Let the pressure term go to P=0 in the second of the cosmological equations
where R is the size scale of the universe,
is the curvature of the universe, c is the speed of light, 
is the cosmological constant,and G is the gravitational constant. Integrating gives
where C is a constant of integration. Now note that first of the cosmological equations
where
is the density of the universe gives the constant C as
Now, plugging (4) into (2) gives
For a flat universe,the curvature
,so
If
, making the substitution
gives
If
making the substitution
gives
If
(i.e, the Einstein-de Sitter cosmological model), then
Summarizing, for
For a universe with and 
For a universe with and 
(1)
where R is the size scale of the universe,
is the curvature of the universe, c is the speed of light, is the cosmological constant,and G is the gravitational constant. Integrating gives (2)
where C is a constant of integration. Now note that first of the cosmological equations
(3)
where
is the density of the universe gives the constant C as(4)
Now, plugging (4) into (2) gives
(5)
For a flat universe,the curvature
,so(6)
If
, making the substitution (7)
(8)
gives
(9)
If
making the substitution (10)
(11)
gives
(12)
If
(i.e, the Einstein-de Sitter cosmological model), then(13)
Summarizing, for
(14)
For a universe with
and (15)
For a universe with
and (16)
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