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I was wondering if anybody knows the relation between the photon temperature $T$ and neutrino temperature $T_{\nu}$? And why this can be written as

$$T_{\nu}=\left(\frac{4}{11}\right)^{1 / 3} T \mathcal{S}^{1 / 3}\left(x=\frac{m_{e} c^{2}}{k_{B} T}\right)$$

$$\mathcal{S}(x)=1+\frac{45}{2 \pi^{4}} \int_{0}^{\infty} y^{2}\left(\sqrt{y^{2}+x^{2}}+\frac{y^{2}}{3 \sqrt{y^{2}+x^{2}}}\right) \frac{1}{\exp \left(\sqrt{y^{2}+x^{2}}\right)+1} d y$$

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  • $\begingroup$ Hello! I have edited your question using MathJax (LaTeX) math typesetting. For future questions, you can refer to MathJax basic tutorial and quick reference. Thanks! $\endgroup$
    – jng224
    Commented Apr 20, 2021 at 10:51
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    $\begingroup$ have you read this ? en.wikipedia.org/wiki/… $\endgroup$
    – anna v
    Commented Apr 20, 2021 at 11:17
  • $\begingroup$ Thank you a lot, I will look at it now $\endgroup$
    – Ivan
    Commented Apr 20, 2021 at 11:57
  • $\begingroup$ I understand a little more now, but I do not understand where $S^\frac{1}{3}$ comes from. What exactly is $S$? $\endgroup$
    – Ivan
    Commented Apr 20, 2021 at 12:08

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The ratio of temperatures is discussed in https://en.wikipedia.org/wiki/Cosmic_neutrino_background . The section "Indirect evidence for the CνB" also describes the math for the ratio. Hope this helps you.

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