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Problem 1

Find the lower limit for the mass of the dark matter particles in case they are:
a) bosons;
b) fermions.

Problem 2

Find the lower limit for the mass of fermion particles that constitute a compact spherical object of dark matter with radius $R$ and mass $M$.

Problem 3

Show that particles that contitute a galaxy of mass $M=10^{10}M_\odot$ and radius $R=3kpc$ must be non-relativistic.

Problem 4

In assumption that neutrinos have mass $m_\nu$ and temperature of decoupling $T_D\simeq1MeV$, determine their contribution to the presently observed energy density.

If WIMPs represent dark matter, they will not just generate the background density of Universe, but they will also cluster together with the usual baryon matter in the galactic halo. In particular, they will be presented in our own galaxy - the Milky Way. It gives hope to detect the relic WIMPs immediately on the Earth. What is these hope related to? By definition, WIMPs do not interact with photons. However, WIMPs could annihilate into usual matter (quarks and leptons) in the early Universe. Otherwise they would have too high relative abundance today. Due to the crossover symmetry, the temperature of annihilation of e.g. WIMPs into quarks is related to the amplitude of elastic scattering of WIMPs on quarks. Therefore WIMPs should interact, though weakly, with usual matter. Due to this interaction, WIMPs can elastically scatter on target nuclei and lead to a recoil of detector nuclei partially transferring energy to them. Therefore the search for the events of elastic scattering of WIMPs on detector nuclei is one of the prospective ways of galactic dark matter research.

Problem 5

Show that preservation of thermal equilibrium for a certain component in the expanding Universe is possible only under the condition $\Gamma\ll H$, where $\Gamma$ is the rate of reaction needed to support the equilibrium.

Problem 6

Show that at high temperatures $T\gg m_\chi$ ($m_\chi$ is the WIMPs mass) the ratio of equilibrium densities of WIMPs and photons is constant: $n_\chi^{eq}/n_\gamma^{eq}=const$.

Problem 7

Show that the Boltzmann equation describing the evolution of WIMPs number density $n_\chi$ in particle number preserving interactions leads to the usual relation for non-relativistic matter $n_\chi\propto a^{-3}$.

Problem 8

Show that in the early Universe for $T\ge m_\chi$, the WIMPs number density follows its equilibrium value at temperature decreasing.

Problem 9

Show that decreasing of the annihilation cross-section leads to increasing of the relic density: contrary to Charles Darwin, the weakest wins.

Problem 10

What temperature does neutrino's density "freezing" take place at?

Problem 11

Considering the WIMPs as the thermal relics of the early Universe, estimate their current density.