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## Homework Statement

Suppose (incorrectly) that H scales as temperature squared all the way back until the time when the temperature of the universe was 10

^{19}GeV/k

_{B}(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 ρ

_{Λ}today is equal to 0.7*ρ

_{critical}and ρ

_{Λ}remains constant throughout the history of the universe. What was ρ

_{Λ}/ (3H

^{2}/8πG) back then?

(From

*Modern Cosmology*by Dodelson, pg. 25)

## Homework Equations

ρ_critical = (3H

_{0}

^{2}/8πG)

T = 10

^{19}GeV/k

_{B}= 1.16045* 10

^{32}K

T

_{0}= 2.725 K

For a radiation-dominated universe, a ∝ t

^{1/2}.

## The Attempt at a Solution

I understand a part of the solution wherein ρ

_{Λ}/ ρ

_{critical}= 0.7, but I'm supposed to multiply this value by something.

In the answer key, Dodelson multiplies 0.7 by the ratio of (H

_{0}/ 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 / H*

_{0}= a^{-2}."Given this, the inverse of H / H

_{0}would result in a

^{2}, and since H scales as temperature squared, then (a

^{2})

^{2}= a

^{4}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 H

_{0}/ 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.)