Phase Diagram for Nflation
aa r X i v : . [ h e p - t h ] M a r Phase Diagram for Nflation
Iftikhar Ahmad a , Yun-Song Piao a and Cong-Feng Qiao a,b a ) College of Physical Sciences,Graduate University of CAS, YuQuan Road 19A, Beijing 100049, China b ) Theoretical Physics Center for Science Facilities (TPCSF), CAS., Beijing 100049, China
Recently, it was showed that there is a large N phase transition in Nflation, in which when thenumber of fields is large enough, the slow roll inflation phase will disappear. In this brief report, weillustrate 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.
PACS numbers:
I. INTRODUCTION
Recently, Dimopoulos et.al [1] showed that the manymassive axion fields predicted by string vacuum can becombined and lead to a radiatively stable inflation, calledNflation, which is an interesting implement of assisted in-flation mechanism proposed by Liddle et.al [2], see alsoRefs. [3, 4] for many studies, and a feasible embeddingof inflation in string theory. Then Easther and McAl-lister found [5] that for the mass distribution followingMarˇ c enko-Pastur law, the spectral index of scalar per-turbation is always redder than that of its correspondingsingle field. However, this result is actually valid for anymass contribution and initial condition of fields, as hasbeen shown in [6, 7] numerically and in [8] analytically.In addition, it was found for Nflation that the ratio oftensor to scalar is always same as in the single field case[9] and the non-Gaussianity is small [10, 11], see alsoRefs.[12] for relevant studies.In inflation, when the value of field increases up tosome value, the quantum fluctuation of field will be ex-pected to overwhelm its classical evolution. In this case,the inflaton field will undergo a kind of random walk,which will lead to the production of many new regionswith different energy densities. This was called as eter-nal inflation [13, 14]. In principle, it was thought thatdependent on the value of field, there are generally threedifferent phases in single field inflation, i.e. eternal infla-tion phase, slow roll inflation phase and fast roll phase,which should be also valid for Nflation.However, recently, it was found [15] that when thenumber of fields is large enough, the slow roll inflationphase will disappear, which means there exists a large Ntransition for Nflation. The reason is, though the endvalue of slow roll inflation decreases with the increase ofnumber N of fields, the value separating the slow rollinflation phase and the eternal inflation phase, hereaftercalled as the eternal inflation boundary for convenience,decreases more rapidly, thus they will cross inevitably atsome value of N , after this the slow roll inflation phasewill go out of sight. This result means there is a boundfor the number of fields driving the slow roll Nflation.This is also consistent with recent arguments from blackhole physics [16, 17], in which there exists a gravitationalcutoff, whose value equals to our bound, beyond which the quantum gravity effect will become important, seealso Refs. [18, 19] for some similar bounds.In single field inflation, when the inflaton field is in itseternal inflation boundary the primordial density pertur-bation δρ/ρ ∼
1, thus it will be hardly possible for us toreceive 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 relationbetween the entropy and the total efolding number [21],in which when δρ/ρ ∼
1, the entropy in unit of efoldingnumber is less than one, which means we can not obtainany information. Thus it is significant to examine howabove results change for Nflation, especially what occursaround its phase transition point. It can be expectedthat there maybe more general and interesting results.In this paper, we will firstly illustrate the phase diagramfor Nflation, and then give relevant discussions.
