On the information content of the matter power spectrum

TL;DR

This paper analytically examines the matter power spectrum covariance matrix and Fisher information, revealing fundamental bounds on the information content due to non-Gaussian effects, with implications for cosmological parameter estimation.

Contribution

It provides an analytical approximation for the covariance matrix and Fisher information of the nonlinear matter power spectrum, highlighting intrinsic bounds on information content.

Findings

Spectrum information is limited by 'plateaux' caused by the trispectrum.

Effective number of modes is bounded, explaining earlier simulation results.

Measurement precision is limited to percent level by these bounds.

Abstract

We discuss an analytical approximation for the matter power spectrum covariance matrix and its inverse on translinear scales, $k \sim 0.1h - 0.8h/\textrm{Mpc}$ at $z = 0$. We proceed to give an analytical expression for the Fisher information matrix of the nonlinear density field spectrum, and derive implications for its cosmological information content. We find that the spectrum information is characterized by a pair of upper bounds, 'plateaux', caused by the trispectrum, and a 'knee' in the presence of white noise. The effective number of Fourier modes, normally growing as a power law, is bounded from above by these plateaux, explaining naturally earlier findings from $N$-body simulations. These plateaux limit best possible measurements of the nonlinear power at the percent level in a $h^{-3}\textrm{Gpc}^3$ volume; the extraction of model parameters from the spectrum is limited explicitly by their degeneracy to the nonlinear amplitude. The value of the first, super-survey (SS) plateau depends on the characteristic survey volume and the large scale power; the second, intra-survey (IS) plateau is set by the small scale power. While both have simple interpretations within the hierarchical \textit{Ansatz}, the SS plateau can be predicted and generalized to still smaller scales within Takada and Hu's spectrum response formalism. Finally, the noise knee is naturally set by the density of tracers.