Chaplygin Gas of Tachyon Nature Imposed by Symmetry and Constrained via H(z) Data
aa r X i v : . [ g r- q c ] S e p Chaplygin Gas of Tachyon Nature Imposed by Noether Symmetry and Constrainedvia H ( z ) Data
L. G. Collodel ∗ and G. M. Kremer † Departamento de F´ısica, Universidade Federal do Paran´a, 81531-980 Curitiba, Brazil (Dated:)An action of general form is proposed for a Universe containing matter, radiation and darkenergy. The latter is interpreted as a tachyon field non-minimally coupled to the scalar curvature.The Palatini approach is used when varying the action so the connection is given by a more genericform. Both the self-interaction potential and the non-minimally coupling function are obtainedby constraining the system to present invariability under global point transformation of the fields(Noether Symmetry). The only possible solution is shown to be that of minimal coupling andconstant potential (Chaplygin gas). The behavior of the dynamical properties of the system iscompared to recent observational data, which infers that the tachyon field must indeed be dynamical.
PACS numbers: 98.80.-k, 98.80.jk ∗ [email protected] † kremer@fisica.ufpr.br I. INTRODUCTION
Tachyons have been vastly studied in M/String theories. Since the realization of its condensation proprieties,researchers have gained interest in its applications in cosmology. At first, there was the problem of describing thestring theory tachyon by an effective field theory that would lead to the correct lagrangian in classical gravity. Thefirst classical description of the tachyon field ([1, 2]) addressed the lagrangian problem, making way for building thefirst model within tachyon cosmology ([3]).Being a special kind of a scalar field, it present negative pressure, making the tachyon a natural candidate toexplain dark energy ([4–7]). The inflationary period could also be explained if one considers the inflaton to behave asa tachyon field. Many different attempts were made under this assumption, testing a wide variety of self-interactingpotentials such as power-laws, exponentials and hyperbolic functions of the field ([8–15]). The possible case scenariowhere the tachyon plays both roles, inflaton and dark energy, has also been studied in the works ([16, 17]), where thefirst establishes constraints on the potential so the radiation’s era could commence.The studies above introduced a tachyon field which is minimally coupled through the metric, hence providing justanother source for the gravitational field. Nevertheless, such fields might also be considered to be non-minimallycoupled to the scalar curvature, becoming part of the spacetime geometry by generating a new degree of freedom forgravity. In this scenario, the gravitational constant G becomes a variable function of spacetime.Tachyon fields in the non-minimal coupling context were analyzed for both the inflationary period ([18]) and thecurrent era ([19]). In those cases, the coupling functions and the self-interacting potentials were given in an ad-hoc manner, as exponentials and power-law forms.Every time we choose a different coupling or potential function, we create a new cosmological model, or even a newtheory of gravity in the non-minimal case. This is a very difficult task since the lack of experiments and observationsobligates one to find heuristics arguments to support the choice made. The advantages of searching for symmetries insystems where the lagrangian is known is widely entertained, it not only helps us find exact solutions but might alsogive us physically meaningful constants of movement. What is less appreciated is the fact that one can constrain asystem (one that lacks a closed form of the functional) to present symmetry. In what concerns non-minimally coupledtachyon fields, Noether symmetries were used to establish the coupling and self-interaction functions in the papers([20, 21]). The latter makes use of the Palatini approach, in a way to generalize the