# Emergence of Warm Inflation in Curved Space-Time between Accelerating Branes

Authors: Aroonkumar Beesham
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

It appears that having our own brane to somehow interact with other branes could give rise to quite an interesting system and that interaction could lead to some observable effects. We consider the question of whether or not these signatures of interaction between the branes can be observed. To answer this question, we investigate the effect induced by the inflaton in the WMAP7 data using the warm inflationary model. In this model, slow-roll and perturbation parameters are given in terms of the inflaton thermal distribution. We show that this distribution depends on the orbital radius of the brane motion under the interaction potential of other branes in extra dimensions. Thus, an enhancement in the brane inflation can be a signature of an orbital motion in extra dimensions, and consequently, some signals of other branes can be detected by observational data. According to experimental data, theN≃50 case leads to ns≃0.96, where N and ns are the number of e-folds and the spectral index, respectively. This standard case may be found in the range 0.01<Rtensor‐scalar<0.22, where Rtensor‐scalar is the tensor-scalar ratio. We find that at this point, the radial distance between our brane and another brane is R=1.5GeV−1 in intermediate and R=0.02225GeV−1 in logamediate inflation.


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## Body

## 1. Introduction

Recently, it was argued that the boundary conditions to be imposed on the quantum state of the whole multiverse could be such that brane universes could be created in entangled pairs [1]. Also, the consideration of entanglement between the quantum states of two or more brane universes in a multiverse scenario provides us with a completely new paradigm that opens the door to novel approaches for traditionally unsolved problems in cosmology, more precisely, the problems of the cosmological constant, the arrow of time, and the choice of boundary conditions, amongst others [2]. Some authors have tried to find direct evidence of the existence of other brane universes using a dark energy model [3, 4]. Also, some researchers show that other branes are made observable for us through interaction with our own brane [5]. In their paper, the orbital radius of our brane in extra dimensions can be described according to the interaction potential of other branes. In some scenarios, the properties of the interaction potential are calculated for a composite quantum state of two branes whose states are quantum mechanically correlated [1, 2]. It appears that having our own brane to somehow interact with other branes could give rise to quite an interesting system and that interaction could lead to an orbital motion in extra dimensions.This scenario is very close to the well-known DGP model. Dvali et al. [6] proposed a new braneworld model, named the DGP model, having two branches. One branch is known as the accelerating branch, i.e., the accelerating phase of the universe can be explained without adding a cosmological constant or dark energy, whereas the latter one represents the decelerating branch. In this paper, we generalize this model and show that by the acceleration of branes, some extra fields emerge. These fields dissolve into branes and lead to inflation. For this reason, these fields could be considered candidates for inflation.The main question is the possibility of considering the properties of other branes against observational data. The warm inflationary model helps us to perform precision tests of the universal extra dimensional models and explore the new physics against observational data. In this scenario, after the period of inflation, the radiation of the universe becomes dominant and the reheating epoch will not happen. The results of this model are compatible with the WMAP7 and Planck data [7]. In this theory, slow-roll and perturbation parameters are given in terms of the thermal distribution of the inflaton. On the other hand, this distribution is given in terms of the orbital radius of the brane motion [5] in extra dimensions. As the interaction potential increases, the effect of the inflaton radiation from the horizon that appears in the brane-antibrane system on the universe’s inflation becomes systematically more effective because at higher energies, there exist more channels for inflaton production and its decay into particles.The outline of the paper is as follows. In Section2, we consider the effect of the orbital radius of the brane motion under the interaction potential of the other branes on the thermal distribution of inflatons. In Section 3, using the warm inflationary model, we analyze the signature of other branes against observational data. The last section is devoted to a summary and conclusion.


