IPMU-19-0170

Is Trans-Planckian Censorship a Swampland Conjecture?

Ryo Saito,1, 2 Satoshi Shirai,2 and Masahito Yamazaki2

1Graduate School of Sciences and Technology for Innovation,Yamaguchi University, Yamaguchi 753-8512, Japan

2Kavli Institute for the Physics and Mathematics of the Universe (WPI),University of Tokyo Institutes for Advanced Study, University of Tokyo, Chiba 277-8583, Japan

(Dated: November, 2019)

During an accelerated expansion of the Universe, quantum fluctuations of sub-Planckian size canbe stretched outside the horizon and be regarded effectively classical. Recently, it has been conjec-tured that such horizon-crossing of trans-Planckian modes never happens inside theories of quantumgravity (the trans-Planckian censorship conjecture, TCC). We point out several conceptual problemsof this conjecture, which is in itself formulated as a statement on the restriction of possible scenariosin a theory: by contrast a standard swampland conjecture is a restriction of possible theories in thelandscape of the quantum gravity. We emphasize the concept of swampland universality, i.e. that aswampland conjecture constrains any possible scenario in a given effective field theory. In order toillustrate the problems clearly we introduce several versions of the conjecture, where TCC conditionis imposed differently to scenarios realizable in a given theory. We point out that these differentversions of the conjecture lead to observable differences: a TCC violation in another Universe canexclude a theory, and such reduction of the landscape restricts possible predictions in our Universe.Our analysis raises the question of whether or not the trans-Planckian censorship conjecture can beregarded as a swampland conjecture concerning the existence of UV completion.

I. INTRODUCTION

The standard dogma of low-energy effective field the-ory and renormalization is based on the idea of separationof energy scales in physics. This has been a mixed bless-ing for physicists. On the one hand such a hierarchy inenergy scales ensures that, whenever one is interested ina low-energy effective field theory, one can concentrate onthe physics at the relevant energy scale: any ignoranceconcerning higher energy scales (such as whatever hap-pens at the cutoff scale) can be parametrized by the co-efficients of non-renormalizable operators in the effectivetheory. On the other hand, the flip side of the same coin isthat it is in general difficult to probe physics at very highenergy scales, since these effects are typically suppressedby some powers of the cutoff scale. This challenge is ex-tremely sharp for a quantum-gravity theorist who wishesto check their theories experimentally: what are the low-energy consequences of quantum gravity, whose energyscale can be as high as the Planck scale?

Early Universe provides an ideal setup for address-ing this question. It is widely believed in the literaturethat there is a period of accelerated expansion (inflation)in the early Universe [1–5]. During inflation Planck-suppressed corrections to the inflaton potential can spoilthe flatness of the inflaton potential so that the physicscan be very sensitive to Planck-scale physics. The accel-erated expansion in addition stretches the sub-Planckianmodes to macroscopic scales, thus potentially opening upavenues for direct observations of Planck-scale physics. Ifthis is indeed the case, one might argue that the cosmo-logical fluctuations such as the Cosmic Microwave Back-ground (CMB) anisotropies are highly sensitive to un-known physics at the Planck scale. This is the famoustrans-Planckian problem in inflation, which has long been

discussed in the literature, see, e.g. Refs. [6–10]. Thetrans-Planckian problem has also been long discussed forblack holes, see e.g. Refs. [11–15] for early references andRef. [16] for a summary.

Recently, Bedroya and Vafa boldly proposed that nocosmological trans-Planckian problem arises in the land-scape of the quantum gravity [17]—the horizon cross-ing of sub-Planckian modes never happens. This conjec-ture is called the trans-Planckian censorship conjecture(TCC). In Ref. [17], TCC is claimed to be yet anotherexample of a swampland conjecture [18, 19], a necessarycondition for the low-energy effective field theory to havea UV completion inside theories of quantum gravity, suchas string theory (see e.g. Refs. [20–22] for reviews onswampland conjectures).

After the proposal by Ref. [17], a number of papers[23–48] have studied consequence of TCC. It should nev-ertheless be pointed out that there are still conceptualproblems in TCC in itself. In this paper, we would liketo draw one’s attention to problems in formulating TCCas a swampland conjecture. In order to address this ques-tion more precisely, we formulate several versions of TCCand discuss the (de)merits of each in turn.

II. SWAMPLAND CONJECTURES

Let us begin with general remarks on swampland con-jectures.

A swampland conjecture is a conjectural necessary con-dition for embedding a low-energy effective field theoryinto a theory of quantum gravity. A swampland condi-tion should therefore be satisfied for all possible scenariostheoretically realizable in a given effective field theory—it does not matter if the scenario is unlikely, or if it is

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realized in our Universe, as long as the scenario has anon-zero probability of realization. For later reference,let us state this as a principle:

Swampland Universality: A constraint from aswampland condition applies to any possible real-ization inside a given low-energy effective field the-ory.

