Galilean invariance and the consistency relation for the nonlinear squeezed bispectrum of large scale structure

TL;DR

Galilean invariance imposes nonperturbative Ward identities that tightly constrain the nonlinear evolution of Large Scale Structure, yielding a squeezed-limit relation between the nonlinear bispectrum and power spectrum that holds up to $O(f_{ m NL}^2)$. By formulating a path-integral approach with frame fixing, the authors derive these Ward identities and analyze how initial non-Gaussianity modifies the relations to linear order in $f_{ m NL}$. They show that many resummation schemes break GI and incur spurious IR effects, and introduce the eRPT scheme, a GI-invariant resummation that preserves Ward identities at all orders. The work provides a principled framework for GI-consistent nonlinear LSS predictions and offers a tool to disentangle primordial non-Gaussianity from nonlinear gravitational evolution in observations and simulations.

Abstract

We discuss the constraints imposed on the nonlinear evolution of the Large Scale Structure (LSS) of the universe by galilean invariance, the symmetry relevant on subhorizon scales. Using Ward identities associated to the invariance, we derive fully nonlinear consistency relations between statistical correlators of the density and velocity perturbations, such as the power spectrum and the bispectrum. These relations are valid up to O (f_{NL}^2) corrections. We then show that most of the semi-analytic methods proposed so far to resum the perturbative expansion of the LSS dynamics fail to fulfill the constraints imposed by galilean invariance, and are therefore susceptible to non-physical infrared effects. Finally, we identify and discuss a nonperturbative semi-analytical scheme which is manifestly galilean invariant at any order of its expansion.

Figures (7)

• Figure 1: Relation (\ref{['W1']}) at one loop.
• Figure 2: Diagrams contributing to the nonlinear bispectrum in the squeezed limit, in which the top leg carries vanishing momentum. Only terms up to linear order in the initial nongaussianity are included. The external double lines denote the full renormalized PS and propagator, while the field in the upper vertex is still in the linear regime.
• Figure 3: Diagrams contibuting to the full PS.
• Figure 4: Inserting the tadpole in all possible ways in a diagram, integrating over the corresponding times, and summing over all diagrams contributing to a correlator, gives the GT of that correlator.
• Figure 5: One tadpole insertions on a linear power spectrum reproduce the first order term in the expansion in ${\bf{V_0}}$ of the linear part of (\ref{['power-GT']}).