Friedmann–Lemaître–Robertson–Walker metric

Friedmann–Lemaître–Robertson–Walker metric
Name	Friedmann–Lemaître–Robertson–Walker metric
Caption	A visualization of spatial slices in an expanding universe described by the metric.
Field	General relativity, Physical cosmology
Discovered by	Alexander Friedmann, Georges Lemaître, Howard P. Robertson, Arthur Geoffrey Walker

Friedmann–Lemaître–Robertson–Walker metric. The Friedmann–Lemaître–Robertson–Walker metric is the cornerstone of modern Big Bang cosmology, providing the mathematical framework for a homogeneous and isotropic expanding universe. It is an exact solution to Einstein's field equations of general relativity and forms the basis for all quantitative models of cosmic evolution. The metric's parameters describe the universe's geometry and its dynamical expansion through cosmic time.

Definition and general form

The metric defines the spacetime interval in a universe obeying the cosmological principle. In comoving coordinates, its line element is given by \( ds^2 = -c^2 dt^2 + a(t)^2 \left[ \frac{dr^2}{1-k r^2} + r^2 d\Omega^2 \right] \), where \( c \) is the speed of light and \( t \) represents cosmic time. The function \( a(t) \), known as the scale factor, encodes the expansion history of the universe. The constant \( k \) quantifies the spatial curvature, taking values specific to hyperbolic geometry, Euclidean geometry, or spherical geometry. The angular element \( d\Omega^2 \) encompasses coordinates like those from the Schwarzschild metric.

Cosmological principle and homogeneity

The derivation of the metric rigorously applies the cosmological principle, which posits large-scale homogeneity and isotropy. This principle was strongly influenced by the philosophical Copernican principle and gained empirical support from observations like the Hubble-Lemaître law. Isotropy is confirmed by the remarkable uniformity of the cosmic microwave background radiation, first detected by Arno Penzias and Robert Wilson. Homogeneity implies that the universe has no preferred location, a concept tested by large-scale surveys such as the Sloan Digital Sky Survey.

Curvature and scale factor

The curvature parameter \( k \) determines the global geometry of spatial hypersurfaces. For \( k = +1 \), space is finite and positively curved, analogous to the surface of a 3-sphere. The case \( k = 0 \) corresponds to a flat, infinite Euclidean space, a critical geometry predicted by models like cosmic inflation. A value of \( k = -1 \) describes a negatively curved, infinite hyperbolic space. The evolution of the scale factor \( a(t) \) is governed by the matter and energy content of the universe, including contributions from dark energy and cold dark matter.

Solutions and Friedmann equations

Substituting the metric into the Einstein field equations yields the Friedmann equations, which dictate the dynamics of \( a(t) \). These equations relate the Hubble parameter to the total energy density of the universe, comprising components like baryonic matter, radiation, and the cosmological constant. Specific solutions include the Einstein–de Sitter universe for a matter-dominated epoch and the de Sitter universe for a cosmology dominated by a cosmological constant. The equations are foundational for the Lambda-CDM model, the standard model of Big Bang cosmology.

Historical development and naming

The metric emerged from independent work by several key figures in early 20th-century cosmology. Alexander Friedmann first derived the expanding universe solutions from general relativity in 1922, while Georges Lemaître independently rediscovered them and proposed the primeval atom hypothesis. Later, Howard P. Robertson and Arthur Geoffrey Walker rigorously proved the metric's uniqueness under the assumptions of isotropy and homogeneity. The synthesis of their contributions is honored in the metric's name, though it is sometimes referenced as the Robertson–Walker metric or the FLRW metric.

Observational evidence and applications

The FLRW framework is strongly validated by multiple astronomical observations. The expansion is directly measured through the Hubble-Lemaître law using Type Ia supernovae and Cepheid variable stars. The spatial flatness predicted by the metric is confirmed by detailed analysis of anisotropies in the cosmic microwave background by missions like the Wilkinson Microwave Anisotropy Probe and the Planck satellite. The model is essential for interpreting baryon acoustic oscillations, the formation of large-scale structure, and the observed abundance of light elements from Big Bang nucleosynthesis.

Category:General relativity Category:Physical cosmology Category:Metric tensors