CMB distortion anisotropies

# Evolution of CMB spectral distortion anisotropies and tests of primordial non-Gaussianity

## Abstract

Anisotropies in distortions to the frequency spectrum of the cosmic microwave background (CMB) can be created through spatially varying heating processes in the early Universe. For instance, the dissipation of small-scale acoustic modes does create distortion anisotropies, in particular for non-Gaussian primordial perturbations. In this work, we derive approximations that allow describing the associated distortion field. We provide a systematic formulation of the problem using Fourier-space window functions, clarifying and generalizing previous approximations. Our expressions highlight the fact that the amplitudes of the spectral-distortion fluctuations induced by non-Gaussianity depend also on the homogeneous value of those distortions. Absolute measurements are thus required to obtain model-independent distortion constraints on primordial non-Gaussianity. We also include a simple description for the evolution of distortions through photon diffusion, showing that these corrections can usually be neglected. Our formulation provides a systematic framework for computing higher order correlation functions of distortions with CMB temperature anisotropies and can be extended to describe correlations with polarization anisotropies.

###### keywords:
Cosmology: cosmic microwave background – theory – observations
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## 1 Introduction

Measurements of cosmic microwave background (CMB) spectral distortions can teach us about the thermal history of the Universe (see Chluba & Sunyaev, 2012; Sunyaev & Khatri, 2013; Tashiro, 2014; Chluba, 2016, for recent overviews). Aside from the distortion signals imprinted at arcminute angular scales by clusters of galaxies through the Sunyaev-Zeldovich (SZ) effect (Zeldovich & Sunyaev, 1969; Carlstrom et al., 2002), anisotropies in the distortion to the CMB frequency spectrum are expected to be small. However, anisotropic heating caused by non-standard early-universe processes can in principle lead to observable distortion anisotropies (Chluba et al., 2012b). One example is related to the dissipation of small-scale perturbations (wavenumber ) in the photon-baryon fluid in the presence of ultra-squeezed limit non-Gaussianity (Pajer & Zaldarriaga, 2012; Ganc & Komatsu, 2012). In this case, the local acoustic heating rate is modulated by large-scale modes (), such that the CMB spectrum can vary across that sky. This effect can be used to place limits on primordial non-Gaussianity and test its scale-dependence (Biagetti & et al., 2013; Emami et al., 2015; Dimastrogiovanni & Emami, 2016) using future spectrometers like PIXIE (Kogut et al., 2011, 2016).

Several estimates for the observability of these signals through their correlation with large-scale temperature perturbations can be found in the literature (Pajer & Zaldarriaga, 2012; Ganc & Komatsu, 2012; Biagetti & et al., 2013; Emami et al., 2015; Khatri & Sunyaev, 2015; Ota et al., 2015; Creque-Sarbinowski et al., 2016). In addition, new estimates related to damping-induced acoustic reheating (Naruko et al., 2015), correlations with polarization anisotropies (Ota, 2016) and higher order correlation functions (Bartolo et al., 2016; Shiraishi et al., 2016) as a new probe of primordial non-Gaussianity have been studied. All the aforementioned computations are based on different approximations and assumptions. Here, we develop a common formulation of the problem using Fourier-space window functions. These functions allow separating dissipation and thermalization physics from the statistical properties of the primordial perturbations and their spatial evolution in a transparent way with minimal assumptions.

We use our formulation to justify the simple approximations given recently in Emami et al. (2015) for the primordial contributions to the chemical potential () and Compton- correlation functions with the CMB temperature, and . These approximations directly show that the cross power spectra not only depend on the level of non-Gaussianity at small scales (), parametrized by , but also on the amplitude of the average distortion signal created by the homogenous heating term. This shows that there are two independent ways to enhance the amplitude of the primordial distortion fluctuations: by larger non-Gaussianity and/or a modified small-scale power spectrum. This highlights that independent limits on can only be derived in absolute measurements (e.g., PIXIE), for which the average distortion of the CMB monopole is also obtained, while in differential measurements (e.g., CORE, Litebird) the interpretation remains model-dependent.

