# Design of a Cyclic Multiverse 

Yun-Song Piao 

College of Physical Sciences, Graduate School of Chinese Academy of Sciences, Beijing 100049, China 

Recently, it has been noticed that the amplification of the amplitude of curvature perturbation cycle by cycle can lead to a cyclic multiverse scenario, in which the number of universes increases cycle by cycle. However, this amplification will also inevitably induce either the ultimate end of corresponding cycle, or the resulting spectrum of perturbations inside corresponding universe is not scale invariant, which baffles the existence of observable universes. In this paper, we propose a design of a cyclic multiverse, in which the observable universe can emerges naturally. The significance of a long period of dark energy before the turnaround of each cycle for this implementing is shown. 

Recently, a cosmological cyclic scenario, in which the universe experiences the periodic sequence of contrac- tions and expansions [1], has been rewaked [2], and brought the distinct insights into the origin of observ- able universe. There has been lots of studies for cyclic or oscillating universe [3],[4],[5],[6],[7],[8],[9],[10],[11],[12], also [13] for a review. However, the global configuration of cyclic universe is actually more complex than imag- ined. In cyclic universe the amplitude of the curvature perturbation on super horizon scale is generally increas- ing during the contraction of each cycle, while is nearly constant during the expansion of each cycle. Thus the net result is that the amplitude of perturbation is ampli- fied, which occurs cycle by cycle, 

Recently, it has been argued that the amplification of the amplitude of curvature perturbation cycle by cycle can lead to a cyclic multiverse scenario [14] $^{1}$, in which the universe proliferates, i.e. the number of universes increases cycle by cycle. There this was illustrated by including a contracting phase with w ≃ 0 in each cycle of a cycle universe. In principle, since shortly after the end of each cycle the curvature perturbation on super hori- zon scale can be amplified to order one, the universe will be inevitably separated into lots of parts independent of one another after each cycle, each of which actually corre- sponds to a new universe and independently evolves up to succedent cycle, and then proliferates again $^{2}$. This result in some sense incorporates the second law of ther- modynamics in such a fashion that the increase of total entropy in consecutive cycles is explained as or replaced 

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with the increase of the number of new universes, which has been discussed in [14] in detailed. 

The cyclic multiverse scenario might be significant for understanding the origin of observable universe. How- ever, generally the amplitude of perturbation modes that enter into the horizon during the expansion in previous cycle and then leave it during the contraction in cur- rent cycle is generally larger than that induced by the quantum fluctuation of background field in current cy- cle. This will inevitably lead to either the ultimate end of corresponding cycle, or the resulting spectrum of pertur- bations inside each universe is not scale invariant even if they leave the horizon during the contraction with w ≃ 0, which baffles possible existence of observable universes in this scenario. We, in this paper, will discuss this problem in detail and then design a possible solution. 

We begin with an illustration of cyclic universe model and the review of the evolution of perturbation in a cyclic universe with the contraction with w ≃ 0. There have been lots of studies of bounce cosmology [13]. Recently, the nonsingular bounce has been implemented in non- local higher derivative theories of gravity [19], which is ghostfree, while here that provided by us is only a simple example serving the purpose of illustration. We intro- duce a normal field ϕ with its potential M $^{2}$ϕ2 − $^{Λ}$∗, and a field χ with negative ˙ χ $^{2}$, which plays a crucial role in giving a nonsingular bounce [20]. $^{Λ}$∗ is a small posi- tive constant. Thus the minimum of potential is neg- ative, which is responsible for the turnaround of cyclic universe. We show such an illustration of cyclic universe in Fig.1, in which M = 0 . $^{9, Λ}$∗ = 10 − $^{10}$, the initial value 10 $^{4}$ϕ$_{0}$ = 5 M and 10 $^{4}$χ$_{0}$ = 3 M are used and M$_{P}$ = 1 is set. It seems that this model suffers from the quantum instabilities and inconsistencies [19]. However, it can be expected that this effective description is only an approxi- mation of a fundamental theory with the UV completion below certain physical cutoff, while the appearance of ghost terms or the quantum instabilities is only an arte- fact of this approximation. In string theory, a slowly decaying D3 brane can be depicted by an open string tachyon mode, which is described within a open string field theory. This will bring a nonlocal polynomial inter- action. In this case, the behavior of open string tachyon can be effectively simulated by a ghost field, e.g.[21],[22], 

