Reheating in small-field inflation on the brane: The Swampland Criteria and observational constraints in light of the PLANCK 2018 results
aa r X i v : . [ h e p - t h ] J a n Reheating in small-field inflation on the brane: The SwamplandCriteria and observational constraints in light of the PLANCK2018 results
Constanza Osses ∗ and Nelson Videla † Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso,Avenida Brasil 2950, Casilla 4059, Valpara´ıso, Chile.
Grigoris Panotopoulos ‡ Centro de Astrof´ısica e Gravita¸c˜ao-CENTRA, Departamento de F´ısica,Instituto Superior T´ecnico-IST, Universidade de Lisboa-UL,Av. Rovisco Pais, 1049-001 Lisboa, Portugal (Dated: January 25, 2021) bstract We study cosmological inflation and its dynamics in the framework of the Randall-SundrumII brane model. In particular, we analyze in detail four representative small-field inflationarypotentials, namely Natural inflation, Hilltop inflation, Higgs-like inflation, and Exponential SUSYinflation, each characterized by two mass scales. We constrain the parameters for which a viable in-flationary Universe emerges using the latest PLANCK results. Furthermore, we investigate whetheror not those models in brane cosmology are consistent with the recently proposed Swampland Cri-teria, and give predictions for the duration of reheating as well as for the reheating temperatureafter inflation. Our results show that (i) the distance conjecture is satisfied, (ii) the de Sitter con-jecture and its refined version may be avoided, and (iii) the allowed range for the five-dimensionalPlanck mass, M , is found to be 10 TeV . M . TeV. Our main findings indicate thatnon-thermal leptogenesis cannot work within the framework of RS-II brane cosmology, at least forthe inflationary potentials considered here.
PACS numbers: 98.80.Cq ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION Standard hot big-bang cosmology, based on four-dimensional General Relativity (GR) [1]combined with the cosmological principle, is supported by the three main pillars of moderncosmology. Those are i) the Hubble’s law [2],ii) the Primordial big-bang Nucleosynthesis(BBN) [3], and iii) the Cosmic Microwave Background (CMB) Radiation [4]. The emergingcosmological model of the Universe seems to be overall quite successful, however some issuesstill remain regarding the initial conditions required for the big bang model, such as thehorizon, the flatness, and the monopole problem. Cosmological inflation [5–8] provides uswith an elegant mechanism to solve those shortcomings all at once. Moreover, in the infla-tionary Universe, primordial curvature perturbations with an approximately scalar-invariantpower spectrum, which seed CMB temperature anisotropies and the structure formation ofthe Universe, are generated from the vacuum fluctuations of a scalar field, the so calledthe inflaton [9–15]. Therefore, inflationary dynamics is currently widely accepted as thestandard paradigm of the very early Universe, although we do not have a theory of inflationyet. For a classification of all single-field inflationary models based on a minimally coupledscalar field see [16], while for a large collection of inflationary models and their connectionto Particle Physics see e.g. [17, 18].One can test the paradigm of cosmological inflation comparing its predictions on the r − n s plane with current cosmological and astronomical observations, and specially withthose related to the CMB temperature anisotropies from the PLANCK collaboration [19, 20]as well as the BICEP2/Keck-Array data [21, 22]. In particular, currently there only existsan upper bound on the tensor-to-scalar ratio r , since the tensor power spectrum has notbeen measured yet. The PLANCK upper limit on the tensor-to-scalar-ratio, r . < . r . < . φ ofthe field during inflation through the Lyth bound, assuming that r is nearly constant [23]∆ φM pl ≃ r r N k , (1)where N k is the number of e -folds before the end of inflation, and M pl is the reduced Planckmass associated with Newton’s gravitational constant by M pl = 1 / √ πG . Models with∆ φ < M pl and ∆ φ > M pl are called small-field and large-field models, respectively. If next-3eneration CMB satellites, e.g. LiteBIRD [24], COrE [25] and PIXIE [26], are not able todetect primordial B-modes, an upper limit of r < .
002 (95% C.L.) will be reached, implyingthat small-field inflation models will be favored, since a particular feature of these modelsis that tensor modes are much more suppressed with respect to scalar modes than in thelarge-field models. In this type of models, the scalar field is rolling away from an unstablemaximum of the potential, being a characteristic feature of spontaneous symmetry breaking.Let us consider the inflaton potential of the form V ( φ ) = Λ [1 − U ( φ )] , (2)where Λ is a constant having a dimension of a mass and U ( φ ) is a function of φ .Natural Inflation (NI) with a pseudo-Nambu Goldstone boson (pNGB) as the inflaton[27] arises in certain particle physics model [28]. The scalar potential, which is flat due toshift symmetries, has the form V ( φ ) = Λ (cid:20) − cos (cid:18) φf (cid:19)(cid:21) . (3)and it is characterized by two mass scales f and Λ with f ≫ Λ. It is assumed that a globalsymmetry is spontaneously broken at some scale f , with a soft explicit symmetry breakingat a lower scale Λ. Natural Inflation has been already studied in standard cosmology basedon GR [29, 30]. In particular, Natural Inflation is consistent with current data [19, 20] fortrans-Planckian values of the symmetry breaking scale f , for which it may be expected thelow-energy effective theory, on which (3) is based, to break down [31]. Another type of small-field models supported by Planck data are Hilltop inflation models, which are described bythe potentials [32, 33] V ( φ ) = Λ (cid:20) − (cid:18) φµ (cid:19) p (cid:21) , (4)where p is typically an integer power. In order to stabilize the potential from below, theformer potentials are often written down as V ( φ ) = Λ (cid:20) − (cid:18) φµ (cid:19) p (cid:21) + ..., (5)where higher order terms are included in the ellipsis. The fashionable models with p = 2and p = 4 are ruled out by current observations for µ . M pl regardless of the omitted termsdesignated by the ellipsis. However, those models yield predictions favored by PLANCK2018 when µ & M pl for any value of the power p , which becomes indistinguishable from4hose of linear inflation, i.e. V ( φ ) ∼ φ [34]. For numerical as well as analytic treatmentsof Hilltop inflation in the framework of GR, see Refs. [35–38]. A consistent modification ofthe quadratic Hilltop model ( p = 2) yields a Higgs-like potential [34, 35], which is used todescribe dynamical symmetry breaking V ( φ ) = Λ " − (cid:18) φµ (cid:19) , (6)where the extra quartic term prevents the potential from becoming negative beyond thevacuum expectation value (VEV) µ . It has been shown that such a potential remainsfavored by current data as long as the mass scales are high [30, 34]. Another small-fieldmodel, derived in the context of supergravity, corresponds to Exponential SUSY inflation,where the potential is given by [39] V ( φ ) = Λ (cid:0) − e − φ/f (cid:1) (7)which is asymptotically flat in the limit φ → ∞ . This inflaton potential also appears inthe context of D-brane inflation [40] and it predicts a small value of the tensor-to-scalar for f < M pl [41], being inside the (68% C.L.) boundary constrained by PLANCK 2018 data[42]. Another supergravity-motivated model is K¨ahler moduli inflation [43] V ( φ ) = Λ (cid:16) − c φ / e − c φ / (cid:17) , (8)which predicts a very small tensor-to-scalar ratio, well inside the (68% C.L.) contour [42].The inflationary period ends when the equation-of-state parameter (EoS) becomes largerthan w = − /
3, i.e. the slow-roll approximation breaks down, the expansion decelerates,and the Universe enters into the radiation era of standard Hot big-bang Cosmology [44].The transition era after the end of inflation, during which the inflaton is converted into theparticles that populate the Universe later on is called reheating [45, 46] (for comprehensivereviews, see e.g. Refs. [47–49]). As was shown Ref. [50], the EoS parameter presents a sharpvariation during the reheating phase due to the out-of-equilibrium nonlinear dynamics offields. Unfortunately, the underlying physics of reheating is highly uncertain, complicated,and it cannot be directly probed by observations, although some bounds from BBN [51, 52],the gravitino problem [53–57], leptogenesis [58–64], and the energy scale at the end of in-flation do exist [48, 49]. There is, however, a strategy that allows us to obtain indirect5onstraints on reheating. First we parameterize our ignorance assuming for the fluid a con-stant equation-of-state w re during reheating. Next, we find certain relationships between thereheating temperature, T re , and the duration of reheating, N re , with w re and the inflationaryobservables [65–73].Considering that inflation opens up the window to probe physics in the very high energyregime, it is also tempting to construct inflationary models in string theory [74]. Althoughwe do not have a full quantum gravity theory yet, string theory is believed to be a promisingcandidate, which possesses a space of consistent low-energy effective field theories derivedfrom it, called the landscape [75–77]. The landscape consists of a vast amount of vacuadescribed by different effective field theories (EFTs) at low energies. At the same time,there is another set of EFTs, dubbed the swampland , which are not consistent with stringtheory. Accordingly, one can ask the question what criteria a given low-energy EFT shouldsatisfy in order to be contained in the string landscape. In this direction, several criteria ofthis kind, dubbed swampland criteria [78, 79] have been proposed so far, with the followingimplications for inflationary model-building • The distance conjecture: ∆ φM pl < O (1) , (9) • The de Sitter conjecture: M pl | V ′ | V > c ∼ O (1) . (10)The distance conjecture implies that scalar fields cannot have field excursions much largerthan the Planck scale, since otherwise the validity of the EFT breaks down [80]. As it canbee seen from Eq. (1), in the context of inflation, field excursions are related to the tensor-to-scalar ratio. Accordingly, this conjecture limits the possibility of measuring tensor modes andhence primordial B-modes in the CMB. Specifically, for N k &
