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 Hus spectrum response formalism. Finally, the noise knee is naturally set by the density of tracers.
Owing to the mass-sheet degeneracy, cosmic shear maps do not probe directly the Fourier modes of the underlying mass distribution on scales comparable to the survey size and larger. To assess the corresponding effect on attainable cosmological parame
ter constraints, we quantify the information on super-survey modes in a lognormal model and, when interpreted as nuisance parameters, their degeneracies to cosmological parameters. Our analytical and numerical calculations clarify the central role of super-sample covariance (SSC) in shaping the statistical power of cosmological observables. Reconstructing the background modes from their non-Gaussian statistical dependence to small scales modes yields the renormalized convergence. This diagonalizes the spectrum covariance matrix, and the information content of the corresponding power spectrum is increased by a factor of two over standard methods. Unfortunately, careful calculation of the Cramer-Rao bound shows that the information recovery can never be made complete, any observable built from shear fields, including optimal sufficient statistics, are subject to severe information loss, typically $80%$ to $90%$ below $ell sim 3000$ for generic cosmological parameters. The lost information can only be recovered from additional, non-shear based data. Our predictions hold just as well for a tomographic analysis, and/or full sky surveys.
The sufficient statistics of the one-point probability density function of the dark matter density field is worked out using cosmological perturbation theory and tested to the Millennium simulation density field. The logarithmic transformation is rec
overed for spectral index close to $-1$ as a special case of the family of power transformations. We then discuss how these transforms should be modified in the case of noisy tracers of the field and focus on the case of Poisson sampling. This gives us optimal local transformations to apply to galaxy survey data prior the extraction of the spectrum in order to capture most efficiently the information encoded in large scale structures.
A large fraction of this thesis is dedicated to the study of the information content of random fields with heavy tails, in particular the lognormal field, a model for the matter density fluctuation field. It is well known that in the nonlinear regime
of structure formation, the matter fluctuation field develops such large tails. It has also been suggested that fields with large tails are not necessarily well described by the hierarchy of $N$-point functions. In this thesis, we are able to make this last statement precise and with the help of the lognormal model to quantify precisely its implications for inference on cosmological parameters : we find as our main result that only a tiny fraction of the total Fisher information of the field is still contained in the hierarchy of $N$-point moments in the nonlinear regime, rendering parameter inference from such moments very inefficient. We show that the hierarchy fails to capture the information that is contained in the underdense regions, which at the same time are found to be the most rich in information. We find further our results to be very consistent with numerical analysis using $N$-body simulations. We also discuss these issues with the help of explicit families of fields with the same hierarchy of $N$-point moments defined in this work. A similar analysis is then applied to the convergence field, the weighted projection of the matter density fluctuation field along the line of sight, with similar conclusions. We also show how simple mappings can correct for this inadequacy, consistently with previous findings in the literature (Abridged) .
We study the constraining power on primordial non-Gaussianity of future surveys of the large-scale structure of the Universe for both near-term surveys (such as the Dark Energy Survey - DES) as well as longer term projects such as Euclid and WFIRST.
Specifically we perform a Fisher matrix analysis forecast for such surveys, using DES-like and Euclid-like configurations as examples, and take account of any expected photometric and spectroscopic data. We focus on two-point statistics and we consider three observables: the 3D galaxy power spectrum in redshift space, the angular galaxy power spectrum, and the projected weak-lensing shear power spectrum. We study the effects of adding a few extra parameters to the basic LCDM set. We include the two standard parameters to model the current value for the dark energy equation of state and its time derivative, w_0, w_a, and we account for the possibility of primordial non-Gaussianity of the local, equilateral and orthogonal types, of parameter fNL and, optionally, of spectral index n_fNL. We present forecasted constraints on these parameters using the different observational probes. We show that accounting for models that include primordial non-Gaussianity does not degrade the constraint on the standard LCDM set nor on the dark-energy equation of state. By combining the weak lensing data and the information on projected galaxy clustering, consistently including all two-point functions and their covariance, we find forecasted marginalised errors sigma (fNL) ~ 3, sigma (n_fNL) ~ 0.12 from a Euclid-like survey for the local shape of primordial non-Gaussianity, while the orthogonal and equilateral constraints are weakened for the galaxy clustering case, due to the weaker scale-dependence of the bias. In the lensing case, the constraints remain instead similar in all configurations.
