Autodifferentiable likelihood pipeline for the cross-correlation of CMB and large-scale structure due to the kinetic Sunyaev-Zeldovich effect

We develop an optimization-based maximum likelihood approach to analyze the cross-correlation of the cosmic microwave background (CMB) and large-scale structure induced by the kinetic Sunyaev-Zeldovich (kSZ) effect. Our main goal is to reconstruct the radial velocity field of the Universe. While the existing quadratic estimator (QE) is statistically optimal for current and near-term experiments, the likelihood can extract more signal-to-noise in the future. Our likelihood formulation has further advantages over the QE, such as the possibility of jointly fitting cosmological and astrophysical parameters and the possibility of unifying several different kSZ analyses. We implement an autodifferentiable likelihood pipeline in JAX, which is computationally tractable for a realistic survey size and resolution, and evaluate it on the Agora simulation. We also implement a machine-learning-based estimate of the electron density given an observed galaxy distribution, which can increase the signal-to-noise for both the QE and the likelihood method.

Figure 1

Example of power spectra of radial velocity (left) and matter density (right) from Agora simulations (solid blue) compared to theoretical values calculated with camb (dashed black) at $z=1.15$. The power spectra were estimated from the apodized flat-sky projection of $1/12$ of the full-sky AGORA simulation.

Figure 2

Left: halo density (Agora) and galaxy density (LSST Y-10) per sterad per redshift as a function of redshift. Right: power spectra of halo overdensity from Agora and emulated halo overdensity (i.e., Poisson sampled biased matter overdensity) for LSST-Y10 gold sample galaxy density level. At the kSZ length scales of $\ell >4000$ the Agora map is noise dominated, while the emulated LSST map is signal dominated.

Figure 3

Lensed CMB power spectrum along with three different instrumental noise levels and generated kSZ signal (simulation volume only). For CMB-S4, the kSZ (shown only for our simulation volume $0.5<z<3$) is below the noise level, while for the futuristic experiments, it is above the noise level. Here the CMB-S4 noise curve includes ILC foreground deprojection, while the 1, 1 noise means white noise with $N_T=1\mathrm{\mu K}\mathrm{arcmin}$ and ${\theta }_{\mathrm{FWHM}}=1\mathrm{arcmin}$ (without ILC). For either of these experiments, our analysis uses only scales up to $\ell \simeq 6000$ to limit the GPU requirements of this theoretical study.

Figure 4

Velocity reconstruction based on ${\delta }_m$ and ${\theta }^{\mathrm{CMB}}$. Cross-correlation coefficient $r_{\ell }$ (upper row) and noise power spectra (lower row) of QE and MLE as a function of multipole number $\ell$ for three different redshifts. CMB noise and galaxy density are, respectively, $N_{1,1}^{\mathrm{pCMB}}$ ($N_T=1\mathrm{\mu K}\mathrm{arcmin}$ and ${\theta }_{\mathrm{FWHM}}=1\mathrm{arcmin}$, without foregrounds) and $n_{\mathrm{LSST}}^g$ (LSST gold sample). We include data in 60 redshift bins ($0.5<z<3$) with angular resolution ${\ell }_{\max }=6000$ in the reconstruction, which is the resolution limit with our GPU memory.

Figure 5

Left: “de-kSZing” of primary CMB. Cross-correlation coefficient $r(\ell )$ of ${\widehat{\theta }}^{\mathrm{MAP}}$ or ${\theta }^{\mathrm{obs}}$ with ${\theta }^{\mathrm{pCMB}}$ for two noise configurations: $N_{\mathrm{HD}}^{\mathrm{pCMB}},10\times n_g^{\mathrm{LSST}}$ and $N_{1,1}^{\mathrm{pCMB}},n_g^{\mathrm{LSST}}$. As expected, the reconstructed primary CMB cross-correlates better with the true primary CMB than the observed CMB does. The de-kSZing is not large here because we do not include an estimate of small-scale velocities from small-scale galaxies (see main text). Right: power spectra of reconstructed and true kSZ signal (red and purple) and pCMB (green and orange) for the $N_{\mathrm{HD}}^{\mathrm{pCMB}},10\times n_g^{\mathrm{LSST}}$ noise configuration (correspondent to blue curves on the left).

Figure 6

Velocity reconstruction based on ${\delta }_h$ (Agora simulation halos) and ${\theta }^{\mathrm{CMB}}$. Cross-correlation coefficient $r_{\ell }$ (upper row) and noise power spectra (lower row) of QE and MLE as a function of multipole number $\ell$ for two different CMB noise levels: $N_{1,1}^{\mathrm{pCMB}}$ (left) and $N_{S4}^{\mathrm{pCMB}}$ (right) at $z=0.8$. In the reconstruction, we include data in 60 redshift bins ($0.5<z<3$) with angular resolution ${\ell }_{\max }=6000$ in the reconstruction. For $N_{1,1}^{\mathrm{pCMB}}$, we see an improvement by a factor of 2, while for $N_{S4}^{\mathrm{pCMB}}$ (including ILC) we see no improvement.

Figure 7

Cross-correlation $1-r_{e\widehat{e}}^2(\ell )$ of the true electron field with its estimate from the analytic template, Eq. (42), compared to the neural network estimate for two halo density levels: $n_{g,\mathrm{LSST}}$ (left) and $10\times n_{g,\mathrm{LSST}}$ (right). We find a significant improvement in the cross-correlation due to the neural network, which turns out to be larger with higher shot noise (lower galaxy density). Note, however, that in our simulations ${\delta }_e={\delta }_m$ and results may change with realistic hydrodynamic simulations and galaxy properties.

Figure 8

Left: cross-correlation coefficient $r_{v\widehat{v}}(\ell )$ of true and QE-reconstructed velocity field based on the standard and the neural network estimate of the electron density. Right: corresponding power spectra and residuals. The experimental parameters are similar to an $\mathrm{LSST}\times \mathrm{SO}$ analysis. We find a small 25% improvement when the QE uses the neural network template, rather than the analytic template [Eq. (42)].

Figure 9

Power spectra of MAP-estimated velocities ${\widehat{v}}_{1,2}$ and residuals with the true fields ${\epsilon }_{1,2}$ for two levels of generated kSZ signal along with estimated errors from the Hessian. The two errors match very well, i.e., the Hessian errors are identical to the true residual errors.