Friedmann
equations

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Category:
Physical cosmology

v

t

e

Alexander
Friedmann

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[1]
from Einstein's field
equations of gravitation for the Friedmann–Lemaître–Robertson–Walker
metric and a perfect fluid with a given
mass density

and pressure
.
The equations for negative spatial curvature were given by Friedmann in
1924.[2]

Contents

1 Assumptions

2 The
equations

3 Density
parameter

4 Useful solutions

4.1 Mixtures

5 Rescaled Friedmann
equation

6 See also

7 Notes

8 External
links

Assumptions[edit]

Main article: Friedmann–Lemaître–Robertson–Walker
metric

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
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. The parameter
discussed below takes the value 0, 1, −1 in these three cases respectively. It
is this fact that allows us to sensibly speak of a "scale factor",
.

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 Ricci tensor. With stress-energy
tensor for perfect fluid, we plug them into Einstein field equations and the
resulting equations are described below.

The equations[edit]

There are two independent Friedmann equations for modeling a homogeneous,
isotropic universe. The first is:

which is derived from the 00 component of Einstein's field
equations. And the second is:

which is derived from the first together with the trace of
Einstein's field equations.
is the Hubble parameter,
G, Λ, and c are universal constants (G is Newton's gravitational
constant, Λ is the cosmological
constant, c is the speed of light in
vacuum). k is constant throughout a particular solution, but may vary
from one solution to another. a, H, ρ, and p are functions
of time.
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. We see that in the Friedmann equations, a(t) depends
only on ρ, p, Λ, and intrinsic curvature k. It does not depend on which
coordinate system we chose for spatial slices. There are two commonly used
choices for a and k which describe the same physics:

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k = +1, 0 or −1 depending on whether the shape of the
universe is a closed 3-sphere, flat (i.e. Euclidean space) or an
open 3-hyperboloid,
respectively.[3]
If k = +1, then
is the radius of curvature of the universe. If k = 0, then a may
be fixed to any arbitrary positive number at one particular time. If k
= −1, then (loosely speaking) one can say that i·a is the radius of
curvature of the universe.

a is the scale factor
which is taken to be 1 at the present time.
is the spatial curvature when
(i.e. today). If the shape of the
universe is hyperspherical
and
is the radius of curvature (
in the present-day), then .
If
is positive, then the universe is hyperspherical. If
is zero, then the universe is flat.
If
is negative, then the universe is hyperbolic.

Using the first equation, the second equation can be re-expressed as

which eliminates
and expresses the conservation of mass-energy .

These equations are sometimes simplified by replacing

to give:

And 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[edit]

The density parameter, ,
is defined as the ratio of the actual (or observed) density
to the critical density
of the Friedmann universe. The relation between the actual density and the
critical density determines the overall geometry of the universe. In earlier
models, which did not include a cosmological
constant term, critical density was regarded also as the watershed 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 atoms per cubic metre.[4]

Estimated relative distribution for components of
the energy density of the Universe. Dark energy dominates the
total energy (74%) while dark matter (22%)
constitutes most of the mass. Of the remaining baryonic matter (4%), only one
tenth is compact.

A much greater density comes from the unidentified dark matter; both ordinary
and dark matter contribute in favor 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 accelerates its expansion.
Therefore, the universe will expand forever.[citation
needed]

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
is the critical density for which the spatial geometry is flat (or Euclidean).
Assuming a zero vacuum energy density, if
is larger than unity, the space sections of the universe are closed; the
universe will eventually stop expanding, then collapse. If
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
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
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
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
is the radiation density today (i.e. when ),

is the matter (dark plus baryonic) density today,
is the "spatial curvature density" today, and
is the cosmological constant or vacuum density today.

Useful solutions[edit]

The Friedmann equations can be solved exactly in presence of a perfect
fluid with equation of state

where
is the pressure,
is the mass density of the fluid in the comoving frame and
is some constant.

In spatially flat case (k = 0), the solution for the scale
factor is

where
is some integration constant to be fixed by the choice of initial conditions.
This family of solutions labelled by
is extremely important for cosmology. E.g.
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

matter-dominated

Another important example is the case of a radiation-dominated
universe, i.e., when .
This leads to

radiation dominated

Note that this solution is not valid for domination of the cosmological
constant, which corresponds to an .
In this case the energy density is constant and the scale factor grows
exponentially.

Solutions for other values of k can be found at Tersic, Balsa. "Lecture Notes on Astrophysics". Retrieved 20 July 2011. .

Mixtures[edit]

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 a = 1; B is the density of
radiation (w = 1/3) when a = 1; and C
is the density of "dark energy" (w=−1). One then substitutes this
into

and solves for a as a function of time.

Rescaled Friedmann
equation[edit]

Set ,
where
and
are separately the scale factor and
the Hubble parameter today.
Then we can have

where .
For any form of the effective potential ,
there is an equation of state
that will produce it.

See also[edit]

Universe

Mathematics
of general relativity

Solutions
of Einstein's field equations

Warm inflation

Notes[edit]

^
Friedman,
A (1922). "Über die Krümmung des Raumes". Z. Phys. 10 (1):
377–386. Bibcode:1922ZPhy...10..377F. doi:10.1007/BF01332580.  (German) (English translation in: Friedman, A (1999). "On the Curvature of Space".
General Relativity and Gravitation 31 (12): 1991–2000. Bibcode:1999GReGr..31.1991F. doi:10.1023/A:1026751225741. ). The original Russian manuscript
of this paper is preserved in the Ehrenfest
archive.

^
Friedmann,
A (1924). "Über die Möglichkeit einer Welt mit konstanter negativer
Krümmung des Raumes". Z. Phys. 21 (1): 326–332. Bibcode:1924ZPhy...21..326F. doi:10.1007/BF01328280.  (German) (English translation in: Friedmann, A (1999). "On the Possibility of a World
with Constant Negative Curvature of Space". General Relativity and
Gravitation 31 (12): 2001–2008. Bibcode:1999GReGr..31.2001F. doi:10.1023/A:1026755309811. )

^
Ray A d'Inverno, Introducing Einstein's
Relativity, ISBN
0-19-859686-3.

^
Rees, M., Just Six Numbers, (2000) Orion Books,
London, p. 81, p. 82

External links[edit]

A
simple derivation of the Friedmann equations

Retrieved from "http://en.wikipedia.org/w/index.php?title=Friedmann_equations&oldid=555710246"

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