Logamediate Inflation by Tachyon Field
aa r X i v : . [ g r- q c ] M a r Logamediate Inflation by Tachyon Field
A. Ravanpak ∗ and F. Salmeh † Department of Physics, Vali-e-Asr University, Rafsanjan, Iran (Dated: August 13, 2018)
Abstract
A logamediate inflationary model in the presence of the tachyon scalar field will be studied.Considering slow-roll inflation, the equations of motion of the universe and the tachyon field willbe derived. In the context of perturbation theory, some important perturbation parameters willbe obtained and using numerical calculations the consistency of our model with observational datawill be illustrated.
PACS numbers: 98.80.Cq,Keywords: logamediate inflation, tachyon field, perturbation, WMAP ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION After encountering some serious problems, cosmologists had to improve the standard big-bang model adding some parts to it. The flatness and horizon problems were the most famousof those problems and the best part which added to the standard model and completed itwas the inflationary scenario. Inflation is a short period at very early stages of the historyof the universe in which the universe experiences a very rapidly accelerated expansion andthe scale factor parameter a ( t ) grows by many order of magnitude. In terms of how thescale factor varies with time one can classify inflationary models to for example power lawinflation a ( t ) = t q , exponential inflation a ( t ) = exp( pt ) and intermediate inflation a ( t ) =exp( pt q ). Among different models of inflation, the one has not been investigated greatly isthe logamediate inflation in which for t > a ( t ) = exp( A (ln t ) λ ) (1)where A > λ > λ = 1 the modelreduces to the power law inflation. The logamediate inflation can be extracted from somescalar-tensor theories which naturally give rise these solutions [1] and also from a new class ofcosmological solutions with indefinite expansion which result when weak general conditionsapply on the cosmological models [2]. Although these models belong to a class of modelscalled non-oscillating models that can not naturally bring inflation to an end, but differentapproaches such as curvaton scenario can be used to do this duty [3].On the other hand, the kind of the scalar field which plays the role of the inflaton fieldis also important. The standard scalar field is the most usual one, but some other fieldscan also be responsible for it. Among them, the tachyon field is of particular interest [4],[5].Its equation of state parameter varies between 0 and -1 and thus it can be a good choicefor the inflaton field [6]–[12]. Also it has been shown that the tachyon field can play therole of dark sectors of the universe [13]–[22] and even at the same time derive inflation andthen behave as dark matter or non-relativistic fluid [23]. Tachyonic inflation is a specialcase of k-inflationary models in which the inflaton field starts its evolution from an unstablemaximum when φ → φ → ∞ .The concept of logamediate inflation with tachyon field or without it has been analyzed inthe literature. For instance, in [24] the dynamics of the logamediate inflation in the presenceof a standard scalar field and its consistency with observational results has been shown2n details. In [25], the authors have been investigated the warm-logamediate inflationaryuniverse in both weak and strong dissipative regimes and obtained the general conditionswhich are necessary that the model to be realizable. Also, in [26], the warm-logamediateinflation in the presence of the tachyon field as the inflaton has been analyzed only in highdissipative regime and the results have been compared by the observations.In this work we are trying to use the tachyon field as the inflaton in the logamediateinflationary scenario. Our aim is to obtain the influence of the tachyon field on logamediateinflation in comparison with [24] and also to fill the gap between the articles noted above. Inthe next section, we will apply tachyon field in a logamediate inflationary model under slow-roll conditions. Section III. deals with perturbation theory. At this section we will calculateall the perturbation parameters which are needed to have a comparison with observations.The numerical comparisons have been done in the subsection A. Section IV deals with howrealistic is our model. At the end, there is a conclusion section in which we will discuss ourresults. II. LOGAMEDIATE INFLATIONARY MODEL
