The Information Content of Projected Galaxy Fields

The power spectrum of the nonlinearly evolved large-scale mass distribution recovers only a minority of the information available on the mass fluctuation amplitude. We investigate the recovery of this information in 2D "slabs" of the mass distribution averaged over $\approx100$~$h^{-1}$Mpc along the line of sight, as might be obtained from photometric redshift surveys. We demonstrate a Hamiltonian Monte Carlo (HMC) method to reconstruct the non-Gaussian mass distribution in slabs, under the assumption that the projected field is a point-transformed Gaussian random field, Poisson-sampled by galaxies. When applied to the \textit{Quijote} $N$-body suite at $z=0.5$ and at a transverse resolution of 2~$h^{-1}$Mpc, the method recovers $\sim 30$ times more information than the 2D power spectrum in the well-sampled limit, recovering the Gaussian limit on information. At a more realistic galaxy sampling density of $0.01$~$h^3$Mpc$^{-3}$, shot noise reduces the information gain to a factor of 5 improvement over the power spectrum at resolutions of 4~$h^{-1}$Mpc or smaller.