theory, since the non-minimalcoupling can provide a metric-independent connection.The Chaplygin gas was first introduced by Chaplygin in 1902 ([22]) in the realms of aerodynamics. This gasfeatures an exotic equation of state ( p c ∝ − /ρ c ), which was originally used to describe the lifting force on a wingof an airplane. Because its pressure is negative, the Chaplygin gas became a good candidate to explain dark energy([23–28]). The attempts to correlate fields and fluids soon showed that the constant potential tachyon field behavesas a Chaplygin gas ([29–32]). Its equation of state allows generalizations, giving rise to the so called GeneralizedChaplygin Gas , or just GCG. This gas exerts a negative pressure proportional in moduli to the inverse of some powerof its energy density and was investigated in works such as ([33–37]), including its relationship to a - now, not constantpotential - tachyon field ([38]). Originally, the equation of state of a Chaplyigin gas was so simple that even with theexhausted studies about the GCG there was still plenty of room for further generalizations. Endowing the EoS with alinear barotropic term, which alone would describe an ordinary fluid, enriched the GCG which under this assumptionis called the
Modified Chaplygin Gas , MCG. Its motivations lie precisily on the possible field nature of the gas ([39]),and its parameters have been constrained via observational analysis ([40]). Further generalizations account for higherorder energy density terms in the EoS of a Chaplygin Gas, the
Extended Chaplyigin Gas , ECG ([41, 42]).In this work, we start from a very general lagrangian for a tachyon field non-minimally coupled to the scalarcurvature. Matter and radiation fields are also included in the system as perfect fluids from the beginning. Theconnection is initially taken to be metric independent and the action is also varied with respect to it, a processknown as the Palatini approach. Since we consider a flat, homogeneous and isotropic Universe, the flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric is used to rewrite our functional in the form of a point-like lagrangian.This presents an extra term than usual, which comes from the independent connection. The system is constrained tothat one which presents invariance under continuous point transformations, or a Noether symmetry. The coupling andself-interaction potential functions of the tachyon field are then determined. Every new field added to the lagrangianclearly influences these point transformations. For that matter, it is important to start off from a complete system(including the radiation fields) if one takes symmetry as a first principle. We show that for this system to be Noethersymmetrical, the non-minimal coupling must vanish and the self-interaction potential must be constant, hence thetachyonic Chaplygin gas. The system is initially composed of five free parameters, namely the Hubble constant, thethree density parameters for recent times and the normalized constant potential. The radiation parameter is thenestablished in a ad-hoc way so we are left with four different free parameters. These are determined via the χ analysisfor the recent H ( z ) data from SNe and gamma-ray bursts. We show that although dark energy tends asymptoticallyto a cosmological constant, any small discrepancies make the tachyon field dynamically active, so the Chaplygin gaspresents property of transition from pressureless matter to dark energy.In order to clarify the typos and notations here used, we remark: the metric signature is (+ , − , − , − ); the Levi-Civita connection is written with a tilde, ˜Γ λµν = (cid:8) λµν (cid:9) while the independent connection is given without it Γ λµν .Natural constants were rescaled to the unity (8 πG = c = 1). Throughout the whole paper, derivatives in equationsare presented as follows: dots represent time derivatives, while ∂ q i ≡ ∂∂q i and ∂ ˙ q i ≡ ∂∂ ˙ q i stand for partial derivativeswith respect to the generalized coordinate q i and velocity ˙ q i , respectively. II. ACTION AND POINT LAGRAGIAN
