Non-minimal kinetic coupling and Chaplygin gas cosmology
NNon-minimal kinetic coupling and Chaplygin gascosmology
L. N. Granda , ∗ , E. Torrente-Luj´an , † and J. J. Fernandez-Melgarejo , ‡ Departamento de F´ısica, Universidad del Valle, 25360 Cali Colombia Departamento de F´ısica, Universidad de Murcia , Campus Espinardo, E-30100 Murcia Spain
Abstract
In the frame of the scalar field model with non minimal kinetic coupling togravity, we study the cosmological solutions of the Chaplygin gas model ofdark energy. By appropriately restricting the potential, we found the scalarfield, the potential and coupling giving rise to the Chaplygin gas solution. Ex-tensions to the generalized and modified Chaplygin gas have been made.
A wide range of cosmological observations indicate that the universe has entered aphase of accelerating expansion, which becomes one of the important puzzles of thecontemporary physics. Those observations include the type Ia supernovae (SnIa)standard candles [1], [2], the angular location of the first peak in the CMB powerspectrum [3] and baryon acoustic oscillations of the matter density power spectrum ∗ [email protected], [email protected] † [email protected], [email protected] ‡ [email protected] a r X i v : . [ h e p - t h ] J un N = 4 conformal supergravity [19, 20]. A model with non-minimal deriva-tive couplings was proposed in [21], [22], [23] in the context of inflationary cosmology,and recently, non-minimal derivative coupling of the Higgs field was considered in[24], also as inflationary model. In [25] a derivative coupling to Ricci tensor has beenconsidered to study cosmological restrictions on the coupling parameter, and the role2f this coupling during inflation. Some asymptotical solutions for a non-minimal ki-netic coupling to scalar and Ricci curvatures were found in [26], and quintessenceand phantom cosmological scenarios with non-minimal derivative coupling have beenstudied in [27]. A scalar field with kinetic term coupled to a product of Einsteintensors has been considered in [28]. Non-minimal coupling of scalar fields (includingkinetic terms) with modified f ( R ) theories have been also considered to solve the DEproblem in [29], [30], [31], [32], [33].In this paper we consider an explicit coupling between the scalar field the kinetic termand the curvature [34, 35, 36], as the source of DE and will establish the connectionwith the Chaplygin gas, by obtaining the solution of the field equations that repro-duce the cosmological evolution as given by the perfect fluid obeying the equationof state of the Chaplygin gas [16]. The Chaplygin gas has attracted much attentionin cosmology, as it allows to interpolate between a dust dominated phase of the evo-lution of the Universe in the past, and an accelerated one at recent time. On theother hand, as in the case of the Chaplygin gas, the scalar field with the non-minimalkinetic coupling to the curvature, has been shown to be successful in the descriptionof the DM and DE without introducing separately the DM term (see [35, 36]), i.e.it describes the DM and DE as manifestations of a common scalar field at differentepochs. For this reason, it would be interesting to analyze the connection of thescalar field with non-minimal kinetic couplings with Chaplygin gas, which will be thesubject of study in the present paper. A potential and coupling function that maygive a dynamical description to the Chaplygin gas, have been found. The generalizedand modified version of the Chaplygin gas have been also considered. The scalar field with kinetic couplings to curvature is given by [35]: S = (cid:90) d x √− g (cid:104) πG R − ∂ µ φ∂ µ φ − ξR ( F ( φ ) ∂ µ φ∂ µ φ ) − ηR µν ( F ( φ ) ∂ µ φ∂ ν φ ) − V ( φ ) (cid:105) . (2.1)3nd we will use the flat FRW background metric given by the line element ds = − dt + a ( t ) (cid:2) dr + r (cid:0) dθ + sin θdφ (cid:1)(cid:3) (2.2)where a is the scale parameter. The dimensionality of the coupling constants ξ and η depends on the type of function F ( φ ). Taking the variation of action (2.1) withrespect to the metric, we obtain a general expression of the form R µν − g µν R = κ T µν (2.3)where κ = 8 πG and the tensor T µν represents the variation of the terms whichdepend on the scalar field φ and can be written as T µν = T φµν + T ξµν + T ηµν (2.4)where T φµν , T ξµν , T ηµν correspond to the variations of the minimally coupled terms, the ξ and the η couplings respectively. Due to the interaction between the scalar fieldand the curvature, the derived expressions for the density and pressure for the scalarfield can be regarded as effective ones. From now on, and in order to simplify theequations (the field equations will contain only second order derivatives) we will usethe restriction η = − ξ , which is equivalent to a coupling of the kinetic term to theEinstein tensor G µν (see [22], [23]). Evaluating the 00 and 11 components of the Eq.