Natural double inflation - Occhionero, Franco; Amendola, Luca (Vistas in astronomy, 2002)
The astronomical interest of double inflation stems from the possibility it gives of inserting a feature in an otherwise featureless (or scale invariant) perturbation spectrum, precisely at the scale (100Mpc, say) that goes through the horizon at the (sharp) separations between the two successive inflations. Double inflation occurs when two scalar fields (or inflatons) dominate sequentially the cosmic expansion or when vacuum polarization — Ricci scalar R squared added to the Lagrangian — is taken into account and only one inflaton ψ is present. (This perhaps is more natural as it exploits quantum effects to reduce to one the number of the ad hoc ingredients.) In that case we know from Starobinsky's pioneering work that the first inflation is driven by R — rightly called then scalaron — under the rules of Fourth Order Gravity, (FOG), while the second is driven by ψ under the rules of ordinary General Relativity, (GR). Unfortunately most of the appeal of the scalaron-inflaton scenario in relation to the feature in the perturbation spectrum, is lost because a delicate fine tuning of the value of the (second) inflaton at the beginning of the second inflation is required, in the absence of which the two inflations merge in one and no scale is singled out. In order to overcome this difficulty, we introduce in the Lagrangian density a new scalar coupling between ψ and R2, analogous to the well known non minimal coupling between ψ and R of canonical GR. We show that in this way the two inflationary episodes of FOG and GR may be neatly distinguished from each other, regardless of the initial value of ψ. This is due to the influence of the coupling on the shape of the conformal potential, in which one can easily carve a channel of evolution, consisting in fact of two orthogonal valleys. Then, for most of phase space the attractor is this doubly inflationary trajectory that lies at the bottom of the two valleys (Fig. 1). In fact, in this case the Universe first rolls down to the bottom of the first valley where the inflaton takes the desired value, and then goes on the course sketched above, consisting of the FOG inflation driven by the scalaron and the GR inflation driven by the inflaton. Numerical examples are given (Fig.2). In conclusion it is pointed out that this engineering of the conformal potential may also be exploited, perhaps more interestingly, far beyond the original motivation of this work: for example, to generate bubbles of astrophysical interest in a sort of first-order inflation, which does not have the graceful exit problem of old inflation and has been accordingly nicknamed by us getaway inflation (Fig. 3).