Detecting the polarization induced by scattering of the microwave background quadrupole in galaxy clusters

We analyze the feasibility of detecting the polarization of the cosmic microwave background (CMB) caused by scattering of the remote temperature quadrupole by galaxy clusters with forthcoming CMB polarization surveys. For low-redshift clusters, the signal is strongly correlated with the local large-scale temperature and polarization anisotropies, and the best prospect for detecting the cluster signal is via cross-correlation. For high-redshift clusters, the correlation with the local temperature is weaker and the power in the uncorrelated component of the cluster polarization can be used to enhance detection. We derive linear and quadratic maximum-likelihood estimators for these cases, and forecast signal-to-noise values for the Sunyaev-Zeldovich (SZ) surveys of a Planck-like mission and SPTPol. Our estimators represent an optimal “stacking” analysis of the polarization from clusters. We find that the detectability of the effect is sensitive to the cluster gas density distribution, as well as the telescope resolution, cluster redshift distribution, and sky coverage. We find that the effect is too small to be detected in current and near-future SZ surveys without dedicated polarization follow-up, and that a rms noise on the Stokes parameters of roughly $1\mu \mathrm{K}$-arcmin for each cluster field is required for a $2\sigma$ detection, assuming roughly 550 clusters are observed. We show that ACTPol is in a better position to observe the effect than SPTPol due to its advantageous survey location on the sky, and we discuss the novel spatial dependence of the signal. We discuss and quantify potential biases from the kinetic part of the signal caused by the relative motion of the cluster with respect to the CMB, and from the background CMB polarization behind the cluster, discussing ways in which these biases might be mitigated. Our formalism should be important for next-generation CMB polarization missions, which we argue will be able to measure this effect with high signal-to-noise. This will allow for an important consistency test of the $\mathrm{\Lambda }\mathrm{CDM}$ model on scales that are inaccessible to other probes.

Figure 1

Autospectrum of the polarization due to the primordial quadrupole at $z=0.025$ (red, dot-dashed), $z=0.425$ (green, dashed) and $z=0.975$ (blue, solid).

Figure 2

Left: Cross spectrum of the local CMB temperature and the cluster polarization due to the primordial quadrupole at $z=0.025$ (red, dot-dashed), $z=0.425$ (green, dashed) and $z=0.975$ (blue, solid). Right: Correlation coefficients of the cross spectra on the left. Note that in our polarization convention, the cross spectrum is negative over this range of $l$.

Figure 3

As Fig. 2, but for the cross spectrum of the local CMB $E$ mode and cluster polarization.

Figure 4

Correlation function ${\mathrm{\Xi }}^{+}$ at zero lag ($\beta =0$) and equal redshift $z$.

Figure 5

Scaled optical depth profiles as a function of the scaled angular radius $\theta /{\theta }_s$, where ${\theta }_s=r_s/d_A$, for the $\beta$ model (blue, top solid), and the cNFW model (green, lower solid). We have assumed a cluster with $R_{500}=1\mathrm{Mpc}$. Note that the optical depth profiles used in the $S/N$ calculation are truncated at $\theta /{\theta }_s=5c_{500}=5.885$.

Figure 6

Redshift distribution of SZ clusters from the Planck 2013 catalog.

Figure 7

Forecast redshift distribution of SZ clusters for SPTPol.

Figure 8

Magnitude of the correlated part of the polarization field sourced at $z=0.05$ (top) and $z=1.35$ (bottom) in Galactic coordinates. Units are $\mu \mathrm{K}$.

Figure 9

$S/N$ for the Planck-like survey with the $\beta$ model (green, lower solid) and cNFW model (green, lower dashed), and SPTPol with the $\beta$ model (blue, upper solid) and cNFW model (blue, upper dashed), assuming that all clusters in each survey are located in a single redshift bin.