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# Friedmann-Lemaître-Robertson-Walker

The Friedmann-Lemaître-Robertson-Walker (FLRW) metric describes a homogeneous, isotropic expanding/contracting universe. Depending on geographical/historical preferences, this may be referred to under the names of a preferred subset of the four scientists Alexander Friedmann, Georges Lemaître, Robertson and Walker, e.g. Friedmann-Robertson-Walker (FRW) or Robertson-Walker (RW).

The FLRW metric is used as a first approximation for the standard big bang cosmological model of the universe. Because the FLRW assumes homogeneity, some popular accounts mistakenly assert that the big bang model cannot account for the observed lumpiness of the universe. In actuality, the FLRW is used as a first approximation for the evolution of the universe because it is simple to calculate, and models which calculate the lumpiness in the universe are added onto FLRW as extensions. As of 2003, the theoretical implications of the various extensions to FLRW appear to be well understood, and the goal is to make these consistent with observations from COBE and WMAP.

The metric can be written as

$ds^2 = c^2 dt^2-a(t)^2[dr^2+\bar{r}^2 d\Omega^2]$
where:
a(t) = the scale factor of the universe at time t
$\bar{r} =\begin{cases} R_C \sinh(r/R_C), &\mbox{for negative curvature} \\ r, &\mbox{for zero curvature} \\R_C \sin(r/R_C), &\mbox{for positive curvature} \end{cases}$
where RC = the absolute value of the radius of curvature
dΩ2 = dθ2 + sin2θdφ2
In this formulation of the metric,
r gives the comoving distance from the observer
$\bar{r}$ gives the proper motion distance .

The solution to the FLRW metric for a fluid with constant density and pressure is given by the Friedmann equations. Most cosmologists agree that the observable universe is well approximated by an almost FLRW model, that is, a model which follows the FLRW metric apart from primordial density fluctuations. In a strictly FLRW model, there are no clusters of galaxies, stars or people, since these are objects much denser than a typical part of the universe.

However, for brevity, the almost FLRW model is often referred to simply as the FLRW model (or the FRW model).