Non-minimally coupled tachyon field with Noether symmetry under the Palatini approach
aa r X i v : . [ g r- q c ] N ov Non-minimally coupled tachyon field with Noether symmetryunder the Palatini approach
Lucas G. Collodel a) and Gilberto M. Kremer b) Departamento de F´ısica, Universidade Federal do Paran´a, 81531-980 Curitiba,Brazil
A model for a homogeneous, isotropic, flat Universe composed by dark energy and matter is investigated. Darkenergy is considered to behave as a tachyon field, which is non-minimally coupled to gravity. The connectionis treated as metric independent when varying the action, providing an extra term to the Lagrangian density.The self-interaction potential and coupling are naturally found by imposing a Noether symmetry to the system.We analyze the evolution of the density parameters and we compare the results obtained for the decelerationparameter, luminosity distance and Hubble parameter with those found in literature from observational data.PACS numbers: 98.80.-k, 95.35.+d, 95.36.+x
I. INTRODUCTION
Dark energy has played a central role in cosmolog-ical researches ever since 1998 with the discovery ofthe currently accelerated expansion of our Universe (seee.g. ). New theories containing dark components, at-tempt not only to explain this expansion’s feature butalso how structures are formed through the evolutionof anisotropies, the age of the Universe, the flatnessproblem and so on. Many different approaches to de-scribe the nature of dark energy have been made withinthe last fifteen years, among the most popular ones arethe cosmological constant, scalar fields, fermionic fields,aether fields and possibly tachyon fields, a special kindof scalar field with its grounds in string theory, butwhich can be easily generalized within the framework ofclassical gravity. Tachyon fields have also been consid-ered to be the inflaton in the early stages of our Uni-verse. The papers consider the minimally cou-pled tachyon field to behave as dark energy, whereas forthe inflaton, minimally coupled tachyon fields were stud-ied with a great variety of self-interaction potentials suchas power-laws, exponential, hyperbolic functions of thefield (e.g. ). In the papers the tachyonfield was considered to be the responsible for both infla-tionary period and the currently accelerated expansion.Constraints on the behaviour of the potential were de-veloped in the work , where it is shown that potentialspresenting V ( φ → ∞ ) → to explain the inflationaryperiod where the potential was given in the exponen-tial form and the coupling by a power series of the field.In the work it is also described the nature of dark en-ergy with potential and coupling in the form of power-lawfunctions. Also within this context, a derivative coupling a) Electronic mail: [email protected] b) Electronic mail: kremer@fisica.ufpr.br was analyzed in the paper. .All the works listed above constrained the dynam-ics to the potentials and couplings inserted into thesystems in an ad-hoc way. The Noether symmetryoffers great advantage in this matter, as one con-strains the solutions only to be compatible with sym-metries and the functions are obtained naturally. In theworks , the approach of symmetrywas used to construct different models concerning f ( R )gravity, scalar field theories, fermionic fields and finally,the latter one treats a non-minimal coupling between thetachyon field and gravity with Noether symmetry.In this paper we intend to give a Palatini treatment tothe symmetrical constrained coupled tachyon field. Al-though the scientific community seem to have lost in-terest in the Palatini approach, the idea of consideringa metric independent connection beautifully generalizesthe model as one makes no restrictions when varying theaction with respect to the dynamical variables. We con-sider a homogeneous, isotropic flat expanding Universe,described by the Friedmann-Lemaˆıtre-Robertson-Walker(FLRW) metric and composed of dark energy and pres-sureless matter (both baryonic and dark). Dark energyis described as a tachyon field, which is coupled to thecurvature scalar. We narrow the solutions of our systemby imposing a compatibility with the Noether symme-try, that naturally provides which sample of functions isallowed for the self-interaction potential of the field, aswell as for the coupling. Also, when varying the action,we consider the connection to be metric independent, indoing so, our point-like Lagrangian is granted an extraterm. The model reproduces satisfactorily the recent be-haviour of the Universe when compared to observationaldata.The metric adopted is the FLRW for the flat spacewith signature (+ , − , − , − ). The Levi-Civita connectionis written with a tilde, ˜Γ λµν = (cid:8) λµν (cid:9) while the independentconnection is given without it Γ λµν . Natural constantswere rescaled to the unity (8 πG = c = 1). Throughoutthe whole paper, derivative in equations are presented asfollows: dots represent time derivatives, while ∂ q i ≡ ∂∂q i and ∂ ˙ q i ≡ ∂∂ ˙ q i stand for partial derivatives with respect tothe generalized coordinate q i and velocity ˙ q i , respectively.Furthermore, as reminded in section V, primes denotedifferentiation with respect to redshift z . II. ACTION
