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

References 333Finally, equation (16.207) provides the constraint2β − β 2 − dα 2 = 0. (16.214)We then have a system of two equations for the two parameters α, β (note that,if α is a solution for a given γ , then also −α is a solution, associated to −γ ).We have, in general, two solutions. The trivial flat space solution β = 2,α = 0,corresponds to dust matter (γ = 0) according to equation (16.206). For γ ̸= 0we obtain insteadα =2γ1 + dγ 2 , β = 21 + dγ 2 , (16.215)which fixes the time evolution of a and φ:2γ2a ∼ t 1+dγ 2 , φ =− ln t, (16.216)1 + dγ 2and also of the more conventional variables ρ,φ:ρ = ρa −d = ρ 0 a −d(1+γ) , φ = φ + d ln a =2(dγ − 1)ln t. (16.217)1 + dγ 2This particular solution reproduces the small curvature limit of the generalsolution with perfect fluid sources (see the last two papers of [8]), sufficientlyfar from the singularity. As in the vacuum solution (16.159) there are fourbranches, related by time-reversal and by the duality transformation (16.208), andcharacterized by the scale factorsa ± (±t) ∼ (±t) ±2γ/(1+dγ 2) . (16.218)The duality transformation that preserves φ and ρ, and inverts the scale factor,in this case is simply represented by the transformation γ → −γ . Considerfor instance the standard radiation-dominated solution, corresponding to d = 3,γ = 1/3, and t > 0, and associated to a constant dilaton, according toequation (16.217). A duality transformation gives a new solution with γ =−1/3,namely (from (16.216) and (16.217)):a ∼ t −1/2 , ρ ∼ a −2 , φ ∼−3lnt. (16.219)By performing an additional time reflection we then obtain the pre-big bangsolution ‘dual to radiation’, already reported in eq. (16.16).References[1] Weinberg S 1972 Gravitation and Cosmology (New York: Wiley)[2] Kolb E W and Turner M S 1990 The Early Universe (Redwood City, CA: Addison-Wesley)