Renormalized Halo Bias

TL;DR

The paper demonstrates that a consistent, renormalized halo-bias model must include non-local and higher-derivative terms, elevating halo bias to an effective theory with a double expansion in density/velocity fluctuations and spatial derivatives. It provides a complete basis of renormalized operators up to quartic order for Gaussian initial conditions and shows Galileon operators are not renormalized at leading order in derivatives, organizing the bias expansion accordingly. At one loop, it derives the renormalized bias parameters and shows how they feed into halo power spectra, bispectra, and trispectra, including degeneracies and potential observational handles. The work thus clarifies how short-scale physics is systematically removed and how the renormalized parameters govern halo statistics across two- and higher-point functions, with implications for EFT-of-LSS analyses and data interpretation.

Abstract

This paper provides a systematic study of renormalization in models of halo biasing. Building on work of McDonald, we show that Eulerian biasing is only consistent with renormalization if non-local terms and higher-derivative contributions are included in the biasing model. We explicitly determine the complete list of required bias parameters for Gaussian initial conditions, up to quartic order in the dark matter density contrast and at leading order in derivatives. At quadratic order, this means including the gravitational tidal tensor, while at cubic order the velocity potential appears as an independent degree of freedom. Our study naturally leads to an effective theory of biasing in which the halo density is written as a double expansion in fluctuations and spatial derivatives. We show that the bias expansion can be organized in terms of Galileon operators which aren't renormalized at leading order in derivatives. Finally, we discuss how the renormalized bias parameters impact the statistics of halos.

Figures (6)

• Figure 1: Non-linearities in the gravitational evolution, in the biasing and in redshift space distortions (RSD) complicate the relationship between the primordial initial conditions and large-scale structure observables.
• Figure 2: Diagram $(a)$ is not a (partially) 1PI diagram, as it does not contain contractions between different fields within the composite operator. On the other hand, diagrams $(b)$ and $(c)$ are examples of a partial 1PI graph and a full 1PI graph, respectively.
• Figure 4: Numerical results for the ${\cal I}$-terms of halo-halo power spectrum, $P_{hh}(q)$, in the real universe.
• Figure 5: Illustration of the degeneracy of the different contributions to the halo-matter power spectrum. For each curve, $b_{\cal F}$ and $b_{\cal I}$ are kept fixed, but $b_{{\cal G}_2}^{(R)}$ is varied between $-3$ and $+3$.
• Figure 6: Illustration of how the degeneracy presented in fig. \ref{['fig:Phm-Degeneracy']} is broken by the halo-halo power spectrum. While $P_{hm}$ (black band) mostly depends only on the effective bias parameters $b_{\cal F}$ and $b_{\cal I}$, $P_{hh}$ (gray band) is sensitive to $b_{{\cal G}_2}^{(R)}$.