We consider a Universe composed by both (pressure-less) ordinary and dark matter and dark energy, whichis here described as a tachyon field ( φ ) that is non-minimally coupled to the curvature. The action is writtenas S = Z d x √− g h F ( φ ) R − V ( φ ) p − ∂ µ φ∂ µ φ − L m i . (1)where F ( φ ) is the coupling, V ( φ ) is the self-interactionpotential, L m is the matter field’s Lagrangian density and R is the scalar curvature (given in terms of the indepen-dent connection Γ λµν ). Under the Palatini approach, themetric and the affine connection are taken initially to beindependent dynamical variables, meaning that the ac-tion’s variation shall be done separately with respect tothese two quantities. The variation with respect to theconnection yields the well known expression,Γ ρµν = ˜Γ ρµν + 12 F (cid:0) δ ρν ∂ µ F + δ ρµ ∂ ν F − g µν ∂ ρ F (cid:1) . (2) III. EQUATIONS OF MOTION
We rewrite the Lagrangian density on the spatiallyflat FLRW metric and, after integrating the second or-der terms by parts, we get the first order point-like La-grangian density L = − F ˙ a a − a ˙ a∂ φ F ˙ φ − a V q − ˙ φ − a F ( ∂ φ F ˙ φ ) − ρ m . (3)where a is the scale factor and ρ m is a constant andrepresents the current energy density of the matter fields.Comparing this Lagrangian density with the one ob-tained by the metric approach , we notice an extra term a F ( ∂ φ F ˙ φ ) . The Friedmann equation is then obtainedthrough the energy equation, E L = ˙ a∂ ˙ a L + ˙ φ∂ ˙ φ L − L = 0 , (4)and reads H = (cid:18) ˙ aa (cid:19) = ρ F . (5)Here H is the Hubble parameter and ρ = ρ φ + ρ m the total energy density. Furthermore, ρ φ = V q − ˙ φ − H∂ φ F ˙ φ − (cid:16) ∂ φ F ˙ φ (cid:17) F ; ρ m = ρ m a , (6)denote the tachyon field’s and matter’s energy density, re-spectively. We may now apply the Euler-Lagrange equa-tions for both degrees of freedom ( a, φ ). From the scalefactor variation, the acceleration equation follows in theform ¨ aa = − F ( ρ + 3 p φ ) , (7)where p φ = − V q − ˙ φ − (cid:16) ∂ φ F ˙ φ (cid:17) F + 2 ∂ φ F ˙ φ + 2 ∂ φ F ¨ φ + 4 H∂ φ F ˙ φ (8)is the tachyon field’s pressure. Similarly, the Euler-Lagrange equation is applied to the tachyon field, leadingto the Klein-Gordon equation, which reads¨ φ V (1 − ˙ φ ) / − ∂ φ F ) F ! + ˙ φ
32 ( ∂ φ F ) F − ∂ φ F ∂ φ FF ! + ˙ φ ˙ aa V q − ˙ φ − ∂ φ F ) F − ∂ φ F (cid:18) ¨ aa + ˙ a a (cid:19) + ∂ φ V q − ˙ φ = 0 . (9) IV. NOETHER SYMMETRY
The coupling and the self-interaction potential haveyet to be specified. It is clear that by choosing them, onenarrows the solutions for the equations of motions, asthe dynamical system becomes more restricted. Such achoice cannot be made arbitrarily, and when setting thefunctions in an ad-hoc manner, it shall be substantiatedby reasonable arguments, often found analyzing observa-tional data. The coupling cannot vary harshly for exam-ple, or there would be an over/under production of Heat the time of nucleosynthesis . As for the poten-tial, most of them are motivated on the grounds of quan-tum field theory. When applying the Noether symmetryon the other hand, one gets the coupling and potentialfunctions naturally, without having to make any extraimpositions about them. Of course it does not make themodel more general, but it restricts the solutions to thosethat exhibit symmetry. If a dynamical system is symme-try compatible, there will always be a conserved quantity,also called constant of motion or Noether charge. Thereis a special class of vector fields that perform a variationalsymmetry, and for that reason they are often called gen-erators of symmetry (or complete lift). This vector fieldis written as X ≡ n X i α i ∂∂q i + (cid:18) ddλ α i (cid:19) ∂∂ ˙ q i , (10)where the coefficients α i are functions of the the gener-alized coordinates ( a , φ ), and λ is the independent vari-able, thus representing the time in our system. Accordingto the Noether theorem, if the Lie derivative of the La-grangian density along X vanishes, the system carries aconserved quantity. Mathematically speaking X L = L X L = 0 → L ∆ h θ L , X i = 0 , (11)where ∆ = d/dt is the dynamical vector field and θ L = ∂ L ∂ ˙ q j dq j (12)is a Cartan one-form defined locally. It is then clear thatthe constant of movement is giving by the inner productΘ ≡ h θ L , X i = α i ∂ L ∂ ˙ q i . (13)Evaluating the lift on our Lagrangian, yields 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 = 0 . (14)The above equation is a homogeneous polynomial of de-gree 2 in the generalized velocities, and since it must beidentically zero, every coefficient must vanish. This leadsto six partial differential equations, namely6 αF + 6 β∂ φ F a + 12
