Analytic Model for Covariance Matrices of the 2-, 3-, and 4-Point Correlation Functions in the Gaussian Random Field Approximation

Abstract: Analyses of the galaxy N-Point Correlation Functions (NPCFs) have a large number of degrees of freedom, meaning one cannot directly estimate an invertible covariance matrix purely from mock catalogs, as has been the standard approach for the 2PCF and power spectrum. Instead, templates are used based on assuming a Gaussian Random Field density with the true, Boltzmann-solver-computed power spectrum. The resulting covariance matrices are sparse but have notable internal structure. To understand this structure better, we seek a fully analytic, closed-form covariance matrix template, using a power law power spectrum $P(k) \propto 1/k$ and including shot noise. We obtain a simple closed-form solution for the covariance of the 2PCF, as well as closed-form solutions for the fundamental building blocks (termed ``$f$-integrals'') of the covariance matrices for the 3PCF, 4PCF, and beyond. We achieve single-digit percent level accuracy for the $f$-integrals, confirming that our power spectrum model is a suitable alternative to the true power spectrum. In the $f$-integrals, we find that the greatest contributions arise when closed triangles may be formed. When $f$-integrals are multiplied together, as needed for the covariance, the number of non-vanishing configurations reduces. We use these results to present a clearer picture of the covariance matrices' structure and sparsity, which correspond to triangular and non-triangular regions. This will be useful in guiding future NPCF analyses with spectroscopic galaxy surveys such as DESI, Euclid, Roman, and SPHEREx.