Ghost Inflation and De Sitter Entropy Monodromy Arthur Hebecker, Joerg Jaeckel, Fabrizio Rompineve Et Al

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This content was downloaded from IP address 86.27.89.240 on 24/03/2020 at 09:02 JCAP08(2016)002 hysics P ond law of black le ic t s of the early universe, d only black holes but also , ar niversity of Tokyo, i-Hamed, et al. is satisfied if rse (WPI), ology, r fields such as those in the Sciences (IPM), 10.1088/1475-7516/2016/08/002 ondensation. In particular we etter understanding of gravity, able assumption that couplings oretical Physics, e a notion of cosmological Page d if there remains a tiny positive doi: strop Rio Saitou . A b,c ion to the author(s) c,b . Content from this work may be used 3 osmology and and osmology [email protected] [email protected] Creative Commons Attribution 3.0 License , , Shinji Mukohyama, C a 1602.06511 inflation, initial conditions and eternal universe, physic

In the setup of ghost condensation model the generalized sec [email protected] rnal of rnal Article funded by SCOAP under the terms of the ou An IOP and SISSA journal An IOP and [email protected] School of Astronomy, Institute for Research in Fundamental P.O. Box 19395-5531, Artesh Highway,Center Tehran, Iran for Gravitational Physics,Kyoto Yukawa Institute University, for The 606-8502, Kyoto, Japan Kavli Institute for theThe Physics University and of Mathematics Tokyo of InstitutesKashiwa, the for Chiba Unive Advanced 277-8583, Study, Japan TheSchool U of Physics, HuazhongWuhan University 430074, of China Science and Techn E-mail: b c d a cosmological perturbation theory ArXiv ePrint: hole thermodynamics can bebetween respected the under a field radiativelyStandard responsible st Model for are ghostcosmology suppressed condensate are by expected and to the matte we play Planck important consider scale. roles a towards cosmological ourshall Since setup b show to not that test the theghost de inflation theory Sitter happened of entropy in ghost the boundcosmological early c proposed epoch constant by of in our Arkan the universetime an future after infinity. inflation. We then propos Keywords: Received March 2, 2016 Revised June 7, 2016 Accepted July 1, 2016 Published August 1, 2016 Abstract.

Sadra Jazayeri, J Ghost inflation and de Sitter entropy and Yota Watanabe and the title of the work, journal citation and DOI. Any further distribution of this work must maintain attribut JCAP08(2016)002 4 6 7 1 3 4 A 10 12 12 10 14 (1.2) (1.1) recise ]. It is the rst law re- 2 does not decrease; and , A d that the horizon area wn as black hole thermo- ynamic equilibrium are also imilarly to ordinary thermo- re of a black hole to be the oles that are expected to play k holes have only a small num- y radiation and hydrogen atoms ]: the zeroth law states that the ary thermodynamics [ 1 , B κ ~ πck 2 – 1 – = T + (rotation & charge terms) N πG κdA 8 in the space of stationary black holes to that of the horizon = bh Λ cannot become zero within finite affine time. M bh κ dM ] of thermal radiation from black holes that determined the p 3 of a stationary black hole is constant over the horizon; the fi κ is the Newton’s constant; the second law states that the N as G From the four laws of black hole thermodynamics it is expecte Black holes have properties similar to thermodynamics, kno A 4.1 Black hole4.2 evaporation and information Definition4.3 of cosmological Page time Cosmological Page time after ghost inflation 3.1 Ghost3.2 condensation Ghost3.3 inflation Lower bound on Ω or its increasing function plays the role of entropy in ordin the third law states that where 5 Summary and discussions 1 Introduction In the history of quantumplayed mechanics, important researches on roles. blackbod Itsimilar is roles probably towards our cosmology better and understanding black of h gravity. 4 Cosmological Page time 2 De Sitter entropy bound conjecture 3 Implications to ghost inflation Contents 1 Introduction lates variation of the mass surface gravity form of thetemperature black of hole Hawking entropy: radiation, by identifying the temperatu dynamics. Because ofber the no-hair of theorems, parameters stationaryparametrized while blac by a a non-gravitating smalldynamics, number system the of in black macroscopic hole thermod parameters. thermodynamics S has four laws [ Hawking’s discovery [ area JCAP08(2016)002 use (1.3) vanish S ) for the black hole 1.3 ond law of black hole ), quantum mechanics ing radiation and thus N econd law since a black G nd the entropy of matter tional interaction through ravity. In particular, it is mber of microscopic states ]. The point is that under ze again that it is possible ntioned at the beginning of iptions, the generalized sec- iven by ]. We then seek implications 9 in the presence of the ghost ng radiation, the generalized re the accretion of the ghost te is only responsible for the rresponding holographic dual since such interactions should e expected to play important ponsible for ghost condensate ent paper we thus consider a we briefly review the de Sitter , st condensation, which is the 14 ro by demanding that 2 8 at least. A specific interaction l from violating the generalized 2 Pl neralized second law is violated 2 / avity ( se. he SM fields to the field causing led by radiative corrections since roposals for violation of the gen- 1 . In particular, our main focus in l, but as far as the natrual setup ost condensate and the SM fields ], there are some proposals to vi- 2 Pl ) he formula ( /M 4 N 2 is the speed of light, one concludes /M c ]. However, it was later shown that if M πG 7 – , (8 5 / A N 3 G c = 1 , where ~ B 2 4 . Once we admit this at the classical level, the k Pl c – 2 – bh M = M S → ∞ 2 Pl ] is suppressed by M 9 ). For this reason, many people believe that the black hole , B 8 k 100GeV) is the scale of spontaneous Lorentz violation. Beca . ( M = 0. It is also the Hawking radiation that invalidates the sec The rest of the paper is organized as follows. In section In theories and states that allow for holographic dual descr A ) and statistical mechanics ( ~ natural assumption of the Planckiancontributions suppression from is graviton not loops spoi are suppressed by 1 that was assumed to exist in [ of the strong suppression,eralized the second physical processes law in arecondensate those too to p slow the toto black decrease violate hole the the increases generalized entropy it.ghost second befo condensate law We with by want strength directlyis to stronger coupling concerned emphasi than t we gravitationa steerthe away present from paper, such not couplings.roles only towards As black our already holes deepcosmological me but setup understanding also to of cosmology test gravity. ar the the theory present In of paper the ghost is condensation pres de Sitter entropy in inflationary univer from the first law that the entropy of the black hole should be g and the energy of the black hole to be disappear in the decoupling limit, olate the generalized secondwe law propose in weak the enoughand literature direct [ couplings the between Standard the Modelthe field minimal (SM) res coupling, fields then orsecond it if is law possible they in to have athe prevent only the natural natural gravita mode effective assumption field thatgravity sector, theory the all (EFT) field the setup causingshould interactions be [ ghost between suppressed the condensa by field the Planck for scale, gh entropy bound conjecture proposed by Arkani-Hamed, et al. [ entropy, known as Bekenstein-Hawking formula, involves gr for up to addition of an arbitrary constant, which canthermodynamics: be set to a ze blackthe hole horizon area may may losesecond decrease. its law Taking then mass into account statesoutside due the the that to Hawki black the the hole sum should Hawk not of decrease. the Intriguingly, black t hole entropy a condensate, where ( entropy should be a key towards our better understanding of g expected that theassociated entropy with of the a black black hole. hole should reflect the nu ond law ishole expected is to probably dual bein to dual a a to thermal certain state. setup thedescription then Therefore, ordinary in if it thermodynamic the the probablysimplest ge s conventional implies realization sense. that of there the In is Higgs no the mechanism co context for of gravity [ gho JCAP08(2016)002 is ion, with δρ (2.5) (2.2) (2.3) (2.1) (2.4) (2.6) , where tot N xtended to is understood A we propose a notion of pace is unstable on the is the Hubble expansion 4 ), where erated during the period H 1.3 al inflation. For this reason, ( no post-inflationary observer . In particular, we show that Sitter horizons have properties is the energy density and ction . 3 . onally collapse to a black hole ry epoch is written in the form ρ h by simply replacing 2 g horizon has temperature given ˙ , , φ end H n driven by a canonical scalar field, 2 ), where − ∼ < S Hdt π  . ˙ 2 , φ (2 . = ρ δφ ˙ δρ φ 1 N  H H/ is bounded from above as end . dN S ∼ ∼ = πG beginning 1 and thus tot ]. In the following we thus consider a de Sitter 4 2 . S dS ˙ dN N T – 3 – φ 4 . − 13 ]. Again by adopting the natural unit, we obtain 2 − | Hδt , as obs H < π = 11 8 N ∼ 0 12 ˙ end [ obs H S δρ/ρ = a | S δa N e . 1 is devoted to a summary of the paper and some discussions. ∼ dS − dN 5 tot H ρ ]. By applying this to the de Sitter case and adopting the δρ N 10 = 1), we obtain ) [ are the de Sitter entropy at the end and the beginning of inflat B ) can be violated if we include the regime of eternal inflation k 1.2 2.5 ) was derived for non-eternal slow-roll inflation, it can be e = ~ 2.5 beginning = S ). c 2 H and N is the number of e-foldings defined by G end ( N S becomes of order unity or larger. However, perturbation gen The inequality ( In quantum gravity, however, it is believed that a de Sitter s π/ = respectively. perturbation of the energy density estimated as meaning that the total number of e-folding Here, as the area of the cosmological horizon [ where of the de Sitter entropy bound on ghost inflation in section Excluding eternal inflation, we have general slow-roll inflation with or without the eternal epoc the de Sitter entropycosmological bound Page time. is satisfied Section in our universe. In se S we have δρ/ρ of eternal inflationlater is at highly the time nonlinearwho of and can the observe would horizon the reentry. gravitati while modes Hence the generated there during bound would the ( be epoch of etern natural unit ( space as a part of slow-roll inflation. For slow-roll inflatio 2 De Sitter entropyIt bound has been conjecture known that notsimilar only to black hole thermodynamics. horizons butby also Indeed, de the any same bifurcating formula Killin ( rate. Also, theof Friedmann the equation first in law a with slow-roll the de inflationa Sitter entropy given by the formula Poincar´erecurrence time scale This is rewritten as the observable number of e-foldings JCAP08(2016)002 ty , as and (3.1) end X S = 0 and l possible es from a ′ V epending on ectured that s mechanism, let ], can we stabilize so that the origin s a result, the gauge ··· (1) numerical factors, ··· ]. This is the de Sitter + O 2 + 14 [ ls of inflation that violate r of spontaneous symmetry , X he de Sitter entropy bound X etry breaking is the vacuum  nded from above by 4 + ) is that the logarithm of the is the inverse of the spacetime ecting ··· f the force law in the infrared. ations from inflation. M est in the sense that the number w ghost condensation and then r the gauge group of interest. It ernal inflation without violation X variant under a constant shift of 2.6 ion, the