(2.3) in the spatially-flat Friedmann-Robertson-Walker (FRW) background (2.2), it isobtained (with the Hubble parameter H = ˙ a/a , and for homogeneous time-dependingscalar field) H = κ (cid:20)
12 ˙ φ + 9 ξH F ( φ ) ˙ φ + V ( φ ) (cid:21) (2.5)and − H − H = κ (cid:20)
12 ˙ φ − V ( φ ) − ξ (cid:16) H + 2 ˙ H (cid:17) F ( φ ) ˙ φ − ξH (cid:18) F ( φ ) ˙ φ ¨ φ + dFdφ ˙ φ (cid:19)(cid:21) (2.6)where “dot” represents the derivative with respect to the cosmological time t . Takingvariation in (2.1) with respect to the scalar field in the FRW background, gives theequation of motion as follows¨ φ + 3 H ˙ φ + dVdφ + 3 ξH (cid:18) F ( φ ) ¨ φ + dFdφ ˙ φ (cid:19) + 18 ξH F ( φ ) ˙ φ + 12 ξH ˙ HF ( φ ) ˙ φ = 0 (2.7)4here the first three terms describe the minimally coupled field. In what follows wewill study cosmological solutions to Eqs. (2.6) and (2.7) giving rise to acceleratedexpansion, and according to the cosmological scenario described by the Chapkygingas solutions.The Chaplygin gas is described by the following equation of state (EoS) p = − Aρ (2.8)where p and ρ are respectively the pressure and density, and B is a positive constant.The continuity equation takes the form˙ ρ + 3 H (cid:18) ρ − Aρ (cid:19) = 0 (2.9)This equation can be easily integrated in the variable x = log a , yielding ρ = (cid:2) A + Be − x (cid:3) / (2.10)which according to the Friedmann equation gives the following Hubble parameter H = κ (cid:2) A + Be − x (cid:3) / = κ (cid:20) A + Ba (cid:21) / (2.11)This solution has the known advantages of describing the presureless matter domi-nance stage (epoch) of the universe at early times ( a <<
1, normalizing the currentvalue of a to 1), and the future universe dominated by the cosmological constant,entering in a de Sitter phase at a >>
1. In the next section we will consider thissolution to integrate the equations (2.5) and (2.7) with respect to the scalar field φ and the coupling F , and in this manner we will obtain a description of the Chaplygingas cosmology in the frame of the scalar field with non-minimal kinetic coupling tocurvature. Then, we will follow the same procedure with the generalized and modifiedChaplygin gas models. In order to integrate the Eqs. (2.5) and (2.7) for a given Hubble parameter, we shouldimpose additional restrictions on the scalar field potential in order to consistently find5he rest of the variables, as follows (see [35]).In terms of the variable x = log a , and defining the function θ ( x ) = φ (cid:48) , the Eq. (2.7)can be written as (after multiplying by ˙ φ )12 ddx (cid:0) H θ (cid:1) + 3 H θ + dVdx + 9 ξH dH dx F θ + 3 ξH ddx ( F θ ) + 18 ξH F θ = 0 (3.1)From Eq. (2.5), changing to the variable x , we can write the product F φ (cid:48) = F θ asfollowing
F θ = 13 ξκ H − θ ξH − V ξH (3.2)taking the derivative of Eq. (3.2) and replacing F θ and d ( F θ ) /dx into Eq. (3.1), wearrive at the following equation involving θ , H and V H dθdx + H (cid:18) H + dH dx (cid:19) θ + 4 H dVdx − (cid:18) H + dH dx (cid:19) V + 12 H κ (cid:18) H + dH dx (cid:19) = 0 (3.3)In this manner, we obtain a first order differential equation for the functions θ , H and V . In order to integrate the equation (3.3), and thanks to the fact that the functions θ and V are separated, we can impose a restriction on the scalar field potential givenby the equation2 H dVdx − (cid:18) H + dH dx (cid:19) V + 6 H κ (cid:18) H + dH dx (cid:19) = 0 (3.4)which simplifies the Eq. (3.3):2 H dθdx + (cid:18) H + dH dx (cid:19) θ = 0 (3.5) The Chaplygin gas solution
