Brane inflation and Trans-Planckian censorship conjecture
aa r X i v : . [ h e p - t h ] F e b Brane inflation and Trans-Planckian censorship conjecture
Abolhassan Mohammadi, ∗ Tayeb Golanbari, † and Jamil Enayati ‡ Department of Physics, Faculty of Science,University of Kurdistan, Sanandaj, Iran.Physics Department, College of Education,University of Garmian, Iraq. (Dated: February 16, 2021)The constraint of trans-Planckian censorship conjecture on the brane inflation model is considered.The conjectures put an upper bound on the main parameter including temperature, inflation time,potential, and the tensor-to-scalar ratio parameter r . It is determined that the resulting constraintcould be stronger than what we have for the standard inflationary models. The constraint, in general,depends on the brane tension and it is concluded that the conjecture also confined the value of branetension to have consistency for the model. Confining the brane tension turns into a determiningvalue for the five-dimensional Planck mass. The case of slowly varying Hubble parameter gives moreinteresting results for the ǫ and r , which indicates that the value of the tensor-to-scalar perturbationis not required to be extremely small. PACS numbers:
I. INTRODUCTION
Understanding the origin of the initial condition of theuniverse is one of the main purpose of the cosmology.Inflationary scenario not only solves the problems of thestandard big-bang theory, it also predicts the quantumperturbations which are the main seeds for large scalestructure of the universe [1–4]. The scenario has receivedhuge interest since its introduction [5–9] and it has beengeneralized in many differential ways [10–19].The scenario of inflation has been extremely supportedby the observational data [20–22], and many of differ-ent inflationary models could successfully pass the ob-servational test [23–38, 38, 39]. Besides, there are someother conjectures that it is expected that they shouldbe satisfied by an inflationary model. The story is thatthe general theory of relativity is a low-energy effectivefield theory where the scale of energy is the Planck mass.Based on string theory, which is known as the best can-didate for quantum gravity, every effective field theoryshould possess some features and meets some conjecture.One series of these conjectures for dividing the consis-tent effective field theory from the inconsistent ones isthe swampland conjectures proposed by [40–42] whichhas been the topic of many research works [43–57]. Theswampland conjectures concern field distance and de Sit-ter vacuum. An effective field theory that has a de Sit-ter vacuum is not consistent with quantum gravity andstands in the swampland zone. The swampland criteriahave been considered for different inflationary models,and it seems that the standard inflationary model witha canonical scalar field is in direct tension with these ∗ Electronic address: [email protected]; [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] conjectures. However, modified models of inflation, e.g.k-essence model [58], and warm inflation [59] have thechance to successfully pass the test.At the time of inflation, the universe is dominated by ascalar field, named inflaton, which produces a quasi-deSitter expansion. Then, the universe undergoes an ex-treme expansion in a short time. As inflation expands thespacetime, the quantum fluctuations are also stretchedout and their wavelengths grow and cross the Hubblehorizon, while the Hubble horizon remains almost un-changed. At the time that the quantum fluctuationscross the horizon, they freeze and lose their quantumnature. Standing on the same logic as the last para-graph, the Trans-Planckian censorship conjecture (TCC)has been proposed by Bedroya and Vafa [60] stating thatno quantum fluctuation with a wavelength shorter thanthe Planck length is allowed to cross the Hubble horizon,freezes and become classical. The conjecture could beformulated as e N l p < H − e (1)where l p is the Planck length and H − e is the Hubblehorizon at the end of inflation.The TCC imposes a constraint on the model which in-cludes spacetime expansion, like inflation, and it im-plies no limit on the cosmological evolution of standardbig-bang cosmology where the fluctuations never crossthe horizon. For the inflationary model, the TCC leadsto some severe constraints [61, 62] which will challengemany of the current models of inflation. In [61], it isshown that the consequence of TCC on the energy scaleof inflation is V e < GeV , which results to an upper bound on the