## 2. The Thermal Distribution of Inflatons near the Appeared Horizon in the Brane-Antibrane System

Previously, the dynamical behavior of a pair of Dp and anti-Dp branes which move parallel to each other in the region that the brane and antibrane annihilation will not occur was considered [5]. Also, the orbital radius of the brane motion due to the interaction potential in extra dimensions was studied. Using these results, we calculate the thermal distribution of inflatons near the horizon that appears in the brane-antibrane system and show that the thermal distribution of inflatons can be given in terms of the orbital radius of the brane motion in extra dimensions.Thed-dimensional metric in the brane-antibrane system is expressed as(1)ds2=gµνdxµdxν+gρσdxρdxσ+gabdxadxb,where gµν and gρσ are the p-dimensional metrics along the Dp and anti-Dp branes, respectively, and gab is the d‐2p dimensional metric along the transverse coordinates.Now, let us consider the wave equation of the inflaton in extra dimensions between two branes:(2)−∂2c2∂χ2+∂2∂r2B=0,where χ and r are the transverse coordinates between the two branes. This equation corresponds to flat space-time. The interaction potential between the Dp brane and the anti-Dp brane in extra dimensions is of the type [5](3)VR∼64π2μ427,where μ4=27/32π2T3h4, hR=b4/R4, R is the orbital radius distance between the two branes, T3 the brane tension, and b the curvature radius of the AdS5 throat. This potential leads to curved space-time.There are more models for interbrane potential; however, we make use of a symmetric model to explain the interaction between two parallel and similar branes. This helps us to understand the model better, and we could generalize it by adding more corrections due to the nonsymmetric part of potential. However, this is not the main thrust of this work.Thus, to write the inflaton wave equation in curved space-time, we should use the following reparameterizations:(4)r⟶ρr,χ,χ⟶τr,χ,that lead to the following inflaton wave equation:(5)∂τ∂r2−∂τ∂χ2∂2c2∂τ2+∂ρ∂r2−∂ρ∂χ2∂2∂ρ2B=0.We can normalize the distance between the two branes to unity by making the following choices:(6)ρr,χ=rRχ,τ=βc2∫0χdt′Rχ′Ṙχ′−βr22.With the above considerations, the wave equation is written as(7)−g1/2∂∂xμgμν−g1/2∂∂xμB=0,where x5=τ and x4=ρ, and the metric elements are obtained as(8)gττ=−1β2c2RṘ1−Ṙ2/c2ρ21+Ṙ/c2ρ2,g44=R21+Ṙ2/c2ρ21−Ṙ/c2ρ2.The horizon of this system is located at(9)rhorizon=cRṘ,where c is velocity of light. In Kruskal coordinates, the metric of the system becomes [8, 9](10)ds2=gμνdxμdxν+gρσdxρdxσ−rhorizone−r/rhorizonrdu¯dv¯+r2dθ2,u¯=−2rhorizone−u/2rhorizon,v¯=−2rhorizone−v/2rhorizon,u=χ−r∗,v=χ+r∗,r∗=−r−rhorizonlnr−rhorizon.Since the Killing vector in Kruskal coordinates is given by∂/∂u¯ on the past horizon H−, the positive frequency normal mode solution in Kruskal coordinates is approximated by(11)B∝e−iωu¯,where ω is the inflaton energy in extra dimensions. Using this fact that v¯=0 on H− [9], we can estimate the original positive frequency normal mode on the past horizon as(12)B∝e−iωu¯=u¯2rhorizon−i2rhorizonω=−u¯/2rhorizon−i2rhorizonωregionI,−u¯/2rhorizon−i2rhorizonrhorizonregionII.In Equation (12), we can use the fact that −1−i2rhorizonω=e2rhorizonω. Using Equation (12), we observe that the inflaton states in the horizon satisfy the following condition [9, 10]:(13)Bout−tanhrωBinsystemin⊗out=0,tanhrω=e−2rhorizonω,which actually constitutes a boundary state. In fact, we can view Hawking radiation as the pair creation of a positive energy field that goes to infinity and a negative energy field that falls into the horizon of the brane-antibrane system. The pair is created in a particular entangled state. So the Unruh state can be viewed as an entangled thermal state. The above definition of the positive frequency solution in terms of Bout and Bin leads to the Bogoliubov transformation [8–10] for the particle creation and annihilation operators in the brane-antibrane system and Minkowski space-times in the exterior region of the system:(14)d=coshrωαout−sinhrωαin†,d†=coshrωαout†−sinhrωαin,tanhrω=e−2πrhorizonω,where d† and d are the creation and annihilation operators, respectively, acting on the Minkowski vacuum, αout† and αout are the respective operators acting on the brane-antibrane vacuum outside the event horizon, and αin† and αin are the respective operators acting on the brane-antibrane vacuum inside the event horizon.Thus, we can write the Bogoliubov transformation between the Minkowski and curved creation and annihilation operators as(15)dsystemout⊗in=αout−tanhrωαin†systemout⊗in=0,which actually constitutes a boundary state. Now, we assume that the system vacuum systemout⊗in is related to the flat vacuum 0flat by(16)systemout⊗in=F0flat,where F is a function to be determined later.Fromαout,αout†=1, we obtain αout,αout†m=∂/∂αout†αout†m and αout,F=∂/∂αout†F. Then, using Equations (15) and (16), we get the following differential equation for F:(17)∂F∂αout†−tanhrωαin†F=0,and the solution is given by(18)F=etanhrωαout†αin†.By substituting Equation (18) into Equation (16) and by properly normalizing the state vector, we get(19)systemout⊗in=Netanhrωαin†αout†0flat=1coshrω∑mtanhmrωmout⊗m¯in,where min and m¯out are the orthonormal bases (normal mode solutions) for a particle that acts on Hin and Hout, respectively, and N is the normalization constant.Equation (19) expresses that the states inside and outside the horizon are entangled. However, this entanglement depends on the event horizon and the horizon is given in terms of R, the orbital radius of the brane motion in the interaction potential of the other brane, rhorizon=cR/R, and consequently, the entanglement changes with the orbital radial distance between the two branes. We derive the thermal distribution for inflatons in extra dimensions as the following:(20)B=out⊗insystemαin†αinsystemout⊗in=e−2πrhorizonω1−e−2πrhorizonω.The above equation shows that different numbers of inflatons are produced with different probabilities inside and outside of the apparent horizon in the brane-antibrane system. These probabilities are related to the orbital radial distance of the two branes and the energy of the inflatons.


## 3. Considering the Effect of Other Branes on Cosmic Inflation by Using the Warm Inflationary Model

In this section, we enter the effects of the interaction potential between the branes on the results of the derivation of slow-roll and perturbation parameters and other important parameters in the inflationary model [7]. We show that these parameters are given in terms of the orbital radial distance between the two branes and describe the shape of the interaction potential between branes. Also, using the inflationary model, we discuss the signature of interaction between branes against observational data.Previously, it has been shown that in the FRW brane with the metric(21)ds2=gµνdxµdxν=−dt2−a2tdxidxi,the dynamics of warm inflation is presented by these equations [7]:(22)ρ̇+3HP+ρ=−ΓḂ2,ρ̇γ+4Hργ=−ΓḂ2,H2=12Ḃ2+VB+13ργ,VB=m2B2,where ρ is the energy density, p is the pressure, ργ is the energy density of the radiation, Γ is the dissipative coefficient, B is the thermal distribution of the inflaton, and the overdot (˙) is the derivative with respect to cosmic time. In the previous section, we discussed that the thermal distribution of the inflaton can be given as a function of the orbital radial distance between branes. Using this fact, we can rewrite the above equations as(23)ρ̇+3HP+ρ=−ΓR¨R−Ṙ2πωR22,ρ̇γ+4Hργ=−ΓR¨R−Ṙ2πωR22,H2=12R¨R−Ṙ2πωR22+VR,Ṙ+13ργ,VR,Ṙ=m21−Ṙ2πωR2.Using quantum field theory methods [11, 12], the dissipation coefficient (Γ) in the above equations could be calculated as(24)Γ=Γ0T3B2~Γ04π2ω2T3R2Ṙ2,where T is the temperature of the thermal bath. During