Hot and cold dark matter

Which is the nature of this non-baryonic dark matter (DM) is still a mystery.

Among the known weak interacting particles, the obvious candidates are neutrinos. They are a form of hot dark matter (HDM) because they are light particles which were certainly relativistic when their thermal equilibrium was frozen, $i.e.$, when the rate of their reactions became smaller than the Hubble parameter which occurred at $T\sim 1MeV.$ Therefore, except for the fact that now the temperature of neutrinos is slightly smaller than that of photons of CBR, the density of neutrinos in the universe is certainly not much lower than that of photons, i.e., around 100 per $cm^3$ for each species.

Since it has been determined by the accelerator LEP that there are 3 kinds of light neutrinos, if their masses were of a few $eV.$ , neutrinos might explain all DM. However this seems to be unlikely because if DM was relativistic when the structures started to be formed, the small scale inhomogeneities would be washed out and the large scale ones would be the first to appear, contrary to the observations that have shown that galaxies appeared at $z\sim 5$, i.e., earlier than clusters.

The question if neutrinos have mass is one of the most important issues of present elementary particles physics. Of great importance has been the observation of neutrino oscillation at Superkamiokande(6) indicating that there is a difference of mass between two kinds of neutrinos of around 0.1 $%% eV$. This leads to a lower bound of one type of neutrino $m_\nu \geq 0.1\ eV$ and therefore, even though neutrinos cannot explain all DM, it must be

$$\Omega _v\geq \Omega _L.$$ (23)

Therefore, one must look for a cold dark matter (CDM) candidate nonrelativistic particle when structures began.

One possibility is that when thermal equilibrium froze its mass was larger than temperature so that the density was suppressed in comparison with that of photons. Such particles are thought to emerge from supersymmetric (SUSY) extensions of the standard model which assume that for each fermion there is a boson (i) giving stability to quantum corrections of Higgs mass, (ii) predicting high energy unification of coupling constants for electromagnetic, weak and strong interactions and (iii) being an ingredient of string theories which include gravity. The lightest supersymmetric particle would be the neutralino, fermion mixture of photino, zino, and higgsino. Since the scale for breaking of SUSY should be $\sim 1\ TeV$ to avoid excessive quantum corrections to the Higgs mass, it is reasonable to expect a neutralino mass $m_x\sim 100\ GeV$.

It is possible to estimate the density of neutralinos compared to that of photons from the freeze-out condition at temperature $T_x<m_x$.

$$n_x \langle \sigma \upsilon \rangle = H(T_x )\simeq \frac{T_x^2}{m_{pl}}\simeq \frac{n_\gamma }{m_{pl}T_x}.$$ (24)

Considering that the thermal averaged cross-section $\langle \sigma \upsilon \rangle$ at freeze-out is due to annihilation with a coupling $\alpha \sim 10^2$, the assumption

$$\langle \sigma \upsilon \rangle \sim \frac{\alpha ^2}{m_x^2}\sim 10^{-36}cm^2$$ (25)

corresponds to a weak interaction. Taking $m_x\sim 10\ T_x$, from Eqs.(24) and (25) it turns out that

$$n_x\sim 10^{-12}n_\gamma ,$$ (26)

and being at present $m_x$ around 15 orders of magnitude larger than the energy of a photon, recalling Eq.(11) it is possible to fit

$$\Omega _x\sim 0.35,$$ (27)

explaining all DM, as it was done many years ago(7).

Accelerators have so far excluded the lightest supersymmetric particle with $%% m_x<50\ GeV$. A very intense search of these weak interacting massive particles (WIMP) is pursued both through direct detection by the recoil of $%% Ge$ nucleus or indirect detection by high energy neutrinos coming from the annihilation $\chi \chi \rightarrow \nu \vec{\nu}$ possible if $\chi$ are gravitationally captured by sun, which is one of the aims of the South Pole Amanda neutrino telescope.

Another candidate for CDM, which was never in thermal equilibrium, is the axion: a theoretical particle introduced(8) to avoid the CP violation in strong interactions. It is a neutral pseudoscalar particle which, when the temperature of universe falls below the confinement scale $\wedge _{QCD}\sim 200\ MeV$ acquires a mass because of its mixture with the state of pion. The mass of the axion is

$$m_a\sim \frac{\wedge _{QCD}^2}{f_a},$$ (28)

where $f_a$ is of the order of a superheavy quark mass to which it is coupled. Because of this interaction, the coupling of axion with electromagnetic fields is of the form

$${{\mathcal L} _{a \gamma\gamma}}\sim{\alpha\over{f_a}}a\vec E \cdot \vec B$$ (29)

which gives an extremely large lifetime for the decay $a\rightarrow 2\gamma$.

With $,$ the choice $f_a\sim 10^{12}GeV$, from Eq.(28), $m_a\sim 10^{-5}eV.$ How is it possible that with such a tiny mass the axion is a nonrelativistic particle? This would come from the fact that the equation of motion of the axion field in the expanding universe is

$$\ddot{a}+3H(t)\dot{a}+m_a^2(t)a=0.$$ (30)

When the Hubble parameter is larger than the axion mass the friction term of Eq.(30) dominates, and the solution is $a=$ constant everywhere. Afterwards when $m_a>H$, $a$ starts the oscillation giving rise to the particles which, due to the uniformity of the field in space, have almost vanishing momenta, so that they are nonrelativistic.

It is a delicate matter to evaluate the contribution to universe energy density of an initial non-alignment of the field when the QCD mass effects were very small. But due to the fact that $m_a$ decreases with $f_a$, the potential is flatter and misalignment larger so that it turns out that

$$\Omega _a\sim \left( \frac{10^{-5}eV}{m_a}\right) ^{1.1}.$$ (31)

Therefore, axions may close the universe if they have the correct mass.

Experiments are underway(9) using the interaction Eq.(29) to transform the axion with an intense magnetic field into a wave in a resonant cavity.

On the other extreme of mass scale, it is also possible that superheavy relics of the age of the grand unification theories (GUT), $m_{GUT}\geq 10^{15}GeV$, form a part, not necessarily very large, of the galactic halo.

The interest of these objects is that they might be the origin, through a very slow decay, of the ultra-high energy cosmic rays (UHECR)(10). Of course the question is how could they survive till the present epoch. One possibility is that they are particles(11) which interact with the known ones only through forces of gravitational order. Another (12) is that they are cosmic strings stabilized by superconducting currents. This possible origin of UHECR may be elucidated by the Auger observatory which is beginning to be built.