Suppose (incorrectly) that H scales as temperature squared all the way back until the time when the temperature of the universe was 1019 GeV/kB (i.e., suppose the universe was radiation dominated all the way back to the Planck time).
Also suppose that today the dark energy is in the form of a cosmological constant Λ, such that ρΛ today is equal to 0.7*ρcritical and ρΛ remains constant throughout the history of the universe. What was ρΛ / (3H2/8πG) back then?
(From Modern Cosmology by Dodelson, pg. 25)
ρ_critical = (3H02/8πG)
T = 1019 GeV/kB = 1.16045* 1032 K
T0 = 2.725 K
For a radiation-dominated universe, a ∝ t1/2.
The Attempt at a Solution
I understand a part of the solution wherein ρΛ / ρcritical = 0.7, but I'm supposed to multiply this value by something.
In the answer key, Dodelson multiplies 0.7 by the ratio of (H0 / H)2. The text states:
"By assumption, the universe is forever radiation dominated (clearly not true today, but a good approximation early on), so H / H0 = a-2."
Given this, the inverse of H / H0 would result in a2, and since H scales as temperature squared, then (a2)2 = a4 which can then be applied to the ratio of the temperature. That latter part makes sense. However, I'm not quite understanding where Dodelson pulled the ratio of H0 / H from to get things started.
Could anyone provide any insight on this? Thank you very much for your help.
(This question is being attempted via an independent study and not a homework question. Additionally, there are no cosmology specialists at my university who could provide any useful feedback on how to attack this situation.)