Fig. 5.30 In this picture the plot of the function f (t) defined in (5.8.45) is presented for two different solutions of the same equations (5.8.40) corresponding to different boundary conditions. In the first plot where the initial conditions saturate the bound (5.8.43) we clearly see the slow role phase characterized by f (t) 1. The second plot where the initial conditions violate in a severe way the bound (5.8.43), f (t) is never approximately constant and equal to one. No slow roll phase is therefore present

5.8.2.1 Number of e-Folds

The primary motivation to consider the inflationary scenario is the solution of the paradox provided by (5.7.6). If in that equation we regard t0 not as our present time rather as the time at which inflation ended, then the paradox can be solved in case the exponential increase of the scale factor is of order 1060. The success of an inflationary model can be measured by estimating the so called number of e-folds:

where aend is the scale factor at the end of inflation and ainit is the scale factor at its beginning. The number of e-folds necessary to solve the horizon problem is Ne 60–70.