A generalization of the general theory of relativity is proposed through a non-minimal coupling of a function of thetachyon field. The general action for both geometry and source is written S = Z d x √− gf ( φ ) R − V ( φ ) p − ∂ µ φ∂ µ φ − L s , (1)where φ is the tachyon field, f ( φ ) is the non-minimal coupling function, V ( φ ) is the self-interaction potential and L s is the lagrangian density of other sources (matter and radiation).In order to attain a more general theory we allow the connection to be metric independent. The variation of theaction with respect to the connection Γ ρµν results in the well-known formΓ ρµν = ˜Γ ρµν + 12 f (cid:0) δ ρν ∂ µ f + δ ρµ ∂ ν f − g µν ∂ ρ f (cid:1) . (2)where ˜Γ ρµν is the Levi-Civita connection.Usually, the self-interaction potential and the coupling function are set in a ad-hoc manner. Instead of approachingthe problem this way, we would like to constrain the system to that which has a Noether symmetry. This is done byoperating a variational vector field on the point-like lagrangian, and for this, we need to rewrite it on a specific metric.For a flat, homogeneous and isotropic Universe, spacetime is described by the flat FLRW metric. The point-likefunctional in (1) then becomes L = 6 f (¨ aa + ˙ a a ) − a f ( ∂ φ f ˙ φ ) + 3 a ∂ φ f ˙ φ + 3 a ∂ φ f ¨ φ + 9 ˙ aa ∂ φ f ˙ φ − a V q − ˙ φ − a ρ s , (3)and ρ s is point-like lagrangian for a perfect fluid ([43]).In this system, besides dark energy, the Universe is composed by matter (ordinary and dark) and radiation. Bothdark matter and ordinary matter are treated as dust, hence represented by the same entity here. As the Universeexpands, matter’s density decrease with a − while radiation’s with a − . The lagrangian above contains second-orderterms which are more tedious to deal with. Since the action limits are fixed, we can integrate these terms by parts,without loss of generality, so we can work with a first-order lagrangian, which reads L = − f ˙ a a − a ˙ a∂ φ f ˙ φ − a V q − ˙ φ − a f ( ∂ φ F ˙ φ ) − ρ m − ρ r a , (4)where ρ m and ρ r are the recent values of the total density of matter and radiation, respectively, in the Universe. III. NOETHER SYMMETRIES
Our system may now be constrained to that which is endowed with a Noether symmetry by finding the forms of f ( φ ) and V ( φ ) that allow symmetrical point transformation. This means that our lagrangian shall have such a formthat a specific continuous transformation of the generalized coordinates a → ¯ a and φ → ¯ φ preserves the general formof the functional, L (¯ a, ¯ φ ) = L ( a, φ ) . (5)In order to find the function forms of V ( φ ) and f ( φ ) that allow such transformation, we need to apply a certainvector field on the lagrangian (4). This vector field, X , is then called a variational symmetry, or complete lift , andreads X ≡ α i ∂ q i + ˙ α i ∂ ˙ q i , (6)where the coefficients α i are functions of the generalized coordinates a, φ . The operation of X on the lagrangian issimply the Lie derivative of L along this vector field ( L X L ). According to the Noether theorem, if this derivativevanishes, there will be a conserved quantity named Noether charge . Hence, this will be a variational symmetry if X L = L X L = 0 , (7)such that L ∆ h θ L , X i = 0 , (8)where ∆ = d/dt is the dynamical vector field and θ L = ∂L∂ ˙ q i dq i , (9)is the locally defined Cartan one-form. The brackets represent the scalar product between vector field and one-form,in the Dirac notation. Thus, the Noether charge readsΣ ≡ h θ L , X i = α i ∂L∂ ˙ q i . (10)The condition (7) reads in full form,0 = XL (11)= α∂ a L + β∂ φ L + (cid:16) ˙ a∂ a α + ˙ φ∂ φ α (cid:17) ∂ ˙ a L + (cid:16) ˙ a∂ a β + ˙ φ∂ φ β (cid:17) ∂ ˙ φ L, which for our system becomes0 = α − f ˙ a − a ˙ a∂ φ f ˙ φ − a V q − ˙ φ − a ( ∂ φ f ) ˙ φ f + ρ r a ! + β − ∂ φ f ˙ a a − a ˙ a∂ φ f ˙ φ − a ∂ φ V q − ˙ φ + 3 a ( ∂ φ f ) ˙ φ f − a ∂ φ f ∂ φ f ˙ φ f ! + (cid:16) ∂ a α ˙ a + ∂ φ α ˙ φ (cid:17) (cid:16) − f ˙ aa − a ∂ φ f ˙ φ (cid:17) + (cid:16) ∂ a β ˙ a + ∂ φ β ˙ φ (cid:17) × − a ˙ a∂ φ f + a V ˙ φ q − ˙ φ − a ( ∂ φ f ) ˙ φf , (12)where α = α and β = α .The equation above must hold for any value of ˙ a and ˙ φ . If it were a polynomial equation for these dynamicalvariables one could simply make all coefficients equal to zero, but the different powers of the square roots turn thetask a little more complicated. We shall differentiate with respect to these quantities and evaluate the resultingequations for different values of ˙ a and ˙ φ , then we get the solutions for α ( a, φ ), β ( a, φ ), V ( φ ) and f ( φ ).Making ˙ a = ˙ φ = 0 in (12) we get 3 αa V − α ρ r a + βa ∂ φ V = 0 . (13)Differentiating equation (12) three times with respect to ˙ φ and evaluating at ˙ φ = 0 and ˙ a = 1 gives3 a V ∂ a β = 0 , (14)hence β = β ( a ). Similarly, differantitating (12) once with respect to ˙ φ , multiplying it by (1 − ˙ φ ) / and evaluatingat ˙ φ = 1 and ˙ a = 0 we get V ∂ φ β = 0 , (15)and we conclude that β = β is a constant, since the potential must be non-zero. The fourth derivative of (12) withrespect to ˙ φ , evaluated at ˙ φ = 0 and taking into account that ∂ φ β = 0 leads to9 αa V + 3 βa ∂ φ V = 0 . (16)Since ρ r = 0, dividing (16) by 3 and equating with (13) results in α = 0 and V = V , constant potential. Thus,equation (12) reduces to − ∂ φ f ˙ a a − a ˙ a∂ φ f ˙ φ + 3 a ( ∂ φ f ) ˙ φ f − a ∂ φ f ∂ φ f ˙ φ f = 0 , (17)and it is clear that ∂ φ f = 0. The coupling must then be minimal. IV. EQUATIONS OF MOTION
The lagrangian (4), for constant self-interaction potential and f = 1 / L = − a ˙ a − V a q − ˙ φ − ρ m − ρ r a . (18)The Friedmann equation is obtained through the energy equation E L = ˙ a ∂L∂ ˙ a + ˙ φ ∂L∂ ˙ φ − L , which gives H = 13 ρ, (19)where H = ˙ a/a is the Hubble parameter and ρ = ρ m + ρ r + ρ φ is the total energy density of the fields, being ρ φ = V q − ˙ φ , (20)the energy density of the tachyon field, ρ m = ρ m /a and ρ r = ρ r /a the matter’s and the relativistic material’sdensities, respectively.The Euler-Lagrange equation for the scalar factor, together with (19), provides the acceleration equation, being¨ aa = −
16 ( ρ + 3 p ) , (21)where p = p r + p φ is the pressure of the fields (as usual, matter behaves as dust so p m = 0), and p r = ρ r /
3. Thepressure exerted by the tachyon field is p φ = − V q − ˙ φ . (22)The Euler-Lagrange equation for the tachyon field gives the generalized Klein-Gordon equation for the field, whichis the same as the fluid equation for dark energy when written in terms of its energy density and pressure˙ ρ φ + 3 H ( ρ φ + p φ ) = 0 . (23)An equation of state in the form p ( ρ ) can now be written for the tachyon field. From equation (20) we see that q − ˙ φ = V /ρ φ , which when substituted in (22) yields p φ = − V ρ φ . (24)The Chaplygin gas is a fluid described by an equation of state of the kind p = − Aρ , (25)where A is a positive defined constant, which is precisely the same as (24) for A = V . Thus, as widely know fromthe literature, see e.g. ([29–32]), a tachyon field only minimally coupled to the scalar curvature, of constant potential,behaves as a Chaplygin gas. V. NOETHER CONSTANT
Any lagrangian system endowed with a Noether symmetry will present a constant of motion, as stated by Noether’stheorem. The Noether charge (10) here becomes Σ = α ∂L∂ ˙ a + β ∂L∂ ˙ φ = V a ˙ φ q − ˙ φ , (26)which is simply the first integral of (23). VI. SOLUTIONS
The energy density of the Chaplygin gas, and its pressure, can be rewritten as functions of the scale factor, usingequation (10). These forms are well known from literature and read ρ φ = r Σ a + V ; p φ = − V q Σ a + V , (27)where Σ is the Noether constant. From this equation, we see the dual nature of the Chaplygin gas, which behavesas dust matter for a ≤ ρ φ ∼ Σ a ; p φ ∼ , (28)and as a cosmological constant for a ≥ ρ φ ∼ V ; p φ ∼ − V . (29)In order to obtain our parameters’ curves with respect to the redshift, we use the relationship a = 1 / (1 + z ). TheFriedmann equation (19) then becomes H = 13 (cid:18)q Σ (1 + z ) + V + ρ m (1 + z ) + ρ r (1 + z ) (cid:19) . (30)The equation above can be written in a dimensionless form by dividing it by the Hubble constant, H ≡ H (0) = ρ /
