Rejuvenating the Hope of a Swampland Consistent Inflated

Home , Cumrun Vafa, Eternal inflation, Nathan Seiberg, Thomas Hertog

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expansion happened in a non standard inflationary 2 Swampland De Sitter Conjecture (SDSC) : This regime, considering that inflation was the cause of the Conjecture states that it is not possible to create dS expansion (and also with the added consideration of Vacua in String Theory [62]. The conjecture is a result only single field models, because multi field models have of the observation that it has been very hard to generate been shown to be on a very good standing with the dS Vacua in String Theory [67, 68]( While it has been conjectures in even the standard paradigm [98, 99]). shown that creating dS Vacua in String Theory is possible in some schemes ,like the KKLT Construction While it has been shown that some paradigms of [69]). The Conjecture sets a lower bound on the gradient inflation can be rescued from the swampland with of Scalar Potentials in an EFT , different schemes, eternal inflation has been one peculiar inflationary regime which has not been fully consistent V ′ with all the swampland conjectures till now. The mp | | c (1) (2) V ≥ ∼ O conflicts of eternal inflation and the swampland were where c is some constant of (1) , and V is the scalar first highlight by Matsui and Takahashi in [100], where Field Potential. Another ” refinedO ” form of the Swamp- they showed that one of the core requirements for eternal land De Sitter Conjecutre (RSDSC) places constraints on inflation was in direct conflict with the dS conjecture (2). the hessian of the scalar potential (a finding which first Dimopoulos then considered eternal inflation on steep φ˙ φ appeared in [64] and later in [70] ) and is given by potentials with a turning point at = 0 (where is the inflaton field) and found that these models of eternal

