The first scientific evidence for a non-static universe came with the formulation of the theory of General Relativity by Einstein in 1915, following a completely new interpretation of the nature of the gravitational force. The basic idea is that the presence of matter alters the geometry of space-time. Test-particles follows the geodesics in space-time, which determine their orbits.

Quantitatively this idea is expressed by equating a *geometry tensor*
*G*_{µ}
(which incorporates the ten curvature terms required to
specify the geometry) to a *matter tensor*
*T*_{µ}
which contains all the
information (ten terms) about the mass-energy present in the
manifold. (*G* is Newton's constant)

(1) |

In a simplified model in which the universe is represented as an
homogeneous isotropic fluid (an approximation suggested by the
homogeneous distribution of galaxies in space) this equation reduces
to a much simpler equation for *R*: the distance between any two
objects
(galaxies for instance) in the universe. (Here (*R*^{0}) is
the time derivative of *R*).

(2) |

The space-time curvature terms are on the left-hand side of the
equation. The first term: (*R*^{0}/*R*)^{2} is
the *time curvature* while the
second one: *k*/*R*^{2} is the space curvature. The
value of *k* is normalized
to take one of the three values: *k* = 0 (flat universe), +1 (closed
universe) or -1 (open universe). The gravitational effect of matter is
represented by the energy density
. The last term
is the famous
*cosmological constant*. Mathematically it is a sort of a priori
undetermined constant of integration.

It is this equation which historically gave the first indication
that our universe may not be static. *It equates the time derivative of
the scale factor R(R^{0}) to a sum of terms which
has no obvious reason to be zero.*

Einstein decided to give to
the exact
numerical value required to
obtain *R*^{0} = 0. This decision was most
unfortunate. Einstein failed to
make the important discovery that the universe is not static.
Furthermore it was soon shown that his solution is not stable. Even if
is (arbitrarily)
chosen to neutralize the time derivative of *R*
today, any density perturbation would suffice to bring the fluid out
of this unstable equilibrium and induce a general motion one way or
the other (contraction or expansion).

After 1930, thanks to the work of Edwin Hubble, the systematic
motion (recession) of the galaxies was detected. The observations gave
the relation
*R*^{0}/*R* = *cst* = *H*, where *H* is
the Hubble *constant*. The value
of *H* is still uncertain by a factor of two. It lies between 50
and 100
km/sec/megaparsec. The inverse of this parameter has the dimension of
a time: the time scale of the universal expansion. Its value lies
between 10 and 20 billion years.

Soon after this detection, models of the expanding universe were
presented by Friedman, Lemaitre, and Alpher, Herman, Gamov. Assuming
an homogeneous fluid of ideal gases, the
*T*_{µ}
simplifies to its diagonal form containing
(, - *P*,
- *P*, - *P*) where *P* is the pressure
and
the density of the fluid. The dynamics is given by eq. 2. The
energy-momentum conservation expression,
*T*_{µ;} = 0,
simplifies to the following equation:

(3) |

The equation
*T*_{µ;} = 0 implies that the entropy per
comoving volume *S* = *sR*^{3} remains constant which, in turn, requires that:

(4) |

The entropy per unit volume *s*, given by
*s* = ( +
*P*)/*T*, decreases
with *R*^{-3}, just as the number-density of particles.

One should be careful about these two last equations. They assume that entropy generating processes are operating in the fluid. In the standard mode this is guaranteed by the hypothesis of a fluid of pure non-interacting particles (except by gravitational interactions). In the real world these equations are valid only when entropy generating processes are negligibly small. This is correct most of the time. There are important chapters of cosmic expansion when this is not true, and appropriate corrections have to be made.

Eqs. (3, 4) need to be completed by an equation of state for the
fluid. Three regimes are important during cosmic expansion: radiation,
matter, and *vacuum*.

A) When the universe is *radiation dominated*, i.e., when the largest
contribution to the energy density is due to relativistic particles
(*kT* >> *Mc*^{2}), we have
*T*^{4}
and the pressure term is one-third the density term (*P* =
1/3). From
eqs. (3, 4) we get:
*R*^{-4}
or *R* 1/*T* as
expected from entropy conservation. The entropy density *s* is
proportional to the number-density of relativistic particles, with a
numerical factor of 7.1.

This regime applies to the early hot universe. Consider eqs. ((3, 4)
again. Through the work of Hubble we know that R increases with time.
At early moments, the density term
*R*^{-4}
dominates the *k*/*R*^{2} and the
cosmological constant. Thus we have
(*R*^{0}/*R*)
*R*^{-2}
or *R*
*t*^{1/2} where *t* is the timescale of the universe.

B) For a matter dominated (cold) universe the pressure of the
non-relativistic particles is negligible compared to their mass
density. *P* = 0.

Using eqs. (3, 4) again, we find
*R*^{-3}
*T*^{3}. This time we have
*R*
*t*^{2/3}.

C) In recent works, a third regime has been shown to be of great
importance in the early universe. At certain times, the energy-density
is dominated by the so-called *vacuum energy terms* associated with
various physical phase transitions. These terms, which do not appear
in the standard classical fluid model described before, are related to
the quantum field description of matter. The undetermined cosmological
constant of Einstein has become a convenient way to introduce these
terms in the formalism. The equation of state of this regime is
(*P* =
-), hence
= *cst*
from eqs. (3, 4) (note the strange result that the
vacuum density does not change during the expansion ...). And we have
*R*
exp(*t*/*t*_{0}); an exponential rate of expansion
called *inflation.*