Interacting Chaplygin gas revisited
aa r X i v : . [ phy s i c s . g e n - ph ] J u l Interacting Chaplygin gas revisited
Subhajit Saha a Department of Mathematics,Panihati Mahavidyalaya,Sodepur 700110, West Bengal, India.
Saumya Ghosh b and Sunandan Gangopadhyay c Department of Physical Sciences,Indian Institute of Science Education and Research Kolkata,Mohanpur 741246, West Bengal, India. (Dated: The 23 rd June, 2017)The implications of considering interaction between Chaplygin gas and a barotropic fluid withconstant equation of state have been explored. The unique feature of this work is that assuming aninteraction Q ∝ Hρ d , analytic expressions for the energy density and pressure have been derivedin terms of the Hypergeometric F function. It is worthwhile to mention that an interactingChaplygin gas model was considered in 2006 by Zhang and Zhu, nevertheless, analytic solutions forthe continuity equations could not be determined assuming an interaction proportional to H timesthe sum of the energy densities of Chaplygin gas and dust. Our model can successfully explain thetransition from the early decelerating phase to the present phase of cosmic acceleration. Arbitrarychoice of the free parameters of our model through trial and error show at recent observational datastrongly favors w m = 0 and w m = − over the w m = case. Interestingly, the present model alsoincorporates the transition of dark energy into the phantom domain, however, future decelerationis forbidden.Keywords: Chaplygin gas; Barotropic fluid; Interaction; Analytic solutionPACS Numbers: 98.80.-k, 95.35.+d, 95.36.+x a Electronic Address: [email protected] b Electronic Address: [email protected] c Electronic Address: [email protected]
1. INTRODUCTION
Chaplygin gas (CG) is a class of dark energy (DE) models (for a review on DE, see [1, 2]) which mimics theobserved late-time acceleration of the Universe as was reported in [3–5]. Kamenshchik, Moschella, and Pasquier[6] were the first to consider CG in the context of Cosmology. Such a fluid is assumed to have the followingequation of state (EoS) : p = − Aρ , (1)where p and ρ are, respectively, pressure and energy density in a comoving reference frame, and A is a positiveconstant. It is named after Sergey Chaplygin who introduced the above EoS [11] as a suitable mathematicalapproximation for calculating the lifting force on a plane wing in aerodynamics. CG can also be considered asa special case of a tachyon with a constant potential [1, 12].The convenience of CG is due to the fact that the corresponding Euler equations have a very large groupof symmetry, which implies their integrability. The relevant symmetry group has been recently described inmodern terms [13]. CG has a remarkable connection with string theory since it can be obtained from theNambu-Goto action for D -branes moving in a ( D + 2)-dimensional spacetime in the light-cone parametrization[14]. It can also be derived for the moving brane via the Born-Infeld Lagrangian [15]. Further, CG is alsoknown to admit a supersymmetric generalization [16, 17]. Certain effects in deformable solids, of stripe statesin the context of the quantum Hall effect and of other phenomena, can be explained by the negative pressurearising from the CG EoS [18]. It is also useful when studying the stabilization of branes in black hole bulks[19–21]. A microscopic description of CG suggests that it can be regarded (phenomenologically) as the effectof the immersion of our four-dimensional world into some multidimensional bulk [22, 23]. Another intriguingproperty of CG is that it gives positive and bounded square of sound velocity c s = Aρ , which is a non-trivialfact for fluids with negative pressure [6, 24]. This sound velocity is negligible at early times and approaches thespeed of light in the late-time limit (for a detailed discussion on this topic, see [25]).The advantages [6, 24] of the Chaplygin class of cosmological models is three-fold. Firstly, they describe asmooth transition from the decelerating phase of the Universe to the present phase of cosmic acceleration andsuch a transition is achieved with only one fluid. Secondly, a unified macroscopic phenomenological descriptionof