V ′′ inflation were also in conflict with the dS conjecture. It m c′ (1) (3) p V ≤− ∼ O was then shown by Kinney [101] that eternal inflation can be consistent with the Refined dS conjecture (3), and this could prevent eternal inflation from being in the swampland( as it has been shown that consistency These conjectures have pretty interesting implica- with either of (2) or (3), if not both, can prevent a tions on cosmology and in particular on single field paradigm from plunging into the swampland [64] ). But inflation. It was shown in [71] that single field inflation it was then shown by Brahma and Shandera in [102] that in a GR based cosmology is not consistent with these eternal inflation can actually not be consistent with the conjectures, considering the data on Inflation [6, 7]. A Refined dS conjecture as well. In addition to this, Wang lot of work has been done to alleviate this conflict [72– et.al [103] showed that the Gibbons-Hawking entropy 75]. It has ,however, been shown that if the background bounds for eternal inflation are also not consistent with cosmology for single field inflation is not GR based then the swampland conjectures , which is again a very this regime of inflation can still be consistent with the serious issue for eternal inflation. Hence, it does look swampland criterion [76–79]. Particularly, in [80] it was like eternal inflation has some unavoidable conflicts with shown that (cold) single field inflation can be consistent the conjectures. However, a key point all of the ventures with these criterion in a large class of non-GR based discussed above is that they considered eternal inflation cosmologies. It’s also worth noting that the paradigm in a GR based cosmology.The situation does look a tad of warm inflation [13] has also been shown to be quite bit better in more general cosmologies , as Lin et.al consistent with the swampland criterion for single showed in [104] that certain hilltop models of eternal models in both GR and non-GR based cosmologies inflation in a braneworld cosmology can be consistent [81–87]. Another swampland conjecture which has with the dS conjecture and its refined form. This is gathered quite an immediate interest with regards to encouraging and could point towards the idea that in inflationary cosmology is the recently proposed ”Trans order for eternal inflation to be consistent with the Planckian Censorship Conjecture”(TCC) [66]. It was swampland conjectures, one should look towards regimes shown in [88] that single field GR based inflationary which are motivated by String theory (as braneworld models can only be consistent with the TCC if they are cosmology also has its roots in string theory [105–107]). severely fine tuned, which is quite ironic considering that inflation was made to solve the fine tuning problem in A regime of inflation which was inspired from string standard big bang cosmology. A lot of work has further theory itself is tachyonic inflation [21], where the infla- been done to understand the issues of the TCC with ton field is considered to be of tachyonic nature. This single field inflation [76, 89–94]. In [95], it also shown paradigm of inflation has been shown to be consistent that the TCC can actually be derived from the distance with the swampland criterion in a GR based cosmology conjecture (1) considering that latter criterion is true. for a warm inflationary setup [28] under some condi- Single field models in non standard inflationary regimes tions. Tachyonic inflation has also been studied in a have been shown to not be severely fine tuned due to the braneworld background cosmology [24, 29, 108], so it TCC, unlike their counterparts models in a GR based would be quite interesting to see if tachyonic inflation cosmology [96, 97]. The current literature on inflation on the RS-II Brane is consistent with the conjectures, and the swampland criterion hence suggests that these as these scenarios for single field cases have always been conjectures support the notion that early universe 3 on relatively amicable terms with the conjectures as on this, please refer to [109–112]] compared to GR based paradigms. Further, eternal 2 ρ ρ Λ4 ω inflation with tachyonic fields in the braneworld can H = 2 1+ + + 4 (8) present a great chance of being consistent with the 3mp 2λ 3 a h i conjectures and the reasons for that are two-fold. First, where we are working in c = ~ = 1 units, with ρ being the braneworld inflationary scenarios have proven to be a energy density, λ being the brane tension , Λ4 being the great fit with the swampland conjectures in a diverse 5D cosmological constant and ω being the so called ”dark range of regimes and this cosmological setup, as noted radiation ” term. As we are considering the very early above, traces its roots in string theory itself. Secondly, universe, we can ignore the cosmological constant term, tachyonic models of inflation are also inspired by string whereas the dark radiation term vanishes rapidly due to theoretic properties and hence, one can be optimistic 4 the a− dependence. So, we can write the friedmann in exploring the consistency of such a string influenced equation during inflation as regime with the conjectures for eternal inflation. Indeed, in our paper we show that this is indeed the case and ρ ρ 1 V (φ) 1 V (φ) hence, this paper is structured as follows. In section H2 = φ 1+ φ = 1+ 3m2 2λ 3m2  2λ  II, we discuss some of the basic features of tachyonic p p 1 φ˙2 1 φ˙2 h i − − inflation after which we show that tachyonic inflation- q  q (9) ary models in the high energy RS-II Braneworld are From the action, one can arrive at the field equation of virtually unscathed by the swampland conjectures and motion as [23], are consistent with all of the criterion individually too. Then, in Section III we show that eternal inflation with φ¨ V +3Hφ˙ + ′ = 0 (10) tachyonic scalar fields in a braneworld scenario suffers 1 φ˙2 V no issues with the swampland conjectures and finally in − Section IV, we conclude with some final remarks on our while one can also use energy conservation to write , explorations. ρ˙φ +3H(ρφ + pφ)=0 (11) In the slow roll limit of inflation with tachyonic scalar fields, we have φ˙ << 1 and φ¨ << 3Hφ˙ , which allows us to write the Friedmann equation and the field equation II. TACHYONIC INFLATION IN THE RSII of motion as, BRANEWORLD AND THE SWAMPLAND CRITERION 2 V V H = 2 1+ (12) 3mp 2λ h i We start with the 4-dimensional action of the tachyon field minimally coupled to gravity, which can be written V 3Hφ˙ + ′ ≅ 0 (13) as [21, 23] V R 2 µν 4 it’s immediately apparent that (12) reduces to the usual S = mp V (φ) 1 g ∂µφ∂ν φ √ gd x (4) 2 V 2 − − − Friedmann equation for a GR based cosmology H = 3 2 Z mp h p i in the low energy limit λ >> V . Hence, to better illus- The stress energy tensor in a spatially flat FLRW metric 2 2 2 2 trate the effects of the braneworld scenario for inflation, ds = dt + a(t) dx is given by − one can instead consider the high energy limit of the picture V >>λ, which allows us to now write the Fried- µ ∂ µ Tν = L ∂ν φ gν = diag( ρφ, p˜φ) (5) mann equation (12) as, ∂(∂µφ) − L − 2 2 V where = √ g m 2 R V (φ) 1 gµν ∂ φ∂ φ is the H = (14) p 2 µ ν λm2 LagrangianL density− of the− Tachyon− field , V (φ) is the 6 p p potential, while ρφ and pφ are the energy and pressure The Number of e-folds can be written using it’s usual densities of the field given by definition for some initial field value φi to some final value φ as, V (φ) e ρφ = (6) φ 2 e H 1 φ˙ N = dφ (15) − φ˙ q Zφi And using the approximation (13), we can then write the ˙2 pφ = V (φ) 1 φ (7) e-fold number as − − q φi 3H2V The Friedmann equation for a Randall-Sundrum II N = dφ (16) Braneworld cosmological scenario is given by [ for details V Zφe ′ 4

