Astroparticle Physics

200 9 The Early Universewhich relates the time derivative of the energy density ϱ andthe pressure P . One can suppose that ϱ is dominated by radiation,so that the equation of state (9.21) can be used,P = ϱ 3 . (9.27)fluid equationradial dependence of ϱSubstituting this into the fluid equation (9.26) gives˙ϱ + 4ϱṘR = 0 . (9.28)The left-hand side is proportional to a total derivative, so onecan write1R 4 ddt(ϱR 4) = 0 . (9.29)This implies that ϱR 4 is constant in time, and thereforeϱ ∼ 1 R 4 . (9.30)If, instead, one would have assumed that ϱ was dominatedby non-relativistic matter, one would have used the equationof state P = 0, and in a similar way one would have found(see Problem 1 in this chapter)ϱ ∼ 1 R 3 . (9.31)In either case the dependence of ϱ on R is such that forvery early times, that is, for sufficiently small R, theterm8πGϱ/3 on the right-hand side of the Friedmann equationwill be much larger than k/R 2 . One can then ignore thecurvature parameter and effectively set k = 0; this is def-initely valid at the very early times that will be consideredin this chapter and it is still a good approximation today, 14billion years later. The Friedmann equation then becomescurvature parametermodified FriedmannequationṘ 2R 2 = 8π Gϱ . (9.32)3In the following (9.32) will be solved for the case whereϱ is radiation dominated. One can write (9.30) as( ) 4 R0ϱ = ϱ 0 , (9.33)Rwhere here ϱ 0 and R 0 represent the values of ϱ and R atsome particular (early) time. One can guess a solution of