# Massive Fermi gas in the expanding universe

## Abstract

The behavior of a decoupled ideal Fermi gas in a homogeneously expanding three-dimensional volume is investigated, starting from an equilibrium spectrum. In case the gas is massless and/or completely degenerate, the spectrum of the gas can be described by an effective temperature and/or an effective chemical potential, both of which scale down with the volume expansion. In contrast, the spectrum of a decoupled massive and non-degenerate gas can only be described by an effective temperature if there are strong enough self-interactions such as to maintain an equilibrium distribution. Assuming perpetual equilibration, we study a decoupled gas which is relativistic at decoupling and then is red-shifted until it becomes non-relativistic. We find expressions for the effective temperature and effective chemical potential which allow us to calculate the final spectrum for arbitrary initial conditions. This calculation is enabled by a new expansion of the Fermi-Dirac integral, which is for our purpose superior to the well-known Sommerfeld expansion. We also compute the behavior of the phase space density under expansion and compare it to the case of real temperature and real chemical potential. Using our results for the degenerate case, we also obtain the mean relic velocity of the recently proposed non-thermal cosmicmore »

- Authors:

- Bethe Center for Theoretical Physics and Physikalisches Institut der Universität Bonn, Nussallee 12, 53115 Bonn (Germany)

- Publication Date:

- OSTI Identifier:
- 22679983

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Cosmology and Astroparticle Physics; Journal Volume: 2017; Journal Issue: 03; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; COMPARATIVE EVALUATIONS; COSMIC NEUTRINOS; DECOUPLING; DENSITY; DISTRIBUTION; EQUILIBRIUM; EXPANSION; FERMI GAS; INTERACTIONS; PHASE SPACE; RED SHIFT; RELATIVISTIC RANGE; SPECTRA; THREE-DIMENSIONAL CALCULATIONS; UNIVERSE; VELOCITY

### Citation Formats

```
Trautner, Andreas, E-mail: atrautner@uni-bonn.de.
```*Massive Fermi gas in the expanding universe*. United States: N. p., 2017.
Web. doi:10.1088/1475-7516/2017/03/019.

```
Trautner, Andreas, E-mail: atrautner@uni-bonn.de.
```*Massive Fermi gas in the expanding universe*. United States. doi:10.1088/1475-7516/2017/03/019.

```
Trautner, Andreas, E-mail: atrautner@uni-bonn.de. Wed .
"Massive Fermi gas in the expanding universe". United States.
doi:10.1088/1475-7516/2017/03/019.
```

```
@article{osti_22679983,
```

title = {Massive Fermi gas in the expanding universe},

author = {Trautner, Andreas, E-mail: atrautner@uni-bonn.de},

abstractNote = {The behavior of a decoupled ideal Fermi gas in a homogeneously expanding three-dimensional volume is investigated, starting from an equilibrium spectrum. In case the gas is massless and/or completely degenerate, the spectrum of the gas can be described by an effective temperature and/or an effective chemical potential, both of which scale down with the volume expansion. In contrast, the spectrum of a decoupled massive and non-degenerate gas can only be described by an effective temperature if there are strong enough self-interactions such as to maintain an equilibrium distribution. Assuming perpetual equilibration, we study a decoupled gas which is relativistic at decoupling and then is red-shifted until it becomes non-relativistic. We find expressions for the effective temperature and effective chemical potential which allow us to calculate the final spectrum for arbitrary initial conditions. This calculation is enabled by a new expansion of the Fermi-Dirac integral, which is for our purpose superior to the well-known Sommerfeld expansion. We also compute the behavior of the phase space density under expansion and compare it to the case of real temperature and real chemical potential. Using our results for the degenerate case, we also obtain the mean relic velocity of the recently proposed non-thermal cosmic neutrino background.},

doi = {10.1088/1475-7516/2017/03/019},

journal = {Journal of Cosmology and Astroparticle Physics},

number = 03,

volume = 2017,

place = {United States},

year = {Wed Mar 01 00:00:00 EST 2017},

month = {Wed Mar 01 00:00:00 EST 2017}

}