DE and dark matter (DM) can be given by such a class of models. Finally, they represent the simplestdeformations of the concordance ΛCDM model. In spite of these remarkable advantages, CG is in disagreementwith the observational data obtained from CMB anisotropies [15, 26, 36] which is attributed to the fact that theJeans instability of perturbations in CG models behaves similarly to cold dark matter (CDM) fluctuations inthe dust-dominant stage but disappears in the acceleration stage. A strong integrated Sachs-Wolfe (ISW) effectarises due to the dual effect of the suppression of perturbations and the presence of a non-zero Jeans length[1, 25]. To remedy the situation, the generalized Chaplygin gas (GCG) model was proposed [15] which has anEoS given by p = − Bρ α with B > < α <
1, nevertheless, the parameter α is severely constrained, i.e.,0 < α < . p = Cρ − Dρ n , C & D are positive constants and n ≥ p = wρ , w constant. Our investigationhas been primarily focussed on the dynamics of the coexistence of the fluids in the presence of an interactionterm proportional to the Hubble parameter times the DE density. This class of interaction terms generallyappears in the interacting holographic dark energy (HDE) model [33, 34]. In the absence of interaction, thereexists no scaling solutions owing to the fact that the EoS of CG decreases with scale factor while the dark matter(DM) EoS remains constant. It is worthwhile to mention that Zhang and Zhu [35] have earlier considered akind of interacting Chaplygin gas model in which the Chaplygin gas plays the role of DE and interacts withCDM particles via an interaction term of the form Γ = 3 cH ( ρ d + ρ m ), where ρ d and ρ m respectively denotethe energy densities due to DE and CDM, H is the Hubble parameter, and c is the coupling constant. Theyfound a stable scaling solution at late times with the Universe evolving into a phase of steady state. Moreover, An “anti-Chaplygin” EoS (Eq. (1) with a negative A ) can be found in the description of wiggly strings [9, 10]. Sergey Chaplygin was a Russian mathematician, physicist, and engineer who is very likely the only scientist who has a lunarcrater, a city, and a cosmological model named after him. their effective EoS could also cross the phantom barrier. However, their form of interaction failed to producean analytic solution which is necessary to obtain in order to have a clear and a nice picture of the cosmologicalmodel concerned.Our paper is organized as follows — Section 2 describes the basic equations that govern a flat FLRW universefilled with CG and a fluid with EoS p = wρ , w constant. Section 3 is concerned with the cosmological implicationsof considering interaction between the two sectors. Finally, a short discussion and scope of future work appearsin Section 4.
2. BASIC EQUATIONS OF OUR MODEL
As stated, we consider a flat FLRW universe governed by the metric ds = − dt + a ( t ) (cid:2) dr + r (cid:0) dθ + sin θdφ (cid:1)(cid:3) , (2)where a ( t ) is the scale factor of the Universe. Assuming a perfect fluid having energy-momentum tensor givenby ( u µ is the 4-velocity of the fluid) T µν = ( ρ + p ) u µ u ν + pg µν , (3)the Friedmann and the acceleration equations can be obtained as3 H = ρ, (4)2 ˙ H = − ( ρ + p ) (5)respectively, where H is the Hubble parameter defined by H = ˙ a ( t ) a ( t ) , ρ is the total energy density of the Universeand p is the pressure term. We have also assumed that 8 πG = c = 1, without any loss of generality. Using theabove equations, one can obtain the energy-momentum conservation equation˙ ρ + 3 H ( ρ + p ) = 0 . (6)Since we shall be working with a two-fluid system, the total energy density ρ and the total pressure p can bewritten as ρ = ρ m + ρ d (7) p = p m + p d , (8)where ρ d and p d represent, respectively, the energy density and pressure due to DE which is considered to beCG (EoS: p d = − Aρ d , A is a positive constant), and the corresponding quantities with suffix m are due to thematter field which we shall assume to be a barotropic fluid with EoS given by p m = w m ρ m , w m ≥ − is aconstant. The lower bound on w m assures that the barotropic fluid does not violate the strong energy condition.