We explicitly model the transition between and -type distortions (Sunyaev & Zeldovich, 1970b; Zeldovich & Sunyaev, 1969) using distortion visibility functions (Chluba, 2013b, 2016). We also include the effect of photon diffusion to the transfer functions for the distortion anisotropies. We highlight that, in contrast to temperature perturbations, no pressure waves appear for distortions. Distortion anisotropies simply smear out before propagating wave fronts can develop, with a damping scale that is times smaller than the Silk-damping scale for temperature perturbations. This modification is usually not included and only becomes noticeable at multipoles .

We compare the results for different approximations and also improve previous analytic expressions capturing the -dependence of the correlation function caused by temperature transfer effects more accurately. We provide simple analytic expressions for the distortion signals, which can be easily evaluated by specifying the small-scale power spectrum and scale-dependence of . Overall the main goal is to clarify and simplify the formulation of the problem for anisotropic heating processes. The expressions can be extended to the case of polarization and higher order correlation functions in a straightforward manner, but this is left to future work.

The paper is structured as follows: in Sect. 2, we give the formulation of the problem and compute the temperature-distortion correlation functions. We directly compare with previous numerical estimates in Sect. 2.6. In Sect. 3, we discuss distortion transfer effects, showing that they are small at . Our conclusions are presented in Sect. 4.

## 2 Anisotropic spectral distortions through damping of small-scale perturbations

Following the evolution of the CMB spectrum with anisotropies in general is a hard problem. Not only spatial photon diffusion, but also redistribution of photons in energy (Compton scattering) and photon production (Bremsstrahlung and double Compton emission) have to be included for an anisotropic medium. A formulation of the required evolution equations was given by Chluba et al. (2012b). Here, we shall start by neglecting spatial transfer effects for the distortion evolution before decoupling at . We will return to the more general problem in Sect. 3. Under this assumption, we only need to specify the spatially varying heating rate caused by the damping of primordial temperature perturbations, which is given by the angular average

 ddt[Q({x},t)ργ]≈−4⟨Θ˙Θ⟩Ω≡−4∫Θ˙Θd2^n4π (1)

over different directions . Here, denotes the CMB temperature fluctuation at any location x. We only compute the monopole of the heating rate (angle-average), but do not consider the dipolar and quadrupolar heating rate, which cause tiny corrections1.

In the tight coupling limit (Hu & Sugiyama, 1996), the multipole hierarchy can be truncated after the quadrupole and corrections sourced by the damping of pure polarization terms may be neglected (Chluba et al., 2015). The contributions proportional to the monopole drop out of the final result, as well as the dipole parts after including second-order scattering terms (Chluba et al., 2012b). Thus, only the quadrupole contribution is left, for which the relevant time derivative is given by (Hu & White, 1997). The final result for the anisotropic heating rate thus is

 ddt[Q({x},t)ργ] ≈15˙τΘ22({x},t) (2) =15˙τ∫d3k(2π)3d3k′(2π)3ei{x}⋅({k}+{k}′)R({k})R({k}′)T2(k,t)T2(k′,t),

where is the time-derivative of the Thomson optical depth, and denotes the radiation transfer functions in -space. The latter only depends on and map the initial curvature perturbation amplitude, , into temperature anisotropies at a later stage. In the tight coupling regime (), we have (Hu & Sugiyama, 1996)

 T2(k,t) ≈8kDc15˙τasin(krs)√3e−k2/k2D (3)

where with for three massless neutrinos; denotes the sound horizon; is conformal time; is the scale factor and is the standard photon damping scale, which in the radiation-dominated era is determined by , or .

For the final type of the distortion fluctuations, it is important when the energy is released. Here, we shall describe the final signal as a superposition of and distortions with an additional temperature shift. The and contributions can be modeled using energy branching ratios, and , for the and distortions (Chluba, 2013b). This approximation assumes that the distortion shape only depends on the total number of Compton scatterings but not on the anisotropies themselves. Different approximations for the distortion visibilities are summarized in Chluba (2016). Unless stated otherwise, here we will use (Chluba, 2013b)

 Jy(z) ≈⎧⎪⎨⎪⎩(1+[1+z6×104]2.58)−1forzrec≤z0otherwise (4a) Jμ(z) ≈Jbb(z)[1−exp(−[1+z5.8×104]1.88)], (4b)

where and accounts for the effect of thermalization, which becomes very efficient at redshifts and erases the distortions.