In order to consistently solve the Eqs. (3.4) and (3.5), we propose the expressionfor the Hubble parameter H , given by that of the Chaplygin gas (2.11) H = κ (cid:2) A + Be − x (cid:3) / (3.6)6efining the scaled Hubble parameter ˜ H = H/H , Eq. (3.6) can be written as˜ H = (cid:104) ˜ A + ˜ Be − x (cid:105) / (3.7)where the parameters ˜ A and ˜ B are now dimensionless and are given by˜ A = (cid:18) κ H (cid:19) A, ˜ B = (cid:18) κ H (cid:19) B. (3.8)where ˜ A and ˜ B satisfy the flatness condition (considering the Chaplygin gas domi-nance) ˜ A + ˜ B = 1 (3.9)Replacing ˜ H in (3.4), and defining the dimensionless scalar potential ˜ V = κ V /H ,after changing the Eq. (3.4) to the “tilde” variables and integration we obtain thescalar field potential˜ V ( x ) = Ce x (cid:16) ˜ A + ˜ Be − x (cid:17) / − A ˜ B e x (cid:16) ˜ A + ˜ Be − x (cid:17) / F (cid:34) , , , − ˜ Ae x ˜ B (cid:35) (3.10)where C is the integration constant. Replacing ˜ H in (3.5) we get the followingexpression for θ θ ( x ) = φ (cid:48) = θ e − x (cid:16) ˜ A + ˜ Be − x (cid:17) / , (3.11)where θ is the integration constant. Integrating the square root of this last equation,we obtain the scalar field as (considering the ( − ) sign root) φ ( x ) = 4 θ / e − x/ B / F (cid:34) , − , , − ˜ Ae x ˜ B (cid:35) (3.12)Finally, the coupling function F is found by replacing the Eqs. (3.7)-(3.11) in theFriedmann Eq. (3.2), giving the result F ( x ) = e x g ( x ) − / ξκ H θ (cid:104) g ( x ) / − κ θ e − x − C e x + 2 ˜ A ˜ B e x g ( x ) / F (cid:34) , , , − ˜ Ae x ˜ B (cid:35) (cid:105) (3.13)where g ( x ) = ˜ A + ˜ Be − x . Although we can not have an analytic expression for thepotential in terms of the scalar field, we can illustrate the behavior of the potential as7howed in fig.1, for a given set of parameters. The constant C is selected so that thetime variation of the gravitational coupling does not exceed the observational limits(see below). As can be seen from Fig. 1, for the selected values of the parameters,the potential is a monotonic decreasing function of the scalar field. The decreasingrunaway behavior of the potential describing dark energy, is a key fact for a realisticcosmological model [37], [38]. Φ (cid:144) M p V (cid:72) Φ (cid:76) (cid:144) M p H Fig. 1
The evolution of the potential for the Chaplygin gas solution with the scalarfield for ˜ A = 0 . , C = 11 . The generalized Chaplygin gas solution
A generalization of the Chaplygin gas has also been considered to describe the darkmatter and dark energy. It’s equation of state is given by p G = − Aρ αG (3.14)where ρ G and p G are the energy density and pressure of the generalized Chaplygingas, A is a positive constant and α is considered to lie in the range 0 < α ≤
1, whichguarantees the stability and causality (see below) [15], [16], [39]. Note that α = 1corresponds to the original Chaplygin gas. The EoS (3.14) has an equivalent fieldtheory representation in a generalization of the Born-Infeld theory [39] (in the scalarfield representation used in [39], the Born-Infeld Lagrangian density is reproduced for α = 1). solving the continuity equation ( ˙ ρ G + 3 H ( ρ G + p G ) = 0), leads to the energydensity in terms of x = log a ρ G = (cid:0) A + Be − α +1) x (cid:1) α (3.15)8here B is a positive integration constant. This density for α > − a << a >> α →
0, Eq. (3.15) reproducesthe ΛCDM model. This model also captures the attention because of it’s connectionwith string theory and supersymmetry: the EoS (3.14) can be obtained in the Nambu-Goto action for d -branes moving in a ( d + 2) dimensional space time [40], and hassupersymmetric generalization [41]. As a criteria to constraint the constant α we canuse the sound speed for the fluid described by Eq. (3.14), given by c s = dp G dρ G = αAA + Be − α ) x = αAA + B (1 + z ) α ) (3.16)where the last is written in terms of the redshift z ( e − x = (1 + z )). At future z → − c s → α . Therefore, stability requires α >
0, and causalityrequires α < α to the interval0 ≤ α ≤ α at high redshift, during thematter dominated epoch). However some authors have considered the “forbidden”region α > c s is treated as group velocity [42], (see also [43] forsuperluminal sound speed). According to some observational studies, for the case ofthe pure GCG the α < − values are favored, which is very close to the ΛCDM limit[44]. This tight restriction may be increased if we consider additionally the barionicmatter component ( α < − ) [45], or adding cold dark matter and barionic mattercomponents ( α < .
