Swampland, Trans-Planckian Censorship and Fine-Tuning Problem for Inflation: Tunnelling Wavefunction to the Rescue
aa r X i v : . [ h e p - t h ] F e b Swampland, Trans-Planckian Censorship andFine-Tuning Problem for Inflation:Tunnelling Wavefunction to the Rescue
Suddhasattwa Brahma ∗ , Robert Brandenberger † and Dong-han Yeom , ‡ Department of Physics, McGill University,Montr´eal, QC H3A 2T8, Canada Department of Physics Education, Pusan National University,Busan 46241, Republic of Korea Research Center for Dielectric and Advanced Matter Physics,Pusan National University, Busan 46241, Republic of Korea
Abstract
The trans-Planckian censorship conjecture implies that single-field models of inflationrequire an extreme fine-tuning of the initial conditions due to the very low-scale ofinflation. In this work, we show how a quantum cosmological proposal – namely thetunneling wavefunction – naturally provides the necessary initial conditions withoutrequiring such fine-tunings. More generally, we show how the tunneling wavefunctioncan provide suitable initial conditions for hilltop inflation models, the latter beingtypically preferred by the swampland constraints.
The trans-Planckian censorship conjecture (TCC), proposed recently [1], aims to resolve anold problem for inflationary cosmology. The idea that observable classical inhomogeneitiesare sourced by vacuum quantum fluctuations [2, 3] lies at the heart of the remarkablesuccess of inflationary predictions. On the other hand, if inflation lasted for a long time,then one can sufficiently blue-shift such macroscopic perturbations such that they end upas quantum fluctuations on trans-Planckian scales [4]. This would necessarily require thevalidity of inflation, as an effective field theory (EFT) on curved spacetimes, beyond thePlanck scale. This is what the TCC aims to prevent by banishing any trans-Planckianmode from ever crossing the Hubble horizon and, thereby, setting an upper limit on theduration of inflation. Yet, in order to explain current horizon scale inhomogeneities asbeing sourced by vacuum quantum fluctuations, one needs a sufficient amount of inflation,putting a lower bound on the number of e -folds. These conditions combine to set an upper ∗ e-mail address: [email protected] † e-mail address: [email protected] ‡ e-mail address: [email protected] H inf < √ × − M Pl , (1)which can only be realized by a model of low-scale inflation . This conclusion is independentof any assumptions on how one obtains inflation. In this note, however, we will be workingin the context of the usual assumption that it is a canonically normalized scalar matterfield ϕ with potential energy V ( ϕ ) which is responsible for leading to inflation.Taking into account the observed scalar power spectrum, one also gets a bound on theslow-roll parameter ǫ := ( M /
2) ( V ′ /V ) , such that it is negligibly small [5] ǫ < − . (2)If one assumes the standard single-field consistency relation, it is easy to see that thisimplies that the primordial tensor-to-scalar ratio ( r ) for inflation must be unobservablytiny [5] r < − . (3)However, since primordial B -modes are yet to be observed, this does not necessarily implydoom for inflation. It might be conceivable that, provided one does not invoke alternatemechanisms for the production of primordial tensor modes [11], a tiny r is part of thepredictions of inflationary models.Even if a small r does not require one to revisit inflation as the preferred model ofearly-universe cosmology, there are other potentially troubling consequences of the TCCfor inflation. But, first, let us note another interesting observational fact. From theexpression for the running of the scalar power spectrum, or the spectral tilt, we know that n s − η − ǫ , (4)where the second slow-roll parameter η := M ( V ′′ /V ). The spectral tilt is quite tightlyconstrained from observations, and the mean of its measured value from PLANCK is givenby n s = 0 .
