Gμ/c² = 1.17 × 10^-6, 1.89 × 10^-6 and 1.9 × 10^-6 for the NAMBU, AH-mimic and AH models, respectively. The reasons for differences between the spectra for these two approaches are discussed in Battye & Moss (2010). Briefly, the main reasons for the differences are twofold: First, the overall normalisation, which is due to the NAMBU models having smaller values of ξ, more strings per horizon volume, and larger values of β, with each of the string segments being heavier, than the two AH models. Both these effects mean that a lower value of Gμ/c² is required to achieve the same amplitude for the anisotropies. Secondly, the enhanced peak at small angular scales, which is caused by the value of ξ being smaller in the radiation era than in the matter era, meaning that there are more strings per horizon volume in the radiation era when the small-scale anisotropy is imprinted, and hence more anisotropy on those scales for a given Gμ/c².

The method used for the semilocal strings (denoted SL) and O(4) global texture (denoted TX) is fundamentally the same as for the AH model: we simulate the field theory on a discretized grid and compute the energy-momentum tensor at regular intervals. From these snapshots we derive the UETCs by correlating the scalar, vector and tensor parts at different times. The only difference is the field-theory action being used in the simulations. In Fig. 4 we present the spectra we used for the semilocal strings and global textures, taken from Urrestilla et al. (2008). These models are also shown with the AH cosmic string model for comparison.

2.5. Maps of CMB anisotropies from cosmic strings

In order to go further than the two-point correlation function, we have used numerical simulations of Nambu-Goto cosmic string evolution in an FLRW spacetime to generate various CMB synthetic maps. The use of simulations is crucial to produce realistic string configurations on our past light cone and have been the subject of various code development in the last twenty years (see Albrecht & Turok 1989; Bennett & Bouchet 1989, 1990; Allen & Shellard 1990; Vincent et al. 1998; Moore et al. 2001; Ringeval et al. 2007; Blanco-Pillado et al. 2011). Until recently, the underlying numerical challenges have limited the resolution of the full sky maps to an angular resolution of 14′ (corresponding to a HEALPix resolution of N_side = 256) in Landriau & Shellard (2003, 2011) (see also early work in Allen et al. 1996). In order to extend the applicability of these maps to the small scales probed by Planck, we have used the maps described in Ringeval & Bouchet (2012) that have an angular resolution of 0.85′ (N_side = 4096). This map is obtained by considering the ISW contribution from (10), sourced by the Nambu-Goto stress tensor, and which can be recast into the form (Stebbins & Veeraraghavan 1995) $\frac{\mathrm{\Delta }\mathit{T}}{\mathit{T}}\mathrm{\left(}{n̂}\mathrm{\right)}\mathrm{=}\mathrm{-}\frac{\mathrm{4}\mathit{G\mu }}{{\mathit{c}}^{\mathrm{2}}}{\mathrm{\int }}_{{X}\hspace{0.17em}\mathrm{\cap }\hspace{0.17em}{{x}}_{\mathit{\gamma }}}\left[{Ẋ}\mathrm{-}\frac{\mathrm{(}{n̂}\mathrm{·}{{X}}^{{\prime }}\mathrm{)}\mathrm{·}{{X}}^{{\prime }}}{\mathrm{1}\mathrm{+}{n̂}\mathrm{·}{Ẋ}}\right]\mathrm{·}\frac{\mathit{X}{n̂}\mathrm{-}{X}}{{\mathrm{(}\mathit{X}{n̂}\mathrm{-}{{X}}^{\mathrm{)}}}^{\mathrm{2}}}\hspace{0.17em}\mathrm{d}\mathit{l}\hspace{0.17em}\mathit{.}$(11)The integral is performed over all string position vectors X = { Xⁱ } intercepting our past line cone (in the transverse temporal gauge). Primes and dots denote differentiation with respect to the spatial and time-like worldsheet coordinates ζ¹ and ζ⁰ respectively, while dl is the invariant string length element. Taking the limit gives back the small angle and flat sky approximation used in Hindmarsh (1994); Bouchet et al. (1988); Fraisse et al. (2008). For generating the full sky map, Eq. (12) has been evaluated without any other approximation and required more than 3000 Nambu-Goto string simulations of various sizes to fill the whole comoving volume between the observer and the last scattering surface. We note that the use of different simulations does not induce any visible artefact, essentially because only the subset of strings intercepting our past light cone leaves imprints in the CMB. As a result, the probability of observing an edge remains very small. Discussions of these effects and details on the numerics can be found in Ringeval & Bouchet (2012).

This method therefore includes all string effects from the last scattering surface till today, but does not include the Doppler contributions induced by the strings into the plasma prior to recombination. As a result, our full sky map represents the ISW contribution from strings, which is dominant at large and small scales but underestimates the signal on intermediate length scales where recombination effects on the surface of last scattering dominate (see the discussion in Sect. 2.4). This can be seen directly by making a qualitative comparison between the ISW string power spectrum, obtained from the simulated maps and shown in Fig. 5, with the full CMB string power spectrum predicted for Nambu strings shown in Fig. 3 (dashed line). We therefore expect the string searches based on the simulated maps to be less constraining than those using the power spectrum, though certainly robust as any line-like gravitating object should generate such a signal.

Fig. 5 Integrated Sachs-Wolfe angular power spectra extracted from the full sky cosmic string maps at different resolutions (labelled by Nside), with or without applying the anti-aliasing procedure (see text). The anti-aliasing filtering gives back the correct power up to ℓmax ≲ 2Nside.

Calibration and training for the non-Gaussian searches of Sect. 4 have required the generation of new full sky and statistically independent cosmic string maps. The numerical challenges underlying the N_side = 4096 map (Ringeval & Bouchet 2012) are such that it was numerically too expensive to create another one of the same kind. At this resolution, the computations typically require 800 000 cpu-hours, so we have chosen to generate three new maps at a lower resolution of 1.7′, i.e., N_side = 2048. Unfortunately, at this lower resolution, the simulated string maps, hereafter referred to as raw maps, exhibit a strong aliasing at small scales that could have induced spurious systematics even after convolution with the Planck beam. This aliasing concerns pixel-sized structures and comes from the method used to numerically evaluate Eq. (12). In order to save computing time, the signal associated with each pixel is only computed at the centroid direction . This has the effect of including some extra power associated with string small-scale structure that is below the pixel angular size, thereby aliasing power into the map. In order to address this problem, we have used semi-analytical methods to design an optimal anti-aliasing filter, both in harmonic space and in real space. As discussed in Fraisse et al. (2008); Bevis et al. (2010), the small scale angular temperature power spectrum slowly decays as a power law ℓ^−p such that any deviations from this behaviour can only come from the aliasing. For each N_side = 2048 raw map, we have performed a multi-parameter fit of the power spectrum, and of the one-point distribution function, to extract, and then removes, its small scale aliasing contribution. In order to validate the procedure, we have checked that the power spectrum of each of the filtered maps matches the one associated of the raw N_side = 4096 map, the latter being also being affected but at half the scale. In Fig. 5, we have plotted the power spectra of one of the N_side = 2048 maps before and after convolution with our anti-aliasing filter. As expected, it matches with the one extracted from the N_side = 4096 map (here truncated at ℓ = 4096). We note that the string bispectra reconstructed from the antialiased N_side = 2048 maps and the N_side = 4096 maps are essentially identical, as discussed in Sect. 4.2.2 and shown in Fig. 11. Finally, in order to include the effects associated with the HEALPix pixelization scheme, the anti-aliased maps have been convolved with the HEALPix pixel window function before being used for further processing.

Fig. 6 All sky Mollweide projection of the simulated cosmic strings CMB sky after convolution by a Gaussian beam of 5′ resolution. The colour scale indicates the range of (ΔT/T)/(Gμ/c2) fluctuations.

Fig. 7 A 20° gnomic projection patch extracted from the full sky map and zooming into string induced temperature steps (see Fig. 6). Applying the spherical gradient magnitude operator enhances the temperature steps, and thus the string locations, even more (right).

In total, this method has provided four theoretical full sky string maps that have been used in the string searches we will discuss in Sect. 4. As an illustration, we have represented in Fig. 6, one of the filtered string map after convolution by a Gaussian beam of FWHM = 5′. The colour scale traces the relative temperature anisotropies ΔT/T, divided by the string tension Gμ/c². The anisotropy patterns may look Gaussian at first because most of the string signatures show up on the smallest length scales. In Fig. 7, we have plotted a gnomic projection representing a field of view of 20°, in which the temperature steps are now clearly apparent. The right panel of Fig. 7 represents the magnitude of the spherical gradient, which enhances the steps.

Finally, in order to provide a much larger statistical sample beyond only four string realisations, we have also produced a collection of 1000 small angle patches (7.2°) of the CMB sky derived in the flat sky approximation (Stebbins 1988; Hindmarsh 1994; Stebbins & Veeraraghavan 1995; Bouchet et al. 1988; Fraisse et al. 2008). Although the large-scale correlations are lost, these maps have been shown to accurately reproduce various analytically expected non-Gaussian string effects such as the one-point and higher n-points functions by Takahashi et al. (2009), Hindmarsh et al. (2009), Hindmarsh et al. (2010), Regan & Shellard (2010), Yamauchi et al. (2010b), Yamauchi et al. (2010a), Ringeval (2010).

