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BOAS
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Homework Statement
According to the standard assumptions, there are three species of (massless) neutrinos. In the temperature range of 1MeV < T < 100MeV, the density of the universe is believed to have been dominated by the black-body radiation of photons, electron-positron pairs, and three neutrinos all of which were in thermal equilibrium.
1. Neglecting any change in the degrees of freedom at T > 100MeV, show using the Friedmann equation for a flat radiation-dominated universe ##H^2 = \frac{8\pi G}{3} \rho_R## that for temperatures T > 1MeV the time since the start of the hot big bang is given by ##t(T) = (\frac{A}{g_∗})^{1/2} \frac{M_P}{T^2}## where ##M_P## is the reduced Planck mass, and A is a constant that you should give explicitly.
Homework Equations
The Attempt at a Solution
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I have found an expression for ##t(T)## but have a factor of ##ln(a)## spoiling my constant ##A## and I don't know how I can get rid of it.
Starting with the Friedmann equation:
##H^2 = \frac{8\pi G}{3} \rho_R##
I know that the energy density for several relativistic species in thermal equilibrium is given by ##\rho_R = \sum_i \rho_i = \frac{\pi^2}{30} g_* T^4_{\gamma}##, where ##g_*## is the effective degrees of freedom.
Substituting this into the Friedmann equation yields
##H^2 = \frac{8\pi G}{3} \frac{\pi^2}{30} g_* T^4_{\gamma}##
I take the square root and use the definition of the Hubble parameter
##\frac{da}{dt} \frac{1}{a} = \sqrt{\frac{8\pi G}{3} \frac{g_*}{30}} \pi T_{\gamma}^2##
Simply integrating this causes the following problem
##t = \frac{ln(a)}{\pi T^2_{\gamma}} \sqrt{\frac{3}{8 \pi G} \frac{30}{g_*}}##
which we can write as
##t = ln(a) \sqrt{\frac{90}{\hbar c g_* \pi^2}} \frac{M_P}{T^2_{\gamma}}##
However, I can't write that in the form I'm asked to, since ln(a) is not a constant.
I think I have made an error by integrating the equation in the way that I did, but I don't know how to go about this.
Thank you for any help you can give!