Astroparticle Physics

176 8 CosmologyFig. 8.2Two galaxies at distances r(t) andR(t) from our owndistance scaleHubble parameterexpansion rateclump together forming clusters and voids. A typical intergalacticdistance is on the order of 1 Mpc, so a cube withsides of 100 Mpc could have a million galaxies. Therefore,one should think of the galaxies as the ‘molecules’ of a gas,which is isotropic and homogeneous when a volume largeenough to contain large numbers of them is considered.If the universe is assumed to be isotropic and homogeneous,then the only possible motion is an overall expansionor contraction. Consider, for example, two randomly chosengalaxies at distances r(t) and R(t) from ours, as shown inFig. 8.2.An isotropic and homogeneous expansion (or contraction)means that the ratioχ = r(t)/R(t) (8.8)is constant in time. Therefore, r(t) = χR(t) andṙ = χṘ = Ṙ r ≡ H(t)r , (8.9)Rwhere dots indicate derivatives with respect to time. The ratioH(t) = Ṙ/R (8.10)is called the Hubble parameter. It is the fractional change inthe distance between any pair of galaxies per unit time. H isoften called the expansion rate of the universe.Equation (8.9) is exactly Hubble’s law, where H(t) atthe present time is identified as the Hubble constant H 0 .Sothe hypothesis of an isotropic and homogeneous expansionexplains why the speed with which a galaxy moves awayfrom us is proportional to its distance.