Rejuvenating the hope of a swampland consistent inflated multiverse with tachyonic inflation in the high energy RS-II Braneworld
aa r X i v : . [ h e p - t h ] J a n Rejuvenating the hope of a swampland consistent inflated multiverse with tachyonicinflation in the high energy RS-II Braneworld
Oem Trivedi ∗ School of Arts and Sciences, Ahmedabad University,Ahmedabad 380009,India (Dated: January 5, 2021)The swampland conjectures from string theory have had some really interesting implications oncosmology, in particular on inflationary models. Some models of inflation have been shown to beincompatible with these criterion while some have been shown to be severely fine tuned, with most ofthese problems arising in single field inflationary models in a General relativistic cosmology. Recentworks have although optimistically shown that single field models in more general cosmologies canbe consistent with these conjectures and hence there is an optimism that not all such models lie inthe swampland. However a paradigm of inflation which has been shown to not be perfectly okaywith the conjectures is eternal inflation. So in this work, we discuss Tachyonic inflation in the highenergy RS-II Braneworld scenario in the context of the swampland conjectures while also consideringthe possibility of swampland consistent eternal inflation. We show that our concerned regime evadesall the prominent swampland issues for single field inflation being virtually unscathed. After this,we show that the main conflicts of eternal inflation with the swampland can easily be resolvedin the considered tachyonic scenario and in particular, we also discuss the exciting prospect of aGeneralized Uncertainty Principle facilitating the notion of Swampland consistent eternal inflation.Our work as a whole reignites the possibility that there can be a swampland (and possibly, quantumgravitationally) consistent picture of a ”Multiverse”.
I. INTRODUCTION
The idea of Cosmic Inflation has achieved a tremen-dous amount of success in describing various propertiesof the early universe [1–5]. Numerous predictions ofInflation for the early universe have been repeatedlyvalidated by various satellite experiments, and the mostrecent data from the Planck experiment follows thistrend [6–9]. Further the observational data supports ahuge variety of Inflationary models which are motivatedfrom vastly different backgrounds, from Modified gravitytheories to quantum gravitational realizations [10–20].Inflation has not only been thoroughly studied in regimeswhere the inflaton is a scalar or a vector field, but also forvery non standard scenarios as well where the Inflaton isa complex scalar, a tensor or even a tachyonic field [21–31] . A form of Inflation which has attained widespreadinterest in the cosmological community in recent decadesis Eternal inflation [32–43]. Perhaps the most starklingoutcome of Inflation continuing eternally is the pro-duction of a ”Multiverse”, as Inflation does not haveto stop everywhere at once and can keep on happeningin some parts of space while it ceases in some other parts.There has been a lot of work dedicated towards a” Theory of Everything ” in recent years, and arguablythe most well known candidate for such a paradigm isString Theory [44–52]. As String theory presents itself insuch a vivid mannerism, one can reasonably expect thistheory to have wide ranging implications for cosmology.Consequently, there is a rich and diverse amount of ∗ [email protected] literature which has explored the cosmological impli-cations of the ideas of String theory [53–60]. Amongstthe many cosmologically intriguing features of stringtheory is the incredibly high amount of possible vacuastates it allows, which goes as high as O (10 ) and thisgoes to constitute what is known as the ”landscape” ofstring theory. A natural question which then arises isexactly what class of low energy effective field theoriesare actually consistent with String theory. In a bidto answer this question, Vafa introduced the term”Swampland” to refer to the class of low energy effectivefield theories which are inconsistent with the frameworkof String theory. [61]. Further, in recent years a numberof field theoretic UV completion criterion from stringtheory called the ”swampland conjectures ” have beenproposed [62–66] which