simplest implementation µν − + due to a non-trivial potential such − g Sitter entropy bound conjecture to [ 2 4 ) = . It is thus supposed that the system φ M i φ  φ L = µ )( is roughly understood as the observable ∂ , where φ h X φ ( L obs ν Q N is some mass scale introduced to make 3 . If we can trust the effective field theory (EFT) φ∂ = 0 fluctuation is unstable due to the existence e = 0 there are stable minima where µ – 4 – i M )+ end i 6 ∂ Φ S Φ h 3 X µν h ( e starts with g = 0? The answer turns out to be yes. However, since P ) holds in many models of inflation that satisfy the null i 6 − φ  φ g = µ 2.6 − ∂ h X √ ]. It is therefore important to identify the regime of validi = 14 φ , be less than L obs through N 3 φ become as important as the linear term, we should consider al ) was derived for slow-roll inflation in the above, it was conj ) states that the observable number of different Hubble volum e X 2.6 2.6 , at least in a range of field space. Let us then consider terms d const. . In the same spirit as in the tachyon instability for the Higg + µν φ g 0. The system is thus stabilized at those stable minima and, a = 0 is unstable due to a ghost. Here The de Sitter entropy bound ( The bound ( In ghost condensation, on the other hand, the order paramete → i > φ φ dimensionless. By adding nonlinear terms as µ ′′ , ∂ ˙ breaking is the vev of derivative of a scalar field, enjoys the shift symmetry, i.e.φ the action of the system is in the first derivative of h metric φ far as the EFT is a valid description of cosmological perturb While the bound ( us suppose that the Lagrangian of entropy bound conjecture. the null energy condition, asof well the as null those energy of condition false [ vacuum et energy condition. However, the bound may be violated in mode the same bound should hold in other models of inflation as well observable number of independent modes from inflation be bou of the de Sitter entropy bound conjecture. 3 Implications to ghostIn inflation this section let usghost investigate inflation, the which implications is of inflationof the driven analogue de by of the ghost Higgs condensat of mechanism in Nambu-Goldstone gravity. (NG) It boson isconjecture the is in simpl only the one. contextghost Before of inflation. studying ghost t inflation, we briefly revie 3.1 Ghost condensation In Higgs mechanism, theexpectation order value parameter (vev) of of spontaneous ais symm scalar therefore supposed field, that say a Φ,as non-vanishing charged vev a unde is double-well developed potential. At the origin single Hubble patch, of a tachyon, but away from the origin based on the semi-classical approximation, the system away from the origin V number of independent modes from inflation. Therefore, negl an intuitive interpretation of the de Sitter entropy bound ( symmetry is broken spontaneously, leading to modification o terms consistent with the symmetry as terms nonlinear in JCAP08(2016)002 0 and → = 0, (3.6) (3.5) (3.3) (3.4) (3.2) ) = 0 ′ ′ . The π 0 0 2˙ P P X X ( − , agreeing P = 00 µν h g e derivatives, X ) 0 ther terms such , X ) in a flat FLRW ( with .  as well as the shift i X P ij ( ξ φ j ··· = 0 and timelike and K → ∂ i 6 gP + φ φ + − ν µ →− ij j and ∂ √ , starting at the quadratic ξ φ K h i φ∂ he equations of motion, the s residual symmetry. Start- imply introducing a Nambu- o that ∂ 00 ij µ = inkowski or de Sitter. We call ∂ h ons that they are irrelevant at , ′ sion to de Sitter background is φ K + pending on whether ariance, i.e. the symmetry under ) ion is 2 y effective field theory describing P L i e hypersurfaces at the linearized ij 2 0 . ) = 0, meaning that either h ˙ is. We thus need to form a linear M ) has a non-zero root α h + φ ′ j 2 is invariant under the residual sym- X ∂ kl → µν P ( ′ − 3 h 2 00 − ij a 2 ij P . h Pg ( j t h ) 0 K ∂ = jl h 2 i δ , h 1 t, ~x ∂ i ( µν ik M α ′ ξ ) is δ 2 T ), is − t 0 ~x ( ). Therefore, the system at the attractor coupled – 5 – ∂ ij 0 ). The latter case is not allowed if the system in- − φ (unitary gauge) t, ~x → h + 2 nor X ( 0 ) = ( i ~ ξ ~x ∂ j 0 . Having constructed the general action in the unitary ( 00 P 0 φ t, h + → ∞ 2 1 h h kl − i ( ~x a = . Concretely, by replacing 0 K → 1 8 = h π i jl φ , the transformation under the residual symmetry transfor- = 0. The system is thus driven to the attractor with  → 0 ij δ ij 4 + i ) components that is invariant under the residual symmetry δ µν ~x ik t φ K h ij δ M µ ∂ , h + → = + h t 00 µν ij h η EH K ) and ( L i = → = 0, provided that the function = 00 µν i 6 and g h φ eff ij µ L ∂ . One might worry about the existence of terms with higher tim K h 2 0. Similar conclusion can be easily drawn in the presence of o ij ) δ φ <  = ) 0 as the universe expands ( )( 0 K X X Let us then assume that a background is characterized by → ( ( ˙ P φ Q spacetime, the equation of motion for where we have assumed the symmetry under the field reflection meaning that where and corresponds to thelevel. extrinsic We curvature of thuslevel constant-tim have as a general Lagrangian in the unitary gauge or Goldstone (NG) boson as gauge, it is easy to obtain the action in the general gauge by s or stress-energy tensor for this simple system is with gravity allows for a Minkowski or de Sitter solution, de cludes a ghost at the origin with that of a cosmological constant as We should write downing most with general Minkowski background actionstraightforward), for invariant simplicity under (though thi exten that the background metricthis is background maximally ghost symmetric, condensation. either M One can choose a gauge s but we shall laterlow see energies by so that a we properfluctuations can of analysis construct the a of system consistent scaling low-energ around dimensi the attractor. symmetry that we have already invoked. For example, for mation at the