Friedmann–Lemaître–Robertson–Walker metric

The Friedmann–Lemaître–Robertson–Walker metric is the spacetime geometry of a homogeneous, isotropic universe, with cosmic expansion or contraction encoded by a time-dependent scale factor.

Scale factor (cosmology): A time-dependent quantity that describes how distances between comoving points change as the universe expands or contracts. Its evolution carries the metric’s expansion or contraction.
Metric tensor: A mathematical field that defines spacetime intervals, angles, and distances. The FLRW metric is a particular metric tensor with cosmological symmetries.
Hubble–Lemaître law: The relation between a galaxy’s distance and its recession velocity in an expanding universe. The scale factor’s expansion produces the large-scale distance–velocity relation.
Bianchi cosmology: A family of homogeneous cosmological models that can have anisotropic spatial expansion. It relaxes FLRW isotropy while retaining spatial homogeneity.
Comoving coordinates: Coordinates that remain fixed for objects moving with the average cosmic expansion. They separate the universe’s expansion from the motion of individual objects.
General relativity: A theory of gravity in which matter and energy shape spacetime geometry. The FLRW geometry is used with Einstein’s equations to model cosmic evolution.
Cosmological redshift: The increase in a photon’s wavelength as the universe expands during its journey. The metric links wavelength stretching directly to the change in scale factor.
Lemaître–Tolman–Bondi model: A spherically symmetric solution of Einstein’s equations with inhomogeneous matter and radial expansion. It relaxes homogeneity, allowing radial cosmic structure absent from FLRW.
Spatial curvature: A measure of how spatial geometry differs locally or globally from Euclidean geometry. The metric allows positively curved, flat, or negatively curved spatial slices.
Cosmological principle: The assumption that the universe is homogeneous and isotropic on sufficiently large scales. It supplies the symmetry assumptions that single out the FLRW form.

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