the inflationary epoch, the energy density ρ is more than the radiation energy density ρ>ργ; however, it is comparable with the potential energy density VB2 (ρ~V) [7]. The slow-roll approximation (B¨≤3H+Γ/3Ḃ) [13, 14] with the condition that inflation radiation production be quasistable (ρ̇γ≤4Hργ,ρ̇γ≤ΓḂ) leads to the following dynamic equations [7]:(25)3H1+r3Ḃ=−12V′,ργ=34rḂ2=r1+r/32V′2V=CT4,H2=12V,where r=Γ/3H and C=π2g∗/30 (g∗ are the number of relativistic degrees of freedom). In the above equations, a prime (′) denotes a derivative with respect to the field B. Using this equation and the thermal distribution of the inflaton in Equation (20), we can obtain the dynamic equations with respect to R, the orbital radial distance between the two branes:(26)3H1+r3R¨R−Ṙ2πωR2=−12V′R,Ṙ,ργ=34rR¨R−Ṙ2πωR22=r1+r/32V′2R,ṘVR,Ṙ=CT4,H2=12VR,Ṙ,where the prime (′) denotes a derivative with respect to R. From the above equations, the temperature of the thermal bath is given by [7](27)T=−rḢ2C1+r/31/4=−rR¨R−Ṙ4CπωR21+r/31/4.This temperature depends on the orbital radial distance between the two branes. As the branes come close to each other, the temperature of the thermal bath increases. The reason for this is as follows: with decreasing distance between the two branes, the interaction potential increases and more inflatons radiate from the apparent horizon of the brane-antibrane system.At this stage, we tend to calculate the dependency of slow-roll parameters on the orbital radial distance between different branes. These parameters in warm inflation are [7](28)∈=−1HddtlnH,η=−H¨HḢ,where H=ȧ/a and a is the scale factor. To calculate these parameters, we should determine the explicit form of the scale factor.Until now, eight possible asymptotic solutions for cosmological dynamics have been proposed [15]. Three of these solutions have a noninflationary scale factor, and another three solutions give de Sitter, intermediate, and power law inflationary expansion. Finally, two cases of these solutions have asymptotic expansion with the scale factor (a=a0expAlntλ. This version of inflation is named logamediate inflation [16]. In this paper, we will study the warm tachyon inflationary model in the scenarios of intermediate and logamediate inflation.Firstly, let us consider intermediate inflationary expansion. In this model, the expansion of the universe is between standard de Sitter inflation with the scale factorat=a0expH0t and power law inflation with the scale factor at=tp,p>1 (slower than the first one) [17, 18]. The scale factor of this model has the form below [19–21]:(29)a=a0expAtf,0<f<1,where A is a positive constant. The number of e-folds in this case is [7](30)N=∫t1tHdt=Atf−t1f,where t1 is the begining time of inflation. From Equations (20), (24), (25), (26), (27), and (29), we obtain the Hubble parameter as(31)H=fAlnB−lnB0ω¯8f−1/5f+2=fAlne−2πrhorizonω/1−e−2πrhorizonω−lne−2πr0,horizonω/1−e−2πr0,horizonωω¯8f−1/5f+2∼fA−2πωrhorizon−r0,horizon+ln1−e−2πr0,horizonω/1−e−2πrhorizonωω¯8f−1/5f+2∼fA2πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω¯8f−1/5f+2,B=B0expω¯5f+2/8,where ω¯=6/Γ02C/33/41/28fA5/81−f1/8/5f+2 and Γ0 is constant. This equation insists that the evolution of our brane universe is affected by the number of inflatons that are radiated from the apparent horizon of the brane-antibrane system and it changes with an increase or decrease in the orbital radial distance between the two branes.The important slow-roll parametersϵ and η are given by(32)∈=1−ffAlnB−lnB0ω¯−8f/5f+2=1−ffAlne−2πrhorizonω/1−e−2πrhorizonω−lne−2πr0,horizonω/1−e−2πr0,horizonωω¯−8f/5f+2∼1−ffA−2πωrhorizon−r0,horizon+ln1−e−2πr0,horizonω/1−e−2πrhorizonωω¯−8f/5f+2∼1−ffA2πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω¯−8f/5f+2,η=2−ffAlnB−lnB0ω¯−8f/5f+2=2−ffAlne−2πrhorizonω/1−e−2πrhorizonω−lne−2πr0,horizonω/1−e−2πr0,horizonωω¯−8f/5f+2∼2−ffA−2πωrhorizon−r0,horizon+ln1−e−2πr0,horizonω/1−e−2πrhorizonωω¯∼2−ffA2πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω¯−8f/5f+2,respectively. These