Friedmann Equations and Hubble Parameter

Friedmann Equations:
Inserting both the stress energy tensor and the FLRW Metric and taking the  fluid to be comoving with the expansion of the universe, we arrive at equations of motion for scale factor a, known as the Friedmann
equation as shown in Olive (1999),

where a -dot is the first derivative of scale function a with respect to cosmological proper time t and is the density. The equation of acceleration a (Double dot)

The third equation is called the conservation equation and can also be derived from the stress energy tensor.

If pressure and density are non-negative which means ( ro + 3p) > 0 and the cosmological constant is negligible then the a is always negative. This implies is that _a(dot) must be either negative or positive and the universe must be either expanding or contracting. Assuming it is expanding the minus sign of the right hand side of second equation above implies that the expansion of the universe is slowing down with

time. However research papers published by Perlmutter et al. (1999) for the supernova Cosmology Project, Brian Schmidt from High-Z Supernova research Team and Riess (2000) cast doubt on this claim. They have received the Nobel prize for proving that the expanding universe accelerating and a small cosmological constant is needed to explain this. To discuss the evolution of the scale factor further, aided by above equations if we go back in time, _a must increase and so the curve a(t) must concave downwards towards the time (t) axis. As a result, there should be some nite time in the past, scale factor must have been equal to zero. The event when scale factor a towards 0 and consequently when the time is defined to be zero is

a singularity at which the density of the universe become in nite. This is often referred to as the big bang.

Hubble Parameter

In 1914 Vesto Slipher found that light from most galaxies is redshifted. By taking this work further Edwin Hubble measured the recession of speed of distant galaxies from their observed redshift and established their distance from earth. Then Hubble derived a law that said, the recession velocity, v, of a galaxy is directly proportional to its distance, d, from the earth. Therefore

where the constant of proportionality, H0 is known as Hubble's constant with the dimensions of [Time]-1 (per). For the present time H0 = 72 +/-  8 kms per second per Mpc (Mega Parsec). However there is an greater uncertainty in the value of H0 and it is conventional to parameterise it's value in terms of the parameter h de ned as follows;

Current observations indicate a range of values for h, 0:6 < h < 0:75. As we can see, then the rate of the expansion of the universe is determined by the Hubble parameter as

What the Hubble parameter has given us is the observational proof of the expansion of the universe. The observations of the universe continue to constrain the expansion rate and the universe is expanding with time as general relativity originally predicted.

Reference:

1. K. A. Olive. Primordial big bang nucleosynthesis. ArXiv Astrophysics e-prints, Jan. 1999. URL http://arxiv.org/abs/astro-ph/9901231.

2. S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro,S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M. Newberg, W. J. Couch, and Supernova Cosmology Project. Measurements of Omega and Lambda from 42 High-Redshift Supernovae. ApJ, 517:565{586, June 1999. doi: 10.1086/307221.

3. A. G. Riess. The Case for an Accelerating Universe from Supernovae. PASP, 112: 1284{1299, Oct. 2000. doi: 10.1086/316624.