Astroparticle Physics

396 B Results from Statistical Physics: Thermodynamics of the Early Universeresulting momentumdistributioninternal degrees of freedomUsing these modified names for the Lagrange multipliersgives the desired result for the momentum distribution,f a (p) = g a(L/2π) 3e (E a−µ a )/T ∓ 1 ,(B.25)where one uses the minus sign if particle type a is a bosonand plus if it is a fermion. The number of internal degrees offreedom, g a , is usually 2J + 1 for a particle of spin J ,butitcould include other degrees of freedom besides spin such ascolour.B.2 Number and Energy Densities“There are 10 11 stars in the galaxy.That used to be a huge number. But it’sonly a hundred billion. It’s less than thenational deficit! We used to call themastronomical numbers. Now we shouldcall them economical numbers.”Richard P. Feynmannumber density nFrom the Planck distribution given by (6.81), (B.1) one canproceed to determine the number and energy per unit volumefor all of the particle types. The function (B.25) gives thenumber of particles of type a in a momentum-space volumed 3 p. The number density n is obtained by integrating thisover all of momentum space and dividing by the volumeV = L 3 , i.e.,n = 1 V∫f(p) d 3 p =∫g(2π) 3d 3 pe (E−µ)/T ± 1 , (B.26)n: energy integraln: relativistic limitwhere for clarity the index indicating the particle type hasbeen dropped. Since the integrand only depends on the magnitudeof the momentum through E = √ p 2 + m 2 , one cantake the element d 3 p to be a spherical shell with radius p andthickness dp, sothatd 3 p ̂= 4πp 2 dp.FromE 2 = p 2 + m 2one gets 2E dE = 2p dp and therefore√E 2 − m 2 E dEn =g ∫ ∞2π 2 m e (E−µ)/T . (B.27)± 1The integral (B.27) can be carried out in closed form onlyfor certain limiting cases. In the limit where the particles arerelativistic, i.e., T ≫ m,andalsoifT ≫ µ, one finds