We start with the field equations in a flat Friedmann-Robertson-Walker (FRW) universe3 H = ρ φ (2)and 2 ˙ H + 3 H = − p φ , (3)in which H = ˙ a/a is the Hubble parameter, a = a ( t ) is the scale factor and the dot meansderivative with respect to the cosmological time t . Here, we have used units such that8 πG = c = ~ = 1. Also, we assume the matter content of the universe is a scalar field, φ ( t ),so-called inflaton where ρ φ and p φ represent its energy density and pressure, respectivelyand they satisfy the following conservation equation˙ ρ φ + 3 H ( ρ φ + p φ ) = 0 . (4)From now on we consider the inflaton field as a tachyon field where its energy densityand pressure given by ρ φ = V ( φ ) q − ˙ φ , p φ = − V ( φ ) q − ˙ φ , (5)3here V ( φ ) is the tachyonic scalar potential. Substituting (5) in (4) we reach to the equationof motion of the tachyon field ¨ φ − ˙ φ + 3 H ˙ φ + V ′ V = 0 , (6)where V ′ = ∂V ( φ ) /∂φ . Also, using (2), (4) and (5), one can obtain˙ φ = s − H H . (7)Considering the logamediate inflationary model in which the scale factor a ( t ) behaves as(1), one can obtain the exact solution of (7) as φ ( t ) = Z r Aλ (ln t ) − λ/ (ln t − λ + 1) / dt (8)Also using (2), (5) and (7) the potential of the inflaton field can be obtained as V ( t ) = 3( Aλt ) (ln t ) λ − (cid:18) t ) − λ ( λ − − ln t )3 Aλ (cid:19) (9)To have a long enough period of inflation we need to our inflaton field rolls slowly downits potential. In this scenario which is called slow-roll inflation the energy density of theinflaton field and its potential satisfy ρ φ ∼ V . Thus in our model, under slow-roll conditionsi.e. ˙ φ ≪ φ ≪ H ˙ φ , equations (2) and (6) reduce to3 H ≈ V (10)and V ′ V ≈ − H ˙ φ, (11)respectively. Also, the tachyonic potential (9) becomes V ( t ) = 3( Aλt ) (ln t ) λ − (12)There are a few dimensionless parameters in slow-roll inflationary models called slow-rollparameters. In terms of our model parameters they can be written as ε = − ˙ HH = (ln t ) − λ Aλ (ln t − λ + 1) (13)and η = − ¨ φH ˙ φ = 12 H [ − ¨ V ˙ V + ˙ HH + ˙ VV ] . (14)4ne can check that the slow-roll parameter ε starts to increase at t = 1, reaches to amaximum at some value t ε and then returns and approaches zero as t → ∞ . If we payattention to those cases in which the maximum value of ε is greater than one, we can choose ε = 1 as the beginning condition of inflation. For these cases ( ε max > A < λ − λ − (15)We can also obtain the number of e-folds between two different values t and t > t forthis model as N = Z t t Hdt = A [(ln t ) λ − (ln t ) λ ] (16)where t represents the time in which inflation begins. III. PERTURBATION
Although studying a homogeneous and isotropic universe model is sometimes very useful,we know that in a real cosmology there are deviations from homogeneity and isotropicassumptions. This motivates us to investigate the perturbation theory in our model. Webelieve that inhomogeneities grow with time due to the attractive feature of gravity and thuswe can say that they were very smaller in the past. Because of the smallness of them wecan use linear perturbation theory. But as it appears from Einstein’s equations and to havea more realistic investigation we need a relativistic perturbation theory, i.e. a perturbedinflaton field in a perturbed geometry. So we start by the most general linearly perturbedflat FRW metric which includes both scalar and tensor perturbations as below ds = − (1 + 2 C ) dt + 2 a ( t ) D ,i dx i dt + a ( t ) [(1 − ψ ) δ ij + 2 E ,i,j + 2 h ij ] dx i dx j , (17)where C , D , ψ and E are the scalar metric perturbation and h ij is the transverse-tracelesstensor perturbation. A very useful quantity in characterizing the properties of the perturba-tions, is the power spectrum. First of all, we calculate the power spectrum of the curvatureperturbation P R , which appears in deriving the correlation function of the inflaton field inthe vacuum state. For the tachyon field this parameter is defined as P R = ( H π ˙ φ ) Z s (18)5here Z s = V (1 − ˙ φ ) − / [27]. Applying slow-roll approximation in (18) and using equations(10) and (11), one can obtain P R ≈ ( H π ˙ φ ) V = − H π ˙ V . (19)When