In a radiation dominated era, how do I show this?

[Phil1093]

As above. In a radiation dominated how do I show this equation?
http://i.imgur.com/nMwxvD5.png (It's easier for me to post the image to imgur than use latex.

I know g is the relativistic degrees of freedom.
I think that the scale factor a(t) is proportional to 1/t
and that a(T) is proportional to 1/T

I think it might have something to do with,

s= s/v = (E+P)/KbT
and s=(1+w) Pc^2/KbTbut other than that, I don't know where to start :/ Any help would be appreciated :)

 

[Asim Riaz]

The equation you provided is a general equation of state for a radiation dominated universe and therefore is not dependent on the scale factor a(t).To show this, we need to look at the energy density ρ and pressure p of radiation in a Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. In a FLRW universe, the energy density of radiation is given by:ρ = 3gπ2T^4/15c^5where g is the number of relativistic degrees of freedom and T is the temperature. Similarly, the pressure of radiation is given by:p = gπ2T^4/45c^5Combining these two equations gives us:s = (ρ + p)/kBT = (3gπ2T^4/15c^5 + gπ2T^4/45c^5)/kBT = (4gπ2T^4/15c^5)/kBT = (1 + w)Pc^2/kBTwhere w = 1/3 is the equation of state parameter for radiation.Therefore, we can see that the equation of state for a radiation dominated universe is indeed given by s = (1 + w)Pc^2/kBT.