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

Appendix B. Duality symmetry 323we have (H i =ȧ i /a i ):By noting thatddt[2 e−φ(∇φ) 2 = ˙φ 2N 2 ,Ɣ 0 00 = ṄN ≡ F,R = 1 [N 2 2F ∑ i( d∏ )( ∑a kNk=1 i( d∏ )[= e−φa k 2 ∑ Nk=1 i√d∏−g = N a i ,Ɣ 0i j = H i δ ji ,H i − 2 ∑ iH i)]Ḣ i − ∑ ii=1Ɣ ij 0 = a iȧ iN 2 δ ij,( ∑) 2Hi 2 − H i]. (16.142)iḢ i − 2F ∑ H i − 2 ˙φ ∑ ( ∑) 2H i + 2 H i],iii(16.143)the action (16.140), modulo a total derivative, can be rewritten as:S =− 1 ∫ d∏2λ d−1 d d e −φ [x dt a i˙φ 2 − ∑ ( ∑) 2Hi 2 + H i − 2 ˙φ ∑ H i].sNi=1iii(16.144)We now introduce the so-called shifted dilaton φ, defined by∫ de −φ d xd∏= a i e −φ , (16.145)λ d sfrom whichφ = φ + ∑ ln a i , ˙φ = ˙φ + ∑ H i (16.146)ii(by assuming spatial sections of finite volume, ( ∫ d d x √ |g|) t=constant < ∞,we have absorbed into φ the constant shift − ln(λ −d ∫s d d x), required to securethe scalar behaviour of φ under coordinate reparametrizations preserving thecomoving gauge). The action becomes:S =− λ ∫ (sdt e−φ ∑)˙φ2 − Hi2 . (16.147)2 NBy inverting one of the d scale factors the corresponding Hubble parameterchanges sign,a i →ã i = ai −1 , H i → ˜H i = ˙ã i dai−1= a i =−H i , (16.148)ã i dti=1i