- #1
bananabandana
- 113
- 5
Homework Statement
Show that the time of matter-radiation equality, t_{eq} can be written:
$$ t_{eq} =\frac{a_{eq}^{\frac{3}{2}}}{H_{0}\sqrt{\Omega_{m}}} \int_{0}^{1} \frac{x}{\sqrt{x+1}} dx $$
Homework Equations
$$ t = \int_{0}^{t} dt = \int_{0}^{a} \frac{1}{H(a)} \frac{da}{a} $$ [Given]
$$ H^{2}(a) \approx H^{2}_{0} \bigg( \frac{\Omega_{m}}{a^{3}} + \frac{\Omega_{r}}{a^{4}}\bigg)$$
The Attempt at a Solution
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I won't write it out here - it's just a lot of algebra - but substitute the definition of ##H(a)## into the equation for the time and rearrange is clearly what you need to do.
I got stuck however, because apparently you are meant to say: [this is in the solution set provided by my lecturer]
$$ \bigg( \frac{\Omega_{m}}{\Omega_{r}} \bigg) = \frac{1}{1+z_{eq}} = a_{eq}$$
This makes no sense to me at all! At matter-radiation equality, we could expect, by definition:
$$ \frac{\Omega_{m}}{\Omega_{r}}=1 \implies z_{eq} = 0$$
i.e matter-radiation equilibrium is occurring right now, which is obviously nonsense. [and would conflict completely with the result we are trying to show]
Have I misunderstood something?
Thanks!