My question refers to the derivation of the energy density $p_{\nu,i}$ of a neutrino species given in the footnote on page 16 of Lecture Notes on Cosmology by Komatsu. I have reproduced this below:

$$p_{\nu, i} = 2\int \frac{d^3 p}{(2\pi)^3} \frac{E_i(p)}{e^{p/T_{\nu}}+1} = \int \frac{p^2 dp }{\pi^2} \frac{\sqrt{p^2 + m_{\nu,i}^2}}{e^{p/T_{\nu}}+1}$$

Can somebody please explain how we have gone from having $(2\pi)^3$ in the denominator here to having $\pi^2$? Likewise, how have we gone from $d^3p$ to $dp$. An explicit mathematical answer is needed to help me understand this step.


1 Answer 1


It appears that it's simply using spherical coordinates for the integration measure:

$$d^3p = 4\pi p^2 dp$$

Together with the overall numerical factors (including the factor of $2$ outside of the integral), that gives

$$ 2\frac{d^3p}{(2\pi)^3} = \frac{p^2dp}{\pi^2}$$


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