2) [46]. On the other hand, based on observations of the barionicpower spectrum, the GCG plus barion matter is favored for α ≥ H = (cid:104) ˜ A + ˜ Be − α +1) x (cid:105) α (3.17)where the dimensionless ˜ A and ˜ B are given by˜ A = (cid:18) κ H (cid:19) α A, ˜ B = (cid:18) κ H (cid:19) α B (3.18)and satisfy the flatness condition (for pure generalized Chaplygin gas content) ˜ A + ˜ B =1. Replacing (3.17) in (3.4) and solving the Eq. (3.4) in “tilde” variables we obtain9he following solution for the scalar potential˜ V ( x ) = Ce x (cid:16) ˜ A + ˜ Be − α ) x (cid:17) α ) − A (2 α −
1) ˜
B e α ) x (cid:16) ˜ A + ˜ Be − α ) x (cid:17) α F (cid:34) , α α , − α ) , − ˜ Ae α ) x ˜ B (cid:35) (3.19)where C is the integration constant. Solving (3.5) with ˜ H given by (3.17), we findthe expression for θ θ ( x ) = φ (cid:48) = θ e − x (cid:16) ˜ A + ˜ Be − α ) x (cid:17) α ) , (3.20)where θ is the integration constant. Integrating the square root of (3.20) it follows φ ( x ) = 4 θ / e − x/ B α ) F (cid:34) α ) , − α ) , α α ) , − ˜ Ae x ˜ B (cid:35) (3.21)And the corresponding coupling function, as follows from (3.2) and (3.17-3.20) is F ( x ) = e x g α ( x ) − ξκ H θ (cid:104) g α ( x ) / − κ θ e − x − C e x +2 ˜ A (2 α −
1) ˜
B e α ) x g α ( x ) / F (cid:34) , α α , − α ) , − ˜ Ae α ) x ˜ B (cid:35) (cid:105) (3.22)where g α ( x ) = (cid:16) ˜ A + ˜ Be − α ) x (cid:17) α . In all equations the dependence on the redshift z or in the scale factor a is obtained by replacing e − x = 1 + z = a − . In fig.2 weplot the behavior of the potential in terms of the scalar field for two values of α corresponding to the “physical” region 0 ≤ α ≤
1, and the superluminal region α ∼ Φ (cid:144) M p V (cid:72) Φ (cid:76) (cid:144) M p H Fig. 2
The potential for the GCG versus the scalar field for ˜ A = 0 . , α = 0 . , C = 25 . (dashed), and ˜ A = 0 . , α = 3 , C = 3 . . Note thedecreasing behavior, which is an important characteristic for dark energy potentials The scalar field potentials for the non-minimally coupled scalar field, that reproducethe dynamics of the Chaplygin and generalized Chaplygin gas are decreasing functionsof the scalar field. Note the runaway behavior for all curves, which are characteristicof well behaved dark energy potentials [37], [38]. The parameters are chosen in sucha way that V ( φ ) is definite positive, at least for z > −
1. For other choices of theparameters, it can be shown numerically that the curves can have one maximum. C is used to accomplish the observational restrictions on the time variation of thegravitational coupling.So, we have reconstructed the scalar model with kinetic couplings to curvature (2.1),for a given Hubble parameter describing the CG and GCG cosmologies. In both caseswe exploited the additional degree of freedom represented in the coupling function,to constraint the scalar potential in a way consistent with the Friedmann equations.A more general formulation of cosmological reconstruction method (in time and x variables) for a number of modified gravities including scalar tensor theories, havebeen performed in [47], [48]. In the scalar tensor theories considered in [47], [48], thescalar potential and couplings are reconstructed by using an apropriate redefinitionof the scalar field, and giving the particular type of cosmological evolution encodedin H . The reconstruction was considered in the cosmological time and the e-foldingvariable x , and concrete examples of accelerated late time cosmologies have been11rovided. In this works for the case of f ( R ) gravity, the reconstruction was achievedby introducing an auxiliary scalar field. The modified Chaplygin gas solution