An example of a swampland conjecture is the swamp-land de Sitter conjecture (dSC) [49] (see also Refs. [50,51]). This conjecture states that the total scalar poten-tial V of a low-energy effective field theory satisfies theinequality

MPl|∇V | ≥ c V , (1)

where c is an O(1) constant and MPl ' 2 × 1018GeV isthe reduced Planck mass. Since this condition appliesto the potential of the theory and not to any particularrealization, this condition indeed satisfies the swamplanduniversality.

After the proposal of Ref. [49], it has subsequently beenpointed out that dSC causes severe problems with thepresence of the Higgs field [52–54], axion fields [53, 55]as well as cosmic inflation models. 1 For these reasons,there have been attempts to refine dSC [53, 59, 72–75].While the inequality (1) is speculative at a general pointin the configuration space, it is better motivated in theasymptotic regions of the moduli space [72, 76–79]. Forthis reason, one possible approach for refining the conjec-ture is to try to formulate another swampland conjecturewhich relaxes the constraint from (1) at a general pointin the moduli space, while preserving the essence of (1)in the asymptotic regions. TCC can be regarded as anexample for such an attempt.

III. TRANS-PLANCKIAN CENSORSHIPCONJECTURE

Let us consider a quantum fluctuation of sub-Planckiansize (of length scale l . lPl, where lPl is the Plancklength). The expansion of the Universe stretches thephysical scale of this fluctuation by a factor a/aini, whereaini and a denote the scale factor at the initial and giventime respectively. When the expansion accelerates, thisfactor grows monotonically with respect to the Hubblehorizon ∼ H−1. Therefore, the physical scale will even-tually exit the Hubble horizon if the accelerated expan-sion continues for a sufficiently long time. TCC claims

1 The literature is too vast to be exhaustively summarized here.See e.g. Refs. [53, 56–61] for early references based on dSC, ande.g. Refs. [62–71] for examples of similar analysis based on therefined dSC of Ref. [72].

that this horizon exit of the trans-Planckian quantumfluctuation never occurs [17]:

lPla

aini.

1

H, i.e.

a

aini.MPl

H. (2)

We will hereafter refer to the inequality of Eq. (2) as theTCC condition.

When a quantum fluctuation exits the Hubble horizon,it is widely believed that the quantum state transits intoan effectively classical state (see e.g. Refs. [80–85] for asample of early references). The typical narrative is thatthe state is in a highly-squeezed state after the horizonexit and then quickly decoheres due to interactions withthe environment. We do not need detailed mechanismsfor this quantum-to-classical transition in the following,except we should keep in mind that the point of TCCconcerns this classicalization, rather than the horizon exititself.

As already stated, TCC can be regarded as an up-grade of dSC. TCC is in general weaker than dSC ata generic point of the configuration space (thus avoid-ing the problems of dSC discussed above), while preserv-ing the essence of dSC at asymptotic regions [17] (cf.[72, 76, 79]).

Despite these similarities, there is actually a huge dif-ference between TCC and dSC. While dSC is a constrainton the potential of a theory, and hence of the theory itself,TCC is a constraint on a realization (a scenario) insidethe theory, at least on a first look. This is in some tensionwith the expectation that a swampland condition shouldbe a constraint for the low-energy effective field theories.A given low-energy effective field theory in general allowsfor many realizations, some of which might violate TCCeven when others are consistent with TCC. The swamp-land universality implies that such an effective field the-ory is in the swampland, irrespective of whether our par-ticular Universe satisfies the TCC condition (2) (Fig. 1).The possible existence of a TCC-violating Universe mightseem like an academic problem and irrelevant to us. Ithas, however, an observable impact: a stronger reductionof the quantum-gravity landscape leads to more restric-tions on possible predictions in our Universe. In order tosee if this is indeed the case, we first need to be more ex-plicit on what TCC means, since TCC can be interpretedin several different manners. In the following we discusseach interpretation in turn (see Fig. 2). We will in par-ticular see that some versions put the Standard Model ofparticle physics and cosmology into the swampland.

IV. UNIVERSAL TCC

One natural interpretation of TCC is the followingversion, which straightforwardly satisfies the swampland

3

EFT

ai(t)a2(t)Our Universe

aour(t)

a1(t)

Our Universeaour(t)

TCC-Violating Realizations

?

?

FIG. 1. An example of EFT in the swampland when theswampland universality is applied for TCC. This theory isexcluded because it allows realizations that do not satisfy theTCC condition (2) (TCC-violating realizations), irrespectiveof whether our Universe is in them or not. Such reductionof the landscape will also restrict possible predictions in ourUniverse.

TCC

Universal?

Selection Rule?

Universal TCC

Classical TCC

Probabilistic TCCObservational TCC

No

Yes

Yes

ProbabilityObserved Universe

No

FIG. 2. For precise discussions of TCC, it is important todistinguish between different variations of TCC listed in thisfigure, since each leads to different conclusions and interpre-tations. We can first ask if TCC applies universality, whichis the case for the universal TCC (UTCC). Another possibil-ity is to impose some selection rule and apply TCC to somesubsector, e.g. by restricting to classical solutions. This givesthe classical TCC (CTCC). Another option is to require thatthere is at least one TCC-consistent Universe realizable insidethe theory (observational TCC, OTCC). This is minimallyenough for the assumption that TCC applies to our observ-able Universe, irrespective of what happens in other possiblescenarios. Finally, we can adopt probabilistic interpretations,and allow for TCC violations with “small’ probabilities (prob-abilistic TCC, PTCC).

universality: Universal TCC (UTCC): the TCC condition (2)is obeyed for classically-observable quantities in anypossible realization allowed in a given effective fieldtheory. Note that here we included the phrase “classically-

observable quantities” in the formulation of UTCC, sincean intrinsically-quantum Universe does not have a well-defined concept of the classical spacetime (and hence ofthe classical scale factor), in which case it is not clearhow the TCC condition (2) should be formulated.

EFT

ai(t)a2(t)Our Universe

aour(t)

a1(t)

TCC-violating Realizations

FIG. 3. UTCC claims that the TCC condition (2) is obeyedfor classically-observable quantities in any possible realizationallowed in a given effective field theory. This condition isviolated if the theory admits even one TCC-violating Universeas a possibility.

This version of TCC is rather strong and essentiallyexcludes any inflationary scenario, while the condition(2) by itself allows for a finite period of inflation. Thisis because, in this version of TCC, the TCC condition(2) should be applied to all possible trajectories of thescale factor in a given theory—one can consider a TCC-violating realization in any inflationary model by tak-ing into account the backreaction of the fluctuations tothe background expansion. After the horizon exit, asmentioned above, the fluctuations are classicalized andlocally indistinguishable from the corresponding back-ground. As known in the stochastic inflation formalism[80, 86–98], the backreaction of the fluctuations changesthe background trajectory stochastically. It is then pos-sible, for example, that the background inflaton staysat the same point in the potential for an arbitrary longtime due to the backreaction effect, making the value ofthe e-folding number of the inflation arbitrary large (seeFig. 4). The TCC condition (2) is clearly violated in thissituation. Such a large backreaction effect is familiar inthe eternal inflation scenario [86, 88, 99–102], except herewe do not necessarily assume that the inflation continuesin most part of our Universe. In any inflationary model,a realization of large quantum fluctuations exists with anon-zero probability and their backreaction effect leadsto the violation of the TCC condition (2). It is also easyto understand that any inflation model with a metastablevacuum is excluded by a similar argument. To elude thisconclusion, an inflationary model should strictly forbidall TCC-violating realizations, while allowing realizationsof a finite period of inflation with the e-folding number(2) and the production of the density fluctuations in ourUniverse. To our knowledge, such inflationary model hasnot been known. It should be remarked that this argu-ment is applicable to all scalar fields with a positive andflat potential.

Moreover, there is a more serious objection that UTCCis incompatible with any scalar field whose potential hasa local maximum with positive energy. This is because

4

a

ainilPl

t

φ

lPl

Distance Scale

H−1

φend

typical trajectory

exceptional trajectory

classical trajectory

FIG. 4. Schematic picture of stochastic trajectories of theinflaton field φ due to the backreaction effect. The inflationends when the field value crosses φ = φend. The exceptionaltrajectory (red line) violates the TCC condition (2), while thetypical trajectory (blue line) satisfies it.

we can consider a realization where the scalar field isplaced on the top of the potential and stays there for-ever. In this solution, the accelerated expansion occurseternally, leading to the violation of the TCC condition(2). This is indeed the case for the Higgs potential inthe Standard Model of particle physics. As in the caseof the de Sitter swampland conjecture, one can try torectify this problem by e.g. modifying the electroweaksector by including extra fields or by coupling the Higgsfield to the quintessence field [103–105]. However, eachhas its phenomenological problems as already pointed outin Refs. [52–54] (save for rather fine-tuned possibilities).The same applies to local maxima of mesons or axions[53, 55].

We therefore conclude that, while theoretically appeal-ing as a swampland conjecture, UTCC is in practice toostrong to be imposed.

