Finding the distribution of matter using lenses - I: deconvolution-based reconstruction with CMB lensing

Abstract

The matter power spectrum is one of the primary statistical descriptors of the large-scale distribution of matter in the Universe and provides a powerful probe of cosmic structure formation. Measurements of cosmic microwave background (CMB) lensing offer an integrated view of the matter distribution over a wide range of redshifts, enabling the reconstruction of the underlying matter power spectrum. In this work, we reconstruct the reference linear matter power spectrum $ P_\text{lin}(k,0)$ from the baseline joint CMB lensing measurements of Planck PR4, ACT DR6, and SPT-3G using a covariance-weighted modified Richardson-Lucy(MRL) deconvolution algorithm. The reconstructed spectrum is found to be consistent with the fiducial linear prediction on large scales, while exhibiting a systematic enhancement for $k \gtrsim 0.1\,{\rm Mpc}^{-1}$, where nonlinear gravitational evolution becomes important. To investigate this behavior, we introduce a scale-dependent correction factor, $A(k)$, defined through $P(k)=A(k)\,P_{\rm nl}(k),$ where $P_{\rm nl}(k)$ is the fiducial nonlinear matter power spectrum obtained from 2LPT simulations. The reconstructed correction factor remains consistent with unity within $2\sigma$ confidence over the reconstructed range, indicating that the observed enhancement is well explained by the standard nonlinear evolution of the matter power spectrum. In addition, the reconstruction shows agreement with the fiducial BAO template around the BAO feature at $k\sim(0.04-0.06)\ {\rm Mpc}^{-1}$, indicating that some BAO-scale information survives the lensing projection.