50, it is found r . O (10 − ),which lies on the edge of detectability for future experiments [24–26]. In addition, the deSitter conjecture states that slope of the scalar field potential satisfies a lower bound when V > refined de Sitter swamplandconjecture , proposed in [83, 101], sates that:6
Refined de Sitter conjecture: M pl | V ′ | V > c ∼ O (1) or M V ′′ V < − c ′ ∼ O (1) . (11)With this refinement, which allows for a scalar field potential with maxima (hilltop) to exist,the conflicts with some small-field potentials, such as Higgs-like, QCD axion [84, 102, 103]and Hilltop [104, 105], are resolved.Additionally, there is another Swampland conjecture proposed recently in the literature,known as the Trans-Planckian Censorship Conjecture (TCC). Roughly speaking, the TCCclaims that in a consistent quantum gravity theory, quantum fluctuations at sub-Planckianlevel are forbidden to exit the Hubble horizon during inflation. As a consequence, cosmicinflation is in direct conflict with this conjecture in regards to the upper bound on thetensor-to-scalar ratio, number of e -folds and energy scale of inflation [106, 107].A novel way to satisfy the refined swampland criteria is to consider inflation on thebrane [88] and related works [92, 97, 98, 104]. Furthermore, considering inflation in non-standard cosmologies is motivated by at least two facts, namely i) deviations from thestandard Friedmann equation arise in higher-dimensional theories of gravity, and ii) thereis no observational test of the Friedmann equation before the BBN epoch. A well-studiedexample of a novel higher-dimensional theory is the brane-world scenario, which inspiredfrom M/superstring theory. Although brane models cannot be fully derived from the fun-damental theory, they contain at least the key ingredients found in M/superstring theory,such as extra dimensions, higher-dimensional objects (branes), higher-curvature correctionsto gravity (Gauss-Bonnet), etc. Since superstring theory claims to give us a fundamentaldescription of Nature, it is important to study what kind of cosmology it predicts.Since there is a growing interest in studying inflationary models that meet both obser-vational data and Swampland Criteria, the main goal of the present work is to study therealization of some representative small-field inflation models, namely Natural inflation, Hill-top inflation, Higgs-like inflation and Exponential SUSY inflation, in the framework of theRS-II brane model, in light of the recent PLANCK results and their consistency with theswampland criteria. Furthermore, we give predictions regarding the duration of reheatingas well as the reheating temperature after inflation.We organize our work as follows: After this introduction, in the next section we summarizethe basics of the brane model as well as the dynamics of inflation and the basic formulas7or determining the duration of reheating as well as for the reheating temperature afterinflation. In sections from III to VI we analyze each of the proposed small-field inflationmodels in the framework of RS-II model and present our results. Finally, in the last sectionwe summarize our findings and exhibit our conclusions. We choose units so that c = ~ = 1. II. BASICS OF BRANEWORLD INFLATIONA. Braneworld cosmology
In brane cosmology our four-dimensional world and the Standard Model (SM) of particlephysics are confined to live on a 3-dimensional brane, whereas gravitons are allowed topropagate in the higher-dimensional bulk. Here we shall assume that only one additionalspatial dimension, perpendicular to the brane, exists. Since the higher-dimensional Plankmass, M , is the fundamental mass scale instead of the usual four-dimensional Planck mass, M , the brane-world idea has been used to address the hierarchy problem of particle physics,first in the simple framework of a flat (4+ n ) space-time with 4 large dimensions and n smallcompact dimensions [108], and later it was refined by Randall and Sundrum [109, 110]. Forexcellent introduction to brane cosmology see e.g. [111]. In the RS-II model [110], thefour-dimensional effective field equations are computed to be [112] (4) G µν = − Λ g µν + 8 πM τ µν + (cid:18) πM (cid:19) π µν − E µν , (12)where Λ is the four-dimensional cosmological constant, τ µν is the matter stress-energytensor on the brane, π µν = (1 / τ τ µν + (1 / g µν τ αβ τ αβ − (1 / τ µα τ αν − (1 / τ g µν , and E µν = C αβρσ n α n ρ g βµ g σν is the projection of the five-dimensional Weyl tensor C αβρσ on thebrane, where n α is the unit vector normal to the brane. E µν and π µν are the new terms,not present in standard four-dimensional Einstein’s theory, and they encode the informationabout the bulk. The four-dimensional quantities are given in terms of the five-dimensionalones as follows [113] M = r π (cid:18) M √ λ (cid:19) M , (13)Λ = 4 πM (cid:18) Λ + 4 πλ M (cid:19) , (14)where M pl = M / √ π ≃ . × GeV is the reduced Planck mass, and λ is the branetension. 8he Friedmann-like equation describing the backround evolution of a flat FRW Universeis found to be [114] H = Λ π M ρ (cid:16) ρ λ (cid:17) + E a . (15)where a is the scale factor, H is the Hubble parameter, ρ is the total energy density ofthe cosmological fluid, and E is an integration constant coming from E µν . The term E a isknown as the dark radiation, since it scales with a the same way as radiation. However,during inflation this term will be rapidly diluted due to the quasi-exponential expansion,and therefore in the following we shall neglect it. The five-dimensional Planck mass isconstrained by the standard big-bang nucleosynthesis to be M &
10 TeV [115], implyingthat λ & (1 MeV) ∼ (10 − M pl ) . A stronger constraint on M , namely M & TeV,results from current tests for deviations from Newton’s gravitational law on scales largerthan 1 mm [116].In the discussion to follow we shall set the four-dimensional cosmological constant Λ tozero, i.e. we shall adopt the RS fine tuning Λ = − πλ / (3 M ), so that the model canexplain the current cosmic acceleration without a cosmological constant. Finally, neglectingthe term E a the Friedmann-like equation (15) takes the final form H = 8 π M ρ (cid:16) ρ λ (cid:17) , (16)upon which our study on brane inflation will be based. B. Inflationary dynamics
At low energies, i.e., when ρ ≪ λ , inflation in the brane-world scenario behaves in exactlythe same way as standard inflation, but at higher energies we expect inflationary dynamicsto be modified.We consider slow-roll inflation driven by a scalar field φ , for which the energy density ρ and the pressure P are given by ρ = ˙ φ + V ( φ ) and P = ˙ φ − V ( φ ), respectively, where V ( φ )is the scalar potential. Assuming that the scalar field is confined to the brane, the usualfour-dimensional Klein-Gordon (KG) equation still holds¨ φ + 3 H ˙ φ + V ′ = 0 , (17)where a prime denotes differentiation with respect to φ , while an over dot denotes differ-entiation with respect to the cosmic time. In the slow-roll approximation the cosmological9quations take the form (16) and (17) H ≃ π M V (cid:18) V λ (cid:19) , (18)and ˙ φ ≃ − V ′ H . (19)The brane-world correction term
V /λ in Eq. (18) enhances the Hubble rate for a givenpotential. Thus there is an enhanced Hubble friction term in Eq. (19), as compared to GR,and brane-world effects will reinforce slow-roll for the same potential.That way, using those two equations, it is possible to write down the expression for theslow-roll parameters on the brane as [113] ǫ ≡ ǫ V V /λ (1 +
V / λ ) , (20) η ≡ η V
11 +
V / λ , (21)where ǫ V = M π (cid:0) V ′ V (cid:1) and η V = M π V ′′ V are the usual slow-roll parameters of standardcosmology for a canonical scalar field. Considering the definition of ǫ V for standard inflation,the de Sitter swampland conjecture Eq. (10) and the first equation of its refined version in(11) imply ǫ V ∼ c / ∼ O (1) , (22)which rules out slow-roll inflation, since the former is in conflict with ǫ V ≪
1. Slow-rollinflation on the brane implies that ǫ ≪ | η | ≪
1, which can be achieved in the high-energy regime, i.e., V ≫ λ , despite the fact that both ǫ V and η V are large due to the largeslope of the potential. This feature is crucial for avoiding the refined swampland criteria[88]. In the high-energy limit, Eqs. (20) and (21) become ǫ ≃ ǫ V (cid:18) λV (cid:19) , (23) η ≃ η V (cid:18) λV (cid:19) , (24)while in the low-energy limit V ≪ λ , Eqs. (20) and (21) are reduced to the usual slow-rollparameters of standard cosmology. Clearly, the deviations from standard slow-roll inflationcan be seen in the high-energy regime, as both parameters are suppressed by a factor V /λ .10he number of e -folds in the slow-roll approximation, using (16) and (17), yields N k ≃ − πM Z φ end φ k VV ′ (cid:18) V λ (cid:19) dφ, (25)where φ k and φ end are the values of the scalar field when the cosmological scales cross theHubble-radius and at the end of inflation, respectively. As it can be seen, the number of e -folds is increased due to an extra term of V /λ . This implies a more amount of inflation,between these two values of the field, compared to standard inflation.