Now, we know that the swampland universality of TCCcannot be satisfied in its literal form. In the follow-ing, then, we will examine three possibilities to formulateTCC in an acceptable form: (1) TCC-violating solutionsare not realized due to (unknown) quantum gravity ef-fects, (2) TCC-violating solutions can exist but are neverobserved by us, (3) TCC violations with a small proba-

bility are tolerated.

V. CLASSICAL TCC

The first approach is to introduce a selection rule: oneassumes that only a limited class of realizations is actu-ally allowed in a given theory. In order that this formu-lation of TCC qualifies as a swampland conjecture, thisselection rule should be uniquely determined given an ef-fective field theory. However, as we will see, it would bedifficult to find a reasonable selection rule. To clarify thispoint, let us consider a particular selection rule that onlypicks up classical realizations:

Classical TCC (CTCC): the TCC condition (2)is obeyed for any possible classical realization (i.e.classical solution) allowed in a given effective fieldtheory.

EFT

ai(t)

aclassical(t)Our Universe

aour(t)

TCC-violating Realizations

FIG. 5. An example of EFT in the swampland for CTCC.CTCC claims that the TCC condition (2) is obeyed for anypossible classical realization allowed in a given effective fieldtheory. This condition is therefore violated only when the the-ory contains a TCC-violating classical realization, as shownin this figure.

In this conjecture, only the classical realizations are as-sumed to be possible and the swampland universality isapplied to them. CTCC gives a proper swampland condi-tion because, for a given effective field theory, its possibleclassical solutions are unambiguously defined as the ex-trema of its action integral. It is obvious that CTCCis unacceptable without any proviso because quantumeffects have been observed in experiments. A possibleexcuse is to assume that CTCC is only applied to cos-mological solutions, where quantum gravity comes intoplay.

However, there remain problems with this CTCC.First, CTCC has to at least forbid the inflaton’s fluc-tuations that cause the backreaction. Therefore, CTCCforces us to abandon the standard inflationary paradigmof the structure formation from the quantum fluctua-tions. One might argue that the cosmological fluctua-tions can be seeded by other scalar fields such as the cur-vaton [106–109], but the swampland universality (even itsclassical version) implies that their fluctuations shouldbe also forbidden because any such scalar field can be

5

the inflaton—there is a classical solution where the fielddrives inflation.

Another point is that CTCC misses the motivationsof TCC. Recall that TCC is meant to preclude the clas-sicalization of the trans-Planckian quantum fluctuationsin order to preserve a self-consistent classical picture ofspacetime on larger scales. On the other hand, CTCCforbids these trans-Planckian quantum fluctuations aswell as the problematic fluctuations that cause the back-reaction effect. Therefore, it is unclear what is problem-atic when CTCC is violated.

Furthermore, the problem of the local maxima of theHiggs and meson potentials, as pointed out above forUTCC, still remain even in this formulation.

From the arguments above, the selection rule shouldstrictly forbid large quantum fluctuations that stop afield rolling down its potential slope but should allowsmall quantum fluctuations that kick a field from a localmaximum of the potential, and those that source the cos-mological fluctuations for the structure formation if onewishes to keep the standard inflationary paradigm. Ifwe brutally forbid large quantum fluctuations, it is verylikely to be in sharp conflict with basic principles of thequantum mechanics—since the Schrodinger equation isdiffusive, the wavefunction spreads over the field spaceunless confined by an infinite potential barrier. One canstill be very speculative and argue that it might be possi-ble to justify such a selection rule in the grand frameworkof quantum gravity—a possible analogy is that the quan-tization condition in the Bohr’s old quantum theory, aselection rule of classical trajectories, was justified even-tually in the modern quantum mechanics in a frameworkcompletely different from classical mechanics. While thiscould be possible in principle, one should note that thisis a dramatically more radical requirement than that re-quired by the standard swampland conjectures.

VI. OBSERVATIONAL TCC

We have discussed difficulties in introducing a selectionrule for realizations. Then, a next possible approach is toallow all realizations, but forbid that the TCC violationis observed, at least, by us:

Observational TCC (OTCC): the TCC condi-tion (2) is obeyed in our Universe. This is the version implicitly used in many of the re-

cent papers discussing phenomenological consequencesof TCC. OTCC provides the minimum setup to discusspractical consequences of TCC. As we will see, however,some results from the literature are reproduced here butsome are not in OTCC.