Further, using the Friedmann equation (14), one can Considering the dS conjecture(2) in this scenario, we can write N as write

φi 3 V 2λ 2 N = dφ (17) ǫ> c (22) 2λm2V V 2 Zφe p ′ A particularly important class of parameters for any in- As c (1) , in order for ǫ << 1 , one would require ∼ O flationary model are the slow roll parameters [10] and in particular, the ǫ and η parameters. One can define these V >> √2λ (23) parameters for tachyonic brane inflation through their usual definitions in the high energy limit as , Hence, it would appear that the dS conjecture implies a constraint on the energy scale of inflation ( as during in- 2 2 H˙ 2λm V ′ flation ρ = ρφ ≅ V ) for tachyonic inflation on the RS-II ǫ = = p (18) −H2 V 4 braneworld. But, it is far from the case as we recall that the formulation built above is in the high energy limit of the braneworld, which characterized by V >>λ>>. So in reality, the tachyonic inflation in the braneworld in the H¨ 1 mpV ′′ 1 mpV ′ η = =2√6λ (19) high energy limit is not strong constrained by the dS con- −HH˙ V V − V V ′ jecture and is virtually unscathed. Similarly, the change Further the power-spectrum of the curvature perturba- in the e-fold number during inflation can be written using (17) as, tion PR that is derived from the correlation of first order scalar field perturbation in the vacuum state can be written as [113] V 2 ∆φ ∆N ≅ (24) m V ′ 2 2 2λ  m p  H 1 p V PR(k)= (20) 2πφ˙ V (1 φ˙2)   − Now considering both the distance (1) and dS conjecture where in the slow roll limit, one can ignore the contribu- , we see that term in the square brackets in (24) has tion of φ˙. One can then further find out other observa- to be less than unity. This shows that in order to have tionally revelant perturbation parameters like the scalar sufficient amount of e-folds, one again needs and tensor spectral index, tensor-to-scalar ratio etc. using their standard definitions [21, 23, 113]. We will not V >> √2λ (25) illustrate that here as we have built enough groundwork to start seeing why tachyonic brane inflation is consis- and once again as we are in the high energy limit tent with the swampland in the high energy regime. The already V >>λ the requirement (25) does pose a very prominent issues of conflict between the swampland con- strong constraint on the energy scale of inflation in such jectures and single field inflation which were discussed in a regime which might have otherwise ruled out some length in [71] can be briefed as follows. One of the issues inflationary models in this scenario. So this makes it concerns the dS conjecture and inflationary requirement clear that there is no issue of a insufficient e-fold number for the ǫ parameter, as it was shown that if one con- for this inflationary regime. Finally we note that even siders the dS conjecture seriously than the requirement if one considers both the dS conjecture and the refined that ǫ << 1 is categorically not satisfied. Another issue dS conjecture (3) to simultaneously hold true and concerns the fatal constraints that the dS and distance applies them the on η parameter (19), the observational conjecture applied together have for the e-fold number, requirement for η 0 is still identically satisfied as both as they constrain N << 1 which is quite clearly in vio- terms in the square≤ bracket in (19) become negative lation of the bare minimum inflationary requirements on . In conclusion the only ”constraint” applied by the the number of e-folds.Finally, the third main issue which swampland conjectures on tachyonic inflation in the high was elaborated in [71] is the inconsistency between con- energy RS-II Braneworld is a small lower limit on the straints set on the η parameter from observational data energy scale of inflation, which in the high energy limit on inflation [7] and the implications that the refined dS is quite intrinsically satisfied. With this we see that conjecture has on the same parameter.We will now elab- this regime of inflation is strictly not in the swampland orate under what conditions Tachyonic Inflation on the for effectively all kinds of potentials. As we have now RS-II Braneworld evades all of the issues of single field in- demonstrated that our concerned inflationary paradigm flation with the swampland conjectures. Firstly, focusing is quite consistent with the swampland conjectures, we on. The ǫ parameter (18), can be written as now proceed towards showing how eternal inflation in

2 this regime is also consistent with these criterion. 2λ m V ǫ = p ′ (21) V 2 V 5

