Einstein Equations and FLRW Metric

Any modern treatment of cosmology, the study of the structure and evolution of the universe on the largest scales, begins with a number of assumptions. The first is that physical laws, which have been tested in our local environment, are universal, hold- ing at all times and in all places in the universe. Importantly this includes the theory of General Relativity, which is our best theory of gravity at the present time. The second major assumption is that we will need to assume some symmetries for the universe. These symmetries are motivated both by simplicity and by observations of the cosmos. The assumed symmetries are that the universe is is homogeneous and isotropic on the largest scales, which states that we do not occupy a special place in the universe. In Einstein gravity, spacetime is described by four-dimensional metric, gµν (χk ), that satisfies the Einstein equations.

where Rµν the Ricci tensor, R is the Ricci scalar, Tµν is the stress-energy tensor for all the fields present and Λ is the cosmological constant. FLRW Metric The current understanding we have of the evolution of the universe is based upon the Friedmannn-Leimatre-Robertson-Walker metric which has the symmetries of homogeneity and isotropy, motivated by observations. It’s validity supported by direct and indirect observational evidence extends back to the beginning of the epoch of primordial nucleosynthesis. The high degree of symmetry of the FLRW metric is considered as the cornerstone of the standard cosmological model. It depends on only one dynamical variable cosmic scale factor a(t) (in some literature this is symbolised as R(t)). The FLRW metric is given explicitly as

where spherical polar coordinates r, θ, and φ are comoving coordinates, t is the proper time, a(t) is the scale factor, k is the curvature parameter which can take values of +1, 0 and -1 for spaces of constant positive curvature, flat or negative spatial curvature, respectively. The scale factor a(t) has dimensions of length. If k = +1 then r ranges from 0 to 1. In below figure the two dimensional analogue of the three dimensional curvature manifolds of homogeneous and isotropic universe are illustrated.

An observer in a homogeneous and isotropic universe, moving so the universe is observed to be isotropic, would measure the stress-energy tensor to be

as stated in Peebles and Ratra (2003) where the diagonal form is a consequence of the symmetry and the diagonal components define the pressure and energy density. We assume that the fuid is comoving with the expansion of the universe. According to Kolb and Turner (1994), from the energy conservation
law for adiabatic expansion, the change in energy in a comoving volume element v = a3 is equal to minus the pressure times the change in volume i.e,

where p is the pressure and is the density. The parameter w is a constant and lies in the range of 0 <= w <=1, which is called the Zel dovich interval. This equation of state is the most general form for

the stress energy in a FLRW space-time.

 Reference:
1. P. J. Peebles and B. Ratra. The cosmological constant and dark energy. Reviews of Modern Physics, 75:559{606, Apr. 2003. doi: 10.1103/RevModPhys.75.559.
2. P. Coles and F. Lucchin. Cosmology: The Origin and Evolution of Cosmic Structure. John Wiley & Sons, 2002. ISBN 9780471489092. URL http://books.google. co.uk/books?id=uUFVb-DHtCwC.
3. E. Kolb and M. Turner. The Early Universe. Frontiers in Physics. Westview Press, 1994. ISBN 9780201626742. URL http://books.google.co.uk/books id=Qwijr-HsvMMC.