FIG. 3: In a cyclic universe, the sketch of ln ( 1 a | h $_{|}$) vs. ln( t ) for the case that there is a period of inflation before bounce in j cycle. The blue lines denote the evolutions of perturbation modes, and the red regions denote the contracting phases in each cycle. The dashed line denotes the inflation, and the following solid line denotes the evolution after inflation. 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 $^{6}$. 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 FIG. 4: The Penrose diagram of a cyclic multiverse. This is obtained by gluing the corresponding parts of diagrams of contracting universe, expanding universe and dS universe, where the dashed lines denoting the turnaround surface and the bounce surface have been signed respectively. Before each turnaround surface, the universe will experience the contrac- tion, bounce, expansion successively and then enter into a period of dark energy domination. After this, the parent universe is separated into lots of parts independent of one another, each of which corresponds to a new universe and repeats above evolution. In principle, since the experience of each universe after the proliferation is generally different, they will not be expected to be synchronous in cycling. This means that generally when we are in a period of dark energy domination, it is possible that there are many other universes which are in the period of contraction or bounce or others. 6 The inflation might occur after the bounce in some cycles, see Fig.3, since the energy scale of bounce is generally high, it can be expected that after the bounce the field might land on a higher plain of effective potential. In this case, ‘b’ mode will be pushed outside of observable universe by the inflation in j cycle, and the primordial perturbations suitable for observable universe, such as ‘c’ and ‘d’ modes, can emerge during inflation, which are in- duced by the quantum fluctuation of inflaton field. The inflation after bounce has been originally studied in Refs. [33, 34], in which the imprint of bounce on cosmic microwave background has been pointed out, see also for latest studies [35],[36]. Here for cyclic multiverse, the occurrence of inflation has similar effect as that of a period of dark energy domination. However, both can be different in the theoretical model building and observational signals, which might be interesting for studying. 7 However, in slow roll inflation, the multiverse can be also induced by the classical rolling of inflaton along a web of branches of its effective potential [42]. 4 ln 1 a|h| ln(t) t$_{0}$ a b cycle cycle perturbation modes c d contraction expansion r = 0 evolution of  1 a|h| k mode bounce surface turnaround  surface