Cosmology Fine Tuning Problem

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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 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)

Homework Equations


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.)
 

Answers and Replies

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George Jones
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Homework Statement


Suppose (incorrectly) that H scales as temperature squared ...
Multiply the expression in the exercise by one in the form ##1 = \rho_{cr} / \rho_{cr}##, then use equation (1.3) to substitute for ##\rho_{cr}## in the numerator (but not in the denominator).
 
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George, your input was incredibly helpful and brought the entire picture together. Thank you so much for your help! I appreciate that.
 

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