Reconstruction of the remote dipole and quadrupole fields from the kinetic Sunyaev Zel’dovich and polarized Sunyaev Zel’dovich effects

The kinetic Sunyaev Zel’dovich (kSZ) and polarized Sunyaev Zel’dovich (pSZ) effects are temperature and polarization anisotropies induced by the scattering of cosmic microwave background (CMB) photons from structure in the post-reionization universe. In the case of the kSZ effect, small angular scale anisotropies in the optical depth are modulated by the CMB dipole field, i.e., the CMB dipole observed at each spacetime point, which is sourced by the primordial dipole and especially the local peculiar velocity. In the case of the pSZ effect, similar small-scale anisotropies are modulated by the CMB quadrupole field, which receives contributions from both scalar and tensor modes. Statistical anisotropies in the cross-correlations of CMB temperature and polarization with tracers of the inhomogeneous distribution of electrons provide a means of isolating and reconstructing the dipole and quadrupole fields. In this paper, we present a set of unbiased minimum variance quadratic estimators for the reconstruction of the dipole and quadrupole fields, and forecast the ability of future CMB experiments and large-scale structure surveys to perform this reconstruction. Consistent with previous work, we find that a high fidelity reconstruction of the dipole and quadrupole fields over a variety of scales is indeed possible, and demonstrate the sensitivity of the pSZ effect to primordial tensor modes. Using a principle component analysis, we estimate how many independent modes could be accessed in such a reconstruction. We also comment on a few first applications of a detection of the dipole and quadrupole fields, including a reconstruction of the primordial contribution to our locally observed CMB dipole, a test of statistical homogeneity on large scales from the first modes of the quadrupole field, and a reconstruction technique for the primordial potential on the largest scales.

Figure 1

Left: Signal to noise per dipole field mode ${\overline{v}}_{\mathrm{eff},\ell m}$ for a full sky experiment with experimental sensitivity oriented at CMB S4 and LSST (see main text), using the redshift binning given in Table 1. Right: Same with 12 bins, dividing each previous bin in two.

Figure 2

Signal to noise from scalar perturbations per quadrupole field mode $a_{\ell m}^{q,E}$ for a full sky experiment with experimental sensitivity oriented at CMB S4 and LSST, using the redshift binning in Table 1. Left: Using $E$ and $B$ modes. Right: Using only $E$ modes. This illustrates the power of $B$ modes for this application.

Figure 3

Top: Signal to noise for tensors with $r=0.1$ per quadrupole field mode $a_{\ell m}^{q,E}$ (left) and $a_{\ell m}^{q,B}$ (right) for a full sky experiment with experimental sensitivity oriented at CMB S4 and LSST, using the redshift binning in Table 1, using $E$ and $B$ modes. Bottom: Using the same configuration but reducing the CMB noise by a factor of 10.

Figure 4

Explained variance as a function of the number of principal components. Left: Dipole field ${\overline{v}}_{LM}$ with ${\ell }_{\max }=50$ and 6 redshift bins. Right: Quadrupole field $a_{LM}^{q,E}$ with ${\ell }_{\max }=4$ and 6 redshift bins.

Figure 5

The correlation coefficient equation (64) between the CMB temperature quadrupole and the $E$-mode type remote quadrupole as a function of bin. We assume a 6-bin configuration with reconstruction noise on the remote quadrupole expected by CMB S4 combined with LSST. The error bars are computed from the Fisher estimate in Eq. (65).

Figure 6

Ratio of the binned power spectrum $C_l^{\overline{v}\overline{v}}$ of the dipole field computed using only the peculiar velocity term [the third term in Eq. (a3)] to the full remote dipole power spectrum.

Figure 7

The correlation function of the dipole and quadrupole field is plotted at $({\chi }_{\alpha },{\chi }_{\alpha }+\delta \chi )$, where ${\chi }_{\alpha }$ is fixed to be the midpoint of a redshift bin. The redshift values are given across the top axis, and the corresponding comoving distance values along the bottom axis, consistent with the binning in Table 1. Specifically, the different curves from left to right fix ${\chi }_{\alpha }=0.16$, 0.48, 0.79, 1.11, 1.43, 1.75, in units of $H_0^{-1}$. Top: Correlation of the quadrupole field $q_{2M}(\chi )$, which has a very large correlation length. Middle: Correlation of the dipole field $v_{1M}(\chi )$. The correlations are smaller than in the quadrupole case because the effective velocities are dominated by the Doppler effect. Bottom: Correlation of the dipole field $v_{5M}(\chi )$. The correlation length is much smaller than the bin size, in particular for low redshift bins.

Figure 8

Power spectrum $C_l^{vv}$ of the dipole field as a function of redshift $\chi$. We compare $C_l^{vv}$, the power evaluated in the bin center, with $C_l^{\overline{v}\overline{v}}$, the power of the average field. At $L=1$ the average field is comparable in size to the unaveraged field, while for $L=5$ the cancellations are important. The effect of cancellations is expectedly larger for larger bins.

Figure 9

Power spectra of primordial tensor modes contributions to the CMB temperature quadrupole from $E$ modes [$C_{\ell }^{T,E}(\chi ,{\chi }^{\prime })$, plain lines) and $B$ modes ($C_{\ell }^{T,B}(\chi ,{\chi }^{\prime })$, dashed lines], evaluated at the same redshift, $\chi ={\chi }^{\prime }={\chi }_{\alpha }$, taken to be the midpoints of the bins given in Table 1. The power increases with redshift, and is roughly similar for $E$ and $B$ modes.