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Biased Tracers and Time Evolution

Mehrdad Mirbabayi, Fabian Schmidt, Matias Zaldarriaga

TL;DR

This work addresses how time evolution shapes galaxy bias within perturbation theory by formulating a local-in-space, time-nonlocal relation along fluid trajectories and constructing a finite, non-redundant basis of operators built from spatial derivatives of the Newtonian potential and convective time derivatives. A Green's-function perspective is developed: instantaneous formation at $\tau_*$ yields $\delta_g(\mathbf{x},\tau_*)$, and later-time bias is obtained by superposing conserved tracers, yielding a time-evolving but local $\delta_g(\mathbf{x},\tau) = \sum_O b_O(\tau) O(\mathbf{x},\tau)$. The authors provide an explicit third-order calculation to determine the time dependence of Eulerian bias coefficients $b_O^E(\tau)$ as functions of $D_*(\tau)=D(\tau_*)/D(\tau)$, and they show that a complete basis of local operators built from convective derivatives of $\partial_i\partial_j\Phi$ (or equivalently the Lagrangian distortion tensor $M^{ij}$) suffices for a consistent, renormalization-safe description; new operators first appear at fourth order. They also connect Eulerian and Lagrangian pictures, examine formation history and velocity bias, and discuss implications for the EFT of large-scale structure. Overall, the framework yields a compact, physically motivated bias parametrization that remains closed under renormalization and captures time evolution systematically for modeling galaxy clustering.

Abstract

We study the effect of time evolution on galaxy bias. We argue that at any order in perturbations, the galaxy density contrast can be expressed in terms of a finite set of locally measurable operators made of spatial and temporal derivatives of the Newtonian potential. This is checked in an explicit third order calculation. There is a systematic way to derive a basis for these operators. This basis spans a larger space than the expansion in gravitational and velocity potentials usually employed, although new operators only appear at fourth order. The basis is argued to be closed under renormalization. Most of the arguments also apply to the structure of the counter-terms in the effective theory of large-scale structure.

Biased Tracers and Time Evolution

TL;DR

This work addresses how time evolution shapes galaxy bias within perturbation theory by formulating a local-in-space, time-nonlocal relation along fluid trajectories and constructing a finite, non-redundant basis of operators built from spatial derivatives of the Newtonian potential and convective time derivatives. A Green's-function perspective is developed: instantaneous formation at yields , and later-time bias is obtained by superposing conserved tracers, yielding a time-evolving but local . The authors provide an explicit third-order calculation to determine the time dependence of Eulerian bias coefficients as functions of , and they show that a complete basis of local operators built from convective derivatives of (or equivalently the Lagrangian distortion tensor ) suffices for a consistent, renormalization-safe description; new operators first appear at fourth order. They also connect Eulerian and Lagrangian pictures, examine formation history and velocity bias, and discuss implications for the EFT of large-scale structure. Overall, the framework yields a compact, physically motivated bias parametrization that remains closed under renormalization and captures time evolution systematically for modeling galaxy clustering.

Abstract

We study the effect of time evolution on galaxy bias. We argue that at any order in perturbations, the galaxy density contrast can be expressed in terms of a finite set of locally measurable operators made of spatial and temporal derivatives of the Newtonian potential. This is checked in an explicit third order calculation. There is a systematic way to derive a basis for these operators. This basis spans a larger space than the expansion in gravitational and velocity potentials usually employed, although new operators only appear at fourth order. The basis is argued to be closed under renormalization. Most of the arguments also apply to the structure of the counter-terms in the effective theory of large-scale structure.

Paper Structure

This paper contains 19 sections, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A Green's function for galaxy formation
  3. Flow of dark matter
  4. Flow of galaxies
  5. Bias evolution at third order: explicit calculation
  6. Formation history and age
  7. Velocity bias
  8. Connection with Lagrangian perturbation theory
  9. Invariant density
  10. Listing the bias parameters
  11. Renormalization
  12. Renormalization and counter-terms
  13. Conclusions
  14. Connection between velocity dispersion and tidal tensor
  15. Tracer with instantaneous formation at third order
  16. ...and 4 more sections

Figures (1)

  • Figure 1: (a) A vertex with large incoming and outgoing momenta $p\sim \Lambda$. Dots represent omitted lines whose momenta add up to $\mathbf{k}-\mathbf{k}_1$. (b) A composite operator with two of the incoming momenta large and nearly opposite. All interactions along the large-momentum line yield local expressions in terms of the low-momentum incoming lines.