II. PHASE DIAGRAM FOR NFLATION
In the Nflation model, the inflation is driven by manymassive fields. For simplicity, we assume that the massesof all fields are equal, i.e. m i = m , and also φ i = φ , whichwill also be implemented in next section. Following Ref.[15], the end value of slow roll inflation phase and theeternal inflation boundary with respect to N are givenby φ ≃ M p √ N , (1) φ ≃ N / r M p m , (2)respectively. It can be noticed that the end value goesalong with √ N , it decreases slower than the eternal in-flation boundary with N , since the latter changes with N / . Thus when we plot the lines of the end value andthe eternal inflation boundary moving with respect to N ,respectively, there must be a point where these two linescross, see Fig.1. This crossing point is N ≃ M p m , (3) φ∼ 1Ν (M /m) p 2 N φ m eternal inflation critical point φ∼ 1Ν fast roll slow roll ∆ N N FIG. 1: The φ − N phase diagram for Nflation. The uppersolid line is the eternal inflation boundary and the lower solidline is the end value of the slow roll inflation. These two linessplit the region into three phases, i.e. eternal inflation phase,slow roll inflation phase and fast roll phase. There is a criticalpoint, beyond which the slow roll inflation phase disappears. beyond which the slow roll inflation phase will disappear.Thus here we call this point as the critical point. It seemsbe expected that after the critical point is got across, theline denoting the eternal inflation boundary will not ex-tend downwards any more, the line left is that denotingthe end value, which still obeys Eq.(1), see the dashedline of Fig.1. The reason is the calculation of the eternalinflation boundary is based on the slow roll approxima-tion, while below the end value the slow rolling of field isactually replaced by the fast rolling, in this case the quan-tum fluctuation is actually suppressed, thus it is hardlypossible that the quantum fluctuation of field will over-whelm its classical evolution. However, the case maybenot so simple. In next section, we will see there is an en-tropy bound for the number of fields, and at the criticalpoint this bound is saturated. This means that beyondthe critical point our above semiclassical arguments cannot be applied. Thus in this sense in principle what isthe diagram beyond the critical point remains open.The value of fields at critical point can be obtained bysubstituting Eq.(3) into any one of Eqs.(1) and (2), whichis φ ≃ m . This indicates that if initially φ < m , no mat-ter what N is, the slow roll inflation will not occur. Theexistence of slow roll inflation is important for solving theproblems of standard cosmology and generating the pri-mordial perturbation seeding large scale structures. Inthe phase diagram Fig.1, we can see that the slow roll in-flation phase is in a limited region, which means in orderto make Nflation responsible for our observable universe,the relevant parameters must be placed suitably.We assume that all mass are equal only for simplic-ity. For the case that not all mass are equal, the resultis also similar, as has been shown in Ref. [15], in which the mass distribution following Marˇ c enko-Pastur law [5]is taken for calculations. Thus the phase diagram is stillFig.1, the only slight difference is replacing m with theaverage mass ¯ m . It should be noted that here in thephase diagram the number N of fields dose not includemassless scalar fields. The reason is when the massesof fields are negligible, they will not affect the motion ofmassive fields dominating the evolution of universe, whilethe perturbations used to calculate the quantum jump offields are those along the trajectory of fields space, sincethe massless fields only provide the entropy perturbationsorthogonal to the trajectory, which thus are not consid-ered in the calculations deducing Eqs.(1) and (2). Thusif there are some nearly massless fields and some massivefields with nearly same order, it should be that there isa bound N . M p / ¯ m , in which only massive fields areincluded in the definition of ¯ m and N . III. DISCUSSIONA. On primordial density perturbation at theeternal inflation boundary
In single field inflation, when the inflaton field is inits eternal inflation boundary, the primordial perturba-tion δρ/ρ ∼
1. The primordial density perturbationduring Nflation can be calculated by using the formulaof Sasaki and Stewart [22]. In slow roll approximation, (cid:16) δρρ (cid:17) ∼ m N φ M p [6, 23]. The motion of the eternal in-flation boundary obeys Eq.(2). Thus substituting Eq.(2)into it and then cancelling the variable φ , we can obtain δρρ ≃ √ N , (4)where the factor with order one has been neglected, whichhereafter will also be implemented. We can see that δρ/ρ is decreased with respect to the increase of N , and foreach value of N , δρ/ρ is always less than one. This resultis obviously different from that of single field. The rea-son leading to this result is, in single field inflation theeternal inflation boundary and the point that the den-sity perturbation equals to one are same, however, thechanges of both with N are different, one is ∼ /N / and the other is ∼ / √ N . Intuitively, the eternal infla-tion means that the quantum fluctuations of fields leadto the production of many new regions with different en-ergy densities, thus it seems that when we approach theeternal inflation boundary the density perturbation willbe expected to near one. Thus in this sense our resultlooks like unintuitive. However, in fact what the eter-nal inflation phase means should be a phase in which thequantum fluctuation of field overwhelms its classical