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 N ≡ ln a ( t$_{end}$ ) a ( t ) = ∫ t$_{end}$ t Hdt ≈ 1 M 2 P ∫ φ φ$_{end}$ V V ′ dφ , (5) 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] N ( k ) = 62 − ln k a$_{0}$H$_{0}$ − ln 10 16 GeV V 1 / 4 k + ln V 1 / 4 k V 1 / 4 end − 1 3 ln V 1 / 4 end ρ 1 / 4 rh , (6) 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$_{L}$ ≤ 4 A phase of fast roll inflation prior to a slow roll phase of inflation has been invoked in order to suppress the power spectrum on large scales [152–154]. 5 A more rigorous set of parameters are presented describing slow-roll inflation in Eq. (41), which is inde- pendent of slow-roll conditions or the number of fields. 6 For the particular scale of today’s Hubble length, we define: N$_{Q}$ ≡ N ( k = a$_{0}$H$_{0}$ ). The corresponding slow-roll parameters at that scale will be denoted by ϵ$_{Q}$, η$_{Q}$ . 13