Astroparticle Physics

370 17 Solutionsb) At last scattering (t = 380 000 a, temperature at the time of last scattering: T dec =0.3 eV, see Chap. 11):T dec = 0.3eV/k ≈ 3500 K ,ϱ dec = ϱ 0(TdecT 0) 4≈ 0.26 eV/cm 3 ×( ) 3500 4≈ 0.7TeV/cm 3 .2.7256. Naïvely one would expect the fraction of neutral hydrogen to become significant whenthe temperature drops below 13.6 eV. But this happens only at much lower temperaturesbecause there are so many more photons than baryons, and the photon energy distribution,i.e., the Planck distribution, has a long tail towards high energies. The baryon-tophotonratio, η ≈ 5 × 10 −10 , is extremely small. Therefore, the temperature must besignificantly lower than this before the number of photons with E>13.6 eV is comparableto the number of baryons. Furthermore, interaction or ionization can take place inseveral steps via excited states of the hydrogen atom, the H 2 molecule, or the H + 2 ion.One finds that the numbers of neutral and ionized atoms become equal at a recombinationtemperature of T rec ≈ 0.3 eV (3500 K). At this point the universe transforms froman ionized plasma to an essentially neutral gas of hydrogen and helium.17.12 Chapter 121. The Friedmann equation for k = 0, corresponding to the dominance of Λ, reads, seealso Problem 11.2,H 2 = 8πG3 (ϱ + ϱ v).WithΛ = 8πGc 2 ϱ v ⇒ H 2 − 1 3 Λc2 = 8πG ϱ.3Since ϱ>0 one has the inequalityH 2 − 1 3 Λc2 ≥ 0 or Λ ≤ 3H 2c 2 ≈ 2 × 10 −56 cm −2 .This is just a reflection of the fact that in the visible universe there is no obvious effectof the curvature of space. The size of the visible flat universe being 10 28 cm can beconverted intoΛ 1 ≤ 10 −56 cm −2 .If one assumes on the other hand that Einstein’s theory of relativity is valid down to thePlanck scale, then one would expectΛ 2 ≈ (l 2 Pl )−1 ≈ 10 66 cm −2 .The difference between the two estimates is 122 orders of magnitude.