Kinetic Sunyaev Zeldovich effect in an anisotropic CMB model:
measuring low multipoles of the CMB at higher redshifts using intensity and polarization spectral distortions

The specific intensity and polarization of the CMB are commonly described using the set of Stokes parameters $(I_{\nu},Q_{\nu},U_{\nu})$ (Chandrasekhar, 2013). The change in these parameters due to Thomson scattering with the electrons in the cluster is conveniently expressed in terms of the radiative transfer equation (Rybicki and Lightman, 1986)

where $\theta_{sc}$ and $\phi_{sc}$ are the polar and azimuthal angles of scattering between $\hat{\bm{\gamma}}$ and $\hat{\bm{\gamma}}^{\prime}$, and $\tau=\int n_{e}\sigma_{T}ds$ is the optical depth of the cluster with respect to Thomson scattering. Here $n_{e}$ is the number density of the electrons, $\sigma_{T}$ is the Thomson cross section and $s$ is the length of the cluster along the line of sight.

The thermal motion of the electrons (which causes the tSZ distortion) and multiple scattering effects (Chluba et al., 2014; Chluba and Dai, 2014) are neglected to simplify the calculations. The scattering events are assumed to be elastic with no energy transfer between the electrons and photons. In the limit of cold clusters, since the scatterings do not change the bulk motion of the cluster, it is safe to use Thomson scattering (Portsmouth and Bertschinger, 2004a). For high temperature clusters, since the electrons upscatter the CMB photons, the elasticity assumption does not hold anymore and Compton scattering should be used instead (Portsmouth and Bertschinger, 2004b). The tSZ corrections in this limit are of the order of the perturbative parameter $\theta_{e}\equiv kT_{e}/m_{e}c^{2}\approx 0.01$ for a typical cluster, so if calculated, they can be linearly added to equations (1) and (2). The initial polarization of the CMB is also neglected, but the polarization-induced effects also linearly couple to equations (1) and (2), so the extra terms can be simply added at any point. It is important to mention that although the polarization-induced effects are typically smaller than the temperature-induced ones, due to their different spatial morphology they can be comparable or even dominant over them in certain directions over the sky.

In order to integrate the radiative transfer equations for an anisotropic incoming intensity with the harmonics expansion

we rewrite the integrand of equations (1) and (2) in terms of spin-weighted spherical harmonics using the generalized addition theorem (Hu and White, 1997; Ng and Liu, 1999; Bars, )

After integrating over all the incoming photons $\hat{\bm{\gamma}}^{\prime}$ these equations simplify to

with each term defined as

and for $\ell\neq 2$

The right hand side (RHS) of equation (8) is induced by the photons that scatter into the line of sight of the observer, while the RHS of equation (7) is a combination of the photons that scatter into and out of the line of sight. We refer to these terms as the Temperature-Induced Polarization (TinPol) and Temperature-Induced Intensity (TinIn) effects. Notice that both these effects depend on the quadrupole of the CMB as observed by the cluster and therefore are highly correlated with each other. Equation (9) is basically the harmonic expansion of the photons that scatter out of the line of sight. We will refer to these terms as the blurring Sunyaev Zeldovich effect (bSZ) (Hernandez-Monteagudo and Sunyaev, 2010). Although the temperature quadrupole-induced intensity $\delta I_{\nu}^{(2)}$ is a combination of both in-scattered and out-scattered photons, since the frequency dependence of these scattering events are identical they are not distinguishable from each other. Therefore the overall effect is written as one single term, which we have called the TinIn effect. However, it is important to remember that the nature of this effect is different from the bSZ, which is only caused by out-scattering of the photons.

Note that the monopole term that survives the radiative transfer integral, has been canceled out with the monopole term of the bSZ expansion. This means that in a completely isotropic CMB model where all the $a_{lm}$s for $l>0$ vanish, $\Delta I_{\nu}(\hat{\bm{\gamma}})$ and $\Delta(Q_{\nu}\pm iU_{\nu})(\hat{\bm{\gamma}})$ will be equal to zero. This is because in an isotropic model the number of scatterings that deflect the photons towards and out of the line of sight are equal to each other. However it is evident from equations (7) and (8) that after including the anisotropies, the quadrupole moment of the incident radiation induces a change in the observed intensity and polarization of the CMB in the direction of the cluster $\hat{\bm{\gamma}}$. It is important to note that there is an implicit location/redshift dependence in all the $a_{\ell m}$s on the RHS of these equations. Therefore the TinIn and TinPol signals give us access to the observed quadrupole moment of the CMB at the redshift of the cluster $z$ as first noticed in (Kamionkowski and Loeb, 1997).