classify whether a given regimelies in the swampland or not. As string theory is alsoseen by many as a viable paradigm of quantum gravity,if a low energy EFT satisfies these criterion then itcould also potentially be on amicable terms with a self-consistent theory of quantum gravity. While there havebeen a considerable number of swampland conjectures(and their modifications) proposed in recent times, theconjectures which have had very telling cosmologicalimplications are :1 : Swampland Distance Conjecture (SDC) : Thisconjecture limits the field space of validity of anyeffective field theory [63] . This sets a maximum rangetraversable by the scalar fields in an EFT as∆ φm p ≤ d ∼ O (1) (1)where m p is the reduced Planck’s constant, d is someconstant of O (1) , and φ is the Scalar Field of the EFT.2 Swampland De Sitter Conjecture (SDSC) : ThisConjecture states that it is not possible to create dSVacua in String Theory [62]. The conjecture is a resultof the observation that it has been very hard to generatedS Vacua in String Theory [67, 68]( While it has beenshown that creating dS Vacua in String Theory ispossible in some schemes ,like the KKLT Construction[69]). The Conjecture sets a lower bound on the gradientof Scalar Potentials in an EFT , m p | V ′ | V ≥ c ∼ O (1) (2)where c is some constant of O (1) , and V is the scalarField Potential. Another ” refined ” form of the Swamp-land De Sitter Conjecutre (RSDSC) places constraints onthe hessian of the scalar potential (a finding which firstappeared in [64] and later in [70] ) and is given by m p V ′′ V ≤ − c ′ ∼ O (1) (3)These conjectures have pretty interesting implica-tions on cosmology and in particular on single fieldinflation. It was shown in [71] that single field inflationin a GR based cosmology is not consistent with theseconjectures, considering the data on Inflation [6, 7]. Alot of work has been done to alleviate this conflict [72–75]. It has ,however, been shown that if the backgroundcosmology for single field inflation is not GR based thenthis regime of inflation can still be consistent with theswampland criterion [76–79]. Particularly, in [80] it wasshown that (cold) single field inflation can be consistentwith these criterion in a large class of non-GR basedcosmologies. It’s also worth noting that the paradigmof warm inflation [13] has also been shown to be quiteconsistent with the swampland criterion for singlemodels in both GR and non-GR based cosmologies[81–87]. Another swampland conjecture which hasgathered quite an immediate interest with regards toinflationary cosmology is the recently proposed ”TransPlanckian Censorship Conjecture”(TCC) [66]. It wasshown in [88] that single field GR based inflationarymodels can only be consistent with the TCC if they areseverely fine tuned, which is quite ironic considering thatinflation was made to solve the fine tuning problem instandard big bang cosmology. A lot of work has furtherbeen done to understand the issues of the TCC withsingle field inflation [76, 89–94]. In [95], it also shownthat the TCC can actually be derived from the distanceconjecture (1) considering that latter criterion is true.Single field models in non standard inflationary regimeshave been shown to not be severely fine tuned due to theTCC, unlike their counterparts models in a GR basedcosmology [96, 97]. The current literature on inflationand the swampland criterion hence suggests that theseconjectures support the notion that early universe expansion happened in a non standard inflationaryregime, considering that inflation was the cause of theexpansion (and also with the added consideration ofonly single field models, because multi field models havebeen shown to be on a very good standing with theconjectures in even the standard paradigm [98, 99]).While it has been shown that some paradigms ofinflation can be rescued from the swampland withdifferent schemes, eternal inflation has been one peculiarinflationary regime which has not been fully consistentwith all the swampland conjectures till now. Theconflicts of eternal inflation and the swampland werefirst highlight by Matsui and Takahashi in [100], wherethey showed that one of the core requirements for eternalinflation was in direct conflict with the dS conjecture (2).Dimopoulos then considered eternal inflation on