# Could not slow neutrinos orbit galaxies and clusters, thus comprising a large component of even COLD dark matter?

Cold dark matter is the type of dark matter that is most eminently neutrino-free. But neutrinos themselves suffer a large survivorship detection bias (“all the neutrinos you can detect necessarily have to have relativistic speeds”) https://physics.stackexchange.com/questions/267035/where-are-all-the-slow-neutrinos

By the same principle that there are more pebbles than boulders, and more red dwarfs than blue giants, slow neutrinos should be more abundant than the fast, detectable ones.

Non-relativistic (slow) neutrinos - in particular those going at below the galactic escape velocity - could be a major competent of cold dark matter: slow neutrinos could be slow enough to orbit galaxies and galactic clusters and thus form a significant amount of cdm, which should be shaped as a galactic halo.

If neutrinos are produced at all speeds, those stellar (modern) neutrinos below galactic escape velocity will accumulate indefinitely. This should generate a halo-shaped cloud, and this could comprise a large part (of course 15% are MACHOs etc https://ned.ipac.caltech.edu/level5/Sept17/Freese/Freese4.html - I am assuming here a multi-component DM) of cold dark matter.

• Regarding MACHOS - a more modern limit (from EROS and OGLE) is <10% of the dark halo can be composed of objects between about $10^{-6}$ and 1 solar masses (e.g. planets, brown dwarfs, cold white dwarfs etc.). arxiv.org/pdf/2002.12778.pdf Nov 23, 2020 at 15:19

These neutrinos would have to be really cold. The cosmic neutrino background is at 1.9K, and they are considered hot dark matter, because they would have been highly relativistic at the epoch of structure formation. To be considered cold dark matter, and also to be captured in orbits in galaxies, the neutrinos would have to be much colder - totally non-relativistic now.

Let's assume an average energy then of about 0.1 eV for each neutrino (similar to their likely rest mass energy). In order to account for $$\Omega_{\rm CDM}\sim 0.3$$ there would need to be $$5\times 10^{10}$$ per cubic metre, or about $$10^{10}$$ per cubic metre per flavour, on average over the universe.

To account for dark matter in galaxies (e.g. the Milky Way) we need $$\sim 10^{12} M_\odot$$ within about 100 kpc, meaning a number density of neutrinos of $$10^{14}$$ per cubic metre.

These neutrinos are spin 1/2 fermions and thus would have a Fermi energy of about $$5\times 10^{-3}$$ eV. That would mean if they were any colder than 6 K, they would be degenerate and exert a degeneracy pressure. This would be sufficient to prevent the halos forming - Tremaine & Gunn (1979) showed that cold dark matter halos cannot be made from low mass leptons like standard neutrinos.

Edits:

A back of the envelope Tremaine-Gunn limit (see also Boyarsky et al. 2009) is to assume that the escape speed of the galaxy is $$v$$, it's dark matter halos has radius $$r$$, the total mass is $$M$$ and the neutrino mass is $$m$$.

The number of available quantum states to spin half fermions in this volume, up to a momentum $$mv$$ is $$N = \left(\frac{4\pi r^3}{3}\right) \left(\frac{8\pi}{3}\right) \left(\frac{mv}{h}\right)^3$$ We can write $$v = \left(\frac{2GM}{r}\right)^{1/2}.$$ The mass contained in these particles cannot be greater than if every quantum state is filled by one fermion of mass $$m$$ and if this is to explain dark matter, then this mass must be $$\sim M$$. Thus $$M < m\left(\frac{4\pi r^3}{3}\right) \left(\frac{8\pi}{3}\right) \left(\frac{m\sqrt{2GM/r}}{h}\right)^3$$ and $$mc^2 > 8.9\left(\frac{r}{\rm 100 kpc}\right)^{-3/8} \left(\frac{M}{10^{12}M_\odot}\right)^{-1/8}\ {\rm eV}\ .$$ Thus there are not enough quantum states to accommodate a halo of fermions unless their rest mass energies exceed about 10 eV. For neutrinos, there are 3 flavours and anti particles, which reduces this number by $$6^{1/4}$$, but conversely it must be increased because particles in the halo cannot be uniformly distributed in velocity between 0 and $$v$$.

10 ev is about two orders of magnitude larger than the likely rest masses of the known neutrinos.

The idea that neutrinos from stars can make any contribution to dark matter halos is untenable. The vast majority of solar neutrinos have energies above 0.1 MeV, and so for an assumed neutrino rest mass energy of $$\sim 0.1$$ eV, they have Lorentz factors that exceed $$10^6$$ - i.e. they travel exceedingly close to the speed of light and are not confined to galaxies. The neutrinos emitted during supernova explosions are even more energetic. Secondly, even if you invent a magical process that can produce neutrinos with kinetic energies less than 0.1 eV, you still have the Tremaine-Gunn limit to overcome and even if all the rest mass of all the stars in a galaxy was converted into neutrinos, that mass would still fall short by more than an order of magnitude of that required to explain dark matter in galaxies. You cannot imagine that the neutrinos would "pile up" over time because the vast majority of the mass that has ever been turned into stars is still in the form of stars today and the stars that have already lived and died are a tiny percentage of the required dark matter mass.

