Friedmann equations

Friedmann equations
NASA / WMAP Science Team · Public domain · source
Name	Friedmann equations
Field	Cosmology
Introduced by	Alexander Friedmann
Year	1922

Friedmann equations The Friedmann equations are a pair of ordinary differential equations governing the expansion of a homogeneous and isotropic Universe described by the Friedmann–Lemaître–Robertson–Walker metric within General relativity. They relate the scale factor, spatial curvature, energy density and pressure of the cosmos, and underpin models used to interpret observations from projects such as Hubble Space Telescope, Planck, and WMAP. These equations connect theoretical frameworks from Albert Einstein, Alexander Friedmann, and Georges Lemaître to empirical constraints provided by surveys like Sloan Digital Sky Survey and experiments including Baryon Oscillation Spectroscopic Survey.

Introduction

The Friedmann equations arise by applying the Einstein field equations of General relativity to a maximally symmetric metric, yielding relations between the scale factor a(t), the Hubble parameter H(t), spatial curvature k, cosmological constant Λ, and the stress–energy content characterized by energy density ρ and pressure p. They are central to the Big Bang paradigm, inflationary scenarios associated with Alan Guth and Andrei Linde, and the concordance ΛCDM model constrained by missions such as Planck and observatories like Keck Observatory.

Derivation

Derivation typically begins with the Friedmann–Lemaître–Robertson–Walker metric and a perfect-fluid stress–energy tensor motivated by thermodynamic considerations used in analyses by Richard C. Tolman and Stephen Hawking. Substituting this metric into the Einstein field equations with cosmological constant Λ yields two independent equations: one relating H^2 to ρ, k and Λ, and another involving the acceleration ä/a tied to ρ and p. Conservation of stress–energy, ∇_μ T^μν = 0, leads to the continuity equation linking ρ̇, H and p, an approach employed in studies by George Gamow and Ralph Alpher in primordial nucleosynthesis contexts assessed by instruments such as CMB-S4.

Solutions and cosmological models

Solutions to the Friedmann equations produce distinct cosmological models depending on parameters and equations of state w = p/ρ. For w = 0 one obtains matter-dominated expansion used in early analyses by Edwin Hubble; for w = 1/3 radiation-dominated solutions connect to predictions of Big Bang nucleosynthesis tested against observations from WMAP and Planck. Inclusion of Λ yields de Sitter solutions relevant to inflation and late-time acceleration confirmed by the Supernova Cosmology Project and the High-Z Supernova Search Team. Spatial curvature k distinguishes open, flat, and closed models studied in reviews by John Peacock and constrained by datasets from Sloan Digital Sky Survey and 2dF Galaxy Redshift Survey. Exotic components such as dark matter (e.g., Weakly interacting massive particle candidates) and dark energy alter expansion histories and structure formation analyses led by collaborations like Dark Energy Survey.

Cosmological parameters and observational constraints

Key parameters derived from Friedmann dynamics include the Hubble constant H0, density parameters Ω_m, Ω_r, Ω_Λ, and curvature parameter Ω_k, which are estimated by combining probes: Type Ia supernova, Baryon acoustic oscillation, Cosmic microwave background, and gravitational lensing. Tensions such as the H0 discrepancy between local distance ladder results from SH0ES and CMB-inferred values from Planck have stimulated work involving teams at European Space Agency and National Aeronautics and Space Administration. Measurements of primordial element abundances tested against predictions by George Gamow and collaborators use observations at facilities like Very Large Telescope and Atacama Cosmology Telescope to constrain radiation density and neutrino properties associated with collaborations such as IceCube Neutrino Observatory.

Extensions and generalizations

Extensions generalize the Friedmann framework to incorporate modifications of gravity, anisotropies, or inhomogeneities. Examples include Brans–Dicke theory and f(R) gravity motivated by attempts from researchers linked to institutions like Institute for Advanced Study, as well as anisotropic models such as Bianchi cosmologies. Inhomogeneous models like the Lemaître–Tolman metric relax homogeneity assumptions used in void and backreaction studies related to observatories including Subaru Telescope. Quantum cosmology approaches—canonical quantization and loop quantum cosmology pursued at Perimeter Institute and Penn State University—apply modifications near singularities, while inflationary model-building by Andrei Linde and groups at CERN explores scalar-field-driven dynamics that extend simple perfect-fluid equations of state.

Historical context and significance

The equations were first derived by Alexander Friedmann in the early 1920s, independently of contemporaneous contributions by Georges Lemaître and following foundational work by Albert Einstein on General relativity. Their adoption transformed cosmology from static-universe debates, notably involving the Einstein–de Sitter model and discussions at institutions such as Princeton University and Cambridge University, into a predictive physical science capable of confronting data from observers like Edwin Hubble and experiments including COBE. The Friedmann equations remain a cornerstone linking theoretical physics, observational astronomy, and large collaborations—e.g., Planck, Dark Energy Survey, Sloan Digital Sky Survey—and continue to guide research into dark matter, dark energy, inflation, and fundamental tests of gravity at facilities such as Large Hadron Collider and LIGO Scientific Collaboration.

Category:Cosmology