steeppotentials with a turning point at ˙ φ = 0 (where φ is theinflaton field) and found that these models of eternalinflation were also in conflict with the dS conjecture. Itwas then shown by Kinney [101] that eternal inflationcan be consistent with the Refined dS conjecture (3),and this could prevent eternal inflation from being inthe swampland( as it has been shown that consistencywith either of (2) or (3), if not both, can prevent aparadigm from plunging into the swampland [64] ). Butit was then shown by Brahma and Shandera in [102] thateternal inflation can actually not be consistent with theRefined dS conjecture as well. In addition to this, Wanget.al [103] showed that the Gibbons-Hawking entropybounds for eternal inflation are also not consistent withthe swampland conjectures , which is again a veryserious issue for eternal inflation. Hence, it does looklike eternal inflation has some unavoidable conflicts withthe conjectures. However, a key point all of the venturesdiscussed above is that they considered eternal inflationin a GR based cosmology.The situation does look a tadbit better in more general cosmologies , as Lin et.alshowed in [104] that certain hilltop models of eternalinflation in a braneworld cosmology can be consistentwith the dS conjecture and its refined form. This isencouraging and could point towards the idea that inorder for eternal inflation to be consistent with theswampland conjectures, one should look towards regimeswhich are motivated by String theory (as braneworldcosmology also has its roots in string theory [105–107]).A regime of inflation which was inspired from stringtheory itself is tachyonic inflation [21], where the infla-ton field is considered to be of tachyonic nature. Thisparadigm of inflation has been shown to be consistentwith the swampland criterion in a GR based cosmologyfor a warm inflationary setup [28] under some condi-tions. Tachyonic inflation has also been studied in abraneworld background cosmology [24, 29, 108], so itwould be quite interesting to see if tachyonic inflationon the RS-II Brane is consistent with the conjectures,as these scenarios for single field cases have always beenon relatively amicable terms with the conjectures ascompared to GR based paradigms. Further, eternalinflation with tachyonic fields in the braneworld canpresent a great chance of being consistent with theconjectures and the reasons for that are two-fold. First,braneworld inflationary scenarios have proven to be agreat fit with the swampland conjectures in a diverserange of regimes and this cosmological setup, as notedabove, traces its roots in string theory itself. Secondly,tachyonic models of inflation are also inspired by stringtheoretic properties and hence, one can be optimisticin exploring the consistency of such a string influencedregime with the conjectures for eternal inflation. Indeed,in our paper we show that this is indeed the case andhence, this paper is structured as follows. In sectionII, we discuss some of the basic features of tachyonicinflation after which we show that tachyonic inflation-ary models in the high energy RS-II Braneworld arevirtually unscathed by the swampland conjectures andare consistent with all of the criterion individually too.Then, in Section III we show that eternal inflation withtachyonic scalar fields in a braneworld scenario suffersno issues with the swampland conjectures and finally inSection IV, we conclude with some final remarks on ourexplorations. II. TACHYONIC INFLATION IN THE RSIIBRANEWORLD AND THE SWAMPLANDCRITERION
We start with the 4-dimensional action of the tachyonfield minimally coupled to gravity, which can be writtenas [21, 23] S = Z h m p R − V ( φ ) p − g µν ∂ µ φ∂ ν φ i √− gd x (4)The stress energy tensor in a spatially flat FLRW metric ds = − dt + a ( t ) d x is given by T µν = ∂ L ∂ ( ∂ µ φ ) ∂ ν φ − g µν L = diag ( − ρ φ , ˜ p φ ) (5)where L = √− g (cid:2) m p R − V ( φ ) p − g µν ∂ µ φ∂ ν φ (cid:3) is theLagrangian density of the Tachyon field , V ( φ ) is thepotential, while ρ φ and p φ are the energy and pressuredensities of the field given by ρ φ = V ( φ ) q − ˙ φ (6) p φ = − V ( φ ) q − ˙ φ (7)The Friedmann equation for a Randall-Sundrum IIBraneworld cosmological scenario is given by [ for details on this, please refer to [109–112]] H = ρ m p h ρ λ i + Λ ωa (8)where we are working in c = ~ = 1 units, with ρ being theenergy density, λ being the brane tension , Λ being the5D cosmological constant and ω being the so called ”darkradiation ” term. As we are considering the very earlyuniverse, we can ignore the cosmological constant term,whereas the dark radiation term vanishes rapidly due tothe a − dependence. So, we can write the friedmannequation during inflation as H = ρ φ m p h ρ φ λ i = 13 m p V ( φ ) q − ˙ φ λ V ( φ ) q − ˙ φ (9)From the action, one can arrive at the field equation ofmotion as [23], ¨ φ − ˙ φ + 3 H ˙ φ + V ′ V = 0 (10)while one can also use energy conservation to write ,˙ ρ φ + 3 H ( ρ φ + p φ ) = 0 (11)In the slow roll limit of inflation with tachyonic scalarfields, we have ˙ φ << φ << H ˙ φ , which allows usto write the Friedmann equation and the field equationof motion as, H = V m p h V λ i (12)3 H ˙ φ + V ′ V ≅ H = V m p in the low energy limit λ >> V . Hence, to better illus-trate the effects of the braneworld scenario for inflation,one can instead consider the high energy limit of the pic-ture V >> λ , which allows us to now write the Fried-mann equation (12) as, H = V λm p (14)The Number of e-folds can be written using it’s usualdefinition for some initial field value φ i to some final value φ e as, N = Z φ e φ i H ˙ φ dφ (15)And using the approximation (13), we can then write thee-fold number as N = Z φ i φ e H VV ′ dφ (16)Further, using the Friedmann equation (14), one canwrite N as N = Z φ i φ e V λm p V ′ dφ (17)A particularly important class of parameters for any in-flationary model are the slow roll parameters [10] and inparticular, the ǫ and η parameters. One can define theseparameters for tachyonic brane inflation through theirusual definitions in the high energy limit as , ǫ = − ˙ HH = 2 λm p V ′ V (18) η = − ¨ HH ˙ H = 2 √ λ (cid:20) V ′ m p V ′′ V − V m p V ′ V (cid:21) (19)Further the power-spectrum of the curvature perturba-tion P R that is derived from the correlation of first orderscalar field perturbation in the vacuum state can be writ-ten as [113] P R ( k ) = (cid:18) H π ˙ φ (cid:19) V (1 − ˙ φ ) (20)where in the slow roll limit, one can ignore the contribu-tion of ˙ φ . One can then further find out other observa-tionally revelant perturbation parameters like the scalarand tensor spectral index, tensor-to-scalar ratio etc. us-ing their standard definitions [21, 23, 113]. We will notillustrate that here as we have built enough groundworkto start seeing why tachyonic brane inflation is consis-tent with the swampland in the high energy regime. Theprominent issues of conflict between the swampland con-jectures and single field inflation which were discussed inlength in [71] can be briefed as follows. One of the issuesconcerns the dS conjecture and inflationary requirementfor the ǫ parameter, as it was shown that if one con-siders the dS conjecture seriously than the requirementthat ǫ << N << η parameter from observational dataon inflation [7] and the implications that the refined dSconjecture has on the same parameter.We will now elab-orate under what conditions Tachyonic Inflation on theRS-II Braneworld evades all of the issues of single field in-flation with the swampland conjectures. Firstly, focusingon. The ǫ parameter (18), can be written as ǫ = 2 λV (cid:18) m p V ′ V (cid:19) (21) Considering the dS conjecture(2) in this scenario, we canwrite ǫ > λV c (22)As c ∼ O (1) , in order for ǫ << V >> √ λ (23)Hence, it would appear that the dS conjecture implies aconstraint on the energy scale of inflation ( as during in-flation ρ = ρ φ ≅ V ) for tachyonic inflation on the RS-IIbraneworld. But, it is far from the case as we recall thatthe formulation built above is in the high energy limit ofthe braneworld, which characterized by V >> λ >> . Soin reality, the tachyonic inflation in the braneworld in thehigh energy limit is not strong constrained by the dS con-jecture and is virtually unscathed. Similarly, the changein the e-fold number during inflation can be written using(17) as, ∆ N ≅ V λ ∆ φm p (cid:16) m p V ′ V (cid:17) (24)Now considering both the distance (1) and dS conjecture, we see that term in the square brackets in (24) hasto be less than unity. This shows that in order to havesufficient amount of e-folds, one again needs V >> √ λ (25)and once again as we are in the high energy limitalready V >> λ the requirement (25) does pose a verystrong constraint on the energy scale of inflation in sucha regime which might have otherwise ruled out someinflationary models in this scenario. So this makes itclear that there is no issue of a insufficient e-fold numberfor this inflationary regime. Finally we note that evenif one considers both the dS conjecture and the refineddS conjecture (3) to simultaneously hold true andapplies them the on η parameter (19), the observationalrequirement for η ≤ III. SWAMPLAND CONSISTENT ETERNALINFLATION