evo-lution, which is not certain to suggest that the densityperturbation is about one.Thus different from single field inflation, in which weare impossible to receive the information from the eter-nal inflation phase since in that time the black hole hasswallowed us due to the primordial density perturbationwith near one, it seems that when N is large, we may ob-tain some information from the eternal inflation phase,at least in principle we can obtain those from the bound-ary of eternal inflation phase. Beyond this boundary, thefields are walked randomly, thus the slow roll approxi-mation is broken and the results based on the slow rollapproximation are not robust any more. In principle, forthe eternal inflation phase of Nflation we need to calcu-late the density perturbation in a new way to know howmuch it is actually, which, however, has been beyond ourcapability. The eternal inflation phase for single fieldshas been studied by using the stochastic approach [24]. B. On entropy bound
The entropy during Nflation can be approximatelygiven by dS entropy S ∼ M p H . Here we regard S as theentropy at the eternal inflation boundary. Thus we have S ∼ M p H ∼ M p N m φ ∼ √ N M p m , (5)where Eq.(2) has been used. It is interesting to find that S is proportional to √ N , which means the entropy in-creases with the number of field. Here the case is slightlysimilar to that of the entanglement entropy for a blackhole, in which there seems be a dependence of the entan-glement entropy on N , which conflicts the usual result ofblack hole entropy, since each of fields equally contributesto the entropy [17]. However, this problem may be solvedby invoking the correct gravity cutoff Λ ∼ M p √ N [16], ashas 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 ∼ M p m , which is just the result for single field,i.e. S ∼ M p m at the eternal inflation boundary. Thus itseems that the argument in Ref. [17] is universal for therelevant issues involving N species.It can be noticed that the efolding number N ∼ Nφ M 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 par-alleling the φ axis in Fig.1, N obtained will be the totalefolding number along corresponding line in slow roll in-flation phase, hereafter called ∆ N , see Fig.1. Thus withEq.(2), we can have ∆ N ∼ M p m √ N . Then we substitute itinto Eq.(5), and thus for the eternal inflation boundary,we have N · ∆ N ≃ S, (6)which is a general entropy bound including N , and is alsoour main result. It means that below the eternal inflationboundary, we have the bound N · ∆ N . S . This resultindicates that for fixed N , i.e. along the line parallelingthe 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, thenumber N of fields is bounded by S , and at the eternalinflation boundary, the entropy bound is saturated.There are two special cases, corresponding to the re-gions around red points in Fig.1. For details, one is thatfor 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. Bymaking the derivatives of N and S with respect to thetime, respectively, we can have d N dS ≃ M p m S , (7)where S is the function of φ , see the second equation inEq.(5), and thus can be used to cancel φ . By integratingthis equation along the line paralleling the φ axis in Fig.1,where the lower limit is the eternal inflation boundaryand the upper limit is the end value of slow roll infla-tion phase, and then applying approximation condition φ e ≪ φ , where φ and φ e represent the values of eternalinflation boundary and the end of slow roll inflation, re-spectively, which actually implies that S e ≫ S and thus( S e − S ) /S e ≃
1, we have∆
N ≃ ( δρρ ) S, (8)where (cid:16) δρρ (cid:17) ∼ M p m S has been applied, which can beobtained since both δρρ 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 singlefield inflation, δρρ ∼ N . S for slow roll phase, i.e. thetotal efolding number is bounded by the entropy, which issaturated 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 fromthat in Ref. [25]. Thus combining Eqs.(4) and (8), wecan find Eq.(6) again. This also indicates the result ofEq.(4) is reliable.The other is that for N being near its critical point,in which approximately we have ∆ N ≃
1, thus we canobtain N ≃ S , i.e. S is saturated by the number N offields. This can also be seen by combining Eq.(3) for thecritical point and Eq.(5), in which we can find S ≃ N atthe critical point.Thus below the critical point, N . S . From Eq.(5), S ∼ √ N M p m & N can be obtained. This means N . ( M p m ) . In Refs. [16], it was argued that M p is renor-malized in the presence of N fields at scale m so that M p & N m , in other words, N > ( M p m ) is inconsistent.Here, if N > ( M p m ) , then combining it and Eq.(5), wewill have N > S , i.e. the number N of fields is largerthan the dS entropy of critical point. This is certainlyimpossible, since intuitively it may be thought that thereis at least a freedom degree for each field, thus the to-tal freedom degree of N fields system, i.e. the entropy,should be at least N , while dS entropy is the maximalentropy of a system. Thus we arrive at same conclusionwith Ref. [16] from a different viewpoint. This againshows the consistence of our result. Acknowledgments
We thank Y.F. Cai for helpfuldiscussions and comment. I.A thanks the support of(HEC) Pakistan. This work is supported in part byNSFC under Grant No: 10491306, 10521003, 10775179,10405029, 10775180, in part by the Scientific ResearchFund of GUCAS(NO.055101BM03), in part by CAS un-der Grant No: KJCX3-SYW-N2. [1] S. Dimopoulos, S. Kachru, J. McGreevy and J.G.Wacker, arXiv:hep-th/0507205.[2] A.R. Liddle, A. Mazumdar and F.E. Schunck, Phys. Rev.