linearized level, Since we assumed thataction Minkowski should is start a at background the satisfying quadratic t order. While The residual symmetry is then thetime-dependent spatial spatial diffeomorphism coordinate inv transformations metry transformation, neither combination of the (0 transformation, and then square it. The invariant combinat JCAP08(2016)002 le = 0 (3.7) (3.8) (3.9) i (3.10) χ el is of h and has 4 as the is locally / 2 φ on and thus . It is then /M . π 2 4 d to do proper ]. If we assume ) around y effective field + ∆, where we # / π 1 and thus are not c 16 χ r ~ ∇ ··· ( is well described by ˙ M π l operators such as for → π ~x )+ 2 χ 3 φ>φ π is not 1 but 1 ij m , , K π dtd 2 s of symmetry for i ) ological upper bound, d~x R + 2 K eld theory, provided that we . / ckground (CMB) photons by ··· ] π 1 gher than j + − + 17 perficial new degrees of freedom. ~ ∇ near dynamics [ r e can ignore the mixing between or is π i 2 inear operators that can enter the ll dark matter in the universe then 2 , that is responsible for the ghost ) ning that they are irrelevant at low ~ ∇ as [ φ π ~ ∇ → 2 ( )( 2 ~ ∇ ij M , 1 ( d~x 2 K M α , 2 α + M dt 0. We thus expect to have upper bounds on − 1 π − j 2 − → r 2 ~  ∇ 100 GeV ˙ i – 6 – (model-dependent) π ). Those operators that exist but are irrelevant 00 h M ~ . ∇ → h ( M ~x , 1 2 3 2 , that plays the role of the waterfall field in hybrid M 2 dt χ ≪ − M +∆ so that the mass squared α ∆ and a negative constant for dtd 2 c ˙ π − 10 eV Z  − c 4 2 1 & 2 " M 4 M <φ<φ = M φ<φ 4, which is positive. One can easily see that all other possib π ∆ / + I − c EH φ . It is obvious that the scaling dimension of L ] is a model of inflation driven by ghost condensation. The mod sets the order parameter of the spontaneous Lorentz violati 2 α = 15 M + eff 1 L as the cutoff scale. We have thus constructed a good low-energ . We now show that they are irrelevant. For this purpose we nee α , respectively, we obtain 2 π j = M ∂ i /M α 2 ∂ ) In the effective theory there are of course higher dimensiona The scale π + from observations and experiments. The strongest phenomen ∇ ij ( ˙ K condensation, but also another field that the excitation of ghostone condensate can is obtain responsible a for (model-dependent) a lower bound on hybrid type in the sense that the system has not only π inflation models. For example, one can suppose that the shift is a positive constant for broken in the range where 3.2 Ghost inflation Ghost inflation [ analysis of scaling dimensions.the NG In boson the and decouplingthe the effective limit, metric action on perturbation of and the thus form the dynamic the scaling dimension 1 action is invariant under the scaling nonlinear operators have positive scalingenergy dimensions, (with mea energies and momenta straightforward to calculate theeffective scaling action. dimension of For example, nonl the leading nonlinear operat Those superficial new degreespresent of in freedom the have regimeconsider frequencies of hi validity oftheory the around the low ghost energy condensation. effective fi include terms with higher time derivatives, which lead to su comes from thelumps stochastic of scattering positive and of negative cosmic energies created microwave due ba to nonli M general relativity is recovered in the limit JCAP08(2016)002 (3.13) (3.11) (3.12) (3.15) , meaning 2 ] and trispec- M be well above try in the field 15 = 0, generating i χ . The Planck 2015 is arbitrary. Hence h 2 nteractions, the non- tot + ∆, the waterfall field /M ), the Hubble expansion 2 c pectrum [ N ) π 3.9 e is low, the energy density ~ ∇ ( was normalized by a power of φ>φ n. While the de Sitter entropy perator as π al- and orthogonal-types, while rowave background anisotropies, on, we are now ready to consider nd the prediction for the corre- φ wer spectrum. By requiring that t symmetry introduces deviation he low-energy effective field theory. . , the strongest bound on nonlinear . 4 5 , . / shown in ( . Therefore, the amplitude of density 5 0 5 4 / /  4 . Since . . 1 M 4 ) − 4, the amplitude of quantum fluctuations 4 / 5 H M / 1 / − ) 4 βα  − 10 (68% CL statistical) (3.14) 10MeV), meaning that the amplitude of tensor is 1 82 ∼ H/M ( ∼ ] is is much lower than the cutoff scale – 7 – βα . 1 is dimensionless. Thus the amplitude of quantum ≃ , H/M ˙ 2 δφ φ 20 − . H Hπ M H H 43 6 = M . equil Mπ NL ∼ 0 ± ∼ π f = 0, ending the inflation. 4 ρ − i 6 ]. On the other hand, for δρ − χ h 18 ), the total number of e-foldings = 2 H N Λ equil NL G f GeV or so). ( 8 ∆, the background is essentially de Sitter with π/ − = c (1) is dimensionless, S O 0 without loss of generality (because of the reflection symme does not have to be low and thus the reheating temperature can = φ<φ , one thus obtains 2 ˙ 5 φ − ˙ H φ > is the coefficient of the leading nonlinear operator ˙ ]. Because of the higher derivative nature of the nonlinear i 2 10 Pl β 19 M ∼ One can calculate not only the power spectrum but also the bis Since the scaling dimension of so that develops a non-vanishing vev, = 3 where Gaussianity generated by ghostlocal-type inflation non-Gaussianity is is rather ofinteractions small. the comes equilater In from particular sponding the nonlinear equilateral-type parameter bispectrum is a M the TeV scale (up to 10 perturbation is easily estimated as space). For during ghost inflation is proportional to ( fluctuation is expected to be of order (temperature & polarization) constraint [ scale-invariant perturbations. Afrom soft the exact breaking scale-invariance of [ the shif have set χ rate during ghost inflation ismode rather is low ( too lowρ to be observed. While the Hubble expansion rat trum [ This estimate agrees withthis the be explicit responsible calculation for of theδρ/ρ the observed amplitude po of cosmic mic that ghost inflation is