When reading Sterile neutrino hot, warm, and cold dark matter I came across the following momentum distribution function for a neutrino species $\alpha$:

$$\tag{5.8} f(p,t) = \frac{1}{e^{E(p)/T + \eta_{\nu_\alpha}}+1}$$

where $\eta_{\nu_\alpha}= \mu_{\nu_\alpha }/T$ and $E(p) \approx p$.

Why is the function $f$ a function of variables $p$ (momentum) and $t$ (time) when the RHS shows that the function only depends on $p$ and on $T$ (temperature)?

On another article Dodelson-Widrow production of sterile neutrino Dark Matter with non-trivial initial abundance it is stated that $f(p,t)$ and $f(p,T)$ are equivalent.

But how can this be so? How can temperature and time be equivalent?

  • $\begingroup$ I'd wager that it's just a typo, and that they meant to put $T$ instead of $t$. Is the equation written the same way in the version published in PRD? $\endgroup$ Feb 18, 2021 at 13:47

1 Answer 1


I am too low on reputation to comment, so I post it as an answer.

This is not my field of expertise (at all!) but I came across an idea that seems similar.

Also here:

Wick rotation connects statistical mechanics to quantum mechanics by replacing inverse temperature $1 / ( k_B T )$ with imaginary time $it/\hbar$."

Maybe this can lead someone to an answer.


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