tensor-to-scalarparameter r [61], r < − , which anticipates an extremely small amplitude for theprimordial gravitational waves. Note that the resulthas been obtained by taking an almost constant Hubbleparameter during inflation and it is assumed that rightafter inflation we have a radiation dominant phase andreheating occurs very fast. For the k-essence model, ageneralized version of TCC is proposed which involvesthe sound speed of the model [58] in which for c s < r depends on the Friedmann equation of the model.The equation is modified in higher dimension modelsof cosmology, in which in RS brane-world and extraquadratic terms of energy density also appears in theFriedmann equation which dominates the linear term inhigh energy regime. Due to this, the TCC imposes astronger constraint on the parameter of brane inflationthan the standard inflation. II. TCC IN BRANE INFLATION
This conjecture could be stated in the following form a e a i l p ≤ H − e (2)where the subscribes e and i stand for end and beginningof inflation and l p indicates the Planck length; l − p = m p .Following [61], this condition could take a stronger form.Suppose that there is an expanding phase for the universein the pre-inflationary phase. Then, it is reasonable toassume that there might be some modes with a physicalwavelength equal to or shorter than the Planck length, l p , between the time t p (Planck time)and t i . This leadone to an stronger version of the above conjecture as a e a p l p ≤ H − e (3)Because of having an inverse relationship between thescale factor and the temperature in the radiation domi-nant phase, the above condition is expressed in terms ofthe temperature as well T p T i e N l p ≤ H − e (4)in which T p is the temperature at the Planck time and T i is the temperature at the beginning of inflation.Same as [61], it is assumed that the pre-inflationary phaseis a radiation dominant era where the scale factor be-haves as a ( t ) ∝ t / . On the other hand, there is another condition related to the possibility of producing a causalmechanism for the observed structure of the universe. Itstates that the current comoving Hubble radius must beoriginated inside the comoving Hubble radius at the on-set of inflation. In a mathematical language, it means( a H ) − ≤ ( a i H i ) − which after some manipulation could be rewritten as H − e − N T T e ≤ H − i = H − e (5)the last equality on the right hand side of the equation isbased on our assumption for this section that the Hubbleparameter remains constant during inflation. T and T e are the temperature at the present time and at the endof inflation. And H is the present Hubble parameter.Combining the two conditions, Eq.(4) and (5), results in H e H T T e ≤ H i l p T i T p . (6)Just before and right after inflation, there is an inflation-ary phase. Then, the energy density in the Friedmannequation is a thermal bath of radiation, so H = 8 π m p ρ r (cid:16) ρ r λ (cid:17) (7)where ρ r is the energy density of the thermal bath givenby ρ r = π g ⋆ ( T ) T . (8)We restrict the situation to the high energy regime where ρ r ≫ λ . In this case, the Friedmann equation is simpli-fied to H = s π m p λ π g ⋆ ( T ) T (9)Evaluation H i and H e from the above equation and sub-stituting the result in Eq.(6), there is T i T e ≤ . × π m p λg ⋆ ( T i ) g ⋆ ( T e ) H T . (10)Taking a matter dominant phase for the present time,the current Hubble parameter is read as H = T eq m p T (11)where T eq is the temperature at the time when energydensity of matter and radiation are equal. Also, in thelate time, the universe is in low energy regime, and themodified Friedmann equation (7) comes back to the stan-dard form. Applying this equation on Eq.(10) leads to T e ≤ . × π √ m p λg ⋆ ( T i ) g ⋆ ( T e ) p T eq T (12)note that here it is assumed that T i = T e . Taking g ⋆ ( T i ) = g ⋆ ( T e ) ≃ , it turns to T e ≤ . × λ GeV (13)which depends on the values of the brane tension; notethat the brane tension λ has the dimension M , so thedimension of temperature is right. The brane tensionhas not been determined accurately but there are someestimations about it. To reproduce the nucleosynthesisas in standard cosmology λ ≥ and also variousastrophysical applications implies that λ ≥ × GeV .There is some understanding about the reheating tem-perature, which here is shown by T e . After inflation, theuniverse is very cold and almost empty of any particle.The reheating phase is an explanation (and a possiblescenario) for creating particles warming up the universeafter inflation. However, there are some restrictionsabout the