Equations of Friedmann

The equations of Friedmann correspond to the equations of the General relativity (called equations of Einstein) written in the context of a cosmological Modèle homogeneous and isotropic. They thus govern the evolution of growth the Rate of the Univers and in consequence of the distance between two remote stars (the Scale factor) and according to the time called in this context cosmic Temps. The evolution of these quantities is determined by the properties of the material contents of the universe (Rayonnement, Atomes, black Matière, cosmological Constante, etc), thus possibly that the theory of the Gravitation considered: it is indeed possible to replace general relativity by another relativistic theory of the Gravitation. It can for example be a question of a Théorie tensor-scalar. Another way of changing the model is to consider standard general relativity but in a universe having one or more additional Dimensions. It is the case of the models of Cosmologie branaire.

The equations of Friedmann draw their name from the Russian Physicien Alexandre Friedmann which was the first to write them in the current of the Années 1920, with a first article treating of spaces with positive space curve in 1922, then more general in 1924 including the case of a negative space curve. It was followed closely by Georges Lemaître which found these equations in 1927, predicting or explaining the Loi of Hubble before this one is not discovered in 1929. Previously, in 1917, Albert Einstein had written them in the particular case of a static universe, like Willem de Sitter in the case of an empty matter universe, but with a cosmological Constante. The equations of Friedmann under their current form were found and presented in a form unified by Howard P. Robertson in 1929, then independently by Arthur G. Walker in 1936. For all these reasons, the cosmological type of model describes by these equations is called universe of Friedmann-Lemaître-Robertson-Walker (shortened in FLRW). The name of Lemaître is often absent, thus more rarely than that of Friedmann, these two people étants nevertheless considered as truths discoverers of these equations.

The equations of Friedmann are at the base of the quasi totality of the cosmological models, of which of course the Big Bang.

Two equations of Friedmann

There exist two equations, the first connecting the Rate growth H , the space Courbure K and the Scale factor has with the Densité of energy ρ , the second connecting the Pression P to the temporal derivative of the growth rate. The variable of time used is the cosmic Temps, which corresponds primarily to the time measured on Ground. Within the framework of general relativity, these two equations are written:

$3\; \backslash \; left\; (\backslash \; frac\; \{H^2\}\; \{c^2\}\; +\; \backslash \; frac\; \{K\}\; \{a^2\}\; \backslash \; right)\; =\; \backslash \; frac\; \{8\; \backslash \; pi\; G\}\; \{c^4\}\; \backslash \; rho$,

$-\; 2\; \backslash \; frac\; \{\backslash \; dowry\; H\}\; \{c^2\}\; -\; 3\; \backslash \; frac\; \{H^2\}\; \{c^2\}\; -\; \backslash \; frac\; \{K\}\; \{a^2\}\; =\; \backslash \; frac\; \{8\; \backslash \; pi\; G\}\; \{c^4\}\; P$,

where G is the Constante of Newton and C the Speed of light (the quantity $8\; \backslash \; pi\; G/c^4$ is sometimes called Constante of Einstein). These two equations are not independent: the second is obtained by taking the temporal derivative of the first, and by using the conservation equations, connecting the derivative of the density of energy to the pressure. When one considers an extension of general relativity or another theory, these equations are modified. For this reason, the term of equation of Friedmann is sometimes employed in the singular, in which case the only equation considered is the first.

It can arrive (for example in the models with dimensions additional) which they are not more consequences one of the other. In the other cases, it is possible to combine the two equations in order to modify the second, either not to reveal a certain combination of the pressure and density of energy in the member of right-hand side, or to reveal only the derivative second scale factor in the member of left (see paragraph below). It is also possible to carry out a change of variable to use the Temps conforms rather than cosmic time. The solution of these equations is once carried out the dependence of the density of energy and the pressure compared to time or with the known scale factor. A certain number of exact solutions are known.

Some particular solutions

When the member of right-hand side comprises only one species, and in the absence of space curve, the equations of Friedmann can be solved without difficulty. In the presence of several types of matter and/or curve, exact analytical solutions can sometimes be found. In the other cases, a numerical Résolution of the equations is done without difficulty.

