the_nu_wgv

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 the_nu_wgv [2014/06/09 16:53]gastis created the_nu_wgv [2014/06/18 19:08]willcox 2014/07/01 14:24 vassh 2014/07/01 14:24 vassh 2014/06/18 19:08 willcox 2014/06/18 19:08 willcox 2014/06/18 19:07 willcox 2014/06/18 15:46 cyburt 2014/06/09 16:56 gastis created2014/06/09 16:53 gastis removed2014/06/09 16:53 gastis created Next revision Previous revision 2014/07/01 14:24 vassh 2014/07/01 14:24 vassh 2014/06/18 19:08 willcox 2014/06/18 19:08 willcox 2014/06/18 19:07 willcox 2014/06/18 15:46 cyburt 2014/06/09 16:56 gastis created2014/06/09 16:53 gastis removed2014/06/09 16:53 gastis created Next revision Both sides next revision Line 1: Line 1: - * [[Don Willcox]] + ​Members:  ​Don Willcox, Nicole Vassh, Panos Gastis - * [[Nicole Vassh]] + - * [[Panos Gastis]] + + ===== Decoupling Temperature Calculation ===== + + + We would like to find the decoupling temperature,​ such that: + $$\Gamma= G_F^2T^5 + \mu_\nu^2T^3 \sim H = \frac{T^2}{M_{pl}}$$ + + Let's replace with some more illustrative constants:​ + $$\Gamma = \frac{T^5}{M_W^4} + g_\nu^2\frac{T^3}{m_p^2} \sim H = \frac{T^2}{M_{pl}}$$ + + For $g_\nu\equiv 0$, we have: + $$T_0^3 = \frac{M_W^4}{M_{pl}}$$ + + We can use this to rewrite the cubic equation, in dimensionless form: + $$\left(\frac{T}{T_0}\right)^3 = 1 - g_\nu^2 \frac{M_{pl}^{2/​3}M_W^{4/​3}}{m_p^2}\frac{T}{T_0}$$ + + We can see that the last term is of order unity when $g_\nu\sim \frac{m_p}{M_{pl}^{1/​3}M_W^{2/​3}}\sim 10^{-8}$, allowing us to rewrite the equation as: + $$\left(\frac{T}{T_0}\right)^3 = 1 - \left(\frac{g_\nu}{10^{-8}}\right)^2 \frac{T}{T_0}$$ + + + ===== Link to Git Repository: ===== + + * [[https://​github.com/​dwillcox/​bbn-numu]] 