D58 , 061301 (1998).[3] E.J. Copeland, A. Mazumdar and N.J. Nunes, Phys. Rev.
D60 , 083506 (1999); A.M. Green and J.E. Lidsey, Phys.Rev.
D61 , 067301 (2000); K.A. Malik and D. Wands,Phys. Rev.
D59 , 123501 (1999); A. Mazumdar, S. Pandaand A. Perez-Lorenzana, Nucl. Phys.
B614 , 101 (2001).[4] Y.S. Piao, R.G Cai, X. Zhang and Y.Z. Zhang Phys.Rev.
D66 , 121301 (2002);R. Brandenberger, P. Hoand H. Kao, JCAP , 011 (2004) ; M. Majum-dar and A.C. Davis Phys. Rev.
D69 , 103504 (2004);J. Ward, Phys. Rev.
D73 , 026004 (2006); H. Singh,arXiv:hep-th/0608032; A. Jokinen and A. Mazumdar,Phys. Lett.
B597 , 222 (2004); K. Becker, M. Becker andA. Krause, Nucl. phys.
B715 (2005) 349-371; J.M. Clineand H. Stoica, Phys. Rev.
D72 , 126004 (2005).[5] R. Easther and L. McAllister, JCAP , (2006) 018.[6] S.A. Kim and A.R. Liddle, Phys. Rev.
D74 , 023513(2006).[7] S.A. Kim and A.R. Liddle, arXiv:0707.1982.[8] Y.S. Piao, Phys. Rev.
D74 , (2006) 047302.[9] L. Alabidi, and D.H. Lyth, JCAP , 016 (2006).[10] S.A. Kim and A.R. Liddle, Phys. Rev.
D74 , 063522(2006).[11] D. Battefeld and T. Battefeld, JCAP , 012 (2007).[12] D. Seery, J.E. Lidsey and M.S. Sloth, JCAP , 027(2007); D. Seery and J.E. Lidsey, JCAP , 008(2007); T. Battefeld and R. Easther, JCAP , 020 (2007); J.O. Gong, Phys. Rev.
D75 , 043502 (2007); M.E.Olsson, JCAP , 019 (2007); K.L. Panigrahi and H.Singh, arXiv:0708.1679; J. Ward, arXiv:0711.0760.[13] A. Vilenkin, Phys. Rev.
D27 , 2848 (1983).[14] A.D. Linde, Phys. Lett.
B175 , 395 (1986).[15] I. Ahmad, Y.S. Piao and C.F. Qiao, JCAP , 023(2008).[16] G. Dvali, arXiv:0706.2050; G. Dvali and M. Redi,arXiv:0710.4344.[17] G. Dvali, S.N. Solodukhin, arXiv:0806.3976.[18] Q.G. Huang, Phys. Rev.
D77 , 105029 (2008).[19] L. Leblond and S. Shandera, JCAP , 007 (2008).[20] R. Bousso, B. Freivogel and I.S. Yang, Phys. Rev.
D74 (2006) 103516.[21] N. Arkani-Hamed, S. Dubovsky, A. Nicolis, E.Trincheriniand G. Villadoro, JHEP (2007) 055.[22] M.Sasaki and E.D. stewart, Prog. Theor. Phys. , 71.[23] D.H. Lyth and A. Riotto, Phys. Rep. , 1 (1999).[24] A.A. Starobinsky, in “Current Topics in Field The-ory, Quantum Gravity and Strings,” edited by H.J. deVega and N. Sanchez, Lecture Notes in Physics, Vol.26(Springer, Heidelberg, 1986), 107; A.S. Goncharov, A.D.Linde, V.F. Mukhanov, Int. J. Mod. Phys. A2 , 561(1987); A.D. Linde, “Particle Physics and InflationaryCosmology”, (Harwood, Chur, Switzerland, 1990); Con-temp. Concepts Phys. , 1 (2005), arXiv:hep-th/0503203;A.D. Linde, Nucl. Phys. B372 , 421 (1992).[25] Q.G. Huang, M. Li and Y. Wang, JCAP0709