965 [12]. Since ǫ is constrained to be negligibly small from the TCC, this impliesthat η ∼ − . . (5)This conclusion is particularly interesting when one recalls the de-Sitter (dS) swamplandconjecture [13]. The dS conjecture states that for an EFT to not be in the swampland ,one must have (for ( c, c ′ ) ∼ O (1)) | V ′ | V > cM Pl or V ′′ V < c ′ M , (6) This bound can be alleviated if one assumes a non-standard cosmology after the end of inflation [6, 7]or if the equation of state parameter during the bulk of inflation deviates significantly from w = − The swampland of effective field theories is the set of EFTs which are not consistent with superstringtheory - see [14, 15] for reviews. .e., either the potential is very steep or, when near its maxima, must have large tachyonicinstabilities. For inflationary models, this implies that either ǫ ∼ O (1), or, if ǫ ≪
1, then | η | ∼ O (1). Since the TCC implies that ǫ ≪
1, observations (4) seem to require that η ∼ . i.e. , it is an O (1) number . It might seem that this range of values of η mightnot be enough for avoiding stochastic eternal inflation [16, 17]. On the other hand, severalstudies have shown that stochastic eternal inflation is in severe conflict with the swamplandconstraints [17, 18]. Note that this apparent contradiction is easily resolved when one alsotakes into account our required initial value for ϕ , as spelled out later in (12). The regionof parameter space corresponding to this value, as parametrized by (10) is incompatiblewith eternal inflation [17], thus confirming the intuition that the our model is well out ofthe regime of stochastic eternal inflation.Given these constraints (1), (2) and (4), models of inflation which can be made viablewith the TCC and, more generally, the swampland constraints (see Sec-5), seem to behilltop ones . This is so because for hilltop potentials, one generically gets a very tiny ǫ and a large η , as is seen to be preferred by the swampland criteria. In the next section, wefirst give the phenomenological parameters for a hilltop potential given the TCC bounds.However, even allowing for such a model, another potential problem for such a small-fieldinflationary model lies in the initial condition fine-tuning problem. We shall describe thisin Sec-3 before going on to give a solution for it in Sec-4 in the form of the tunnellingwavefunction, which gives a quantum completion for inflation. Finally, in Sec-5, we showhow such a hilltop model, and thus the tunnelling wavefunction, is preferred when all ofthe swampland constraints are taken into account, before concluding in Sec-6. We consider a potential which has the form of a hilltop near ϕ = 0, i.e. V ( ϕ ) = V " − (cid:18) ϕµ (cid:19) . (7)This form of the potential must break down for values of | ϕ | comparable or larger than µ . We will assume that the potential asymptotes to zero for large values of | ϕ | . For thispotential, the second slow-roll parameter, for field values | ϕ | ≪ µ , takes the form η := M (cid:18) V ′′ V (cid:19) ≃ − (cid:18) M Pl µ (cid:19) . (8)Using (5), one gets µ ∼ M Pl . (9) This may be seen as an amelioration of the eta -problem since we do not need a value η ≪ Such potentials can also be more tractable for holographic cosmology [19]. In fact, as we will argue later, the breakdown of the potential must occur in fact at much smallervalues of | ϕ | .
3n the above, we have used the fact that ϕ/µ ≪ ǫ := M (cid:18) V ′ V (cid:19) ≃ (cid:18) M Pl µ (cid:19) (cid:18) ϕµ (cid:19) , (10)where we, once again, use ϕ/µ ≪
1. From the expression of the dimensionless scalar powerspectrum, P s = 18 π ǫ (cid:18) HM Pl (cid:19) , (11)one can get the value of ϕ i at the beginning of the inflationary phase by using (10) above,as ϕ i ∼ V / M Pl . (12)In the above, we have used P s ≃ . × − and the Friedmann equation3 M H ≃ V . (13)Since our goal later on is to show that there exists a mechanism which allows for thequantum mechanical tunnelling of the inflaton to such a position on field space, we firstneed to make sure that such a tunnelling is safe from quantum fluctuations. Note that ifindeed the field can tunnel to this position, it is safe from quantum fluctuations displacingit from this position since the amplitude of quantum fluctuations is given by h δϕ i / ≃ H inf π ∼ V / √ πM Pl , (14)clearly demonstrating that h δϕ i / ≪ ϕ i .Thus, we can fix the two parameters of this model µ ∼ M Pl from observations (andthe TCC, implying ǫ ≪ η ), and V / < × − M Pl [5]. The final thing to check is thatthe classical drift of the field, ∆ ϕ is much smaller than µ , so that one can get enoughe-folds of inflation. The TCC implies that the field traverses a very small