Ernesto Lorenzo

In Cosmology, the metric Friedmann-Lemaître-Robertson-Walker (often shortened FLRW ) is metric making it possible to describe a homogeneous Univers locally , locally isotropic expanding or contraction. This model is used like a first approximation of the cosmological model standard of the universe, the Big Bang.

According to the geographical or historical preferences, model FLRW is sometimes indicated according to the names of part of the four scientists Alexander Friedmann, Georges Lemaître, Howard Percy Robertson and Arthur Geoffrey Walker; for example: Friedmann-Robertson-Walker (FRW) or Robertson-Walker (RW).

Use of the metric FLRW to describe the universe

Because model FLRW supposes that the universe is homogeneous, one could conclude from it that the model of Big Bang cannot take into account the fluctuations of density present in the universe. Indeed in a model strictly FLRW, there is no Galaxy cluster, neither star, neither Planet, nor biological beings, since these objects are denser by far than the Universe on average.

It of it is nothing, because actually, model FLRW is only used as first approximation because of the simplicity which it brings to calculations; the models taking account of the fluctuations of density are then added with model FLRW. The majority of the cosmologists agree so that the part of the observable Univers is well approximated by a model almost FLRW , i.e. a model which follows the metric FLRW to share of the paramount fluctuations of density. In 2003, the theoretical implications of these various extensions seem included/understood well and it goal is to make them coherent with the observations carried out by the satellites COBE and WMAP.

However, with the risk to forget the differences between model perfectly FLRW and the disturbed model, the model almost FLRW is normally called simply the model FLRW'

Mathematical formulation

In Coordinated polar $(R, \ theta, \ phi) \;$ , it can be written like:

or, using a change of coordinates to reveal the Distance comobile $\ chi \;$ :

where

$a (T) \;$ is the Scale factor of the Univers at the time T.
the parameter $k \;$ expresses the space Courbure and can take three values: +1, 0 or -1, respectively characterizing a closed curved space (correspondent with a spherical geometry), a flat space (corresponding to the Euclidean Geometry usual) and a space curves open (correspondent with a hyperbolic geometry)
$\ textstyle {\ rm D} \ Omega^2 = {\ rm D} \ theta^2 + \ sin^2 \ theta \; {\ rm D} \ phi^2$ expresses the contributions of metric related to the " direction" $(\ theta, \ phi) \;$ . For the study of the expansion of the universe, one will often take ${\ rm D} \ Omega^2 = 0 \;$ , since one will consider the radial trajectories of the photons according to a Géodésique.
$\chi \;$ is defined such as:

\ begin {boxes} R = \ sin \ chi & \ textrm {if \} K = 1 \ \ R = \ chi & \ textrm {if \} K = 0 \ \ R = \ sinh \ chi & \ textrm {if \} K = -1 \ \ \ end {boxes} where

\ chi \;

makes it possible to determine the Distance comobile.

$S_k (\ chi) = R \;$ (and can be expressed according to $\ chi$ directly starting from the definition above).

Metric FLRW according to the values of the curve

Metric FLRW in a flat space

If $k = 0 \;$ , one can rewrite the metric one:

{\ rm D} s^2 = c^2 {\ rm D} t^2 - has (T) ^2 ({\ rm D} r^2 + r^2 {\ rm D} \ Omega^2) \;

One finds here the traditional value of metric of a usual space equipped with a scale factor, expressed in radial coordinates $(R, \ theta, \ phi) \;$

Metric FLRW in a space of positive curve

If $k = +1 \;$ , one has

${\ rm D} s^2 = c^2 {\ rm D} t^2 - has (T) ^2 \ left (\ frac {1 - r^2} + r^2 {\ rm D} \ Omega^2 \ right)$

It is seen that in $r=1$ one has a singularity: one thus will seek a change of coordinates on the interval $] - 1; 1 \;$ allowing to reveal easily it [[distance comobile]. By noticing that $\ int \ frac {\ sqrt {1 - r^2}} = \ arcsin r$ , one chooses $\ chi \;$ such as $r = \ sin (\ chi) \;$ . One then obtains the new form of the metric one:

{\ rm D} s^2 = c^2 {\ rm D} t^2 - has (T) ^2 ({\ rm D} \ chi^2 + \ sin^2 \ chi \; {\ rm D} \ Omega^2) \;

Metric FLRW in a space of negative curve

If $k = -1 \;$ , one has

${\ rm D} s^2 = c^2 {\ rm D} t^2 - has (T) ^2 \ left (\ frac {1 + r^2} + r^2 {\ rm D} \ Omega^2 \ right)$

By noticing that $\ int \ frac {\ sqrt {1 + r^2}} = \ operatorname {arcsinh} \ r$ , one chooses like change of coordinates $\ chi \;$ such as $r = \ sinh (\ chi) \;$ . One then obtains the new form of the metric one:

{\ rm D} s^2 = c^2 {\ rm D} t^2 - has (T) ^2 ({\ rm D} \ chi^2 + \ sinh^2 \ chi \; {\ rm D} \ Omega^2) \;

See too

Alexandre Friedmann
Big Bang
Georges Lemaître
General relativity
Universe finished of Friedmann
George Gamow

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