3. COSMOLOGICAL DYNAMICS FOR CG + BAROTROPIC FLUID WITH INTERACTION Q ∝ Hρ d We shall now study the implications of considering interaction between matter (barotropic fluid with constantEoS) and DE (CG) sectors. It has already been mentioned earlier that an interacting CG model was consideredin the literature [35], nevertheless, analytic solutions for the continuity equations could not be determinedassuming an interaction of the form Γ = 3 cH ( ρ d + ρ m ). The authors [35] performed a phase-space analysis andobtained a stable scaling solution at late times with the Universe evolving into a phase of steady state. Sincethere is no microphysical hint on the nature of interaction between Chaplygin gas and matter, we are boundto consider a phenomenological form of the interaction term. In what follows, it can be seen that the non-conservation equations for CG and the barotropic fluid arising due to an interaction of the form Q = 3 b Hρ d ( b is the coupling parameter) produces analytic expressions for ρ and p .Interaction between DE and DM has some important consequences such as in alleviating the coincidenceproblem [36–41], among others. The coincidence problem can be solved, or atleast alleviated if DE decays intoDM [35], thus diminishing the difference between the densities of the two components through the evolution ofthe Universe. Therefore, if CG is assumed to decay into matter, then under the said form of interaction, thenon-conservation equations for the two sectors can be written as˙ ρ d + 3 H ( p d + ρ d ) = − Q = − b Hρ d , (9)˙ ρ m + 3(1 + w m ) Hρ m = + Q = +3 b Hρ d . (10)Plugging in the EoS of CG in Eq. (9), and evaluating Z d ρ d ρ d (cid:16) b − Aρ d (cid:17) = ln (cid:18) B ′ a (cid:19) , we obtain ρ d as ρ d = 1 √ b " A + (cid:18) B ′ a (cid:19) b , (11)where B ′ is the constant of integration. Note that when there is no interaction, i.e., b = 0, we get back theCG density obtained in [6]. Now, putting the above expression for ρ d in Eq. (10) and multiplying both sides ofthe equation by a w m ) , the matter density ρ m can be evaluated as ρ m = 1 a w m ) " C ′ + b ( b − w m )(1 + w m ) r (1 + b ) h A + (cid:0) B ′ a (cid:1) b i ( a w m ) − (1 + w m ) ( A + (cid:18) B ′ a (cid:19) b ) + √ A (1 + b ) s A + (cid:18) B ′ a (cid:19) b × F " , − w m b ) , − w m b ) , − A (cid:18) B ′ a (cid:19) b (12)from the integral a ρ m = 3 b √ b Z a s A + (cid:18) Ba (cid:19) b d a + C ′ , with the constant of integration as C ′ . Few comments regarding the free parameters of our model are in order.Note that for a prescribed matter EoS w m , our model consists of four free parameters — the CG parameter A ,the coupling parameter b , and the two constants of integration B ′ and C ′ . If one restricts to the spatially flatcase, then ΛCDM has only one free parameter (Ω m ), while the most discussed dynamical DE model, φ CDMhas two free parameters (Ω m and α ) [7, 8]. In the latter case, one has the attractor solution, so there is nodependence on initial conditions. In our interacting CG model, the two constants arising due to integrationcan be fixed so that we shall also be left with only two free parameters, A and b . However, due to a highdegree of nonlinearity in the expressions, it is quite difficult to identify the relations of the parameters withthose occurring in the more well-known DE models. Now, the explicit expressions for the total energy density ρ and the total pressure p can be written as ρ = 1 a w m ) " C ′ + b ( b − w m )(1 + w m ) r (1 + b ) h A + (cid:0) B ′ a (cid:1) b i ( a w m ) − (1 + w m ) ( A + (cid:18) B ′ a (cid:19) b ) + √ A (1 + b ) s A + (cid:18) B ′ a (cid:19) b × F " , − w m b ) , − w m b ) , − A (cid:18) B ′ a (cid:19) b + 1 √ b " A + (cid:18) B ′ a (cid:19) b , (13) Mathematica software was used to evaluate the integral obtained in Eq. (10). The authors are grateful to the anonymous reviewer for raising this very important issue. p = w m a w m ) " C ′ + b ( b − w m )(1 + w m ) r (1 + b ) h A + (cid:0) B ′ a (cid:1) b i ( a w m ) − (1 + w m ) ( A + (cid:18) B ′ a (cid:19) b ) + √ A (1 + b ) s A + (cid:18) B ′ a (cid:19) b × F " , − w m b ) , − w m b ) , − A (cid:18) B ′ a (cid:19) b − A p (1 + b ) " A + (cid:18) B ′ a (cid:19) b − (14)respectively, where F [ y , y , y , x ] is known as the Gauss’s hypergeometric function [42]. The effective EoSparameter w eff = pρ and the deceleration parameter q = (cid:16) pρ (cid:17) − w d , w eff , and q against the redshift z have been presented in Figure 1. In doing so, we have assumed the following values fordifferent free parameters: A = 5, b = 0 . B ′ = 0 .
1, and C ′ = 1. These choices lead to a CG EoS of − .
983 atthe present epoch which is nearly consistent with recent observations [43]. The values of the other parameters,particularly, q , Ω d , and Ω m at the present epoch (for the three chosen values of w m ) have been presented inTable I. Among the three cases, the values corresponding to w m = 0 and w m = − show a better consistencywith the latest observational data [43–48] compared to the w m = case. We have also calculated the redshiftof transition from deceleration to acceleration ( z da ) for w m = , w m = 0, and w m = − to be ≈ . ≈ . ≈ .
14 respectively. The dust case shows a better agreement with previous analyses [49, 50] as comparedto the other two cases. A chi-square analysis could have been more fruitful in this situation but due to a largenumber of free parameters, the software packages were not able to produce interesting results.
FIG. 1. The variations of the DE density w d , the effective EoS w eff , and the deceleration parameter q against theredshift z . The solid, dashdot, and dashed curves in the middle and the right panels correspond to w m = , w m = 0,and w m = − respectively.TABLE I. Present values of the cosmological parameters with A = 5, b = 0 . B ′ = 0 .
1, and C ′ = 1Parameter w m = w m = 0 w m = − q − . − . − . d .
692 0 .
692 0 . m .
308 0 .
308 0 . Unlike the non-interacting CG models, this interacting scenario incorporates the transition of DE into thephantom regime. Since w d = − Aρ d = − A (1 + b ) A + (cid:0) B ′ a (cid:1) b , (15)we observe that as the scale factor becomes very large, the second term inside the denominator can be ignoredand w d reduces to w d = − − b which always has a value less than the phantom barrier of −
1, for a non-vanishing coupling parameter b . FIG. 2. The variations of the fractional energy densities of DE (Ω d ) and matter (Ω m ) against the redshift z . The left,middle, and right panels correspond to w m = , w m = 0, and w m = − respectively. Using the same set of values for the free parameters, we have plotted the variations of the fractional energydensities of DE (Ω d ) and matter (Ω m ) and presented them in Figure 2. All the three panels show that CG hasstarted dominating over the matter sector in recent past which has led to the observed late time accelerationof the Universe. It is also evident that the energy density of DE will steadily increase with the evolution ofthe Universe and will lead to complete evaporation of matter in some future epoch. However, there is somepeculiarity in the w m = − case, or in a more general sense, the negative w m case. The energy densities Ω d andΩ m form a ”knot” in some past redshift interval (the position of the knot depends upon the values of the freeparameters chosen) which implies that the CG was the dominant force in the early Universe. However, it is notcorrect to speculate that it could explain the inflationary stage because in that case, q should have shown twotransitions, which is of course not evident from the variation of q in Figure 1. Therefore, we can conclude thatin the presence of an interaction of the form Q = 3 b Hρ d , CG and the barotropic fluid with EoS p m = w m ρ m (cid:0) w m ≥ − (cid:1) can be considered as suitable candidates for DE and matter respectively. In other words, this isa suitable model to explain the medieval deceleration phase as well as the late-time acceleration phase of theUniverse. Nevertheless, this type of interaction does not support future deceleration as has been reported incertain particle creation and backreaction models [51, 52].
