An Effective Theory for Biased Tracers via the Boltzmann-Equation Approach

TL;DR

The paper develops a Boltzmann-equation–based effective theory (BEA) for biased tracers that unifies density and velocity bias through a general tracer collision term. In the linear regime, it yields time- and scale-dependent bias parameters and recovers peak-bias as a special case, while naturally generating velocity bias and higher-derivative effects. When applied to redshift-space distortions, the BEA power spectrum reproduces the EFTofLSS result up to $k^4$ with fewer free parameters and includes a built-in $k^4$ contribution from the collision-term structure. This provides a self-consistent framework linking small-scale tracer dynamics to large-scale observables and suggests extensions to richer collision operators and nonlinear regimes.

Abstract

We develop an effective theory for biased tracers formulated at the level of the Boltzmann equation, providing a unified description of density and velocity bias. We introduce a general effective collision term in the tracer Boltzmann equation to encode tracer dynamics that are intrinsically different from those of dark matter. This collision operator leads to modified continuity and Euler equations, with source terms reflecting the collision-term physics. At linear order, this framework predicts time- and scale-dependent bias parameters in a self-consistent manner, encompassing peak bias as a special case while clarifying how velocity bias and higher-derivative effects arise. Applying the resulting bias model to redshift-space distortions, we show that the Boltzmann-equation approach reproduces the power spectrum of biased tracers obtained in the Effective Field Theory of Large-Scale Structure up to $k^4$ terms with fewer independent parameters.

Figures (2)

• Figure 1: The time and $k$-dependence of the bias parameters $b(a,k)$ and $b_v(a,k)$ are shown in the left and right panels, respectively. The parameters are set to $C_1=-1/2,~C_4H_0^2/H^2=-10^{-3},~C_2=C_3=0$ for which the simple solutions in Eqs. \ref{['b sol']} and \ref{['bv sol']} are applicable. We also set $A(k)$ and $B(k)$ so that $b=b_v=1$ at the initial time $a_\mathrm{ini}=10^{-3}$ for all $k$. The initial $k$-dependent contributions of $A(k)$, $B(k)$, and $C_4$ cancel each other. As the system evolves, however, the $A(k)$ and $B(k)$ terms decay while the $C_4$ term persists, giving rise to the $k^2$ dependence seen at late times.
• Figure 2: $b(a,k)$ and $b_v(a,k)$ are shown for the parameter set in Eqs. \ref{['Fig2 para1']} and \ref{['Fig2 para2']}. Unlike Fig. \ref{['fig:b-and-bv_eff-const']}, $b(a,k)$ and $b_v(a,k)$ exhibit non-trivial $k$-dependence at the initial time $a_{\mathrm{ini}}$, and the diffusion terms $C_{gk}$ and $C_{vk}$ are assumed to decay as $a^{-3/2}$. As seen in Eqs. \ref{['b sol']} and \ref{['bv sol']}, the coefficients $A(k)$ and $B(k)$ always decay with time, and together with the diminishing diffusion terms, the initial $k$-dependence is eventually washed out. Consequently, the bias parameters gradually approach scale-independent behavior on large scales.