Astroparticle Physics

B.2 Number and Energy Densities 397⎧ζ(3)⎪⎨πn =2 gT 3 for bosons,3 ζ(3) ⎪⎩4 π 2 gT 3 for fermions.(B.28)Here ζ is the Riemann zeta function and ζ(3) ≈ 1.20 206 ....Notice that in particle physics units the number density hasdimension of energy cubed. To convert this to a normalnumber per unit volume, one has to divide by (¯hc) 3 ≈(0.2GeVfm) 3 .In the non-relativistic limit (T ≪ m), the integral (B.27)becomes( ) mT 3/2n = g e −(m−µ)/T , (B.29)2πn: non-relativistic limitwhere the same result is obtained for both the Fermi–Diracand Bose–Einstein distributions. One sees that for a nonrelativisticparticle species, the number density is exponentiallysuppressed by the factor e −m/T , the so-called Boltzmannfactor. This may seem counter intuitive, since the densityof air molecules in a room is certainly not suppressed bythis factor, although they are non-relativistic. One must takeinto account the fact that the chemical potentials depend ingeneral on the temperature, and this dependence is exactlysuch that all relevant quantities are conserved. In the case ofthe air molecules, µ varies with temperature so as to exactlycompensate the factor T 3/2 e −m/T .For very high temperatures, to good approximation allof the chemical potentials can be set to zero. The total numberof particles will be large compared to the net values ofany of the conserved quantum numbers, and the constraintseffectively play no rôle.To find the energy density ϱ one multiplies the numberof particles in d 3 p by the energy and integrate over all momenta,E d 3 pϱ =e (E−µ)/T ± 1= g ∫ ∞√E 2 − m 2 E 2 dE2π 2 m e (E−µ)/T . (B.30)± 1As with n, the integral can only be carried out in closed formfor certain limiting cases. In the relativistic limit, T ≫ m,one findsg ∫(2π) 3very high temperaturesenergy density ϱϱ: relativistic limit