Let us first discuss phenomenological predictions ofOTCC. The first example is an upper limit on the pri-mordial tensor amplitude, or the energy scale of inflation,

EFT

ai(t)a2(t)Our Universe

aour(t)

a1(t)

· · · · · ·

TCC-violating Realizations

FIG. 6. An example of EFT in the swampland for OTCC.OTCC claims that the TCC condition (2) is obeyed in ourUniverse, and an EFT is excluded only when our Universeviolates the TCC condition, irrespective of what happens inother possible realizations. In other words, an EFT is allowedif it has at least one TCC-consistent solution as a possibility.The swampland universality is maximally violated for OTCC.

as already discussed elsewhere [26–28, 30–32, 35, 36, 39].Applying the TCC condition (2) to the end time of in-flation, we find that the expansion rate at this time He

is limited by the e-folding number of inflation Ninf as,

He

MPl. e−Ninf . (3)

When the expansion rate is approximately constant dur-ing inflation He ' e−εHNinfHinf (εH 1), this imme-diately implies that the longer inflation leads to moresuppression of the primordial tensor fluctuations:

PT =2

π2

(Hinf

MPl

)2

< 0.2 e−(2−εH)Ninf , (4)

or equivalently the energy scale of inflation, V/M4Pl '

(3π2/2)PT . On the other hand, the inflation should belong enough for the CMB modes to cross the horizonduring inflation as,

eNinf >

(Hinf

H0

)e−Nafter , (5)

in order that the fluctuations are created from the quan-tum fluctuations. Here, H0 and Nafter are the Hubbleparameter at the present time and the e-folding numberafter the inflation, respectively. Therefore, since Nafter

cannot be arbitrarily large, it is possible to find an upperlimit on the primordial tensor amplitude.

The precise value of the upper limit depends on thehistory of Our Universe. The e-folding number Nafter isdecomposed into those of the inflaton oscillation Nosc andof the Big Bang Universe,

eNBB =

(g∗s(TR)

g∗s(T0)

) 13(TRT0

)= Ω

14

rad

(g∗s(TR)

g∗s(T0)

) 13(g∗(T0)

g∗(TR)

) 14(HR

H0

) 12

, (6)

where the subscripts R and 0 indicate quantities at thereheating and the present time, respectively, and g∗s(T )

6

(g∗(T )) is the effective degrees of freedom for the entropydensity (energy density) at temperature T . When theinstantaneous reheating is assumed, we have Nosc = 0and HR = He ' Hinf . Then, from Eqs. (4) and (5), onecan find an extremely small upper limit on the tensor-to-scalar ratio [23],

r ≡ PTPR<

2

π2PR

(1

Ωrad

H20

M2Pl

) 13

= O(10−30) , (7)

or V 1/4 < O(108)GeV on the energy scale of inflation,for the values of the cosmological parameters reportedin Ref. [110]. Here, we have omitted the effective de-grees of freedom. This tight constraint can be relaxedby considering, e.g., multi-stage inflation He 6= Hinf ornon-standard history of the Universe Nafter 6= NBB [26–28, 30–32, 35, 36, 39]. A general argument with themleads to [27],

r <g∗(T0)

45ΩradPR

(T0TR

)2

= O(10−8)

(TR

1 MeV

)−2. (8)

A tighter constraint can be obtained when one seri-ously considers the flatness problem. Imposing the ob-served upper bound on the curvature density parameterΩK0 < ΩK,obs, its initial value ΩK,ini should satisfy,

eNinf >

∣∣∣∣ ΩK,iniΩK,obs

∣∣∣∣ 12(Hinf

H0

)e−Nafter , (9)

which is obtained from Eq. (5) with the replacementsHinf → |ΩK,ini|1/2Hinf and H0 → |ΩK,0|1/2H0. From asimilarity to Eq. (5), it is easy to find that

r <

∣∣∣∣ΩK,obsΩK,ini

∣∣∣∣ rupper , (10)

where rupper is the upper limit in Eq. (8). Therefore,unless one fine-tunes the value of ΩK,ini, the present con-straint ΩK0 < ΩK,obs = O(10−3) [110] leads to a tighterconstraint on r. We also conclude that the spatial curva-ture ΩK0 should be detectable in future [111] when theprimordial tensor modes with r ∼ rupper are observed.Note that a negative value of ΩK is claimed to arisegenerically in the string theory landscape [112] (see alsoRef. [113]), when we consider a quantum tunneling as inthe open inflation scenario [114–117]. In this respect, itwould be interesting to further study constraints on thespatial curvature.

Here, it should be noted that these constraints are onlyapplied to the primordial tensor modes generated by thestandard mechanism. First, for the tensor-to-scalar ratiowith such a small amplitude, it is necessary to take intoaccount non-linearly generated tensor modes [118–121] aswell as foreground contamination. Moreover, several non-standard mechanisms are known to generate large ten-sor modes [122–126]. These non-standard mechanismsshould be also tested with the swampland conjectures

as well as observations of e.g. the scalar perturbations[127–130]. A possible viable example is the generationfrom spectator non-Abelian (e.g. SU(2)) gauge fields cou-pled to a pseudo scalar axion [123, 131]. In this mech-anism, quantum fluctuations of the gauge fields can beenhanced through a transient instability and source largetensor modes. It has been argued in Ref. [130] that thetensor-to-scalar ratio can reach to r ∼ 10−3 at a spe-cific scale even for a very low energy scale of inflationV 1/4 ' 30 MeV, which is well below the upper limit(7). This scenario should be further investigated if itis consistent with the other swampland conjectures andcan be realized in the quantum-gravity landscape (seee.g. Ref. [132] for discussion). It would be interesting toexplore this point further.

