Nonlinear Perturbation Theory Integrated with Nonlocal Bias, Redshift-space Distortions, and Primordial Non-Gaussianity

TL;DR

The paper develops a comprehensive nonlinear perturbation framework that unifies Eulerian and Lagrangian treatments of nonlocal bias, redshift-space distortions, and primordial non-Gaussianity. By deriving interconnections between Eulerian and Lagrangian bias kernels, introducing diagrammatic rules, and implementing vertex resummations, it provides a consistent method to predict polyspectra (power, bispectrum, trispectrum) in redshift surveys for a broad class of bias models. It covers local and multivariate Lagrangian biases, halo/peaks-inspired schemes, and universal mass-function-based biases, including stochasticity interpretations and the impact of nonlinear evolution on bias. The framework enables principled, scalable modeling of scale-dependent bias and FoG effects, with direct applicability to BAO analyses and precision cosmology studies of large-scale structure.

Abstract

The standard nonlinear perturbation theory of the gravitational instability is extended to incorporate the nonlocal bias, redshift-space distortions, and primordial non-Gaussianity. We show that local Eulerian bias is not generally compatible to local Lagrangian bias in nonlinear regime. The Eulerian and Lagrangian biases are nonlocally related order by order in the general perturbation theory. The relation between Eulerian and Lagrangian kernels of density perturbations with biasing are derived. The effects of primordial non-Gaussianity and redshift-space distortions are also incorporated in our general formalism, and diagrammatic methods are introduced. Vertex resummations of higher-order perturbations in the presence of bias are considered. Resummations of Lagrangian bias are shown to be essential to handle biasing schemes in a general framework.

Figures (17)

• Figure 1: The relation between Eulerian and Lagrangian biases. The Eulerian bias is expressible by the Lagrangian bias (dotted arrow) and vice versa (dashed arrow), only when the biases are allowed to be nonlocal. Note that nonlinear evolutions and formation of objects are nonlocal processes in nonlinear regime.
• Figure 2: Diagrammatic rules for the Eulerian perturbation theory in real space. In each vertex, $\bm{k} = \bm{k}_1 + \cdots + \bm{k}_n$ should be satisfied. The upper and lower rules correspond to the expansions in Eq. (\ref{['eq:1-2']}) and (\ref{['eq:1-3']}), respectively. A dashed line should be "internal": one end of a dashed line should be connected to a vertex with double solid line, and the other end should be connected to a vertex with solid lines.
• Figure 3: Diagrammatic rules for contributions from the primordial polyspectra. All the free ends of solid lines in Fig. \ref{['fig:EPTreal']} should be connected to each other by these rules. When the initial density field is random Gaussian, the lower graph with $n\geq 3$ does not exist. In the lower graph, $\bm{k}_1 + \cdots + \bm{k}_n = \bm{0}$ should be satisfied. The case of $n=2$ in the lower graph is equivalent to the upper graph.
• Figure 4: Shrunk vertex. The triangle represents all the possible tree graphs constructed by the rules in Fig. \ref{['fig:EPTreal']} of EPT. The shrunk vertex can also be expressed by LPT diagrams.
• Figure 5: Shrunk vertices in Eulerian perturbation theory in real space.