Friedmann - Robertson - Walker - Lemaitre Models:
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Start with the Einstein field eqs:


where



is the Einstein tensor (geometry), and            is the 
stress-energy tensor (mater).

There are 10 independent Einstein eqs. in a general 
case (tensor symmetry).

Choose the metric tensor            such that the metric 
in comoving coordinates is:



This is the Robertson-Walker metric, and it is the most 
general metric for a homogeneous, isotropic universe.  
The isotropy gets rid of the off-diagonal elements of
            and            (only four  eqs. left), and the 
homogeneity further reduces the number of independent eqs. 
to two (energy and pressure).

These two are the Friedmann (without the       ), or the
Friedmann-Lemaitre (with the       ) equations.