Matter Power Spectrum Covariance Research Articles

CARPool Covariance: Fast, unbiased covariance estimation for large-scale structure observables

Abstract The covariance matrix $\boldsymbol{\Sigma }$ of non-linear clustering statistics that are measured in current and upcoming surveys is of fundamental interest for comparing cosmological theory and data and a crucial ingredient for the likelihood approximations underlying widely used parameter inference and forecasting methods. The extreme number of simulations needed to estimate $\boldsymbol{\Sigma }$ to sufficient accuracy poses a severe challenge. Approximating $\boldsymbol{\Sigma }$ using inexpensive but biased surrogates introduces model error with respect to full simulations, especially in the non-linear regime of structure growth. To address this problem we develop a matrix generalization of Convergence Acceleration by Regression and Pooling (CARPool) to combine a small number of simulations with fast surrogates and obtain low-noise estimates of $\boldsymbol{\Sigma }$ that are unbiased by construction. Our numerical examples use CARPool to combine GADGET-III N-body simulations with fast surrogates computed using COmoving Lagrangian Acceleration (COLA). Even at the challenging redshift z = 0.5, we find variance reductions of at least $\mathcal {O}(10^1)$ and up to $\mathcal {O}(10^4)$ for the elements of the matter power spectrum covariance matrix on scales 8.9 × 10−3 < kmax < 1.0 hMpc−1. We demonstrate comparable performance for the covariance of the matter bispectrum, the matter correlation function and probability density function of the matter density field. We compare eigenvalues, likelihoods, and Fisher matrices computed using the CARPool covariance estimate with the standard sample covariance and generally find considerable improvement except in cases where $\boldsymbol{\Sigma }$ is severely ill-conditioned.

Cosmological model parameter dependence of the matter power spectrum covariance from the DEUS-PUR Cosmo simulations

ABSTRACT Future galaxy surveys will provide accurate measurements of the matter power spectrum across an unprecedented range of scales and redshifts. The analysis of these data will require one to accurately model the imprint of non-linearities of the matter density field. In particular, these induce a non-Gaussian contribution to the data covariance that needs to be properly taken into account to realize unbiased cosmological parameter inference analyses. Here, we study the cosmological dependence of the matter power spectrum covariance using a dedicated suite of N-body simulations, the Dark Energy Universe Simulation–Parallel Universe Runs (DEUS-PUR) Cosmo. These consist of 512 realizations for 10 different cosmologies where we vary the matter density Ωm, the amplitude of density fluctuations σ8, the reduced Hubble parameter h, and a constant dark energy equation of state w by approximately $10{{\ \rm per\ cent}}$. We use these data to evaluate the first and second derivatives of the power spectrum covariance with respect to a fiducial Λ-cold dark matter cosmology. We find that the variations can be as large as $150{{\ \rm per\ cent}}$ depending on the scale, redshift, and model parameter considered. By performing a Fisher matrix analysis we explore the impact of different choices in modelling the cosmological dependence of the covariance. Our results suggest that fixing the covariance to a fiducial cosmology can significantly affect the recovered parameter errors and that modelling the cosmological dependence of the variance while keeping the correlation coefficient fixed can alleviate the impact of this effect.

Response approach to the matter power spectrum covariance

We present a calculation of the matter power spectrum covariance matrix Cov(k1, k2) that uses power spectrum responses to accurately describe the coupling between large- and small-scale modes beyond the perturbative regime. These response functions can be measured with (small-volume) N-body simulations, which is why the response contributions to the covariance remain valid and predictive at all orders in perturbation theory. A novel and key step presented here is the use of responses to compute loop contributions with soft loop momenta, which extends the application of the response approach beyond that of previously considered squeezed n-point functions. The calculation presented here does not involve any fitting parameters. When including response-type terms up to 1-loop order in perturbation theory, we find that our calculation captures the bulk of the total covariance as estimated from simulations up to values of k1, k2 ∼ 1 h Mpc-1. Moreover, the prediction is guaranteed to be accurate whenever the softer mode is sufficiently linear, {k1, k1} ≲ 0.08 h Mpc-1. We identify and discuss straightforward improvements in the context of the response approach, which are expected to further increase the accuracy of the calculation presented here.

Responses in large-scale structure

We introduce a rigorous definition of general power-spectrum responses as resummed vertices with two hard and n soft momenta in cosmological perturbation theory. These responses measure the impact of long-wavelength perturbations on the local small-scale power spectrum. The kinematic structure of the responses (i.e., their angular dependence) can be decomposed unambiguously through a ``bias'' expansion of the local power spectrum, with a fixed number of physical response coefficients, which are only a function of the hard wavenumber k. Further, the responses up to n-th order completely describe the (n+2)-point function in the squeezed limit, i.e. with two hard and n soft modes, which one can use to derive the response coefficients. This generalizes previous results, which relate the angle-averaged squeezed limit to isotropic response coefficients. We derive the complete expression of first- and second-order responses at leading order in perturbation theory, and present extrapolations to nonlinear scales based on simulation measurements of the isotropic response coefficients. As an application, we use these results to predict the non-Gaussian part of the angle-averaged matter power spectrum covariance CovNGℓ=0(k1,k2), in the limit where one of the modes, say k2, is much smaller than the other. Without any free parameters, our model results are in very good agreement with simulations for k2 ≲ 0.06 h Mpc−1, and for any k1 ≳ 2k2. The well-defined kinematic structure of the power spectrum response also permits a quick evaluation of the angular dependence of the covariance matrix. While we focus on the matter density field, the formalism presented here can be generalized to generic tracers such as galaxies.

