Master's Thesis in Theoretical Physics - Universiteit Utrecht

B.3 General Relativity in a conformal FLRW universeAll the above equations are valid in a regular FLRW spacetime. However, we can simplifyand generalize many of our equations by performing the conformal transformationdt = adη.(B.19)The metric now takes the simple formg µν = a 2 (η)diag(−1,1,1,1) = a 2 (η)η µν ,(B.20)where η µν is the Minkowski metric and η is conformal time. The Christoffel connectiontakes the simple formΓ α µν = a′ ()δ α µaδ0 ν + δα ν δ0 µ − δα 0 η µν , (B.21)where a ′ = da/dη, or explicitly,Γ 0 00= a′aΓ i j0= a′a δi jFor convenience, we now list certain relationsΓ 0 i j= − a′a η i j = a′a δ i j.ȧ = dadt = 1 daa dη = a′aä = dȧdt = 1 d(a ′ /a)= a′′a dη a 2 − (a′ ) 2a 3H = ȧa = a′Ḣ = 1 da dηa 2( a′a 2 )= a′′a 2 − 2 a′a 4 .(B.22)Note that the above connections are valid in D space time dimensions. The non-vanishingcomponents of the Ricci tensor can now easily be derived, and in D dimensions they areR αβ = a 2 ( H 2 (D − 1)η αβ + Ḣ(η αβ − (D − 2)δ 0 α δ0 β ) ). (B.23)Note that Ḣ isThe Ricci scalar R in Eq. (B.7) is thenḢ = d dt H = 1 addη H = 1 a H′ .R = H 2 (D + 2Ḣ)(D − 1).(B.24)(B.25)In 4 dimensions we again recover the Ricci scalar from Eq. (B.17). As a reference, we nowcalculate the components of the Ricci tensor and Ricci scalar in terms of the scale factor aand derivatives of the scale factor with respect to conformal time,[a′′( a′) ] 2R 00 = 3a − aR i j = −[a′′a + ( a′a) 2]η i j = −[a′′a + ( a′a) 2]δ i j .(B.26)