Unified treatment of cosmological perturbations from super-horizon to small scales

TL;DR

The authors address the problem of accurately modeling cosmological perturbations across linear to highly non-linear scales in a Universe with a cosmological constant and pressureless matter. They develop a hybrid post-Newtonian and second-order perturbation framework in the Poisson gauge, expressing metric perturbations through a single gravitational potential $\varphi$ sourced by Newtonian matter variables $\rho$ and $v^{\alpha}$, and derive the complete set of evolution equations for scalars, vectors, and tensors. Key contributions include a unified set of equations that reproduce linear and second-order GR as well as Newtonian theory in their respective limits, and leading PN expressions for vector and tensor modes with explicit quadrupole-like gravitational-wave sources. The formalism enables direct numerical implementation from N-body simulations and has broad applications to secondary CMB anisotropies, gravitational lensing, and stochastic gravitational-wave backgrounds from non-linear structure formation, thus bridging scales from super-horizon to small scales. Its significance lies in providing a practical, consistency-checked toolkit for modeling relativistic effects in structure formation across all cosmologically relevant regimes.

Abstract

We study the evolution of cosmological perturbations, using a hybrid approximation scheme which upgrades the weak-field limit of Einstein's field equations to account for post-Newtonian scalar and vector metric perturbations and for leading-order source terms of gravitational waves, while including also the first and second-order perturbative approximations. Our equations, which are derived in the Poisson gauge, provide a unified description of matter inhomogeneities in a Universe filled with a pressureless and irrotational fluid and a cosmological constant, ranging from the linear to the highly non-linear regime. The derived expressions for scalar, vector and tensor modes can have a wide range of cosmological applications, ranging from secondary CMB anisotropy and polarization effects, cosmographic relations in a inhomogeneous Universe, gravitational lensing effects and the stochastic gravitational-wave backgrounds generated by non-linear cosmic structures.