final temperature of the universe at the endof reheating. The reheating temperature should be highenough to recover the hot big bang nucleosynthesis, andon the other hand, it should be small enough to preventthe creation of unwanted particles. These conditions arecombined as 10 − GeV < T r = T e < GeV . Tosatisfy the condition, the brane tension should stand inthe range 10 − GeV < λ < GeV , which is verywide range.The upper bound on temperature T e also implies anupper bound for the potential of the inflation as V e < (cid:16) . × GeV / (cid:17) λ / (14)in which for λ = 5 × GeV the potential shouldsatisfy the upper bound V e < . × GeV , meaningthat the potential could have the same order as thebrane tension or it should be lower. The important pointis that it was assumed that the whole process of inflationoccurs at high energy limit where λ ≪ V . From Eq.(26)it is realized that as the brane tension gets bigger theHigh energy regime assumption are more likely to beviolated, for example for λ = 10 GeV the potentialshould satisfy the condition V e < . × GeV which clearly violated the high energy assumption.However, there is chance for preserving the high energyregime assumption for lower magnitude of the branetension. For instance, by taking λ = 10 GeV thereis V e < . × GeV , which is consistent with theassumption. Therefore, it seems that the Eq.(26) appliesa condition for the magnitude of brane tension as well.How much expansion do we have in the inflationaryphase? Substituting Eq.(26) in the TCC condition (2),one finds a bound for the number of e-fold as e N < . × GeV / λ / . (15)Based on the wide studies of inflation, it is expectedto have about 55 −
65 number of e-fold expansion. Also Eq.(21) implies that lower brane tension leads toa bigger upper bound, in which for λ = 10 GeV ,we have e N < . × meaning that N <
74 thatclearly satisfies the aforementioned e-folding assumption.The result for the temperature determines the values ofthe Hubble parameter H i which appears in the amplitudeof the scalar perturbations as [18, 71] P s = 325 π s π m p λ H i ǫ , (16)where ǫ is the first slow-roll parameter. According to thePlanck data, the amplitude of the scalar perturbation isof the order of P s ∝ − . To satisfy this observationalconstraint and at the same time hold the condition (12),the slow-roll parameter should be about ǫ ≤ . × − . (17)The above constraint on the slow-roll parameter ǫ , has adirect impact on the tensor-to-scalar parameter r whichis related to ǫ as [72–74] r ≤ . × − . (18)which states that r is extremely small.The obtained result could be utilized to have some un-derstanding about the start time of inflation. Supposethat after Planck time and before inflation the universestands in radiation dominant phase. In this phase, thescale factor depends on time as a ( t ) ∝ t / , and since thescale factor is written as the inverse of the temperature,the beginning time of inflation is achieved as t i = T p T i t p ≥ . × GeV / λ / t p (19)It means that if one wants to have inflation started atthe time about 10 − s, then the brane tension shouldtake a huge value as λ ∼ GeV . This value ofthe brane tension clearly violate the discussed highenergy regime assumption. Preserving the assumptionrestricts the brane tension to almost be of the orderof λ ≤ GeV . By inserting this value in Eq.(19),the start point of inflation is obtained to be about t i > s, which is absolutely unacceptable. III. TCC FOR VARYING HUBBLEPARAMETER
In this section, we relax the assumption of having aconstant Hubble parameter during inflation. The Hubbleparameter is assumed to vary slowly as Eq.(20) whichappears in the power-law models [58] H ( N ) = H i e αN (20)where N = 0 is associated to the start point of infla-tion and α is taken as constant. The TCC condition (3)remains unchange and it is extracted that e (1+ α ) N ≤ T i T p m p H i (21)The condition regarding the causal mechanism of the uni-verse structure is read as H e H T T e ≤ e (1+ α ) N (22)Utilizing Eq.(21), one arrives at H e H T T e ≤ H i l p T i T p . (23)Same condition as we had in the previous case for con-stant H . Using the Friedmann equation (9), the abovecondition is given by Eq.(10). The temperature T i is theradiation temperature at the start time of inflation. Af-ter inflation, we assumed that the reheating occurs fastand the universe is warm up to the temperature T reh andwe take this temperature as the temperature of the uni-verse at the end of inflation, T e = T reh . In the previoussection, it was assumed that the temperatures at the endand beginning of inflation are the same. Here we supposethe relation T i = βT e for these two temperature where β is a constant. For the current Hubble parameter, Eq.