III. SWAMPLAND CONSISTENT ETERNAL INFLATION Another very severe issue faced by eternal inflation is the inconsistency its entropy bounds have with the dS As discussed in section I, recent works have shown that conjecture, something which was highlighted in [103]. eternal inflation is not on an amicable standing with the This issue is in particular focused for large field infla- swampland conjectures . One of the key requirements tionary models and can be understood by considering in order for eternal inflation to take place in any kind the Bekenstein-Gibbons-Hawking entropy of the local of regime that quantum fluctuations of the inflaton field event horizon of an expanding cosmology [115–117] should dominate over the classical field evolution. The 4πm2 amplitude of quantum fluctuations on scales of order the S = p (31) Hubble length is given by (the following expression is BGH H2 independent of the underlying gravitational theory [23]) During a period in stochastic inflation when the inflaton H moves up the potential due to quantum fluctuations, H <δφ> = (26) Q 2π increases and thus the entropy of the area bounded by the horizon decreases. For the process above to be consistent 1 while the classical field variation in Horizon time H− is with the second law of thermodynamics even after taking given by into account quantum gravitational effects, the variation of the entropy above is bounded as [118] φ˙ δφ = (27) cl H δS > 1 (32) − The requirement that quantum fluctuations dominate the Now if one considers a usual scalar field inflationary classical field evolution hence translates into model based in a General Relativistic cosmology, then <δφ> H2 bound (31) eventually turns out to be in violation of Q = > 1 (28) the dS conjecture and thus leading to serious issue for δφ ˙ cl 2πφ the entropy bounds for eternal inflation which was dis- In the usual case of single scalar field inflation, the crite- cussed in [103]. This issue of entropy bounds, however, rion (28) transforms to the following requirement on the will not persist for Tachyonic inflation in a high energy amplitude of curvature perturbations at horizon crossing RS-II braneworld which is what we now show. Using the Friedmann equation (14), one can write the variation of PR(k) > 1 (29) the entropy (31) as 4 The above requirement can eventually be shown to be 48πm V ′λ δS = p δφ (33) inconsistent with the dS conjecture [100, 114]. This was − V 3 the first conflict highlighted between the swampland criterion and eternal inflation. It was then shown in [71] further, considering the change in entropy during a time H that the curvature perturbation amplitude requirement in which using δφ = 2π , we have

(29) can be amicable with the refined dS conjecture but 2 it was further shown in [102] that the perturbation re- 24mpλ mpV ′ 1 δS = (34) quirements cannot be consistent with the refined dS con- − V V √6λ jecture as well . Now writing the amplitude of curvature perturbations for tachyonic inflation (20) in the slow roll Now, employing the criterion (32) leads to the following limit φ˙ << 1, one gets inequality