distance [5]∆ ϕ ∼ √ ǫ M Pl N ∼ V / √ πP / s M Pl N . (15)Using the bound on the number of e -folds imposed by the TCC, N < ln ( M Pl /H inf ), onegets an estimate for the classical drift as∆ ϕ < V / M Pl ≪ µ . (16)4his is the same bound as one got in [5] written in a different way. This shows that ϕ i . ∆ ϕ ≪ µ and thus it is possible for inflation to last a sufficient time, if there is asuitable mechanism to begin inflation at ϕ i . There are two extreme fine-tuning problems for inflation, as implied by the TCC: • Inflation needs to start near the hilltop, at a value of the scalar field ϕ i , as mentionedin the previous section. From the point of view of classical dynamics, there is nocanonical explanation why the field should start at such a small value. • From the point of view of classical dynamical systems, it is natural to assume thatthe kinetic energy of the inflaton is comparable to the potential energy before in-flation starts. However, considering the value of ǫ we require given the TCC andobservations, it displays an extreme fine-tuning for the initial velocity of the inflatonas compared to this natural value [5]. This is the usual fine-tuning problem for theinitial kinetic energy in small field models of inflation [20].For large-field inflation models, as is well-known (see e.g. [21] for a recent review and [22]for some initial references), the inflationary slow-roll solution is a local attractor in initialcondition space, and the above problems do not show up. However, both these problemsappear as the TCC implies that only a small-field model of inflation is allowed, given currentbounds on observations. In the next section, we show how the tunnelling wavefunction cansolve both of these problems at one go .In short, the initial condition problems for inflation arise if one considers inflation fromthe classical physics dynamical systems point of view. However, quantum effects will likelybe very important in the early universe. A goal of the field of quantum cosmology has beento develop a theory for the initial conditions which apply once classical dynamics takes over.According to quantum cosmology (see e.g. [24,25] for an overview), the state is described bya wavefunction , but there are various proposals for how to obtain this wavefunction. Thetwo most popular proposals are the Hartle-Hawking wavefunction [26] and the tunnellingwavefunction [27], although recently potential problems for both approaches have been putforward [28], with several potential resolutions [29]. In the following section, we will studythe implications of the tunnelling wavefunction for the initial condition issue in hilltopmodels of inflation, making use of its most recent reincarnation [30, 31]. However, one needs to add an additional mechanism to the construction to ensure that inflation doesnot last too long. The simplest way to achieve this is by a sudden steepening of the potential after thevalue ϕ i + ∆ ϕ . In [23], it has been conjectured that some sort of tunnelling might alleviate these problems althoughthe “tunnelling” referred in it has nothing to do with the tunnelling wavefunction. The Tunnelling Wavefunction for the Hilltop Po-tential
In the case of homogeneous and isotropic cosmology, the tunnelling wavefunction Ψ forEinstein gravity minimally coupled to a canonically normalized scalar field ϕ (evaluatedat the values a and ϕ of the scale factor and scalar field, respectively) is given by afunctional integral (following the notation of [30])Ψ T ( a , ϕ ) := Z ∞ d N Z D [ a ] D [ ϕ ] e iS (0) [ a,ϕ,N ] , (17)where η is conformal time, a ( η ) is the scale factor, N ( η ) is the lapse function, and S (0) [ a, ϕ, N ] = 6 π Z η −∞ (cid:20) − ˙ a N + a ˙ ϕ N + N ˜ U ( a, ϕ ) (cid:21) d η , (18)is the action. Here, ˜ U ( a, ϕ ) = a − a V ( ϕ )3 , (19)is the superpotential (we have used ˜ U instead of V for the superpotential, as comparedto [30], so as not to cause confusion with the scalar potential). In the above, an overdotstands for a derivative with respect to η .It is convenient to choose a time coordinate t , such thatd η = 1 a d t (20)with the line-element taking the formd s = − N q ( t ) d t + q ( t ) dΩ , (21)where q ≡ a . In these coordinates, the (background) action becomes S (0) [ q, ϕ, N ] = 6 π Z (cid:20) − ˙ q N + q ˙ ϕ N + N (cid:18) − qV ( ϕ )3 (cid:19)(cid:21) d t , (22)where the overdots now refer to derivatives with respect to t .This action is minimized by the solutions of the following classical equations¨ q − N V ( ϕ ) + 4 q ˙ ϕ = 0 , (23)¨ ϕ + 2 (cid:18) ˙ qq (cid:19) ˙ ϕ + 16 N q (cid:18) d V d ϕ (cid:19) = 0 . (24) Note that t is not the usual physical time of homogeneous and isotropic cosmology.