With these definitions, we can write the spatially varying distortions caused by the dissipation process as

 y({x},z) ≈14∫∞zddz′[Q({x},z′)ργ]Jy(z′)dz′ (5a) μ({x},z) ≈1.401∫∞zddz′[Q({x},z′)ργ]Jμ(z′)dz′, (5b)

where we used the redshift as the time coordinate2. These expressions are very similar to those for uniform energy release (e.g., Eq. (6) in Chluba & Jeong, 2014). The only difference is that here the energy release rate varies spatially. This causes anisotropic and distortions with the degree of anisotropy depending on the curvature power spectrum at small scales.

Similar to the uniform and distortions from the dissipation of acoustic modes, we can introduce -space window functions that link the distortion and curvature perturbations (see Chluba et al., 2012a; Chluba & Grin, 2013). From Eqs. (2) and (5) we then have

 X({x},z)≈∫d3k(2π)3d3k′(2π)3ei{x}⋅({k}% +{k}′)R({k})R({k}′)WX(k,k′,z), (6)

where . The -space window functions capture all the thermalization physics and mode coupling. In compact form they can be rewritten as

 WX(k,k′,z) =15∫∞z˙τaHGX(z′)T2(k,z′)T2(k′,z′)dz′ (7) ≈3245∫∞zc2D2˙τaHGX(z′)2kk′sin(krs)sin(k′rs)e−k2+k′2k2D(z′)dz′,

with and .

When using the transfer function approximation, Eq. (3), for the computation of the window function, Eq. (7), the momentum integrals in Eq. (6) are carried out at because horizon scale modes at do not dissipate. Alternatively, one can use more elaborate expressions for the transfer function or full numerical results to capture the super-horizon evolution (Chluba et al., 2012b). However, we find that for our purposes the simpler approximation usually is sufficient.

### 2.1 Properties of the window functions, WX(k,k′,z)

The window functions, , have a few simple properties. First, by symmetry , so that one only has to consider cases . For , the window function reduces to the one for uniform heating, with no coupling between modes (Chluba et al., 2012a; Chluba & Grin, 2013). The only difference is due to the approximations of the energy branching ratios, . In earlier works, and , where for . The effect of this approximation is illustrated in Fig. 1. We set , which is relevant to our discussion. Using the step-function approximation, both the and distortion window functions pick up contributions from a narrower range of scales. With the expressions in Eq. (4), the transition around is smoother. This is because modes of a given scale dissipate energy over a range of redshifts (e.g., Fig. 1 in Chluba et al., 2012a).

When considering non-Gaussianity, the mode coupling also has to be included. Inspecting Eq. (7), it becomes clear that for strongly disparate and the window functions should decrease notably, mainly due to the exponential factor. For , one can also set without affecting the result for the window function significantly (it basically removes the small wiggles seen in Fig. 1). Similarly, for one has , with the substitution and . In the last step, we replaced terms that vary fast with time with their scale-averaged values. We confirmed that this approximation works extremely well, affecting the shapes of the window functions negligibly. However, this approximation eases the numerical computation significantly.

In Fig. 2, we illustrate the numerical results for the and window functions for . For , we found as long as . For larger difference in the wavenumber, the amplitude of drops strongly. At one can simply set (confirmed numerically). Similarly, we have for and for (confirmed numerically). This behavior of the -space window functions eases the numerical computation of the distortion correlation functions significantly.

We also mention that while the heating rate for a single -mode shows oscillatory behavior in time (which means that heating occurs at different phases of the wave propagation, when the spatial gradients in the temperature are largest), the window-function becomes quite smooth, representing the time-averaged heating rate for each mode. This point is often confused in the literature. In particular, the time (redshift) average is required and does not automatically drop out.