3. Power spectrum constraints on cosmic strings and other topological defects

In order to compute constraints on cosmic string scenarios we just add the angular power spectrum to that for an simple adiabatic model – which assumes that they are uncorrelated – with the fraction of the spectrum contributed by cosmic strings being f₁₀ at ℓ = 10. This parameter is then added as an extra parameter to the standard six parameter fit using COSMOMC and the Planck likelihood described in Planck Collaboration XV (2014). We use a Flat ΛCDM cosmology defined through the physical densities of baryons, Ω_bh², and cold dark matter, Ω_ch², the acoustic scale, θ_MC, the amplitude, A_s and spectral index, n_s of density fluctuations and the optical depth to reionization τ. The Hubble constant is a derived parameter and is given by H₀ = 100 h km s^-1 Mpc^-1. We use the same priors on the cosmological and nuisance parameters as are used in Planck Collaboration XVI (2014) and use WMAP polarisation data to help fix τ. In addition to just using the Planck data, we have also added high-ℓ CMB data from SPT and ACT to obtain stronger constraint(Sievers et al. 2013; Hou et al. 2014).

For the USM-based models we use the approach used in Battye et al. (2006) and Battye & Moss (2010). We find that the constraints on the standard six parameters are not significantly affected by the inclusion of the extra string parameter and that there are no significant correlations with other parameters (see Table 3). For the case of Planck data only and using the NAMBU model we find that Gμ/c² < 1.5 × 10^-7 and f₁₀ < 0.015, whereas for the AH-mimic model we find that Gμ/c² < 3.6 × 10^-7 and f₁₀ < 0.033, with all the upper limits being at 95% confidence level. The 1D marginalised likelihoods for f₁₀ are presented in the upper panels of Fig. 8. The differences between the upper limits for the NAMBU and AH-mimic models is compatible with those seen previously using WMAP 7-year and SDSS data (Battye & Moss 2010). The upper limits from this version of the Planck likelihood are better than those computed from WMAP7+SPT (Dvorkin et al. 2011) and WMAP7+ACT (Dunkley et al. 2011) and are significantly better than those from WMAP7+SDSS (Battye & Moss 2010). Based on the Planck “Blue Book” values for noise levels we predicted (Battye et al. 2008) a limit of Gμ/c² < 6 × 10^-8, while the present limit is about a factor of two worse than this. The main reason for this is that the projected limit ignored the need for nuisance parameters to model high ℓ foregrounds and that not all the frequency channels have been used. The corresponding limits for the AH model are f₁₀ < 0.028 and Gμ/c² < 3.2 × 10^-7.

There is now very little degeneracy between the f₁₀ and n_S parameters, something that was not the case for WMAP alone (Battye et al. 2006; Bevis et al. 2008; Urrestilla et al. 2011). This has implication for supersymmetric hybrid inflation models as discussed in Battye et al. (2010) that typically require n_S> 0.98. The simplest versions of these models appear to be ruled out. The strongest correlation using the NAMBU and AH mimic models is between f₁₀ and Ω_bh² as illustrated in Fig. 10. For the AH model the correlation between f₁₀ and the amplitude of primordial perturbations, A_s, is of a comparable magnitude, for all other parameters (including the nuisance parameters) the correlations are even less important. In addition, we find in agreement with Lizarraga et al. (2012), that there are significant correlations between the amount of strings f₁₀ in the AH model and the number of relativistic degrees of freedom N_eff as well as between f₁₀ and the primordial helium abundance Y_He. We leave a detailed study of these correlations to later work.

Fig. 8 Marginalised constraints on f10 for topological defects from Planck data plus polarisation from WMAP (Planck+WP). The left panel show constraints on cosmic strings, with NAMBU in black dashed, AH-mimic in blue dotted and AH in red solid. The right panel show the constraints on SL (blue dotted) and TX (black dashed) compared to AH (again solid red).

Fig. 9 Marginalised constraints on f10 for topological defects with high-ℓ CMB data from SPT and ACT added to the Planck + WP constraints data (compare with constraints shown in Fig. 8). The left panel show constraints on cosmic strings, with NAMBU in black dashed, AH-mimic in blue dotted and AH in red solid. The right panel show the constraints on SL (blue dotted) and TX (black dashed) compared to AH (solid red).

In Fig. 8 we also present the 1D marginalised likelihoods for the texture and semilocal string models (compared to the AH field theory strings). The resulting constraints on the f₁₀ parameter are given in Table 2 as well. For the conversion into constraints on Gμ/c² we have that for semilocal strings Gμ₁₀/c² = 5.3 × 10^-6 and for global texture Gμ₁₀/c² = 4.5 × 10^-6, cf. Urrestilla et al. (2008). We notice that, as expected for a fixed Gμ, semilocal strings lead to significantly less anisotropies than cosmic strings (a factor of about 8 in the C_ℓ), and texture are similar to the semilocal strings. We thus expect significantly weaker constraints on Gμ for the SL and TX models, especially since in addition the constraints on f₁₀ for these models are weaker. Indeed we find a 95% limit of Gμ/c² < 1.10 × 10^-6 for semilocal strings and Gμ/c² < 1.06 × 10^-6 for global textures.

Fig. 10 Marginalised likelihoods in the f10-Ωbh2 plane for the NAMBU model in blue and the AH mimic model in red using Planck +WP. This is the strongest correlation with any of the standard cosmological parameters.

4. Non-Gaussian searches for cosmic strings

Cosmic strings and other topological defects generically create non-Gaussian signatures in the cosmic microwave sky, counterparts of their inevitable impact on the CMB power spectrum. This is a critical test of differentiating defects from simple inflation, while offering the prospect of direct detection. Searches for these non-Gaussian defect signatures are important for two key reasons: on the one hand, constraints from the CMB power spectrum can be susceptible to degeneracies with cosmological parameters in the standard concordance model; on the other hand, any apparent defect detection in the power spectrum should have a well-defined prediction in higher-order correlators or other non-Gaussian signals, and vice versa. Non-Gaussian tests can also be used to distinguish cosmic defects from residual foregrounds or systematic contributions. Below we will present results from NG tests that seek strings in multipole space (bispectrum) and in real space (Minkowski functionals), as well as hybrid methods (wavelets).

4.1. Foregrounds, systematics and validation

It is well-known that the microwave sky contains not only the CMB signal but also emission from different astrophysical contaminants. In particular, point source emission is expected to be a special cause of confusion for cosmic defects, notably those with high resolution signatures, such as cosmic strings. In addition, systematic effects may also be present in the maps at a certain level. Therefore, before claiming a cosmological origin of a given detection, alternative extrinsic sources should be investigated and discarded. This can be done by performing a number of consistency checks in the data, most of which are common to the other non-Gaussianity papers, where they are discussed in greater detail. Here, we provide a brief summary of the main issues.

Foreground-cleaned CMB maps are provided using four different component separation techniques (for further details, see Planck Collaboration XII 2014): SMICA (semi-blind approach); NILC (internal linear combination in needlet space);SEVEM (internal template fitting); and Commander/Ruler (C-R, parametric method). These four foreground-separation methods are complimentary in that some work in pixel space while others in harmonic space (or a combination with wavelets). They were tested using the most realistic Planck simulations available, i.e. the Full Focal Plane (FFP) version 6, including demonstration that they preserved non-Gaussian signals as we will discuss below. In determining the robustness of a particular constraint, we note that it should be replicated with at least two different cleaned CMB maps. The adoption of different masks that exclude different regions of the sky (ranging from more aggressive to more conservative) has also been used to test the stability of non-Gaussian estimators. Further tests entail the use of cleaned maps at different frequencies (for instance, those provided by the SEVEM foreground separation technique). A given detection should be consistent at all frequencies, since the behaviour of contaminants and systematic effects will, in general, vary with frequency. A further test is the study of noise maps constructed from the difference between two Planck maps (either at the same or at different frequencies) smoothed to the same resolution. These maps will not contain the CMB signal and, therefore, any NG detection should vanish on them. The opposite would indicate that the claimed result is due to foreground residuals or to the presence of systematic effects.

The methodologies described below were tested using several approaches, passing through the non-Gaussian validation suite, involving realistic FFP6 simulations, and culminating in a series of Planck String Challenges instituted for this purpose. The non-Gaussianity validation tests are described in detail elsewhere (see Planck Collaboration XXIV 2014) and the realistic FFP6 simulations also (see Planck Collaboration 2013), nevertheless we include a summary here because of their importance also for the validation of cosmic string searches. A set of 96 non-Gaussian maps, with given local, equilateral and orthogonal signals, were created using the methods described in Fergusson et al. (2010a), to which was added coloured and anisotropic noise as specified from the SMICA cleaned map. All NG estimators, notably the modal bispectrum method, were required to identify these (unknown) input signals within the expected precision. Realistic Planck FFP6 simulations, including an unknown non-Gaussian signal, were then used for the next level of validation. These simulations are intended to provide a complete description of the full Planck mission for both HFI and LFI instruments, including the best current estimates of foreground templates (CIB, CO, free-free, thermal dust etc.), the lensed primordial CMB signal (using lensPIX), timestream effects, such as satellite pointing, focal plane geometries, and ring-by-ring detector noise spectra, and finally the simulated time-ordered data was synthesised together using the Planck map-making pipeline at each frequency. The FFP6 maps were then passed through the four component-separation pipelines and NG estimators (both modal and MF) were able to detect the (unknown) f_nl = 20.4 signal (see Planck Collaboration XXIV 2014). We note the caveat that the FFP simulations continue to evolve and improve by incorporating systematic effects more completely and realistically, including “de-glitching” and polarisation effects in the near future. Given the available time and resources, the FFP6 simulation set (on which 15M core hours were expended) represents the current state-of-the-art for this first Planck defects analysis, as well as other non-Gaussian analysis.

