Friedmann equations

The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann in 1922 from Einstein's field equations of gravitation for the Friedmann–Lemaître–Robertson–Walker metric and a perfect fluid with a given mass density formula_1 and pressure formula_2. The equations for negative spatial curvature were given by Friedmann in 1924.
Assumptions.
The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and isotropic, i.e. the cosmological principle; empirically, this is justified on scales larger than ~100 Mpc. The cosmological principle implies that the metric of the universe must be of the form
where formula_4 is a three-dimensional metric that must be one of (a) flat space, (b) a sphere of constant positive curvature or (c) a hyperbolic space with constant negative curvature. This metric is called Friedmann–Lemaître–Robertson–Walker (FLRW) metric. The parameter formula_5 discussed below takes the value 0, 1, −1, or the Gaussian curvature, in these three cases respectively. It is this fact that allows us to sensibly speak of a "scale factor" formula_6.
Einstein's equations now relate the evolution of this scale factor to the pressure and energy of the matter in the universe. From FLRW metric we compute Christoffel symbols, then the Ricci tensor. With the stress–energy tensor for a perfect fluid, we substitute them into Einstein's field equations and the resulting equations are described below.
Equations.
There are two independent Friedmann equations for modelling a homogeneous, isotropic universe. The first is:
which is derived from the 00 component of Einstein's field equations. The second is:
which is derived from the first together with the trace of Einstein's field equations (the dimension of the two equations is time−2). 
formula_9 is the scale factor, "G", Λ, and "c" are universal constants ("G" is Newton's gravitational constant, Λ is the cosmological constant (its dimension is length−2) and "c" is the speed of light in vacuum). ρ and "p" are the volumetric mass density (and not the volumetric energy density) and the pressure, respectively. "k" is constant throughout a particular solution, but may vary from one solution to another.
In previous equations, formula_9, ρ, and "p" are functions of time. formula_11 is the spatial curvature in any time-slice of the universe; it is equal to one-sixth of the spatial Ricci curvature scalar R since 
in the Friedmann model. formula_13 is the Hubble parameter.
We see that in the Friedmann equations, a(t) does not depend on which coordinate system we chose for spatial slices. There are two commonly used choices for formula_9 and "k" which describe the same physics:
Using the first equation, the second equation can be re-expressed as 
which eliminates formula_28 and expresses the conservation of mass–energy formula_29
These equations are sometimes simplified by replacing
to give:
The simplified form of the second equation is invariant under this transformation.
The Hubble parameter can change over time if other parts of the equation are time dependent (in particular the mass density, the vacuum energy, or the spatial curvature). Evaluating the Hubble parameter at the present time yields Hubble's constant which is the proportionality constant of Hubble's law. Applied to a fluid with a given equation of state, the Friedmann equations yield the time evolution and geometry of the universe as a function of the fluid density.
Some cosmologists call the second of these two equations the Friedmann acceleration equation and reserve the term "Friedmann equation" for only the first equation.
Density parameter.
The density parameter formula_34 is defined as the ratio of the actual (or observed) density formula_1 to the critical density formula_36 of the Friedmann universe. The relation between the actual density and the critical density determines the overall geometry of the universe; when they are equal, the geometry of the universe is flat (Euclidean).
In earlier models, which did not include a cosmological constant term, critical density was initially defined as the watershed point between an expanding and a contracting Universe.
To date, the critical density is estimated to be approximately five atoms (of monatomic hydrogen) per cubic metre, whereas the average density of ordinary matter in the Universe is believed to be 0.2–0.25 atoms per cubic metre. 
A much greater density comes from the unidentified dark matter; both ordinary and dark matter contribute in favour of contraction of the universe. However, the largest part comes from so-called dark energy, which accounts for the cosmological constant term. Although the total density is equal to the critical density (exactly, up to measurement error), the dark energy does not lead to contraction of the universe but rather may accelerate its expansion. Therefore, the universe will likely expand forever.
