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Estimate of time of the universe

  1. Apr 12, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that the time of matter-radiation equality, t_{eq} can be written:
    $$ t_{eq} =\frac{a_{eq}^{\frac{3}{2}}}{H_{0}\sqrt{\Omega_{m}}} \int_{0}^{1} \frac{x}{\sqrt{x+1}} dx $$

    2. Relevant equations
    $$ t = \int_{0}^{t} dt = \int_{0}^{a} \frac{1}{H(a)} \frac{da}{a} $$ [Given]
    $$ H^{2}(a) \approx H^{2}_{0} \bigg( \frac{\Omega_{m}}{a^{3}} + \frac{\Omega_{r}}{a^{4}}\bigg)$$

    3. The attempt at a solution

    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!
     
  2. jcsd
  3. Apr 12, 2017 #2
    Problem solved!

    The point is that at matter radiation equality, we must have that:

    $$ \frac{\rho_{M}}{\rho_{R}} = 1 $$

    This does NOT mean that:

    $$ \frac{\Omega_{M}}{\Omega_{R}} = 1 $$ Since :

    $$ \frac{\rho_{M}}{\rho_{R}} = \frac{\frac{\Omega_{M}}{a^{3}}}{\frac{\Omega_{R}}{a^{4}}} = \frac{1}{a}\frac{\Omega_{M}}{\Omega_{R}} $$

    So at radiation matter equality, we have:

    $$ \frac{1}{a_{eq}}\frac{\Omega_{M}}{\Omega_{R}} = 1 \implies \frac{\Omega_{M}}{\Omega_{R}} = \frac{1}{1+z_{eq}} $$ as req'd.
     
  4. Apr 13, 2017 #3
    Opps, I posted an incorrect statement and don't know how to delete this post... sorry.

    Correction:
    I don't think the 3rd equation is correct? is it?
     
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