Similar to the non-Gaussian validation suite, the Planck String Challenges were blind tests employing post-recombination string simulation maps, with an unknown Gμ/c² (as described in Sect. 2.5), co-added to a Gaussian CMB map (created using WMAP7 cosmological parameters). The effect of a 5 arcmin beam was added, together with anisotropic SEVEM noise, and the same mask was employed as for the non-Gaussian analysis (f_sky = 0.73) For calibrating the string searches, a further three string simulations were also provided (without noise). A further 1000 ΛCDM Gaussian maps with SEVEM noise and beam effects were also provided for validation and calculating linear correction terms. The second challenge incorporated improvements in the coloured SEVEM noise model and lensed Gaussian CMB maps. The aim was to determine the sensitivity of the proposed non-Gaussian tests and to see if the Gμ/c² in the challenge map could be measured accurately. The results from these challenges were an important part of the validation for each of the methods described below. In addition, Gaussian FFP6 simulations pipelines for each of the component separation methods were available for determining whether expected foreground residuals were correlated with the string signal.

4.2. Cosmic string bispectrum

4.2.1. Modal bispectrum methods

The CMB bispectrum is the three point correlator of the a_lm coefficients, ${\mathit{B}}_{{\mathit{m}}_{\mathrm{1}}{\mathit{m}}_{\mathrm{2}}{\mathit{m}}_{\mathrm{3}}}^{{\mathit{\ell }}_{\mathrm{1}}{\mathit{\ell }}_{\mathrm{2}}{\mathit{\ell }}_{\mathrm{3}}}\mathrm{=}{\mathit{a}}_{{\mathit{\ell }}_{\mathrm{1}}{\mathit{m}}_{\mathrm{1}}}{\mathit{a}}_{{\mathit{\ell }}_{\mathrm{2}}{\mathit{m}}_{\mathrm{2}}}{\mathit{a}}_{{\mathit{\ell }}_{\mathrm{3}}{\mathit{m}}_{\mathrm{3}}}$. For the purposes of a search for cosmic strings we assume the network cumulatively creates a statistically isotropic signal, that is, we can employ the angle-averaged reduced bispectrum b_ℓ1ℓ2ℓ3, defined by $\begin{array}{ccc}{\mathit{b}}_{{\mathit{\ell }}_{\mathrm{1}}{\mathit{\ell }}_{\mathrm{2}}{\mathit{\ell }}_{\mathrm{3}}}& \mathrm{=}& \sum _{{\mathit{m}}_{\mathit{i}}}{\mathit{h}}_{{\mathit{\ell }}_{\mathrm{1}}{\mathit{\ell }}_{\mathrm{2}}{\mathit{\ell }}_{\mathrm{3}}}^{-2}{\mathrm{𝒢}}_{{\mathit{m}}_{\mathrm{1}}{\mathit{m}}_{\mathrm{2}}{\mathit{m}}_{\mathrm{3}}}^{{\mathit{\ell }}_{\mathrm{1}}{\mathit{\ell }}_{\mathrm{2}}{\mathit{\ell }}_{\mathrm{3}}}{\mathit{B}}_{{\mathit{m}}_{\mathrm{1}}{\mathit{m}}_{\mathrm{2}}{\mathit{m}}_{\mathrm{3}}}^{{\mathit{\ell }}_{\mathrm{1}}{\mathit{\ell }}_{\mathrm{2}}{\mathit{\ell }}_{\mathrm{3}}}\mathit{,}\end{array}$(12)where h_ℓ1ℓ2ℓ3 is a weakly scale-dependent geometrical factor and ${\mathrm{𝒢}}_{{\mathit{m}}_{\mathrm{1}}{\mathit{m}}_{\mathrm{2}}{\mathit{m}}_{\mathrm{3}}}^{\hspace{0.17em}\hspace{0.17em}{\mathit{l}}_{\mathrm{1}}{\mathit{l}}_{\mathrm{2}}{\mathit{l}}_{\mathrm{3}}}$ is the well-known Gaunt integral over three Y_ℓms that can be expressed in terms of Wigner-3j symbols. The CMB bispectrum b_ℓ1ℓ2ℓ3 is defined on a tetrahedral domain of multipole triples { ℓ₁ℓ₂ℓ₃ } satisfying both a triangle condition and ℓ<ℓ_max set by the experimental resolution. When seeking the string bispectrum ${\mathit{b}}_{{\mathit{\ell }}_{\mathrm{1}}{\mathit{\ell }}_{\mathrm{2}}{\mathit{\ell }}_{\mathrm{3}}}^{\mathrm{string}}$ in the Planck data, we employ the following estimator to find or limit its amplitude: $\begin{array}{ccc}\mathrm{ℰ}\mathrm{=}\frac{\mathrm{1}}{\begin{array}{c}\mathrm{˜}\\ {\mathit{N}}^{\mathrm{2}}\end{array}}\sum _{{\mathit{l}}_{\mathit{i}}{\mathit{m}}_{\mathit{i}}}\frac{{\mathrm{𝒢}}_{{\mathit{m}}_{\mathrm{1}}{\mathit{m}}_{\mathrm{2}}{\mathit{m}}_{\mathrm{3}}}^{{\mathit{\ell }}_{\mathrm{1}}{\mathit{\ell }}_{\mathrm{2}}{\mathit{\ell }}_{\mathrm{3}}}\hspace{0.17em}{\mathit{b}}_{{\mathit{\ell }}_{\mathrm{1}}{\mathit{\ell }}_{\mathrm{2}}{\mathit{\ell }}_{\mathrm{3}}}^{\mathrm{string}}}{\begin{array}{c}\mathrm{˜}\\ {\mathit{C}}_{{\mathit{\ell }}_{\mathrm{1}}}\end{array}\begin{array}{c}\mathrm{˜}\\ {\mathit{C}}_{{\mathit{\ell }}_{\mathrm{2}}}\end{array}\begin{array}{c}\mathrm{˜}\\ {\mathit{C}}_{{\mathit{\ell }}_{\mathrm{3}}}\end{array}}\hspace{0.17em}{\mathit{a}}_{{\mathit{\ell }}_{\mathrm{1}}{\mathit{m}}_{\mathrm{1}}}{\mathit{a}}_{{\mathit{\ell }}_{\mathrm{2}}{\mathit{m}}_{\mathrm{2}}}{\mathit{a}}_{{\mathit{\ell }}_{\mathrm{3}}{\mathit{m}}_{\mathrm{3}}}\mathit{,}& & \end{array}$(13)where we assume a nearly diagonal covariance matrix C_{ℓ1m1,ℓ2m2} ≈ C_ℓδ_ℓ1ℓ2δ_{m1 − m2} and we modify C_ℓ and b_ℓ1ℓ2ℓ3 appropriately to incorporate instrument beam and noise effects, as well as a cut-sky. To simplify Eq. (14), we have ignored the “linear term” (which is included in the analysis). A much more extensive introduction to bispectrum estimation can be found in Planck Collaboration XXIV (2014).

A key step in observational searches for non-separable bispectra, such as those induced by cosmic strings (denoted by ${\mathit{b}}_{{\mathit{\ell }}_{\mathrm{1}}{\mathit{\ell }}_{\mathrm{2}}{\mathit{\ell }}_{\mathrm{3}}}^{\mathrm{string}}$), is to expand it into separable modes (Fergusson & Shellard 2009; Fergusson et al. 2010a) taking the signal-to-noise-weighted form: $\begin{array}{ccc}\frac{{\mathit{b}}_{{\mathit{\ell }}_{\mathrm{1}}{\mathit{\ell }}_{\mathrm{2}}{\mathit{\ell }}_{\mathrm{3}}}^{\mathrm{string}}}{\sqrt{{\mathit{C}}_{{\mathit{\ell }}_{\mathrm{1}}}{\mathit{C}}_{{\mathit{\ell }}_{\mathrm{2}}}{\mathit{C}}_{{\mathit{\ell }}_{\mathrm{3}}}}}\hspace{0.17em}\mathrm{=}\sum _{\mathit{n}}{\mathit{\alpha }}_{\mathit{n}}^{\mathit{Q}}\hspace{0.17em}{\mathit{Q}}_{\mathit{n}}\mathrm{\left(}{\mathit{\ell }}_{\mathrm{1}}\mathit{,}{\mathit{\ell }}_{\mathrm{2}}\mathit{,}{\mathit{\ell }}_{\mathrm{3}}\mathrm{\right)}\mathit{,}& & \end{array}$(14)where the modes ${\mathit{Q}}_{\mathit{n}}\mathrm{\left(}{\mathit{\ell }}_{\mathrm{1}}\mathit{,}{\mathit{\ell }}_{\mathrm{2}}\mathit{,}{\mathit{\ell }}_{\mathrm{3}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{6}}\mathrm{\left[}\mathit{q̅}{}_{\mathit{p}}\mathrm{\right(}{\mathit{l}}_{\mathrm{1}}\mathrm{)}\hspace{0.17em}\mathit{q̅}{}_{\mathit{r}}\mathrm{(}{\mathit{l}}_{\mathrm{2}}\mathrm{)}\hspace{0.17em}\mathit{q̅}{}_{\mathit{s}}\mathrm{(}{\mathit{l}}_{\mathrm{3}}\mathrm{)}\mathrm{+}\mathrm{perms}\mathrm{.}\mathrm{]}$ are constructed from symmetrized products (the n label a distance-ordering for the triples { prs }). The product basis functions Q_n(ℓ₁,ℓ₂,ℓ₃) are not in general orthogonal, so it is very useful to construct a related set of orthonormal mode functions R_n(ℓ₁,ℓ₂,ℓ₃) such that ⟨ R_n,R_p ⟩ = δ_np. Substituting the separable mode expansion (15) reduces the estimator (14) to the simple form $\begin{array}{ccc}\mathrm{ℰ}\mathrm{=}\frac{\mathrm{1}}{{\mathit{N}}^{\mathrm{2}}}\sum _{\mathit{n}}{\mathit{\alpha }}_{\mathit{n}}^{\mathit{Q}}\hspace{0.17em}{\mathit{\beta }}_{\mathit{n}}^{\mathit{Q}}\mathit{,}& & \end{array}$(15)where the ${\mathit{\beta }}_{\mathit{n}}^{\mathit{Q}}$ coefficients are found by integrating products of three Planck maps filtered using the basis function ${\mathit{\alpha }}_{\mathit{n}}^{\mathit{Q}}$ (an efficient product with each map multiplied by the separable q_r(ℓ)). We can validate this estimator by using the modal methodology to create CMB map realisations for cosmic strings from the predicted ${\mathit{\beta }}_{\mathit{n}}^{\mathit{Q}}$ with a given Gμ/c² (see Fergusson et al. 2010a). It is easy to show that the expectation value for ${\mathit{\beta }}_{\mathit{n}}^{\mathit{R}}$ for such an ensemble of maps in the orthogonal basis should be $\begin{array}{ccc}\mathrm{⟨}{\mathit{\beta }}_{\mathit{n}}^{\mathit{R}}\mathrm{⟩}\mathrm{=}{\mathit{\alpha }}_{\mathit{n}}^{\mathit{R}}\hspace{0.17em}\mathit{.}& & \end{array}$(16)Alternatively, we can exploit this fact by reconstructing the ${\mathit{\alpha }}_{\mathit{n}}^{\mathit{R}}$ from given CMB map realisations created directly from string simulations, an approach we will adopt here.

