The currently accepted paradigm describing our homogeneous and isotropic Universe is based on the Robertson-Walker metric

(1) |

and Einstein's covariant formula for the law of gravitation,

(2) |

In Eq. (1) d*s* is the line element in four-dimensional
spacetime, *t* is the time, *R*(*t*) is the cosmic scale,
is the comoving distance as measured by an observer who follows
the expansion, *k* is the curvature parameter, *c* is the velocity
of light, and ,
are comoving
angular coordinates. In Eq. (2)
*G*_{µ} is
the Einstein tensor describing the curved geometry of spacetime,
*T*_{µ} is
the energy-momentum tensor, and *G* is Newton's constant.

From these equations one derives Friedmann's equations which can be put into the form

(3) |

(4) |

Here are
energy densities, the subscripts *m* and
refer to matter and
cosmological constant (or dark energy), respectively;
*p*_{m} and
*p*_{} are the
corresponding pressures of matter and dark energy, respectively.
Using the expression for the critical density today,

(5) |

where *H*_{0} is the Hubble parameter at the present time,
one can define density parameters for each energy component by

(6) |

The total density parameter is

(7) |

In what follows we shall ignore the very small radiation density parameter
_{r}. The
matter density parameter
_{m} can
further be divided into a cold dark matter (CDM) component
_{CDM}, a
baryonic component
_{b} and a
neutrino component
_{}.

The pressure of matter is certainly very small, otherwise one
would observe the galaxies having random motion similar to that of
molecules in a gas under pressure. Thus one can set *p*_{m}
= 0 in Eq. (4) to a good approximation. If the expansion is adiabatic so
that the pressure of dark energy can be written in the form

(8) |

and if dark energy and matter do not transform into one another, conservation of dark energy can be written

(9) |

One further parameter is the deceleration parameter
*q*_{0}, defined by

(10) |

Eliminating between
Eqs. (4) and (10) one can
see that *q*_{0} is not an independent parameter.

The curvature parameter *k* in Eqs. (1), (3) and (4) describes the
geometry of space: a spatially open universe is defined by *k* = -1,
a closed universe by *k* = + 1 and a flat universe by *k* = 0. The
curvature parameter is not an observable, but it is proportional
to _{0} - 1, so
if _{0} is
observed to be 1, the Universe is spatially flat.