The specific intensity of the CMB has an almost perfect black body spectrum which can be described as $I_{\nu}(\hat{\bm{\gamma}}^{\prime})=B_{\nu}(T(\hat{\bm{\gamma}}^{\prime}))$ where $B_{\nu}$ is the Planck function defined as $B_{\nu}(T)\equiv\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{h\nu/kT}-1}$ and $T(\hat{\bm{\gamma}}^{\prime})$ is the thermodynamic temperature of the CMB in the $\hat{\bm{\gamma}}^{\prime}$ direction. If we assume that the frequency spectrum of the CMB is isotropic (by negliecting anisotropic effects such as patchy reionization (Santos et al., 2003; Dore et al., 2007)), we only need a single number, namely the thermodynamic temperature of the blackbody $T$, and an appropriate frequency function (here the Planck Function $B_{\nu}$) to describe the intensity of the CMB in every direction $\hat{\bm{\gamma}}$. Consequently the intensity multipole coefficients $a^{I}_{\ell m}(\nu)$, can be expressed as the combination of a frequency function which describes the spectrum of the anisotropies, and the thermodynamic temperature multipole coefficients $a^{T}_{\ell m}$ defined as

Using the Planck frequency function it is easy to show that the relationship between these coefficients up to first order in temperature anisotropies is (see appendix A and (Chluba et al., 2012b))

where $\overline{T}=\frac{a^{T}_{00}}{2\sqrt{\pi}}$ is the thermodynamic temperature of the CMB monopole, $\tilde{B}_{\nu}(\overline{T})\equiv\overline{T}^{-1}B_{\nu}(\overline{T})$ and $\tilde{F}_{\nu}(\overline{T})\equiv\overline{T}^{-1}B_{\nu}(\overline{T})f(h\nu/k\overline{T})$ are respectively the frequency functions of the monopole and higher multipoles of the anisotropies, with $f(x)\equiv\frac{xe^{x}}{e^{x}-1}$. The tilde over the frequency functions denotes the normalization by the temperature monopole $\overline{T}$. The $a^{I}_{\ell m}(\nu)$ coefficients have units of radiative intensity ($\text{W/m}^{2}\text{Hz sr}=10^{26}$Jy/sr) and $a^{T}_{\ell m}$s have units of temperature (Kelvin).

Using equations (11) in (7) and (8) will give us the advantage of separating the frequency dependence of the TinIn and TinPol signals from their spatial morphology over the sky, simply described by the $a^{T}_{\ell m}$s. For a non-moving cluster this separation is trivial, but in the case of a moving cluster it will help us to easily calculate the frequency weights of the leakage of the low multipoles into the quadrupole due to the Doppler effect. After substitution, equations (7) and (8) can be rewritten as

and similarly for the bSZ terms (eq. (9))

where the $a^{T_{z}}_{\ell m}$s are the harmonic coefficients of the temperature anisotropies of the CMB observed by the cluster at redshift $z$ and $\overline{T}_{z}$ is the average temperature at that redshift. As we will see in §III, the bSZ term $\delta I_{\nu}^{(1)}$ will give rise to the kSZ intensity distortion (Sunyaev and Zeldovich, 1980).

Here, since the frequency dependence of these signals are the same as that of the CMB temperature anisotropies, we can also write the distortions in terms of the thermodynamic temperature defined as

as is common practice in tSZ and kSZ calculations. A similar conversion can be used for the polarization parameters $Q_{T}$ and $U_{T}$ as well. After plugging in equations (12)-(14) into (15), it would be evident that the thermodynamic temperature (and polarization) distortions induced by the quadrupole in the direction of non-moving clusters are not frequency-dependent. However, this is not the case for the distortions induced by the motion of the cluster. As will be shown later, due to the addition of the motion-induced effects, the total change in intensity and polarization will have a nontrivial frequency function that cannot be described by the blackbody spectrum anymore. Therefore, although equation (15) can be used to convert all the results in terms of temperature fluctuations, in order to see the complete frequency dependence of the signal and for the sake of conceptual clarity we refrain from using this conversion throughout the calculations.

