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mode denotes the modes that leave the horizon in i cycle but can not enter into the horizon during the expansion of i cycle. These modes will destine to stay on super horizon scale all along up to j cycle. Their amplitudes will be inevitably amplified to order one around the be- ginning time of j cycle. This leads to that the universe is separated into lots of independent new universes, each of which will evolve independently up to the succedent cycle. ‘b’ mode denotes the modes that enters into the horizon during the expansion of i cycle and then leaves it during the contraction of j cycle. These modes gen- erally have amplitudes larger than that induced only by the fluctuation of background field in j cycle, which thus is baneful for the existence of observable universe in j cycle. The dashed line denotes the period of dark en- ergy domination, which can push those baneful modes, such as ‘b’ mode, outside of observable universe. In this case, ‘d’ mode will provide the primordial perturbation of observable universe, which is induced only by the quan- tum fluctuation of background field in j cycle, and its amplitude is given by (2) and thus can be suitable for observations $^{6}$. 

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The causal patch diagram of cyclic multiverse can be plotted in Fig.4, which is obtained by gluing the corre- sponding parts of diagrams of contracting phase, expand- ing phase and dS phase. This causal diagram is slightly similar to that of inflationary multiverse, e.g.[37]. The reason is that before the turnaround in each cycle the universe is in a dS phase. In some sense, such a pe- riod leads that the universe has an average positive en- ergy density for each cycle. In inflationary multiverse, the new universe is generated either in nucleated bubble, e.g. [38],[39], or in the region that the inflaton field is in stochastic walking [40],[41], both are induced by quantum effects $^{7}$. Here, however, the mechanism resulting in mul- tiverse is the amplification of the amplitude of curvature perturbation cycle by cycle, which thus occurs in classi- cal sense, though it initially originates from the quantum fluctuation of field. 

We can have a concrete implement to this multiverse scenario, like in [5]. In this implement, the potential of 