C. Perturbations
In the following we shall briefly review cosmological perturbations in brane-world infla-tion. We consider the gauge invariant quantity ζ = − ψ − H δρ ˙ ρ . Here, ζ is defined on slicesof uniform density and reduces to the curvature perturbation at super-horizon scales. Afundamental feature of ζ is that it is nearly constant on super-horizon scales [117], and infact this property does not depend on the gravitational field equations [118]. Therefore, forthe spatially flat gauge, we have ζ = H δφ ˙ φ , where | δφ | = H/ π . That way, using the slow-rollapproximation, the amplitude of scalar perturbations is given by [113] P S = H ˙ φ (cid:18) H π (cid:19) ≃ π M V V ′ (cid:18) V λ (cid:19) . (26)On the other hand, the tensor perturbations are more involved since the gravitons canpropagate into the bulk. The amplitude of tensor perturbations is given by [113] P T = 64 πM (cid:18) H π (cid:19) F ( x ) , (27)where F ( x ) = " √ x − x ln x + r x ! = (cid:20) √ x − x sinh − (cid:18) x (cid:19)(cid:21) − / , (28)and x is given by x = HM r πλ . (29)The expressions for the spectra are, as always, to be evaluated at the Hubble radiuscrossing k = aH . As expected, in the the low-energy limit the expressions for the spectra11ecome the same as those derived without considering the brane effects. However, in thehigh-energy limit, these expressions become P S ≃ π M λ V V ′ , (30) P T ≃ V M λ . (31)The scale dependence of the scalar power spectra is determined by the scalar spectralindex, which in the slow-roll approximation obeys the usual relation n s = 1 + d ln P S d ln k ,n s ≃ − ǫ + 2 η. (32)The amplitude of tensor perturbations can be parameterized by the tensor-to-scalar ratio,defined to be [44] r ≡ P T P S , (33)which implies that in the low-energy limit this expression becomes r ≃ ǫ V , where ǫ V isthe standard slow-roll parameter, whereas in the high-energy limit we have [119] r ≃ ǫ, (34)with ǫ corresponding to Eq. (23).As we have seen, at late times the brane-world cosmology is identical to the standardone. During the early Universe, particularly during inflation, there may be changes to theperturbations predicted by the standard cosmology, if the energy density is sufficiently highcompared to the brane tension. D. Reheating
Here we shall briefly describe how to compute the number of e -folds of reheating N re aswell as the reheating temperature T re in terms of the scalar spectral index for single-fieldinflation in the high-energy regime of RS-II brane-world scenario. For the derivation of themain formulas, we mainly follow Refs. [66, 68, 70].Reheating after inflation is important for itself as a mechanism achieving what we knowas the hot big-bang Universe. The energy of the inflaton field becomes in thermal radiation12uring the process of reheating through particle creation while the inflaton field oscillatesaround the minimum of its potential. If one considers that during reheating phase themain contribution to the energy density of the Universe comes from a component having aneffective equation-of-state parameter (EoS) w re , and its energy density can be related to thescale factor through ρ ∝ a − w re ) , we can write down the following relation ρ end ρ re = (cid:18) a end a re (cid:19) − w re ) , (35)where the subscripts end and re denote the end of inflation and the end of reheating phase,respectively.The number of e -folds of reheating are related to the scale factor both at the end ofinflation and reheating according to e − N re = a end a re . (36)Then, by combining Eqs. (35) and (36), we can write the number of e -folds of reheatingas N re = 13(1 + w re ) ln (cid:18) ρ end ρ re (cid:19) . (37)On the other hand, we consider the Friedmann-like equation (16) in the high-energy limit ρ ≫ λ H ≃ π M λ ρ , (38)and the slow-roll parameter ǫ , defined as ǫ = − ˙ HH . (39)By combining the time derivative of Eq. (16) with the continuity equation for the scalarfield ˙ ρ = − H ( ρ + P ), ǫ is expressed as follows ǫ = 6 ˙ φ / φ / V . (40)From the last equation, we solve for the kinetic term ˙ φ , yielding˙ φ V ǫ − ǫ . (41)13o, we can write down the expression for the energy density of the scalar field ρ = ˙ φ + V in terms of the slow-roll parameter ǫ and the scalar field potential V as follows ρ = V ˙ φ V + 1 ! , (42) ρ = V (cid:18) ǫ − ǫ + 1 (cid:19) . (43)Accordingly, the relationship between the energy density and the potential at the end ofinflation ( ǫ ( φ end ) = 1) is given by ρ end = 65 V ( φ end ) = 65 V end , (44)which is slightly different from those already obtained in [121], where ρ end = V end . Other-wise, in GR it is found that ρ end = V end [66, 68, 70].Replacing (44) in Eq. (37) we obtain N re = 13(1 + w re ) ln (cid:18) V end ρ re (cid:19) . (45)At the end of reheating phase, the energy density of the universe is assumed to be ρ re = π g re T re , (46)where g re is the number of internal degrees of freedom of relativistic particles at the end ofreheating. Assuming that the degrees of freedom come from the particles in the StandardModel, g re = O (100) for &
175 GeV [48, 49], while for a Minimal Supersymmetric StandardModel ()MSSM, g re = O (200) [97, 122].On the other hand, the entropy is defined as s = 2 π gT , (47)where the temperature is inversely proportional to the scale factor for radiation ( T ∝ a − ).Then, by replacing the temperature in Eq. (47), we have that s ∝ a − . Assuming theconservation of entropy, it yields gT a = const. Now, if we apply the entropy conservationbetween reheating and today g re T re a re = g T a , (48)where g denotes the number of internal degrees of freedom of relativistic particles today,which comes from photons and neutrinos. Then, Eq. (48) becomes g re T re = (cid:18) a a re (cid:19) (cid:20) T + 214 T ν (cid:21) . (49)14or the contribution coming from neutrinos at the right-hand side of (49), we use T ν = (cid:0) (cid:1) / T , where T = 2 .
725 K is the temperature of the universe today. The ratio a a re canbe written as a a re = a a eq a eq a re , (50)where we introduce e − N RD = a re a eq , with N RD being the duration in e -folds of the radiationdominated epoch. Accordingly, Eq. (49) is rewritten as T re = T (cid:18) a a eq (cid:19) e N RD (cid:18) g re (cid:19) . (51)Furthermore, we may compare the wavelength ( λ ≃ a k ) with the Hubble radius ( d H ≃ H ) today , so d H λ = ka H (52)= a k H k a H , (53)where the subscript k denotes when the scale crosses the Hubble radius. Incorporating theintermediates eras, for the ratio a a eq we have (see, e.g. [70]) a a eq = a H k k e − N k e − N re e − N RD . (54)Using this result in (51) we find T re = (cid:18) g re (cid:19) / (cid:18) a T k (cid:19) H k e − N k e − N re . (55)Upon replacement of Eq. (55) in Eq. (45), one solves for N re giving N re = 41 − w re " −
14 ln (cid:18) π g re (cid:19) −
13 ln (cid:18) g re (cid:19) − ln (cid:18) ka T (cid:19) − ln V / end H k ! − N k . (56)By assuming g re ∼ O (100) and using the pivot scale ka = 0 .
05 Mpc − from PLANCK, wearrive to the final expression for the number of e -folds of reheating N re = 41 − w re " . − N k − ln V / end H k ! , (57)where H k can be written down using the definition of the tensor-to-scalar ratio r = P T /P S .Taking P S at the pivot scale and using the expression for P T in the high-energy limit (31),one finds H k = π rP S M r λ π ! / . (58)15inally, combining Eqs. (46) and (45) the reheating temperature is computed as follows T re = (cid:18) π (cid:19) / (cid:18) V end (cid:19) / e − (1+ w re ) N re . (59)Here, the model-dependent expressions are the Hubble rate at the instant when thecosmological scale crosses the Hubble radius, H − k , the number of e -folds N k , and the inflatonpotential at the end of the inflationary expansion, V end . Thus, it is implicit that N re , T re depend on the observables P s , n s and r that we have already discussed. It is also remarkablethe dependence of N re and T re on the 5-dimensional Planck mass, which enters into V end and H k . III. NATURAL INFLATION ON THE BRANEA. Dynamics of inflation
The Natural inflation potential is given by Eq. (3) V ( φ ) = Λ (cid:20) − cos (cid:18) φf (cid:19)(cid:21) . (60)Applying Eqs. (23) and (24) to this potential, we obtain the slow-roll parameters in thehigh-energy regime ǫ = α (1 + cos( y ))(1 − cos( y )) , (61) η = α cos( y )(1 − cos( y )) , (62)where y and α are dimensionless parameters, which by definition are given by y ≡ φf , (63) α ≡ M λ πf Λ , (64)respectively. An important quantity to be computed is the field at Hubble horizon crossing φ k , at which observables, such as the scalar power spectrum, the spectral index and thetensor-to-scalar ratio, are evaluated. In doing so, we first impose the condition ǫ ≡
1, whichallows us to compute the value of the inflaton field at the end of inflationcos( y end ) = cos (cid:18) φ end f (cid:19) = 12 (cid:16) α − √ α √ α (cid:17) . (65)16eplacing this value of the field and the potential in Eq. (25) and performing the integral,the number of e -folds N k is computed to be N k = 1 α (cid:20) cos( y k ) − cos ( y end ) − (cid:18) y k )1 + cos( y end ) (cid:19)(cid:21) . (66)Then, we solve for y k = φ k f , yieldingcos( y k ) = cos (cid:18) φ k f (cid:19) = − − W − [ z ( N k , α )] , (67)where W − denotes the negative branch of the Lambert function [123], and its argument isgiven by z ( N k , α ) ≡ r e − − √ α ( √ α +2 N k √ α −√ α ) (cid:16) α − √ α √ α (cid:17) . (68) B. Cosmological perturbations
Using the potential (60) in the expression for the scalar power spectrum (Eq. (30)), itleads to P S = 112 π α γ (1 − cos( y )) (1 + cos( y )) , (69)where γ = Λ f is a new dimensionless parameter. If we replace Eqs. (61) and (62) into (32),we obtain the expression for the spectral index n s = 1 − α (3 + 2 cos( y ))(1 − cos( y )) . (70)The tensor-to-scalar ratio as a function of the scalar field is obtained after replacing (60)in Eq. (33) r = 24 α (1 + cos( y ))(1 − cos( y )) . (71)After evaluating those observables at the Hubble radius crossing with (67), we find P S = 43 πα γ (1 + W − [ z ( N k , α )]) ( − W − [ z ( N k , α )]) , (72) n s = 1 − α − W − [ z ( N k , α )])(1 + W − [ z ( N k , α )]) , (73) r = 12 α ( − W − [ z ( N k , α )])(1 + W − [ z ( N k , α )]) . (74)The predictions for Natural Inflation regarding the n s − r plane may be generated plottingEqs. (73) and (74) parametrically, varying simultaneously the dimensionless parameter α in17 wide range and the number e -folds N k within the range N k = 60 −