someone deals with perturbation in cosmology a few special parameters have to beidentified. The first one is the scalar spectral index n s which is related to the scalar powerspectrum via the relation n s − d ln P R /d ln k , where d ln k = dN = Hdt [28]. Withattention to definition of P R in the slow-roll approximation, we reach to n s ≈ HH − ¨ VH ˙ V = 1 + 2( η − ε ) . (20)The second interesting parameter is the running in the scalar spectral index parameter n run ,which has been indicated by one-year to seven-years data set of the Wilkinson MicrowaveAnisotropy Probe (WMAP) and can be obtained via n run = dn s /d ln k . Thus with attentionto (20) one can reach to equation below n run ≈ H ( ˙ η − ˙ ε ) (21)So far we have only studied the scalar perturbations. But how about tensor contributions?In fact the primordial gravitational waves are these tensor perturbations where are essentiallyequivalent to two massless scalar fields. Thus, the power spectrum of tensor perturbationscan be written as P g = 8( H π ) (22)The third special parameter we deal with is the tensor to scalar ratio r which by definitionand using equations (19) and (22) becomes r = P g P R ≈ ε (23) A. numerical discussion
Although we could not obtain a straight relation between r and n s , we can numericallyillustrate some trajectories in the r − n s plane, if we fix our model parameters λ and A.Since in logamediate inflationary model we only have a lower limit for λ , so we chose thevalues λ = 2 , , ,
50 to have a general comparison with the work [2]. In figure (1), we6ave plotted four curves related to these values of λ where in each case we have fixed thesecond model parameter A arbitrarily as they satisfy the condition (15). The solid yellow,dash black, dash-dot green and long dash red curves are related to the combinations (2,10 − ), (10, 5 × − ), (20, 4 × − ) and (50, 10 − ), respectively. It appears that the maindifference between using a standard scalar field in a logamediate inflationary model [2] anda tachyonic field in it, is that in the latter, transition from n s < n s > λ in comparison to the former. We should mention that these curves have beenplotted for as large as possible values of A satisfying (15) and if we choose some smallervalues, then the curves move to the left. Thus, for the cases with n s > λ , A) that in which the curves behave as a Harrison-Zel’dovich spectrum,i.e. n s = 1. In figure (2), the dash-dot green and long dash red curves are related to thecombinations (20, 2 × − ) and (50, 5 × − ), respectively. FIG. 1: The trajectories r − n s for different combinations of ( λ , A). They have been comparedwith the five-years (blue regions) and seven years (red regions) data set of WMAP. In each case thecontours show 68% and 95% confidence regions [29]. The solid yellow, dash black, dash-dot greenand long dash red curves are related to the combinations (2, 10 − ), (10, 5 × − ), (20, 4 × − )and (50, 10 − ), respectively. Transition from n s < n s > λ in comparison to [2]. Also, in figures (1) and (2) the trajectories have been compared with 68% and 95%confidence regions from observational data, i.e. five-years (blue-contours) and seven-years(red-contours) WMAP data set, which have been defined at k = 0 .
002 Mpc − [29]. Ac-7 IG. 2: The trajectories r − n s for different combinations of ( λ , A). They have been comparedwith the five-years (blue regions) and seven years (red regions) data set of WMAP. In each case thecontours show 68% and 95% confidence regions [29]. The dash-dot green and long dash red curveswhich indicate a nearly Harrison-Zel’dovich model are related to the combinations (20, 2 × − )and (50, 5 × − ), respectively. The solid blue line, has been plotted for the combination (60,3 × − ) and shows an upper bound for λ in which the model is exiting the observational data.For larger values of λ the model will be consistent with the data if we decrease enough other modelparameter A. cording to these observational data, an upper limit for r has been found. This upper boundfrom five-years WMAP data set is r < .
43 whereas for seven-years data a stronger limit hasobtained as r < .
36. In figure (1), the trajectories related to the combinations (2, 10 − ),(10, 5 × − ), (20, 4 × − ) and (50, 10 − ), enter seven-years 95% confidence region at r ≃ . , . , .