well withinBecause the of regime the of validity phenomenological of t upper bound on Therefore, the background curvature leading to the constraint on the coefficient of the nonlinear o 3.3 Lower boundHaving on briefly Ω reviewed ghost condensationimplications and of ghost inflati the deis Sitter still entropy bound given on by ghost inflatio JCAP08(2016)002 ) eq 2 0 a on. ) is , we H eq t (3 ≫ / aH Λ ≫ a ( = c ≡ k t Λ , our universe matter and the ) is not the total end t ] then argued that 2.6 = , where c t 14 t and that superhorizon , that exit the horizon ing scales are plotted in = obs t min n at tion of the waterfall field N k parameter values). Just for . There are the maximum and inated by dark energy. wer bound on Ω ne cosmological constant and 0) starts well after the matter- d CMB anisotropies. What is ion be satisfied, where Λ is the e time universe. The comoving opy bound ( and the de Sitter entropy and thus is ¨ a> is the present value of the Hubble max subsection, on the contrary, we shall stand for their logarithms. 0 k c a H and . In the late time epoch with exits/reenters the horizon when . In this situation under consideration, there eq k min 2 a k , – 8 – reenters the horizon at ) is indeed satisfied in our universe very well when , that exit the horizon during ghost inflation and then end . After that, the universe is supposed to evolve a and then a radiation dominated (RD) epoch until eq 2.6 min t min ) and k k c reh in figure t = t ( ˙ a a t ) is threaten to be violated. Ref. [ and / = t ≡ 2.6 max c k exits and reenters the horizon just at the end of ghost inflati ) = 1 a but the observable number of e-foldings aH max ( k / tot N )=0. c t = t ( )), we can safely assume that the universe is dominated by the a eq t . Evolutions of the Hubble horizon scale and comoving scales . Also, since the comoving momentum = 1 t Let us now estimate The overall evolutions of the Hubble horizon scale and comov For simplicity we suppose that after the end of ghost inflatio ( a , followed by a reheating at ≡ Figure 1 we assume that ghostimportant inflation here is is that responsible thenumber of for left e-foldings hand the side observe of the de Sitter entr minimum comoving momenta, are the maximum and minimum comoving momenta, ghost inflation spoils theincompatible thermodynamic with interpretation basic of gravitational principles.show that In the this de Sitter entropy bound ( the de Sitter entropy bound ( The comoving scale corresponding to defined by ¨ expansion rate. by requiring that theeffective de Sitter cosmological entropy constant boundthe (that for ghost contribution is inflat from the the sum ghost of condensation) the and genui reenter the horizon. On the horizontal axis, show the behavior of 1 radiation equality as in our universe. We shall then seek a lo figure during ghost inflation andscale then corresponding reenter to the horizon in the lat modes never reenter the horizon in the late time universe dom ( experienced a matterχ dominated (MD) epoch due to the oscilla according to the ΛCDMsimplicity cosmology we (but also assume with that a the priori late-time arbitrary acceleration ( the radiation-matter equality at JCAP08(2016)002 reh t and ing. (3.16) (3.20) (3.18) (3.19) (3.17) = reh t s, ∗ g is the energy , Λ c , Ω a 2 0 c Λ ρ H H m for energy density 2 Pl = + 2 ) = 0, we then obtain M ng scale corresponding to 3 c t , respectively,  = 0. On the horizontal axis, min om for entropy density at eq c , = eq t a a a t . = Λ = 3 ( eq a , k =  reh | 2 4 a Pl T T / a Λ t n of the waterfall field, we have eq m 1 M production between the reheating 3 ρ /  is the Hubble expansion rate at the 1 − = 0 and 2  inf , = 3Ω Λ = H 3 H 2 ρ can be rewritten as eq reh are the photon temperature at / reh reh ¨ a a 2 ∗  1 Pl s, t s, ∗ g c 0 ∗ 2 Pl  eq reh g M g and = H H T T 3 M inf  reh t ) is defined by ¨ 6  eq  ρ ρ t eq 4 ∼ a /  and – 9 – , , 3 = 1 / t Λ ∼ ≫ 1  Λ ( reh ρ c  Ω 4 eq T . Since we assumed that the universe between the end inf a 2 reh 0 + reh T end ρ ρ eq reh a 3 a H s eq s max   ∗ 2 Pl k g  eq ∼ a M a , where ∼ c  and ∼ reh a T eq eq m reh Λ = a ρ eq m a end ρ a ρ a 2 = 2 = is the density parameter for Λ. Setting ¨ are the energy density at the end of inflation and at the reheat 3 H , respectively, and are the entropy density at Λ  2 Pl eq c reh t eq eq M a ρ a stand for logarithms of them. s 3 = c  a t and is the total number of effectively massless degrees of freedo and , respectively. In the last expression, is the matter energy density at . eq . Evolution of the comoving Hubble horizon scale. The comovi eq and t inf , and reh ∗ eq m eq ρ s g t ρ eq are the total number of effectively massless degrees of freed = Next let us estimate a reh reenters the horizon at t = t , eq t s, = end ∗ min present time and Ω Assuming for simplicity that thereand is the no significant matter-radiation entropy equality, we have density corresponding to the cosmological constant Λ, k Figure 2 where cosmological constant so that a g t where where where at of inflation and the reheating is dominated by the oscillatio and JCAP08(2016)002 . n or and . = ρ (3.21) (3.24) (3.22) (3.25) (3.23) rad bh ρ end bh H a ln Tr inf as . rad H ≡ ρ Λ 6 . = / ) 1 rad nd is satisfied in rad , and dim ρ ! end , respectively, and Tr max 2 m eq S 3 − inf T 6 rad = 2 0 H − H , k = H Pl rad would happen in such a 100 GeV Λ 4 nciple be violated in the H / eed satisfies the de Sitter M Ω 1 exp( . rad and e. However, if the observed −

S becomes . × o zero or an extremely small  4 bh / # ! 1 2 2 inf H ) about the black hole evaporation, − , M 3 inf 2 H eq osmic microwave background anisotropies with dim  reh reh 2 2 T H ∗ Pl ∗ 2 − 0 g g bh Pl ( M  H ation to the comoving size of the last scattering 3 3 M he logarithm of the ratio of the comoving size of / M ⊗H 1  (1) factor.

