(ed.). Gravitational waves (IOP, 2001)(422s).

Appendix C. The string cosmology equations 331For this background:Ɣ 0i j = H i δ ji , Ɣ ij 0 = a i ȧ i δ ij , R 0 0 =− ∑ i(Ḣ i + H 2i ),R j i =−Ḣ i δ ji− H i δ ji(∇φ) 2 = ˙φ 2 ,∑H k ,kR =− ∑ i∇ 2 φ = ¨φ + ∑ iH i ˙φ,(2Ḣ i + H 2i ) − ( ∑i∇ 0 ∇ 0 φ = ¨φ,H i) 2,∇ i ∇ j φ = H i ˙φδ ji . (16.198)The dilaton equation (16.193) gives then2 ¨φ + 2 ˙φ ∑ iH i − ˙φ 2 − ∑ i(2Ḣ i + H 2i ) − ( ∑iH i) 2= 0. (16.199)The (00) component of the equation (16.189) gives˙φ 2 − 2 ˙φ ∑ iH i − ∑ i( ∑Hi 2 +iH i) 2= e φ ρ. (16.200)The diagonal, spatial components (i, i) of equation (16.189) (the off-diagonalcomponents are trivially satisfied) give∑∑Ḣ i + H i H k − H i ˙φ − 1 2(2Ḣ i + Hi 2 )ki− 1 2( ∑iH i) 2− 1 2 ˙φ 2 + ¨φ + ˙φ ∑ iH i = 1 2 eφ p i . (16.201)The last five terms on the left-hand side add to zero because of the dilaton equation(16.199), and the spatial equations reduce to(Ḣ i − H i˙φ − ∑ )H k = 1 2 eφ p i . (16.202)kThe above equations are clearly invariant under time-reversal, t →−t. Inorder to make explicit also their duality invariance, let us introduce again theshifted dilaton (see equation (16.146)), such thate φ = e φ / √ −g,˙φ = ˙φ − ∑ iH i , (16.203)and defineρ = ρ √ −g = ρ ∏ ia i ,p = p √ −g = p ∏ ia i . (16.204)