(Updated: 7/14/2015)

In previous comment section of a post, I had written that I had not seen an article that conducted a Cosmological Simulation of Structure formation when including the quantum degenerate nature of fermionic, warm dark matter.

I'm happy to say that an article has recently been submitted to arxiv that does just this.

"

Structure formation in warm dark matter cosmologiesTop-Bottom Upside-Down"

by Sinziana Paduroiu, Yves Revaz, and Daniel Pfenniger

Basically, the gist of the paper is the following: everybody who has modeled warm dark matter previously has had to make assumptions that (a) are not valid, and (b) simplify the simulations too much. As such, Paduroiu

*et al*. argue that modeling warm dark matter is extremely complicated, and nobody is doing it correctly.

They have posted a YouTube site with

videos of their simulations.

One problem with the manuscript is that there is no comparison to actual data.

One thing that I want to add to the discussion (which I've mentioned previously) is that non-thermal warm dark matter does not act exactly the same as a thermal warm dark matter, but with a rest mass lower than the rest mass of the actual rest mass. Too many times, proponents of Cold Dark Matter make simplification of how warm dark matter interacts (such as assuming that non-thermal WDM acts like a thermal WDM particle of lower rest mass), and then show that the rest mass of the particle that explains the "small-scale crisis" is in compatible with the rest mass required to explain the Lyman-alpha forest data. And therefore this CDM proponents claim that WDM fails, and therefore CDM is still king.

The logic here is silly. In particular, I'll thinking of the paper by Schneider et al. "

Warm dark matter does not do better than cold darkmatter in solving small-scale inconsistencies."

The logic in this paper is absurd.

(1) We acknowledge that CDM has a small scale crisis

(2) So we poorly model WDM, and find that it can't solve both the small-scale crisis and fit with data from Lyman-Alpha Forest

(3) "Hence, from an astrophysical
perspective, there is no convincing reason to favour WDM
from thermal or thermal-like production (i.e. neutrinos oscillations)
over the standard CDM scenario."

Let's not ignore the small-scale crisis (as was done by all but one of the spearkers that the recent CMB@50 event at Princeton a few weeks ago.) There is a real problem with CDM, and it won't be solved by Self-interacting Cold Dark matter. The plots below from the Schneider et al. paper and from the

Weinberg et al. paper are two ways of visualizing that there is a small-scale crisis for CDM.

There's also the too high reionization optical depth problem and the core/cusp problem of cold dark matter. There is no "Dark Matter" crisis. It's just a question of what is the rest mass and thermal distribution of dark matter.

Basically, the small-scale crisis is best solved by having a thermal rest mass of ~2 keV and the

Lyman-Alpha forest is best fit with a thermal particle with rest mass of 32 keV. (Though, it should be noted that this best fit was done before the Planck 2015 data release. It will be interesting to see how this "best-fit" changes with updated data from Planck, as well as updated BAO, SZ, & lensing data.)

Because it's the velocity (i.e. v = (4/11)^(1/3) 3kTo/mc * (1 + z) ≈ 151 (1 + z) / m (in eV)

km/s ) that's important for the Lyman-Alpha Forest, this means that the Lyman-Alpha Forest can also be equally explained by a 16 keV rest mass particle that is born with a thermal energy one half as large as a 32 keV thermally-born particle. Or a 8 keV rest mass particle that is born with a thermal energy one fourth as large as a 32 keV thermally-born particle.

(For more information on calculating the temperature/velocity of fermi particles vs. z, that transition between
being relativistic to being non-relativistic at time z=z_critical, is given by equations (10)-(15) in

Pfenniger and Muccione 2008.)

Because it's the number density times deBroglie wavelength cubed that's important for quantum degeneracy at the small-scale, i.e. lambdaDM * hbar^3 / (m^4*v^3) ~ lambdaDM / (m*T^3), then quantum degeneracy remains the same provided that m*T^3 = constant = (2 keV)*(T_thermal)^3

Note that the fact that a resonant sterile neutrino can be born with a temperature less than the temperature of the photons/neutrinos in the surrounding thermal bath means that the sterile neutrinos become non-relativistic sooner than you would expect, and hence their temperature starts dropping as (1+z)^2 sooner than for a thermal particle. The net effect is that, quantum mechanically, a sterile neutrino born with half of the thermal energy and 8 times the rest mass has the same degeneracy parameter as a thermal sterile neutrino. In other words, the number density times deBroglie wavelength cubed is the same for a 2 keV, thermal sterile neutrino as it is for a 16 keV sterile neutrino born with half of the thermal energy.

Solving the two equations, you get the following answers:

Rest Mass = 16 keV

Temperature at birth = 0.5 * Global thermal Temperature at the time of birth

So, this means that a fermion dark matter particle with a rest mass of ~16 keV that is born with half of the thermal energy of the photons and neutrinos (and which can't interact with other particles after decoupling) should be able to explain the small-scale crisis, the missing satellite, and the Lyman-Alpha forest. It should be noted that there are large error bar on the numbers listed above because of the uncertainty of the best fit for Lyman Alpha and uncertainty on the best fit for a quantum particle to explain dwarf galaxies. So, the rest mass of dark matter is likely between 10-20 keV with temperatures of 0.4-0.6 compared with the surrounding media.

Sterile neutrinos generated by the Shi-Fuller mechanism often have distribution function such that they are effectively colder than the surrounding photons/active-neutrinos by a factor of 0.6.

The main point here is that a non-thermal sterile neutrino can explain both the Lyman Alpha forest and the small-scale crisis, the missing satellite, as well as the Lyman-Alpha forest. What we are looking for is a non-thermal particle with a rest mass ~10-20 keV.