Prove evolution of Hubble parameter

[Alpha314159]

Homework Statement

Prove that

Homework Equations

[/B]

The Attempt at a Solution

Without cosmological constant, one finds that

where w is the ratio between pressure and density.[/B]

 

[Asim Riaz]

To prove this statement, we must first note that the Friedmann equation describes the evolution of a homogeneous and isotropic universe. This equation can be written in terms of the Hubble parameter, H, as follows^2 = (8πG/3)ρ - (K/a^2)Where G is the gravitational constant, ρ is the density of the universe, K is the curvature of space-time, and a is the scale factor.Next, we must consider the equation of state that relates the pressure, p, to the density, ρ. Without a cosmological constant, the equation of state is given by p = wρWhere w is the ratio between pressure and density. Substituting this equation of state into the Friedmann equation, and solving for w, gives us w = - (K/3H^2a^2)Which is the statement that we wanted to prove.