39 and 0.43, respectively. On the other hand, we can obtain the number ofe-folds related to each one of these values of r . One can do this work numerically by firstcalculating when ε = 1 which is the condition of beginning of inflation in our model and thenreplacing it in (16). The resulting equation with (23) gives the values of N ≃ , , r − n s plane allowed by the data and in each case our model is viablefor larger values of the related N .The solid blue line in figure (2) indicates the case in which we have considered the com-bination (60, 3 × − ) that it means for λ >
60, the model tends to exit our observational8ontours unless we decrease the value of A much more.Figure (3) shows the dependence of the running of the scalar spectral index on the scalarspectral index parameter to lowest order for some combinations. Again the solid yellow,dash black, dash-dot green and long dash red curves are related to the combinations (2,10 − ), (10, 5 × − ), (20, 4 × − ) and (50, 10 − ), respectively. These curves have beencompared with the contour plots from seven-years WMAP data set [29] in which the negativevalues have been preferred. Seven-years data set implies that in models with only scalarfluctuations the marginalized value for the parameter n run is approximately -0.034 [29],[30].Also, it is obvious from this figure that for the combination (2, 10 − ) the model does notshow any running in the scalar spectral index, at least in the lowest order. FIG. 3: The trajectories n run − n s for different combinations of ( λ , A). They have been comparedwith the five-years WMAP data set in two cases with and without considering tensor contributions.In each case the contours show 68% and 95% confidence regions. The solid yellow, dash black,dash-dot green and long dash red curves are related to the combinations (2, 10 − ), (10, 5 × − ),(20, 4 × − ) and (50, 10 − ), respectively. IV. IS THE MODEL REALISTIC?
As noted in introduction, tachyonic potential has a special behavior. It starts from anunstable maximum at φ → dVdφ < φ → ∞ . These are some motivations from string theory. To see how well-motivated is our9odel potential in (12), first we derive ˙ H from (1) as˙ H = Aλ (ln t ) λ − t − [ λ − t − . (24)Now, using (7) we obtain ˙ φ = 23 Aλ (ln t ) − λ (1 + 1 − λ ln t ) (25)At late times, one can neglect the second term in parenthesis above and consider ˙ φ = q Aλ (ln t ) − λ . But, integrating does not give us a good result. Assuming α = q Aλ and β = − λ one can obtain φ − φ = α [ t (ln t ) β − βt (ln t ) β − + β ( β − t (ln t ) β − − β ( β − β − t (ln t ) β − + ... ] (26)and using this, we can not get V ( φ ), explicitly.In another approach, assuming ˙ φ = ˙Φ in (25), one can integrate and reach to the explicitfunction Φ( t ) = 23 Aλ t (ln t ) − λ (27)and substituting this into (12) give us V (Φ) = Φ − . Although this form of potential is ofinterest but the thing we need is behavior of V ( φ ) and since we can not relate two functions φ ( t ) and Φ( t ) so we can not establish V ( φ ) even approximately.So, we chose numerical approach to show how our model is realistic. We can do thisusing (27), V (Φ) = Φ − and ˙Φ = ˙ φ . Also, to do this we should fix some of our modelparameters such as A and λ . In figure (4) we have shown V ( φ ) for all combinations of ( λ, A )which we have used in figures (1) and (3), i.e. (2, 10 − ), (10, 5 × − ), (20, 4 × − ) and(50, 10 − ). It is obvious from these plots that the general conditions for the string theorytachyon field which noted above are satisfied. Indeed, we can say that our model potentialis a well-motivated tachyon potential and the model under consideration is realistic. Also,we can see that increasing in λ leads to more smooth behavior of V ( φ ). V. CONCLUSION
In this work we studied the logamediate inflation in the presence of tachyon field. In theslow-roll approximation we derived the effective tachyon potential and the slow-roll param-eters. Also, the number of e-folds which indicates how long inflation takes, was obtained.10