3, where ρ is the current density of all fluids in the Universe, giving H H = (cid:18)q ¯Σ (1 + z ) + ¯ V + Ω m (1 + z ) + Ω r (1 + z ) (cid:19) , (31)where Ω i ≡ ρ i /ρ is the current density parameter of the i -th component. The bars indicate that the constantshave also been divided by the current density, i.e., ¯Σ = Σ /ρ and ¯ V = V /ρ , then the density parameter for theChaplygin gas is simply Ω φ = q ¯Σ + ¯ V . (32)This last relationship allows us to investigate the evolution of the Hubble parameter in terms of dark energy’sdensity parameter, instead of the Noether charge. Finally, we write H = H hq [(Ω φ ) − ¯ V ](1 + z ) + ¯ V + Ω m (1 + z ) + Ω r (1 + z ) i / . (33)Recent observations ([44, 45]) limit the range of values associated to these parameters. In particular, there is greatconfidence that Ω r ∼ , × − , so we may adopt this result but will constrain the four remaining parameters (namely H , Ω m , Ω oφ , and ¯ V ) via H ( z ) data. z H obs σ H ( z ) and their respective errors ([46–50]). Table I presents 25 measurements of the Hubble parameter from SNe and gamma-ray burst ([46–50]). Thefunction built for the Hubble parameter (33) depends on the redshift, plus four different parameters. Hence, H = H ( z, H , Ω m , Ω φ , ¯ V ). The values assumed by these parameters that best fit the observational data are theones the minimize the function χ = X i =1 " H obs ( z i ) − H ( z i , H , Ω m , Ω φ , ¯ V ) σ i . (34)A primary condition for a good fit is that χ /dof ≤
1, where dof stands for degrees of freedom and in this case isgiven by the number of data points, dof = 25. Our minimized χ is given by H = 69 . , Ω m = 0 . , Ω φ =0 . V = 0 . χ = 12 . χ /dof = 0 . H ( z ) does not seem to imposevery strict constraints for our current matter density, but for dark energy we see that within 3- σ all points lie in therange 0 . ≤ Ω φ ≤ . φ and ¯ V is much stronger, as one would expect since thepotential defines the energy density. Nevertheless, it is interesting enough to see the form these ellipses take in Fig.2 whilst the current density parameter for dark energy is given by (32). The case Ω φ = ¯ V is just the cosmologicalconstant scenario. From the figure, we see very thin ellipses with a slope close to the unity. The best fit parameterslisted above show a very small difference between the two of them, and as we will see this difference grows bigger inthe past, but there is a high tendency for the cosmological constant.The evolution of the density parameters for different redshift scales are shown below. In Fig. 3 radiation is neglectedfor its energy density is too small to be observed. As the redshift increases dark energy falls but ever more slowly,and for values z ≥ W m W Φ FIG. 1. Confidence intervals for 1-,2- and 3- σ for the density parameters Ω m and Ω φ .FIG. 2. Confidence intervals for 1-,2- and 3- σ for the density parameter Ω φ and the constant potential ¯ V . difference between Ω φ and ¯ V that makes the tachyon field dynamical and the Chaplygin gas property thrive, fromequations (27) and (28) it becomes clear that dark energy decays into matter fields as the redshift grows and the term¯Σ outpaces ¯ V . Furthermore, we are now looking at the matter era, hence the almost constat behavior. In Fig. 4we see the evolution of the density parameters for radiation and the combination of the Chaplygin gas and matter,since the first behaves as the latter. In this scenario, the densities equality happen a bit earlier in out history thanexpected from the cosmological constant case. For instance, we have z eq = 3968 .