70. In Fig. 1, we haveconsidered the two-dimensional marginalized joint confidence contours for ( n s , r ) at the 68%(blue region) and 95% (light blue region) C.L., from the latest PLANCK 2018 results.The allowed values for α are found when a given curve, for a fixed number of e -folds,enters (from above) and leaves (from below) the 2 σ region. One obtains that for N k = 65,the predictions of the model are within the 95% C.L. region from PLANCK data, for α being in the range 3 . × − . α . . × − . (75)In that case, the prediction for the tensor to scalar ratio is the following0 . & r & . . (76) Primordial tilt ( n s ) T e n s o r - t o - sca l a rr a t i o ( r . ) TT + lowE + lensingTT,TE,EE + lowE + lensingV (cid:1) - Cos (cid:2) ϕ f N k = N k = N k = FIG. 1: We show the plot of the tensor-to-scalar ratio r versus the scalar spectral index n s for Nat-ural inflation on the brane along with the two-dimensional marginalized joint confidence contoursfor ( n s , r ) at the 68% (blue region) and 95% (light blue region) C.L., from the latest PLANCK2018 results. Accordingly, for N k = 70, the predictions are within the 95% C.L. for the following rangeof α . × − . α . . × − , (77)18hile r is found to be in the range 0 . & r & . . (78)Thus, combining the previous constraints on α with Eq.(72) and the amplitude of thescalar spectrum P S ≃ . × − , we obtain the corresponding allowed ranges for the dimen-sionless parameter γ . × − . γ . . × − , (79)6 . × − . γ . . × − , (80)for N k = 65 and N k = 70, respectively. The allowed ranges for α and γ are summarized inTable I. N k Constraint on α Constraint on γ
65 0 . . α . . . × − . γ . . × −
70 0 . . α . . . × − . γ . . × − TABLE I: Results for the constraints on the parameters α and γ for Natural inflation in thehigh-energy of Randall-Sundrum brane model, using the last data of PLANCK. After replacing the relation between the 4-dimensional and 5-dimensional Planck masses(Eq. (13)) into the definition of α (Eq.(64)) and using the fact that Λ = γf , the followingexpressions for the mass scales f and Λ are derived f = (cid:18) π αγ (cid:19) / M , (81)Λ = γf = γ (cid:18) π αγ (cid:19) / M . (82)Evaluating those expressions at the several values for α and γ (Table I), we may obtaina value for the brane tension λ as well as the allowed ranges for the mass scales f and Λ forany given value of M . If we consider the lower limit for the five-dimensional Planck mass, M = 10 TeV [116], it yields λ = 1 . × − TeV , while the corresponding constraints onthe mass scales (in units of TeV) are shown in Table II.19 k Constraint on f [TeV] Constraint on Λ [TeV]65 1 . × & f & . × . × & Λ & . ×
70 1 . × & f & . × . × & Λ & . × TABLE II: Results for the constraints on the mass scales f and Λ for Natural inflation in thehigh-energy of Randall-Sundrum brane model M = 10 TeV, using the last data of PLANCK.
In order to obtain an upper bound for the 5-dimensional Planck mass, we take intoaccount that the inflationary dynamics takes places in the high-energy regime, V ≫ λ . Indoing so, we realize that during inflation V ≃ Λ , then if we solve Eq. (13) for the branetension λ , the condition for the high-energy regime imposes the following constraint on theamplitude of the potential Λ ≫ M πM . (83)Combining Eqs. (82) and (83), one finds the following upper bound for the 5-dimensionalPlanck mass M ≪ r π γ (cid:18) π αγ (cid:19) / M . (84)If we replace the allowed values for α and γ in the last equation, we find that the 5-dimensional Planck mass is such M ≪ TeV. So, if we assume that the maximumallowed value for M is two orders of magnitude less, i.e. M = 10 TeV, the brane tensionis computed to be λ = 1 . × TeV , while the constraints on f and Λ are displayed inTable III. N k Constraint on f [TeV] Constraint on Λ [TeV]65 1 . × & f & . × . × & Λ & . ×
70 1 . × & f & . × . × & Λ & . × TABLE III: Results for the constraints on the mass scales f and Λ for Natural inflation in thehigh-energy of Randall-Sundrum brane model M = 10 TeV, using the last data of PLANCK.
From Tables II and III, the mass scales f and Λ take sub-Planckian values and thereis a hierarchy between them consistent with f ≫ Λ, achieving an almost flat potential.Moreover, the constraints already found on α and Eqs. (65) and (67) imply that during20nflation the dynamics is such that φ ∼ f , therefore Natural Inflation in the high-energyregime of the RS-II brane-model takes place at sub-Planckian values of the scalar field. It isworth mentioingn that our results for the mass scales differ almost by one order or magnitudein comparison to those already found in Ref. [124] when using M = 10 TeV. In addition,our results with the upper limit M = 10 TeV are similar to those found in Ref. [98] sofar, where the authors used M = 5 × TeV.After obtaining the allowed parameter space where Natural Inflation in the high-energylimit of Randall-Sundrum brane model is viable, we want to see if the Swampland Criteria aremet in this model. Fig. 2 shows the distance conjecture (9) and the de Sitter conjecture (10)of the Swampland Criteria: the top and bottom panels shows the behaviour of ∆ φ/M pl ≡ ∆ φ and M pl | V ′ | /V ≡ ∆ V , respectively, against the number of e -folds N k for some values of α and the lower (left) and upper (right) limits of M . We note that for the distance conjecture∆ φ , it increases as both N k and the 5-dimensional Planck mass increase, but the curves arealways less than the unity since the scale mass f is always sub-Planckian, so the distanceconjecture is fulfilled. For the de Sitter conjecture, we note that ∆ V decreases with N k ,but it increases as M increases. In this case, we must be careful because ∆ V is relatedto the slow-roll parameter ǫ V in General Relativity, yielding values much larger than thisconjecture requires. Nevertheless, as we discussed in Section II, slow-roll inflation on thebrane implies that ǫ ≪ | η | ≪
1, which can be achieved in the high-energy limit, i.e., V ≫ λ despite the fact that both ǫ V and η V are large. In this way, the de Sitter conjectureand its refined version are avoiding. Additionally, our results for the distance conjectureare similar to those found in Ref. [98] while although our plots have the same behavior forthe de Sitter conjecture, the values of ∆ V differ by several order of magnitude when we use M = 10 TeV.
C. Reheating
We now investigate the predictions regarding the number of e -folds as well as the tem-perature associated with the reheating epoch N re and T re , respectively. In doing so, we plotparametrically Eqs. (73), (57), and (59) with respect to the number of e -folds N k for severalvalues of the effective equation-of-state parameter w re over the range − ≤ w re ≤
1, as wellas α , which encodes the information about the mass scales f and Λ, and the brane tension21 . In Fig. 3, we show the plots for reheating when using the lower limit of the 5-dimensionalPlanck mass, namely M = 10 TeV and two allowed values of α at N k = 65. On the otherhand, in Fig. 4 we use the upper limit on M , M = 10 TeV for the same values of α . Forthe other values of α , the prediction of reheating has the same behaviour, however we willshow the plots that fit better with current observational data. Firstly, we must note that,for the two values of M , the point at which the curves converge (implying instantaneousreheating, i.e. N re →
0) is gradually shifted to the left when we increasing the dimensionlessparameter α . Another important finding is that the temperature at which all curves inter-sect, i.e. the maximum reheating temperature, increases as the 5-dimensional Planck mass M increases. In particular, for M = 10 TeV the maximum reheating temperature is about10 GeV, while for M = 10 TeV, it is about 10 GeV. Then, a new phenomenology arisesin comparison to the former analysis within the standard scenario, in which the maximumreheating temperature (if reheating is instantaneous) is T re . × GeV. This dependencereheating temperature on five-dimensional Planck mass has been realised in Ref. [121] sofar, where the authors reconstructed the inflationary potential in the RS-II brane-world.Furthermore, our reheating temperature, which depends strongly on the five-dimensionalPlanck mass, for M = 10 TeV is at least two orders of magnitude greater than thosefound in [98] in which case the temperature is more sensitive to the number of e -folds.If we assume that during reheating, the universe is governed by an effective equation-of-state of the form P = w re ρ , where P and ρ denote the pressure and the energy density,respectively, of the fluid in which the inflaton decays. Then, it becomes important to findwhat EoS parameter w re is preferred by current observational bounds. For doing that, weanalyze when each curve of the reheating temperature plots against the scalar spectral indexenters to the purple region (at 1 σ of n s ) and meets the point at which all curves converges.Therefore, an allowed range for the scalar spectral index n s as well as N k is found whenfixing α . For consistency, we display the results for the plots of Fig. 4 in Table IV. It isworth noting that these values for the duration of reheating differ from those obtained in [98]which are found to be N re ≈
20 for w re = 1 and N k = 65 when they use M = 5 × TeV.On the other hand, it should be noted that for α = 0 . α = 0 . α = 0 . M = 10 TeV onlyone curve, corresponding to w re = 1 is inside but for M = 10 TeV two curves ( w re = 2 / w re = 1) are inside. For α = 0 .
