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An efficient model of cosmology dependence in the covariance matrix of the matter power spectrum

Theodore Steele, Robert Smith, Roisin O'Connor

TL;DR

The paper addresses the computational challenge of estimating cosmology-dependent covariance matrices for the matter power spectrum, introducing a Taylor-expansion framework around a fiducial cosmology. It defines a response matrix that captures how the covariance changes with cosmological parameters and reconstructs the true covariance at new parameter points using a small set of input cosmologies. Three reconstruction variants are developed (full response, approximate response, and zero-response with interpolation), and validated against QUIJOTE simulations, achieving percent-level accuracy up to k around 0.3 h/Mpc, especially when SSC is included. The approach offers orders-of-magnitude speedups for exploring high-dimensional cosmological parameter spaces, enabling robust inference for current and future large-scale structure surveys, with future work planned to add biasing and alternative shot-noise modeling.

Abstract

Covariance matrices are essential cosmological probes of fundamental physics, providing information on numerous fundamental physical parameters and varying with any change in the underlying cosmology. However, this cosmology dependence, while providing excellent information, also makes them computationally intensive to compute, as a new covariance matrix must explicitly be calculated for every variation in cosmology before comparisons to observational data can be made. In this paper, we develop an efficient model for estimating the parameter dependence of the covariance matrix of the matter power spectrum by Taylor expanding around a known value of the parameter space. This method allows us to use a relatively small number of input cosmologies, specifically one fiducial cosmology and two further cosmologies for each parameter. We explicitly calculate the covariance matrices for these cosmologies and then develop a new model that allows us to interpolate from these the form of the covariance matrix with a cosmology that is located elsewhere in that given parameter space without explicit perturbation theory calculations. This method speeds up covariance matrix calculations in new cosmologies by orders of magnitude compared to explicit perturbation theory calculations at each point in a given parameter space. Using different approximations, we develop three versions of our interpolated covariance matrix and validate the model by recreating all of our input cosmologies using all three forms, both with and without super-sample covariance corrections in each case, and show that the models provide robust recreations of the original results, with the different approximations being valid in certain regimes.

An efficient model of cosmology dependence in the covariance matrix of the matter power spectrum

TL;DR

The paper addresses the computational challenge of estimating cosmology-dependent covariance matrices for the matter power spectrum, introducing a Taylor-expansion framework around a fiducial cosmology. It defines a response matrix that captures how the covariance changes with cosmological parameters and reconstructs the true covariance at new parameter points using a small set of input cosmologies. Three reconstruction variants are developed (full response, approximate response, and zero-response with interpolation), and validated against QUIJOTE simulations, achieving percent-level accuracy up to k around 0.3 h/Mpc, especially when SSC is included. The approach offers orders-of-magnitude speedups for exploring high-dimensional cosmological parameter spaces, enabling robust inference for current and future large-scale structure surveys, with future work planned to add biasing and alternative shot-noise modeling.

Abstract

Covariance matrices are essential cosmological probes of fundamental physics, providing information on numerous fundamental physical parameters and varying with any change in the underlying cosmology. However, this cosmology dependence, while providing excellent information, also makes them computationally intensive to compute, as a new covariance matrix must explicitly be calculated for every variation in cosmology before comparisons to observational data can be made. In this paper, we develop an efficient model for estimating the parameter dependence of the covariance matrix of the matter power spectrum by Taylor expanding around a known value of the parameter space. This method allows us to use a relatively small number of input cosmologies, specifically one fiducial cosmology and two further cosmologies for each parameter. We explicitly calculate the covariance matrices for these cosmologies and then develop a new model that allows us to interpolate from these the form of the covariance matrix with a cosmology that is located elsewhere in that given parameter space without explicit perturbation theory calculations. This method speeds up covariance matrix calculations in new cosmologies by orders of magnitude compared to explicit perturbation theory calculations at each point in a given parameter space. Using different approximations, we develop three versions of our interpolated covariance matrix and validate the model by recreating all of our input cosmologies using all three forms, both with and without super-sample covariance corrections in each case, and show that the models provide robust recreations of the original results, with the different approximations being valid in certain regimes.
Paper Structure (14 sections, 42 equations, 47 figures, 1 table)

This paper contains 14 sections, 42 equations, 47 figures, 1 table.