Universe of dust

When the universe is filled up of nonrelativistic matter (i.e. whose pressure is negligible, from where the term of “dust”), the density of energy decrease only because of dilution due to the expansion. The scale factor evolves/moves then according to the law

$a\; (T)\; =\; a\_0\; \backslash \; left\; (\backslash \; frac\; \{T\}\; \{t\_0\}\; \backslash \; right)\; ^\; \backslash \; frac\; \{2\}\; \{3\}$,

the quantities has and T corresponding to the scale factor and the times evaluated at a given base period. Moreover, the age of the universe at that time, T results from the growth rate at the same time by $t\_0\; =\; \backslash \; frac\; \{2\}\; \{3\}\; \backslash \; frac\; \{1\}\; \{H\_0\}$. Such a solution is called, for historical reasons Univers of Einstein-in Sitter, although the latter are not the first to have exhibé this solution.

In such a model, the universe is thus younger than the Temps of Hubble. If one takes the measured value of the Constante of Hubble, to approximately 70 Kilomètre S by second and Mégaparsec, the numerical application gives an age hardly higher than 9 billion years. This age being significantly lower than the estimated age of many astrophysical objects (some star S, dwarf white, and the Milky Way as a whole), it is considered that such a universe of dust cannot correspond to the observable universe, which is one of the indications among others of the existence of the black energy (see below). It is as to note as the dependence of the scale factor compared to time is such as its derivative second is negative, which corresponds to a phase of decelerated expansion, compatible with the intuition that the gravitational nature of gravity tends to slow down the movement of expansion. The observations, in particular that of the Supernova E of the Ia type, suggest that the current phase of expansion is accelerated, which is another indication of the need for the presence of black energy. It is on the other hand certain that the universe knew in the past a phase where its expansion was dominated by nonrelativistic matter, because it is the only time when the mechanism of gravitational Instabilité can occur.

Universe of radiation

When the universe is filled up of relativistic matter or radiation, the density of energy ρ decrease more quickly than previously because in addition to dilution due to the expansion, the energy of the individual particles decrease with the expansion (it is anything else only the effect of Décalage towards the red). The density decreasing more quickly, the growth rate decrease also more quickly, and the expansion decelerates more quickly than in the case of a universe of dust. The scale factor evolves/moves according to the law

$a\; =\; a\_0\; \backslash \; left\; (\backslash \; frac\; \{T\}\; \{t\_0\}\; \backslash \; right)\; ^\; \backslash \; frac\; \{1\}\; \{2\}$.

The age of the universe in such a model is written

$t\_0\; then\; =\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; frac\; \{1\}\; \{H\_0\}$.

As in the case of the universe of dust, the expansion is decelerated and the age given by this model cannot correspond to the observations. This model corresponds on the other hand well to the moved back times of the history of the universe, when the near total of the matter was in relativistic form. In particular the time of the nucleosynthesis occurred when the density of energy was mainly due to relativistic matter, assumption that one tests directly by the measurement of the Abondance of the light elements. Other indications owing to the fact that the universe knew one period dominated by radiation is the existence of the cosmological diffuse Fond.

Cosmological constant

A cosmological constant can be interpreted like the matter shape of density of constant energy and pressure opposed exactly to this one. When the density of energy is positive, the equations of Friedmann admit, in the absence of space curve, the exponential solution

$a\; \backslash \; propto\; e^\; \{H\_0\; T\}$,

the parameter of Hubble being constant during time. Such a model represents a universe eternal, without beginning nor end. It is the Univers of Sitter, obeying the perfect cosmological Principe. There exists another configuration giving place to the same dynamics of the expansion: the two matter shapes are then of the nonrelativistic ordinary matter, and the extremely exotic matter shape, called Champ C, person in charge of a continuous creation of matter compensating for exactly dilution due to the expansion. It is the base of the Théorie of the stationary state, now abandoned because of the complete absence of motivation theoretical or observational of the field C. Lastly, in the presence of a scalar Champ, this one can, temporarily to behave in a way extremely similar to a cosmological constant, behavior which it can give up thereafter. Thus, it is possible that under the effect of such a scalar field, the expansion of the universe knows an exponential phase. It is the base of the models of cosmic Inflation which predict the existence of such a phase at one very moved back time of the history of the universe.

