Evolution of second-order cosmological perturbations

TL;DR

The paper develops a framework to define gauge-invariant cosmological perturbations at second order around a FRW background and provides an explicit gauge-invariant curvature perturbation on uniform density hypersurfaces, $\zeta$. By leveraging only the local energy conservation equation and neglecting spatial gradients on large scales, it shows that $\zeta$ remains conserved at second order for adiabatic perturbations, implying that primordial Gaussianity is preserved in the large-scale limit for these perturbations. The authors furthermore formulate second-order gauge-invariant quantities on both uniform density and uniform curvature hypersurfaces, offering explicit expressions and transformation rules, and demonstrate how these constructions underpin the interpretation of CMB non-Gaussianity and early-universe perturbations. Overall, the work provides a robust method to handle non-linear perturbations and highlights the conditions under which large-scale curvature perturbations remain constant, with potential extensions to higher orders.

Abstract

We present a method for constructing gauge-invariant cosmological perturbations which are gauge-invariant up to second order. As an example we give the gauge-invariant definition of the second-order curvature perturbation on uniform density hypersurfaces. Using only the energy conservation equation we show that this curvature perturbation is conserved at second order on large scales for adiabatic perturbations.