# Exact joint likelihood of pseudo-$C_\ell$ estimates from correlated Gaussian cosmological fields

## TL;DR

This paper derives an exact joint likelihood for pseudo-$C_5$ power spectrum estimates from Gaussian cosmological fields, applicable to various fields and masks, enhancing the robustness of cosmological parameter inference.

## Contribution

It introduces a method to compute the exact joint likelihood of pseudo-$C_5$ estimates for correlated Gaussian fields with arbitrary masks, without assumptions on mask geometry.

## Key findings

- Likelihood accurately recovers the multivariate distribution of $EE$, $BB$, and $EB$ spectra.
- Method is applicable to CMB, weak lensing, and galaxy clustering analyses.
- Simulations confirm the method's effectiveness in realistic scenarios.

## Abstract

We present the exact joint likelihood of pseudo-$C_\ell$ power spectrum estimates measured from an arbitrary number of Gaussian cosmological fields. Our method is applicable to both spin-0 fields and spin-2 fields, including a mixture of the two, and is relevant to Cosmic Microwave Background, weak lensing and galaxy clustering analyses. We show that Gaussian cosmological fields are mixed by a mask in such a way that retains their Gaussianity, without making any assumptions about the mask geometry. We then show that each auto- or cross-pseudo-$C_\ell$ estimator can be written as a quadratic form, and apply the known joint distribution of quadratic forms to obtain the exact joint likelihood of a set of pseudo-$C_\ell$ estimates in the presence of an arbitrary mask. Considering the polarisation of the Cosmic Microwave Background as an example, we show using simulations that our likelihood recovers the full, exact multivariate distribution of $EE$, $BB$ and $EB$ pseudo-$C_\ell$ power spectra. Our method provides a route to robust cosmological constraints from future Cosmic Microwave Background and large-scale structure surveys in an era of ever-increasing statistical precision.