Constant equation of state

As long as the pressure is proportional to the density, with a constant of constant proportionality, one can solve the equations of Friedmann. One notes W the report/ratio of the pressure to the density,

$w\; =\; \backslash \; frac\; \{P\}\; \{\backslash \; rho\}$.

For any reasonable matter shape, the factor W is included/understood between -1 and 1. The case where W is lower than -1 corresponds to what is called phantom energy, a form of extremely speculative energy giving place to a strange cosmological scenario, the Big RIP. The cases where W is worth 0,1/3, -1 correspond respectively to the cases of dust, the radiation and of the cosmological Constante. When W is different from -1, one obtains

$a\; =\; a\_0\; \backslash \; left\; (\backslash \; frac\; \{3\; (1\; +\; W)\}\{2\}\; H\_0\; |T|\; \backslash \; right)\; ^\; \backslash \; frac\; \{2\}\; \{3\; (1\; +\; W)\}$.

When W is higher than -1, the scale factor tends towards 0 when T tends towards 0. There is thus a Big Bang, and the expansion continues indefinitely. Contrary, when W is lower than -1, then this time, the scale factor tends towards 0 when T is negative and tends towards the infinite one, and it tends towards the infinite one when T tends towards 0. In this case, there are an existing universe of any eternity, but having one lifespan finished in the future (see below).

When W is higher than -1, the age of the universes is written

$t\_0\; =\; \backslash \; frac\; \{2\}\; \{3\; (1+w)\}\; \backslash \; frac\; \{1\}\; \{H\_0\}$.

The value of W equal to 0 and 1/3 give again the preceding results of course. When W is equal to -1, there are a universe without beginning nor end, and not presenting evolution. When W is lower than -1, there are a universe without beginning, but reaching a gravitational Singularité in a finished time, although always remaining expanding. Indeed, when the scale factor tends towards the infinite one, then the density of energy diverges, and this in a finished time: it is the Big RIP ( litt. “large tear”). Time that it remains with the universe before this singularity is given by

$t\_f\; =\; \backslash \; frac\; \{2\}\; \{3|1+w|\}\; \backslash \; frac\; \{1\}\; \{H\_0\}$.

Solutions with several species

If several the matter species or shapes fill up the universe and take part notably in the equations of Friedmann, it is necessary to take account of their equations of state respective and their relative abundance. In the case of two species, for example, the first equation of Friedmann rewrites

$3\; \backslash \; left\; (\backslash \; frac\; \{H^2\}\; \{c^2\}\; +\; \backslash \; frac\; \{K\}\; \{a^2\}\; \backslash \; right)\; =\; \backslash \; frac\; \{8\; \backslash \; pi\; G\}\; \{c^4\}\; \backslash \; left\; (\backslash \; rho\_1\; +\; \backslash \; rho\_2\; \backslash \; right)$.

If one calls W and W the reports/ratios of the pressure to the density of energy of each of the two species and that one supposes them constant with the court of time, one can rewrite this equation using various adimensionnées variables, in the form

$\backslash \; left\; (\backslash \; frac\; \{X\; H\_0\; \{\backslash \; rm\; \{D\}\}\; T\}\; \backslash \; right)\; ^2\; =\; \backslash \; left\; x^\; \{-\; 3\; (1\; +\; w\_1)\}+\; \backslash \; Omega\_2^0\; x^\; \{-\; 3\; (1\; +\; w\_2)\}\backslash \; right$,

where one defined, as from one base period noted with the index or the exhibitor the 0 Densité criticizes $\backslash \; rho^0\_\; \{\backslash \; rm\; \{C\}\}$ by

$\backslash \; rho^0\_\; \{\backslash \; rm\; \{C\}\}\; =\; \backslash \; frac\; \{3\; c^2\; H^2\_0\}\; \{8\; \backslash \; pi\; G\}$,

as well as the quantities without dimension called parameters of density $\backslash \; Omega\_i^0$ by