10 20
30 40 50 60 7002. × - × - × - × - N k Δ ϕ M = TeV α = α = α = α = N k Δ ϕ M = TeV α = α = α = α = × × × × × × N k Δ V M = TeV α = α = α = α = N k Δ V M = TeV α = α = α = α = FIG. 2: Plots of the Swampland criteria for M = 10 TeV (left) and M = 10 TeV (right) interms of the number of e -folds for the different values of α . Top panels show the behaviour of thedistance conjecture where ∆ φ/M pl ≡ ∆ φ , while the bottom panels show the de Sitter conjecturewhere M pl | V ′ | /V ≡ ∆ V . w re N k e -folds for each EoS parameter w re for M = 10 TeV when the dimensionless parameter α is fixed to be α = 0 . We also want to know what are the allowed values for the tensor-to-scalar ratio in terms ofthe reheating temperature for some values belonging the allowed range of the 5-dimensional23 .94 0.95 0.96 0.97 0.98 0.99 1.00020406080 n s N r e f = * TeV λ = * - TeV w re = - / w re = w re = / w re = n s N r e f = * TeV λ = * - TeV w re = - / w re = w re = / w re = - n s L(cid:0)(cid:3) T r e G(cid:4)(cid:5) f = * TeV λ = * - TeV w re = - / w re = w re = / w re = - n s (cid:6)(cid:7)(cid:8) T r e (cid:9)(cid:10)(cid:11) f = * TeV λ = * TeV w re = - / w re = w re = / w re = FIG. 3: Plots of N re and T re as functions of n s using the lower limit of the five-dimensional Planckmass ( M = 10 TeV) for Natural inflation. The blue, pink, green and black curves correspondsto the following values of the EoS parameter: w re = − /
3, 0, 2 / n s = 0 . ± . T < GeV. Also, in order to beconsistent with BBN, it is required T re &
10 MeV. The two plots on the left and on the rightcorrespond to α = 0 . α = 0 . Planck mass. In doing so, we plot parametrically Eqs. (74) and (59) with respect to thenumber of e -folds, which varies according to the available found so far with M and α fixed (see Table IV). The values obtained for r must be consistent with the upper limit byPLANCK 2018 data in combination with the BICEP2/Keck Array (BK14) data. In thisway we can discard those values of α , T re and w re for which r does not meet this bound.We emphasize that this method must be consistent with the previous analysis. The onlyvalues of α and w re in consistency with the former, correspond to α = 0 . w re = 1,24 .94 0.95 0.96 0.97 0.98 0.99 1.00020406080 n s N r e f = * TeV λ = * TeV w re = - / w re = w re = / w re = n s N r e f = * TeV λ = * TeV w re = - / w re = w re = / w re = - n s Log T r e G e V f = * TeV λ = * TeV w re = - / w re = w re = / w re = - n s Log T r e G e V f = * TeV λ = * TeV w re = - / w re = w re = / w re = FIG. 4: Same as Fig. 3 but for the upper limit of the five-dimensional Planck mass, M = 10 TeV.
Log T re GeV r w re = M = TeV
FIG. 5: Plot of the tensor-to-scalar ratio r against the reheating temperature for Natural inflationfor w re = 1 and α = 0 . M = 10 TeV. as it is shown in Fig. 5. Firstly, we observe that the curves starts at T re ≈ GeV which is25onsistent with the previous plots of reheating. In principle, temperatures within the range10 MeV . T re . GeV (gray region from Figs. 3 and 4 may not be discarded, but wouldbe interesting for baryogenesis [125]. Next, we note that the curve for the lower limit of M is well outside the upper limit on r , so we can discard it in principle. Consequently, it isfound that for w re = 1, the reheating temperature must be in the range of10 GeV . T re . GeV , (85)when M = 10 TeV.
IV. HILLTOP INFLATION ON THE BRANEA. Dynamics of inflation
Quadratic Hilltop inflation is driven by the potential (4) V ( φ ) = Λ " − (cid:18) φµ (cid:19) (86)In this case, the slow-roll parameters in the high-energy limit are given by ǫ = α x (1 − x ) , (87) η = − α − x ) , (88)where the dimensionless parameters are defined as follows x ≡ φµ , (89) α ≡ M λπ µ Λ . (90)Unlike Natural Inflation, for our quadratic Hilltop potential, the scalar field at Hubble-radius crossing φ k is found by means numerically. In that case, we start with the definitionof the number of e -folds in terms of the Hubble rate dN = Hdt. (91)Then, using the slow-roll approximation 3 H ˙ φ + V ≈ x and φ , we have a differential expression which gives us x ( N ) dN ≃ − πµV λM V ′ dx. (92)26e obtain the numerical solution for x k by means introducing the initial condition x ( N =0) = x end , where x end = φ end /µ is obtained from the condition at the end of inflation, i.e. ǫ ( x end ) = 1 from Eq. (87). B. Cosmological perturbations
Replacing the potential (86) into Eq. (30) we found the following expression for the scalarpower spectrum as a function of the scalar field P S = 4 γ (1 − x ) π α x , (93)where γ = Λ µ is a dimensionless parameter. To obtain the scalar spectral index and thetensor to scalar ratio, both evaluated at the Hubble-radius crossing, one first replaces thesolution for x k in ǫ and η and uses Eqs. (32) and (33). Next, we plot parametrically n s and r , varying simultaneously α in a wide range and N k within the range N k = 55 −
65. Fig.6 shows the tensor-to-scalar ratio against the spectral index plot using the two-dimensionalmarginalized joint confidence contours for ( n s , r ) at the 68% (blue region) and 95% (lightblue region) C.L., from the latest PLANCK 2018 results. Using the same method as inNatural inflation to found the allowed values of α , one obtains that the predictions of themodel are within the 95% C.L. region from PLANCK data if, for N k = 55, α lies in therange 1 . × − . α . . × − , and the corresponding prediction for the tensor-to-scalar ratio is 0 . & r & . N k = 60, the allowed range of thedimensionless parameter α is 1 . × − . α . . × − , while r is found to be in therange 0 . & r & . N k = 65, α is found to be in range range9 . × − . α . . × − , while the prediction for r is 0 . & r & . α , and the amplitude of the scalarspectrum P S ≃ . × − , the allowed values for γ are found to be in the range3 . × − . γ . . × − , (94)2 . × − . γ . . × − , (95)2 . × − . γ . . × − , (96)for N k = 55, N k = 60 and N k = 65, respectively. The allowed ranges for α and γ aresummarized in Table V. 27 .94 0.95 0.96 0.97 0.98 0.99 1.000.000.050.100.150.20 Primordial tilt ( n s ) T e n s o r - t o - sca l a rr a t i o ( r . ) TT + lowE + lensingTT,TE,EE + lowE + lensingV ∝ - ϕ μ N k = N k = N k = FIG. 6: Plot of the tensor-to-scalar ratio r versus the scalar spectral index n s for quadratic Hilltopinflation on the brane along with the two-dimensional marginalized joint confidence contours for( n s , r ) at the 68% (blue region) and 95% (light blue region) C.L., from the latest PLANCK 2018results. N k Constraint on α Constraint on γ
55 0 . . α . . . × − . γ . . × −
60 0 . . α . . . × − . γ . . × −
65 0 . . α . . . × − . γ . . × − TABLE V: Results of the constraints on the parameters α and γ for Hilltop inflation in the high-energy of Randall-Sundrum brane model, using the last data of PLANCK. The expressions for the mass scales µ and Λ are obtained after replacing Eq. (13) intothe definition of α (Eq. (90)), yielding µ = (cid:18) π αγ (cid:19) / M , (97)Λ = γµ = γ (cid:18) π αγ (cid:19) / M . (98)28fter evaluating those expressions at the several values for α and γ (Table V) and con-sidering the lower limit for the five-dimensional Planck mass, M = 10 TeV, the branetension is found to be λ = 1 . × − TeV while the corresponding constraints on the massscales (in units of TeV) are shown in the top panel of Table VI. Using the same methodto found an upper limit of the five-dimensional Planck mass as in Natural inflation, we ob-tain that M ≪ TeV. Assuming as maximum allowed value M = 10 TeV, we obtain λ = 1 . × TeV and the corresponding values for mass scales are shown in the bottompanel of Table VI. N k Constraint on µ [TeV] Constraint on Λ [TeV]55 2 . × & µ & . × . × & Λ & . ×
60 3 . × . µ & . × . × & Λ & . ×
65 3 . × & µ & . × . × & Λ & . × N k Constraint on µ [TeV] Constraint on Λ [TeV]55 2 . × & µ & . × . × & Λ & . ×
60 3 . × & µ & . × . × & Λ & . ×
65 3 . × & µ & . × . × & Λ & . × TABLE VI: Results for the constraints on the mass scales µ and Λ for quadratic Hilltop inflationin the high-energy limit of Randall-Sundrum brane model using the last data of PLANCK. Thetop table shows the results using M = 10 TeV while the bottom table shows the results using M = 10 TeV.
