1. We can assume instant reheating after the last phase of inflation, which yields, ρ$_{L,end}$ = ρ$_{rh,L}$ . Concentrating on the present horizon scale N$_{Q}$ ≡ N$_{I}$ ( a$_{0}$H$_{0}$ ), we obtain [160] One can impose new constraints arising from the fact that there should be no reprocessing of modes in between the two phases of inflation on the scales probed by CMB experiments. This requirement constraints [160] Hence, in general we obtain the following set of constraints [160]: In the special case n$_{i}$ = 1, it is easy to see that these constraints are trivially satisfied ( ∑ $_{j}$N$_{j}$ > 0). The details of the thermal history of the universe determine the precise number of e-foldings required to solve the horizon problem, but for most practical purposes (for high scale inflation with large reheating temperature) it is sufficient to assume that N$_{Q}$ ≈ 50 − 60, keeping all the uncertainties such as the scale of inflation and the end of inflation within a margin of 10 e-foldings. A significant modification can take place only if there is an epoch of late inflation such as thermal inflation [161, 162], or in theories with a low quantum gravity scale [163–165], or if there are phase transitions during inflation [166–168]. Initially, the theory of cosmological perturbations has been developed in the context of FRW cosmology [169], and for models of inflation in [24, 170–184]. For a complete review on this topic, see [12]. For a real single scalar field there arise only adiabatic density perturbations. In case of several fluctuating fields there will in general also be isocurvature perturbations. Irrespective of the nature of perturbations, any light scalar field obtains quantum fluctuations in a de Sitter space, which has many applications. B. Primordial density perturbations L ∑ i =1 N$_{i}$ = 62 + 1 4 ln ( V 2 I V$_{L}$M 4 P ) + L − 1 ∑ i =1 n$_{i}$ 2 ln ( V$_{i}$$_{+1}$ V$_{i}$ ) . (7) ln ( 1 a$_{k}$$_{c}$H$_{i}$ ) > ln ( 1 a$_{rh,i}$H$_{i}$$_{+1}$ ) , i = 2 , ..., L − 1 (8) N$_{Q}$ > 6 . 9 + 1 − n$_{1}$ 2 ln ( V$_{I}$ V$_{II}$ ) , i ∑ j =2 N$_{j}$ > 1 2 ln ( V$_{II}$ V$_{i}$$_{+1}$ ) − i ∑ j =2 n$_{j}$ 2 ln ( V$_{i}$ V$_{i}$$_{+1}$ ) , i = 2 , ..., L − 1 . (9) 14