2 2 mpV ′ V ≅ H 1 < (35) PR(k) (30) V √ 2 πφ˙ V 4 6λmp 2 ′ From (30), it is immediately apparent that the condition mpV The dS conjecture sets the limit V (1), hence in (28) does not set a direct requirement on the amplitude order for the above inequality to be consistent≥ O with the of curvature perturbations for a tachyonic field like it dS conjecture we need does for a usual scalar field. Further for a large enough √ 2 energy scale , just like the high energy limit of the V >> 4 6λmp (36) RS-II Braneworld considered here, one can easily have PR(k) < 1 whilst still preserving the condition (28). Hence the dS conjecture sets a lower limit on the en- Hence the conflicts of the swampland conjectures with ergy scale of tachyonic inflation in the high energy RS-II eternal inflation in the presence of the requirement braneworld, for it to be on amicable terms with the en- of PR(k) 1 [100–102] are not encountered for our tropy bounds. We note that the bound to bypass the concerned≥ tachyonic regime. entropy issue (36) is in the range with lower limit on 6 the inflation energy scale(25) needed to evade the pri- Using the Friedmann equation (14), we can now write mary issues of the swampland and single field inflation the entropy in terms of the potential discussed in Section II. So the crux of the matter here 2 2 is that in order for the eternal inflation entropy bounds mpπ√6λ 4mp√6λ αo S = + (41) to be consistent with the dS conjecture in our concerned V V 2 tachyonic regime, we need to set a higher lower bound on h i This finally allows us to express the bound (32) in terms the energy scale of inflation which is still consistent with of the GUP corrected entropy as the bounds needed to sort out the primary inflationary issues with the swampland. There is, however, another 2 m V 1 π√6λ 4m √6λ α way to go about the entropy bound issue. The event p ′ p + o horizon entropy considered here is through the standard V 2π√6λ" V V 2 ! Bekenstein-Gibbons-Hawking entropy formulation, while 2 4m √6λ m π√6λ in recent years there has been a lot of work which has ex- + p p < 1 (42) plored corrections needed in (31) in order to accommo- V V ! # date the newly emerging physics from string theory and loop quantum gravity (LQG) [119–130]. Several of these Rewriting the inequality above, we have approaches have the quantum-corrected entropy-area re- √ lation in a general form in terms of (31) as mpV ′ 2π 6λ < 48 2 (43) V πmpλ π√6λαo V 2 + 2V ∞ n S = SBGH + Co ln(SBGH )+ Cn (SBGH )− (37) ′ mpV =1 n The dS conjecture constraints V c (1), hence X in order to maintain consistency with≥ it∼ the O following where Cn are parameters which are model dependent (like recent works have determined C = 1 for LQG inequality will have to hold o − 2 [124]). It was also interestingly pointed out in [131] that 2 48πm λ π√ λα Heisenberg’s Uncertainity principle might be affect by √ p 6 o 2π 6λ> 2 + quantum gravitational effects. Since then, a significant V 2V V α amount of work has gone into finding a Generalized Un- = 2V 2 o 8m2√6λ> 0 (44) certainty Principle(GUP) and looking for the quantum ⇒ − 2 − p corrections it provides to the usual Bekenstein-Gibbons- The last inequality is a bit non-trivial but can be calcu- Hawking formulation of local event horizon entropy. This lated using computing systems like Mathematica. This eventually led to the following form of the quantum cor- inequality sets a lower bounds on λ given by rected entropy due to a GUP, [123, 125, 128, 129] 4 2 αo √παo πα 1 λ> S S S o S 4 (45) = BGH + BGH ln( BGH )+ ( 3 ) 393216mp 4 − 64 O mp p (38) And also a lower bound on the inflation energy scale, where αo is dimensionless constant which is found in the which keeping in mind the brane tension bound can be deformed commutation relations and can be usually con- written approximately as sidered to be positive in nature [130]. The leading order α contribution to the GUP corrected entropy formalism is V >> o (46) due to the √SBGH term, which is an extra term to the al- 8 ready existing logarithmic correction to entropy derived Hence the dS conjecture can be consistent with the GUP from the quantum gravity effects. Considering this form corrected eternal inflation entropy bounds given that the of entropy relation as the one for entropy of a local event brane tension and the inflation energy scale abide by the horizon for an expanding cosmology has provided pretty above mentioned lower limits. The most interesting out- interesting insights (see for example, [132]). Hence, we come of considering the GUP corrected entropy bounds is now consider the form (38) of the entropy for the bound in the nature of limits implied in (45-46). If these bounds (32) and as we would like to entertain in particular the are satisfied (these are in line with the basic requirements implications of the very interesting GUP effects , we only for the energy scale for consistency with the conjectures), consider the leading order GUP correction to the entropy then the famous issues which contribute to the failure of a which provides us with swampland consistent eternal inflation picture are taken care of. Hence the direct relation between the inflaton √παo SGUP ≅ SBGH + SBGH (39) energy scale and the constant α which is found from the 4 o p deformed commutation relations for a GUP is incredibly which can be written in terms of the Hubble Parameter exciting. In this scenario, the existence of a Generalized as Uncertainty Principle facilitates the creation of a swamp- mpπ 4mp αo land consistent eternal inflationary scenario and conse- S = + (40) H H 2 quently, a swampland consistent multiverse. To make 7 this peculiar relation even more clear, we would like to IV. CONCLUDING REMARKS AND demonstrate that the consideration of a GUP corrected DISCUSSION entropy formalism would also allow us to evade the dS conjecture-entropy bound conflict in a simple scalar field In conclusion, in this work we have discussed tachy- model in a GR based scenario. For GR based single field models , the Friedmann equation during the expansion onic inflation in the