If neutrinos are disfavoured as DM candidates why aren't axions?

Numerical simulations of observed large-scale structure formation work best with Cold Dark Matter (CDM; see the answer here). Neutrinos are candidates for Hot Dark Matter (HDM), and hence they cannot account for the total dark matter abundance of the Universe. By the same token, axions are also relativistic because they have very tiny masses. Aren't they also candidates of HDM like neutrinos? Shouldn't they also be disfavoured for the same reason?

• It comes down to the production mechanism, which is not thermal. See chapter 10 of Kolb and Turner for more details. – knzhou Dec 16 '18 at 15:24
• As far as I know the reason that neutrinos cannot account for cold dark matter is that for neutrinos to form structures i.e. be cold they need to be slower than e.g. the escape velocity of the galaxy they are in. Now since neutrinos are fermions only two neutrinos per unit phase space. If you do the calculation you find that with the current bounds on neutrino masses they cannot account for the necessary DM mass without exceeding the escape velocity. This is know as the Tremaine-Gunn bound. Since axions are bosons you can put infinitely many into the same unit phase space however. – Katermickie Dec 16 '18 at 15:29
• If you want to I can search my notes and post the calculation as an answer. The rest comes down to knzhou's comment – Katermickie Dec 16 '18 at 15:30

The answer is that the axions are not relativistic, but rather extremely cold. Neutrinos are hot because they were in thermal equilibrium with the standard model heat bath before they decoupled.

This is not the case for axions, which needs some form of non-thermal production mechanism. Otherwise they could only form hot dark matter, as you say. If the axions wre thermally produced their relic density would also be far too low to explain dark matter (even if hot dark matter was not ruled out).

Following the solution of an exercise sheet in my Dark Matter course on the Tremaine-Gunn bound:

One explanation why neutrinos cannot be cold dark matter is that although non relativistic neutrinos should exist (it is expected that the cosmic neutrino background is non-relativistic although it has not been measured yet) they would not be massive enough to explain the dark matter that is bound in structures like galaxies.

To see this we first need that since neutrinos are fermions the number of states per unit phase space is limited and given by $$n = \frac{g}{(2\pi \hbar)^3}$$ For massless Standard Model neutrinos $$g=1$$

However for massive Neutrinos $$g=2$$

Now taking a typical galactic rotation curve with a rotation velocity of $$v(R)=220\frac{km}{s}$$ at $$R=12$$ kpc we find that we need a mass of the galaxy inside the 12 kpc Radius of $$M=2.69\cdot 10^{41}kg$$ Since we know that most of the mass inside a galaxy consists of Dark Matter we might aswell take this to be the Mass of the Dark Matter. We will later see that this approximation is valid enough to rule out neutrinos as CDM candidates.

Of course in this case the number of neutrinos is given by $$N_\nu = M/m_\nu$$ where $$m_\nu$$ is the mass of the neutrinos. Since we know from oscillation data that the squared mass differences of the neutrinos are very small we might aswell assume only one massive neutrino with $$m_\nu=\sum m_{\nu_i}$$

Now the number of neutrinos is also given by $$N_\nu= V\int n\;d^3p= \frac{4}{3}\pi V p_{\text{max}}^3n$$ with $$V=\frac{4}{3}\pi R^3$$ For neutrinos to be bound we need them to be slower than the escape velocity. Thereby we get $$v_{\text{max}}=v_{\text{esc}}=\sqrt{\frac{2GM}{R}}=3.1\cdot10^5\frac{\text{m}}{\text{s}}$$ which is way below the speed of light i.e. non relativistic.

Therefore we can take $$p_{\text{max}}=m_\nu v_{\text{max}}$$ Now we can calculate the necessary neutrino mass to account for the Dark Matter $$m_{\nu,\text{min}} = \left(\frac{3M}{4Vnv_{\text{esc}}^3\pi}\right)^{1/4}$$ and find that $$m_{\nu,\text{min}}\approx 19.5\,\text{eV}$$ is necessary to account for the observed velocity curve to be explained by Neutrino Dark Matter. The current bound on the sum of all neutrino masses from the particle data group are $$\sum m_i<0.2\,\text{eV}$$ Hence we find that neutrinos cannot account for the observed Dark Matter inside galaxies.

Notice that all of this explanation builds up on the fact that neutrinos are fermions and not bosons i.e. you can only pack a limited amount of neutrinos into one unit phase space. For bosons like the axion this does not hold so even very light bosons can be cold dark matter if the production mechanism allows for them to be non relativistic.

• this bound on the sum of all neutrino masses from the particle data group cannot possibly account for sterile neutrinos, correct? – lurscher Dec 17 '18 at 10:49
• Yes as far as I know this only takes active neutrinos into account. Sterile neutrinos are not excluded as Dark Matter candidates. – Katermickie Dec 17 '18 at 11:03
• Possible heavy sterile neutrinos would/could however change the effective mass of flavor neutrinos via mixing. The bounds on those are not that strong though. – Katermickie Dec 17 '18 at 11:04
• Yes that is exactly what I am saying and when I plug in the numbers I get 19.5 eV as a limit with $m>(3M/4Vv^3n)^{1/4}$ you can check with wolframalpha.com/input/?i=(3%2F4*2.69e%2B41+kg%2F(4%2F3*pi*(12+kpc)%5E3*(3.11e%2B5+m%2Fs)%5E3*pi*2%2F(2*pi*hbar)%5E3))%5E1%2F4+in+eV%2Fc%5E2" – Katermickie Dec 17 '18 at 15:09
• I see. I had made a couple of mistakes in my calculation. Thanks for clarifying. Wow, it's strongly counterintuitive to me that this can give a useful bound. – user4552 Dec 17 '18 at 19:10