parameters depend on the orbital radial distance between the branes. With a decrease in this distance, more inflatons are radiated from the apparent horizon of the system, the slow-roll parameters increase, and, as a result, the universe inflates more.The energy density of radiation in this case has the following form:(33)ργ=31−ffAlnB−lnB0ω¯8f−2/5f+2=31−ffAlne−2πrhorizonω/1−e−2πrhorizonω−lne−2πr0,horizonω/1−e−2πr0,horizonωω¯8f−2/5f+2∼31−ffA−2πωrhorizon−r0,horizon+ln1−e−2πr0,horizonω/1−e−2πrhorizonωω¯8f−2/5f+2∼31−ffA2πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω¯8f−2/5f+2.According to this result, the radiation energy density is given in terms of the orbital radius of the brane motion in extra dimensions. As the interaction potential increases, the effect of the inflaton radiation from the apparent horizon in the brane-antibrane system on cosmic inflation becomes systematically more effective because at higher energies, there exist more channels for inflaton production.Using Equations (30) and (31), the number of e-folds between the two fields B1 and B is given by(34)N=AlnB−lnB0ω¯−8f/5f+2−lnB1−lnB0ω¯−8f/5f+2=Alne−2πrhorizonω/1−e−2πrhorizonω−lne−2πr0,horizonω/1−e−2πr0,horizonωω¯−8f/5f+2−lne−2πr1,horizonω/1−e−2πr1,horizonω−lne−2πr0,horizonω/1−e−2πr0,horizonωω¯−8f/5f+2∼A−2πωrhorizon−r0,horizon+ln1−e−2πr0,horizonω/1−e−2πrhorizonωω¯−8f/5f+2−−2πωr1,horizon−r0,horizon+ln1−e−2πr0,horizonω/1−e−2πr1,horizonωω¯−8f/5f+2∼A2πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω¯−8f/5f+2−2πωR0/Ṙ0−R1/Ṙ1+ln1−e−2πR0/Ṙ0ω/1−e−2πR1/Ṙ1ωω¯−8f/5f+2.This equation depends onB1 and B0. To obtain the explicit form of the number of e-folds in terms of the orbital radius distance between the branes, we should find the relation between B1 and B0. At the begining of the inflation period where ε=1, the inflaton in terms of constant parameters of the model is(35)B1=B0expω¯1−ffA5f+2/8f⟶e−2πr1,horizonω1−e−2πr1,horizonω=e−2πr0,horizonω1−e−2πr0,horizonωexpω¯1−ffA5f+2/8f⟶r1,horizon−1∼r0,horizon−1expω¯1−ffA5f+2/8f−1.From the above equations, we obtain the inflaton (Bt) and the distance between the two branes (Rt) in terms of the number of e-folds:(36)Bt=B0expω¯NA+1−ffA5f+2/8f⟶e−2πrhorizonω1−e−2πr1,horizonω=e−2πr0,horizonω1−e−2πr0,horizonωexpω¯NA+1−ffA5f+2/8f⟶rhorizon−1∼r0,horizon−1expω¯NA+1−ffA5f+2/8f−1⟶Rt=R0exp−∫dtrhorizonN,t.This equation shows that the orbital radial distance between the brane universes depends on the number ofe-folds. This means that as the distance between the branes decreases, more inflatons are created near the apparent horizon of the brane-antibrane system, and the number of e-folds increases.In Figure1, we present the number of e-folds N for the intermediate scenario as a function of R−1, where R is the orbital radial distance between branes. In this plot, we choose R0=0.45GeV−1, ω=4.6GeV, Ṙ0=0.01, Ṙ=0.1, A=1, and f=1/2. It is clear that the number of e-folds N is much larger for a smaller orbital radial distance between the branes. This is because as the distance between the branes becomes smaller, the temperature becomes larger and the thermal radiation of the inflatons enhances.Figure 1
The number ofe-folds N for the intermediate scenario as a function of the R−1 for R0=0.45GeV−1, ω=4.6GeV, Ṙ0=0.01, Ṙ=0.1, A=1, and f=1/2.Now, we will consider tensor and scalar perturbations that appear during the inflationary period for the warm inflation model. These perturbations may leave an imprint in the CMB anisotropy and on the LSS [22–26]. The power spectrum and spectral index are characteristic of each fluctuation: ΔR2k and ns for scalar perturbations and ΔT2k and nT for tensor perturbations. In warm and cool inflation models, the scalar power spectrum is given by [7](37)ΔR2=HḂδB2,where the thermal fluctuation in the warm