As discussed in section I, recent works have shown thateternal inflation is not on an amicable standing with theswampland conjectures . One of the key requirementsin order for eternal inflation to take place in any kindof regime that quantum fluctuations of the inflaton fieldshould dominate over the classical field evolution. Theamplitude of quantum fluctuations on scales of order theHubble length is given by (the following expression isindependent of the underlying gravitational theory [23]) < δφ > Q = H π (26)while the classical field variation in Horizon time H − isgiven by δφ cl = ˙ φH (27)The requirement that quantum fluctuations dominate theclassical field evolution hence translates into < δφ > Q δφ cl = H π ˙ φ > P R ( k ) > φ <<
1, one gets P R ( k ) ≅ (cid:18) H π ˙ φ (cid:19) V (30)From (30), it is immediately apparent that the condition(28) does not set a direct requirement on the amplitudeof curvature perturbations for a tachyonic field like itdoes for a usual scalar field. Further for a large enoughenergy scale , just like the high energy limit of theRS-II Braneworld considered here, one can easily have P R ( k ) < P R ( k ) ≥ S BGH = 4 πm p H (31)During a period in stochastic inflation when the inflatonmoves up the potential due to quantum fluctuations, Hincreases and thus the entropy of the area bounded by thehorizon decreases. For the process above to be consistentwith the second law of thermodynamics even after takinginto account quantum gravitational effects, the variationof the entropy above is bounded as [118] δS > − δS = − πm p V ′ λV δφ (33)further, considering the change in entropy during a timein which using δφ = H π , we have δS = − m p λV m p V ′ V √ λ (34)Now, employing the criterion (32) leads to the followinginequality m p V ′ V < V √ λm p (35)The dS conjecture sets the limit m p V ′ V ≥ O (1), hence inorder for the above inequality to be consistent with thedS conjecture we need V >> √ λm p (36)Hence the dS conjecture sets a lower limit on the en-ergy scale of tachyonic inflation in the high energy RS-IIbraneworld, for it to be on amicable terms with the en-tropy bounds. We note that the bound to bypass theentropy issue (36) is in the range with lower limit onthe inflation energy scale(25) needed to evade the pri-mary issues of the swampland and single field inflationdiscussed in Section II. So the crux of the matter hereis that in order for the eternal inflation entropy boundsto be consistent with the dS conjecture in our concernedtachyonic regime, we need to set a higher lower bound onthe energy scale of inflation which is still consistent withthe bounds needed to sort out the primary inflationaryissues with the swampland. There is, however, anotherway to go about the entropy bound issue. The eventhorizon entropy considered here is through the standardBekenstein-Gibbons-Hawking entropy formulation, whilein recent years there has been a lot of work which has ex-plored corrections needed in (31) in order to accommo-date the newly emerging physics from string theory andloop quantum gravity (LQG) [119–130]. Several of theseapproaches have the quantum-corrected entropy-area re-lation in a general form in terms of (31) as S = S BGH + C o ln( S BGH ) + ∞ X n =1 C n ( S BGH ) − n (37)where C n are parameters which are model dependent(like recent works have determined C o = − for LQG[124]). It was also interestingly pointed out in [131] thatHeisenberg’s Uncertainity principle might be affect byquantum gravitational effects. Since then, a significantamount of work has gone into finding a Generalized Un-certainty Principle(GUP) and looking for the quantumcorrections it provides to the usual Bekenstein-Gibbons-Hawking formulation of local event horizon entropy. Thiseventually led to the following form of the quantum cor-rected entropy due to a GUP, [123, 125, 128, 129] S = S BGH + √ πα o p S BGH − πα o
