  We investigate the amount of primordial information that can be reconstructed
from spectroscopic galaxy surveys, as well as what sets the noise in
reconstruction at low wavenumbers, by studying a simplified universe in which
galaxies are the Zeldovich displaced Lagrangian peaks in the linear density
field. For some of this study, we further take an intuitive linearized limit in
which reconstruction is a convex problem but where the solution is also a
solution to the full nonlinear problem, a limit that bounds the effectiveness
of reconstruction. The linearized reconstruction results in similar cross
correlation coefficients with the linear input field as our full nonlinear
algorithm. The linearized reconstruction also produces similar cross
correlation coefficients to those of reconstruction attempts on cosmological
N-body simulations, which suggests that existing reconstruction algorithms are
extracting most of the accessible information. Our approach helps explain why
reconstruction algorithms accurately reproduce the initial conditions up to
some characteristic wavenumber, at which point there is a quick transition to
almost no correlation. This transition is set by the number of constraints on
reconstruction (the number of galaxies in the survey) and not by where shot
noise surpasses the clustering signal, as is traditionally thought. We further
show that on linear scales a mode can be reconstructed with precision well
below the shot noise expectation if the galaxy Lagrangian displacements can be
sufficiently constrained. We provide idealized examples of nonlinear
reconstruction where shot noise can be outperformed.
