Phase Diagram for Nflation

Recently, it was showed that there is a large N phase transition in Nflation, in which when the number of fields is large enough, the slow roll inflation phase will disappear. In this brief report, we illustrate the phase diagram for Nflation, and discuss the entropy bound and some relevant results. It is found that near the critical point the number of fields saturates dS entropy.

the quantum gravity effect will become important, see also Refs. [18,19] for some similar bounds.
In single field inflation, when the inflaton field is in its eternal inflation boundary the primordial density perturbation δρ/ρ ∼ 1, thus it will be hardly possible for us to receive the information from the eternal inflation phase, since in that time we will be swallowed by black hole [20]. This result may be actually assured by a relation between the entropy and the total efolding number [21], in which when δρ/ρ ∼ 1, the entropy in unit of efolding number is less than one, which means we can not obtain any information. Thus it is significant to examine how above results change for Nflation, especially what occurs around its phase transition point. It can be expected that there maybe more general and interesting results. In this paper, we will firstly illustrate the phase diagram for Nflation, and then give relevant discussions.

II. PHASE DIAGRAM FOR NFLATION
In the Nflation model, the inflation is driven by many massive fields. For simplicity, we assume that the masses of all fields are equal, i.e. m i = m, and also φ i = φ, which will also be implemented in next section. Following Ref. [15], the end value of slow roll inflation phase and the eternal inflation boundary with respect to N are given by respectively. It can be noticed that the end value goes along with 1 √ N , it decreases slower than the eternal inflation boundary with N , since the latter changes with 1 N 3/4 . Thus when we plot the lines of the end value and the eternal inflation boundary moving with respect to N , respectively, there must be a point where these two lines cross, see Fig.1. This crossing point is beyond which the slow roll inflation phase will disappear. Thus here we call this point as the critical point. It seems be expected that after the critical point is got across, the line denoting the eternal inflation boundary will not extend downwards any more, the line left is that denoting the end value, which still obeys Eq.(1), see the dashed line of Fig.1. The reason is the calculation of the eternal inflation boundary is based on the slow roll approximation, while below the end value the slow rolling of field is actually replaced by the fast rolling, in this case the quantum fluctuation is actually suppressed, thus it is hardly possible that the quantum fluctuation of field will overwhelm its classical evolution. However, the case maybe not so simple. In next section, we will see there is an entropy bound for the number of fields, and at the critical point this bound is saturated. This means that beyond the critical point our above semiclassical arguments can not be applied.  1) and (2), which is φ ≃ m. This indicates that if initially φ < m, no matter what N is, the slow roll inflation will not occur. The existence of slow roll inflation is important for solving the problems of standard cosmology and generating the primordial perturbation seeding large scale structures. In the phase diagram Fig.1, we can see that the slow roll inflation phase is in a limited region, which means in order to make Nflation responsible for our observable universe, the relevant parameters must be placed suitably.
We assume that all mass are equal only for simplicity. For the case that not all mass are equal, the result is also similar, as has been shown in Ref. [15], in which the mass distribution following Marčenko-Pastur law [5] is taken for calculations. Thus the phase diagram is still Fig.1, the only slight difference is replacing m with the average massm. It should be noted that here in the phase diagram the number N of fields dose not include massless scalar fields. The reason is when the masses of fields are negligible, they will not affect the motion of massive fields dominating the evolution of universe, while the perturbations used to calculate the quantum jump of fields are those along the trajectory of fields space, since the massless fields only provide the entropy perturbations orthogonal to the trajectory, which thus are not considered in the calculations deducing Eqs. (1) and (2). Thus if there are some nearly massless fields and some massive fields with nearly same order, it should be that there is a bound N M 2 p /m 2 , in which only massive fields are included in the definition ofm and N .

A. On primordial density perturbation at the eternal inflation boundary
In single field inflation, when the inflaton field is in its eternal inflation boundary, the primordial perturbation δρ/ρ ∼ 1. The primordial density perturbation during Nflation can be calculated by using the formula of Sasaki and Stewart [22]. In slow roll approximation, where the factor with order one has been neglected, which hereafter will also be implemented. We can see that δρ/ρ is decreased with respect to the increase of N , and for each value of N , δρ/ρ is always less than one. This result is obviously different from that of single field. The reason leading to this result is, in single field inflation the eternal inflation boundary and the point that the density perturbation equals to one are same, however, the changes of both with N are different, one is ∼ 1/N 3/4 and the other is ∼ 1/ √ N . Intuitively, the eternal inflation means that the quantum fluctuations of fields lead to the production of many new regions with different energy densities, thus it seems that when we approach the eternal inflation boundary the density perturbation will be expected to near one. Thus in this sense our result looks like unintuitive. However, in fact what the eternal inflation phase means should be a phase in which the quantum fluctuation of field overwhelms its classical evolution, which is not certain to suggest that the density perturbation is about one.
Thus different from single field inflation, in which we are impossible to receive the information from the eter-nal inflation phase since in that time the black hole has swallowed us due to the primordial density perturbation with near one, it seems that when N is large, we may obtain some information from the eternal inflation phase, at least in principle we can obtain those from the boundary of eternal inflation phase. Beyond this boundary, the fields are walked randomly, thus the slow roll approximation is broken and the results based on the slow roll approximation are not robust any more. In principle, for the eternal inflation phase of Nflation we need to calculate the density perturbation in a new way to know how much it is actually, which, however, has been beyond our capability. The eternal inflation phase for single fields has been studied by using the stochastic approach [24].