4.2.2. Post-recombination string bispectrum

Fig. 11 Coefficients ${\mathit{\alpha }}_{\mathit{n}}^{\mathit{R}}$ Eq. (15) for the hybrid Fourier mode expansion of the cosmic string bispectrum Eq. (15). The average value ${\mathit{\alpha }}_{\mathit{n}}^{\mathit{R}}\mathrm{=}\mathrm{⟨}{\mathit{\beta }}_{\mathit{n}}^{\mathit{R}}\mathrm{⟩}$ (black line) is in remarkable agreement with all four string simulations as can be seen for n < 30 (lower panel), with each exhibiting better than a 97% correlation overall.

In order to estimate the string bispectrum at Planck resolution we employed the modal reconstruction method Eq. (17) on the post-recombination string simulations described in Sect 2.5. These string maps include the accumulated line-like discontinuities induced by the string network on CMB photons propagating from the surface of last scattering to the present day. This work does not include recombination physics, that is, contributions from the surface of last scattering that will increase the string anisotropy signal substantially. As discussed previously, there are four full-sky maps of two different resolutions, which were provided for the purpose of calibrating Planck searches for cosmic strings (Ringeval & Bouchet 2012). For the modal analysis, we have adopted the hybrid polynomial basis augmented with a local shape mode (in total with n_max = 600 modes), as well as a hybrid Fourier basis (n_max = 300), which are both described in Planck Collaboration XXIV (2014).

To extract the string bispectrum in a Planck-realistic context, we chose a fairly high non-Gaussian signal with Gμ/c² = 1 × 10^-6. As for the Planck String Challenges, described in Sect. 4.1, the normalised string maps were added to noise maps generated by the component separation pipelines of SMICA, NILC and SEVEM, creating twelve sets of 200 simulated string realisations. Each of these maps was then filtered using the modal estimator to find the ${\mathit{\beta }}_{\mathit{n}}^{\mathit{R}}$ coefficients appropriate for each component-separation method. After averaging each set of modal coefficients ${\mathit{\alpha }}_{\mathit{n}}^{\mathit{R}}\mathrm{=}\mathrm{⟨}{\mathit{\beta }}_{\mathit{n}}^{\mathit{R}}\mathrm{⟩}$ over the different (unmasked) noise realisations, we found remarkable consistency between the estimated ${\mathit{\beta }}_{\mathit{n}}^{\mathit{R}}$ for the four string simulations as shown in Fig. 11 for the Fourier modes. Agreement was good across all the 300 ${\mathit{\alpha }}_{\mathit{n}}^{\mathit{R}}$ modes determined, as shown in detail for n = 1−30 (see the lower panel of Fig. 11).

Quantitatively, the different string simulations produced bispectrum shapes that had above 97% correlations with each other (i.e., ${\sum }_{\mathit{n}}{\mathit{N}}^{-1}{\mathit{\alpha }}_{\mathit{n}}^{\mathrm{\left(}\mathrm{1}\mathrm{\right)}}\hspace{0.17em}{\mathit{\alpha }}_{\mathit{n}}^{\mathrm{\left(}\mathrm{2}\mathrm{\right)}}\mathit{>}\mathrm{0.97}$ for ${\mathit{N}}^{\mathrm{2}}\mathrm{=}{\sum }_{\mathit{n}}{\mathit{\alpha }}_{\mathit{n}}^{\mathrm{\left(}\mathrm{1}\mathrm{\right)}\hspace{0.17em}\mathrm{2}}\hspace{0.17em}{\sum }_{\mathit{n}}{\mathit{\alpha }}_{\mathit{n}}^{\mathrm{\left(}\mathrm{2}\mathrm{\right)}\hspace{0.17em}\mathrm{2}}$). The overall integrated bispectrum amplitudes was consistent to within 4%. Despite only four string map simulations, these are small errors relative to experimental and theoretical uncertainties. This robustness indicates that the overall string bispectrum signal at Planck resolution is a statistical summation of very many contributions from the millions of strings between the observer and the last scattering surface. To ensure the bispectrum C_l weighting was not significantly affected by the presence of the large string signal, we repeated the modal extraction procedure for Gμ/c² = 5 × 10^-7 (the string bispectrum amplitude reduced by a factor of 8). For the same string simulation, the shape correlations for different Gμ/c² were 99.4% or above and the amplitude scaled as expected with (Gμ/c²)³ to within 2%. The string bispectrum shown in Fig. 11 is well converged with random errors from the averaging procedure small relative to the actual signal. We conclude that, assuming the physics and numerical accuracy of the string simulations that are available, we have extracted a string bispectrum of sufficient accuracy for the present non-Gaussian analysis.

The overall three-dimensional reconstruction of the string bispectrum shape is shown in Fig. 12, normalised in the usual way to approximately illustrate the signal-to-noise expected (that is, removing an overall l^-4 scaling, by dividing by the constant Sachs-Wolfe bispectrum shape). The first point to note is that the bispectrum is negative, reflecting the underlying string velocity correlations and curvature correlations that have created it; in the expanding universe, curved strings preferentially collapse, creating a negative temperature fluctuation towards the centre and a positive signal outside. In the overall spectrum, the n = 0 mode is dominant, but it is modulated by other modes providing further interesting structure that could be described as broad arms extending along each axis (see Fig. 11); although somewhat “squeezed”, the correlation with the local model is low. The string simulation power spectrum shown in Fig. 5 can be understood to be quantitatively modulating the string bispectrum away from the axes, with the signal slowly decaying beyond (say) l₁,l₂>500 in the l₃ direction.

The CMB bispectrum and trispectrum induced by the post-recombination gravitational effects of cosmic strings have been estimated analytically (Hindmarsh et al. 2009, 2010; Regan & Shellard 2010). With simplifying assumptions, these predicted that the constant mode would be dominant with a broad central “equilateral” peak, but not the substructure observed in Fig. 12. In terms of missing physics in this post-recombination string bispectrum, we expect the recombination signal to lie in the range ℓ = 200−1000 (shown in the full NAMBU power spectrum in Fig. 3) and to significantly enhance the overall amplitude of the bispectrum (see also Landriau & Shellard 2011, where recombination physics is included).

The correlation of the post-recombination string bispectrum with standard primordial shapes is small, because it does not contain an oscillatory component from the transfer functions. The local, equilateral and orthogonal bispectrum models correlate with strings at about 6%, 11% and 12%, respectively. There is also a CMB ISW-lensing bispectrum ${\mathit{b}}_{{\mathit{\ell }}_{\mathrm{1}}{\mathit{\ell }}_{\mathrm{2}}{\mathit{\ell }}_{\mathrm{3}}}^{\mathrm{ISW}}$ arising from the correlation between gravitational lensing and the late-time integrated Sachs-Wolfe effect arising from the underlying dominant Gaussian fluctuations (see, for example, the discussion in Planck Collaboration XXVI 2014). Like the string bispectrum, the ISW-lensing bispectrum is also mainly negative, but it is much more squeezed/flattened and correlates at only about 11% with the present string bispectrum. So the predicted ISW signal should provide a small positive bias for string detection of about 0.44σ. Diffuse point-source contamination is of greater concern because at ℓ_max = 2000 this has a 40% anti-correlation with strings (for the simple Poisson-distributed point source template with ${\mathit{b}}_{{\mathit{\ell }}_{\mathrm{1}}{\mathit{\ell }}_{\mathrm{2}}{\mathit{\ell }}_{\mathrm{3}}}^{\mathrm{PS}}$= constant). This close relationship with point sources requires a joint analysis (see below). Other foreground contamination must also be considered, as we shall discuss, and for this we rely on realistic simulated FFP6 foreground residuals provided by the Planck component separation pipelines (see Planck Collaboration XII 2014).

