Maximum likelihood kinetic Sunyaev-Zel’dovich velocity reconstruction

The kinetic Sunyaev Zel’dovich (kSZ) effect, cosmic microwave background (CMB) temperature anisotropies induced by the scattering of CMB photons from free electrons, will be measured by near-term CMB experiments at high significance. By combining CMB temperature anisotropies with a tracer of structure, such as a galaxy redshift survey, previous literature introduced a number of techniques to reconstruct the radial velocity field. This reconstructed radial velocity field encapsulates the majority of the cosmological information contained in the kSZ temperature anisotropies, and can provide powerful new tests of the standard cosmological model and theories beyond it. In this paper, we introduce a new estimator for the radial velocity field based on a coarse-grained maximum likelihood fit for the kSZ component of the temperature anisotropies, given a tracer of the optical depth. We demonstrate that this maximum likelihood estimator yields a higher fidelity reconstruction than existing quadratic estimators in the low-noise and high-resolution regime targeted by upcoming CMB experiments. We describe implementations of the maximum likelihood estimator in both harmonic space and map space, using either direct measurements of the optical depth field or a galaxy survey as a tracer. We comment briefly on the impact of biases introduced by imperfect reconstruction of the optical depth field.

Figure 1

The primary CMB power spectrum and the kSZ power spectrum using 32 radial bins of equal comoving size between $0.2<z<5$. At large-$\ell$, the kSZ power dominates over the CMB power. It is only once we access these angular scales that we expect the MaxL estimator to show improvements over the QE.

Figure 2

We show the true velocity power spectrum as well as the power of the residuals $v-{\widehat{v}}^{\mathrm{QE}}$ and $v-{\widehat{v}}^{\text{MaxL}}$. On the left is a low-$z$ bin and on the right is a high-$z$ bin. We show ${\widehat{v}}^{\text{MaxL}}$ for increasing resolution $N_{\mathrm{side}}^{\mathrm{out}}$. At low resolution, the “coarse-graining” bias $\beta$ is significant and ${\widehat{v}}^{\text{MaxL}}$ performs worse than ${\widehat{v}}^{\mathrm{QE}}$, but as we increase $N_{\mathrm{side}}^{\mathrm{out}}$ this bias decreases and we get a better reconstruction than ${\widehat{v}}^{\mathrm{QE}}$.

Figure 3

We show the true velocity power spectrum as well as the power of the residuals $v-{\widehat{v}}^{\mathrm{QE}}$ and $v-{\widehat{v}}^{\text{MaxL}}$. On the left is a low-$z$ bin and on the right is a high-$z$ bin. We show ${\widehat{v}}^{\text{MaxL}}$ for an increasing output resolution. Here we see less of an improvement over the QE than we did in Fig. 2 where we did not include the primary CMB. The MaxL residual power is 75% of the QE residual power on small angular scales, and significantly better on large angular scales.

Figure 4

The ratio of the total signal to noise per mode of the MaxL and quadratic estimators. We find that at $N_{\mathrm{side}}^{\mathrm{in}}$ of 1024, the estimators perform similarly; however, as we go to $N_{\mathrm{side}}^{\mathrm{in}}$ of 2048 and higher, the MaxL estimator has higher signal to noise than the QE. In both cases we have chosen a different $N_{\mathrm{side}}^{\mathrm{out}}$ at which the MaxL performs best: if $N_{\mathrm{side}}^{\mathrm{out}}$ is too low, the coarse-graining bias is too large, and if it is too high, the operator $\mathcal{W}$ becomes uninvertible. These $N_{\mathrm{side}}^{\mathrm{out}}$ corresponded to $N_{\mathrm{side}}^{\mathrm{out}}=\{16,64\}$ for $N_{\mathrm{side}}^{\mathrm{out}}=\{1024,2048\}$ respectively.

Figure 5

The true velocity power spectrum $C_L^{vv}$ and the power of the residuals $v-{\widehat{v}}^{QE}$ and $v-{\widehat{v}}^{MaxL}$, using galaxies to estimate the optical depth field. We do this for the MaxL by using the procedure defined in (40). For the QE we use the simulated galaxy maps directly and modify the theoretical weights in the estimator to ensure an unbiased output.