On the information content of the matter power spectrum: Figure 1.

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.

Matter power spectrum covariance matrix from the DEUS-PUR ΛCDM simulations: mass resolution and non-Gaussian errors

The upcoming generation of galaxy surveys will probe the distribution of matter in the universe with unprecedented accuracy. Measurements of the matter power spectrum at different scales and redshifts will provide stringent constraints on the cosmological parameters. However, on non-linear scales this will require an accurate evaluation of the covariance matrix. Here, we compute the covariance matrix of the 3D matter density power spectrum for the concordance $\Lambda$CDM cosmology from an ensemble of N-body simulations of the Dark Energy Universe Simulation - Parallel Universe Runs (DEUS-PUR). This consists of 12288 realisations of a $(656\,h^{-1}\,\textrm{Mpc})^3$ simulation box with $256^3$ particles. We combine this set with an auxiliary sample of 96 simulations of the same volume with $1024^3$ particles. We find N-body mass resolution effect to be an important source of systematic errors on the covariance at high redshift and small intermediate scales. We correct for this effect by introducing an empirical statistical method which provide an accurate determination of the covariance matrix over a wide range of scales including the Baryon Oscillations interval. Contrary to previous studies that used smaller N-body ensembles, we find the power spectrum distribution to significantly deviate from expectations of a Gaussian random density field at $k\gtrsim 0.25\,h\,\textrm{Mpc}^{-1}$ and $z<0.5$. This suggests that in the case of finite volume surveys an unbiased estimate of the ensemble averaged band power at these scales and redshifts may require a careful assessment of non-Gaussian errors more than previously considered.

Thinking outside the box: effects of modes larger than the survey on matter power spectrum covariance

Accurate power spectrum (or correlation function) covariance matrices are a crucial requirement for cosmological parameter estimation from large scale structure surveys. In order to minimize reliance on computationally expensive mock catalogs, it is important to have a solid analytic understanding of the different components that make up a covariance matrix. Considering the matter power spectrum covariance matrix, it has recently been found that there is a potentially dominant effect on mildly non-linear scales due to power in modes of size equal to and larger than the survey volume. This beat coupling effect has been derived analytically in perturbation theory and while it has been tested with simulations, some questions remain unanswered. Moreover, there is an additional effect of these large modes, which has so far not been included in analytic studies, namely the effect on the estimated average density which enters the power spectrum estimate. In this article, we work out analytic, perturbation theory based expressions including both the beat coupling and this local average effect and we show that while, when isolated, beat coupling indeed causes large excess covariance in agreement with the literature, in a realistic scenario this is compensated almost entirely by the local average effect, leaving only ∼ 10% of the excess. We test our analytic expressions by comparison to a suite of large N-body simulations, using both full simulation boxes and subboxes thereof to study cases without beat coupling, with beat coupling and with both beat coupling and the local average effect. For the variances, we find excellent agreement with the analytic expressions for k < 0.2 hMpc−1 at z = 0.5, while the correlation coefficients agree to beyond k = 0.4 hMpc−1. As expected, the range of agreement increases towards higher redshift and decreases slightly towards z = 0. We finish by including the large-mode effects in a full covariance matrix description for arbitrary survey geometry and confirming its validity using simulations. This may be useful as a stepping stone towards building an actual galaxy (or other tracer's) power spectrum covariance matrix.

Baryon oscillations in galaxy and matter power-spectrum covariance matrices

We investigate large-amplitude baryon acoustic oscillations (BAOs) in off-diagonal entries of cosmological power-spectrum covariance matrices. These covariance-matrix BAOs describe the increased attenuation of power-spectrum BAOs caused by upward fluctuations in large-scale power. We derive an analytic approximation to covariance-matrix entries in the BAO regime, and check the analytical predictions using N-body simulations. These BAOs look much stronger than the BAOs in the power spectrum, but seem detectable only at about a 1σ level in gigaparsec-scale galaxy surveys. In estimating cosmological parameters using matter or galaxy power spectra, including the covariance-matrix BAOs can have a several per cent effect on error bar widths for some parameters directly related to the BAOs, such as the baryon fraction. Also, we find that including the numerous galaxies in small haloes in a survey can reduce error bars in these cosmological parameters more than the simple reduction in shot noise might suggest.