Table of Contents

  1. Introduction
  2. Perturbatively Modeling the covariance matrix
  3. Band Averaging
  4. Modeling the parameter dependencies of the covariance matrix
  5. Developing the response matrix $\textbf{R}$
  6. Reconstructing Simulation Covariance Matrices
  7. Discussion
  8. Standard Perturbation Theory
  9. The Power Spectrum
  10. The Bispectrum
  11. The Trispectrum
  12. Simulation Parameters
  13. Response Matrices
  14. Reconstructed Matrix Accuracy

Figures (47)

  • Figure 1: The covariance matrices for the fiducial cosmology. The left hand pair of columns omit SSC corrections, while the right hand pair of columns include them. In each pair of columns, the top left panel shows the SPT model covariance matrix, the top right panel shows the non-linear covariance matrix from the simulations, the bottom left panel shows the reconstructed covariance matrix from Eq. \ref{['eq:Cparallel']} with the full response matrix as given in Eq. \ref{['eq:fullR']}, and the bottom right panel shows the reconstructed covariance matrix with the approximated response matrix as given in Eq. \ref{['eq:aR']}.
  • Figure 2: The covariance matrices for cosmologies with the increased value of the parameter $\Omega_{\mathrm{m}}$ and other parameters kept as in the fiducial case. The left hand pair of columns omit SSC corrections, while the right hand pair of columns include them. In each set of five panels, the top left panel shows the SPT model covariance matrix, the top right panel shows the non-linear covariance matrix from the simulations, the centre left panel shows the reconstructed covariance matrix from Eq. \ref{['eq:Cparallel']} with the full response matrix as given in Eq. \ref{['eq:fullR']}, the centre right panel shows the reconstructed covariance matrix with the approximated response matrix as given in Eq. \ref{['eq:aR']}, and the bottom panel shows the results of Eq. \ref{['eq:Cparallel']} with the response matrix set to zero.
  • Figure 3: The covariance matrices for cosmologies with the decreased value of the parameter $\Omega_{\mathrm{m}}$ and other parameters kept as in the fiducial case. The left hand pair of columns omit SSC corrections, while the right hand pair of columns include them. In each set of five panels, the top left panel shows the SPT model covariance matrix, the top right panel shows the non-linear covariance matrix from the simulations, the centre left panel shows the reconstructed covariance matrix from Eq. \ref{['eq:Cparallel']} with the full response matrix as given in Eq. \ref{['eq:fullR']}, the centre right panel shows the reconstructed covariance matrix with the approximated response matrix as given in Eq. \ref{['eq:aR']}, and the bottom panel shows the results of Eq. \ref{['eq:Cparallel']} with the response matrix set to zero.
  • Figure 4: The covariance matrices for cosmologies with the increased value of the parameter $\Omega_{\mathrm{b}}$ and other parameters kept as in the fiducial case. The left hand pair of columns omit SSC corrections, while the right hand pair of columns include them. In each set of five panels, the top left panel shows the SPT model covariance matrix, the top right panel shows the non-linear covariance matrix from the simulations, the centre left panel shows the reconstructed covariance matrix from Eq. \ref{['eq:Cparallel']} with the full response matrix as given in Eq. \ref{['eq:fullR']}, the centre right panel shows the reconstructed covariance matrix with the approximated response matrix as given in Eq. \ref{['eq:aR']}, and the bottom panel shows the results of Eq. \ref{['eq:Cparallel']} with the response matrix set to zero.
  • Figure 5: The covariance matrices for cosmologies with the decreased value of the parameter $\Omega_{\mathrm{b}}$ and other parameters kept as in the fiducial case. The left hand pair of columns omit SSC corrections, while the right hand pair of columns include them. In each set of five panels, the top left panel shows the SPT model covariance matrix, the top right panel shows the non-linear covariance matrix from the simulations, the centre left panel shows the reconstructed covariance matrix from Eq. \ref{['eq:Cparallel']} with the full response matrix as given in Eq. \ref{['eq:fullR']}, the centre right panel shows the reconstructed covariance matrix with the approximated response matrix as given in Eq. \ref{['eq:aR']}, and the bottom panel shows the results of Eq. \ref{['eq:Cparallel']} with the response matrix set to zero.
  • ...and 42 more figures