Friedmann–Lemaître–Robertson–Walker metric
The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric is an exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected. The general form of the metric follows from the geometric properties of homogeneity and isotropy; Einstein's field equations are only needed to derive the scale factor of the universe as a function of time. Depending on geographical or historical preferences, the set of the four scientists – Alexander Friedmann, Georges Lemaître, Howard P. Robertson and Arthur Geoffrey  Walker – are customarily grouped as Friedmann or Friedmann–Robertson–Walker (FRW) or Robertson–Walker (RW) or Friedmann–Lemaître (FL). This model is sometimes called the "Standard Model" of modern cosmology, although such a description is also associated with the further developed Lambda-CDM model. The FLRW model was developed independently by the named authors in the 1920s and 1930s. General metric. The FLRW metric starts with the assumption of homogeneity and isotropy of space. It also assumes that the spatial component of the metric can be time-dependent. The generic metric which meets these conditions is where formula_2 ranges over  a 3-dimensional space of uniform curvature, that is, elliptical space, Euclidean space, or hyperbolic space. It is normally written as a function of three spatial coordinates, but there are several conventions for doing so, detailed below. formula_3 does not depend on "t" — all of the time dependence is in the function "a"("t"), known as the "scale factor". Reduced-circumference polar coordinates. In reduced-circumference polar coordinates the spatial metric has the form "k" is a constant representing the curvature of the space. There are two common unit conventions: A disadvantage of reduced circumference coordinates is that they cover only  half of the 3-sphere in the case of positive curvature—circumferences beyond that point begin to decrease, leading to degeneracy. (This is not a problem if space is elliptical, i.e. a 3-sphere with opposite points identified.) Hyperspherical coordinates. In "hyperspherical" or "curvature-normalized" coordinates the coordinate "r" is proportional to radial distance; this gives where formula_7 is as before and As before, there are two common unit conventions: Though it is usually defined piecewise as above, "S" is an analytic function of both "k" and "r". It can also be written as a power series or as where  sinc is the unnormalized sinc function and formula_12 is one of the imaginary, zero or real square roots of "k". These definitions are valid for all "k". Cartesian coordinates. When "k" = 0 one may write simply This can be extended to "k" ≠ 0 by defining where "r" is one of the radial coordinates defined above, but this is rare. Curvature. Cartesian coordinates. In flat formula_17 FLRW space using Cartesian coordinates, the surviving components of the Ricci tensor are and the Ricci scalar is Spherical coordinates. In more general FLRW space using spherical coordinates (called "reduced-circumference polar coordinates  above), the surviving components of the Ricci tensor are and the Ricci scalar is Solutions. Einstein's field equations are not used in deriving the general form for the metric: it follows from the geometric properties of homogeneity and isotropy. However, determining the time evolution of formula_25 does require Einstein's field equations together with a way of calculating the density, formula_26 such as a cosmological equation of state. This metric has an analytic solution to Einstein's field equations formula_27 giving the Friedmann equations when the energy–momentum tensor is similarly assumed to be isotropic and homogeneous. The resulting 	Friedmann–Lemaître–Robertson–Walker metric
2868382	equations are: These equations are the basis of the standard Big Bang cosmological model including the current ΛCDM model. Because the FLRW model assumes homogeneity, some popular accounts mistakenly assert that the Big Bang model cannot account for the observed lumpiness of the universe. 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. Nonetheless, the FLRW model is used as a first approximation for the evolution of the real, lumpy universe because it is simple to calculate, and models which calculate 	Friedmann–Lemaître–Robertson–Walker metric
2868383	the lumpiness in the universe are added onto the FLRW models as extensions. Most cosmologists agree that the observable universe is well approximated by an almost FLRW model", i.e., a model which follows the FLRW metric apart from primordial density fluctuations. , the theoretical implications of the various extensions to the FLRW model appear to be well understood, and the goal is to make these consistent with observations from COBE and WMAP. If the spacetime is multiply connected, then each event will be represented by more than one tuple of coordinates. Interpretation. The pair of equations given above is equivalent to  the following pair of equations with formula_32, the spatial curvature index, serving as a constant of integration for the first equation. The first equation can be derived also from thermodynamical considerations and is equivalent to the first law of thermodynamics, assuming the expansion of the universe is an adiabatic process (which is implicitly assumed in the derivation of the Friedmann–Lemaître–Robertson–Walker metric). The second equation states that both the energy density and the pressure cause the expansion rate of the universe formula_33 to decrease, i.e., both cause a deceleration in the expansion of the universe. This is a  consequence of gravitation, with pressure playing a similar role to that of energy (or mass) density, according to the principles of general relativity. The cosmological constant, on the other hand, causes an acceleration in the expansion of the universe. Cosmological constant. The cosmological constant term can be omitted if we make the following replacements Therefore, the cosmological constant can be interpreted as arising from a form of energy which has negative pressure, equal in magnitude to its (positive) energy density: which is an equation of state of vacuum with dark energy. An attempt to generalize this to would not have  general invariance without further modification. In fact, in order to get a term which causes an acceleration of the universe expansion, it is enough to have a scalar field which satisfies Such a field is sometimes called quintessence. Newtonian interpretation. This is due to McCrea and Milne, although sometimes incorrectly ascribed to Friedmann. The Friedmann equations are equivalent to this pair of equations: