The Likelihood for LSS: Stochasticity of Bias Coefficients at All Orders

Abstract

In the EFT of biased tracers the noise field ${\epsilon }_g$ is not exactly uncorrelated with the nonlinear matter field $\delta$. Its correlation with $\delta$ is effectively captured by adding stochasticities to each bias coefficient. We show that if these stochastic fields are Gaussian (the impact of their non-Gaussianity being subleading on quasi-linear scales anyway) it is possible to resum exactly their effect on the conditional likelihood $\mathcal{P}[{\delta }_g|\delta ]$ to observe a galaxy field ${\delta }_g$ given an underlying $\delta$. This resummation allows to take them into account in EFT-based approaches to Bayesian forward modeling. We stress that the resulting corrections to a purely Gaussian conditional likelihood with white-noise covariance are the most relevant on scales where the EFT is under control: they are more important than any non-Gaussianity of the noise ${\epsilon }_g$.

15 pages. v2: very minor change in Eqs. (2.6), (2.8), (2.17), (2.19), (3.16), (3.18), (3.19) to show factorization in real space of the normalization of the likelihood, updated acknowledgements. v3: added section on renormalization, matches published version

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