On the primordial information available to galaxy redshift surveys

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 to those of reconstruction algorithms that have been applied to N-body simulations (as well as of our nonlinear algorithms), 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 galaxy Lagrangian displacements can be sufficiently constrained. We provide idealized examples of nonlinear reconstruction where shot noise can be outperformed.