Soft-Pion theorems for large scale structure

Abstract

Consistency relations -- which relate an N-point function to a squeezed\n(N+1)-point function -- are useful in large scale structure (LSS) because of\ntheir non-perturbative nature: they hold even if the N-point function is deep\nin the nonlinear regime, and even if they involve astrophysically messy galaxy\nobservables. The non-perturbative nature of the consistency relations is\nguaranteed by the fact that they are symmetry statements, in which the velocity\nplays the role of the soft pion. In this paper, we address two issues: (1) how\nto derive the relations systematically using the residual coordinate freedom in\nthe Newtonian gauge, and relate them to known results in $\\zeta$-gauge (often\nused in studies of inflation); (2) under what conditions the consistency\nrelations are violated. In the non-relativistic limit, our derivation\nreproduces the Newtonian consistency relation discovered by Kehagias \\& Riotto\nand Peloso & Pietroni. More generally, there is an infinite set of consistency\nrelations, as is known in $\\zeta$-gauge. There is a one-to-one correspondence\nbetween symmetries in the two gauges; in particular, the Newtonian consistency\nrelation follows from the dilation and special conformal symmetries in\n$\\zeta$-gauge. We probe the robustness of the consistency relations by studying\nmodels of galaxy dynamics and biasing. We give a systematic list of conditions\nunder which the consistency relations are violated; violations occur if the\ngalaxy bias is non-local in an infrared divergent way. We emphasize the\nrelevance of the adiabatic mode condition, as distinct from symmetry\nconsiderations. As a by-product of our investigation, we discuss a simple fluid\nLagrangian for LSS.\n