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

330 Elementary introduction to pre-big bang cosmologywhose variation with respect to g exactly cancels the contribution of the totaldivergence appearing in the last integral of equation (16.188):∫ √|g|e∫ √|g|eδ −φ g K α d α =−φ (g µν ∇ α δg µν −∇ ν δg να ) d α . (16.191)Here K α is a geometric term representing the so-called extrinsic curvature on thed-dimensional closed hypersurface, of infinitesimal area d α , bounding the totalspacetime volume over which we are varying the action. Note that the integral(16.190) differs from the usual boundary term, used in general relativity [89]to derive the Einstein equations, only by the presence of the tree-level dilatoncoupling e −φ to the extrinsic curvature.Let us now perform the variation with respect to the dilaton, again in units2λ d−1s= 1. We get the Euler–Lagrange equations:∂ µ [−2 √ −ge −φ ∂ µ φ] = e −φ√ −g[R + (∇φ) 2 −12 1 H 2 + V ]− e −φ√ −gV ′ + √ −gσ (16.192)(where V ′ = ∂V/∂φ), from whichR + 2∇ 2 φ − (∇φ) 2 + V − V ′ − 1 12 H 2 + e φ σ = 0. (16.193)The variation with respect to B µν ,δ B∫d d+1 x √ |g|e −φ (∂ µ B να )H µνα = 0, (16.194)gives finally∂ µ ( √ |g|e −φ H µνα ) = 0 =∇ µ (e −φ H µνα ). (16.195)Equations (16.189), (16.193) and (16.195) are the equations governingthe evolution of the string cosmology background, at low energy. Note thatequation (16.189) can also be given in a simplified form: if we eliminatethe scalar curvature present inside the Einstein tensor, by using the dilatonequation (16.193), we obtain:R ν µ +∇ µ ∇ ν φ − 1 2 δν µ V ′ − 1 4 H µαβ H ναβ = 1 2 eφ (T ν µ − δµ ν σ). (16.196)For the purpose of these lectures, it will be enough to derive some simplesolution of the string cosmology equations in the absence of the potential (V = 0),of the antisymmetric tensor (B = 0), and with a perfect fluid, minimally coupledto the dilaton (σ = 0), as the matter sources. Assuming for the background aBianchi I type metric, we can work in the synchronous gauge, by settingg µν = diag(1, −ai 2 δ ij), a i = a i (t), φ = φ(t),T ν µ = diag(ρ, −pi 2 δ ji ), p i/ρ = γ i = constant, ρ = ρ(t). (16.197)