Fig. 12 Modal reconstruction of the post-recombination string bispectrum Eq. (15) extracted from Planck resolution map simulations. This is a 3D view of the allowed tetrahedral set of multipoles (ℓ1,ℓ2,ℓ3) showing isosurfaces of the bispectrum density with darker blue for more negative values (it is normalised relative to the constant SW bispectrum).

4.2.3. Planck string bispectrum results

Using the modal bispectrum estimator, we have searched for the string bispectrum in the Planck CMB maps obtained using the foreground-separation techniques SMICA, NILC and SEVEM. We note that the modal estimator has passed through the full validation suite of NG tests described in the Planck Collaboration XXIV (2014), where further details about the analysis can be found. In summary, we have used the standard U73 mask, which includes a Galactic cut and a conservative point source mask, together with “inpainting” as in the f_NL analysis (essentially apodizing the mask). Together with the SMICA, NILC and SEVEM foreground separated maps, realistic noise simulations as specified for the particular foreground-cleaned map, were used to determine the estimator’s linear correction term and to determine the bispectrum variance, which was very nearly optimal. For calibration purposes, we always compare to the string model with tension Gμ/c² = 1 × 10^-6 defining a string bispectrum parameter and normalising it to have f_NL = 1 in this case. The standard deviation Δf_NL = 0.2 obtained from this Planck analysis would imply a 5σ detection of Gμ/c² = 1 × 10^-6 strings. The strong scaling of the bispectrum amplitude on the string tension ∝ (Gμ/c²)³, implies a given measurement yields Gμ/c² = (f_NL)^1/3 × 10^-6.

The results of the string bispectrum estimation for each of the SMICA, NILC and SEVEM maps are shown in Table 4. As shown by the non-Gaussian analysis (see Planck Collaboration XXIV 2014), the Planck data exhibits significant detections of both the predicted ISW-lensing bispectrum (above 2σ) and a residual point source signal (over 3σ). Since these may be confused with the string signal, we also quote their measurements in this Fourier mode basis. The ISW-lensing signal detection is described at length in Planck Collaboration XXVI 2014. The residual point source (PS) signal is discussed in Planck Collaboration XXIV 2014 where it is modelled as a simple Poissonian distribution. In principle, the PS amplitude can be extrapolated from source count models, but direct comparisons remain a goal for future Planck analysis. An independent analysis of the ISW-lensing, point source and string contributions to each map showed no evidence for a cosmic string signal (all estimates were were within 1σ, refer to the third column of Table 4). However, given the large contribution of diffuse point sources and their anti-correlation with the string bispectrum, we have also undertaken a joint analysis of the Planck data (which in this case is the same as marginalising over the point source signal). Before doing so, we subtract the expected ISW lensing signal that provides a f_NL = 0.09 string bias. The fifth column in Table 4 shows that marginalising point sources enhances the string signal up to the 2σ level for all component separation methods; essentially the constant mode becomes more strongly negative once the measured point sources are removed.

Other foregrounds, such as dust emission, could potentially produce spurious string signals if not subtracted properly. Foreground contamination has been extensively studied within the Planck simulation pipeline (FFP6) and foreground residual maps have been provided by each component separation team (including those for thermal dust, CO emission, free-free, synchrotron, spinning dust, kinetic and thermal SZ etc). We have analysed these residual FFP6 maps provided by both SMICA and NILC to seek evidence of correlations with the string bispectrum; individually these foregrounds make only small contributions (apart from PS), but cumulatively their effect cannot be neglected. The SMICA combined-residual map, without point sources and analysed with realistic noise, produces a string bias of f_NL = 0.23, which after ISW subtraction becomes f_NL = 0.14 (relative to a variance Δf_NL = 0.20). After both ISW and foreground residual subtraction, a joint analysis with point sources yields a SMICA string signal f_NL = 0.37 ± 0.21 (Col. 5 in Table 4). The apparent string bias in NILC from residual foregrounds was even higher f_NL = 0.22 (after ISW subtraction), meaning a joint analysis obtained f_NL = 0.23 ± 0.21 (see Table 4).

We conclude, given our present understanding of point sources and foregrounds, that there does not appear to be significant evidence for a string bispectrum signal in the Planck nominal mission maps, so we infer the following post-recombination bispectrum constraint on strings: $\begin{array}{ccc}\mathit{G\mu }\mathit{/}{\mathit{c}}^{\mathrm{2}}\mathit{<}\mathrm{9.0}\mathrm{\times }{\mathrm{10}}^{-7} \mathrm{\left(}\mathrm{95}\mathrm{\%confidence}\mathrm{\right)}\hspace{0.17em}\mathit{.}& & \end{array}$(17)Here, we have taken the more conservative SMICA result f_NL = 0.37 ± 0.21, because of its preferred status for foreground separation techniques in Planck Collaboration XII (2014). The susceptibility of the string bispectrum to point source and other foreground contamination deserves further investigation and will require improved characterisation of the diffuse point source bispectrum (beyond the simple Poisson model), as well as identification of the specific foreground residuals which primarily contribute to the small string bias found in the FFP6 simulations.

The string bispectrum constraint Eq. (18) is a conservative upper limit on the string tension Gμ/c² because we have not included recombination contributions. Although this constraint is weaker than that from the power spectrum, it is an independent test for strings and the first quantitative string bispectrum limit to date. This should be considerably improved in future by inclusion of recombination physics and more precise foreground analysis. A comparison with the power spectrum amplitude indicates the string bispectrum should rise by (2)^3/2, which, together with the full mission data, would see the sensitivity improve by a factor of two (allowing constraints around Gμ/c² < 4 × 10^-7). We note that the bispectrum is not the optimal non-Gaussian test for strings, because the string signal is somewhat suppressed by symmetry (the bispectrum cancels for straight strings). This fact motivates further study of the trispectrum, for which the Planck sensitivity is forecast to be ΔGμ/c² ≈ 1 × 10^-7 (Fergusson et al. 2010b), as well as joint analysis of polyspectra.

4.3. Steerable wavelet searches for cosmic strings

Wavelets offer a powerful signal analysis tool due to their ability to localise signal content in scale (cf. frequency) and position simultaneously. Consequently, wavelets are well-suited for detecting potential CMB temperature contributions due to cosmic strings, which exhibit spatially localised signatures with distinct frequency content. Wavelets defined on the sphere are required to analyse full-sky Planck observations (see, for example, Freeden & Windheuser 1997; Wiaux et al. 2005; Sanz et al. 2006; McEwen et al. 2006; Starck et al. 2006; Marinucci et al. 2008; Wiaux et al. 2008a).

We perform an analysis using the steerable wavelets on the sphere constructed by Wiaux et al. (2005). Here we exploit steerability to dynamically adapt the orientations analysed to the underlying data, performing frequentist hypothesis testing. We apply the first (1GD) and second (2GD) Gaussian derivative steerable wavelets, defined on the sphere through a stereographic projection, in order to search for cosmic strings in the Planck data. A steerable wavelet is a directional filter whose rotation by χ ∈ [ 0,2π) about itself can be expressed in terms of a finite linear combination of non-rotated basis filters. Thus, the analysis of a signal with a given steerable wavelet Ψ naturally identifies a set of wavelet coefficients, W_Ψ(ω₀,χ,R), which describe the local features of the signal at each position ω₀ on the sphere, for each orientation χ and for each physical scale R. Several local morphological properties can be defined in terms of the wavelet coefficients (Wiaux et al. 2008b), including the signed-intensity, $\mathit{I}\mathrm{(}{\mathit{\omega }}_{\mathrm{0}}\mathit{,}{\mathit{R}}^{\mathrm{)}}\mathrm{\equiv }{\mathit{W}}_{\mathrm{\Psi }}\left({\mathit{\omega }}_{\mathrm{0}}\mathit{,}{\mathit{\chi }}_{\mathrm{0}}\mathit{,R}\right)\hspace{0.17em}\mathit{.}$(18)This quantity represents the value of the wavelet coefficient at the local orientation ${\mathit{\chi }}_{\mathrm{0}}{}^{\mathrm{(}}\mathit{\omega }_{\mathrm{0}}\mathit{,}{\mathit{R}}^{\mathrm{)}}$ that maximizes the absolute value of the wavelet coefficient itself. Let us emphasise that the signed-intensity morphological property is a highly non-linear quantity, i.e., ${\mathit{I}}_{\mathit{a}\hspace{0.17em}\mathrm{+}\hspace{0.17em}\mathit{b}}{}^{\mathrm{(}}\mathit{\omega }_{\mathrm{0}}\mathit{,}{\mathit{R}}^{\mathrm{)}}\ne {\mathit{I}}_{\mathit{a}}{}^{\mathrm{(}}\mathit{\omega }_{\mathrm{0}}\mathit{,}{\mathit{R}}^{\mathrm{)}}\mathrm{+}{\mathit{I}}_{\mathit{b}}{}^{\mathrm{(}}\mathit{\omega }_{\mathrm{0}}\mathit{,}{\mathit{R}}^{\mathrm{)}}$, where a and b are two signals on the sphere.