6e now need to solve this set of equations for our hilltop potential with the parametersdescribed in the Sec-2.Our goal is to first solve (23), with the boundary conditions (some of which are just theregularity conditions for getting a smooth initial wavefunction) for the tunnelling wave-function q (0) = 0 and q (1) = a (25)˙ ϕ (0) = 0 and ϕ (1) = ϕ . (26)For our purposes, it is sufficient to show that the tunnelling wavefunction prefers a largepotential energy or, in other words, it would be more probable to tunnel to the hilltopof our given potential. This way we shall be able to explain why the inflaton beginsat our required value of ϕ i , due to the measure defined by the tunnelling wavefunction.In this case, it is sufficient to get an analytical solution even if we have to make someapproximations to derive it.Firstly, we assume the usual slow-roll relation N V ≫ q ˙ ϕ and the fact that thepotential is sufficiently flat. For our form of the potential, this can be easily justified asfollows V ( ϕ ) = V " − (cid:18) ϕµ (cid:19) ≃ V , (27) V ′ ( ϕ ) = − V (cid:18) ϕµ (cid:19) ≃ , (28)which are both valid for ϕ/µ ≪
1, as is the case for our allowed choice of parameter space.In this case, for our (25), the solutions of (23) take the form q ( t ) = V N t + (cid:18) a − V N (cid:19) t , (29) ϕ ( t ) = ϕ . (30)The action can now be evaluated for this soluion and yields S (0) [ a , ϕ , N ] ≃ π (cid:20) N V
108 + N (cid:18) − V a (cid:19) − a N (cid:21) , (31)with V ≃ V " − (cid:18) ϕ µ (cid:19) . (32)We can now perform the lapse integration using the saddle point approximation. Thesaddles for the lapse are given by the following equation: V N s + (cid:0) − V a (cid:1) N s + 9 a = 0 . (33)7e are interested in the values of N in the underbarrier regions which are given by thesolutions N ± = 3 iV ∓ r − V a ! , (34)with the corresponding values of the saddle-point action S (0) (cid:2) a , ϕ , N ± (cid:3) = 12 π iS ± [ a , ϕ ] , (35)with S ± [ a , ϕ ] := ∓ Z a ∗ a p U ( a ′ , ϕ ) d a ′ + Z a ∗ p U ( a ′ , ϕ ) d a ′ , (36)where U = a ˜ U . Here a ∗ is defined as the turning point where U ( a ∗ , ϕ ) = 0. Theimportant thing for us is that under our given approximations, ϕ is a constant, and thetunnelling wavefunction is essentially the same as that for the cosmological constant case.In this case, the amplitude of nucleation is given by the solution for the only saddle-pointwhich contributes to the classically allowed regime U ( a , ϕ ) < N = 3 V i + r V a − ! . (37)The probability of nucleation is given by P ∝ exp (cid:0) − π S ( a , ϕ ) (cid:1) , (38)with S ( a ) = i Z a a ∗ p − U ( a ′ , ϕ ) d a ′ + Z a ∗ p U ( a ′ , ϕ ) d a ′ . (39)The semiclassical factor contributing to the probability comes from the second term above.After a little algebra, one can see that the probability of nucleation for the tunnellingwavefunction is given by P ∝ e − /V , (40)ignoring some numerical factors. This is the same probability one gets for the tunnellingwavefunction by solving the Wheeler-de Witt equation with the tunnelling boundary con-ditions (in the canonical formulation as opposed to the path integral version followed here).This result clearly demonstrates that the tunnelling wave function prefers a nucleationto the top of the hilltop since the probability decreases sharply as the potential energy getssmaller. The exact top of the hill occurs at ϕ top ≡
0. We are interested in the probabilitythat the field nucleates at the specific value of ϕ ∼ ϕ i . However, given the value of µ which8e are using, ϕ i is close to the top of the hill in terms of the value of the potential energy.If we compare the probability to nucleate at a value ϕ ∼ ϕ i to its maximum value, we havelog (cid:18) PP max (cid:19) ∝ − V ( ϕ ) + 1 V ≃ − ϕ V µ , (41)where this approximation is true if ϕ/µ ≪
1. Therefore, the probability difference betweennucleating near ϕ i and ϕ top for the hilltop potential is negligibly small. Since, in order notto obtain too long a period of inflation, the value of the potential has to decrease sharplybeyond the value of ϕ i + ∆ ϕ , the tunnelling probability to a value of | ϕ | larger than thatone is negligible. Hence, the probability of nucleating with | ϕ | ∼ ϕ i is of the order P ∼ ϕ i ϕ i + ∆ ϕ , (42)which is of the order 10 − given the numbers we have used in Sec-2. We thus see that thetunnelling wavefunction gives a simple explanation why inflation should begin at ϕ i for theform of the potential which we are using.The second fine-tuning requirement for inflation from the TCC is that the initial fieldvelocity for the inflation is extremely small. However, for the tunnelling wavefunction, thiscondition is automatically realized since the velocity at the “South Pole” of the Euclideaninstanton has to be precisely zero due to regularity. In other words, we can see from (25)that ˙ ϕ (0) = 0 and for a constant potential, we get the “no-roll” solution. However, oncewe introduce our (slow-roll) hilltop potential, there would be a very small velocity for theinflaton after it tunnels to the hilltop. Nevertheless, this velocity would be extremely tinyand can naturally explain the slow-velocity required for the inflaton given the