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

Kinematics: shrinking horizons 289state with decreasing curvature is associated a dual partner with growingcurvature. On the other hand, our universe, at present, is in the standard postbigbang phase, with decreasing curvature. If we believe that duality has to beimplemented, even approximately, in the course of the cosmological evolution,we can then predict the existence of a phase, in the past, characterized by growingcurvature and by a typical pre-big bang evolution.16.3 Kinematics: shrinking horizonsIf we accept, at least as a working hypothesis, the possibility that our universe hadin the past a ‘dual’ complement, with growing curvature, we are led to the secondof the three questions listed in section 16.1: is the kinematics of the pre-big bangphase still appropriate to solve the problems of the standard inflationary scenario?The answer is positive, but in a non-trivial way.Consider, for instance, the present cosmological phase. Today the dilaton isexpected to be constant, and the universe should be appropriately described byEinstein equations. The gravitational part of such equations contains two types ofterms: terms controlling the geometric curvature of a space-like section, evolvingin time like a −2 , and terms controlling the gravitational kinetic energy, i.e. thespacetime curvature scale, evolving like H 2 . According to present observationsthe spatial curvature term is non-dominant, i.e.r = a−2H 2 ∼ spatial curvatureº 1. (16.18)spacetime curvatureAccording to the standard cosmological solutions, on the other hand, theabove ratio must grow in time. In fact, by putting a ∼ t β ,r ∼ȧ −2 ∼ t 2(1−β) , (16.19)so that r keeps growing both in the matter-dominated (β = 2/3) and in theradiation-dominated (β = 1/2) era. Thus, as we go back in time, r becomessmaller and smaller, and when we set initial conditions (for instance, at the Planckscale) we have to impose an enormous fine tuning of the spatial curvature term,with respect to the other terms of the cosmological equations. This is the so-calledflatness problem.The problem can be solved if we introduce in the past a phase (usually calledinflation), during which the value of r was decreasing, for a time long enough tocompensate the subsequent growth during the phase of standard evolution. It isimportant to stress that this requirement, in general, can be implemented by twophysically different classes of backgrounds.Consider for simplicity a power-law evolution of the scale factor in cosmictime, with a power β, so that the time-dependence of r is the one given inequation (16.19). The two possible classes of backgrounds corresponding to adecreasing r are then the following: