Research Article | Open Access
V. Kamali, M. R. Setare, "Tachyon Warm Intermediate and Logamediate Inflation in the Brane World Model in the Light of Planck Data", Advances in High Energy Physics, vol. 2016, Article ID 9682398, 18 pages, 2016. https://doi.org/10.1155/2016/9682398
Tachyon Warm Intermediate and Logamediate Inflation in the Brane World Model in the Light of Planck Data
Tachyon inflationary universe model on the brane in the context of warm inflation is studied. In slow-roll approximation and in longitudinal gauge, we find the primordial perturbation spectrums for this scenario. We also present the general expressions of the tensor-scalar ratio, scalar spectral index, and its running. We develop our model by using exponential potential; the characteristics of this model are calculated in great detail. We also study our model in the context of intermediate (where scale factor expands as ) and logamediate (where the scale factor expands as ) models of inflation. In these two sectors, dissipative parameter is considered as a constant parameter and a function of tachyon field. Our model is compatible with observational data. The parameters of the model are restricted by Planck data.
Inflation as a theoretical framework presents the better description of the early phase of our universe. Main problems of Big Bang model (horizon, flatness, etc.) could be solved in the context of inflation scenario [1, 2]. Lagrangian formalism in terms of scalar fields can explain this scenario. Quantum fluctuations of the scalar field provide a description of anisotropy of cosmic microwave background (CMB) and origin of the distribution of large scale structure (LSS) [3–6]. Standard model of inflation, “cold inflation,” has two regimes: slow-roll and (p)reheating. In the slow-roll limit kinematic energy is small compared to the potential energy term and the universe expands. Interaction between scalar field (inflation) and other fields (massive and radiation fields) is neglected. After this period, kinetic energy is comparable to the potential energy in (p)reheating epoch. In this era inflation oscillates around the minimum of the potential while losing their energy to other fields (radiation, massless fields) which are presented in the theory. After reheating, the universe is filled by radiation. In (p)reheating epoch, observed universe attaches to the end of inflationary period. Another view of reheating is based on quantum mechanical production of massive particles in classical background inflation [7, 8]. Preheating is probably the most efficient and plausible bridge that could connect inflation to a hot radiation dominated universe [9, 10].
In warm inflationary scenario radiation production occurs during the slow-roll inflation epoch and (p)reheating is avoided [11, 12]. In this scenario thermal fluctuations could play a dominant role to produce initial fluctuations which are necessary for LSS formation [13, 14]. Warm inflationary period ends when the universe stops inflating. After this period the universe enters in the radiation phase [11, 12]. Some extensions of this model are found in [15–19].
In warm inflation there has to be continuously particle production. For this to be possible, then the microscopic processes that produce these particles must occur at a timescale much faster than Hubble expansion. Thus the decay rates (not to be confused with the dissipative coefficient) must be bigger than . Also these produced particles must thermalize. Thus the scattering processes amongst these produced particles must occur at a rate bigger than . These adiabatic conditions were outlined since the early warm inflation papers, such as [20, 21]. More recently there has been considerable explicit calculation from Quantum Field Theory (QFT) that explicitly computes all these relevant decay and scattering rates in warm inflation models [22, 23].
The inflation era in the early evolution of the universe could be described by tachyonic field, associated with unstable D-brane, because of the tachyon condensation near the maximum of the effective potential [24–26]. At the late times, tachyonic fields may add a nonrelativistic fluid or a new form of cosmological dark matter to the universe . The tachyon inflation is a k-inflation model , for scalar field with a positive potential . Tachyon potentials have two special properties; firstly a maximum of this potential is obtained where and second property is the minimum of these potentials which is obtained where . If the tachyon field starts to roll down the potential, then universe dominated by a new form of matter will smoothly evolve from inflationary universe to an era which is dominated by a nonrelativistic fluid . So, we can explain the phase of acceleration expansion (inflation) in terms of tachyon field. Tachyon fields in the ordinary (cold) tachyon inflation framework, after slow-roll epoch, evolve towards minimum of the potential without oscillating about it , so, the (p)reheating mechanism in cold tachyon inflation does not work. Warm tachyon inflation is a picture, where there are dissipative effects playing during inflation. As a result of this the inflation evolves in a thermal radiation bath; therefore the reheating problem of cold tachyon inflation  can be solved in the framework of warm tachyon inflation. We note that the cold tachyonic inflation era can naturally end with the collision of the two branes. In this situation we do not need warm inflation. If the collision of two branes does not arise naturally, warm inflation is perfectly good scenario that can solve the problem of end of thachyon inflation.