 4 1 O 2 / 100 GeV rad M / 1 eq 3  ∗ H − ) g , one obtains a lower bound on Ω 100 GeV eq  42 = reh – 10 – s, 2 inf  ∗ ∗ ]. reh 10 H inf reh g g 42 ( s, ρ ρ − /H ∗ 21 2 " g  10 2 represent trace over Pl   eq ∼ M eq T 2 exp rad ∗ 12 2 3 / π g ) eq 2 / 1 Pl & 1 s,  M ∗ Λ M reh  = 8 g 4 2 Ω s, inf reh − ∗ π eq and Tr ρ reh ρ ), one can define the reduced density matrices g in the main text up to an end s, s, 48 ∗  ∗ S  (10 bh g g obs 2 ∼ / . ∈ H N ∼  1 ( 3  i / obs 1 min end max φ | k N inf k reh S  ρ 6 ρ ∼ inf reh  , where Tr ρ ) ρ ρ 100GeV, the right hand side is extremely tiny and thus the bou . The entanglement entropy is then defined as &  | obs φ rad Λ . ∼ N Ω ih Tr φ M | eq The observable number of e-foldings is then end ≡ a exp( a For example, if we observe the inflationary modes through the c 1 = bh then the observable number ofthe e-foldings Hubble should horizon be (or given the by apparent t horizon) at the end of infl surface. This agrees with We then obtain one can safely neglect the prefactor and the lower bound then ρ Demanding that 4.1 Black holeLet evaporation us and consider a information Hilbert space of the form Since Since ρ For a pure state our universe very well. 4 Cosmological Page time In the previousentropy section bound we if have iteffective happened shown cosmological in that constant the ghostpositive Λ early inflation is value epoch in ind not ofdistant eternal the our future univers but future (but decays thenuniverse not t in the yet the entropy distant now). future, boundfollowing let the may In us argument first in made order remind by pri to ourselves Page [ understand what JCAP08(2016)002 ] . y 22 n (4.2) (4.3) (4.1) (4.4) as i are, re- rad I bh , Page [ . By using h ρ of a radiation n H i ≤ rad . and I m h 1 evaporating black rad ≪ ≪ ρ ]. For the early stage Pl bh , 80 he black hole loses about ) 21 M M [ ) black hole, one obtains n . Pl , ≡ 2 matches with that based on Pl ly earlier than the Page time, ≤ . For 1 ) 70 ) M ut. Haar measure on m bh m em at one time then the amount m /M S n is instead encoded in the entire ≤ , y ≤ ≫ 2 bh ≪ 60 − n n bh 2 Pl n (1 bh πM M ≪ M ≪ 2 M . Information in and the average information 50 , (1 (1 bh = ln i = n S m 2 and the average information , rad bh , bh i I 40 = S . The dimension of the entire Hilbert space is fixed h m n m n rad i≃ m 2 2 – 11 – S , r rad h and rad + S − 30 .  S i ). We thus have 2 n n π m rad 2 y Pl 4 rad −h S ln I ln − 20 h /M m −  one can clearly see that the first bit of information comes i≃ 2 bh i≃ m 3 and rad exp 10 = ln rad i πM S I i h 4 h = ln rad i∼ − and thus one can never recover information by perturbation i rad S 0 I , it is then obvious that h 0 y h rad rad 40 30 20 10 70 60 50 I I bh , and they agree, exp( h to be the black hole entropy, 4 S bh d dt m n ρ = bh ∼ ln r ) of an evaporating semi-classical ( ) rad . n bh n S ρ 80 (2 e bh ≪ shows plots of = . The average entanglement entropy m/ Tr 3 m − mn For the early stage of black hole evaporation, i.e. sufficient From this result one can extract important implication on an i≃ = ≪ rad bh I Figure where the averaging is defined using the unitarily invariant Figure 3 the fact that the above result based on the average information perfectly out from a blackhalf hole of at its the entropy. so After the called Page Page time, time, information the comes time o when t (1 estimated the average entanglement entropy hole by setting ln h and that This is not analytic in That is, if one observesof only information a to small subsystem be ofsystem. recovered a is large tiny. Actually, syst The from total figure informatio to be S subsystem versus its thermodynamic entropy ln spectively, defined as JCAP08(2016)002 ) in 2 2.6 / (4.6) (4.5) 1 in our − a . ∞ odes that ∝ 2 / = 1 are shown in  aH Page as the time when Page t 3. Combining this Page decay / H a a Λ responsible for ghost o an extremely small Page  . Since t universe is metastable φ p Page a eq Page a decay t = )= ective field theory starts to a ass changes from a positive eans that tes the effective field theory behind the horizon. Extrap- , 2 k Pl llatory phase. Alternatively, = il to reconstruct information, 4 is the de Sitter entropy at the eavy fields), the universe after Λ ge time edicts recovery of information. / t o ar field es, the overall evolutions of the M 1 tural to define the cosmological ided that the oscillation of the e de Sitter entropy bound ( H ! (3 end at . On the other hand, starting at to matter. One can think of this 4 / Λ S t Λ that we observe today is eternal. Λ 2 inf H H decay ρ a H 2 Pl decay q 2 4 eq t / M and 1 T = 3 =  a Λ

), where t 4 H / 1 end Page ) decay S a a it is concluded that effective field theory should – 12 – , we obtain reh  ∗ , and g 3.3 ( 4.1 = 3 / (1) deviation from effective field theory prediction nei- decay (1) at the Page time. 1 t , if ghost inflation happened in the early epoch of our O  Page O = H ) starts to be violated. 3.3 eq are the values of reh t s, s, ∗ ∗ 2.6 at g Page g a Page a  in subsection H 12 / 1 end  and a inf reh ρ ρ as the time (after inflation) when the number of inflationary m Page  a ∼ is the value of Page t inf Page decay , where H H a 4 Needless to say, the expected On the other hand, if the observed Λ is not eternal but decays t For example, imagine that our current dark energy dominated end a Page a reenter the horizon becomes as large as exp( end of inflation. In other words, we define the cosmological Pa effective field theory:the both Page fail time, to theA result reconstruct based problem information on with the effectiveeven average field information after theory pr the is Pagedeviate that time. from it the continues This correct to means result fa that by the prediction of eff the matter dominated epoch, we obtain the de Sitter entropy bound ( figure with the estimate of universe then the de Sitter entropy bound is satisfied. This m break down when it reliesolating on this about conclusion half to ofPage the the time de number Sitter of horizon, states we find it na 4.3 Cosmological Page timeAs after we ghost have inflation seen in subsection may be violated in the distant future. ther implies the enddescription of of the the world early at stage the of black Page hole time evaporation. 