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 $z=1$ 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 five improvement over the power spectrum at resolutions of 4~$h^{-1}$Mpc or
smaller.
