1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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?

  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.

    I don't think the 3rd equation is correct? is it?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted