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

String cosmology 22913.3 String cosmologyThe basic mechanism of generation of relic gravitational waves in cosmology hasbeen discussed in the preceding section about inflation where the main conceptshave been exposed. The crucial point of the outcome was the flatness of thespectrum which, combined with the COBE bound at very low frequency, givesa very strong constraint on the spectrum even at high frequency. To satisfy theCOBE bound and still have a chance of being observable at VIRGO or LIGO thespectrum must grow with frequency. A spectrum of this kind has been found in acosmological model suggested by string theory [48].In string theory the fundamental objects are one-dimensional extendedentities, i.e. strings. Their fundamental excitation of given energy and angularmomentum are particles of given mass and spin. String theory has one onlyfundamental (dimensionful) constant: the string tension T which can be tradedfor a fundamental length λ s ≡ √¯hc/T . The mass scale of the excitation of thestring is therefore √ T which, as string theory includes gravity, should be near thePlanck mass 2 . The gauge couplings are not constant (neither at classical level)but depend on the expectation value of a scalar field, the dilaton. For example,Newton’s gravitational constant G is given byG ∼ λ s28π eφ ,while all the gauge couplings are proportional to e φ/2 . In the regime where allcouplings and derivatives are small string physics can be suitably described by afield theory action which describes the dynamics of the light (massless) fields ofthe string (the graviton g µν and the dilaton φ), which isS =− 1 ∫2λ 2 d 4 x √ −g[e −φ (Ê + ∂ µ φ∂ µ φ) + higher derivatives]s+ [higher order in e φ ]. (13.30)where g = det ‖g µν ‖, and Ê is the Ricci scalar. Higher derivative termsare relevant whenever spacetime derivatives become of the order of one (in λ sunits), whereas the higher orders in e φ acquire importance when e φ ∼ 1, thussignalling the beginning of a full stringy-quantum regime. The first terms withoutcorrections reproduce Einsteinian gravity for constant dilaton.To analyse a situation of cosmological interest it suffices to study spatiallyhomogeneous fields, i.e. φ = φ(t) and the metric is chosen so that the line elementds 2 can be written asds 2 = dt 2 − R 2 (t) d⃗x 2 , (13.31)2 This is not a compelling argument as models have been presented in which √ T can be consistentlyput some orders of magnitude below the Planck mass (see, e.g., [49]), but we will not treat thesemodels.