inflation model yields [22–26](38)δB=ΓHT24π31/4.Using Equations (20), (36), (37), and (38), we calculate the scalar power spectrum as(39)ΔR2=−Γ03364π31/2H3/2Ḣ=Γ03364π31/2311fA151−f32C111/8B3lnB−lnB0ω¯−15f−18/5f+2=Γ03364π31/2311fA151−f32C111/8e−2πrhorizonω1−e−2πrhorizonω3×lne−2πrhorizonω/1−e−2πrhorizonω−lne−2πr0,horizonω/1−e−2πr0,horizonωω¯−15f−18/5f+2∼Γ03364π31/2311fA151−f32C111/8e−2πrhorizonω1−e−2πrhorizonω3×−2πωrhorizon−r0,horizon+ln1−e−2πr0,horizonω/1−e−2πrhorizonωω¯−15f−18/5f+2∼Γ03364π31/2311fA151−f32C111/8e−2πR/Ṙω1−e−2πR/Ṙω3×2πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω¯−15f−18/5f+2,where k is the comoving wavenumber. With the wavenumber k=k0=0.002Mpc−1, the combined measurement from WMAP+BAO+SN of ΔR2 is reported by the WMAP7 data [27–29] as(40)ΔR2=2.455±0.096×10−19.Using this equation and Equation (39), and choosing (A=1, f=1/2, Ṙ=0.1, ω=4.6GeV, and Γ0=1), we obtain the radial distance between our brane and another brane, R=1.5GeV−1.This result is consistent with previous calculations [30].Another important perturbation parameter is the spectral indexns which is given by(41)ns−1=−dlnΔR2dlnk=15f−188fAlnB−lnB0ω¯−8f/5f+2=15f−188fAlne−2πrhorizonω/1−e−2πrhorizonω−lne−2πr0,horizonω/1−e−2πr0,horizonωω¯−8f/5f+2∼15f−188fA−2πωrhorizon−r0,horizon+lne−2πr0,horizonω/1−e−2πrhorizonωω¯−8f/5f+2∼15f−188fA−2πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω¯−8f/5f+2,where we have used the thermal distribution in Equation (20). In Figure 2, we show the results for the spectral index in the intermediate scenario as a function of R−1, where R is the orbital radial distance between the branes. In this plot, we choose R0=0.45GeV−1, ω=4.6GeV, Ṙ0=0.01, Ṙ=0.1, A = 1, and f=1/2. As can be seen from Figure 2, the spectral index decreases rapidly when the distance between the branes increases. By comparing Figures 1 and 2, we find that the N≃50 case leads to ns≃0.96. This result is compatible with the observational data [7, 27–29, 31, 32]. At this point, the radial distance between our brane and another brane is R=1.5GeV−1.Figure 2
The spectral index in the intermediate scenario as a function ofR−1 for R0=0.45GeV−1, ω=4.6GeV, Ṙ0=0.01, Ṙ=0.1, A=1, and f=1/2.Using Equation (20), we can calculate the tensor power spectrum and its spectral index as(42)ΔT2=2H2π2=2fA2π2lnB−lnB0ω¯16f−1/5f+2=2fA2π2lne−2πrhorizonω/1−e−2πrhorizonω−lne−2πr0,horizonω/1−e−2πr0,horizonωω¯16f−1/5f+2∼2fA2π2−2πωrhorizon−r0,horizon+ln1−e−2πr0,horizonω/1−e−2πrhorizonωω¯16f−1/5f+2∼2fA2π22πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω¯16f−1/5f+2,(43)nT=−2ε=−2−2ffAlnB−lnB0ω¯−8f/5f+2=−2−2ffAlne−2πrhorizonω/1−e−2πrhorizonω−lne−2πr0,horizonω/1−e−2πr0,horizonωω¯−8f/5f+2∼−2−2ffA−2πωrhorizon−r0,horizon+ln1−e−2πr0,horizonω/1−e−2πrhorizonωω¯−8f/5f+2∼−2−2ffA2πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω¯−8f/5f+2.These perturbations depend on the orbital radial distance between the branes. As we discussed before, these perturbations have a direct effect on the cosmic microwave background (CMB). Thus, we can observe the signature of interaction between the branes by means of observational data.Another important parameter is the tensor-scalar ratio that has the following form:(44)Rtensor‐scalar=−1444π3fA4Γ03π4T21/2ḢH1/2=1444π3fA4Γ03π41/2311fA151−f32C111/8×B3lnB−lnB0ω¯f+2/5f+2=1444π3fA4Γ03π41/2311fA151−f32C111/8e−2πrhorizonω1−e−2πrhorizonω3×lne−2πrhorizonω/1−e−2πrhorizonω−lne−2πr0,horizonω/1−e−2πr0,horizonωω¯f+2/5f+2∼1444π3fA4Γ03π41/2311fA151−f32C111/8e−2πrhorizonω1−e−2πrhorizonω3×−2πωrhorizon−r0,horizon+ln1−e2πr0,horizonω/1−e2πrhorizonωω¯f+2/5f+2∼1444π3fA4Γ03π41/2311fA151−f32C111/8e−2πR/Ṙω1−e−2πR/Ṙω3×2πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω¯f+2/5f+2.In Figure3, we present the tensor-scalar ratio in the intermediate scenario as a function of R−1, where R is the orbital radial distance between the branes. In this plot, we choose R0=0.45GeV−1, ω=4.6GeV, Ṙ0=0.01, Ṙ=0.1, C=70, Γ0=1, A=1, and f=1/2. We observe that as the orbital radius distance between branes increases, the tensor-scalar ratio increases. By comparing Figures 2 and 3, we notice that the standard case ns≃0.96 may be found in 0.01<Rtensor‐scalar<0.22, which agrees with the observational data [7, 27–29, 31, 32]. At this stage, the radial distance between our brane and another brane is R=1.5GeV−1.Figure 3
