it should be noticed that when P 1 / 2 ζ ( ji ) approaches 1, the enhancement of nonlinear effect will make the required N j less. Thus this actually means that nearly at the beginning time of j cycle, the curvature perturbation on super horizon scale will have the amplitude be in order one. This will lead to the density perturbation is obtained, where h ∼ ( ah ) 3 ∼ k 3 for w ≃ 0 has been applied, and the subscript ‘ je ’ denotes the end time of the contracting phase in j cycle. We can see that if P 1 / 2 i ∗ ∼ 10 − $^{5}$, P 1 / 2 je will be larger than that induced by the quantum fluctuation of background field in j cycle since $^{k$_{j}$}$∗ ′ ≪ k$_{je}$ and $^{a$_{i}$}$∗ ≃ $^{a$_{j}$}$∗ ′ , and also the spectrum is quite red since P 1 / 2 je ∼ $_{k}$− 3 j ∗ ′ . This result implies that there can hardly an observable universe after the bounce of j cycle. In addition, it can be noticed that the amplitudes of these modes may be also amplified up to order one, i.e. after the contraction lasts some times, P 1 / 2 j ∼ 1. In this case, the universe will be possibly split into smaller and smaller fragments, which will renders the cycle ultimately end [14], see also [32]. Thus in this sense the feasibility of such a cyclic multiverse scenario is questionable. The perturbation modes that enter into the horizon during the expansion of i cycle and then leave it during the contraction of j cycle are baneful. Their amplitudes can hardly be suppressed by certain mechanism. Thus a reasonable solution is to push these baneful modes to larger scale, i.e. outside of observable universe in cor- responding cycle. This can be implemented by intro- ducing a period of accelerated expansion in each cycle. This period can be set before the turnaround of each cycle. In this case, there will be a period of dark en- ergy domination in corresponding cycle, hereafter the universe collapses and the next cycle begins. The per- turbation modes that enter into the horizon during the expansion of i cycle will possibly leave it during the dark energy domination of i cycle, but not during the con- traction of j cycle. Thus the amplitude of these modes will not increase, up to the turnaround. In this case, P 1 / 2 je = ( h$_{je}$ h 0 )( a$_{i}$$_{∗}$ a 0 )( a 0 a$_{j}$$_{∗}$ $_{′}$) P 1 / 2 j ∗ ′ , where t$_{0}$ in Fig.2 is the time when the period of dark energy domination begins, thus Eq.(6) is changed as on corresponding super horizon scale. In this case, it is obviously impossible that the different regions of global universe will evolve synchronously, even if they are syn- chronous in i cycle. This means that the global universe at the beginning time of j cycle will be inevitably sep- arated into many different parts, each of which actually corresponds to a new universe and will evolve indepen- dently of one another, up to succedent cycle $^{3}$. Therefore, it is evident that the global universe will proliferate af- ter each cycle. In this sense, a cyclic multiverse actually comes into being. However, inside each new universe, generally the am- plitude of primordial perturbation, see ‘b’ mode in Fig.1, is contributed by not only that of perturbations induced by the quantum fluctuation of background field in j cy- cle, but also that of perturbations that enter into the horizon during the expansion of i cycle and then leave it during the contraction of j cycle $^{4}$. In general, the latter amplitude can be larger, and dependent of the expansion behavior and the matter contents in i cycle, its spectrum is also not scale invariant. This can be seen as follows. When $^{k}$∗η ≃ 1 during i cycle, $^{k}$∗ mode just enters into the horizon. Hereafter, its evolution obeys Eq.(1), but the term z ′′ z is negligible, since k ≫ ah . In this case, u$_{k}$ is approximately constant, up to the time when the corre- sponding mode leaves the horizon $^{5}$, which occurs during the contraction in j cycle. Thus we have | $^{u$_{i}$}$∗ | = | $^{u$_{j}$}$∗ ′ | , where ∗ ′ is the time when the corresponding mode leaves the horizon in j cycle, which implies P 1 / 2 j ∗ ′ = ( a$_{i}$$_{∗}$ a$_{j}$$_{∗}$ $_{′}$) P 1 / 2 i ∗ . However, after this mode leaves the horizon in j cycle, its amplitude will increase like Eq.(4). Thus where a ∼ ( ah ) n n$_{−}$ 1 ∼ k n n$_{−}$ 1 has been applied, which gives that for n = 2 $_{3}$, a ∼ 1 k 2 and for n ≫ 1, a ∼ k . k$_{je}$/k$_{j}$$_{0}$ denotes the efolding number of primordial per- turbation contributed by the contraction in j cycle, i.e. N j = ln ( k$_{je}$ k$_{j}$ $_{0}$). The larger it is, the larger the ampli- tude of perturbation is amplified during the contraction is. This point is essentially the same as that obtained in Eq.(4). While k$_{0}$$^{/k$_{i}$}$∗ is the ratio of the wavenumber of the mode that corresponds to the beginning time of dark energy domination to that of the given mode, which ac- tually corresponds to the efolding number that the dark energy phase lasts before the given mode leaves the hori- zon during the dark energy domination of i cycle, and is an exponentially suppressed factor to the amplitude P 1 / 2 je . This result means that the period of dark energy domination not only helps to the continuance of cycling but also the emergence of observable universe. This design can be explained detailed in Fig.2. ‘a’ 3 It has been argued that if a region with super horizon scale has the density perturbation on corresponding scale larger than 1, such a region will correspond to a separated close universe [31]. In principle, the initial conditions of each local universe can be obtained by rescaling the background of parent universe. 4 We thank R. Brandenberger for talking this point to us. 5 We neglected the effect of the transfer function on the perturba- tion spectrum after the corresponding perturbation mode enters into the horizon, which is dependent of the matter content in cor- responding cycle. However, the behaviors of spectrum discussed here are not altered qualitatively P 1 / 2 je = ( h$_{je}$ $^{h$_{j}$}$∗ ′ )$_{P}$ 1 / 2 j ∗ ′ = ( k$_{je}$ $^{k$_{j}$}$∗ ′ ) $^{3}$( $^{a$_{i}$}$∗ $^{a$_{j}$}$∗ ′ )$_{P}$ 1 / 2 i ∗ (6) P 1 / 2 je = ( k$_{je}$ k$_{j}$$_{0}$ ) $^{3}$( k$_{0}$ k$_{i}$ $_{∗}$) 3 P 1 / 2 i ∗ , (7) δρ ρ ∼ P 1 / 2 ζ ( ji ) ∼ 1 (5) 3