Friedmann equations

The Friedmann equations, also known as the Friedmann–Lemaître (FL) equations, are a set of equations in physical cosmology that govern cosmic expansion 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 ρ and pressure p. The equations for negative spatial curvature were given by Friedmann in 1924.

The physical models built on the Friedmann equations are called FRW or FLRW models and form 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.

Assumptions

The Friedmann equations use three assumptions:

the Friedmann–Lemaître–Robertson–Walker metric,

Einstein's equations for general relativity, and

a perfect fluid source.

The metric in turn starts with the simplifying assumption that the universe is spatially homogeneous and isotropic, that is, the cosmological principle; empirically, this is justified on scales larger than the order of 100 Mpc.

The metric can be written as:

c

2

d

τ

2

=

c

2

d

t

2

−

R

2

(

t

)

(

d

r

2

+

S

k

2

(

r

)

d

ψ

2

)

{\displaystyle c^{2}d\tau ^{2}=c^{2}dt^{2}-R^{2}(t)\left(dr^{2}+S_{k}^{2}(r)d\psi ^{2}\right)}

where

S

−

1

(

r

)

=

sinh

⁡

(

r

)

,

S

0

=

1

,

S

1

=

sin

⁡

(

r

)

.

{\displaystyle S_{-1}(r)=\sinh(r),S_{0}=1,S_{1}=\sin(r).}

These three possibilities correspond to parameter k of (0) flat space, (+1) a sphere of constant positive curvature or (−1) a hyperbolic space with constant negative curvature.

Here the radial position has been decomposed into a time-dependent scale factor,

R

(

t

)

{\displaystyle R(t)}

, and a comoving coordinate,

r

{\displaystyle r}

.