64 ln( S BGH ) + O ( 1 m p )(38)where α o is dimensionless constant which is found in thedeformed commutation relations and can be usually con-sidered to be positive in nature [130]. The leading ordercontribution to the GUP corrected entropy formalism isdue to the √ S BGH term, which is an extra term to the al-ready existing logarithmic correction to entropy derivedfrom the quantum gravity effects. Considering this formof entropy relation as the one for entropy of a local eventhorizon for an expanding cosmology has provided prettyinteresting insights (see for example, [132]). Hence, wenow consider the form (38) of the entropy for the bound(32) and as we would like to entertain in particular theimplications of the very interesting GUP effects , we onlyconsider the leading order GUP correction to the entropywhich provides us with S GUP ≅ S BGH + √ πα o p S BGH (39)which can be written in terms of the Hubble Parameteras S = m p πH (cid:20) m p H + α o (cid:21) (40) Using the Friedmann equation (14), we can now writethe entropy in terms of the potential S = m p π √ λV h m p √ λV + α o i (41)This finally allows us to express the bound (32) in termsof the GUP corrected entropy as m p V ′ V π √ λ " π √ λV m p √ λV + α o ! + 4 m p √ λV m p π √ λV ! < m p V ′ V < π √ λ πm p λV + π √ λα o V (43)The dS conjecture constraints m p V ′ V ≥ c ∼ O (1), hencein order to maintain consistency with it the followinginequality will have to hold2 π √ λ > πm p λV + π √ λα o V = ⇒ V − V α o − m p √ λ > λ given by λ > α o m p (45)And also a lower bound on the inflation energy scale,which keeping in mind the brane tension bound can bewritten approximately as V >> α o α o which is found from thedeformed commutation relations for a GUP is incrediblyexciting. In this scenario, the existence of a GeneralizedUncertainty Principle facilitates the creation of a swamp-land consistent eternal inflationary scenario and conse-quently, a swampland consistent multiverse. To makethis peculiar relation even more clear, we would like todemonstrate that the consideration of a GUP correctedentropy formalism would also allow us to evade the dSconjecture-entropy bound conflict in a simple scalar fieldmodel in a GR based scenario. For GR based single fieldmodels , the Friedmann equation during the expansionphase is H = V m p (47)The GUP corrected entropy (39) for this regime becomes S = m p π r V m p r V + α o ! (48)And hence, the bound (32) in this case takes the form m p V ′ V < Vm p H h m p + α o √ V i (49)It is worth mentioning the difference the GUP correctedhas already made in eradicating the entropy bound is-sue for this inflationary regime. The above inequalityclearly shows that it quite possible to satisfy the con-straints due to the dS conjecture on the left hand side ofthe inequality, given that there is some minimum limiton the energy scale of inflation to make the right handside of the inequality sufficiently large. Doing this wouldmean that the numerator on the right hand side far dom-inates over the denominator which using the Friedmannequation (47), can be written as the following inequality V / > m p − α o √ (50)Hence one can eradicate the issue between the dS con-jecture and entropy bounds by considering a GUP cor-rected entropy formalism by constraining the energy scaleduring inflation in accordance with the above bound for α o < √ . Once again, we notice how crucial a rolethe Generalized Uncertainty Principle based correctionsmake in allowing us to evade the entropy bound issueseven in a usual scalar field model based in a GR basedcosmology. This in no way shows that eternal inflation insuch a regime is consistent with the swampland, however,as other issues regarding the conjectures still hold true[100–102, 114]. But this does illustrate that GUP basedquantum corrected entropy does facilitate positively inremoving the entropy bound-dS conjecture conflict inmultiple regimes of inflation. Thus we can draw a veryexciting conclusion from this analysis that consideringa generalized uncertainty principle makes it more easierfor one to have a swampland consistent (and consideringthe premise of the conjectures, a quantum gravitationallyconsistent) picture of a multiverse generated by eternalinflation. IV. CONCLUDING REMARKS ANDDISCUSSION