In order to get an estimate of the amplitude of the TinPol and TinIn signals for non-moving clusters, we set the bSZ terms equal to zero and use the numerical values of the $a^{T}_{2m}$s from the de-boosted Planck SMICA temerature map (Polastri et al., 2015) as an estimate of the quadrupole moment that a galaxy clusters at $z\approx 0$ would observe. Figure 2 shows the angular distribution of these signals over the whole sky. Both signals have similar frequency dependencies but they have different angular distributions over the sky. The TinPol and TinIn signals reach the maximum values of $\Delta(Q+iU)^{\text{max}}_{\nu}=3.6\tilde{F}_{\nu}(\overline{T}_{0})\tau\mu\textnormal{K}$ and $\Delta I^{\text{max}}_{\nu}=18.6\tilde{F}_{\nu}(\overline{T}_{0})\tau\mu\textnormal{K}$. At $\nu=\text{217 GHz}$ where the tSZ signal vanishes, these values correspond to $\Delta(Q+iU)^{\text{max}}_{\text{217 GHz}}\approx 1.7\tau$kJy/sr and $\Delta I^{\text{max}}_{\text{217 GHz}}\approx 9.1\tau$kJy/sr. The reader should keep in mind that at higher redshifts the amplitude and angular dependence of the signal will be different due to the change in $a^{T}_{2m}$ coefficients. However, this change is very small up to $z\sim 1$ Seto and Pierpaoli (2005) so for close by clusters we expect the distortion maps to be highly correlated with the ones in figure 2.

In actual observations, the intensity distortion map in the direction of galaxy clusters contains both TinIn and bSZ effects. Since these effects have the same frequency dependence, it is not easy to separate them without prior knowledge of the temperature anisotropies at higher redshifts. Therefore, the TinIn effect would not be directly observable as opposed to the TinPol. However, by looking at the definition of the TinIn effect in equation (12) and its angular distribution in figure 2, it is evident that at $z=0$ this effect is equal to our local quadrupole rescaled by a factor of $0.9\tau$ (compare with figure 17 of (Ade et al., 2014)). Therefore, we expect the cluster intensity distortion map over the whole sky at low redshifts to be highly correlated with the local quadrupole observed at $z=0$. This correlation can be used to amplify the TinIn signal and separate it from bSZ for clusters at low redshifts. Moreover, using the definition of the spin-2 spherical harmonics through (Goldberg et al., 1967; Hu and White, 1997)

with the operators $\eth^{\pm 2}$ defined as

it is easy to see that the $\eth^{\pm 2}$ derivative of the TinIn map in equation (12) is just a rescaling of the TinPol in (13). Therefore the second derivative of the quadrupole moment of the intensity distortion map is strongly correlated with the polarization distortion map. Furthermore, since TinIn itself is correlated with the local quadrupole at low redshifts, the second derivative of the quadrupole map is also highly correlated with the TinPol. In other words the second derivative of the TinIn map at low redshifts (bottom panel of figure 2) and the local quadrupole, both reproduce a rescaled version of the TinPol map (top panel of figure 2) and can be used to enhance the signal to noise ratio for this effect.

For typical values of $\beta$ the Doppler shift in equation (19) is small. This allows us to use the following Taylor expansion:

where the differential operators $\mathcal{D}_{\nu_{c}}^{(kj)}$ are defined in the appendix (see B.1.1). The superscripts are chosen such that individual terms on the RHS are in $\beta^{k}\mathcal{D}_{\nu_{c}}^{(kj)}Y_{jn}(\hat{\bm{\gamma}}_{c})$ format; they are simply labels to distinguish these operators and they have no tensorial meaning. Each operator $\mathcal{D}_{\nu_{c}}^{(kj)}$ consists of frequency derivatives up to the order $\partial_{\nu_{c}}^{k}\equiv\partial^{k}/\partial\nu^{k}|_{\nu=\nu_{c}}$. The trivial operator $\mathcal{D}_{\nu_{c}}^{(00)}$ only changes the argument of its following function from $\nu$ to $\nu_{c}$ and is defined for consistency in notation. In this equation, since the Doppler factor $(\nu_{c}/\nu_{cmb})^{3}$ only depends on powers of $\mu_{c}=\hat{\bm{\gamma}}_{c}.\hat{\bm{\beta}}$, we have written the expansion in terms of Legendre polynomials and separated the $\hat{\bm{\gamma}}_{c}$ and $\hat{\bm{\beta}}$ dependence using the addition theorem

Equation (22) introduces the leakage of the intensity harmonic coefficients $a^{I}_{\ell m}$s due to the Doppler effect. The second line of this equation shows the Doppler leakage of the first neighbors of the $\ell^{\text{th}}$ multipole into it. Using the addition property of the spherical harmonics $Y_{1m}\times Y_{\ell m}\propto Y_{(\ell\pm 1)m}$, it is easy to see that in the absence of the aberration effect, after substituting this equation back into (21) the observed $a^{I_{c}}_{\ell^{\prime}m^{\prime}}$s in the moving frame have contributions from $a^{I}_{(\ell^{\prime}\pm 1)m^{\prime}}$ coefficients in the CMB frame up to first order in $\beta$. Similarly, the third line in equation (22) brings in contributions from $a^{I}_{(\ell^{\prime}\pm 2)m^{\prime}}$ to second order in $\beta$.

The angle dependence of this expansion is due to the fact that the ratio of the observed frequencies in each frame depends on the observation angle. To first order in $\beta$ the Doppler factor $(\nu_{c}/\nu_{cmb})^{3}$ only depends on the cosine of the angle between the incoming photons and the direction of motion of the cluster so it will distort the $\ell^{\text{th}}$ multipole in the moving frame by a factor of $\mu_{c}=P_{1}(\hat{\bm{\gamma}}_{c}.\hat{\bm{\beta}})$. Therefore it is maximal when the line of sight is parallel to the direction of motion and is zero when it is perpendicular, as there is no Doppler shift to first order in $\beta$ in these directions. It is evident from equation (22) that the Doppler effect distorts both the frequency spectrum of the observed multipoles (frequency leakage), and their angular spectrum (geometrical leakage). Keep in mind that the intensity in every direction is still a blackbody and the Doppler effect only distorts the spectrum of the multipoles. In particular it draws in the first neighbors of each multipole $(\ell\pm 1)$ with the frequency operator $\mathcal{D}_{\nu_{c}}^{(11)}(\propto\partial_{\nu_{c}}^{1})$ to first order in $\beta$, the second neighbors $(\ell\pm 2)$ with the frequency operator $\mathcal{D}_{\nu_{c}}^{(22)}(\propto\partial_{\nu_{c}}^{2})$ to second order in $\beta$, and so on.

We use equation (20) to expand $Y_{\ell m}(\hat{\bm{\gamma}}_{cmb})$ in terms of $Y_{\ell m}(\hat{\bm{\gamma}}_{c})$ (see appendix B.1.2)

The rotation operators $\Omega^{(k,j)}_{\ell mM^{\prime}}(\hat{\bm{\beta}})$ consist of an active and passive rotation on spherical harmonics, performed by two Wigner-D matrices (equation (B.24)). The operators $\Omega^{(1,0)}_{\ell mM^{\prime}}(\hat{\bm{\beta}})$, $\Omega^{(2,+1)}_{\ell mM^{\prime}}(\hat{\bm{\beta}})$ and $\Omega^{(2,-1)}_{\ell mM^{\prime}}(\hat{\bm{\beta}})$ are identically zero. Unlike Doppler effect, the aberration expansion is not frequency dependent and it only induces an angular distortion to the multipoles. The sum limits of the index $j$ in equation (24) show that up to order $\beta^{k}$, aberration distorts the multipole $\ell$ by drawing in its $k^{\text{th}}$ nearest neighbors.