Fig. 13 Deviation of the kurtosis of the signed-intensity as a function of Gμ/c2, normalised to the standard deviation of CMB and noise simulations: ΔK/σ = (K(R,Gμ/c2) − K(R,Gμ/c2 = 0)) /σ. The left panel shows results for the 1GD wavelet and the right shows the 2GD wavelet. Each curve corresponds to a wavelet scale, R (arcmin), included in the analysis. The final sensitivity of the method is determined by combining the two wavelets and all the scales.

The presence of a cosmic string signal in the CMB is expected to leave a non-Gaussian signature that induces a modification in the distribution of I(ω₀,R) with respect to the lensed Gaussian case. We calibrated the dependence of these signatures on the string tension using four simulations of the cosmic string contribution (Ringeval & Bouchet 2012) combined with a set of lensed Gaussian CMB realisations, along with a realistic description of the Planck instrumental properties (refer to Planck Collaboration 2013).

A wide range of string tension values were explored, Gμ/c² ∈ [ 2.0 × 10^-7,1.0 × 10^-6 ], considering several wavelet scales, R = [4.0,4.5,5.0,6.0,8.0,10.0,15.0,20.0] arcmin. We use maps at a HEALPix resolution of N_side = 2048, including multipoles up to ℓ_max = 2500. We analyse the simulations with the same U73 mask of the Planck CMB map (refer to Planck Collaboration XII 2014), which masks both diffuse and compact foregrounds, leaving 73% of the sky remaining for further analysis (refer to discussion in Sect. 4.2.3). When computing the wavelet coefficients of a masked data set, artefacts are introduced close to the mask borders. We therefore define a set of exclusion masks such that, at each scale R, an extra region of the sky is excluded when computing any statistical measure. At each scale the exclusion mask is obtained by expanding the mask borders by three times the width of the corresponding wavelet.

The string non-Gaussian signatures are characterised in terms of the kurtosis of the signed-intensity I(ω₀,R) in Eq. (19) at the different scales R and for both the 1GD and 2GD wavelets. The averaged results from the non-Gaussian simulations were used to model the distribution of the kurtosis as functions of Gμ/c², i.e., K(R,Gμ/c²). Other statistics, such as the skewness and the Higher-Criticism (e.g., Donoho & Jin 2004; Cayón et al. 2005), have also been explored. We found that the kurtosis sensitivity to the string tension is higher than the alternative measures. In Fig. 13 we show the difference between the average kurtosis at several Gμ/c² values and the average kurtosis for Gμ/c² = 0, normalised to the standard deviation of the simulations. On the given range of scales, the 2GD wavelet appears to be more sensitive to the string signal. The final sensitivity of the method in recovering the string tension can be assessed from simulated data. We compare each simulation to the averaged model through a χ² test, thus estimating Gμ/c² for the simulation. The test is performed jointly on the two wavelets for all the scales, and taking into account the correlations by means of a covariance matrix estimated from CMB and noise simulations. From the distribution of the Gμ/c² values recovered from the simulations the null hypothesis can be rejected at 95% CL for Gμ/c²> 7 × 10^-7.

The same χ² test was applied to the SEVEM CMB cleaned map, observing no evidence of a string signal at any of the scales studied. Despite the absence of strings, however, we noted that the χ² of the best fit solution was higher than anticipated, corresponding to a probability to exceed (PTE) of 2%. We identified the cause for such a poor fit to the null result as an incompatibility between the data and simulations at the smallest scales. We note that this incompatibility is not related to the potential presence of a string signal, since it biases the non-Gaussian estimators towards the opposite direction, as reflected also in the χ² values. Similar behaviour is also noticed for the skewness of $\mathit{I}{}^{\mathrm{(}}\mathit{\omega }_{\mathrm{0}}\mathit{,}{\mathit{R}}^{\mathrm{)}}$ and, to a lesser extent, for the Higher-Criticism measure.

We have extensively explored the causes of this incompatibility at small scales. First of all, we verified that such a behaviour is present also in the SMICA CMB solution, as well as SEVEM. We then analysed the half-ring half-difference SEVEM CMB maps, and we compared it to pure noise simulations. Indeed, half-ring maps are made from the data in the first half or second half of each stable pointing period, therefore we can obtain an estimate of the noise in the data by taking half of the difference between the two maps. We conclude that the half-ring noise estimate and noise simulations look compatible. We evaluated the impact on the estimator of unresolved point sources, both from radio galaxies and sub-millimetre star-forming galaxies, using simulations of these astrophysical components processed through the component separation pipelines (Planck Collaboration XII 2014). We found that at a wavelet scale of R = 4 arcmin, these residual foregrounds induce a shift in the kurtosis, ΔK/σ = 0.03, i.e., a bias that is negligible for the present analysis. This shift only increases to more than ΔK/σ = 0.3 when extending the analysis to wavelet scales as small as R = 2.5 arcmin. Finally, we also studied the SEVEM cleaned maps at 143 and 217 GHz, for which we considered wider galactic cuts (mask G56 and G35 from Planck Collaboration XV 2014). Since the inconsistency does not show a significant dependence on the frequency, there is no clear evidence for residual foreground contamination playing a major role.

At this point, it is worth stressing two important properties of the non-Gaussian statistics considered in this section. On the one hand, as mentioned above, it is a very non-linear estimator and, on the other, it depends on the local orientation of the sky signal. In fact, it is the only estimator considered in this paper that has such orientation-dependence. This is a critical point, since it implies that the simulations (lensed CMB, noise and foreground residuals) have to be more sophisticated if they are to capture the corresponding local anisotropic characteristics. This is especially difficult to achieve for the residual foregrounds, which are in part based on foreground models with small-scale fluctuations produced through isotropic and Gaussian extrapolations. As a consequence, we cannot discard completely either instrumental or foreground systematics at the smallest scales causing these incompatibilities. Nevertheless, we note that this lack of anisotropic modelling is not crucial for other estimators, as the angular power spectrum, or even the bispectrum, because these are isotropic quantities.

For the reasons given above, we have restricted our steerable wavelets analysis to scales R ≥ 6 arcmin, where simulations are better understood for our specific purpose and where they appear compatible with the data. Analysing SEVEM on this angular range we find Gμ/c² < 7 × 10^-7 at 95% CL. The next Planck release will continue to improve the modelling of instrumental and foreground properties and, therefore, we expect to be able to fully exploit the steerable wavelet capabilities down to the smallest scales (R = 2.5 arcmin). Based on simulations, this can significantly improve the sensitivity of the estimator, allowing constraints of Gμ/c² < 4 × 10^-7 at 95% CL.

Finally, we note that we have also endeavoured to study the simulated string maps using spherical wavelets, making an extension of previous work (Wiaux et al. 2008a; Hammond et al. 2009) to compute the Bayesian posterior distribution of the string tension. Both the spherical and steerable wavelet methods offer good prospects for improved non-Gaussian string constraints from the Planck full mission data.

4.4. Real space tests for cosmic strings

4.4.1. Minkowski functionals method

Minkowski Functionals (MFs) describe morphological properties of the CMB field, and can be used as generic estimators of non-Gaussianities (Planck Collaboration XXIII 2014). However, they have been considered to constrain specific NG models such as cosmic strings, for example on gradient temperature maps (see e.g., Gott et al. 1990). Indeed they have sensitivity to non-Gaussianity sourced by strings at all orders (i.e., including the kurtosis or trispectrum) and they could prove to be a powerful tool to constrain topological defects in general. For the sake of brevity and conciseness, precise definitions of MFs and analytic formulations are presented in Planck Collaboration XXIII (2014) and, here, we only review how MFs can be used to constrain the string energy density Gμ/c². We follow the Bayesian method discussed in Ducout et al. (2012) for the local model ${\mathit{f}}_{\mathrm{NL}}^{\mathrm{local}}$ but which can quite similarly be applied to the string case.

We measure the four normalised² functionals v_k(k = 0,3) (respectively Area, Perimeter, Genus and N_cluster), computed on n_th = 26 thresholds ν, between ν_min = −3.5 and ν_max = + 3.5 in units of the standard deviation of the map. The four functionals are similarly sensitive to cosmic strings and we build a statistics using the combination of all functionals, forming one vector y (of size n = 104).

The principle of this Bayesian method is to compare measurement of MFs on Planck data ŷ to the MFs model curve for strings at the level Gμ/c², y(Gμ/c²). The Bayes formula is $\mathit{P}\mathrm{(}\mathit{G\mu }\mathit{/}{\mathit{c}}^{\mathrm{2}}\mathrm{|}\hspace{0.17em}\mathit{ŷ}\mathrm{)}\mathrm{=}\frac{\mathit{P}\mathrm{(}\mathit{ŷ}\hspace{0.17em}\mathrm{|}\mathit{G\mu }\mathit{/}{\mathit{c}}^{\mathrm{2}}\mathrm{\left)}\mathit{P}\mathrm{\right(}\mathit{G\mu }\mathit{/}{\mathit{c}}^{\mathrm{2}}\mathrm{)}}{\int \mathit{P}\mathrm{(}\mathit{ŷ}\hspace{0.17em}\mathrm{|}\mathit{G\mu }\mathit{/}{\mathit{c}}^{\mathrm{2}}\mathrm{\left)}\mathit{P}\mathrm{\right(}\mathit{G\mu }\mathit{/}{\mathit{c}}^{\mathrm{2}}\mathrm{)}\mathrm{d}\mathit{G\mu }\mathit{/}\mathit{c}2}\mathrm{·}$(19)We take a flat prior for the parameter Gμ/c², with P(Gμ/c²) being a constant over a reasonable range of values for Gμ/c², mostly determined from previous experiments (between 0 and 1e-6). The evidence ^∫P(ŷ | Gμ/c²)P(Gμ/c²)dGμ/c² is being considered as a normalisation factor.