An expression for the critical density is found by assuming Λ to be zero (as it is for all basic Friedmann universes) and setting the normalised spatial curvature, "k", equal to zero. When the substitutions are applied to the first of the Friedmann equations we find:
The density parameter (useful for comparing different cosmological models) is then defined as:
This term originally was used as a means to determine the spatial geometry of the universe, where formula_36 is the critical density for which the spatial geometry is flat (or Euclidean). Assuming a zero vacuum energy density, if formula_34 is larger than unity, the space sections of the universe are closed; the universe will eventually stop expanding, then collapse. If formula_34 is less than unity, they are open; and the universe expands forever. However, one can also subsume the spatial curvature and vacuum energy terms into a more general expression for formula_34 in which case this density parameter equals exactly unity. Then it is a matter of measuring the different components, usually designated by subscripts. According to the ΛCDM model, there are important components of formula_34 due to baryons, cold dark matter and dark energy. The spatial geometry of the universe has been measured by the WMAP spacecraft to be nearly flat. This means that the universe can be well approximated by a model where the spatial curvature parameter formula_5 is zero; however, this does not necessarily imply that the universe is infinite: it might merely be that the universe is much larger than the part we see. (Similarly, the fact that Earth is approximately flat at the scale of the Netherlands does not imply that the Earth is flat: it only implies that it is much larger than the Netherlands.)
The first Friedmann equation is often seen in terms of the present values of the density parameters, that is
Here formula_46 is the radiation density today (i.e. when formula_47), formula_48 is the matter (dark plus baryonic) density today, formula_49 is the "spatial curvature density" today, and formula_50 is the cosmological constant or vacuum density today.
Useful solutions.
The Friedmann equations can be solved exactly in presence of a perfect fluid with equation of state
where formula_2 is the pressure, formula_1 is the mass density of the fluid in the comoving frame and formula_54 is some constant.
In spatially flat case ("k" = 0), the solution for the scale factor is
where formula_56 is some integration constant to be fixed by the choice of initial conditions. This family of solutions labelled by formula_54 is extremely important for cosmology. E.g. formula_58 describes a matter-dominated universe, where the pressure is negligible with respect to the mass density. From the generic solution one easily sees that in a matter-dominated universe the scale factor goes as
Another important example is the case of a radiation-dominated universe, i.e., when formula_60. This leads to
Note that this solution is not valid for domination of the cosmological constant, which corresponds to an formula_62. In this case the energy density is constant and the scale factor grows exponentially.
Solutions for other values of "k" can be found at .
Mixtures.
If the matter is a mixture of two or more non-interacting fluids each with such an equation of state, then
holds separately for each such fluid "f". In each case,
from which we get
For example, one can form a linear combination of such terms
where: "A" is the density of "dust" (ordinary matter, "w" = 0) when formula_9 = 1; "B" is the density of radiation ("w" = 1/3) when formula_9 = 1; and "C" is the density of "dark energy" ("w"=−1). One then substitutes this into
and solves for formula_9 as a function of time.
Detailed derivation.
To make the solutions more explicit, we can derive the full relationships from the first Friedman equation:
with
Rearranging and changing to use variables formula_78 and formula_79 for the integration
Solutions for the dependence of the scale factor with respect to time for universes dominated by each component can be found. In each we also have assumed that formula_81, which is the same as assuming that the dominating source of energy density is formula_82.
For Matter dominated universes, where formula_83 and formula_50, as well as formula_85.
which recovers the aforementioned formula_89
For Radiation dominated universes, where formula_90 and formula_50, as well as formula_92
For formula_28 dominated universes, where formula_97 and formula_48, as well as formula_99, and where we now will change our bounds of integration from formula_100 to formula_101 and likewise formula_102 to formula_9.
The formula_28 dominated universe solution is of particular interest because the second derivative with respect to time is positive, non-zero; in other words implying an accelerating expansion of the universe, making formula_108 a candidate for dark energy:
Where by construction formula_111, our assumptions were formula_99, and formula_113 has been measured to be positive, forcing the acceleration to be greater than zero.
Rescaled Friedmann equation.
Set formula_114, where formula_56 and formula_113 are separately the scale factor and the Hubble parameter today.
Then we can have
where formula_118. For any form of the effective potential formula_119, there is an equation of state formula_120 that will produce it.