The modified Chaplygin gas (MCG) is defined for the equation of state p = Bρ − Aρ α (3.23)A generalized version of an equation of state that includes CG, GCG and MCGhave been considered in [49] and [50]. Integrating the continuity equation gives thefollowing energy density ρ = (cid:18) A B + Ce − βx (cid:19) α = (cid:18) AB + 1 + Ca β (cid:19) α (3.24)where β = 3(1 + B )(1 + α ) and C is the integration constant. This equation hasthe same functional dependence on x or a , as the case of the GCG, except that ad-ditional limits can be obtained due to additional factor (1 + B ) in the exponent ofEq. (3.24), with respect to the corresponding one in Eq. (3.15). Therefore, the re-sulting potential, scalar field and coupling function corresponding to the MCG showsthe same dependence as given in Eqs. (3.19-3.22), and it can be shown that all theequations of the MCG, at B = 0 become the corresponding equations for the GCG.Additionally to the GCG, the MCG model reproduces the radiation dominated phaseof the universe at high redshift, as can be seen from (3.24) for B = 1 / a << ρ ∼ C / (1+ α ) a − B ) ) at earlier epochs when theconstant term may be neglected. Resumming, the non-minimally coupled scalar fieldalso reproduces the dynamics of the modified Chaplygin gas cosmology, with wellbehaved scalar field, potential and coupling function. The time variation of the gravitational coupling
We can meet the constraints on the current value and the time variation of thegravitational coupling [51], by appropriately defining or constraining the constants C and θ . The effective gravitational coupling from (2.5) is given by G eff = G − ξκ F H θ (3.25)12here we used κ = 8 πG and ˙ φ = H φ (cid:48) H θ . In terms of x , the time variationof the gravitational coupling can be written as˙ G eff G eff = 3 ξκ − ξκ F H θ ddx ( F H θ ) H. (3.26)Replacing the product F H θ from Eq. (3.2), and evaluating at the present time( x = 0), the Eq. (3.26) can be written as˙ G eff G eff (cid:12)(cid:12)(cid:12) x =0 = 3 f ( C, θ )1 − g ( C, θ ) H (3.27)where f ( C, θ ) = ξκ d ( F H θ ) /dx , g ( C, θ ) = ξκ F H θ valuated at x = 0 and theparameters of the model appearing in ˜ H have been fixed, so that the resulting ex-pression (3.27) depends on the constants of integration C and θ . We can meet therestrictions imposed by the current observations on the value and the time varia-tion of the gravitational coupling [51], if f ( C, θ ) and g ( C, θ ) satisfy the constraints: f ( C, θ ) ≈ g ( C, θ ) ≈ f ( C, θ ) ≤ − and g ( C, θ ) ≤ − ). Thus, for the Chaplygin gas solution given by (3.6), (3.10-3.13)with ˜ A = 0 . B = 0 . C ≈
11 and κ θ ≈ . C ≈ . κ θ ≈ . A = 0 . B = 0 . α = 0 .
6) and C ≈ . κ θ ≈ . A , ˜ B ) and α = 3, with therespective potentials plotted in fig. 2. The cosmological implications of the Chaplygin gas model have been intensively in-vestigated in recent literature. We considered the model of scalar field with kineticterms coupled non-minimally to the scalar field and to the curvature, to give a dynam-ical description of the Chaplygin gas model of dark energy. We have found analyticalexpressions for the reconstructed scalar field and potentials that describe the stan-dard, the generalized and modified Chaplygin gas models of dark energy dark materunification. Thanks to the presence of the coupling function F ( φ ), we could impose a13estriction on the potential through Eq. (3.4), which allowed us to find the solutionsthat lead to dynamical description of the Chaplygin gas cosmology. The results showthat the obtained potentials V ( φ ) decrease with the evolving scalar field φ . FromEqs. (3.10,3.12) it follows that φ is an increasing function of the redshift and V isa decreasing function of the redshift, which means that the scalar potential is a de-creasing function of the scalar field. In Fig 1. we show the V ( φ ) dependence for thestandard Chaplygin gas in the redshift interval [0 , C and θ , in order to control the actualvalue of G and it’s time variation. These conditions can be satisfied by imposing theinequalities f ( C, θ ) ≤ − and g ( C, θ ) ≤ − .The above results show that the scalar field model with derivative couplings to cur-vature considered here, provide a dynamical scenario to describe the Chaplygin gascosmology. The wide variety of phenomenologically acceptable solutions [34, 35, 36],support the capability of this model to explain the current status of the acceleratedexpansion of the universe, through different cosmological scenarios. Acknowledgments
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