Difference between revisions of "ANONIMOUS"

If the dark matter consists of bosons then the de Broglie wavelength of the particles must be shorter than the galactic size ($\sim 1 \mbox{ kpc}$). For typical velocities $v\sim 250${ km/s}, one obtains $m \leq 10^{-22} \mbox{ eV}$. This inequality does not pose any restriction on mass of the boson as a candidate for the dark matter particle.

The bound $m_{lb}$ can be determined from the condition that maximum speed of degenerated fermi gas with gravitational interaction does not exceed the escape velocity: $v_\infty = \left(2GM/R\right)^{1/2} $ : \[\hbar \left( {\frac{9\pi M}{2gm^4 R^3 }} \right) \le \sqrt {\frac{2GM}{R}} \Rightarrow m \ge m_{lb} = \left( \frac{9\pi \hbar ^3 }{4\sqrt 2gM^{1/2} R^{3/2} G^{3/2}}\right)^{1/4}. \] Here $g$ is the number of internal degrees of freedom for the dark matter particles.

Escape velocity $v_\infty$ for particles in the galaxy can be found from the condition \[ \frac{mv_\infty ^2 }{2} = G\frac{mM}{R} \Rightarrow v_\infty = \sqrt {\frac{2GM}{R}} \approx 10^5 \mbox{ m/s} \] As the particle velocities in the galaxies cannot exceed the escape velocity $v < v_\infty $, then the particles must be non-relativistic.

$\Omega_\nu h^2 = {m_\nu}/{91.5\mbox{eV}}$.

Preservation of the thermal equilibrium requires that the rate of the supporting reactions is much higher than the relative velocity of the temperature change: $$ \Gamma\gg \left|\frac{\dot T}{T} \right|. $$ Taking into account that $aT = const$ in the expanding Universe, one obtains $\dot T/T = - \dot a/a = - H.$ Thus the condition $\Gamma\gg \left|{\dot T}/{T} \right|$ is equivalent to the requirement \( \Gamma\gg H. \)

In thermal equilibrium \[ n_\chi ^{eq} = \frac{g}{{\left( {2\pi } \right)^3 }}\int {f(\vec p)d\vec p}, \] where $g$ is the number of internal degrees of freedom, and $f(\vec p)$ is either Bose-Einstein or Fermi-Dirac distribution function. At high temperatures ($T \gg m_\chi $ ) $n_\chi ^{eq} \propto T^3$. For photons \[n_\gamma ^{eq} \simeq 0.244\left( {\frac{T}{\hbar c}} \right)^3. \] Therefore $n_\chi ^{eq} /n_\gamma ^{eq} = const$.

The Boltzmann equation for WIMP number density reads \begin{equation}\label{11_10} \frac{dn_\chi }{dt} + 3Hn_\chi = - \left\langle {\sigma _A v} \right\rangle \left[ {n_\chi ^2 - \left( {n_\chi ^{eq} } \right)^2 } \right]. \end{equation} Here $n_\chi ^{eq} $ is the equilibrium WIMP number density. For the case $ n_\chi = n_\chi ^{eq}$ one immediately obtains the solution of the Boltzmann equation in the form $n_\chi\propto a^{ - 3} $.

For early times $H\propto T^2 $, while $n_\chi\propto T^3 $, so the expansion rate decreases slower than the density. The term $3Hn_\chi$ can be neglected in the Boltzmann equation (\ref{11_10}) so that the density can follow its equilibrium value.

High detection ability of a candidate particle in general does not assist, and even rather exclude, that the particle is the dominant component of the dark matter. It is due to the fact that the detection rate is proportional to the cross-section magnitude of the scattering (or annihilation) process, while the relic abundance is inversely proportional to it instead. Thus there is obvious anti-correlation between the detection rate and the cosmological abundance. It follows so that, in spite the natural desire of the experimenters, there is little hope to have a relic with both high abundance and high detection rate. The explanation is illustrated by the picture showing the evolution of particle number in early Universe.