In conclusion, in this work we have discussed tachy-onic inflation in the high energy RS-II Braneworld sce-nario in the light of the swampland conjectures and con-sidered the possibility of swampland consistent eternalinflation in this regime. We started off by describingwhy tachyonic inflation in this particular scenario couldpossibly be one of the most suited arenas for the swamp-land conjectures and hence, might allow for the possi-bility of evading the swampland issues eternal inflationtoo. Then after describing some basics of Tachyonic in-flation in this regime, we showed that this inflationarymodel is virtually unscathed by the swampland conjec-tures and can quite intrinsically evade all the prominentswampland issues which have been discussed for regularsingle field inflation, as they only imply a lower boundon the inflation energy scale which can be easily satis-fied in the considered high energy limit. We then castour glance towards eternal inflation in this scenario, anddiscussed the two main swampland issues in this contextwhere we firstly addressed the curvature amplitude issue.We showed that the basic eternal inflation requirement ofquantum fluctuations dominating over classical field evo-lution does not set unavoidable constraints on tachyonicmodels of any kind and in particular, inflation in high en-ergy RS-II Braneworld. We hence showed that the issuesof inconsistency of the dS and refined dS conjectures withthe perturbation requirement do not hold good for ourtachyonic regime. We then discussed the conflict of en-tropy bounds with the dS conjecture and showed that ourconcerned inflationary regime does not have any issueswith the entropy bound even when the concerned entropyformalism is the standard Bekenstein-Gibbons-Hawkingone, as the lower limit on the energy scale implied for con-sistency with the bound is well within the range neededfor consistency with basic inflationary swampland issues.Then made the case as to why it would be better suitedif we consider quantum corrections to the BGH entropy, and in particular we considered a Generalized Uncer-tainty Principle corrected entropy formalism. In order tofocus on this particular correction to the entropy, we onlyconsidered the leading order GUP corrections to the en-tropy and showed that a GUP formalism of the entropyallows one to evade the entropy bound issues in bothtachyonic inflation in our concerned scenario and even ina usual single field model in a GR based cosmology. Thisallowed us to make a startling observation that a gen-eralized uncertainty principle actually facilitates one tohave a swampland (and hence possibly, a quantum grav-itationally) consistent picture of eternal inflation and aMultiverse. An important thing to note is that we havenot carried out a detailed analysis of eternal inflation inany kind of a tachyonic regime as we have just focused onshowing how the prominent swampland-eternal inflationissues can be resolved in a more quantum gravitation-ally / String theory motivated setting. Indeed, we havenot embarked on exploring the solutions of the Fokker-Planck / Langevin equations for quantum fluctuations ofthe inflaton field for different potentials, which is more ofa standard procedure in the subject. Hence, there mightbe more model dependent requirements for eternal in-flation to take place in a variety for different potentialsthat we have not taken into account here ( like the modeldependent analysis which was done in [102]) as our ex-plorations have been of a more general nature focused onthe prevalent swampland issues currently.
V. ACKNOWLEDGEMENTS
The author would like to thank Dr. SuddhasatwaBrahma (Postdoctoral Research Fellow at McGill Space Institute, McGill University) for his insightful commentswhich led to the initiation of this work. The authorwould also like to thank Dr. Sunny Vagnozzi (Newton-Kavli fellow at at Kavli Insitute of Cosmology (KICC)and the Institute of Astronomy, University of Cambridge)for his very helpful advice during the preparation of thismanuscript. [1] AA Starobinskii. Spectrum of relict gravitational radi-ation and the early state of the universe.
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