An analytic approximation to the covariance between pre- and post-reconstruction galaxy two-point statistics

TL;DR

This paper tackles the problem of joint analysis of pre-reconstruction power spectra and post-reconstruction two-point functions by deriving a simple analytic cross-covariance based on the disconnected Gaussian contribution and BAO reconstruction damping. The authors model the cross-power $P_{\times}$ with a perturbative form $P_{\times} \approx e^{-k^2\sigma^2(\mu)/2}\left[(b+f\mu^2)^2 P_{\rm lin}(k) + \mathrm{SN}\right]$ and compute the cross-covariance between $P_\ell^{\rm pre}(k)$ and $\xi_\ell^{\rm post}(r)$, neglecting the window function in the analytic calculation. Validation against DESI mock catalogs, in both cubic and cut-sky geometries, shows excellent agreement in the cross-covariance blocks and negligible impact on cosmological parameter constraints when using the analytic cross-covariance. This indicates that the pre–post cross-covariance is sufficiently small for approximate treatments to suffice, enabling a path toward fully analytic covariance matrices for next-generation galaxy surveys and potential denoising of covariance estimates from mocks.

Abstract

We present a simple analytic approximation for the covariance between pre-reconstruction galaxy power spectrum measurements and post-reconstruction two-point correlation functions. This cross-covariance is essential for joint analyses that combine full-shape clustering information with baryon acoustic oscillation (BAO) measurements, as commonly performed in modern spectroscopic surveys. Our model builds on the disconnected contribution to the covariance and accounts for the damping of correlations due to the BAO reconstruction process. We validate our analytic prescription against numerical simulations from the Dark Energy Spectroscopic Instrument (DESI), testing both idealized cubic geometries and realistic survey configurations including complex footprints and fiber assignment effects. Despite neglecting survey window functions in the analytic calculation, we find excellent agreement with simulation-based covariances and demonstrate that cosmological parameter constraints are virtually unchanged when using our approximation. Our results show that the pre-post cross-covariance is sufficiently small that even approximate treatments are adequate for cosmological inference, opening a pathway toward fully analytic covariance matrices for next-generation galaxy surveys.

Figures (4)

• Figure 1: Cosmological parameters inferred from mocks using a covariance matrix where the off-diagonal $P_{\ell_1}(k)-\xi_{\ell_2}(r)$ block is determined analytically (orange), numerically (blue), or it is ignored altogether (green). Dashed lines show the true parameter values underlying the mocks. In the left panel, the simulation box is cubic and periodic, whereas in the right the geometry simulates the DESI footprint and lightcone structure and includes also a $\theta$-cut.
• Figure 2: Cross-covariance between $P_{\ell_1}(k)$ and $\xi_{\ell_2}(r)$ calculated from our analytic formula (dashed lines) or from simulations (points) as a function of $k$ for two different values of the pair separation $r$ (colors). The errors on the simulated points are estimated by leave-one-out jackknife. The simulations have cut-sky geometry and a $\theta$-cut, but no window rotation is applied. In the upper right panel we added a small leftward horizontal shift ($\Delta k=-0.0015 \,h{\rm Mpc}^{-1}$) to the orange data points in order to reduce overlap with the blue errorbars.
• Figure 3: Same as figure \ref{['fig:off_diag_blocks_cutsky']}, but with window rotation following the $\theta$-cut. In this plot, we also show the (negligible) impact of nulling negative eigenvalues to ensure positive-definiteness of the composite covariance matrix (solid lines). In the upper right panel we added a small leftward horizontal shift ($\Delta k=-0.0015 \,h{\rm Mpc}^{-1}$) to the orange data points in order to reduce overlap with the blue errorbars (estimated by leave-one-out jackknife.).
• Figure 4: Same as Fig. \ref{['fig:combined']} but with the rotation applied to diagonalize the window after the $\theta$-cut has been applied.