Astroparticle Physics

230 11 The Cosmic Microwave Backgrounduniverse, one needs to determine when this occurred, sincethe composition of the energy density influences the timedependence of the scale factor R(t).Estimates of the time of matter–radiation equality dependon what is assumed for the contents of the universe.For matter it will be seen that estimates based on the CMBproperties as well as on the motion of galaxies in clusterspresent-day energy densities give Ω m,0 = ϱ m,0 /ϱ c,0 ≈ 0.3, where as usual the sub-script 0 indicates a present-day value. For photons it willbe found from the CMB temperature: Ω γ,0 = 5.0 × 10 −5 .Taking into account neutrinos brings the total for radiationto Ω r,0 = 8.4 × 10 −5 , so currently, matter contributes some3 600 times more to the total energy density than does radiation.In Chap. 8 it has been derived by solving the Friedmannequation how to predict the time dependence of theR dependence different components of the energy density. For radiationof energy densities ϱ r ∼ 1/R 4 was obtained, whereas for matter ϱ m ∼ 1/R 3was found. Therefore, the ratio follows ϱ m /ϱ r ∼ R, anditwas thus equal to unity when the scale factor R was 3 600times smaller than its current value.In order to pin down when this occurred, one needs totime dependence of R know how the scale factor varies in time. If it is assumedthat the universe has been matter dominated from the timeof matter–radiation equality up to now, then one has R ∼t 2/3 , and this leads to a time of matter–radiation equality, t mr ,of around 66 000 years. In fact, there is now overwhelmingevidence that vacuum energy makes up a significant portionof the universe, with Ω Λ ≈ 0.7. Taking this into accountleads to a somewhat earlier time of matter–radiation equalityof around t mr ≈ 50 000 years.When the dominant component of the energy densitytransition changes from radiation to matter, this alters the relation be-to a matter-dominated tween the temperature and time. For non-relativistic particleenergy density types with mass m i and number density n i , one now gets forthe energy densityϱ ≈ ∑ im i n i , (11.1)where the sum includes at least baryons and electrons, andperhaps also ‘dark-matter’ particles. The Friedmann equation(neglecting the curvature term) then readsH 2 = 8πG ∑m i n i . (11.2)3i