For this model, the plots for the Swampland criteria, which are not shown, but thesepresent the same behavior that those shown in FIG. 2. For the distance conjecture, ∆ φ increases with the number of e -folds but also increases as the 5-dimensional Planck massgrows, so this conjecture is fulfilled. On the other hand, for the de Sitter conjecture, ∆ V decreases with both the number of e -folds and M which, having in mind the discussion inSection III, it is avoided. 29 . Reheating In the same way as Natural inflation, we can give predictions for reheating plottingparametrically Eqs. (57) and (59) with respect to α and N k over the range of the effectiveEoS − ≤ w re ≤
1. In despite this type of potential is unbounded from below, i.e. doesnot present a minimum around which the inflaton oscillates and reheating is achieved, wemay assume that the details of reheating are encoded in the effective EoS parameter w re .Yet another possibility to achieve reheating is by adding extra terms as those in Eq. (5) for p = 2, which stabilizes the potential and prevent it becomes negative. In FIG. 7, we showthe plots for reheating using M = 10 TeV (left panels) and M = 10 TeV (right panels)for α = 0 . N k = 55. Even though it is not shownin the plots, the behavior of the convergence point is the same as in Natural inflation, i.e.,the point at which the curves converges shifts to the left when α increases. As it can beseen, the maximum temperature of reheating also increases with the five-dimensional Planckmass, giving T re ≈ GeV for M = 10 TeV and T re ≈ GeV for M = 10 TeV.Analyzing when each curve of the reheating temperature plots enters to the purple regionand meets the converge point of instantaneous reheating, an allowed range for N k is foundwhen fixing α . For consistency, we display the results for the plots of Fig. 7 in Table VII.It should be noted that for α = 0 . α = 0 . α = 0 . α = 0 . α = 0 . w re N k -1/3 49 - 560 53 - 562/3 56 - 581 56 - 59 w re N k -1/3 49 - 560 49 - 562/3 56 - 611 56 - 65TABLE VII: Summary of the allowed range for the number of e -folds for each EoS parameter w re when the dimensionless parameter α is fixed to α = 0 . M = 10 TeV and M = 10 TeV respectively.
Plotting parametrically Eqs. (34) and (59), both evaluated at the Hubble radius crossing,30 .94 0.95 0.96 0.97 0.98 0.99 1.00020406080 n s N r e μ = * TeV λ = * - TeV w re = - / w re = w re = / w re = n s N r e μ = * TeV λ = * TeV w re = - / w re = w re = / w re = - n s Log T r e G e V μ = * TeV λ = * - TeV w re = - / w re = w re = / w re = - n s Log T r e G e V μ = * TeV λ = * TeV w re = - / w re = w re = / w re = FIG. 7: Plots of N re and T re as functions of n s for Hilltop inflation. The left panels shows theplots for M = 10 TeV while the right panels shows the plot for M = 10 TeV. The curves andthe shading regions are the same as FIG. 3 and all plots corresponds to α = 0 . with respect to the number of e -folds, it is possible to express the tensor-to-scalar ratio, r , interms of T re . Then, one constrains simultaneously r and T re when α is fixed, and for certainvalues of the EoS parameter and the 5-dimensional Planck mass. From Fig. 8, it is foundthat for w re = − / w re = 0, the available values of T re are 10 GeV, 10 GeV, and 10 GeV, when M is fixed to 10 TeV, 10 TeV, and 10 TeV, respectively. Consequently, theallowed ranges for T re when w re is set to 2/3 and 1, read10 GeV . T re . GeV , (99)10 GeV . T re . GeV , (100)10 GeV . T re . GeV , (101)when M is set to 10 TeV, 10 TeV, and 10 TeV, respectively.31
Log T re GeV r w re = - / M = TeV M = TeV M = TeV
Log T re GeV r w re = M = TeV M = TeV M = TeV
Log T re GeV r w re = / M = TeV M = TeV M = TeV
Log T re GeV r w re = M = TeV M = TeV M = TeV
FIG. 8: Plots for the tensor-to-scalar ratio against the reheating temperature for quadratic Hilltopinflation for w re = − / , , / , α = 0 . M = 10 TeV, M = 10 TeV and M = 10 TeV respectively.
V. HIGGS-LIKE INFLATION ON THE BRANEA. Dynamics of inflation
The potential for Higgs-like inflation is given by Eq. (6) V ( φ ) = Λ " − (cid:18) φµ (cid:19) (102)The slow-roll parameters in the high-energy limit are computed to be ǫ = 4 α x (1 − x ) , (103) η = 2 α x (1 − x ) − α (1 − x ) , (104)32here the dimensionless parameter are defined as x ≡ φµ , (105) α ≡ M λπµ Λ . (106)Similarly to Hilltop inflation, we solve numerically the expression for the scalar fieldat the Hubble-radius crossing. Using the definition of the number of e -folds and the KGequation in the slow-roll approximation, a first order differential equation for x k = φ/f isobtained. The former is solved by using using as initial condition x ( N = 0) = x end , where x end = φ end /µ is obtained from the condition at the end of inflation, i.e. ǫ ( x end ) = 1. B. Cosmological perturbations
The scalar power spectrum is found replacing the potential (102) into Eq. (30), whichyields P S = γ (1 − x ) π α x , (107)where γ = Λ µ . Evaluating the slow-roll parameters ǫ and η at the solution for x k , and usingEqs. (32) and (33), we may obtain both the scalar spectral index and the tensor-to-scalarratio, and generate the n s − r plane. Here, we vary simultaneously the number e -folds N k within the range N k = 60 −
70, and α in a wide range. Fig. 9 shows the tensor-to-scalarratio against the scalar spectral index plot using the two-dimensional marginalized jointconfidence contours for ( n s , r ) at the 68% (blue region) and 95% (light blue region) C.L.,from the latest PLANCK 2018 results.Following the same procedure as before to find the allowed values of α , one obtains thepredictions of the model within the 95% C.L. region from PLANCK data. For N k = 60, α must be within the range 1 . × − . α . . × − . Consequently, the tensor-to-scalarratio lies in the interval 0 . & r & . N k = 65, the predictions are found to be8 . × − . α . . × − and 0 . & r & . N k = 70, α is foundwithin the range 6 . × − . α . . × − , while the corresponding values of thetensor-to-scalar ratio are given by 0 . & r & . α with Eq. (107) and the amplitude of the scalar spectrum33 .94 0.95 0.96 0.97 0.98 0.99 1.000.000.050.100.150.20 Primordial tilt ( n s ) T e n s o r - t o - sca l a rr a t i o ( r . ) TT + lowE + lensingTT,TE,EE + lowE + lensingV ∝ - ϕ μ N k = N k (cid:12) N k (cid:13) FIG. 9: Plot of the tensor-to-scalar ratio r versus the scalar spectral index n s for Higgs-like inflationon the brane along with the two-dimensional marginalized joint confidence contours for ( n s , r ) atthe 68% (blue region) and 95% (light blue region) C.L., from the latest PLANCK 2018 results. P S ≃ . × − we obtain the corresponding allowed ranges for γ . × − . γ . . × − , (108)2 . × − . γ . . × − , (109)2 . × − . γ . . × − , (110)for N k = 60, N k = 65 and N k = 70, respectively. The allowed ranges for α and γ aresummarized in Table VIII. N k Constraint on α Constraint on γ
60 0 . . α . . . × − . γ . . × −
65 0 . . α . . . × − . γ . . × −
70 0 . . α . . . × − . γ . . × − TABLE VIII: Results of the constraints on the parameters α and γ for Higgs-like inflation in thehigh-energy limit of the Randall-Sundrum brane model, using the last data of PLANCK. α (Eq. (106)), and using the fact thatΛ = γ µ , the following expressions for µ and Λ in terms of M are derived µ = (cid:18) π αγ (cid:19) / M , (111)Λ = γµ = γ (cid:18) π αγ (cid:19) / M . (112)Evaluating those expressions at several values for α and γ (Table VIII) and consideringthe lower and upper limit for the five-dimensional Planck mass, the brane tension is foundto be the same as in the previous models, i.e. λ = 1 . × − TeV for M = 10 TeV and λ = 1 . × TeV for M = 10 TeV. The top panels of Table IX show the correspondingvalues of the mass scales for the lower limit of M , while the bottom panels show the valuesof the mass scales for the upper limit. N k Constraint on µ [TeV] Constraint on Λ [TeV]60 3 . × & µ & . × . × & Λ & . ×
65 3 . × & µ & . × . × & Λ & . ×
70 4 . × & µ & . × . × & Λ & . × N k Constraint on µ [TeV] Constraint on Λ [TeV]60 3 . × & µ & . × . × & Λ & . ×
65 3 . × & µ & . × . × & Λ & . ×
70 4 . × & µ & . × . × & Λ & . × TABLE IX: Results for the constraints on the mass scales µ and Λ for Higgs-like inflation inthe high-energy limit of the Randall-Sundrum brane model using the last data of PLANCK. Thetop table shows the results using M = 10 TeV while the bottom table shows the results using M = 10 TeV.
Following the analysis performed for Natural Inflation, it can be shown that the plotsfor the two conjectures of the Swampland Criteria, which are not shown, exhibit a similarbehavior with those shown in Fig. 2. For the distance conjecture, ∆ φ increases with boththe number of e -folds and the 5-dimensional Planck mass, so this conjecture is fulfilled. Onthe other hand, for the de Sitter conjecture, ∆ V decreases with the number of e -folds and35 . For the same arguments given before, the de Sitter Swampland criteria and its refinedversion are avoided. C. Reheating
Following the same method as previous sections, we can give predictions for reheatingby means plotting parametrically Eqs. (57) and (59) with respect to α and N k over therange of the effective EoS − ≤ w re ≤
1. In Fig. 10 we show the plots for reheating using M = 10 TeV (left panels) and M = 10 TeV (right panels) for α = 0 . N k = 55). Our analysis indicates that the behavior of the convergencepoint is the same as in Natural inflation and quadratic Hilltop inflation. As we can see,for M = 10 TeV the maximum reheating temperature is about T re ≈ GeV and for M = 10 TeV is about T re ≈ GeV.Analyzing the curves for the reheating temperature, we found an allowed range for N k for each value of w re when α is fixed. The corresponding intervals are shown in Table X. Forconsistency, we only display the results for the plots of Fig. 10 for M = 10 TeV becausethe allowed range of N k for the lower limit of M is too small. It should be noted that for α = 0 . α = 0 . α = 0 . α = 0 . α = 0 . w re N k e -folds for each EoS parameter w re when the dimensionless parameter α is fixed to be α = 0 . M = 10 TeV.