### 2.2 Free-streaming after recombination

We neglected any spatial evolution of the distortion anisotropies, assuming that they are created in situ and remain there until decoupling at . After decoupling we simply assume that the distortions free-stream to the observer across a distance to the last scattering surface. With the plane wave identity

 ei{x}⋅{k}=4π∑ℓm=ℓ∑m=−ℓiℓjℓ(kx)Y∗ℓm(^%k)Yℓm(^{x}), (8)

the spectral distortion anisotropies in and for an observer at the origin () can thus be expanded into spherical harmonics, , as

 aXℓm ≡∫X({x},zrec)Y∗ℓm(^{n})d2^n =4π(−i)ℓ∫d3k(2π)3d3k′(2π)3Y∗ℓm(^{k}+) ×jℓ(k+rL)R({k}% )R({k}′)WX(k,k′,zrec), (9)

where . This expression can be used to compute approximate correlation functions between distortions and temperature anisotropies (see Sect. 2.3). Corrections due to distortion transfer effects were neglected but will be discussed in Sect. 3.

#### Comparing with previous work

To compare Eq. (2.2) for directly with Pajer & Zaldarriaga (2012) and Ganc & Komatsu (2012), we approximate the window function, Eq. (7). The first simplification is to replace the visibilities by step-functions. In this way, the redshift integral for the -distortion is limited to , where the redshift marks the transition between the and -distortion eras (e.g., Burigana et al., 1991; Hu & Silk, 1993a). If we then also replace the transfer function factor, , we find

 Wμ(k,k′,z) ≈3245∫zμzμ,yc2D2αμ˙τaHkk′e−k2+k′2k2D(z′)dz′ ≈4D2αμkk′k2+k′2∫zμzμ,y∂z′e−(k2+k′2)/k2D(z′)dz′ \lx@stackrel\lx@stackrelk≈k′↓≈2D2αμ[e−(k2+k′2)/k2D(z)]zμzμ,y, (10)

where and . Inserting this back into Eq. (2.2), we obtain

 aμℓm ≈9.1π(−i)ℓ∫d3k(2π)3d3k′(2π)3Y∗ℓm(^{k% }+) ×jℓ(k+rL)R({k}% )R({k}′)[e−(k2+k′2)/k2D(z)]zμzμ,y. (11)

Comparing this with Eq. (40) of Ganc & Komatsu (2012), we find agreement once we set3 in their expression. The only small difference is that, following Pajer & Zaldarriaga (2012), a filter function was added to the -space integral. In our approach, this filter function is directly related to , which vanishes when and differ significantly (see Fig. 2), but was neglected in Eq. (2.2.1). Our approximation also includes a factor of (Chluba et al., 2012b; Inogamov & Sunyaev, 2015) with respect to Pajer & Zaldarriaga (2012), which was based on the classical treatment of the dissipation problem (Sunyaev & Zeldovich, 1970a; Daly, 1991; Hu et al., 1994).

### 2.3 Cross-correlation of μ and y with temperature

We now compute the correlation functions for different combinations of , and temperature perturbations. The temperature anisotropies seen by an observer at the origin can be expanded into spherical harmonics

 aTℓm ≡∫d2nΔT(^{n})TY∗ℓm(^{n}) (12) ≈12π5(−i)ℓ∫d3k(2π)3R({k})Δℓ(k)Y∗ℓm(^{k}),

where is the radiation transfer function in the Sachs-Wolfe limit.4 Alternatively, it is possible to directly use numerical results for the transfer functions at different values of . This is expected to enhance the final correlation function, , at small scales with respect to the Sachs-Wolfe approximation but reduce the overall signal-to-noise ratio due to cancelation effects (e.g., Ganc & Komatsu, 2012).

Using , with curvature power spectrum , the temperature power spectrum in the Sachs-Wolfe limit is given by

 CTT,SWℓ≈4π25∫dkTkTj2ℓ(kTrL)Δ2(kT)≈2π25Δ20ℓ(ℓ+1), (13)

where in the last step we assumed a scale-invariant power spectrum () with amplitude .