TCC. Inother words, following the logic of [5], we provide a quantum mechanical explanation forwhy the velocity of the scalar field should not be large before inflation begins. In fact, forour case, it should be almost zero before inflation starts. Having established how the tunnelling wavefunction can help us attain the initial condition,as required by the TCC, let us point out how the same arguments would also work moregenerally for the other swampland conjectures. Firstly, recall that the swampland distanceconjecture (SDC) [32] states that the field excursion during inflation should be less thansome O (1) number in Planck units for the EFT to be under control. This is easily achievedby the low-scale hilltop model described in this paper.Secondly, the dS conjecture tells us that either the slope of the potential or its secondderivative must be large, as quantified in (6). Unless we invoke additional fields or otherdegrees of freedom in the form of a modified initial state or an action deviating fromGR, it is not possible to have a large slope for an inflationary potential due to the tightobservational bound on the upper limit of r . In other words, for single-field , slow-roll To be more precise, we should use the terminology single-clock in this context. ǫ ). For exceptions, see [33]. On the other hand, it is quite possible to allow for modelswith a large η provided ǫ remains small. The prototype for such an inflationary potentialis indeed the hilltop model with more complicated versions allowing for additional termsin higher polynomial powers of the inflaton. Indeed, in all examples of such inflationarymodels (e.g., natural inflation [34] with a cosine term) the potential can be expanded ina power series which looks like our simple hilltop potential near the maxima. Therefore,if one is to look for a model of single-field, slow-roll inflation, in the spirit of finding thesimplest EFT of a minimally-coupled scalar to GR with some potential, the hilltop modelis the preferred choice in view of all the swampland conjectures. It is not surprising thatboth the TCC as well as the SDC and dS conjecture all prefer the same type of scalarpotential, given the fact that these different swampland conjectures have been shown tobe related (and, indeed, do follow) to each other [1, 35].Given the somewhat special status of the hilltop potential in inflationary model-buildingdue to these theoretical considerations, it is now clear why the tunnelling wavefunctionbecomes important in setting initial conditions for inflation. In this work, we have demon-strated how the tunnelling wavefunction does an excellent job of explaining why in hilltopmodels an initial value of the inflaton close to the top of the hill should be preferred as thestarting point in the phase of classical evolution with a negligibly small velocity. Keepingin mind that this is a truly quantum cosmological boundary condition gives us hope thatthere is a deeper reason to explore such proposals for quantum gravity more seriously inour quest for deriving dS spacetime as a low-energy EFT. We have to add, however, thatfrom the point of view of an EFT it is unnatural to obtain the fairly sharp cutoff of thepotential at a value ϕ ∼ ϕ i + ∆ ϕ which must be added (to make sure that the TCCremains satisfied), and this form of the potential needs to be justified from fundamentalstring theory for satisfactory model-building. In this note we have argued that the tunnelling wavefunction can provide, without too muchtuning, the initial conditions required for hilltop inflation modes which are consistent withthe TCC, initial conditions which from the point of view of a classical dynamical systemsapproach look highly fine tuned. The hilltop potentials we use are consistent with theswampland conjectures on effective field theories which can emerge from superstring theory.On the other hand, rendering a hilltop inflation model consistent with the short durationof inflation which the TCC only allows requires introducing a sharp cutoff in the potentialenergy function which does not look natural.Finally, it is worth pointing out that the conclusions reached from the tunnelling wave-function do not naturally follow from other quantum gravity proposals such as the no-boundary wavefunction. In fact, for the minimal model of a single-field slow-roll potential,the no-boundary wavefunction would prefer tunnelling to the local minima [36] (and cer-tainly not to the top of the hillt). Given the swampland constraints, this shows us that the10unnelling proposal indeed occupies a special place as the preferred boundary condition forthe wavefunction of the universe.As this manuscript was being finalized for submission, a paper [37] appeared which alsodiscusses the initial conditions for plateau models of inflation, but from a very differentpoint of view.
Acknowledgements
This research is supported in part by funds from NSERC, from the Canada Research Chairprogram, by a McGill Space Institute fellowship and by a generous gift from John Greig.DY is supported by the National Research Foundation of Korea (grant no. 2018R1D1A1B07049126).
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