4.2 nor invalida Definition ofFrom cosmological the Page considerations time in subsection universe, provided that the effective cosmological constan value or zero in the future then there is a possibility that th where such that it completely decays after a while at condensation, similarly towaterfall the field case does of not decaythe ghost to phase inflation. radiation transition (but becomes may Prov Hubble matter decay horizon dominated. to scale h In and these the cas comoving scale corresponding t model as aone slow can rolling introduce scalarconstant yet field to another which a waterfall later negative field enters constant whose at an a squared osci certain m value of the scal JCAP08(2016)002 (4.7) (4.8) and thus . On the end , the Hubble Page S a − the number of e 2 / = Page . ted to be a good 1 inf a #  H 2 nor invalidates the . Page 4 eq t − in the case where the end T a>a  ) at 2 a ∼ 4 Λ Page (1) deviation from the Page t t t = end O H S 2 ∼ Pl ≪ M GeV) t t Page M 2 ound 3 H 100 GeV − . bserve unexpected correlations ) (1) deviation from effective field erse reaches the Page time after rees of freedom accessible during  und (10 O Page 42 end able modes cannot be independent les for  a ge as exp( S 2 elation functions among cosmological 10 = / 1 " , we finally obtain k , we obtain − ) 3.3 exp(2 M exp 4 2 reh 2 ∗ −  g  ( 10 3 eq / eq ∼ 2 a – 13 – decay a decay a  a stand for their logarithms. For inf   and the universe becomes matter-dominated. The number eq reh H 1 1 s, s, in subsection − ∗ ∗ − Page g g  a  decay , however, observers start to see  end a 6 S / = and 1 Page M a M t  and using 100 GeV ∼ 100 GeV inf reh decay ρ end  t a ρ  S ,  e × ∼ eq a ∼ , (including the present time), effective field theory is expec (1) deviation from the EFT prediction. O Page decay end a a Page a decay Page a a t ≪ . Evolutions of the Hubble horizon scale and a comoving scale t For As in the case of black hole evaporation, the expected Figure 4 By substituting the estimate of cosmological constant decays at Therefore, it is inthe an extremely observed Λ distant future decays. when the univ description of inflationary modes, predictingobservables. correct corr At around observable modes starts to exceed theinflation, number and of thus physical those deg superficiallyof independent, each other. observ effective field theory description of cosmological observab prediction of effective fieldamong theory. superficially For independent example, modes. one might This o is because at ar of inflationary modes that reenter the horizon becomes as lar theory prediction neither implies the end of the world at aro horizon scale exceeds the comoving scale corresponding to horizontal axis, observers may see By equating this with JCAP08(2016)002 ed value of xample, the , by compar- 4 , the de Sitter 3.3 ion, we can safely use les surrounding us as in , which is defined as the ). This is interpreted as ntropy bound conjecture itter entropy at the end constant at late time by he (effective) cosmological bh -foldings. This means that Page S ome unable to use the EFT mum comoving scale which time, we just become unable EFT description is expected the Page’s argument on the ually violated in the further ays to essentially zero some- cumulates as much as giving t s among inflationary modes with a non-trivial constraint le continues to evaporate, we d expansion of the universe, As the evaporation proceeds sting opportunity in the sense sociated with radiated modes eternal inflation), the total e- al constant at present, and it as large as the black hole’s, or r universe. the lower bound is as weak as in subsection lusion in section bserving the radiation. rved by late time observers. In ation with the use of the semi- e Hubble horizon is constrained potential danger that the model exp( dously large. Thus the de Sitter tion. As we have briefly reviewed ime w-roll eternal inflation, while the nvalid at all time from the begin- he de Sitter entropy, the formation ions among the radiation modes as – 14 – ] that the number of e-foldings that can be observed 14 ). Compared with this extremely tiny lower bound, the observ . This conjecture is supported by a number of evidences. For e 42 end 10 S , it was conjectured in [ − 2 we have then investigated the implications of the de Sitter e 3 exp( Moreover, even if the (effective) cosmological constant dec As an analog of the Page’s argument on the black hole evaporat & (1) deviation from the prediction of the EFT. As a result, the Λ the EFT inthat the reenter the ghost horizon inflation well before model the to cosmological Page calculate t correlation the cosmological constant inentropy our bound present is universe always is satisfied tremen for the ghost inflationtime in in ou the futuredistant so future, that it means thening neither de that to Sitter the the entropy ghost end boundthe inflation as is is eternal the i event slow-roll EFT inflation nor case. that we We will have be drawn this crushed by conc black ho 5 Summary and discussions We have revisited the issue ofin de section Sitter entropy in ghost infla follows: after the Pagemany time, as the the uncertainties dimension inO of correlat the Hilbert space of the black hole ac ison with the caseblack of hole an evaporation, we evaporatingclassical can black EFT, investigate hole. as the far According Hawkingis as radi to sufficiently the less dimensionand than of the that the Hilbert Hilbert associated spacethe space with associated system as with the reaches the the blackto radiation Page hole. calculate becomes time, physical then quantities we involving for