We use the primordial bispectrum in the squeezed limit to describe the scale-dependent5

 ⟨R{k}R{k}′R{k}T⟩ ≈(2π)3δ(3)({k}+{k}′+{k}T)12fNL(k)5PR(k)PR(kT). (14)

Here, was already used, but we confirmed that even more generally this limit for the bispectrum is sufficient. After some algebra (see Appendix B), the spectral distortion-temperature correlation functions take the simple form , where the correlation coefficients are given by

 CXTℓ≈48π25∫dkTkTdkkj2ℓ(kTrL)Δ2(kT) ×fNL(k)Δ2(k)¯WX(kT,k,zrec) (15)

and the azimuthally averaged -space window function is

 ¯WX(kT,k,zrec) =12∫k+kT|k−kT|WX(k,k1,zrec)k1dk1kkT, (16)

with . The averaged window function receives most of its contributions from , so that is preferred. Assuming , one can indeed replace , which gives

 CXTℓ ≈48π25∫dkkdkTkTj2ℓ(kTrL)Δ2(kT)fNL(k)Δ2(k)WX(k,zrec) =124π25∫dkTkTj2ℓ(kTrL)Δ2(kT)×∫dkkfNL(k)Δ2(k)WX(k,zrec) =12CTT,SWℓ∫dkkfNL(k)Δ2(k)WX(k,zrec) =12CTT,SWℓHX≈12CTT,SWℓfNL(kX)⟨X⟩, (17)

where in the last step we assumed that scales slowly with and can simply be evaluated at , respectively relevant to the and eras (see discussion Emami et al., 2015). Explicitly, this means that

 HX =∫dlnkfNL(k)Δ2(k)WX(k,zrec)≈fNL(kX)⟨X⟩ (18)

was assumed. For scenarios with significant scale-dependence of around the distortion pivot scales, , this integral can be evaluated numerically after specifying and . Below we briefly discuss ; however, a more general consideration requires a case-by-case study, which is beyond the scope of this paper.

The approximation was given in Emami et al. (2015) and shows explicitly that the cross-correlation depends on the average value of the distortion parameters,

 ⟨X⟩=∫dlnkΔ2(k)WX(k,zrec). (19)

Equation (2.3) also highlights that for the temperature perturbations and distortions are correlated at the largest scales. On the other hand, for the WMAP convention for the bispectrum, implies that the temperature perturbations and distortions are anti-correlated at the largest scales. Including the full temperature transfer function, the cross power spectrum is expected to change sign at , as previously explained by Ganc & Komatsu (2012). The same statement applies to .

By extracting the data from Fig. 3 of Ganc & Komatsu (2012) using the ADS Dexter tool, we find the ratio of the full radiative transfer result, , in comparison to the Sachs-Wolfe approximation, , within their computation to be well represented by

 ρ(ℓ) ≈1.08[1−0.022ℓ−1.72×10−4ℓ2 +2.00×10−6ℓ3−4.56×10−9ℓ4] (20)

at . Since the large-scale power spectrum parameters are well-known (Planck Collaboration et al., 2016), we can obtain the approximation

 CXTℓ ≈12CTT,SW,1ℓρ(ℓ)fNL(kX)⟨X⟩, (21)

where is given by Eq. (13) with . We will show below that this approximation indeed works extremely well; however, some modifications are required when varies noticeably.

### 2.4 Correlations of μ and y

We carry out the computation for the distortion correlation functions for the Gaussian and non-Gaussian contributions in Appendix C. The Gaussian part is (negligibly) small with a quasi white-noise power spectrum until transfer effects become important. Here we focus on the result for the non-Gaussian contribution . Defining the power spectra, , and using , from Eq. (C.2) we find

 CXYℓ ≈1444π25∫dkkj2ℓ(krL)Δ2(k) ×∫dk1k1dk2k2Δ2(k1)Δ2(k2)fNL(k1)fNL(k2) ×WX(k1,zrec)WY(k2,zrec) =144CTT,SWℓHXHY, ≈144CTT,SWℓfNL(kX)fNL(kY)⟨X⟩⟨Y⟩, (22)

where in the last step we again used Eq. (18). This approximation was given in Emami et al. (2015) and again explicitly illustrates how the correlation function depends on the average values, and . Radiative transfer effects slightly modify the -dependence of the correlation functions, but these effects only become important at (see below) and are not further discussed here.