The likelihood P(ŷ | Gμ/c²) is a multivariate Gaussian: ν_max was chosen not too extreme, to avoid sensitivity to rare events, and the map’s filtering allows to probe a sufficient number of independent harmonic modes to keep fluctuations small. The likelihood’s Gaussianity has also been verified systematically at each point, using simulations. More details on the likelihood are discussed in Gott et al. (1990); Ducout et al. (2012). This allows us to define a simple χ² test for Gμ/c² and the posterior becomes $\mathit{P}\mathrm{(}\mathit{G\mu }\mathit{/}{\mathit{c}}^{\mathrm{2}}\mathrm{|}\hspace{0.17em}\mathit{ŷ}\mathrm{)}\mathrm{\propto }\exp \left[\mathrm{-}\frac{{\mathit{\chi }}^{\mathrm{2}}\mathrm{(}\mathit{ŷ}\mathit{,G\mu }\mathit{/}{\mathit{c}}^{\mathrm{2}}\mathrm{)}}{\mathrm{2}}\right]\mathit{,}$(20)with ${\mathit{\chi }}^{\mathrm{2}}\mathrm{(}\mathit{ŷ}\mathit{,G\mu }\mathit{/}{\mathit{c}}^{\mathrm{2}}\mathrm{)}\mathrm{\equiv }{\mathrm{[}\mathit{ŷ}\mathrm{-}\mathit{y}\mathrm{(}\mathit{G\mu }\mathit{/}{\mathit{c}}^{\mathrm{2}}{\mathrm{\left)}}^{\mathrm{\right]}}}^{\mathit{T}}{\mathit{C}}^{-1}\mathrm{[}\mathit{ŷ}\mathrm{-}\mathit{y}\mathrm{(}\mathit{G\mu }\mathit{/}{\mathit{c}}^{\mathrm{2}}{\mathrm{\left)}}^{\mathrm{\right]}}\mathit{.}$(21)The covariance matrix C is computed from 10⁴ Gaussian simulations³: to verify the Gaussian approximation, tests have been performed using non-Gaussian matrices over a dozen values of Gμ/c² and covariance matrices and χ² remained constant within numerical precision; furthermore the number of Gaussian simulations used to estimate the covariance matrix guarantees the stability of the statistics for these threshold values (see convergence tests in Ducout et al. 2012).

The cosmic string model curve is calibrated on 10³ realistic lensed Planck simulations, to which we have added a string component at a specified level. These simulations take into account the asymmetry of beams and the component separation process (FFP6 simulations, see Planck Collaboration 2013, for a detailed description). For the string component, we had at our disposal only two high resolution string map simulations (Ringeval & Bouchet 2012), so our model is the averaged curve obtained from this combination of Planck and string simulations.

Due to the non-linear dependence of MFs on Gμ/c² and the small number of string simulations, the posterior distribution is quite complex and noisy. For this reason, we evaluated the posterior at n_NL = 51 values of Gμ/c², between 0 and 10 × 10^-7, to obtain our Planck estimate for Gμ/c². This estimate is stable and has been validated in realistic conditions with the Planck String Challenges described above, and for which we found consistent results with the underlying (unknown) Gμ/c².

4.4.2. Minkowski functionals results

For the constraint on Gμ/c², we analysed the foregrounds separated SMICA map at N_side = 2048 and ℓ_max = 2000, using the U73 mask (f_sky = 73% of the sky is unmasked). The smallest point sources holes were inpainted. We applied two specific Wiener filters to the map, designed to enhance the information from the map itself (W_M) and from the gradients of the map (${\mathit{W}}_{\mathrm{D}\mathrm{1}}\mathrm{=}\sqrt{\mathit{\ell }\mathrm{(}\mathit{\ell }\mathrm{+}\mathrm{1}\mathrm{)}}{\mathit{W}}_{\mathrm{M}}$). The filters are shown in Fig. 14.

Additionally, we estimated the average impact of some residual foregrounds and secondaries (FG) on Gμ/c², using the linear properties of MFs and foregrounds models processed through the Planck simulation pipeline (FFP6 simulations). Uncorrelated (Poissonian) unresolved point sources (PS), Cosmic Infrared Background (CIB) and Sunyaev-Zeldovich cluster⁴ (SZ) signals can be introduced as a simple additive bias on MFs’ curves following: $\mathit{ŷ}\mathrm{=}\mathit{ŷ}\mathrm{FGsubtracted}\mathrm{+}\mathrm{\Delta }\mathit{y̅}\mathrm{PS}\mathrm{+}\mathrm{\Delta }\mathit{y̅}\mathrm{CIB}\mathrm{+}\mathrm{\Delta }\mathit{y̅}\mathrm{SZ}\mathit{.}$(22)These biases are obtained as the average obtained on 100 simulations.

Fig. 14 The two Wiener filters, WM and WD1, used to constrain Gμ/c2 with Minkowski functionals.

Fig. 15 Posterior distribution of the parameter Gμ/c2 obtained with Minkowski functionals. This estimate takes into account the lensing of the data, but not the effects of foreground residuals.

We eventually obtain the posterior distribution of Gμ/c² (Fig. 15). As mentioned previously, the posterior curve is complex and noisy and so as to give an estimate of Gμ/c² we report its credible interval at 95%⁵. Results are summarized in Table 5, for raw data (lensing only is accounted for and subtracted) and foreground subtracted data (lensing, PS, CIB and SZ are considered). The discrepancy between the two filters can be explained because the derivative filter W_D1 scans smaller scales than W_M so it is more easily biased by foreground residuals. Given the remaining foreground uncertainties, we take the most conservative of the MFs constraints for the cosmic string contribution to the Planck data to be $\begin{array}{ccc}\mathit{G\mu }\mathit{/}{\mathit{c}}^{\mathrm{2}}\mathit{<}\mathrm{7.8}\mathrm{\times }{\mathrm{10}}^{-7} \mathrm{at}\mathrm{95}\mathrm{\%}\mathrm{C}\mathit{.}\mathrm{I}\mathit{.}& & \end{array}$The corresponding posterior is presented in Fig. 15.

Some caveats need to be mentioned that may influence these results. First, for the MF method itself, an important limitation is the small number of string simulations used to calibrate the estimator. The estimator appears to be mostly sensitive to low-redshift strings (infinite strings, with redshifts between 0 and 30), and this is affected by cosmic variance. As low-redshift string simulations are much faster to produce than complete simulations back to recombination, it should be possible to improve the robustness of the constraint using these relatively soon. Secondly, the impact of Galactic residuals should be assessed in further detail, especially for the filter W_D1 that we have observed to be less robust against residuals than the W_M filter.

With advances in studying these experimental effects there are good prospects for the full mission data, the sensitivity of the MFs estimator should improve substantially, with simulations forecasting possible MF cosmic string constraints of Gμ/c² < 3 × 10^-7 at the 95% C.I.

We note that further real space analysis of string map simulations has been undertaken with scaling indices of the pixel temperature distribution (see, e.g., Räth et al. 2011). Extensions calculating a set of anisotropic scaling indices along predefined directions appear to offer good prospects for string detection.

5. Conclusions

We have reviewed the signatures induced by cosmic strings in the CMB and searched for these in the Planck data, resulting in new more stringent constraints on the dimensionless string tension parameter Gμ/c². A pre-requisite for accurate constraints on cosmic strings is a quantitative understanding of both cosmological string network evolution and the effects they induce in the CMB. These are computationally demanding problems but progress has been made recently on several fronts: first, high resolution simulations of Nambu-Goto strings have yielded robust results for the scale-invariant properties of string networks on large scales, while there has been increasing convergence about small-scale structure and loops (for which the CMB predictions are less sensitive). Secondly, post-recombination gravitational effects of strings have been incorporated into full-sky Planck resolution CMB temperature maps that are important for validating non-Gaussian search methods. Finally, fast Boltzmann pipelines to calculate CMB power spectra induced by causal sources have been developed and tested at high resolution, whether for field theory simulations of strings or textures or for models of Nambu-Goto strings. Three-dimensional field theory simulations of vortex-strings at sufficient resolution should, in principle, converge towards the one-dimensional Nambu-Goto string simulations, but testing this is not numerically feasible at present. For this reason, we believe it is prudent to also include constraints on field theory strings (labelled Gμ_AH), thus encompassing cosmic string models for which radiative effects are important at late times (such as global strings). We believe this brackets the important theoretical uncertainties that remain, that is, we have used the best available information to constrain both Nambu-Goto strings (NAMBU) and field theory strings (AH). This work has also obtained more stringent constraints on semilocal strings and global textures.

5.1. Cosmic string constraints and the CMB power spectrum

Accurate forecasts for the CMB power spectrum induced by cosmic strings are more difficult to compute than their equivalent for simple adiabatic inflationary scenarios. It requires knowledge of the source, quantified by the unequal-time correlator (UETC) of the defect stress-energy tensor, from well before recombination to the present day, which is not computationally feasible. Fortunately, we can exploit scale-invariant string evolution to extrapolate the results of simulations with substantially smaller dynamic range. We use two methods to obtain predictions for the UETCs. An unconnected segment model (USM) is used to model the properties of an evolving string network, determining its density from an analytic one-scale evolution model, and the sources are coupled to the line-of-sight Boltzmann solver CMBACT. A second independent pipeline measures the UETCs directly from string simulations in the Abelian-Higgs field theory, passing these to a modified form of the CMBEASY Boltzmann code. The resulting Nambu-Goto and Abelian-Higgs string CMB power spectra are illustrated in Fig. 3. Free parameters in the USM model can be chosen to phenomenologically match the field theory UETCs (denoted the AH-mimic model) and the comparison is also shown in Fig. 3, validating the two independent pipelines.