B. On entropy bound
The entropy during Nflation can be approximately Here we regard S as the entropy at the eternal inflation boundary. Thus we have where Eq.(2) has been used. It is interesting to find that S is proportional to √ N , which means the entropy increases with the number of field. Here the case is slightly similar to that of the entanglement entropy for a black hole, in which there seems be a dependence of the entanglement entropy on N , which conflicts the usual result of black hole entropy, since each of fields equally contributes to the entropy [17]. However, this problem may be solved by invoking the correct gravity cutoff Λ ∼ Mp √ N [16], as has been argued in Ref. [17]. In Eq.(5) if we replace M p with a same gravity cutoff Λ, then we will obtain S ∼ √ N Λ m ∼ Mp m , which is just the result for single field, i.e. S ∼ Mp m at the eternal inflation boundary. Thus it seems that the argument in Ref. [17] is universal for the relevant issues involving N species.
It can be noticed that the efolding number N ∼ N φ 2 M 2 p . For initial φ being in its eternal inflation boundary, where φ is given by Eq.(2), for fixed N , i.e. along the line paralleling the φ axis in Fig.1, N obtained will be the total efolding number along corresponding line in slow roll inflation phase, hereafter called ∆N , see Fig.1 which is a general entropy bound including N , and is also our main result. It means that below the eternal inflation boundary, we have the bound N · ∆N S. This result indicates that for fixed N , i.e. along the line paralleling the φ axis in Fig.1, the total efolding number ∆N of slow roll inflation phase is bounded by S, while for fixed ∆N , i.e. along the line paralleling the N axis in Fig.1, the number N of fields is bounded by S, and at the eternal inflation boundary, the entropy bound is saturated.
There are two special cases, corresponding to the regions around red points in Fig.1. For details, one is that for N = 1, i.e. single field, we have ∆N ≃ S from Eq.(6), thus the result for single field is recovered [21]. Following [21] to large N, Eq.(6) can be actually also deduced. By making the derivatives of N and S with respect to the time, respectively, we can have where S is the function of φ, see the second equation in Eq. (5), and thus can be used to cancel φ. By integrating this equation along the line paralleling the φ axis in Fig.1, where the lower limit is the eternal inflation boundary and the upper limit is the end value of slow roll inflation phase, and then applying approximation condition φ e ≪ φ, where φ and φ e represent the values of eternal inflation boundary and the end of slow roll inflation, respectively, which actually implies that S e ≫ S and thus (S e − S)/S e ≃ 1, we have where δρ ρ 2 ∼ M 2 p m 2 S 2 has been applied, which can be obtained since both δρ ρ and S are the functions of φ. This result has been showed in Ref. [21] for single field, however, since Eq.(8) is independent on the number N of fields, thus it is still valid for N fields. For single field inflation, δρ ρ ∼ 1 only at eternal inflation boundary, thus we always have ∆N S for slow roll phase, i.e. the total efolding number is bounded by the entropy, which is saturated at eternal inflation boundary. Note that Eq. (8) is an integral result in which δρ/ρ with the change of φ and thus S is condidered, which is slightly different from that in Ref. [25]. Thus combining Eqs. (4) and (8), we can find Eq.(6) again. This also indicates the result of Eq.(4) is reliable.
The other is that for N being near its critical point, in which approximately we have ∆N ≃ 1, thus we can obtain N ≃ S, i.e. S is saturated by the number N of fields. This can also be seen by combining Eq. Here, if N > ( Mp m ) 2 , then combining it and Eq.(5), we will have N > S, i.e. the number N of fields is larger than the dS entropy of critical point. This is certainly impossible, since intuitively it may be thought that there is at least a freedom degree for each field, thus the total freedom degree of N fields system, i.e. the entropy, should be at least N , while dS entropy is the maximal entropy of a system. Thus we arrive at same conclusion with Ref. [16] from a different viewpoint. This again shows the consistence of our result.