In a moving frame, the Doppler and aberration effects both induce geometrical leakage of nearby multipoles into each other, but since their dependence on $\hat{\bm{\beta}}$ is not the same, their contribution to the final observed $a^{I_{c}}_{\ell m}$ can be different. Since the Doppler effect is related to the frequency derivative of the anisotropies, it is usually the dominant effect (appendix A). The derivatives can amplify the Doppler terms by large factors and also shift the peak of the frequency spectrum of the $a^{I_{c}}_{\ell m}$s observed by the cluster.

Using the transformation between the Intensity multipoles in the CMB frame and the cluster’s frame, it is now easy to find the Intensity and polarization induced in the direction of a moving cluster. We simply need to replace the terms on the RHS of equations (7) and (8) with those calculated from equation (25). Although the results derived here will be only valid for an observer in the cluster’s frame, converting it to any arbitrary frame would be trivial using the transformations (19) and (20). One can first transform the signal back to the CMB rest frame and then Lorentz boost it into the frame of a general observer using the same equations. This procedure is outlined in great detail in Refs. (Chluba et al., 2012a) and (Nozawa and Kohyama, 2013).

We start with the change in polarization, because unlike intensity it directly probes the quadrupole observed by the cluster, and it is not distorted by the higher multipoles of the temperature anisotropies that induce the bSZ effect (see equations (5) and (6) in §II.2.1). Equation (8) in the cluster’s frame is

where the quadrupole-induced polarization on the RHS is now proportional to the primordial temperature quadrupole, as well as its first neighboring multipoles (the dipole and the octupole) to first order in $\beta$, and its second neighbors (the monopole and the hexadecapole) to second order in $\beta$ and so on. The first few terms of this expansion are as follows

with the following definitions

The first term in equation (28) is the quadrupole-induced polarization (T₂inPol) that was already present for a non-moving cluster (compare with equation (8)). The monopole-induced term, defined in equation (30), is the well known kSZ polarization effect (kSZPol). In order to make this expression look more familiar, we write the frequency function and geometrical factor explicitly to obtain

where $x_{c}\equiv h\nu_{c}/k\overline{T}_{z}$ and $\vartheta_{\beta}$ and $\varphi_{\beta}$ are the polar and azimuthal Euler angles between $\hat{\bm{\gamma}}_{c}$ and $\hat{\bm{\beta}}$ defined as (Okamoto and Hu, 2002)

In equation (33), $\beta\sin\vartheta_{\beta}=\beta_{\perp}$ is the transverse component of the peculiar velocity with respect to the observer’s line of sight. This expressions generalizes the the results of (Sunyaev and Zeldovich, 1981; Audit and Simmons, 1999) in agreement with (Sazonov and Sunyaev, 1999). The $e^{\pm 2i\varphi_{\beta}}$ factor decomposes the polarization signal into the $Q$ and $U$ components, respectively with factors of $\cos(2\varphi_{\beta})$ and $\sin(2\varphi_{\beta})$. Therefore the combination of the $Q$ and $U$ polarization maps can be used to determine the exact three dimensional direction of the cluster’s velocity vector.

