From Einstein's equation to Friedman equation
The material in this lecture can be found at any introductory book on Cosmology or the Early Universe. For concise versions see
- Rubakov and Gobrunov: sec. 2.2,3.1,3.2,4.1
- Weinberg: sec. 1.5
while a very accessible introduction, without many details, but with the basic concepts elucidated can be found in
Isotropy and Homogeneity dictates that the metric of the Universe should be of the Robertson-Walker (RW) type. There are three possible types of geometry for space, the flat, the spherical and the hyperbolic geometries. Which one is realized depends on the value of the kappa parameter, which is itself NOT evolving in time. Apart from that, the only unknown function in the metric is the a(t) which determines distances. As a(t)grows with t, the Universe expands, in the sense that distances from any point to any other point grow, following a(t). Our goal is to relate a(t) with the matter-energy content of the Universe.
Starting from Einstein's equations of Gerneral Relativity, and using the RW metric we can compute the left hand side of the equations, which determine the geometry of the Universe, evaluating explicitly the Ricci tensor and the curvature scalar.
The right hand side of Einstein's equations is the energy-momentum density tensor, which gets contributions from everything that exists in the Universe. We have introduced it as the Noether current corresponding to space-time translations.
Approximating the Universe as a perfect (isotropic) fluid means that we are examining it at a spatial scale at which the average distance between particles is negligible, and at a time scale in which the mean time between collisions of particles is also negligible. Isotropy and homogeneity then dictates that the energy-momentum tensor is specified by two functions of spacetime, the energy density (ρ(x)) and the pressure (P(x)).
The current conservation equation for the Energy-Momentum tensor, seen as a Noether current, upon using the RW metric, results in a relation between the scaling factor, the energy density and the pressure. This is the so-called continuity equation, and it should be seen as a consistency condition, for the Universe to be invariant under space-time translations.
We need one more equation to relate pressure with density. This depends on the contents of the Universe. There are three different types of contributions:
- Relativistic particles/fields (which we call Radiation, even if it includes relativistically moving fermions): then
P(x)=0
Once we plug this into the continuity equation we see that ρ~ 1/a^3.
- Non-relativistic particles (which we call Matter or Dust): then
ρ(x) = 3 P(x)
which upon using the continuity equation results to ρ ~ 1/a^4
- Dark energy contributions, which we assume to be constant in time, described by a cosmological constant: then
ρ(x) = -P(x) = Λ
resulting to ρ = constant.
We can then insert the different energy-density contributions, as functions of a(t) in the RHS of the 0-0 component of Einstein equation for the RW metric, to arrive at Friedman's equation.
Since the Universe is expanding now, and according to all data we have today, we can assume that the Universe was ever smaller for smaller values of time. The Big Bang, if it happened, corresponds to a boundary value a(0)=0. At that moment all distances are shrunk to zero, the energy density diverges and we have a singularity. We cannot be sure that this was realized, however: before reaching the singularity, the Universe passes from a state where the energy density is so high that Gravity becomes equally strong to the other fundamental interactions (the strong and electroweak force and whatever other yet undiscovered interactions might be there). At that stage we cannot rely on General Relativity to describe the system: we need a quantum theory of gravity.
Nevertheless, this happens at the first 10^-44 of a second. In later times, the scale factor is assumed to approach zero, as time goes to zero. Looking at the way the contributions of radiation, matter and cosmological constant depend on a(t), it is evident that in the first stages of the Universe, radiation dominates. This period lasts for ~47000 years (the exact number depends on one's definition of "domination"). Then there is a transition to the phase where matter dominates, which lasts another 10 Billion years. We now live within the third stage, where the Universe is dominated by the Cosmological constant.
