FLRW

The FLRW metric, named after Alexander Friedmann, Georges Lemaître, Howard P. Robertson, and Arthur G. Walker, is a family of solutions to Einstein's field equations that describes a spatially homogeneous and isotropic universe—the cosmological principle. It forms the basis of the standard model of Big Bang cosmology and is used to model the large-scale dynamics of spacetime.

In comoving coordinates, the line element is ds^2 = -c^2 dt^2 + a(t)^2 [dr^2/(1 - k r^2) + r^2(dθ^2 + sin^2 θ dφ^2)], where a(t) is the scale factor and k is the spatial curvature constant taking values -1, 0, or +1. The scale factor encodes the expansion or contraction of space as a function of cosmic time t.

The dynamics follow from Einstein's equations with a perfect fluid energy-momentum tensor, yielding the Friedmann equations. Key parameters include the Hubble parameter H = ȧ/a and density parameters Ω_m, Ω_r, Ω_Λ, and Ω_k, with Ω_m + Ω_r + Ω_Λ + Ω_k = 1 in the present epoch. The model accommodates a cosmological constant Λ as part of the standard ΛCDM framework.

Applications and limitations: The FLRW framework underpins interpretations of cosmic microwave background anisotropies, large-scale structure, and distance–redshift relations. It assumes homogeneity and isotropy on large scales and is not intended for highly inhomogeneous or small-scale regions, where perturbations or other metric descriptions apply.