$\backslash \; Omega\_i^0\; =\; \backslash \; frac\; \{\backslash \; rho\_i^0\}\; \{\backslash \; rho\_\; \{\backslash \; rm\; \{C\}\}\; ^0\}$,

and the scale factor standardized, X , by $x\; =\; has/a\_0$. This expression is simplified when the total density equalizes the critical density, or, in an equivalent way when the sum of the parameters of density is worth 1. There is then

$\backslash \; left\; (\backslash \; frac\; \{X\; H\_0\; \{\backslash \; rm\; \{D\}\}\; T\}\; \backslash \; right)\; ^2\; =\; \backslash \; left\; x^\; \{-\; 3\; (1\; +\; w\_1)\}+\; \backslash \; Omega\_2^0\; x^\; \{-\; 3\; (1\; +\; w\_2)\}\backslash \; right$.

Each one of these two expressions spreads with an arbitrary number of components. Exact solutions exist for certain values of the parameters W and W .

In particular, in E tallies of the standard Modèle of cosmology, “universe can be described as being filled up of three types of species: relativistic matter (Neutrino S and radiation), nonrelativistic matter (baryon Matter and black Matter), and black energy, which one here will approximate by a cosmological constant. Relative abundances of these species, which vary with time make that it never happens that they coexist with each one of the density of energy significant. At the time current, one finds primarily energy black and nonrelativistic matter, whereas at the moved back times, one found matter relativistic and nonrelativistic, the cosmological constant being negligible. This comes owing to the fact that abundances of these various types of matter vary during time according to $1/x^\; \{3\; (1\; +\; W)\}$: the density of energy associated with the cosmological constant remains constant, whereas that of the or not relativistic relativistic matter grows as one goes up towards the past.

Dust and radiation

When one goes up towards the past, the densities of nonrelativistic matter (“dust”) or relativist (“radiation”) grow like $x^\; \{-\; 3\}$ and $x^\; \{-\; 4\}$ respectively. There is thus

$\backslash \; left\; (\backslash \; frac\; \{X\; H\_0\; \{\backslash \; rm\; \{D\}\}\; T\}\; \backslash \; right)\; ^2\; =\; \backslash \; left\; +\; \backslash \; frac\; \{\backslash \; Omega^0\_\; \{\backslash \; rm\; \{R\}\}\}\; \{x^4\}\; \backslash \; right$,

with

$\backslash \; Omega^0\_\; \{\backslash \; rm\; \{R\}\}\; =\; 1\; -\; \backslash \; Omega^0\_\; \{\backslash \; rm\; \{m\}\}$.

Integration gives

$\backslash \; frac\; \{4\; \backslash \; left\; (1\; -\; \backslash \; Omega^0\_\; \{\backslash \; rm\; \{m\}\}\; \backslash \; right)\; ^\; \backslash \; frac\; \{3\}\; \{2\}\}\; \{3\; \backslash \; left\; (\backslash \; Omega^0\_\; \{\backslash \; rm\; \{m\}\}\; \backslash \; right)\; ^2\}\; \backslash \; left\; -\; \backslash \; sqrt\; \{1\; +\; \backslash \; frac\; \{\backslash \; Omega^0\_\; \{\backslash \; rm\; \{m\}\}\; X\}\; \{1\; -\; \backslash \; Omega^0\_\; \{\backslash \; rm\; \{m\}\}\}\}\; \backslash \; left\; (1\; -\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; frac\; \{\backslash \; Omega^0\_\; \{\backslash \; rm\; \{m\}\}\; X\}\; \{1\; -\; \backslash \; Omega^0\_\; \{\backslash \; rm\; \{m\}\}\}\; \backslash \; right)\; \backslash \; right\; =\; H\_0\; T$.

According to the value of X compared to $\backslash \; frac\; \{1\; -\; \backslash \; Omega^0\_\; \{\backslash \; rm\; \{m\}\}\}\; \{\backslash \; Omega^0\_\; \{\backslash \; rm\; \{m\}\}\}$, which corresponds at the time of transition where density of relativistic and nonrelativistic matter are equal, one finds two modes previously studied. For the values of X small, one obtains

$\backslash \; frac\; \{1\}\; \{2\}\; x^2\; =\; \backslash \; sqrtH\_0\; T$,

what corresponds to the result for the universe of radiation found above, the value of $H\_0$ being replaced by $\backslash \; sqrtH\_0$, corresponds to the value which the constant of Hubble in the absence of nonrelativistic matter would have.