Evaluating Eqs. (34) and (59) at the Hubble radius crossing, and plotting parametricallywith respect to the number of e -folds, we can find the allowed values for the tensor-to-scalarratio in terms of the reheating temperature. The only values of α and w re consistent withthe current bounds on the tensor-to-scalar ratio, correspond to α = 0 . w re = 2 / , M greater than its lower limit, as it is shown in Fig. 11. In this case, it is36 .94 0.95 0.96 0.97 0.98 0.99 1.00020406080 n s N r e μ = * TeV λ = * - TeV w re = - / w re = w re = / w re = n s N r e μ = * TeV λ = * TeV w re = - / w re = w re = / w re = - n s Log T r e G e V μ = * TeV λ = * - TeV w re = - / w re = w re = / w re = - n s Log [ T r e G e V ] μ = * TeV λ = * TeV w re = - / w re = w re = / w re = FIG. 10: Plots of N re and T re as functions of n s for Higgs-like inflation. The left panels shows theplots for M = 10 TeV while the right panels shows the plots for M = 10 TeV. The curves andthe shading regions are the same as FIG. 3 and all plots corresponds to α = 0 . found that for M = 10 TeV and w re = 2 / GeV . T re . GeV , (113)while for w re = 1, one finds that the allowed values for T re are found within the ranges10 GeV . T re . GeV , (114)10 GeV . T re . GeV , (115)when M is fixed to 10 TeV and 10 TeV, respectively.37
Log T re GeV r w re = / M = TeV
Log T re GeV r w re = M = TeV M = TeV
FIG. 11: Plots for the tensor-to-scalar ratio against the reheating temperature for Higgs-like infla-tion for w re = 2 / , α = 0 . M = 10 TeV andthe blue line corresponds to a mass of M = 10 TeV.
VI. EXPONENTIAL SUSY INFLATION ON THE BRANEA. Dynamics of inflation
The last potential we study in the present work is a well motivated one from SUGRA,namely Exponential SUSY inflation, given by Eq. (7) V ( φ ) = Λ (1 − e φ/f ) (116)Replacing this potential into Eqs. (23) and (24) we obtain the set of slow-roll parametersin the high-energy regime as ǫ = α e − x (1 − e − x ) , (117) η = − α e − x (1 − e − x ) , (118)where the dimensionless parameter are defined by x ≡ φf , (119) α ≡ M λ π f Λ . (120)Similarly to the quadratic Hilltop and Higgs-like inflation models, we solve the expressionfor x k ( N k ) numerically, and using as initial condition x ( N k = 0) = x end , where x end = φ end /f is obtained from the condition at the end of inflation, i.e. ǫ end = 1.38 . Cosmological perturbations Replacing the potential (116) into Eq. (30), we obtain the following expression for thescalar power spectrum P S = γ e x (1 − e − x ) π α , (121)where γ = Λ f . Evaluating ǫ and η at the solution for x k and using Eqs. (32) and (33) toobtain n s and r , we plot the predictions on the n s - r . In doing so, we vary simultaneouslythe dimensionless parameter α in a wide range and the number e -folds N k within the range N k = 50 −
60. Fig. 12 shows the tensor-to-scalar ratio against the scalar spectral indexplot using the two-dimensional marginalized joint confidence contours for ( n s , r ) at the 68%(blue region) and 95% (light blue region) C.L., from the latest PLANCK 2018 results. Primordial tilt ( n s ) T e n s o r - t o - sca l a rr a t i o ( r . ) TT + lowE + lensingTT,TE,EE + lowE + lensingV ∝ - ⅇ - ϕ f N k = N k = N k = FIG. 12: Plot of the tensor-to-scalar ratio r versus the scalar spectral index n s for ExponentialSUSY inflation on the brane along with the two-dimensional marginalized joint confidence contoursfor ( n s , r ) at the 68% (blue region) and 95% (light blue region) C.L., from the latest PLANCK2018 results. As we have seen already, the allowed values for α are found when a given curve, for a fixed N k , enters and leaves the 2 σ region. We note that in this model, unlike previous potentialsalready studied, the trajectories never leave the 2 σ region, achieving a very small tensor-39o-scalar ratio, which is well inside the (68% C.L.) contour for large values of α . The latterimplies that we only have a lower bound on α for each value of N k . So, following the samemethod as before, one obtains that the predictions of the model are within the 95% C.L.region from PLANCK data, for N k = 50, if α is such that α & . × − . Therefore, anupper bound for the scalar-to-tensor ratio is achieved, yielding r . . N k = 55, thelower bound on α is α & . × − , while r is found to be r . . N k = 60,the corresponding constraint on α is found to be α & . × − , while the tensor-to-scalarratio is such that r . . α → ∞ , one finds the asymptotic limit of thetensor-to-scalar ratio, that yields r →
0, whereas the asymptotic limit for the spectral indexis found to be n s → .
960 for N k = 50, n s → .
964 for N k = 55 and n s → .
967 for N k = 60.Combining the previous constraints on α with Eq. (121) and the amplitude of the scalarspectrum P S ≃ . × − , we obtain the corresponding allowed ranges for the dimensionlessparameter γ γ & . × − , (122) γ & . × − , (123) γ & . × − , (124)for N k = 50, N k = 55 and N k = 60, respectively. The allowed ranges for α and γ aresummarized in Table XI. N k Constraint on α Constraint on γ α & . γ & . × − α & . γ & . × − α & . γ & . × − TABLE XI: Results of the constraints on the parameters α and γ for Exponential SUSY inflationin the high-energy limit of Randall-Sundrum brane model, using the last data of PLANCK. Replacing Eq. (13) into the definition of α (Eq. (120)) and using the fact that Λ = γ f ,we found the expressions for the mass scales f and Λ as f = (cid:18) π αγ (cid:19) / M (125)40 = γf = γ (cid:18) π αγ (cid:19) / M (126)After evaluating these expressions at several values for α and γ (Table XI), we found that λ = 1 . × − TeV for M = 10 TeV and λ = 1 . × TeV for M = 10 TeV. Thetop panels of Table XII show the corresponding values of the mass scales for the lower limitof M , while the bottom panels shows the values of the mass scales for the upper limit. N k Constraint on f [TeV] Constraint on Λ [TeV]50 f . . × Λ . . × f . . × Λ . . × f . . × Λ . . × N k Constraint on f [TeV] Constraint on Λ [TeV]50 f . . × Λ . . × f . . × Λ . . × f . . × Λ . . × TABLE XII: Results for the constraints on the mass scales f and Λ for Exponential SUSY inflationin the high-energy limit of Randall-Sundrum brane model using the last data of PLANCK. Thetop table shows the results using M = 10 TeV while the bottom table shows the results using M = 10 TeV.
Like previous models, we find numerically that the distance Swampland conjecture, ∆ φ increases as both the number of e -folds and the 5-dimensional Planck mass increase, so thisconjecture is fulfilled, while for the de Sitter conjecture, ∆ V decreases as the number of e -folds increases, and also as M grows. C. Reheating
If one follows the same procedure as in the previous sections, we can give predictionsfor reheating plotting parametrically Eqs. (57) and (59) with respect to α and N k over therange of the effective EoS − ≤ w re ≤
1. Unlike previous models, this kind of potential isderived from SUGRA, hence the corresponding degrees of freedom of relativistic particles41t the end of reheating appearing in the expressions for N re and T re are g re = O (200). InFig. 13 we show the plots of reheating using M = 10 TeV (left panels) and M = 10 TeV(right panels) for α = 0 . N k = 55. As we cansee, the maximum reheating temperature increases with the five-dimensional Planck mass,giving T re ≈ GeV for M = 10 TeV and T re ≈ GeV for M = 10 TeV. n s N r e f = * TeV λ = * - TeV w re = - / w re = w re = / w re = n s N r e f = * TeV λ = * TeV w re = - / w re = w re = / w re = - n s Log T r e G e V f = * TeV λ = * - TeV w re = - / w re = w re = / w re = - n s Log T r e G e V f = * TeV λ = * TeV w re = - / w re = w re = / w re = FIG. 13: Plots of N re and T re as functions of n s for Exponential SUSY inflation. The left panelsshows the plot for M = 10 TeV while the right panels shows the plot for M = 10 TeV.Thecurves and the shading regions are the same as FIG. 3 and all plots corresponds to α = 0 . Analyzing the curves of the plots for reheating, we found the allowed values for numberof e -folds N k when fixing α for a certain value of the EoS parameter w re . For consistency,we display the results for the plots of Fig. 13 in Table XIII. It should be noted that for α = 0 . α = 0 . re N k -1/3 46 - 560 53 - 56 w re N k -1/3 44 - 560 47 - 56TABLE XIII: Summary of the allowed range for the number of e -folds for each EoS parameter w re when the dimensionless parameter α is fixed to α = 0 . M = 10 TeV and M = 10 TeV respectively.