We may live on a brane which is embedded in a higher dimensional universe. This realization has significant implications to cosmology [29–34]. In this scenario, which is motivated by string theory, gravity (closed string modes) can propagate in the bulk, while the standard model of particles (matter fields which are related to open string modes) is confined to the lower-dimensional brane [35–37]. In terms of Randall-Sundrum suggestion, there are two similar but phenomenologically different brane world scenarios [33, 34]. In this paper we will consider that the brane world model corresponds to the Randall-Sundrum II brane world .
The brane world picture is described by the following action [29–34]:In this scenario we have a 3-brane universe which is located in the 5D Anti-de Sitter (AdS) space-time, where this space-time is effectively compactified with curvature scale of AdS space-time. is the Ricci scalar in five dimensions and , where is the 5D Newton’s constant and is Planck scale in five dimensions. is the tension of the brane and if we have no matter on the brane , where the brane becomes Minkowski space-time. In the brane world model the gravity could propagate in the 5D space-time and the Newtonian gravity in four dimensions is reproduced at the scales larger than on the brane. 4D Einstein’s equation projected onto the brane has been found in . Friedmann equation and the equations of linear perturbation theory [39, 40] may be modified by these projections. Einstein’s equations which are projected onto the brane with cosmological constant and matter fields which are confined to 3-brane tension have the following form :where is a projection of 5D weyl tensor, is energy density tensor on the brane, is the Planck scales in 4D, and is a tensor quadratic in
Cosmological constant on the brane in terms of 3-brane tension and 5D cosmological constant is given by 4D Planck scale is determined by 5D Planck scale asThe natural boundary conditions to specify the perturbations of this model are imposed, where the perturbations do not diverge at the horizon of the AdS space-time and we assume that the Weyl curvature may be neglected. On the large scale, the behavior of cosmological perturbations on the brane world models is the same as a closed system on the brane without the effects of the perturbations along the extradimensions in the bulk [41–43]. On the large scale limit, the perturbation parameters of inflation models have a complete set of perturbed equations on the brane which may be solved in quasi-stable and slow-roll limit [41–44]. The study of the perturbation evolution of warm inflation in the brane world model on the large scale, by using equations solely on the brane and without solving the bulk perturbations, is found in . This model has a complete set of perturbed equations on the brane. We would like to study the warm tachyon inflation model on the brane using this approach. Therefore we will consider the linear cosmological perturbations theory for warm tachyon inflation model on the brane. In spatially flat FRW model the Friedmann equation, by using Einstein’s equation (3), has following form :where is scale factor of the model and is Hubble parameter and is the total energy density on the brane. The last term in the above equation denotes the influence of the bulk gravitons on the brane, where is an integration constant which arising from Weyl tensor . This term may be rapidly diluted once inflation begins and we will neglect it. Therefore the projected Weyl tensor term in the effective Einstein equation may be neglected and this term does not give the significant contributions to the observable perturbation parameters. We will also take to be vanished at least in the early universe. So, the Friedmann equation reduces toThe brane tension has been constrained from nucleosynthesis  and a stronger limit of it results from current tests for deviation from Newton’s law, [46, 47].
In the warm inflationary models the total energy density is presented on the brane , where is the energy density of the radiation. The Friedmann equation has this formCosmological perturbations of warm inflation model have been studied in . Warm tachyon inflationary universe model has been studied in [50–53]; also warm inflation on the brane has been studied in . Inflation era is located in a period of dynamical evolution of the universe that the effect of string/M-theory is relevant. On the other hand, string/M-theory is related to higher dimension theories such as space-like branes . Therefore in the present work we will study warm tachyon inspired inflation in the context of a higher dimensional theory instead of General Relativity, that is, Randall-Sundrum brane world and cosmological perturbations of the model by using the above modified Einstein and Friedmann equations.
Recently there has been a new perspective of warm inflation  which is considered warm inflationary era as a quasi-de Sitter epoch of universe expansion; on the other hand as we mentioned it is believed that we may live on the brane; therefore we are interested to study warm tachyon inflation on the brane by using quasi-de Sitter solutions of scale factor.