In the other mode, one finds

$\backslash \; frac\; \{2\}\; \{3\}\; x^\; \backslash \; frac\; \{3\}\; \{2\}\; =\; \backslash \; sqrtH\_0\; T$,

who there still gives again the result of the universe of dust to the correction on the value of the constant of Hubble near.

Dust and cosmological constant

The second important particular case is that of a universe dominated by nonrelativistic matter ( W   =  0) and of which the cosmological constant ( W   =  - 1) becomes dominant. It is according to any probability in a similar situation that trouv our universe, the Accélération of the expansion of the universe attesting existence of the matter shape behaving in a rather close way with a cosmological constant. In this case, the integration of the equations of Friedmann gives, with $\backslash \; Omega\_\; \{\backslash \; rm\; \{m\}\}\; ^0\; =\; 1\; -\; \backslash \; Omega\_\; \backslash \; Lambda^0$,

$\backslash \; frac\; \{has\}\; \{a\_0\}\; =\; \backslash \; left\; -\; \backslash \; Omega\_\; \backslash \; Lambda^0\}\; \{\backslash \; Omega\_\; \backslash \; Lambda^0\}\}\; \backslash \; sinh\; \backslash \; left\; (\backslash \; frac\; \{3\; \backslash \; sqrt\; \{\backslash \; Omega\_\; \backslash \; Lambda^0\}\}\; \{2\}\; H\_0\; T\; \backslash \; right)\; \backslash \; right^\; \backslash \; frac\; \{2\}\; \{3\}$.

For small times, one can carry out a limited development which gives again the behavior in $t^\; \backslash \; frac\; \{2\}\; \{3\}$:

$\backslash \; frac\; \{has\}\; \{a\_0\}\; \backslash \; sim\; \backslash \; left\; \backslash \; sqrt\; \{\backslash \; Omega\_\; \{\backslash \; rm\; \{m\}\}\; ^0\}\; H\_0\; T\; \backslash \; right^\; \backslash \; frac\; \{2\}\; \{3\}$.

One finds exactly the formula already obtained for a universe of dust, with this close it is $\backslash \; sqrt\; \{\backslash \; Omega\_\; \{\backslash \; rm\; \{m\}\}\; ^0\}\; H\_0$ which appears instead of $H\_0$. This comes owing to the fact that in cosmological labsence of constant, the constant of Hubble would be reduced of a factor $\backslash \; sqrt\; \{\backslash \; Omega\_\; \{\backslash \; rm\; \{m\}\}\; ^0\}$. The fact that one finds the formula of the universe of dust comes owing to the fact that at these times, the cosmological constant is negligible compared to the density of matter.

Contrary, for large times, one finds the exponential behavior of the scale factor compared to time:

$\backslash \; frac\; \{has\}\; \{a\_0\}\; \backslash \; sim\; \backslash \; left\; (\backslash \; frac\; \{1\; -\; \backslash \; Omega\_\; \backslash \; Lambda^0\}\; \{4\; \backslash \; Omega\_\; \backslash \; Lambda^0\}\; \backslash \; right)\; ^\; \backslash \; frac\; \{1\}\; \{3\}\; \backslash \; exp\; \backslash \; left\; (\backslash \; sqrt\; \{\backslash \; Omega\_\; \backslash \; Lambda^0\}\; H\_0\; T\; \backslash \; right)$.

In this case, the constant of Hubble tightens indeed towards the asymptotic value $\backslash \; sqrt\; \{\backslash \; Omega\_\; \backslash \; Lambda^0\}\; H\_0$ which enters the exhibitor. The transition between these two modes is carried out at the time when the argument of the hyperbolic Sinus is about 1. When that occurs, the $a\; quantity/a\_0$ is about $\backslash \; left\; -\; \backslash \; Omega\_\; \backslash \; Lambda^0\; \backslash \; right)/\backslash \; Omega\_\; \backslash \; Lambda^0\; \backslash \; right^\; \backslash \; frac\; \{1\}\; \{3\}$. That corresponds to the moment when the density of energy associated with the cosmological constant becomes about that of the matter. Before this time it is that of the matter which dominates, and one finds the results of the universe of dust, after this time, it is the cosmological constant which dominates, and one finds the dynamics of the universe of Sitter.

