content_image/374839.jpg content_image/374841.jpg
in hybrid inflation models [147], where inflation comes to an end via a phase transition, or in oscillatory models of inflation where slow-roll conditions are satisfied only on average [148, 149], or inflation happens on average over every oscillations of a bouncing universe [150], or in fast roll inflation where the slow-roll conditions are never met [151] $^{4}$. The K-inflation where only the kinetic term dominates where there is no potential at all [155]. 

One of the salient features of the slow-roll inflation is that there exists a late time attractor behavior $^{5}$. This means that during inflation the evolution of a scalar field at a given field value has to be independent of the initial conditions. Therefore slow-roll inflation should provide an attractor behavior which at late times leads to an identical field evolution in the phase space irrespective of the initial conditions [156]. In fact the slow-roll solution does not give an exact attractor solution to the full equation of motion but is nevertheless a fairly good approximation [156]. A similar statement has been proven for multi-field exponential potentials without slow-roll conditions (i.e. assisted inflation) [157]. 

The standard definition of the number of e-foldings between time, t , and the end of inflation, t$_{end}$ , is given by 

<img src='content_image/374839.jpg'>

where φ$_{end}$ is defined by ϵ ( φ$_{end}$ ) ∼ 1, provided inflation comes to an end via a violation of the slow-roll conditions. The number of e-foldings can be related to the Hubble crossing mode k = a$_{k}$H$_{k}$ by comparing with the present Hubble length a$_{0}$H$_{0}$ . The final result is [158, 159] 

<img src='content_image/374841.jpg'>

where the subscripts end ( rh ) refer to the end of inflation (end of reheating) $^{6}$. 

A simple generalization of the above formula has been derived in [160], where there are mutiple stages of inflation, with potentials V$_{I}$, V$_{i}$, V$_{L}$ one after the other separated by the matter epochs, whose expansions are parameterized by t n 1 , t n 2 , ..., t $^{n$_{L}$}$, with n$_{1}$, n$_{2}$, ..., n$_{L}$ ≤ 