In one sector of the present work, we would like to consider warm tachyon model on the brane in the context of “intermediate inflation.” This scenario is one of the exact solutions of inflationary field equation in the Einstein theory with scale factor (); this solution of the scale factor in the context of a modified tensor-scalar theory has been found in . The study of this model is motivated by string/M-theory . If we add the higher order curvature correction, which is proportional to Gauss-Bonnet (GB) term and to Einstein-Hilbert action then we obtain a free-ghost action [57, 58]. Gauss-Bonnet interaction is leading order of the “” expansion to low energy string effective action [57, 58] ( is inverse string tension). This theory may be applied for black hole solutions , acceleration of the late time universe [60, 61], and initial singularity problems . The GB interaction in 4D with dynamical dilatonic scalar coupling leads to an intermediate form of scale factor . Expansion of the universe in the intermediate inflation scenario is slower than standard de Sitter inflation with scale factor () which arises as , but faster than power-low inflation with scale factor (). Harrison-Zeldovich [63–65] spectrum of density perturbation, that is, for intermediate inflation models driven by scalar field, is presented for exact values of parameter .
On the other hand we will also study our model in the context of “logamediate inflation” with scale factor () . This model is converted to power-law inflation for cases. This scenario is applied in a number of scalar-tensor theories . The study of logamediate scenario is motivated by imposing weak general conditions on the cosmological models which have indefinite expansion . The effective potential of the logamediate model has been considered in dark energy models . This form of potential is also used in supergravity, Kaluza-Klein theories, and superstring models [68, 70]. For logamediate models the power spectrum could be either red or blue tilted [71, 72]. In , we can find eight possible asymptotic scale factor solutions for cosmological dynamics. Three of these solutions are noninflationary scale factor; another three solutions give power-low, de Sitter, and intermediate scale factors. Finally, two cases of these solutions have asymptotic expansion with logamediate scale factor. We will study our model using intermediate and logamediate scenarios.
Warm inflation models based on ordinary scalar fields have been studied in [15, 44, 73–77]. Particular model of warm inflation which is driven by tachyon field can be found in [50–53]. In , the consistency of warm tachyon inflation with viscous pressure has been studied and the stability analysis for that model has been done. In the present paper we will study warm tachyon inflation without viscosity effect on the brane. We also extended our model by using exact solutions of the scale factor by Barrow , that is, inter(loga)mediate solution.
The paper is organized as follows: in the next section we will describe warm tachyon inflationary universe model in the brane scenario in the background level. In Section 3 we present the perturbation parameters for our model. In Section 4 we study our model using the exponential potential in high dissipative regime and high energy limit. In Section 5 we study the model using intermediate scenario. In Section 6 we develop our model in the context of logamediate inflation. Finally in Section 7 we close by some concluding remarks.
2. The Model
Tachyon scalar field is described by relativistic Lagrangian  asThe stress-energy tensor in a spatially flat Friedmann Robertson Walker (FRW) space-time is presented byFrom the above equation, energy density and pressure for a spatially homogeneous field have the following forms:where is a scalar potential associated with the tachyon field . Important characteristics of this potential are and . In this section, we will present the characteristics of warm tachyon inflation model on the brane in the background level. This model may be described by an effective fluid where the energy-momentum tensor of this fluid was recognized in the above equation.
The dynamic of the warm tachyon inflation in spatially flat FRW model on the brane is described by these equations:where is the dissipative coefficient. In the above equations dots “” mean derivative with respect to cosmic time and prime denotes derivative with respect to scalar field . During slow-roll inflation era the energy density (11) is the order of potential and dominates over the radiation energy . Using the slow-roll limit when and [11, 12], and also when the inflation radiation production is quasi-stable (, ), the dynamic equations (12) and (13) are reduced towhere . In canonical warm inflation scenario the relative strength of thermal damping () should be compared to expansion damping (). We must analyse the warm inflation model in background and linear perturbation levels on our expanding over timescales which are shorter than the variation of expansion rate but large compared to the microphysical processesFor more discussion please see Appendix. Particle production in fact takes place at a constant rate during warm inflation for canonical scalar field where strength of thermal damping dominates over the effect of expansion damping () but for tachyon scalar fields as presented in the above equation . We will study our model in high dissipative regime (). Using these conditions we have which agrees with particle production condition ().
The condition of inflation epoch could be obtained by inequality . Therefore from above equation, warm tachyon inflation on the brane could take place when
Inflation period ends when which implieswhere the subscript denotes the end of inflation. The number of e-folds is given bywhere the subscript denotes the epoch when the cosmological scale exits the horizon.
