The scale factor R at time t for the size of the universe is related to the scale factor 
at time 
by
where a is called the expansion parameter. Then the speed of expansion is
Rearranging gives
which is just the Hubble law with Hubble constant[u/].
The ratio
is related to the [u]redshift z by th equation.
A series expansion of a(t) gives
where q is known as the deceleration parameter. The nonrelativistic time evolution follows the law
[center]
so
[center]
The relativistic time evolution follows
[center]
so
[center]
The dynamical equation for the scale factor R is one of the cosmological equations. If the cosmological constant satisfies
, then
[center]
where G is the gravitational constant,
is the mass density, P is the pressure, and c is the speed of light. Integrating gives
[center]
[center]
But for critical density
[center]
i.e.,
[center]
so
[center]
and
[center]
Rewriting is terms of the parameter
gives
[center]
so
[center]
Solving for q gives
[center]
Now, if P = 0, then
[center]
at time by(1)
where a is called the expansion parameter. Then the speed of expansion is
(2)
Rearranging gives
(3)
which is just the Hubble law with Hubble constant[u/].
(4)
The ratio
is related to the [u]redshift z by th equation.(5)
A series expansion of a(t) gives
(6)
where q is known as the deceleration parameter. The nonrelativistic time evolution follows the law
[center]
(7)
so
[center]
(8)
The relativistic time evolution follows
[center]
(9)
so
[center]
(10)
The dynamical equation for the scale factor R is one of the cosmological equations. If the cosmological constant satisfies
, then[center]
(11)
where G is the gravitational constant,
is the mass density, P is the pressure, and c is the speed of light. Integrating gives [center]
(12)
[center]
(13)
But for critical density
[center]
(14)
i.e.,
[center]
(15)
so
[center]
(16)
and
[center]
(17)
Rewriting is terms of the parameter
gives[center]
(18)
so
[center]
(19)
Solving for q gives
[center]
(20)
Now, if P = 0, then
[center]
(21)
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