Master's Thesis in Theoretical Physics - Universiteit Utrecht

Here, H 0 = H(t = 0) and we have chosen a(t = 0) ≡ a 0 = 1. Note that in the limit where ɛ → 0(de Sitter inflation), the scale factor is a(t) = e H 0t and H(t) = H 0 .Now we switch to conformal time by substituting dt = a(η)dη in the metric, such that theline element is given byds 2 = a 2 (η)(−dη 2 + d⃗x · d⃗x) = a 2 (η)η µν dx µ dx ν . (2.19)Here η is conformal time and η µν = (−1,1,1,1) is the Minkowski metric. This conformallyflat FLRW metric simplifies calculations for the curvature invariants such as the Ricci tensorand Ricci scalar, as is shown in Appendix B.2. We now want to find solutions for thescale factor and Hubble rate for constant ɛ in conformal time. We note the useful relationsȧ = a′a′a, H(η) = and Ḣ = H′a 2 a, where a prime denotes a derivative with respect to η. Thisgives us the following differential equation for a(η)ɛ = − a′′ a(a ′ + 2. (2.20))2Since the relation between real and conformal time is dη = dta(t), i.e. an integral relationbetween η and t, we have a freedom in choosing the zero of conformal time and also of theboundary conditions for a(η). This means we can choose some conformal time η 0 where wehave the boundary conditions a(η 0 ) = 1 and H(η 0 ) = H 0 . We can choose this η 0 such that thesolutions of Eq. (2.20) area(η) =1H 0[−(1 − ɛ)H0 η ] , H(η) =1[1−ɛ−(1 − ɛ)H0 η ] . (2.21)−ɛ1−ɛThis choice reduces for ɛ = 0 to H(η) = H 0 and a(η) = − 1H 0, which is the conformal scaleηfactor in de Sitter space. Also, the universe is expanding (both a and H positive) for eitherɛ < 1 and −∞ < η < 0, or for ɛ > 1 and 0 < η < ∞. A useful relation iswhich gives the simple expression for the scale factor1H(η)a(η) = −(1 − ɛ)η , (2.22)1a(η) = −Hη(1 − ɛ) . (2.23)For ɛ ≪ 1 we are in the so-called quasi de Sitter space, which reduces to de Sitter space inthe limit where ɛ → 0.2.3 Einstein equations from an action principleThe Einstein equations (2.1) can also be derived from an action. We therefore introduce theEinstein-Hilbert action,∫S EH = d D x R−g , (2.24)16πG Nwhere g = det(g µν ) and R is the Ricci scalar defined in Eq. (B.7). We now vary this actionwith respect to the metric g µν . We use thatδ −g = − 1 2−ggµν δg µν , (2.25)