• @MarkBesser But they would have to be really cold. How does their origin change that? The neutrinos emitted by stars are basically travelling at $c$. It is also fairly clear that the rest-mass energy in stellar neutrinos can't exceed the mass of the stars that produced them, and stars are a minor component of the matter density of the universe. Nov 20, 2020 at 17:00
• @MarkBesser - besides ripping up nuclear physics and the Pauli Exclusion Principle - how do you propose that neutrinos emitted from stars can comprise about 20 times the masses of the stars themselves? Most of the stars that were ever born (containing the vast majority of the stellar mass) are still alive now. Nov 23, 2020 at 12:41
• @MarkBesser there weren't enough of these massive stars. Even if they converted ALL their rest mass into neutrinos (which they don't) they would have to have a time-integrated mass that is equal to all the luminous matter that can be seen in the universe today in order to account for 10% of the dark matter, which is nowhere near true. Ultra-cold cosmic neutrinos produced by some other mechanism than the C$\nu$B wouldn't overcome the Tremain-Gunn phase-space argument. They would have to be massive as well. e.g.something like sterile neutrinos. Not the neutrinos we know about. Nov 24, 2020 at 15:53
• @MarkBesser cold DM refers to them being cold at the epoch when structures were formed. It is not a description of their current state. Neutrinos from the big bang are not currently ultrarelativistic and, depending on their exact rest masses, they will have started to "accumulate" or be focused by gravitational wells, but are still moving way to fast to be captured. The neutrinos have a frozen-in momentum distribution, but the momentum (with respect to the local co-moving volme) will decrease as the scale factor expands. Nov 30, 2020 at 14:36
• @MarkBesser see physics.stackexchange.com/a/268348/43351 Nov 30, 2020 at 14:42

It's an interesting but frustrating question. ;)

As you mentioned, we can't detect slow neutrinos. Direct detection of them may never be technically feasible. One answer of the linked question does mentions that there are some possible indirect detection techniques for neutrinos below current thresholds, but detecting the theoretically huge numbers of neutrinos & antineutrinos released & produced during a couple of phases of the Big Bang is a much harder proposition. Those neutrinos have experienced a far greater redshift than the cosmic microwave background. As another answer at that link mentions, the CNB (cosmic neutrino background) redshift is on the order of $$10^{10}$$, compared to the 1100 or so of the CMB.

We can estimate numbers of low energy neutrinos, but there could be some factor that our theories have overlooked, so the numbers might be way off. However, cold slow neutrinos don't have much energy, so even in astronomically huge quantities they don't have much impact on spacetime curvature, certainly not enough to account for all the dark matter that we've indirectly detected via its mass.

According to Wikipedia's article on the Lambda CDM cosmological model, the relic neutrinos could account for as much as 0.5% of the universes energy content. OTOH, that's more than the 0.01% due to EM radiation, which is dominated by CMB photons.

Our current most sensitive neutrino detection reaction, the Alsace-Lorraine technique (so-named because it uses a gallium → germanium → gallium sequence) has a threshold of 233 keV. That is, the kinetic energy of those neutrinos is over a quarter of a million times their (rest) mass energy. And our detectors are lucky to catch about 1 neutrino per billion that pass through them. Note that 233 keV is less than half the rest mass energy of an electron (511 keV).

Neutrinos need to be very cold / redshifted to orbit anything (apart from black holes & possibly neutron stars). Bear in mind that even neutrinos with an eV or so of kinetic energy are still relativistic. So they can be deflected by galaxies and even stars, but they can't get into a closed orbit.

As I said earlier, the CNB neutrinos are highly redshifted, and so (some of them) can be gravitationally bound to galaxies, and maybe even individual stars. So they are a component of dark matter, but a fairly small one.

The bulk of the Big Bang neutrinos (and antineutrinos, the term "neutrino" can cover both types when the difference between them isn't relevant) in the CNB were released during neutrino decoupling, 1 second after the start of the big bang. From Wikipedia:

In Big Bang cosmology, neutrino decoupling was the epoch at which neutrinos ceased interacting with other types of matter, and thereby ceased influencing the dynamics of the universe at early times. Prior to decoupling, neutrinos were in thermal equilibrium with protons, neutrons and electrons, which was maintained through the weak interaction.

Decoupling occurred approximately at the time when the rate of those weak interactions was slower than the rate of expansion of the universe. Alternatively, it was the time when the time scale for weak interactions became greater than the age of the universe at that time. Neutrino decoupling took place approximately one second after the Big Bang, when the temperature of the universe was approximately 10 billion kelvin, or 1 MeV.

After decoupling, some neutrinos and antineutrinos were released as neutrons converted to protons and vice versa. Proton → neutron conversion normally requires a high energy environment, because neutrons have more mass than protons. Conversely free neutrons are unstable, with a half-life of a little over 10 minutes. There were also some neutrinos produced during Big Bang nucleosynthesis (which ended about 20 minutes after the Big Bang), as hydrogen was converted to helium. BB nucleosynthesis cleaned up most of the remaining free neutrons.

• The cosmic neutrino background is non-relativistic (or at least not highly relativistic). Nov 19, 2020 at 23:48