15 (approximately 37.5 thousandsyears since the beginning of the Universe) whereas for a non-dynamical field describing we would expect z eq ∼ ω φ = p φ /ρ φ , between dark energy’s pressure and energy density is show in Fig. 5. As expected fromequation (27), the ratio tends asymptotically to − q is plotted in Fig. 6. The transition from a decelerated to an accelerated expansionhappens at z t = 0 .
65, while for our current time q = − .
56, both results in agreement with observations ([53]).Although the Chaplygin gas has been extensively studied before, new observational data provides great motivationto revisit the model and set new constraints. The evolution of the Hubble parameter described by this model,together with the data we used to define our parameters is shown in Fig. 7. Unfortunately, there are not satisfactorily W i W m W Φ FIG. 3. The dotted lines stand for dark energy while the solid lines represent dark matter. As we enter the matter dominatedera, dark energy decays into matter fields contributing even more for its dominance, with its energy density falling ever moreslowly, assuming an almost constant behavior. W i W r W m + W Φ FIG. 4. Density parameters plotted for high redshift values. The dashed line represent the matter fields, where the Chaplygingas is included once it behaves as dust at this point. Solid line stand for radiation’s energy density parameter. Equality indensities happen at z = 3968 . - - - - - Ω Φ FIG. 5. Ratio between pressure and energy density for the Chaplygin gas. Any small difference between Ω φ and ¯ V growsconsiderably with the redshift and dark energy eventually becomes a pressureless field, hence matter. many measurements to make solid statistics for this parameter as there are for the distance modulus, for instance.Also, the errors associated with the data from gamma-ray bursts are much bigger than one would desire them to be.Nevertheless, these sources provide information from a much younger Universe compared to SNe, making it worthwhilefor testing models and constraining parameters. VII. CONCLUSIONS
In this work, we started from a general action where a tachyon field represents the nature of dark energy. We allowedit to be non-minimally coupled to the scalar curvature and we considered the connection to be independent throughthe Palatini approach. Dark matter, baryonic matter and relativistic material were included in source fields, as ourintention was to build a more complete model. Instead of establishing the self-interaction potential and the couplingfunction in a ad-hoc manner, we stated that symmetry should play a more primary role and only functions capable0 - - FIG. 6. Deceleration parameter. Expansion becomes accelerated at z t = 0 .
65 and the current value of this parameters standsat q = − . H z L FIG. 7. The observational data for the Hubble parameter ([46–50]) favor the dynamical field over the cosmological constant.The more dynamical the tachyon field is, greater is the inclination of the H ( z ) curve. Even with such big error bars, we canidentify the case ¯ V = 0 .
69 as being the one that accommodates the more data. of composing a continuous and symmetric point transformation on the generalized coordinates would be considered.This lead to the simpler case where the tachyon field is only minimally coupled and its potential is constant, behavingas a Chaplygin gas.The theoretical framework of the Chaplygin gas has been deeply investigated and is widely found in the literature.For this reason, we focused on more recent observational data to constrain the dynamics of the system. The Hubbleparameter suggests that, be the Chaplygin gas the underlying nature of dark energy, it shall be slightly dynamical,as opposed to its cosmological constant particular case since, as we see, any small difference between its constantpotential and current density parameter grows considerably with the redshift. As this component presents a dualbehavior, acting as dark energy for small redshifts and decaying into matter fields later on, the matter era beginsearlier in the history of the Universe, what could help explain older structures.
VIII. ACKNOWLEDGMENTS
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