2.7 Constraints on warm dark matter
-body simulations of large-scale structures that assume a CDM cosmology appear to
over-predict the power on small scales when compared to observations : ‘the missing-satellite
problem’ [494, 511, 869, 188], the ‘cusp-core problem’ [568, 833, 974] and sizes of mini-voids
. These problems may be more or less solved by several different phenomena [e.g. 310],
however one which could explain all of the above is warm dark matter (WDM) [143, 248, 159]. If
the dark matter particle is very light, it can cause a suppression of the growth of structures
on small scales via free-streaming of the dark matter particles whilst relativistic in the early
2.7.1 Warm dark matter particle candidates
Numerous WDM particle models can be constructed, but there are two that occur most commonly in
literature, because they are most plausible from particle physics theory as well as from cosmological
- Sterile neutrinos may be constructed to extend the standard model of particle physics. The
standard model active (left-handed) neutrinos can then receive the observed small masses
through, e.g., a see-saw mechanism. This implies that right-handed sterile neutrinos must be
rather heavy, but the lightest of them naturally has a mass in the keV region, which makes
it a suitable WDM candidate. The simplest model of sterile neutrinos as WDM candidate
assumes that these particles were produced at the same time as active neutrinos, but they
never thermalized and were thus produced with a much reduced abundance due to their weak
coupling [see 136, and references therein].
- The gravitino appears as the supersymmetric partner of the graviton in supergravity models.
If it has a mass in the keV range, it will be a suitable WDM candidate. It belongs to a more
general class of thermalized WDM candidates. It is assumed that this class of particles achieved
a full thermal equilibrium, but at an earlier stage, when the number of degrees of freedom was
much higher and hence their relative temperature with respect to the CMB is much reduced.
Note that in order for the gravitino to be a good dark matter particle in general, it must be
very stable, which in most models corresponds to it being the LSP [e.g. 151, 221].
Other possible WDM candidates exist, for example a non-thermal neutralino  or a non-thermal gravitino
2.7.2 Dark matter free-streaming
The modification of the shape of the linear-theory power spectrum of CDM due to WDM can be calculated
by multiplication by a transfer function 
with suitable parameter  and with the scale break parameter, in the case of thermal
This is a fit to the solution of the full Boltzman equations.
There is a one-to-one relation between the mass of the thermalized WDM particle (e.g.,
gravitino), and the mass of the simplest sterile neutrino , such that the two models have an identical
impact on cosmology 
where . The difference comes from the fact that in the gravitino case the particle is fully
thermalized, the number of effective degrees of freedom being determined by mass and energy density of
dark matter, while in the simplest sterile neutrino case the number of degrees of freedom is fixed, while
abundance is determined by mass and energy density of dark matter.
In order to extrapolate the matter power spectrum to later times one must take into account the
nonlinear evolution of the matter density field. This is not straightforward in the WDM case  and most
likely needs to be explored through further simulations .
2.7.3 Current constraints on the WDM particle from large-scale structure
Measurements in the particle-physics energy domain can only reach masses uninteresting in the WDM
context, since direct detectors look mainly for a WIMP, whose mass should be in the GeV – TeV range.
However, as described above, cosmological observations are able to place constraints on light dark matter
particles. Observation of the flux power spectrum of the Lyman- forest, which can indirectly measure
the fluctuations in the dark matter density on scales between and gives the limits
of or equivalently at 95% confidence level [927, 929, 812]. For the
simplest sterile neutrino model, these lower limits are at odds with the upper limits derived from X-ray
observations, which come from the lack of observed diffuse X-ray background from sterile neutrino
annihilation and set the limit at the 95% confidence limit . However, these results do
not rule the simplest sterile neutrino models out. There exist theoretical means of evading
small-scale power constraints [see e.g. 160, and references therein]. The weak lensing power spectrum
from Euclid will be able to constrain the dark matter particle mass to about