### 2.5 Numerical results for the cross-power spectra

To obtain the results for the distortion-temperature correlation function, we assume a power-law power spectrum and non-Gaussianity around pivot scale . We set and (Planck Collaboration et al., 2014a, 2016). The integral over in Eq. (2.3) runs from to . This includes basically all modes that can lead to distortions in the and -eras before recombination ends.

To ease the numerical calculation as a function of , we define the function

 dCXT(kT)dlnkT=∫dlnkfNL(k)Δ2(k)¯WX(k,kT,zrec) (23)

such that . Thus, the -dependence of the correlation function can be obtained after tabulating . For the correlation, we can furthermore directly compare with previous calculations (Pajer & Zaldarriaga, 2012; Ganc & Komatsu, 2012) by setting

 ¯Wμ(k,kT,zrec)≈2.27[e−2k2/k2D(z)]zμzμ,yWf(kT/k2D(zμ,y)), (24)

where is a filter function that was introduced by hand.

In Fig. 3, we show the differential correlation function, for , using our approach and the one of Pajer & Zaldarriaga (2012). The approximation of Pajer & Zaldarriaga (2012) picks up contributions from significantly smaller scales. However, most of the total correlation arises from large scales, with . Thus, the final result for the correlation is hardly affected by the exact shape of the window function at . In this case, one can indeed set , such that with

 dCμT(kT→0)dlnkT≈Hμ≈fNL(kμ)⟨μ⟩ (25)

the simple approximation works extremely well.

In Fig. 3, we also show the differential correlation function for the correlation. Its amplitude at large scales is about four times lower than for the distortion, owing to the different relations of dissipated energy and distortion amplitude [i.e., versus ]. The correlation mainly picks up contributions from , so that the approximation is again well-justified.

#### Simple formulae for varying nS and nfNL

For power-law dependence of , with , one can capture the modifications by computing the average value , Eq. (19), with effective spectral index around modified pivot-scale . We find

 CμTℓ ≈12CTT,SW,1ℓρ(ℓ)fNL(kμ)Δ2(kμ)Iμ(n∗S) lnIμ(n) ≈2.60[1−0.290ξ+0.471ξ2−0.0516ξ3] (26)

with to represent the numerical result very well. The effective spectral index is , which can strongly differ from the values obtained at large angular scales. The expression for gives very similar results as the one presented in Chluba & Grin (2013); however, here we used the slightly improved distortion window-function, with distortion visibility according to Eq. (4), and shifted the pivot-scale to .

Similarly, for we can write

 CyTℓ ≈12CTT,SW,1ℓρ(ℓ)fNL(ky)Δ2(ky)Iy(n∗S) lnIy(n) ≈0.81[1−1.44ξ+2.39ξ2−0.214ξ3] (27)

with and .

The expressions given above clearly show that the temperature-distortion correlations can be enhanced in three main ways: i) by increasing , ii) by increasing the amplitude of the small-scale power spectrum, , or iii) by changing the scaling of non-Gaussianty or the small-scale power spectrum around the distortion pivot scales, captured by . The first two effects affect the results most significantly, while the latter is less important unless deviates strongly from unity.

### 2.6 Estimates for the observability of the μ−T correlation

The -dependence of the correlation functions in both cases is fully determined by . We can thus estimate the expected signal-to-noise ratio using (e.g., Ganc & Komatsu, 2012)

 (SN)2 ≈ℓmax∑ℓ=2(2ℓ+1)(CXTℓ)2CTTℓCXX,Nℓ =144H2Xℓmax∑ℓ=2(2ℓ+1)(CTT,SW,1ℓ)2ρ2(ℓ)CTTℓCXX,Nℓ, (28)

where is the CMB temperature power spectrum and the noise level for the distortions. For PIXIE (see Pajer & Zaldarriaga, 2012) we have , where denotes the smallest detectable distortion monopole signal.

#### Estimate in the Sachs-Wolfe limit

To compare with previous results, we first obtain an estimate for the signal-to-noise ratio in the Sachs-Wolfe limit, which was used in several works (e.g., Pajer & Zaldarriaga, 2012; Emami et al., 2015). For this we assume with and set