4. SHORT DISCUSSION AND SCOPE OF FUTURE WORK
The consequences of considering a particular form of interaction (proportional to the Hubble parameter timesthe DE density) between CG and a barotropic fluid having constant EoS has been discussed. We have obtainedanalytic solutions (in terms of the Hypergeometric F function) for the total energy density and the totalpressure in the presence of such an interaction term. Interacting CG models have occurred in the literature([35] for instance), however, analytic solutions for the continuity equations could not be found assuming aninteraction term proportional to Hubble parameter times the total energy density. Also, all of these modelsconsider only dust as the matter source. Arbitrary choice of the free parameters of our model through trial anderror show that recent observational data strongly favors w m = 0 and w m = − over the w m = case. Thepresent model also shows the transition of DE into the phantom era in the future which is a property sharedby the model of Zhang and Zhu [35]. However, future deceleration is not supported by our model. We reiteratethat the merit of our work is in the analytic solution obtained with our assumption of the form of interaction The fractional energy densities of Ω d and Ω m are given by Ω d = ρ d ρ and Ω m = ρ m ρ respectively. which will provide a deeper picture of the model. This can be achieved by constraining the model parameterswith the help of sophisticated data analysis softwares which can be the basis of a future work. It would also beinteresting to investigate the dynamics of an universe filled with CG and a barotropic fluid and interacting viavarious other forms of the interaction term which occur in the literature. ACKNOWLEDGMENTS
Subhajit Saha was partially supported by SERB, Govt. of India under National Post-doctoral FellowshipScheme [File No. PDF/2015/000906]. Sunandan Gangopadhyay acknowledges the support by DST SERB underStart Up Research Grant (Young Scientist), File No. YSS/2014/000180. [1] E.J. Copeland, M. Sami, and S. Tsujikawa, Int. J. Mod. Phys. D , 1753 (2006), and references therein.[2] L. Amendola and S. Tsujikawa, Dark Energy: Theory and Observations (Cambridge University Press, UK, 2010).[3] A.G. Riess et al., Astron. J. , 1009 (1998).[4] S.J. Perlmutter et al., Astrophys. J. , 565 (1999).[5] N.A. Bachall, J.P. Ostriker, S. Perlmutter, and P.J. Steinhardt, Science , 1481 (1999).[6] A. Kamenshchik, U. Moschella, and V. Pasquier, Phys. Lett. B , 265 (2001).[7] P.J.E. Peebles and B. Ratra, Astron. J. , L17 (1988).[8] B. Ratra and P.J.E. Peebles, Phys. Rev. D , 3406 (1988).[9] B. Carter, Phys. Lett. B , 61 (1989).[10] A. Vilenkin, Phys. Rev. D , 3038 (1991).[11] S. Chaplygin, Uchenie Zapiski Imperatorskogo Moskovskogo Universiteta , 1 (1904).