An interesting point is to calculate the relation between age of the universe and time of Hubble $1/H\_0$. One finds

$t\_0\; thus\; =\; \backslash \; frac\; \{1\}\; \{H\_0\}\; \backslash \; frac\; \{2\}\; \{3\}\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{\backslash \; Omega\_\; \backslash \; Lambda^0\}\}$

\ arg \ sinh \ sqrt {\ frac {\ Omega_ \ Lambda^0} {1 - \ Omega_ \ Lambda^0}} . For small values of $\backslash \; Omega\_\; \backslash \; Lambda^0$, one finds the usual value of $t\_0\; =\; 2/3:00\; \_0$, whereas as $\backslash \; Omega\_\; \backslash \; Lambda^0$ grows (while remaining of course lower than 1), the age of the universe becomes larger than the time of Hubble.

Other writings

In certain cases, one can prefer to express the equations according to the scale factor and one of the parameter of Hubble. It comes then

$3\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{c^2\}\; \backslash \; frac\; \{\backslash \; dowry\; a^2\}\; \{a^2\}\; +\; \backslash \; frac\; \{K\}\; \{a^2\}\; \backslash \; right)\; =\; \backslash \; frac\; \{8\; \backslash \; pi\; G\}\; \{c^4\}\; \backslash \; rho$,

$-\; 2\; \backslash \; frac\; \{1\}\; \{c^2\}\; \backslash \; frac\; \{\backslash \; ddot\; has\}\; \{has\}\; -\; \backslash \; frac\; \{1\}\; \{c^2\}\; \backslash \; frac\; \{\backslash \; dowry\; a^2\}\; \{a^2\}\; -\; \backslash \; frac\; \{K\}\; \{a^2\}\; =\; \backslash \; frac\; \{8\; \backslash \; pi\; G\}\; \{c^4\}\; P$.

If necessary, the second equation can be rewritten while making disappear the term from curve by using the first, which gives

$\backslash \; frac\; \{\backslash \; ddot\; has\}\; \{has\}\; =\; -\; \backslash \; frac\; \{4\; \backslash \; pi\; G\}\; \{3\}\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; rho\; +\; 3\; P\}\; \{c^2\}\; \backslash \; right)$.

This last form, which gives the relative acceleration of two distant objects because of expansion of the universe is a particular case of the equation of Raychaudhuri and for this reason sometimes called thus. More generally, very written second equation of Friedmann revealing the derivative second scale factor (or the derivative first of the growth rate) can be called thus.

Lastly, one can also prefer to extract the cosmological constant from the material contents of the universe, thus giving him a purely geometrical role. This choice is regarded today as not very convenient in cosmology because the true nature of black energy is unknown, but corresponds historically to that of Einstein and Lemaître. One obtains then, by considering that the cosmological constant is homogeneous contrary to the square a length,

$3\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{c^2\}\; \backslash \; frac\; \{\backslash \; dowry\; a^2\}\; \{a^2\}\; +\; \backslash \; frac\; \{K\}\; \{a^2\}\; \backslash \; right)\; -\; \backslash \; Lambda\; =\; \backslash \; frac\; \{8\; \backslash \; pi\; G\}\; \{c^4\}\; \backslash \; rho$,

$-\; 2\; \backslash \; frac\; \{1\}\; \{c^2\}\; \backslash \; frac\; \{\backslash \; ddot\; has\}\; \{has\}\; -\; \backslash \; frac\; \{1\}\; \{c^2\}\; \backslash \; frac\; \{\backslash \; dowry\; a^2\}\; \{a^2\}\; -\; \backslash \; frac\; \{K\}\; \{a^2\}\; +\; \backslash \; Lambda\; =\; \backslash \; frac\; \{8\; \backslash \; pi\; G\}\; \{c^4\}\; P$.