(11)is applied. Therefore, one arrives at T e ≤ . × π √ m p λg ⋆ ( T i ) g ⋆ ( T e ) β p T eq T . (24)or T e ≤ (cid:16) . × GeV (cid:17) λβ . (25)which depends on the brane tension and the constant β .In comparison to Eq.(13), one could get a bigger valuesof temperature when β < T i < T e whichmeans that the temperature after inflation (provided bythe reheating phase) is bigger that the temperature justbefore inflation. If β > T i > T e ), the temperaturereceives a smaller values.The upper bound (25) leads to the following conditionfor the brane inflation V e < (cid:16) . × GeV / (cid:17) λ / β (26)which leads to the initial potential V i < (cid:16) . × GeV / (cid:17) β λ / (27)From the Friedmann equation (7) and Eq.(20), one findsthat V ( N ) = V i e αN . Comparing it with above relation,it is realized that the parameter α and β are related via β = e − αN .Although higher values of brane tension λ leads to thebigger inflation potential, it could weaken the high en-ergy assumption. The difference with the previous caseis the constant β which could play an essential role forpreserving the condition V ≪ λ . Eq.(27) is reflected inthe Hubble parameter H i , which appears in the ampli-tude of the scalar perturbations. To come to an agree-ment with observational data there should be P s ∝ − ,the first slow-roll parameter ǫ should satisfy the followingcondition ǫ ≤ . × − β (28)and this condition is reflected in the parameter r as r ≤ . × − β (29)Again, the large value of the constant β could assuagethe strong condition that we have for the previous case.The constant β could lead to some interesting results.For instance, taking λ = 2 × GeV and β = 5 × indicates that the reheating temperature and the infla-tion energy scale are respectively about T e = 26 .
26 GeVand V i = 9 . × GeV . The reheating temperaturestands in the acceptable range and the magnitude of thepotential perfectly satisfies the high energy assumption.On the other hand, for these values of β , the slow-rollparameter ǫ and tensor-to-scalar ratio are estimated as ǫ = 0 . r = 0 . r is not required to be extremelysmall. IV. CONCLUSION
The recently proposed Trans-Planckian censorshipconjecture seems to impose a strong constraint on stan-dard inflation. The conjecture has been considered inbrane inflation where there is an extra infinite spatialdimensional leading to a modified Friedmann equation.The Friedmann evolution equation of the model showsthat there is an extra quadratic term of the energy den-sity which dominates the linear term in the high energyregime.It was assumed that there is a radiation-dominated epochin the pre-inflationary phase and after inflation, we haveagain another reheating epoch. The reheating phase wasassumed to occur very fast. The conjecture was consid-ered for two cases first by taking the Hubble parameteras a constant during inflation and in the second case,it was taken as a slowly varying function. The TCCforbids any mode with an initial wavelength smaller orequal to the Planck length to cross the Hubble horizonand be classical. Applying the condition led to an upperbound for the temperature T e , which in general dependson the brane tension. The temperature T e is also recog-nized as the reheating temperature, and due to our un-derstanding of the reheating temperature, we found anacceptable range for the brane tension λ . The parameteraffects the value of the potential as well, and preservingthe high energy condition restricts the obtained range ofthe brane tension. Comparing the results with the caseof the standard inflation, there are stronger constraintson ǫ and r as r < − . However, a problem was en-countered regarding the beginning time of inflation. Se-lecting λ = 10 GeV implies a beginning time about t i = 10 s, which is unacceptable. Note that higher val-ues of λ indicate smaller t i , but it violates the high energycondition assumption, and smaller λ predicts bigger t i .Next, we consider the case of slowly varying Hubble pa-rameter. The results for the case were more interestingin which for the obtained range of the brane tension, thehigh energy condition is preserved and at the same timehaving a small initial time, about 10 − s. The interest-ing point is that the tensor-to-scalar ratio parameter is not required to be extremely high, and it could be of theorder of 10 − . acknowledgement The authors thank
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