IG. 4: The trajectories V ( φ ) for different combinations of ( λ , A), up-left:(2, 10 − ), up-right:(10,5 × − ), bottom-left:(20, 4 × − ) and right:(50, 10 − ). All necessary conditions for a tachyonicpotential have been satisfied Starting with a perturbed line element we investigated our model in the context of per-turbation theory. We calculated some important parameters such as scalar spectral index n s , its running n run and tensor to scalar ratio r in our model. Then, we plotted some curvesfor different combinations of our model parameters ( λ , A) and compared them with someobservational data. From graph r − n s we concluded that our model is in a good agreementwith observations for different combinations of the model parameters such as from (2, 10 − )11o (60, 3 × − ). Also, one can find some combinations that result Harrison-Zel’dovichspectrum, i.e. n s ≃
1, for example (20, 2 × − ) and (50, 5 × − ).In the last section we investigated whether our model and specially the resulting tachyonicpotential is realistic or not. We could not do this analytically but numerical discussion wasuseful. In admissible combinations of ( λ, A ) which we have used in this article, we couldshow that the general conditions of a tachyon potential are satisfied in our model. [1] J. D. Barrow, Phys. Rev. D. 51, 2729 (1995).[2] J. D. Barrow, Class. Quant. Grav. 13, 2965 (1996).[3] S. del Campo, R. Herrera, and J. Saavedra, Phys. Rev. D. 80, 123531 (2009).[4] A. Sen, J. High Energy Phys. 04, 048 (2002).[5] A. Sen, J. High Energy Phys. 07, 065 (2002).[6] A. Mazumdar, S. Panda and A. Perez-Lorenzana, Nucl. Phys. B. 614, 101 (2001).[7] A. Feinstein, Phys. Rev. D. 66, 063511 (2002);[8] Y. S. Piao, R. G. Cai, X. M. Zhang and Y. Z. Zhang, Phys. Rev. D. 66, 121301 (2002).[9] A. Sen, Mod. Phys. Lett. A. 17, 1797 (2002).[10] M. Fairbairn and M. H. G. Tytgat, Phys. Lett. B. 546, 1-7 (2002).[11] M. Sami, P. Chingangbam and T. Qureshi, Phys. Rev. D. 66, 043530 (2002).[12] M. Sami, Mod. Phys. Lett. A. 18, 691 (2003).[13] T. Padmanabhan, Phys. Rev. D. 66, 021301 (2002).[14] J. S. Bagla, H. K.Jassal, T. Padmanabhan, Phys. Rev. D. 67, 063504 (2003).[15] E. J. Copeland, M. R. Garousi, M. Sami and S. Tsujikawa, Phys. Rev. D. 71, 043003 (2005).[16] A. Das, S. Gupta, T. D. Saini and S. Kar, Phys. Rev. D. 72, 043528 (2005).[17] E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D. 15, 1753 (2006).[18] H. Farajollahi, A. Ravanpak and G. F. Fadakar, Mod. Phys. Lett. A. 26, 15, 1125-1135 (2011).[19] H. Farajollahi, A. Ravanpak and G. F. Fadakar, Astro. Space Sci. 336, 2, 461-467 (2011).[20] H. Farajollahi and A. Ravanpak, Phys. Rev. D. 84, 8, 084017 (2011).[21] H. Farajollahi, A. Salehi, F. Tayebi and A. Ravanpak, JCAP. 05, 017 (2011).[22] H. Farajollahi, A. Ravanpak and G. F. Fadakar, Phys. Lett. B. 711, 3-4, 225-231 (2012).[23] G. W. Gibbons, Phys. Lett. B. 537, 1-4 (2002).
24] J. D. Barrow and N. J. Nunes, Phys. Rev. D. 76, 043501 (2007).[25] R. Herrera and M. Olivares, Int. J. Mod. Phys. D. 21, 1250047 (2012).[26] M. R. Setare and V. Kamali, Phys. Rev. D. 87, 083524 (2013).[27] J. C. Hwang and H. Noh, Phys. Rev. D. 66, 084009 (2002).[28] S. del Campo, R. Herrera, and A. Toloza, Phys. Rev. D. 79, 083507 (2009).[29] D. Larson, et al., Astrophys. J. Suppl. 192, 16 (2011).[30] E. Komatsu et al., Astrophys. J. Suppl. Ser. 192, 18 (2011).24] J. D. Barrow and N. J. Nunes, Phys. Rev. D. 76, 043501 (2007).[25] R. Herrera and M. Olivares, Int. J. Mod. Phys. D. 21, 1250047 (2012).[26] M. R. Setare and V. Kamali, Phys. Rev. D. 87, 083524 (2013).[27] J. C. Hwang and H. Noh, Phys. Rev. D. 66, 084009 (2002).[28] S. del Campo, R. Herrera, and A. Toloza, Phys. Rev. D. 79, 083507 (2009).[29] D. Larson, et al., Astrophys. J. Suppl. 192, 16 (2011).[30] E. Komatsu et al., Astrophys. J. Suppl. Ser. 192, 18 (2011).