The tensor-scalar ratio in the intermediate scenario as a function ofR−1 for R0=0.45GeV−1, ω=4.6GeV, Ṙ0=0.01, Ṙ=0.1, A=1, and f=1/2.Now, we would like to consider the signature of interaction between branes in the context of logamediate inflation with the scale factor(45)at=a0expAlntλ,where A is a constant parameter. This model is converted to power law inflation for the λ=1 case. This scenario is applied in a number of scalar-tensor theories [16]. The effective potential of this solution is used in dark energy models [33], supergravity, Kaluza-Klein theories, and superstring models [16, 34]. The number of e-folds in this case is given by [7](46)N=∫t1tHdt=Alntλ−lnt1λ,where t1 is the beginning time of inflation. From Equations (20), (24), (25), (26), (27), and (45), we may find the inflaton B and also the orbital radial distance between the two branes:(47)lnB−lnB0=ω~Ξt⟶lne−2πrhorizonω1−e−2πrhorizonω−lne−2πr0,horizonω1−e−2πr0,horizonω=ω~Ξt⟶−2πωrhorizon−r0,horizon+ln1−e−2πr0,horizonω1−e−2πrhorizonω∼ω~Ξt⟶2πωR0Ṙ0−RṘ+ln1−e−2πR0/Ṙ0ω1−e−2πR/Ṙω∼ω~Ξt,where ω~=6/Γ02C/33/41/2−45λ+3λA51/8 and Ξt=γ5λ+3/8,lnt/4 is the incomplete gamma function [35, 36]). The potential in terms of the orbital radial distance between the two branes is presented as(48)V=2λ2A2lnΞ−1lnB−lnB0/ω~2λ−2Ξ−1lnB−lnB0/ω~2=2λ2A2lnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙω/ω~2λ−2Ξ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙω/ω~2.This equation shows that the inflatonic potential on our brane depends on the orbital radial distance and the interaction potential between the two branes. In fact, the interaction between branes causes inflation of our universe.Now, we obtain the slow-roll parameters of the model in this case:(49)∈=lnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙω/ω~λA1−λ,η=2lnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙω/ω~1−λλA.In this case, like the intermediate case, as the distance between the branes decreases, more inflatons are created in the brane-antibrane system, the slow-roll parameters increase, and the universe inflates.Using Equations (20), (46), and (47), the number of e-folds between the two fields B1 and Bt can be obtained as(50)N=AlnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω~λ−lnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR1/Ṙ1ωω~λ=AlnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω~λ−λAλ/1−λ,where R1 is the the orbital radial distance between the two branes at the begining of the inflationary epoch when ϵ=1. Using the above equation, the orbital radial distance between the two branes in the inflationary period could be obtained in terms of the number of e-folds as(51)R−R0=ω~Ṙ2Ṙ022πωΞexpNA+λAλ/1−λ1/λ.This equation shows that, in this case, like the intermediate case, the number ofe-fields depends on the orbital radial distance between the branes. This is because as the distance between the branes decreases, the number of inflatons, which has direct effects on the number of e-folds, increases.Also, the scalar and tensor power spectra in this case are given by(52)ΔR2=Γ03364π31/2311λA152C111/8e−2πrhorizonω1−e−2πrhorizonω−3×exp−158lnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω~×lnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω~λ15λ−1/8λ×exp−3ω~ΞexplnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω~,ΔT2=2λ2AlnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙω/ω~2−2λπ2Ξ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙω/ω~2.These spectra in the context