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

Equation 1 (1)

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

Equation 2 (2)

In Eq. (1) ds is the line element in four-dimensional spacetime, t is the time, R(t) is the cosmic scale, sigma 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 theta, phi are comoving angular coordinates. In Eq. (2) Gµnu is the Einstein tensor describing the curved geometry of spacetime, Tµnu 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

Equation 3 (3)

Equation 4 (4)

Here rho are energy densities, the subscripts m and lambda refer to matter and cosmological constant (or dark energy), respectively; pm and plambda are the corresponding pressures of matter and dark energy, respectively. Using the expression for the critical density today,

Equation 5 (5)

where H0 is the Hubble parameter at the present time, one can define density parameters for each energy component by

Equation 6 (6)

The total density parameter is

Equation 7 (7)

In what follows we shall ignore the very small radiation density parameter Omegar. The matter density parameter Omegam can further be divided into a cold dark matter (CDM) component OmegaCDM, a baryonic component Omegab and a neutrino component Omeganu.

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 pm = 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

Equation 8 (8)

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

Equation 9 (9)

One further parameter is the deceleration parameter q0, defined by

Equation 10 (10)

Eliminating R ddot between Eqs. (4) and (10) one can see that q0 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 Omega0 - 1, so if Omega0 is observed to be 1, the Universe is spatially flat.

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