Astroparticle Physics

11.6 Determination of Cosmological Parameters 239Fig. 11.4CMB power spectrum. The set ofmeasurements with the smallererror bars is from WMAP; thosewith the larger errors represent anaverage of measurements prior toWMAP (from [29])the critical density, Ω, as well as the components of the energydensity from baryons, Ω b , and from all non-relativisticmatter Ω m .As an example, in the following a rough idea will begiven of how the angular power spectrum is sensitive to Ω.Consider the largest region that could be in causal contactat the time of last scattering t ls ≈ t dec ≈ 380 000 years.This distance is called the particle horizon d H . Naïvely onewould expect this to be d H = t (i.e., ct, butc = 1 has beenassumed). This is not quite right because the universe is expanding.The correct formula for the particle-horizon distanceat a time t in an isotropic and homogeneous universeis (see, e.g., [30]),∫ tdt ′d H (t) = R(t)0 R(t ′ ) . (11.16)If the time before matter–radiation equality is considered(t mr ≈ 50 000 years), then R ∼ t 1/2 and d H (t) = 2t. Forthe matter-dominated era one has R ∼ t 2/3 and d H = 3t. Ifa sudden switch from R ∼ t 1/2 to R ∼ t 2/3 is assumed att mr , then one finds from integrating (11.16) a particle horizonat t ls ofd H (t ls ) = 3t ls − t 2/3lstmr 1/3 ≈ 950 000 (light-)years .(11.17)As most of the time up to t ls is matter dominated, the resultis in fact close to 3t ls .Ω determinationparticle horizondistancesin an expanding universe