The observational TCC also constrains the maximumvalue of the reheating temperature, TR. Since the maxi-mum reheating temperature is realized for the instanta-neous reheating, the inequality (7) can be rewritten asthe upper limit on the reheating temperature [23],

TR < O(108) GeV , (11)

or

TR < O(108)

∣∣∣∣ΩK,obsΩK,ini

∣∣∣∣ 23

GeV , (12)

when the flatness problem is taken into account. Theseare in tension with the thermal leptogenesis [133], whichrequires TR & 109GeV [134, 135].

Next, let us point out that OTCC does not reproduceall known predictions in the literature. To clarify thispoint, let us formulate OTCC as a swampland condition.For the TCC condition (2) to be satisfied in our Universe,its minimal requirement for an effective field theory isstated as:

Inside a given low-energy effective theory, there ex-ists at least one realization satisfying the TCC con-dition (2). Let us consider the possibility that we live in a

metastable vacuum with a positive energy. Imposingthe TCC condition (2) on the trans-Planckian modesat the present time, it has been concluded in Ref. [17]that our Universe should terminate within a finite “ter-mination time” tend, whose upper limit was given bytend . H−10 log(MPl/H0).2 A stronger bound of the ter-mination time, tend . H−10 log(MPl/Hinf), is obtained byrequiring the TCC condition (2) on the trans-Planckian

2 This is the cosmological counterpart of the scrambling time, seeRef. [136] for a recent related discussion.

7

modes at the beginning of inflation. 3 Suppose that thepresent vacuum decays into another vacuum with a meanlifetime τ . From the argument above, one may arguethat OTCC excludes the theories with metastable vacuawhose mean lifetime τ is larger than the termination timetend.

This conclusion, however, is not correct — the de-cay of a vacuum, quantum tunneling to another vac-uum, occurs probabilistically and there is always a non-zero probability that a vacuum decays in a short time,pdecay(tend) = 1 − e−tend/τ . In formulating OTCC, weclaimed that our Universe can be maximally atypical.Therefore, as long as the probabilisitic interpretation ofquantum mechanics is taken, it is possible to evade theupper limit on the termination time tend simply by as-suming that we are living in such a short-lived realiza-tion. This means that OTCC only requires a finite meanlifetime τ < ∞ and there is essentially no constraint onit. As far as τ is finite, it is possible for the metastablevacuum to decay before the TCC violation. Interest-ingly, the Standard Model is completely compatible withOTCC, as the electroweak vacuum is unstable. The es-timated lifetime of the electroweak vacuum is extremelyhuge [137–140], but has a finite value.

A similar argument opens up the possibility of theold inflation scenario [3–5]. The old inflation scenario isusually excluded by the graceful-exit problem. However,there is a tiny probability that our patch of the Universesimultaneously decays into the Big Band phase after asufficiently long period of inflation. Therefore, once oneaccepts our atypicality, the old inflation scenario is notexcluded yet.

These discussions illustrate an unsatisfactory point inthe formulation of OTCC. At first sight, OTCC seems togive a proper swampland condition — it selects the the-ories that allow at least one TCC-consistent realization.However, in order not for us to observe TCC violations,we should additionally assume that our Universe is one ofsuch TCC-consistent realizations. It is not clear aprioriwhy this should be satisfied in the absence of any selectionrule of realizations. If it is impossible to justify this ex-treme atypicality of us, only possibility is to allow a smallprobability to observe the TCC violation, at the cost ofmaking the trans-Planckian censorship more loose. Inthe next section, we will consider such a formulation ofTCC, the probabilistic interpretation of TCC.

3 This condition is derived by combining eNinf+Nafter+H0tend 'MPl/H0 with the horizon-crossing condition (5). To be exact,the definition of tend here is different from the former case. Thetermination time tend is measured from the present time here butfrom the beginning of the current accelerated expansion for theformer one. However, this difference is ∼ H−1

0 and negligible.

VII. PROBABILISTIC TCC

Instead of complete eliminations, let us here allow fora small fraction of TCC-violating realizations and assignprobabilities to observe them. One approach to formu-late TCC along these lines is to appeal to our usual in-tuition that our examples of TCC-violating realizationsare probabilistically rare. This motivates us to proposethe following probabilistic interpretation of TCC:

Probabilistic TCC (PTCC): the TCC condition(2) is obeyed in almost all realizations, except thatTCC-violating realizations with “small” probabili-ties are tolerated.

