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

Kinematics: shrinking horizons 293standardevolutionade SitterinflationCONSTANT HORIZONFigure 16.4. Qualitative evolution of the Hubble horizon (broken curve) and of the scalefactor (full curve) in the standard inflationary scenario.In all cases the proper size d e (t) evolves in time like the so-called Hubble horizon(i.e. the inverse of the Hubble parameter), and then like the inverse of the curvaturescale. The size of the horizon is thus constant or growing in standard inflation(class I), decreasing in pre-big bang inflation (class II), both in the S-frame andin the E-frame.Such an important difference is clearly illustrated in figures 16.4 and 16.5,where the broken curves represent the evolution of the horizon, the full curves theevolution of the scale factor. The shaded area at time t 0 represents the portion ofuniverse inside our present Hubble radius. As we go back in time, according tothe standard scenario, the horizon shrinks linearly, (H −1 ∼ t), but the decreaseof the scale factor is slower so that, at the beginning of the phase of standardevolution (t = t 1 ), we end up with a causal horizon much smaller than the portionof universe that we presently observe. This is the well-known ‘horizon problem’of the standard scenario.In figure 16.4 the phase of standard evolution is preceded in time by a phaseof standard de Sitter inflation. Going back in time, for t < t 1 , the scale factorkeeps shrinking, and our portion of universe ‘re-enters’ inside the Hubble radiusduring a phase of constant (or slightly growing in time) horizon.In figure 16.5 the standard evolution is preceded in time by a phase of prebigbang inflation, with growing curvature. The universe ‘re-enters’ the Hubbleradius during a phase of shrinking horizon. To emphasize the difference, we haveplotted the evolution of the scale factor both in the expanding S-frame, a(t), andin the contracting E-frame, ã(t). Unlike in standard inflation, the proper size of