Astroparticle Physics

9.2 Thermodynamics of the Early Universe 195⎧π 4⎪⎨ T ≈ 2.701 T for bosons,30 ζ(3)〈E〉 =7π⎪⎩4(9.12)180 ζ(3) T ≈ 3.151 T for fermions.In the non-relativistic limit one finds for the energy densityenergy densitiesϱ = mn , (9.13)where n is the number density. This is given in the nonrelativisticlimit by( ) mT 3/2n = g e −m/T , (9.14)2πwhere the same result is obtained for both the Fermi–Diracdistribution and Bose–Einstein distribution and, as above, itis assumed that chemical potentials can be neglected. Onetherefore finds that for a non-relativistic particle species, thenumber density is exponentially suppressed by the factore −m/T . In the non-relativistic limit, the average energy isthe sum of mass and kinetic terms,Fermi–Dirac distributionBose–Einstein distribution〈E〉 =m + 3 T ≈ m. (9.15)2The final approximation holds in the non-relativistic limitwhere T ≪ m.9.2.2 The Total Energy DensityThe formulae above give ϱ and n for a single particle type.What is needed for the Friedmann equation, however, is thetotal energy density from all particles. This is simply thesum of the ϱ values for all particle types, where at a giventemperature some types will be relativistic and others not.The expression that one will obtain for the total energydensity is greatly simplified by recalling that the numberdensity of a non-relativistic particle species is exponentiallysuppressed by the factor e −m/T . This will hold as long as thereduction in density is not prevented by a conserved quantumnumber, which would imply a non-zero chemical potential.Assuming this is not the case, to a good approximationthe non-relativistic matter contributes very little to the totalenergy density. At early times in the universe (t