Astroparticle Physics

196 9 The Early Universeyears), it will be seen that the total energy density was dominatedby relativistic particles. This is called the ‘radiationdominatedera’, and in this period one can ignore the con-tribution to ϱ from non-relativistic particles. Then, using theappropriate relativistic formulae from (9.10) for bosons andfermions, one obtains for the total energy densityradiation-dominated eraϱ =∑i=bosonsπ 230 g iT 4 + ∑j=fermions7 π 28 30 g j T 4 , (9.16)where the sums include only those particle types that arerelativistic, i.e., having m ≪ T . This can be equivalentlywritten asϱ = π 230 g ∗T 4 , (9.17)where the effective number of degrees of freedom g ∗ is de-fined aseffective numberof degrees of freedomHere as well, the sums only include particles with m ≪ T .T is always assumed to mean the photon temperature,since its value for the present era is very accurately measuredfrom the cosmic microwave background radiation tobe T ≈ 2.73 K. Some particle types may have a differenttemperature, however, since they may no longer be inthermal contact with photons. Neutrinos, for example, effec-tively decoupled from other particles at a time before mostelectrons and positrons annihilated into photons. As a result,the neutrino temperature today is around 1.95 K. In general,one can modify (9.18) to account for different temperaturesby usingdecoupling of neutrinosneutrino temperatureg ∗ =∑i=bosonsg i + 7 8∑j=fermionsg j . (9.18)g ∗ =∑i=bosons( ) 4 Tig i + 7 T 8∑j=fermions( ) 4 Tjg j .T(9.19)This more general form of g ∗ will be rarely needed exceptfor the example of neutrinos mentioned above.In order to compute g ∗ at a given temperature T , oneneeds to know what particle types have m ≪ T ,andalsothe number of degrees of freedom for these types is required.