Astroparticle Physics

234 11 The Cosmic Microwave Backgroundof a photon follows λ ∼ R. Therefore, the ratio of the scalefactor R at a previous time to its value R 0 now is related tothe redshift z throughR 0R = λ 0= 1 + z. (11.7)λFurthermore, today’s CMB temperature is measured to beT 0 ≈ 2.73 K and it is known that T ∼ 1/R. Therefore onegets1+z = T ≈ 0.3eVT 0 2.73 K × 18.617 × 10 −5 ≈ 1300 ,eV K−1 (11.8)recombination time≈ 300 000 yearsdecoupling of photonsfrom matterdecoupling temperaturewhere Boltzmann’s constant was inserted to convert temperaturefrom units of eV to K. If one assumes that the scalefactor follows R ∼ t 2/3 from this point up to the present,then it is found that recombination was occurring at( ) R 3/2 ( ) 3/2 T0t 0t rec = t 0 = t 0 =R 0 T rec (1 + z rec ) 3/2≈ 1.4 × 1010 years(1300) 3/2 ≈ 300 000 years . (11.9)Shortly after recombination, the mean free path for aphoton became so long that photons effectively decoupledfrom matter. While the universe was an ionized plasma, thephoton scattering cross section was dominated by Thomsonscattering, i.e., elastic scattering of a photon by an electron.The mean free path of a photon is determined by the numberdensity of electrons, which can be predicted as a functionof time, and by the Thomson scattering cross section,which can be calculated. As the universe expands, the electrondensity decreases leading to a longer mean free pathfor the photons. This path length becomes longer than thehorizon distance (the size of the observable universe at agiven time) at a decoupling temperature of T dec ≈ 0.26 eV(3000 K) corresponding to a redshift of 1 + z ≈ 1100. Thiscondition defines decoupling of photons from matter. Thedecoupling time is( ) 3/2 T0t 0t dec = t 0 =≈ 380 000 years .T dec (1 + z dec ) 3/2(11.10)