Renormalization of Lagrangian bias via spectral parameters

TL;DR

This paper introduces a simplified spectral parameter reparametrization method for renormalizing Lagrangian bias, extending previous models to include higher-derivative operators and addressing divergences from contact terms.

Contribution

It proposes a new, simpler renormalization scheme for Lagrangian bias that incorporates higher-derivative operators and handles divergences using stochastic fields.

Findings

The method provides explicit relations between renormalized and unrenormalized biases.

It extends the peak background split framework to include these new bias operators.

Divergences from contact terms are effectively removed using stochastic fields.

Abstract

We extend the definition of Lagrangian local bias proposed by Matsubara (2008) to include curvature and higher-derivative bias operators. Evolution of initially biased tracers using perturbation theory (PT) generates multivariate bias parameters as soon as nonlinear fluctuations become important. We present a procedure that reparametrize a set of spectral parameters, the arguments of the Fourier transformed Lagrangian bias function, from which multivariate renormalized biases can be derived at any order in bias expansion and PT. We find our method simpler than previous renormalization schemes because it only relies on the definition of bias, fixed from the beginning, and in one equation relating renormalized and unrenormalized spectral parameters. We also show that our multivariate biases can be obtained within the peak background split framework, in that sense this work extends that of Schmidt, Jeong and Desjacques (2013); however, we restrict our method to Gaussian initial conditions. Non-linear evolution also leads to the appearance of products of correlators evaluated at the same point, commonly named contact terms, yielding divergent contributions to the power spectrum, in this work we present an explicit method to remove these divergences by introducing stochastic fields.