ANONIMOUS - Revision history

Cosmo All: /* Problem 9 */

2012-10-02T11:58:10Z

‎Problem 9

2012-10-02T11:57:17Z

‎Problem 7

2012-10-02T11:56:56Z

‎Problem 6

2012-10-02T11:56:38Z

‎Problem 4

2012-10-02T11:56:22Z

‎Problem 3

2012-10-02T11:55:55Z

‎Problem 2

2012-10-02T11:54:59Z

‎Problem 4

2012-10-02T11:54:44Z

2012-10-02T10:47:16Z

Created page with "2"

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[[Category:Dark Matter|2]]

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      <p style="text-align: left;">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.
 <gallery widths=600px heights=400px>
-File:f02.JPG|
+File:f02.jpg|
 </gallery>
 </p>
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 === Problem 10 ===
 What temperature does neutrino's density "freezing" take place at?

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      <p style="text-align: left;">The Boltzmann equation for WIMP number density reads
 \begin{equation}\label{11_10}
-\frac{{dn_\chi  }}{{dt}} + 3Hn_\chi   =  - \left\langle {\sigma _A
+\frac{dn_\chi  }{dt} + 3Hn_\chi   =  - \left\langle {\sigma _A
 v} \right\rangle \left[ {n_\chi ^2  - \left( {n_\chi ^{eq} }
 \right)^2 } \right].
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 === Problem 8 ===
 Show that in the early Universe for $T\ge m_\chi$, the WIMPs number density follows its equilibrium value at temperature decreasing.

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 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 \[n_\gamma ^{eq}  \simeq
-.244\left( {\frac{T}{{\hbar c}}} \right)^3. \] Therefore $n_\chi
+.244\left( {\frac{T}{\hbar c}} \right)^3. \] Therefore $n_\chi
 ^{eq} /n_\gamma ^{eq}  = const$.</p>
    </div>
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 === 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 $n_\chi\propto a^{-3}$.

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    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">$\Omega_\nu  h^2 =  {m_\nu}/{91.5\mbox{\it eV}}$.</p>
+     <p style="text-align: left;">$\Omega_\nu  h^2 =  {m_\nu}/{91.5\mbox{eV}}$.</p>
    </div>
 </div></div>

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

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      <p style="text-align: left;">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}}
+\[\hbar \left( {\frac{9\pi M}{2gm^4 R^3 }} \right) \le \sqrt {\frac{2GM}{R}}
-\Rightarrow m \ge m_{lb} = \left( \frac{{9\pi \hbar ^3 }}{{4\sqrt 2
+\Rightarrow m \ge m_{lb} = \left( \frac{9\pi \hbar ^3 }{4\sqrt 2gM^{1/2} R^{3/2} G^{3/2}}\right)^{1/4}.
-gM^{1/2} R^{3/2} G^{3/2}
- }}\right)^{1/4}.
 \]
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 === Problem 3 ===
 Show that particles that contitute a galaxy of mass $M=10^{10}M_\odot$ and radius $R=3kpc$ must be non-relativistic.

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    </div>
 </div></div>
 <p>''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.''</p>
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 <div style="border: 1px solid #AAA; padding:5px;">
 === 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.