Master's Thesis in Theoretical Physics - Universiteit Utrecht

Table 2.1: Scaling of the energy density ρ, scale factor a, Hubble parameter H and thedeceleration parameter ɛ in different era.Era ω ρ a H ɛGeneral ω ∝ a −3(1+ω) ∝ t23(1+ω)23(1+ω)tMatter 0 ∝ a −3 ∝ t 2 321Radiation3∝ a −4 ∝ t 1 2de Sitter inflation −1 constant ∝ e H 0tH 0 = constant 032(1 + ω)323t12t2There is a third equation which is due to the fact that the Einstein tensor is covariantly conserved,i.e. ∇ µ G µν = 0, where ∇ µ is the covariant derivative defined in Eq. (B.2) . Thereforewe also have that ∇ µ T µν = 0, and the zeroth component gives0 = ∇ µ T µ0= g µλ (∂ λ T µ0 − Γ σ µλ T σ0 − Γ σ λ0 T µσ= −∂ 0 ρ − ȧa gi j g i j (ρ + p)= − ˙ρ − 3 ȧ (ρ + p) (2.13)aAn important remark is that the conservation of the stress-energy tensor implies that theEinstein equations are not independent. We have derived three equations (Eqs. (2.11),(2.12) and (2.13)), but only two are independent. We now assume that our perfect fluidobeys the equation of statep = ωρ, (2.14)where ω is a constant. For a matter dominated universe ω = 0, for a radiation dominateduniverse ω = 1 3and for an inflationary universe ω ≃ −1. The equation of state allows us tosolve Eq. (2.13) for ω ≠ −1, and we getwhich gives for the scale factor when we solve Eq. (2.6)Now we plug this in Eq. (2.10) to find that ɛ is a constant,ρ ∝ a −3(1+ω) , (2.15)2a ∝ t 3(1+ω) . (2.16)ɛ = 3 (1 + ω). (2.17)2We see that for a matter dominated universe ɛ = 3 2and for a radiation dominated universeɛ = 2, so in both cases the expansion of the universe decelerates. For a general inflationaryuniverse, ω ≃ −1 and ɛ ≪ 1. For future references we will now summarize the above in Table2.1The fact that ɛ is constant allows us to solve Eq. (2.10) for both H(t) and a(t), and wecan write the solutions asa(t) = (1 + ɛH 0 t) 1 ɛ , H(t) =H 01 + ɛH 0 t . (2.18)