EFT

ai(t)aj(t)Our Universe

aour(t)

a1(t)

· · ·

TCC-violating Realizations

pj

p1

pipour

FIG. 7. An example of EFT in the swampland for PTCC.PTCC claims that the TCC condition (2) is obeyed in almostall realizations except that TCC violating realizations with“small” probabilities are tolerated. In this figure, PTCC isviolated when the probabilities p1, . . . , pj for TCC-violatingpossibilities (given by scale factors a1(t), . . . , aj(t)) are notsufficiently “small” under a given probability measure. Thisinevitably introduces the dependence on the probability mea-sure.

This version of TCC removes the previous objection: itdoes not exclude the Standard Model of particle physicsbecause the examples of TCC-violating realizations dis-cussed above have very small probabilities. This versioncould in principle work as a swampland condition, a re-striction of effective field theories. It is, however, unsat-isfactory as a trans-Planckian “censorship” constraint,since the “censorship” is incomplete. From the phe-nomenological standpoint, PTCC can effectively work asTCC with a selection rule of realizations but no violationof principles of quantum mechanics. For theories allowedin PTCC, cutting off TCC-violating realizations barelychanges predictions for average values of observable, suchas the primordial power spectra, because rare realizationsdo not contribute much to the average.

A problem of this proposal is that it depends on howto define the “small” probabilities — a larger thresholdof probability eliminates a larger number of effective fieldtheories. How rare a realization is accepted, and why?

The threshold is also directly relevant to some ob-servable predictions. For example, as already men-tioned, TCC claims that a metastable de Sitter vac-uum must terminate by a termination time tend < T 'H−10 ln(MPl/Hinf). For the present value of the Hubble

8

parameter H0, the upper limit T is much larger than theobserved duration of the current accelerated expansion.Let us consider the case that the present metastable vac-uum will decay into another vacuum with a mean lifetimeτ . In this case, the probability of survival at t = T , or theTCC violation, is estimated to be psurvival(T ) = e−T/τ '(Hinf/MPl)

1/(H0τ). Therefore, if we take the thresholdlarger than Hinf/MPl = O(10−5)

√r, the lifetime should

be smaller than the observed duration of the current ac-celerated expansion τ . tage ∼ H−10 . Then, we may havealready observed signs of the vacuum decay.

More fundamentally, PTCC depends on the choice ofthe probability, inevitable leading to the measure prob-lem, the non-uniqueness of probabilities in cosmologicalsetups (see e.g. Refs. [141–144] for reviews). It is possibleto come up with some measure by choosing a particulartime slice (e.g., by proper-time or conformal time), how-ever this breaks the covariance of the measure, and thedifferent choices can lead to vastly different probabilities.

Even when the probability is defined, there is a sepa-rate question: typicality of our Universe. 4 In the formu-lation of PTCC, it is implicitly assumed that our Uni-verse is typical — if the probability of the TCC-violatingrealization is small, we rarely observe it.

Is it appropriate to simply assume that our Universe istypical? Let us discuss an example for possible atypical-ity of our Universe. First, it can be argued that PTCCdisfavors the dynamical relaxation of the cosmologicalconstant through quantum tunnelings in the landscapeof metastable de Sitter vacua, because it will experiencea large number of de Sitter phases and eventually vio-late the TCC condition (2). If so, we have to assumethat our Universe has a tiny value of the cosmologicalconstant from the initial time, which would be extremely“unlikely” in the string landscape. This means that if ourtypicality is strictly assumed, we are excluding our ownUniverse —we concluded that we should not be present.

The typicality highly depends on the choice of theprobability measure. 5 One possible way out is to intro-duce anthropic factors to the probability measure. Sup-pose that in formulating PTCC we use the probabilitymeasure which is highly peaked when “we” are present.This anthropic modification could make our presence typ-ical.

However, it introduces another ambiguity in the for-mulation of PTCC. It is far from clear what is meant by“we”—is this an intelligent life, and then what counts as“intelligence life”? We can tentatively propose some cri-teria, however, it is not clear why we should adopt sucha measure. The problem is especially sharp when we

4 Typicality (“principle of mediocracy” in the language ofRef. [145]) in cosmological setups have been discussed in theliterature. See e.g. Refs. [146–150] for a variety of viewpoints ontypicality.

5 Our typicality is not guaranteed under some probability mea-sures, as pointed out in Refs. [141–144].

remember that we wish to formulate PTCC as a swamp-land conjecture. If anthropic considerations are neededfor formulating a swampland conjecture, it seems we arearguing that our existence is strictly required for the con-sistency of UV completion in quantum gravity. This iscertainly a radical idea about quantum gravity.