The writings above use cosmic time. It is possible and sometimes useful to substitute the to him Temps conforms, η , defined by the formula

$a\; (\backslash \; eta)\; \{\backslash \; rm\; D\}\; \backslash \; eta\; =\; C\; \{\backslash \; rm\; D\}\; T\; \backslash ,$

or

$\backslash \; eta\; (t\_0)\; =\; \backslash \; int^\; \{t\_0\}\; C\; \backslash \; frac\; \{has\; (T)\}$

so that the element length becomes proportional to metric of Minkowski (one says conformément minkowskien):

$\{\backslash \; rm\; D\}\; s^2\; =\; a^2\; (\backslash \; eta)\; \backslash \; left\; (\{\backslash \; rm\; D\}\; \backslash \; eta^2\; -\; \backslash \; gamma\_\; \{ij\}\; \{\backslash \; rm\; D\}\; x^i\; \{\backslash \; rm\; D\}\; x^j\; \backslash \; right)$.

In this case, one can define a “parameter of Hubble conforms” $\{\backslash \; mathcal\; H\}$ by

$\{\backslash \; mathcal\; H\}\; =\; \backslash \; frac\; \{1\}\; \{has\}\; \backslash \; frac$.

Such a quantity does not have immediate physical interpretation, but allows the rewriting of the equations of Friedmann in term of time in conformity:

$3\; \backslash \; left\; (\{\backslash \; mathcal\; H\}\; ^2\; +\; K\; \backslash \; right)\; =\; \backslash \; frac\; \{8\; \backslash \; pi\; G\}\; \{c^4\}\; a^2\; \backslash \; rho$,

$-\; 2\; \backslash \; partial\_\; \backslash \; eta\; \{\backslash \; mathcal\; H\}\; -\; \{\backslash \; mathcal\; H\}\; ^2\; -\; K\; =\; \backslash \; frac\; \{8\; \backslash \; pi\; G\}\; \{c^4\}\; a^2\; P$.

The solution of these equations appreciably follows the same stages as in the preceding cases. In particular, one finds the dependences following of the scale factor compared to time in conformity:

• $a\; \backslash \; propto\; \backslash \; eta$ in a universe of radiation
• $a\; \backslash \; propto\; \backslash \; eta^2$ in a universe of dust
• $a\; \backslash \; propto\; -\; \backslash \; eta^\; \{-\; 1\}$ with a cosmological constant
• $a\; \backslash \; propto\; |\backslash \; eta|^\; \backslash \; frac\; \{2\}\; \{1\; +\; 3\; W\}$ with the matter shape whose equation of state is $P\; =\; W\; \backslash \; rho$.

To note that η varies from 0 with $+\; \backslash \; infty$ according to a law of power with positive exhibitor as long as W is higher than -1/3, varies exponentially when W is worth -1/3 and varies $-\; \backslash \; infty$ to 0 according to a law of power with negative exhibitor when W is lower than -1/3.

The interest to solve the equations of Friedmann in term of time in conformity comes owing to the fact that the concept of Horizon of the particles and of Horizon of the events is very narrowly connected to the relation $a\; (\backslash \; eta)$, and in particular with its behavior for smallest and greater values of η . Moreover, the concepts of angular distance and Distance of luminosity depend them also directly on this same relation.

Interpretation

The equations of Friedmann in the presence of nonrelativistic matter can be found (in a heuristic way) by a purely Newtonian reasoning. One can indeed consider the evolution of a matter sphere, of which one supposes the constant density at any moment. This assumption is inaccurate in general, but to do it allows to place itself in a situation rather similar to that of a homogeneous and isotropic universe. in which case, the growth rate of the sphere is connected to its density consequently formula that the first equation of Friedmann. The crucial difference between these two approaches comes from the part played by the curve K (in the relativistic model) which are connected in the Newtonian model with a constant of integration without geometrical significance. In general relativity it determines the geometrical properties of space.

Derivation of the equations of Friedmann

The equations of Friedmann are anything else only the writing of the equations of Einstein describing a homogeneous and isotropic universe. Their derivation does not raise a particular difficulty, and they represent even one of the exact solutions analytical simplest among those known with these equations.

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