The EFT Likelihood for Large-Scale Structure in Redshift Space

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, $\mathcal{P}[{\tilde{\delta }}_g\mathrm{\mid }\delta ,\text{\hspace{0.17em}⁣}\mathbf{v}]$ 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.