VIII. IS TCC A SWAMPLAND CONJECTURE?

In this paper, we discussed the question of whether ornot the Trans-Planckian Censorship Conjecture (TCC)qualifies as a swampland conjecture.

We first emphasized that the concept of swamplanduniversality, i.e. that a swampland conjecture constrainsany possible scenario in a given effective field theory, aslong as it has a non-zero probability of realization, irre-spective of whether or not it is realized in our Universe.This means that TCC can be applied to Universes whichwe do not necessarily observe, and hence can lead to morestringent constraints than are hitherto discussed in theliterature.

We have seen that TCC formulated with the swamp-land universality, Universal TCC (UTCC), leads to con-tradictions with the well-established theory, the Stan-dard Model of particle physics and cosmology. A pos-sible way out is to introduce a selection rule that elim-inates all TCC-violating realizations. For example, wecan restrict to classical configurations, as formulated inClassical TCC (CTCC). However, this leads to contradic-tions again with the Standard Model of particle physics.It would be difficult to find a successful selection rule,keeping the law of quantum mechanics.

We also discussed the possibility of applying TCC toour particular Universe, Observational TCC (OTCC). Itis a swampland conjecture that picks up theories withat least one TCC-consistent Universe. At first sight, itwould give the TCC constraints in the literature. How-ever, we have argued that this is not the case for thedecay of a metastable de Sitter vacuum, or more gener-ally when probabilities inherent in quantum mechanicsare involved with the cosmological evolution. This is be-cause we now allow for maximal atypicality of us — weshould be living in one of the TCC-consistent Universesin order not to observe the violation of the TCC condi-tion (2). In this formulation, there remains a problemhow to justify this atypicality of us.

We have therefore introduced a probabilistic interpre-tation to TCC, Probabilistic TCC (PTCC). In this pro-posal, a small probability of the TCC violation is toler-ated. This version could still work as a swampland con-dition, which eliminates theories with large probabilitiesof the TCC-violating realizations, although it is unsatis-factory as a trans-Planckian censorship constraint.

Note that all these different versions of TCC lead todifferent conclusions for practical problems. For example,concerning the presence and the lifetime of a metastablede Sitter vacuum, the consequences are:

9

UTCC/CTCC: The metastable de Sitter vacuum isnot allowed for any mean lifetime.

OTCC: Any finite value of the mean lifetime is allowed.

PTCC: The mean lifetime has a finite upper limit de-pending on the definition of the probability.

As is clear from this example, PTCC depends non-trivially on the choice of the probability measure. Wepointed out that PTCC suffers from the measure prob-lem, i.e. the non-uniqueness of the probability measure,and that the measure would likely be anthropic. Onemight still hope that unknown (or even known) knowl-edge of quantum gravity naturally and inevitably picksup a unique measure, hence solving the measure problem.If this is the case, then at least the concept of probabilityis well-defined.

Irrespective of the measure problem, once we acceptthe probabilistic interpretation, then it seems very likelythat swampland conjectures in general should be formu-lated probabilistically under the same measure — the vio-lation of the conjectures is allowed with a small probabil-ity. This is a rather dramatic consequence. For example,the swampland conjecture of no exact global symmetry[151–155] is believed to be a statement applicable to anypossible realization in the landscape. However, the prob-abilistic nature of the swampland conjecture, if generallyapplied, suggests that some theories can survive if theypredict a small probability of symmetry non-conserving

processes.

In any formulation, we have seen that there are severalchallenges in considering TCC as a swampland condition— one needs to introduce any one of (unknown) selec-tion rule, our atypicality (i.e. anthropic selection rule),or the TCC violation with a small probability. These donot immediately reject TCC but their justification willrequire deeper understanding of quantum gravity. Onecould also argue that TCC is not a swampland condi-tion after all, i.e. a constraint not on theories but justsolutions. While this could be possible in principle, it re-quires modification of possible predictions extracted froma given theory — this is more radical requirement thanthe standard swampland conjectures.

ACKNOWLEDGEMENTS

We would like to thank Matthew Johnson, MisaoSasaki, Neil Turok, Daisuke Yamauchi and Beni Yoshidafor related discussions. M.Y. would like to thank Perime-ter institute for hospitality during the final stages ofthis project. This research was supported in partby WPI Research Center Initiative, MEXT, Japan,and by the JSPS Grant-in-Aid for Scientific Research(R.S: No. 17K14286, No. 19H01891, S.S: No. 17H02878,No. 18K13535, No. 19H04609, M.Y: No. 17KK0087,No. 19K03820, No. 19H00689).

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