To compute constraints on cosmic string scenarios we have added the angular power spectrum to that for a simple adiabatic model, assuming that they are uncorrelated, with the fraction of the spectrum contributed by cosmic strings being f₁₀ at ℓ = 10. This has been added to the standard 6 parameter fit using COSMOMC with flat priors. For the USM models we have obtained the constraint for the Nambu-Goto string model $\begin{array}{ccc}\mathit{G\mu }\mathit{/}{\mathit{c}}^{\mathrm{2}}\mathit{<}\mathrm{1.5}\mathrm{\times }{\mathrm{10}}^{-7}\mathit{,} {\mathit{f}}_{\mathrm{10}}\mathit{<}\mathrm{0.015}\mathit{,}& & \end{array}$(23)while for the Abelian-Higgs field theory model we find, $\begin{array}{ccc}\mathit{G}{\mathit{\mu }}_{\mathrm{AH}}\mathit{/}{\mathit{c}}^{\mathrm{2}}\mathit{<}\mathrm{3.2}\mathrm{\times }{\mathrm{10}}^{-7}\mathit{,} {\mathit{f}}_{\mathrm{10}}\mathit{<}\mathrm{0.028}\hspace{0.17em}\mathit{.}& & \end{array}$(24)The marginalised likelihoods for f₁₀ and in the f₁₀–Ω_bh² plane were presented in Fig. 10. With Planck nominal mission data these limits are already about a factor of two more stringent than the comparable WMAP 7-year string constraints and these Planck limits improve further with the inclusion of high-ℓ data.

5.2. Non-Gaussian searches for cosmic strings

Complementary searches for non-Gaussian signatures from cosmic strings were performed and we have reported constraints from the string bispectrum, steerable wavelets and Minkowski functionals. These methods participated in the Planck String Challenges and have undergone non-Gaussian validation tests.

The post-recombination string bispectrum has been reconstructed and calibrated from string-induced CMB maps using a modal estimator. String challenge analysis with Planck-realistic noise simulations and mask indicated a nominal mission sensitivity of ΔGμ/c² ≈ 5.8 × 10^-7. Analysis of SMICA, NILC and SEVEM foreground-separated maps has yielded f_NL = 0.37 ± 0.21 for the string bispectrum shape, which translates into a bispectrum constraint on the string tension, Gμ/c² < 9.0 × 10^-7 (95% CL). Steerable wavelet methods have been calibrated on string simulation maps added to Gaussian CMB maps with realistic noise and masking, showing a sensitivity of up to ΔGμ/c² ≈ 4 × 10^-7. The string signal was shown to have greater impact on the kurtosis of the signed-intensity than on its skewness, and no evidence of a string signal was found in the Planck data. Minkowski functionals have been applied to string simulation maps in a Planck-realistic context, computing the four functionals – area, perimeter, genus and N_cluster – after application of Weiner filters. Using these distributions, a Bayesian estimator has been constructed to constrain the string tension. Analysis of the SMICA foreground-cleaned maps yielded a MF constraint of Gμ/c² < 7.8 × 10^-7 (95% CL).

Non-Gaussian searches for strings are complementary to the power spectrum analysis and yield constraints as low as Gμ/c² < 7.8 × 10^-7, though we note the potential impact of foreground residuals in limiting current precision. These are conservative upper bounds because they only include post-recombination string contributions, unlike the string power spectrum analysis. Having such a broad suite of tools, ranging from multipole space, through wavelets, to real space detection methods, allows cross-validation and reinforces the conclusion that there is at present no evidence for cosmic strings in the Planck nominal mission data.

Acknowledgments

The development of Planck has been supported by: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN, JA 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); and PRACE (EU). A description of the Planck Collaboration and a list of its members, including the technical or scientific activities in which they have been involved, can be found at http://www.sciops.esa.int/index.php?project=planck&page=Planck_Collaboration. We also wish to acknowledge the use of the COSMOS supercomputer, part of the DiRAC HPC Facility funded by STFC and the UK Large Facilities Capital Fund, use of the Andromeda cluster of the University of Geneva, and resources of the National Energy Research Scientific Computing Center.

References

All Tables

All Figures

	Fig. 1 The spacetime around a cosmic string is conical, as if a narrow wedge were removed from a flat sheet and the edges identified. For this reason cosmic strings can create double images of distant objects. Strings moving across the line of sight will cause line-like discontinuities in the CMB radiation.
In the text

	Fig. 2 Characteristic CMB temperature discontinuity created by a cosmic string. Here, the simulated Nambu-Goto string has produced a cusp, a small region on the string that approaches the speed of light, which has generated a localised CMB signal.
In the text

Fig. 3 Cosmic string power spectra used in this analysis: NAMBU (black dashed), AH-mimic (blue dotted) and AH (red solid). The spectra have been set to equal power at ℓ = 10. When normalised to 850 μK at ℓ = 10 they correspond to string tensions of Gμ/c2 = 1.17 × 10-6, 1.89 × 10-6 and 2.04 × 10-6 respectively. Note that the limits discussed in this paper mean that the CMB spectra presented here are less than 3% of the overall power spectrum amplitude and hence the differences observed at high ℓ do not have a large effect.

In the text

Fig. 4 Comparison between global texture (black dashed) and semilocal (blue dotted) string power spectra and the AH field theory strings (red solid), normalised to unity at ℓ = 10. As expected, the SL spectrum lies in between the TX and the AH spectra. The AH spectrum was recomputed for the Planck cosmological model with sources from Bevis et al. (2010), and the SL and TX spectra were taken from Urrestilla et al. (2008).

In the text

	Fig. 5 Integrated Sachs-Wolfe angular power spectra extracted from the full sky cosmic string maps at different resolutions (labelled by Nside), with or without applying the anti-aliasing procedure (see text). The anti-aliasing filtering gives back the correct power up to ℓmax ≲ 2Nside.
In the text

	Fig. 6 All sky Mollweide projection of the simulated cosmic strings CMB sky after convolution by a Gaussian beam of 5′ resolution. The colour scale indicates the range of (ΔT/T)/(Gμ/c2) fluctuations.
In the text

	Fig. 7 A 20° gnomic projection patch extracted from the full sky map and zooming into string induced temperature steps (see Fig. 6). Applying the spherical gradient magnitude operator enhances the temperature steps, and thus the string locations, even more (right).
In the text

	Fig. 8 Marginalised constraints on f10 for topological defects from Planck data plus polarisation from WMAP (Planck+WP). The left panel show constraints on cosmic strings, with NAMBU in black dashed, AH-mimic in blue dotted and AH in red solid. The right panel show the constraints on SL (blue dotted) and TX (black dashed) compared to AH (again solid red).
In the text

Fig. 9 Marginalised constraints on f10 for topological defects with high-ℓ CMB data from SPT and ACT added to the Planck + WP constraints data (compare with constraints shown in Fig. 8). The left panel show constraints on cosmic strings, with NAMBU in black dashed, AH-mimic in blue dotted and AH in red solid. The right panel show the constraints on SL (blue dotted) and TX (black dashed) compared to AH (solid red).

In the text

	Fig. 10 Marginalised likelihoods in the f10-Ωbh2 plane for the NAMBU model in blue and the AH mimic model in red using Planck +WP. This is the strongest correlation with any of the standard cosmological parameters.
In the text

Fig. 11 Coefficients ${\mathit{\alpha }}_{\mathit{n}}^{\mathit{R}}$ Eq. (15) for the hybrid Fourier mode expansion of the cosmic string bispectrum Eq. (15). The average value ${\mathit{\alpha }}_{\mathit{n}}^{\mathit{R}}\mathrm{=}\mathrm{⟨}{\mathit{\beta }}_{\mathit{n}}^{\mathit{R}}\mathrm{⟩}$ (black line) is in remarkable agreement with all four string simulations as can be seen for n < 30 (lower panel), with each exhibiting better than a 97% correlation overall.

In the text

	Fig. 12 Modal reconstruction of the post-recombination string bispectrum Eq. (15) extracted from Planck resolution map simulations. This is a 3D view of the allowed tetrahedral set of multipoles (ℓ1,ℓ2,ℓ3) showing isosurfaces of the bispectrum density with darker blue for more negative values (it is normalised relative to the constant SW bispectrum).
In the text

Fig. 13 Deviation of the kurtosis of the signed-intensity as a function of Gμ/c2, normalised to the standard deviation of CMB and noise simulations: ΔK/σ = (K(R,Gμ/c2) − K(R,Gμ/c2 = 0)) /σ. The left panel shows results for the 1GD wavelet and the right shows the 2GD wavelet. Each curve corresponds to a wavelet scale, R (arcmin), included in the analysis. The final sensitivity of the method is determined by combining the two wavelets and all the scales.

In the text

	Fig. 14 The two Wiener filters, WM and WD1, used to constrain Gμ/c2 with Minkowski functionals.
In the text

	Fig. 15 Posterior distribution of the parameter Gμ/c2 obtained with Minkowski functionals. This estimate takes into account the lensing of the data, but not the effects of foreground residuals.
In the text