modes the more first than time bec to ghost inflation.folding In number the is arbitrary ghostmight in inflation break principle, the (as and de in Sitter thusthat entropy there the bound. the is slow-roll This a de may Sitter beon an entropy cosmological intere bound parameters. conjecture Intriguingly,entropy might as bound provide we can us have be shown constant restated at as a the lower presenthad bound been time. on created the in This the valuefrom is of inflationary above era t simply by and because reentered thegives into the an horizon th maxi upper scale bound ofif on the we the (effective) adopt value cosmologic ofwe the the can ghost observable number inflation obtain ofthe model the e de for lower Sitter the bound entropyΩ early bound. on accelerate the Fortunately or (effective) unfortunately, cosmological by late time observers should be bounded from above by the de S of inflation to break down at aroundto the Page use time. the However, EFT aftercan to the make still Page correct watch predictions, the and black if hole the evaporation black perhaps ho through o bound holds in thetotal standard number slow-roll of inflation. e-foldings can Also,of be in significantly a larger slo black than hole t section prevents the extra e-foldings from being obse JCAP08(2016)002 ) end . On S n . since the Page in the ghost t Page t Page t ed in part by World entropy bound if the (1) deviation from the ll before O e Agence Nationale de la se the EFT even after the ANR-10-LABX-63) part of describe physical quantities a of the present paper was . ld conclude that the amount t the EFT does not result in al to the de Sitter entropy at Page time. Nevertheless, it is oration, we are not able to use Paris (IAP). During the visit, scribe those novel correlations modes, even after ing further inflationary modes ated with the part of radiation inistry of Education, Culture, lativity it is not easy to predict , if one observes only a small the Hilbert space for the rest of c Research No. 24540256 (S.M.) ration, it is in principle possible Page reentering the horizon. Without extremely small value sometime ion model we will be crushed by t merely intend to describe part of e patches and/or the inflationary tches and the inflationary modes rrelations between very early and nary modes more than exp( as giving when the de Sitter entropy bound /or inflationary modes more than 4.1 eeps its value for the future. Even Page t ∼ t (1) deviation from the prediction of the EFT. (1) deviation from the EFT prediction, we can O -point correlation functions with too large O – 15 – n , we for the first time become unable to use the EFT to Page t , we become unable to use the EFT for the calculation of Page t ), for example, , we do not expect such a catastrophic event after end S Page t ). After that, although we expect end In summary, the ghost inflation always satisfies the de Sitter Based on Page’s argument reviewed in subsection S still survive the Page time and keep observing further modes knowing the UV-completion for ghostwhat inflation the and future general re observernot will far from detect intuition on thatvery his/her he/she late might sky see horizon-reentered after unforecasted modes co the by and the that EFT he/she that cannot people de today know of. Acknowledgments This work was supported inand part by No. Grant-in-Aid for 16J06266 Scientifi Premier (Y.W.). International The Research workSports, Center of Science Initiative S.M. and (WPI), and Technologydeveloped M (MEXT), Y.W. when Japan. S.M. was was support part The visiting of basic Institut his ide work Astrophysiquethe has de Idex been SUPER, done within and the received Labex financial ILP state (reference aid managed by th time when the observedthe e-folding number end becomes of roughly inflation. equ We can use the EFT in the ghost inflation we quantities involving the Hubble patches and/or the inflatio uncertainties in correlations amonghave the not different yet Hubble accumulated pa as much as giving subsystem of a largeany system contradiction: at both one the timeof EFT then and information it the to is entropy be argument expected recoveredthe wou is tha Hawking tiny. radiation For this aPage reason, one time, if time, as we then far wethat as we may the intend still dimension to of be describesystem. the able is Hilbert In to sufficiently space the smaller u ghost associ thanthe inflation that EFT model, of to as describe in physicalmodes the quantities more black involving hole than the Hubbl evap exp( Only at and after around the other hand, we mayinvolving safely smaller use the number EFT of if Hubble we merely patches intend and to inflationary since the uncertainties in correlations accumulate as much exp( describe physical quantities involving Hubble patches and prediction of the EFT.the Whilst black in hole the after eternal slow-roll inflat inflation model. Therefore, asin in the the case late of timealthough black universe the hole EFT evapo after prediction ghost is inflation not to necessarily keep useful observ after is eventually violated. At around (effective) cosmological constant ofif the the present (effective) universe cosmological k in constant the decays future, to it zero is or in an an extremely distant future JCAP08(2016)002 ]. ]. , ] ]. SPIRE , , IN (2003) 056 SPIRE , ] [ IN , SPIRE 07 (2003) 009 ]. ] [ IN Dynamics of [ A measure of 05 (1975) 199 (2010) 30 ]. 3 SPIRE JHEP 43 (2009) 070 , IN [ pan. He is grateful to hep-th/0603158 Ghost inflation s supported in part by JCAP [ 09 , SPIRE ts to thank H. Firouzjahi (1976) 107 IN ork of Y.W. was supported spitality. S.J. would like to hep-th/0312099 Ghost condensation and a ] [ [ 75065. R.S. thanks people in , JHEP arXiv:0704.1814 nd T. Wiseman, heir kind hospitality during his [ ir under the reference ANR-11- , A 57 darriaga, ]. ]. (1973) 2333 ]. (2006) 509 Open Astron. 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