In this section we will study inhomogeneous perturbations of the FRW background. As we have mentioned in the introduction we ignore the influence of the bulk gravitons on the brane arising from Weyl tensor , so we neglect the back-reaction due to metric perturbations in the fifth dimension. These perturbations in the longitudinal gauge may be described by the perturbed FRW metricwhere and are gauge-invariant metric perturbation variables [80, 81]. The equation of motion is given byWe expand the small change of field into Fourier components asIn warm inflation thermal fluctuations of the inflation dominate over the quantum ones; therefore we have classical perturbation of scalar field . All perturbed quantities have a spatial sector of the form , where is the wave number. Perturbed Einstein field equations in momentum space have only the temporal parts
Note that the effect of the bulk (extradimension) to perturbed projected Einstein field equations on the brane may be found in (29). We will describe the nondecreasing adiabatic and isocurvature modes of our model on large scale limit. In this limit we have obtained a complete set of perturbation equations on the brane. Therefore the perturbation variables along the extradimensions in the bulk could not have any contribution to the perturbation equations on super-horizon scales (see, e.g., [41–44]). The same approach, for nontachyon warm inflation model on the brane, in  is presented. Warm inflation model may be considered as a hybrid-like inflationary model where the inflation field interacts with radiation field [49, 82, 83]. Entropy perturbation may be related to dissipation term . Perturbation of entropy in warm inflation model is given by In this paper we will study potential of the model as a function of scalar field (); therefore the entropy perturbation will be neglected. We will study this important issue (potential as function of temperature, ) in future works.
During inflationary phase with slow-roll approximation, for nondecreasing adiabatic modes on large scale limit , we assume that the perturbed quantities could not vary strongly. So we have , , , and . In the slow-roll limit and by using the above limitations, the set of perturbed equations are reduced toUsing (34), (36), and (37), perturbation variable is determined
We can solve the above equations by taking tachyon field as the independent variable in place of cosmic time . Using (16) we find
From above definition we haveUsing above equation and (40), we find
A solution for the above equation iswhere is integration constant. From above equation and (42) we find small change of variable aswhere
In the above calculations we have used the perturbation methods in warm inflation models [44, 50–53, 84], where the small change of variable may be generated by thermal fluctuations instead of quantum fluctuations , and the integration constant may be driven by boundary conditions for field perturbation. Perturbed matter fields of our model are inflation , radiation , and velocity . We can explain the cosmological perturbations in terms of gauge-invariant variables. These variables are important for development of perturbation after the end of inflation period. The curvature perturbation and entropy perturbation are defied by [87, 88]where . The boundary condition of warm inflation models is found in very large scale limits; that is, where the curvature perturbation and the entropy perturbation vanishes .
For high or low energy limit ( or ) and by inserting , the above equation reduces to which agrees with the density perturbation in cold inflation model [1, 2]. In the warm inflation model the fluctuations of the scalar field in high dissipative regime () may be generated by thermal fluctuation instead of quantum fluctuations  aswhere in this limit freeze-out wave number corresponds to the freeze-out scale at the point when dissipation damps out to thermally excited fluctuations () . in (50) can be found in , where Fourier transformed to momentum space is used (see, e.g., Appendix of  and Section of ); therefore is introduced in Fourier space and we can present spectral index and running in Fourier space. With the help of (49) and (50) in high energy () and high dissipative regime () we findor equivalentlywhere
An important perturbation parameter of inflation models is scalar index which in high dissipative regime is presented bywhere
In (55) we have used a relation between small change of the number of e-folds and interval in wave number (). Running of the scalar spectral index may be found as
This parameter is one of the interesting cosmological perturbation parameters which is approximately , by using observational results [3, 4]. During inflation epoch, there are two independent components of gravitational waves () with action of massless scalar field which are produced by the generation of tensor perturbations. Tensor perturbations do not couple to the thermal background; therefore gravitational waves are only generated by quantum fluctuations, the same as in standard fluctuations . However, if the gravitational sector is modified then the expression for tensor power spectrum changes with respect to General Relativity. In particular, the amplitude of the tensor perturbation on the brane is presented as [91, 92]where the temperature in extra factor denotes the temperature of the thermal background of gravitational wave , , and (in high energy limit, , we have ). Spectral index is presented aswhere . Using (51) and (58) we write the tensor-scalar ratio in high dissipative regimewhere is referred to pivot point  and . An upper bound for this parameter is given by using Planck data, [3, 4].
