A cosmological model formulated by Friedmann in 1922 and independently by Lemaître (1927). It assumes a homogeneous and isotropic universe with positive cosmological constant 
and expansion parameter a governed by the equations of motion
and
where
is the total mass-energy density,
is the mass density, and c is the speed of light. These equations were originally rigorously derived from general relativity, but can also be rigorously derived from Newtonian mechanics and thermodynamics for small length scales, i.e., r such that
where M is the mass
The unintegrated form of (1) is
so
Adding a term taking the cosmological constant into account,
The integrated form of (1) is
Equation (2) can be derived by assuming the expansion of the universe is adiabatic,
where
is the heat change (an inexact differential), dEis the energy change, P is the pressure, and V is the volume. Then
and
so
But, since
The time evolution for a large cosmological constant
is
For a matter dominated universe at initial time with expansion parameter a,
The redshift at a time
with expansion parameter 
is defined as
Plugging in the curvature
yields
Density goes as
so plugging (33) into (32)
For a matter dominated universe,
where q is the deceleration parameter, so (36) simplifies to
The static solution is
If P = 0, then
where H is the Hubble constant. Also
where q is the deceleration parameter.
and expansion parameter a governed by the equations of motion (1)
and
(2)
where
is the total mass-energy density, is the mass density, and c is the speed of light. These equations were originally rigorously derived from general relativity, but can also be rigorously derived from Newtonian mechanics and thermodynamics for small length scales, i.e., r such that (3)
where M is the mass
The unintegrated form of (1) is
(4)
(5)
so
(6)
Adding a term taking the cosmological constant into account,
(7)
The integrated form of (1) is
(8)
(9)
(10)
(11)
(12)
(13)
Equation (2) can be derived by assuming the expansion of the universe is adiabatic,
(14)
where
is the heat change (an inexact differential), dEis the energy change, P is the pressure, and V is the volume. Then(15)
(16)
and
(17)
so
(18)
(19)
(20)
But, since
(21)
(22)
The time evolution for a large cosmological constant
(23)
is
(24)
(25)
(26)
(27)
(28)
For a matter dominated universe at initial time with expansion parameter a,
(29)
The redshift at a time
with expansion parameter is defined as (30)
Plugging in the curvature
(31)
yields
(32)
Density goes as
(33)
so plugging (33) into (32)
(34)
(35)
(36)
For a matter dominated universe,
where q is the deceleration parameter, so (36) simplifies to(37)
The static solution is
(38)
(39)
(40)
(41)
If P = 0, then
(42)
where H is the Hubble constant. Also
(43)
where q is the deceleration parameter.
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