The EFT Likelihood for Large-Scale Structure in Redshift Space

TL;DR

This paper develops an effective field theory likelihood model for biased tracers in redshift space, showing how to incorporate galaxy bias, velocity fields, and redshift distortions into a Gaussian likelihood framework for large-scale structure analysis.

Contribution

It introduces a Gaussian likelihood model for redshift-space galaxy overdensities based on EFT, accounting for bias and velocity fields, and discusses its perturbative matching and numerical implications.

Findings

Likelihood is Gaussian in the difference between observed and modeled overdensities.

Redshift-space distortions modify the covariance depending on matter and velocity fields.

Neglecting field-dependent covariance is justified at second-order bias expansion.

Abstract

We study the EFT likelihood for biased tracers in redshift space, for which the bias expansion of the galaxy velocity field $\mathbf{v}_g$ plays a fundamental role. The equivalence principle forbids stochastic contributions to $\mathbf{v}_g$ to survive at small $k$. Therefore, at leading order in derivatives the form of the likelihood ${\cal P}[\tilde{\delta}_g|\delta,\!\mathbf{v}]$ to observe a redshift-space galaxy overdensity $\tilde{\delta}_g(\tilde{\mathbf{x}})$ given a rest-frame matter and velocity fields $\delta(\mathbf{x})$, $\mathbf{v}(\mathbf{x})$ is fixed by the rest-frame noise. If this noise is Gaussian with constant power spectrum, ${\cal P}[\tilde{\delta}_g|\delta,\!\mathbf{v}]$ is also a Gaussian in the difference between $\tilde{\delta}_g(\tilde{\mathbf{x}})$ and its bias expansion: redshift-space distortions only make the covariance depend on $\delta(\mathbf{x})$ and $\mathbf{v}(\mathbf{x})$. We then show how to match this result to perturbation theory, and that one can consistently neglect the field-dependent covariance if the bias expansion is stopped at second order in perturbations. We discuss qualitatively how this affects numerical implementations of the EFT-based forward modeling, and how the picture changes when the survey window function is taken into account.