4. Exponential Potential
In this section we consider our model with the tachyonic effective potentialwhere parameter is related to mass of tachyon field . The exponential form of the potential has characteristics of tachyon field ( and ). We develop our model in high dissipative regime; that is, and high energy limit; that is, for a constant dissipation coefficient . From (54) slow-roll parameter in the present case has the form
Also the other slow-roll parameter is obtained from (56)
Dissipation parameter in this case is given byWe find the evolution of tachyon field with the help of (16)where . Hubble parameter for our model has this form
From (24) the number of e-folds, at the end of inflation, by using the potential (61) for our inflation model is presented byor equivalentlywhere . Using (51) and (60), we could find the scalar spectrum and scalar-tensor ratiowhere andwhere . In the above equation we have used (53), where
5. Intermediate Inflation
Intermediate inflation is denoted by the scale factorThis model of inflation is faster than power-low inflation and slower than de Sitter inflation. In this section we will study our model in the context of intermediate inflation in two cases: (1) and (2) which have been considered in the literature [50–53].
In high dissipative () and high energy () limits the equations of the slow-roll inflation, that is, (12) and (13), are simplified asInflation field may be derived from above equations in this case ()where . Using above equation and the scale factor of intermediate inflation, tachyonic potential and Hubble parameter are presented as Dissipative parameter is given by using above equationThe slow-roll parameters of the model in the present case may be obtained asWe present the number of e-folds aswhere is the scalar field at the beginning of the inflation. From the above equation we can present the scalar field in terms of number of e-folds and intermediate parametersNow we could find the perturbation parameters of the model. The power spectrum is obtained from (51), (53), and (73)where . We present the spectral index which is one of the important perturbation parameters from (55) and (73)Harrison-Zeldovich spectrum, that is, , is obtained for an exact value of parameter (i.e., ). For we found the cases which is compatible with observational data.
Tensor-scalar ratio of the model in this case is presented by using (60) and (74)where In Figure 2, tensor-scalar ratio in terms of number of e-folds is plotted, where . We could see lead to [3, 5, 6]. The expression for the perturbation given by (43) is valid when . The choice of the parameters of the model has to be consistent with this condition . In Figure 3 we plot in terms of spectral index that shows the model is compatible with observational data in warm inflation limit . We also checked the high dissipative condition in Figure 4 that we can see agreement with observational data.
Dissipative parameter may be considered as a function of scalar field [50–53]. We will study our model in the context of intermediate inflation, where . In this case the scalar field is determined from (74) and (75)Therefor the Hubble parameter and potential of the model in terms of tachyon potential have the following forms:Dissipative parameter is presented by using above equationImportant parameters of the slow-roll inflation in this case are presented asThe number of e-folds is given bywhere is the tachyon field at the beginning of the inflation period. We find this field where the slow-roll parameter is equal to oneFrom above equations we present the scalar field in terms of number of e-folds and intermediate parameters and Spectral index is presented using (55)We can find the scale invariant spectrum (Harrison-Zeldovich spectrum); that is, , where . In Figure 5, we plot the spectral index in terms of number of e-folds where . For we can see the spectral index is confined to which is compatible with Planck data [3, 4]. Power spectrum and scalar-tensor ratio of this model may be obtained from (51) and (60), respectively,whereIn Figure 6 we can see high dissipative condition agrees with Planck data. In Figure 7 tensor-scalar ratio in terms of number of e-folds is plotted, where . We could see lead to [3, 5, 6].
6. Logamediate Inflation
In this section we will study warm tachyon inflation model in the context of logamediate scenario. The scale factor of this model is given bywhere is a positive constant and . We consider this model in two cases: (1) Dissipative parameter is constant. (2) Dissipative parameter is proportional to tachyon field potential .
In this case the scalar field is given by using (75) and (95)where . Using above equation, the Hubble parameter and tachyon potential have the following forms We derive the slow-roll parameters in logamediate scenario The number of e-folds for present model of inflation is presented as: is the inflation at the beginning of the inflation era. From above equation the scalar field is presented in terms of number of e-folds Dissipative parameter is given byPower spectrum and scalar-tensor ratio of logamediate inflation are derived from (51) and (60):whereBy using (55), we could find the spectral index In Figure 8, the dependence of spectral index on the number of e-folds is shown (for and cases). It is observed that the small values of the number of e-folds are assured for large values of parameter. This figure shows the scale invariant spectrum (Harrison-Zeldovich spectrum, i.e., ) could be approximately obtained for . From above equation and (102), a relation between scalar-tensor ratio and spectral index is obtained