Astroparticle Physics

9.3 Solving the Friedmann Equation 1999.2.4 Relation between Temperature and Scale FactorFinally, in this section a general relation between the temperatureT and the scale factor R will be noted. All lengths,when considered over distance scales of at least 100 Mpc,increase with R. Since the de Broglie wavelength of a particle,λ = h/p, is inversely proportional to the momentum,one sees that particle momenta decrease as 1/R. For photons,one has E = p, and so their energy decreases as 1/R.Furthermore, the temperature of photons in thermal equilibriumis simply a measure of the photons’ average energy, soone gets the important relationmomentum and scale factorT ∼ R −1 . (9.25)This relation holds as long as T is interpreted as the photontemperature and as long as the Hubble expansion iswhat provides the change in T . In fact this is not exact,because there are other processes that affect the temperatureas well. For example, as electrons and positrons becomenon-relativistic and annihilate into photons, the photon temperaturereceives an extra contribution. These effects can betaken into account by thermodynamic arguments using conservationof entropy. The details of this are not critical forthe present treatment, and one will usually be able to assume(9.25) to hold.photon temperature9.3 Solving the Friedmann Equation“No one will be able to read the greatbook of the universe if he does not understandits language which is that ofmathematics.”Galileo GalileiNow enough information is available to solve the Friedmannequation. This will allow to derive the time dependence ofthe scale factor R, temperature T , and energy density ϱ. If,for the start, very early times will be considered, one cansimplify the problem by seeing that the term in the Friedmannequation (8.15) with the curvature parameter, k/R 2 ,can be neglected. To show this, recall from Sect. 8.5 the fluidequation (8.28),curvature parameter˙ϱ + 3Ṙ (ϱ + P) = 0 , (9.26)R