Plotting parametrically Eqs. (34) and (59) with respect to the number of e -folds, we ex-press the allowed values for the tensor-to-scalar ratio in terms of the reheating temperature.The only values for α and w re in agreement with current bounds on the tensor-to-scalar ra-tio, correspond to α = 0 . w re = − / ,
0, as it is depicted in Fig. 14. In particular,for w re = − /
3, the reheating temperature must be in the ranges10 GeV . T re . GeV , (127)10 GeV . T re . GeV , (128)10 GeV . T re . GeV , (129)when M takes the values 10 TeV, 10 TeV, and 10 TeV, respectively. On the other hand,for w re = 0, the allowed ranges for T re are found to be10 GeV . T re . GeV , (130)10 GeV . T re . GeV , (131)10 GeV . T re . GeV , (132)when fixing M as 10 TeV, 10 TeV, and 10 TeV, respectively.The production of massive relics, such as gravitinos, is an important issue when discussingsupersymmetric models, since their overproduction might spoil the success of BBN [53–57].In the context of brane-world cosmology, the gravitino problem is avoided provided thatthe transition temperature, T t , is bounded from above, T t ≤ (10 − ) GeV [55]. Thetransition temperature is the temperature at which the evolution of the Universe passesfrom the brane-world cosmology into the standard one, and it is given by [126] T t = 1 . × (cid:18) g re (cid:19) / (cid:18) M GeV (cid:19) / (133)43learly, the upper bound on T t implies an upper bound on M , and therefore in the case ofexponential SUSY inflation the five-dimensional Planck mass is finally forced to take valuesin the range 10 TeV ≤ M ≤ TeV . (134) Log T re GeV r w re = - / M = TeV M = TeV M = TeV
Log T re GeV r w re = M = TeV M = TeV M = TeV
FIG. 14: Plots for the tensor-to-scalar ratio against the reheating temperature for ExponentialSUSY inflation for w re = − / , α = 0 . M = 10 TeV, M = 10 TeV and M = 10 TeV respectively.
VII. BARYOGENESIS VIA LEPTOGENESIS
Finally, let us comment on the generation of baryon asymmetry in the Universe. Any vi-able and successful inflationary model must be capable of generating the baryon asymmetry,which comprises one of the biggest challenges in modern theoretical cosmology. PrimordialBig Bang Nucleosynthesis [127] as well as data from CMB temperature anisotropies [128–132]indicate that the baryon-to-photon ratio is a very small but finite number, η B = 6 . × − [133]. This number must be calculable within the framework of the particle physics we know.Although as of today several mechanisms have been proposed and analysed, perhaps the mostelegant one is leptogenesis [134]. In this scenario a lepton asymmetry arising from the out-of-equilibrium decays of heavy right-handed neutrinos is generated first. Next, the leptonasymmetry is partially converted into baryon asymmetry via non-perturbative ”sphaleron”effects [135]. 44f particular interest is the non-thermal leptogenesis scenario [133, 136–145], since thelepton asymmetry is computed to be proportional to the reheating temperature after infla-tion. Therefore, within non-thermal leptogenesis the baryon asymmetry and the reheatingtemperature, two key parameters of the Big Bang cosmology, are linked together. Further-more, in supersymmetric models the gravitino problem [146, 147] puts an upper bound onthe reheating temperature after inflation [148], and therefore thermal leptogenesis [149, 150],which requires a high reheating temperature [151], is much more difficult to be implemented.Moreover, contrary to thermal leptogenesis where one has to solve the complicated Boltz-mann equations numerically, in the non-thermal leptogenesis scenario one can work withanalytic expressions.The initial lepton asymmetry, Y L = n L /s , is converted into baryon asymmetry Y B = n B /s via sphaleron effects [135] Y B = aY B − L (135)or Y B = aa − Y L ≡ C Y L (136)where n is the number density of leptons or baryons, s is the entropy density of radiation, s = (2 π h ∗ T ) /
45, and the conversion factor a is computed to be a = (24+4 N H ) / (66+13 N H )[152], with N H being the number of Higgs doublets in the model. In SM with only one Higgsdoublet, N H = 1, a = 28 /
79 and C = − /
51, while in MSSM with two Higgs doublets, N H = 2, a = 8 /
23 and C = − / Y L = 32 T re M I X i BR ( φ → N i N i ) ǫ i (137)where ǫ is the CP-violation asymmetry factor, and BR ( φ → N i N i ) is the branching ratio ofthe inflaton decay channel into a pair of right-handed neutrinos φ → N i N i .Moreover, lepton asymmetry is generated by the out-of-equilibrium decays of the heavyright-handed neutrinos into Higgs bosons and leptons N → Hl, N → ¯ lH † (138)provided that T re < M . The CP-violation asymmetry factor is defined by [153] ǫ = Γ − ¯ΓΓ + ¯Γ (139)45here Γ = Γ( N → lH ) and ¯Γ = Γ( N → ¯ lH † ), for any of the three right-handed neutrinos,and it arises from the interference of the one–loop diagrams with the tree level coupling[153]. In concrete SUSY GUT models based on the SO (10) group it typically takes values ǫ ∼ − [154, 155].Assuming the mass hierarchy M ≪ M , , the inflaton is not sufficiently heavy to decayinto N , N , and therefore the channels φ → N N and φ → N N are kinematically closed.Thus, we obtain for baryon asymmetry the final expression Y B = 3 C T re M I ǫ (140)It thus becomes clear that the three relevant mass scales, namely T re , M I , M , must satisfythe following hierarchy T re < M < M I (141)and therefore within non-thermal leptogenesis the inflaton mass must be always larger thanthe reheating temperature.In the models discussed here the inflaton mass is given in terms of the two mass scales, µ, Λ, as follows M I ∼ Λ µ (142)while for any given value of M the allowed range for µ, Λ , T re is known, according to theanalysis presented in the previous sections. Given the numerical results already presented,it is easy to verify that for a given M , the inflaton mass is always lower than T re . Hence,we conclude that in single-field inflationary models with a canonical scalar field in the RS-IIbrane model non-thermal leptogenesis cannot work, at least for the concrete inflationarypotentials considered here.There is another way to see that non-thermal leptogenesis cannot work here. Let us ignorefor a moment the fact that the mass scales violate the required hierarchy mentioned before,and let us show graphically how the CP-violation asymmetry factor depends on the reheatingtemperature after inflation. This is shown in the figures 15 and 16 for M = 10 TeV and M = 10 TeV, respectively. Clearly, it turns out that for the reheating temperatureobtained before, the CP-violation asymmetry factor is many orders of magnitude lower thanwhat typically concrete particle physics models predict, 10 − , as already mentioned before.46 × × × × × × - × - × - × - × - × - T re [ GeV ] ϵ FIG. 15: CP-violation asymmetry factor, ǫ , as a function of the reheating temperature after infla-tion, T re , for the Higgs-like inflationary potential and M = 10 TeV. The solid curve correspondsto the SM, while the dashed curve to the MSSM. × × × × × × × - × - × - × - × - T re [ GeV ] ϵ FIG. 16: Same as previous figure, but for M = 10 TeV.
Therefore, for those two independent reasons we conclude that non-thermal leptogenesiscannot work within the framework of RS-II brane cosmology, at least for the inflationarypotentials considered here. Consequently, one must rely on the mechanism of thermal lep-togenesis, which in the framework of RS brane cosmology has been analysed in [126, 156],and it requires a sufficiently high M . In particular, in the high-energy regime of branecosmology, it is found that M must take values in the range 10 GeV < M < GeV47156].As a final remark, we have not included the TCC in our analysis. We hope to be able toaddress this point in a future work.
Note added:
As our work was coming to its end, another work similar to ours appeared[98]. There, too, the authors have studied different types of inflationary potentials in theframework of five-dimensional RS brane model, and they could determine models that satisfyboth data and swampland criteria at the same time. We find the following differencescompared to our analysis: i) the allowed range for the free parameters of each model is notdetermined, ii) nothing is mentioned about baryon asymmetry, and iii) the tensor-to-scalarratio has been overlooked.
VIII. CONCLUSIONS
We have studied the dynamics of four concrete small-field inflationary models based ona single, canonical scalar field in the framework of the high-energy regime of the Randall-Sundrum II brane model. In particular, we have considered i) an axion-like potential for theinflaton (Natural Inflation), ii) Hilltop potential with a quadratic term (quadratic Hilltopinflation), iii) a potential arising in the context of dynamical symmetry breaking (Higgs-likeinflation), and iv) a SUGRA-motivated potential (Exponential SUSY inflation). Adoptingthe Randall-Sundrum fine-tuning, all the models are characterized by 3 free parameters intotal, namely the 5-dimensional Planck mass, M , and the two mass scales of the inflatonpotential. We have shown in the n s − r plane the theoretical predictions of the modelstogether with the allowed contour plots from the PLANCK Collaboration, and we havedetermined the allowed range of the parameters for which a viable inflationary Universeemerges. The mass scales of the inflaton potential have been expressed in terms of the five-dimensional Planck mass, which remains unconstrained using the PLANCK results only.However, on the one hand current tests for deviation from Newton’s gravitational law atmillimeter scales, and on the other hand the assumption that inflation takes place in thehigh-energy limit of the RS-II brane model force the five-dimensional Planck mass to liein the range 10 TeV . M . TeV, and therefore all parameters are finally known.After that, we have shown that for those types of potentials the inflation incursion is sub-Planckian, then the distance Swampland conjecture is satisfied. Nevertheless, the de Sitter48wampland Criteria and its refined version may be evaded for these potentials in the high-energy regime of the RS-II brane model instead. Finally, we have computed the reheatingtemperature T re as well as the duration of reheating, N re , versus the scalar spectral index n s assuming four different values of the EoS parameter w re = − / , , / , M . Then, by applying the constraint on M already found, an allowed rangefor the reheating temperature as well as for the tensor-to-scalar ratio could be obtained foreach model. Furthermore, we have shown that non-thermal leptogenesis cannot work withinthe framework of RS-II brane cosmology, at least for the inflationary potentials consideredhere. Consequently, one must rely on the mechanism of thermal leptogenesis, which in thehigh-energy regime of the RS brane cosmology requires a sufficiently high five-dimensionalPlanck mass, M > GeV.
Acknowlegements
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