Reconstructing Projected Matter Density from Cosmic Microwave Background

TL;DR

This work develops a direct 2‑D reconstruction of the projected matter density $\kappa$ from gravitational lensing of the cosmic microwave background by using quadratic combinations of CMB temperature derivatives. The authors formulate a formalism with derivative pairs ${\cal S}$, ${\cal Q}$, ${\cal U}$ that couple to $\kappa$ and the shear, and derive estimators for the convergence power spectrum $C_l^{\kappa\kappa}$ in both large‑ and small‑scale limits, including noise and beam effects. Through simulations, they demonstrate cluster mass‑profile recovery when small‑scale power (e.g., OV effect) is present, and show that even without a direct map of $\kappa$, the power spectrum can be recovered or cross‑correlated with external tracers to yield detections. They also analyze practical requirements for MAP/Planck observations and highlight cross‑correlation opportunities with SZ, X‑ray background, and galaxy surveys to enhance the cosmological leverage of the method.

Abstract

Gravitational lensing distorts the cosmic microwave background (CMB) anisotropies and imprints a characteristic pattern onto it. The distortions depend on the projected matter density between today and redshift $z \sim 1100$. In this paper we develop a method for a direct reconstruction of the projected matter density from the CMB anisotropies. This reconstruction is obtained by averaging over quadratic combinations of the derivatives of CMB field. We test the method using simulations and show that it can successfully recover projected density profile of a cluster of galaxies if there are measurable anisotropies on scales smaller than the characteristic cluster size. In the absence of sufficient small scale power the reconstructed maps have low signal to noise on individual structures, but can give a positive detection of the power spectrum or when cross correlated with other maps of large scale structure. We develop an analytic method to reconstruct the power spectrum including the effects of noise and beam smoothing. Tests with Monte Carlo simulations show that we can recover the input power spectrum both on large and small scales, provided that we use maps with sufficiently low noise and high angular resolution.

Figures (9)

• Figure 1: The upper panel shows the correlation function of $\tilde{S}$, $\tilde{Q}$, $\tilde{U}$ for SCDM. The lower panel shows the power spectra of $\cal SS$, $\cal EE$, $\cal BB$ and $\cal SE$.
• Figure 2: Simulation results for the covariances of $N_l^{\cal S S}$, $N_l^{\cal E E}$$N_l^{\cal B B}$ and $N_l^{\cal S E}$ normalized to their values for Gaussian noise given in equation (\ref{['covariances']}). The Gaussian approximation is an excellent one for $l<1000$.
• Figure 3: Top left panel: input cluster on a 6'$\times$6' field. Top right panel: unlensed CMB map. We assumed that the Ostriker-Vishniac effect could be detected with sufficient signal to noise to be used in the reconstruction of $\kappa$. Bottom left: CMB field after being lensed by the cluster. Bottom right: the background shows the $\cal S$ field while the rods represent the shear variables ${\cal Q}$ and ${\cal U}$, both of which can be used to reconstruct the density profile.
• Figure 4: Reconstructed radial profile using the data in Figure 2. The points marked with ${\cal E}$ (${\cal S}$) correspond to the reconstruction of $\kappa$ based on ${\cal E}$ (${\cal S}$). The points marked unlensed correspond to the result of applying the ${\cal E}$ reconstruction to the unlensed field. The input $\kappa$ was taken to have zero mean.
• Figure 5: (a) $N^{\cal S S}_l$, $N^{\cal E E}_l$, $N^{\cal B B}_l$ and $N^{\cal S E}_l$ power spectra for the unlensed CMB field. The lines correspond to the results of equation (\ref{['ps4']}). Also shown is the weighted average of the recovered power spectra from ${\cal S}$, ${\cal E}$ and their cross correlation together with the input $4 C^{\kappa \kappa}_l$. (b) Ratio of the recovered power spectra to the input $C^{\kappa \kappa}_l$. The signal in ${\cal B}$ is also shown.