high energy RS-II Braneworld scenario in the light of the swampland conjectures and con- phase is sidered the possibility of swampland consistent eternal V inflation in this regime. We started off by describing H2 = (47) 3m2 why tachyonic inflation in this particular scenario could p possibly be one of the most suited arenas for the swamp- The GUP corrected entropy (39) for this regime becomes land conjectures and hence, might allow for the possibility of evading the swampland issues eternal inflation too. Then after describing some basics of Tachyonic in- α 2 3 2 3 o flation in this regime, we showed that this inflationary S = mpπ 4mp + (48) r V rV 2 ! model is virtually unscathed by the swampland conjectures and can quite intrinsically evade all the prominent And hence, the bound (32) in this case takes the form swampland issues which have been discussed for regular single field inflation, as they only imply a lower bound m V 8V p ′ < (49) on the inflation energy scale which can be easily satis- V 2 fied in the considered high energy limit. We then cast mpH 48m + αo√3V p our glance towards eternal inflation in this scenario, and h i discussed the two main swampland issues in this context It is worth mentioning the difference the GUP corrected where we firstly addressed the curvature amplitude issue. has already made in eradicating the entropy bound is- We showed that the basic eternal inflation requirement of sue for this inflationary regime. The above inequality quantum fluctuations dominating over classical field evo- clearly shows that it quite possible to satisfy the con- lution does not set unavoidable constraints on tachyonic straints due to the dS conjecture on the left hand side of models of any kind and in particular, inflation in high en- the inequality, given that there is some minimum limit ergy RS-II Braneworld. We hence showed that the issues on the energy scale of inflation to make the right hand of inconsistency of the dS and refined dS conjectures with side of the inequality sufficiently large. Doing this would the perturbation requirement do not hold good for our mean that the numerator on the right hand side far dom- tachyonic regime. We then discussed the conflict of en- inates over the denominator which using the Friedmann tropy bounds with the dS conjecture and showed that our equation (47), can be written as the following inequality concerned inflationary regime does not have any issues 12m2 with the entropy bound even when the concerned entropy V 1/2 > p (50) formalism is the standard Bekenstein-Gibbons-Hawking √3 2 αo one, as the lower limit on the energy scale implied for con- − 4 sistency with the bound is well within the range needed Hence one can eradicate the issue between the dS con- for consistency with basic inflationary swampland issues. jecture and entropy bounds by considering a GUP cor- Then made the case as to why it would be better suited rected entropy formalism by constraining the energy scale if we consider quantum corrections to the BGH entropy during inflation in accordance with the above bound for 8 , and in particular we considered a Generalized Uncer- αo < √3 . Once again, we notice how crucial a role tainty Principle corrected entropy formalism. In order to the Generalized Uncertainty Principle based corrections focus on this particular correction to the entropy, we only make in allowing us to evade the entropy bound issues considered the leading order GUP corrections to the en- even in a usual scalar field model based in a GR based tropy and showed that a GUP formalism of the entropy cosmology. This in no way shows that eternal inflation in allows one to evade the entropy bound issues in both such a regime is consistent with the swampland, however, tachyonic inflation in our concerned scenario and even in as other issues regarding the conjectures still hold true a usual single field model in a GR based cosmology. This [100–102, 114]. But this does illustrate that GUP based allowed us to make a startling observation that a gen- quantum corrected entropy does facilitate positively in eralized uncertainty principle actually facilitates one to removing the entropy bound-dS conjecture conflict in have a swampland (and hence possibly, a quantum grav- multiple regimes of inflation. Thus we can draw a very itationally) consistent picture of eternal inflation and a exciting conclusion from this analysis that considering Multiverse. An important thing to note is that we have a generalized uncertainty principle makes it more easier not carried out a detailed analysis of eternal inflation in for one to have a swampland consistent (and considering any kind of a tachyonic regime as we have just focused on the premise of the conjectures, a quantum gravitationally showing how the prominent swampland-eternal inflation consistent) picture of a multiverse generated by eternal issues can be resolved in a more quantum gravitation- inflation. ally / String theory motivated setting. Indeed, we have 8 not embarked on exploring the solutions of the Fokker- Institute, McGill University) for his insightful comments Planck / Langevin equations for quantum fluctuations of which led to the initiation of this work. The author the inflaton field for different potentials, which is more of would also like to thank Dr. Sunny Vagnozzi (Newton- a standard procedure in the subject. Hence, there might Kavli fellow at at Kavli Insitute of Cosmology (KICC) be more model dependent requirements for eternal in- and the Institute of Astronomy, University of Cambridge) flation to take place in a variety for different potentials for his very helpful advice during the preparation of this that we have not taken into account here ( like the model manuscript. dependent analysis which was done in [102]) as our explorations have been of a more general nature focused on the prevalent swampland issues currently.

V. ACKNOWLEDGEMENTS

The author would like to thank Dr. Suddhasatwa Brahma (Postdoctoral Research Fellow at McGill Space

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