of logamediate inflation, like intermediate inflation, change with an increase or decrease in the orbital radial distance between the branes. The spectral index in this case has the following forms:(53)ns−1=−15λ−18λAlnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω~λ−1,nT=−2lnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙω/ω~1−λλA.In Figures4 and 5, we present the number of e-folds N and the spectral index for the logamediate inflation scenario as a function of R−1, where R is the orbital radial distance between the branes. In these plots, we choose R0=0.45GeV−1, ω=4.6GeV, Ṙ0=0.01, Ṙ=0.1, λ=10, A=1, and f=1/2. In this case, like the intermediate case, we find that the number of e-folds N and the spectral index are much larger for smaller orbital radial distance between branes. This is because as the distance between the branes becomes smaller, the temperature becomes larger and the thermal radiation of the inflatons enhances.Figure 4
The number ofe-folds N in the logamediate inflation scenario as a function of R−1 for R0=0.45GeV−1, ω=4.6GeV, Ṙ0=0.01, Ṙ=0.1, λ=10, A=1, and f=1/2.Figure 5
The spectral index in the logamediate inflation scenario as a function ofR−1 for R0=0.45GeV−1, ω=4.6GeV, Ṙ0=0.01, Ṙ=0.1, λ=10, A=1, and f=1/2.Finally, we could find the tensor-scalar ratio in terms of the orbital radial distance between two branes:(54)Rtensor‐scalar=1444π3Γ03π41/22C11λA3111/8e−2πr0,horizonω1−e−2πr0,horizonω3×exp−18lnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω~×lnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω~λ−31λ−1/8λ×exp3ω~ΞexplnΞ−12πωR0/Ṙ0−R/Ṙ+ln1−e−2πR0/Ṙ0ω/1−e−2πR/Ṙωω~.In Figure6, we present the tensor-scalar ratio in the logamediate scenario as a function of R−1, where R is the orbital radial distance between the branes. In this plot, we choose R0=0.45GeV−1, ω=4.6GeV, Ṙ0=0.01, Ṙ=0.1, C=70, Γ0=1, λ=10, A=1, and f=1/2. In this case, like the intermediate case, with an increase in the orbital radial distance between branes, the tensor-scalar ratio increases. By comparing Figures 5 and 6, we notice that the standard case ns≃0.96 may be found in 0.01<Rtensor‐scalar<0.22, which agrees with the observational data [7, 27–29, 31, 32]. At this stage, the radial distance between our brane and another brane is R=0.02225GeV−1.Figure 6
The tensor-scalar ratio in the logamediate inflation scenario as a function ofR−1 for R0=0.45GeV−1, ω=4.6GeV, Ṙ0=0.01, Ṙ=0.1, λ=10, A=1, and f=1/2.


## 4. Summary and Discussion

In this research, we calculate the thermal distribution of inflatons near the apparent horizon in a brane-antibrane system and show that the energy density, slow-roll parameters, number ofe-folds, and perturbation parameters can be given in terms of the orbital radius of the brane motion in extra dimensions. According to our results, when the distance between branes increases, the number of e-folds and the spectral index for both the intermediate and logamediate models decrease rapidly; however, the tensor-scalar ratio increases. This is because as the separate distance between branes decreases, the interaction potential increases, and at higher energies, there exist more channels for inflaton production near the apparent horizon in the brane-antibrane system; consequently, the effect of inflaton radiation from this horizon on cosmic inflation becomes systematically more effective. We find that the N≃50 case leads to ns ≃ 0.96. This standard case may be found in 0.01<Rtensor‐scalar<0.22, which agrees with the observational data [7, 27–29, 31, 32] (we note that some new observational data has been obtained, but we believe that our models will fit this as well. This work in under progress). At this point, the radial distance between our brane and another brane is R=1.5GeV−1 in the intermediate model and R=0.02225GeV−1 in the logamediate model.



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