[12] A. Frolov, L. Kofman, and A. Starobinsky, Phys. Lett. B , 8 (2002).[13] D. Bazeia and R. Jackiw, Ann. Phys. (N.Y.) , 246 (1998).[14] M. Bordemann and J. Hoppe, Phys. Lett. B , 315 (1993).[15] M.C. Bento, O. Bertolami, and A.A. Sen, Phys. Lett. B , 172 (2003).[16] J. Hoppe, arXiv: hep-th/9311059.[17] R. Jackiw and A.P. Polychronakos, Phys. Rev. D , 085019 (2000).[18] K. Stanyukovich, Unsteady Motion of Continuos Media (Pergamon, Oxford, 1960).[19] A. Kamenshchik, U. Moschella, and V. Pasquier, Phys. Lett. B , 7 (2000).[20] L. Randall and R. Sundrum, Phys. Rev. Lett. , 4690 (1999).[21] M. Banados, M. Henneaux, C. Teitelboim, and J. Zanelli, Phys. Rev. D , 1506 (1993).[22] N. Bilic, G.B. Tupper, and R. Viollier, arXiv: astro-ph/0207423.[23] R. Sundrum, Phys. Rev. D , 085009 (1999).[24] V. Gorini, A. Kamenshchik, U. Moschella, and V. Pasquier, arXiv: gr-qc/0403062.[25] W. Zimdahl and J.C. Fabris, Class. Quantum Grav. , 4311 (2005).[26] N. Bilic, G.B. Tupper, and R. Viollier, Phys. Lett. B , 17 (2002).[27] L. Amendola, F. Finelli, C. Burigana, and D. Carturan, JCAP , 005 (2003).[28] H.B. Benaoum, arXiv: hep-th/0205140.[29] Z.K. Guo and Y.Z. Zhang, Phys. Lett. B , 326 (2007).[30] M.R. Setare, Phys. Lett. B , 329 (2007).[31] M.R. Setare, Phys. Lett. B , 1 (2007).[32] X.H. Zhai, Y.D. Xu, and X.Z. Li, Int. J. Mod. Phys. D , 1151 (2006).[33] H. Kim, H.W. Lee, and Y.S. Myung, Phys. Lett. B , 605 (2006).[34] B. Wang, Y. Gong, and E. Abdalla, Phys. Lett. B , 141 (2005).[35] H. Zhang and Z.H. Zhu, Phys. Rev. D , 043518 (2006).[36] L. Amendola, Phys. Rev. D , 043511 (2000).[37] L.P. Chimento, A.S. Jakubi, D. Pavon, and W. Zimdahl, Phys. Rev. D , 083513 (2003).[38] G. Olivares, F. Atrio-Barandela, and D. Pavon, Phys. Rev. D , 063523 (2005).[39] G. Mangano, G. Miele, and V. Pettorino, Mod. Phys. Lett. A , 831 (2003).[40] G.R. Farrar and P.J.E. Peebles, Astrophys. J. , 1 (2004).[41] D. Pavon and W. Zimdahl, Phys. Lett. B , 206 (2005).[42] E.W. Weisstein, “Hypergeometric function” (From MathWorld — A Wolfram Web Resource).[43] P.A.R. Ade et al. [Planck Collaboration], Astron. Astrophys. , A13 (2016).[44] M.V. dos Santos, R.R.R. Reis, and I. Waga, JCAP , 066 (2015).[45] S. del Campo, I. Duran, R. Herrera, and D. Pavon, Phys. Rev. D , 083509 (2012).[46] O. Akarsu, T. Dereli, S. Kumar, and L. Xu, Eur. Phys. J. Plus , 22 (2014).[47] R. Nair, S. Jhingan, and D. Jain, JCAP , 018 (2012).[48] A.A. Mamon and S. Das, arXiv: 1610.07337 [gr-qc]. [49] O. Farooq and B. Ratra, Astrophys. J. Lett. , L7 (2013).[50] O. Farooq, F.R. Madiyar, S. Crandall, and B. Ratra, Astrophys. J. , 26 (2017).[51] N. Bose and A.S. Majumdar, Mon. Not. R. Astron. Soc. , L45 (2011).[52] S. Chakraborty, S. Pan, and S. Saha, Phys. Lett. B738