Master's Thesis in Theoretical Physics - Universiteit Utrecht

Finally, we use the Bianchi identity∇ λ R ρασβ + ∇ ρ R αλσβ + ∇ α R λρσβ = 0,(B.8)and contract this whole expression twice (i.e. we multiply with g βα g σλ ) to find that∇ α (R αβ − 1 2 R g αβ) = 0,(B.9)which allows us to define the Einstein tensorG αβ = R αβ − 1 2 R g αβ.(B.10)B.2 General Relativity in a flat FLRW universeIn cosmology the universe is best described by an expanding Minkowski spacetime. Themetric that corresponds to such an expanding universe is the flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric,g µν (x) = diag(−1, a 2 , a 2 , a 2 ), a = a(t), (B.11)where a(t) is the scale factor. The line element then takes the formds 2 = g µν (x)dx µ dx ν = −dt 2 + a 2 (t)d⃗x · d⃗x.(B.12)We see that the spatial part of the line element scales with the scale factor a(t). This meansthat if a(t) grows in time, the universe expands. Using the FLRW metric we find that thenonvanishing elements of the Christoffel connection defined in Eq. (B.3) areΓ i j0= ȧa δi j = Hδi jwhere we have defined the Hubble parameterΓ 0 i j= ȧa g i j = Ha 2 δ i j ,H ≡ ȧa .(B.13)(B.14)(B.15)We can now also calculate the nonvanishing components of the Ricci tensor from Eq. (B.6),and in 4 dimensions they areThe Ricci scalar R in Eq. (B.7) is thenR 00 = −3 äa = −3(H2 + Ḣ)[ ( ) ä ȧ 2 ]R i j =a + 2 g i j = [ Ḣ + 3H 2] g i jaR = g µν R µν = 6The components of the Einstein curvature tensor are finally(B.16)[ ( ) ä ȧ 2 ]a + = 6(Ḣ + 2H 2 ). (B.17)a( ) ȧ 2G 00 = 3 = 3H 2a[G i j = − 2 ä ( ) ȧ 2 ]a + g i j = −(2Ḣ + 3H 2 )a 2 .a(B.18)