The second line of equation (28) shows the dipole and octupole-induced polarization (T₁inPol and T₃inPol). We will refer to these terms collectively as T_1,3inPol. The most interesting feature of these new terms is that they induce different frequency and angular distortions to the TinPol at different frequency bands. Note that in equations (29)-(32), the dependence of the signals on the observation direction $\hat{\bm{\gamma}}_{c}$, and the velocity direction of the cluster $\hat{\bm{\beta}}$ are separated as a result of our generalized approach. For the T_1,3inPol effect, changing any of these directions changes the total frequency function of the signal. In figures 5 and 6, by fixing one of these directions, we show the dependence of the signal on the other one. The top panel of figure 5 shows how much the dipole and octupole can change the TinPol signal and its frequency dependence in the direction of the north galactic pole for an exaggerated value of $\beta$. The T₂inPol effect, which is the only term that does not depend on the velocity of the cluster, reaches its maximum at 218 GHz. The peak value of this signal is 1.7 $\tau$kJy/sr $(3.6\tau\mu\textnormal{K})$ which for a cluster with optical depth $\tau\approx 0.02$ is equal to 34 Jy/sr $(0.07\mu\textnormal{K})$. In the bottom panel we have separated the T_1,3inPol for comparison with the T₂inPol and kSZPol for realistic values of $\beta$. Since the leakage due to the Doppler effect is typically dominant over the aberration effect (see figure 3), it shifts the position of the peak of the dipole and octupole-induced polarizations with respect to the quadrupole-induced signal. The direction of motion of the cluster $\hat{\bm{\beta}}$ is chosen to maximize the T_1,3inPol in this specific direction in the sky. For bullet-like clusters ($\beta=0.015$), kSZPol is dominant over the T₂inPol but for low velocities ($\beta=0.001$) it is approximately smaller by a factor of 10. The reason for kSZPol’s fast drop is its dependence on $\beta_{\perp}^{2}$. Although T_1,3inPol is much smaller than kSZPol at high velocities, it induces a 10% correction to T₂inPol at 217 GHz. For smaller velocities this effect is about 5% of the kSZPol at low frequencies. Although kSZPol is typically the dominant effect in equation (28) for larger values of $\beta$, it is nevertheless distinguishable from the other TinPol effects due to its distinct frequency dependence and large amplitude. The leakage of the hexadecapole has been neglected in equation (28) because it only contributes 0.2% (2%) to the total polarization in the Rayleigh-Jeans (Wien) frequencies.

Doppler and aberration leakages of the dipole and octupole also induce an angular distortion to the TinPol signal. Figure 6 shows the shift in the location of the minima and maxima of the signal to represent the angular change of the overall signal at different frequencies, for clusters moving with $\vec{\bm{\beta}}=-0.05\hat{\bm{z}}$. The maxima and minima of the TinPol signal are respectively $32^{\circ}$ and $14^{\circ}$ apart at 217 GHz and 300 GHz. For a more realistic velocity $\beta=0.015$, these angle separations are about $15^{\circ}$ for the maxima and $5^{\circ}$ for the minima. Note that the angular distortions in this plot are calculated for one particular direction of motion of clusters. Figure 6 simply shows that the induced angular distortions are frequency dependent and unless the clusters in some observation direction at a certain redshift bin have a coherent bulk motion, the expected signal would not change as described in this plots. Indeed, for a more realistic map one would need to populate the sky with clusters moving in different directions, but even in that case the angular dependence of the signal will change at different frequencies. This feature can be extremely helpful in a multi-frequency and whole-sky survey to identify and extract the TinPol effects from foreground distortions and even kSZ and tSZ effects which are not expected to have similar distributions over the sky. Also, the angle dependence of this distortion, changes the ratio between T_1,3inPol and T₂inPol effects at different observation directions, and obviously makes T_1,3inPol the dominant effect in the areas where T₂inPol is small.

The top panel of figure 7 shows a similar map to figure 6 at 300GHz, which is the frequency at which the peak of T_1,3inPol typically lies. We can see that T_1,3inPol is larger than 10% (20%) of T₂inPol over roughly 20% (10%) of the sky. Here, the average contribution of the T_1,3inPol over the whole sky is 10%. The bottom panel shows different polarization signals in an arbitrary direction $(l,b)=(-54^{\circ},-27^{\circ})$ which is indicated with a black dot in the top panel. In this direction the amplitude of T_1,3inPol for $\beta=0.015$ is comparable to T₂inPol and even becomes dominant at 386 GHz. The peak value of T_1,3inPol is at 289 GHz and at this frequency it is as large as 70% of the T₂inPol. For the smaller values of $\beta=0.003$ and $\beta=0.001$, T_1,3inPol is respectively 13% and 4% of the T₂inPol at the peak frequency.

Similar to the case of polarization, for intensity we can rewrite equation (7) in the cluster’s frame as

where the new bSZ⁽¹⁾ in the cluster’s frame is

where

is the dipole-induced bSZ term, which was already present even in the case of the non-moving cluster,

is the first order monopole-induced intensity, known as the kSZ intensity effect and

is the leakage of the quadrupole into the dipole-induced bSZ. Similarly, the new TinIn in the cluster’s frame would be equal to

where again

is the quadrupole-induced intensity which was already derived for non-moving clusters,