FRW

FRW stands for the Friedmann–Robertson–Walker metric, a class of solutions to Einstein's field equations in general relativity that models a homogeneous and isotropic universe. Developed from the cosmological principle, FRW metrics assume the same properties at every point in space and in all directions, leading to a spacetime that is spatially homogeneous and isotropic. The line element is ds^2 = -c^2 dt^2 + a(t)^2 [ dr^2/(1-kr^2) + r^2(dθ^2 + sin^2θ dφ^2) ], where t is cosmic time, a(t) is the scale factor, k ∈ {−1, 0, +1} denotes spatial curvature, and c is the speed of light. The metric is a solution for a universe filled with a perfect fluid with energy density ρ and pressure p; the evolution of a(t) is governed by the Friedmann equations: (ȧ/a)^2 = (8πG/3)ρ − k c^2 / a^2 + Λ c^2 / 3 and ä/a = - (4πG/3)(ρ + 3p/c^2) + Λ c^2 / 3, where Λ is the cosmological constant. The Hubble parameter H(t) = ȧ/a describes the rate of expansion, and redshift z relates to the scale factor via z = a0/a − 1. FRW spacetimes underpin modern cosmology, providing the framework for the standard model, including ΛCDM, which combines matter, radiation, and dark energy. While FRW models capture large-scale homogeneity and isotropy, they gloss over small-scale inhomogeneities and anisotropies, which are studied with perturbation theory and numerical simulations.