cosmology - What does the cosmic neutrino background look like today, given that neutrinos possess mass?

This question is inspired by (or a follow-up to) the threads Where are all the slow neutrinos? and Is it possible that all “spontaneous nuclear decay” is actually “slow neutrino” induced?

The cosmic neutrino background (CνB) consists of the "primordial" neutrinos from the time when the universe had cooled/expanded enough for the neutrinos to "decouple" and become free. This of course is similar to the cosmic microwave background except that photons decoupled at a much later time (age $1.2\cdot 10^{13}$ s) than neutrinos (age $1$ s).

My question is, what would we expect the CνB to look like today, given that we know that neutrinos possess mass. According to the linked threads, the neutrinos would have become non-relativistic today because of the great expansion of space since the time when the universe was only one second old.

So will the relic neutrinos from CνB be captured gravitationally by galaxies and stars and similar objects, in a way that today these neutrinos are orbiting galaxy centers, stars etc.? I imagine this is like how most cold/slow hadronic matter (like dust or isolated hydrogen atoms or whatever) will be gravitationally bound to galaxies, stars etc.

Because if that is the case, this neutrino "background" will no longer be have a "background" appearance.

So what to expect: Is the neutrino background still quite isotropic and homogenous and "background-like", maybe even with a temperature associated to it? Or are those particles now captured by all sorts of objects, and no longer constituting a "background" of cold/slow particles?

If one assumes a particular rest mass for each neutrino, say $0.1$ eV, maybe it is even possible to answer this quantitatively?

Answer

As far as theory goes, the Cosmic Neutrino Background (CvB) was created within the first second after the Big Bang, when neutrinos decoupled from other matter. Nevertheless, while the universe was still hot neutrinos stayed in thermal equilibrium with photons. Neutrinos and photons shared a common temperature until the universe cooled down to a point where electrons and positrons annihilated and transfered their temperature to the photons. With the continuing expansion of the universe both the photon background and the neutrino background continued to cool down.

From these assumptions one can derive the properties of the Cosmic Neutrino Background today. The calculations are neither particularly lengthy or difficult, but I'll skip them here. As a result of these calculations one expects the CvB to have a temperature of

$$ T_\nu = 1.95~\mathrm{K} = 1.7\cdot 10^{-4}~\mathrm{eV},$$

an average momentum of

$$ \left< p \right> = 5.314 \cdot 10^{-4}~\mathrm{eV},$$

a root mean squared momentum dispersion of

$$ \sqrt{\left< p^2 \right>} = 6.044 \cdot 10^{-3}~\mathrm{eV}$$

and a density of

$$ 112~\nu/\mathrm{cm}^3 $$

for each of the three neutrino flavors. This density is many orders of magnitude more abundant in that energy range than neutrinos from any other sources. This number is equally divided into neutrinos and antineutrinos.

These are rather hard predictions of Big Bang cosmology. This makes the CvB so important: if we could measure it, any deviation of these numbers cited above would mean that there is a serious and fundamental flaw in our cosmological models.

However, one has to keep in mind that all these numbers are averaged over the whole universe. Since neutrinos do have a non-zero rest mass, they are indeed affected by gravity. They cannot cluster like Dark Matter, because even though CvB Neutrinos are "slow", they are still too fast (many hundreds of km/s) to form clusters and therefore no viable Dark Matter candidate. But they may form gigantic weakly bound halos around galaxies that go far beyond the Dark Matter clusters. This may lead to a local enhancement of the CvB density due to the gravitational attraction of the massive neutrinos to large-scale structures in the universe. Unfortunately, this density enhancement cannot be quantified yet, because it depends very much on the absolute neutrino mass, which is still unknown today. For a mass of 0.1$~$eV, which you assumed in your question, there would probably be no relevant density enhancement of CvB neutrinos near our galaxy. The neutrinos would be too fast and simply stream out of the gravitational potential. If the neutrino masses turn out to be larger, on the other hand, the effect of gravity can become significant and density enhancement factors of $\approx$100 might be possible.

There would also be a "CvB neutrino wind". Just like the Cosmic Microwave Background, the neutrino background is not co-moving with our reference frame. Rather, our galaxy and the Earth are passing through the gigantic cloud of CvB neutrinos, so the neutrino distribution would not appear completely isotropic to us. It would appear a bit blue-shifted in one direction and a bit red-shifted in the other.

I would like to emphasize though, that a possible CvB detection experiment would probably not yield much information about the properties of the CvB. The only feasible method conceived to detect CvB neutrinos uses the neutrino induced $\beta$-decay of unstable nuclei. This process mainly provides us a yes/no-answer about the existence of the Cosmic Background Neutrinos. It does not tell us anything about the temperature of the CvB. In principle it would be possible to determine the density (via the rate) or even the anisotropy (via an annular rate modulation), but I doubt that it we could get anything better than the right order of magnitude. What one can determine from neutrino induced $\beta$-decay is the absolute neutrino mass. But this is not a property specific to the CvB and can be measured in other ways, too.