F a∂ a α + 6 a ∂ φ F ∂ a β = 0 , (15)9 a α ( ∂ φ F ) F − a ( ∂ φ F ) β F + 3 a ∂ φ F ∂ φ F βF +6 ∂ φ F a ∂ φ α + 3 a ( ∂ φ F ) F ∂ φ β = 0 , (16)4 ∂ φ F αa + 2 ∂ φ F βa + 2 ∂ φ F a ∂ a α + 4 F a∂ φ α + a ( ∂ φ F ) F ∂ a β + 2 a ∂ φ F ∂ φ β = 0 , (17) − αa V − a β∂ φ V = 0 , (18) a V ∂ φ β = 0 , a V ∂ a β = 0 . (19)From (18) we infer that α = − βa∂ φ V V , while from (19)we conclude that β = β is a constant. Substitutingthese coefficients in (15), one finds that V ∝ F . Solving (16) and (17) for the self-interaction potential and for thecoupling, we find V ( φ ) = V e kφ ; F ( φ ) = F e kφ , (20)where k , V and F are constants. Note the linear de-pendence between the potential and the coupling are thesame found in . These results replace the former un-known quantities in the Lagrangian, as well as in theequations of motion. V. NUMERICAL SOLUTIONS
In order to integrate our equations of motion numer-ically and plot the curves, we shall change the indepen-dent variable from time to redshift, this will turn out tobe very useful when setting the initial conditions. Fromthe relation z = 1 /a −
1, we infer ddt = − H (1 + z ) ddz , (21)where the primes stand for differentiation with respect tothe redshift ( z ). With this change, we shall find numer-ical solutions for the Hubble parameter and the tachyonfield ( H, φ ). So we make use of the Friedmann equation(5) together with the acceleration equation (7), to giverise to one simple relation4 F e kφ HH ′ (1 + z ) = ρ φ + ρ m + p φ . (22)The Klein-Gordon equation (9) now becomes: (cid:0) H (1 + z ) φ ′′ + HH ′ (1 + z ) φ ′ (cid:1) × V [1 − H (1 + z ) φ ′ ] / − F k ! − F k H (1 + z ) φ ′ + V k p − H (1 + z ) φ ′ − H (1 + z ) V p − H (1 + z ) φ ′ − F k ! − F k (cid:2) H − HH ′ (1 + z ) (cid:3) = 0 . (23)These two differential equations, together with the ini-tial conditions, will give us the complete behaviour ofthe components’ densities and pressure, as well as of thescale factor. Before choosing the initial conditions, wemay rescale the quantities so they become dimensionlessas follows: H → ¯ H = H p ρ , V → ¯ V = Vρ ,φ → ¯ φ = φ p ρ , k → ¯ k = k p ρ . (24)where ρ = ρ φ + ρ m is the total energy density at thepresent time. Accordingly, we are now searching solu-tions for the behaviour of the density parameters, definedas Ω φ = ρ φ /ρ for the tachyon field and Ω m = ρ m /ρ forthe matter field.We need now to consider a few facts concerning theinitial conditions ¯ φ ′ (0), ¯ φ (0) and ¯ H (0). Firstly, forthe tachyon field to exhibit a negative real pressure atpresent times, it is required that ˙ φ (0) ≪
1, so it seemsvery reasonable to set ¯ φ ′ (0) = 10 − . Hence, we have ρ φ (0) ∼ V (0), which is the same as Ω φ (0) = ¯ V (0) = 0 . φ (0) = ¯ k − ln(0 . / ¯ V ). Further-more, the coupling must equal one-half in present time,so F = ¯ V / .
44, in a way that we have now, only two freeparameters, namely ¯ V and ¯ k . Finally, by the Friedmannequation (5), we set ¯ H (0) = 1 / √
3. Fixing ¯ V = 1, wenext plot the results for ¯ k = 0 .
1, ¯ k = 0 .
05 and ¯ k = 0 . W i k ‡ k ‡ k ‡ W m W Φ FIG. 1. Density parameter vs. redshift.
The evolution of the density parameters are shown inFig. 1. The weaker the coupling is, the more rapidlyincreases the dark energy density while the matter field’sdecreases. Contrarily, for the strongest coupling ¯ k = 0 . - - - - - Ω Φ k ‡ k ‡ k ‡ FIG. 2. ω φ vs. Redshift VI. OBSERVATIONAL PARAMETERSA. Deceleration Parameter
The deceleration parameter – given by q = 1 / p/ ρ – is plotted in Fig. 3. As the coupling increases, the tran-sition from decelerated to accelerated expansion happensat lower redshifts. This is already expected and con-firmed from Fig. 2, where we can clearly see that forstronger couplings, the pressure to energy density ratio, ω φ , of the tachyon field decreases much more rapidly, as-suming values of ω φ > − / z eq is the redshift when the densities are equal, z t stands for the redshift of the decelerated-acceleratedtransition and q (0) is the value of the deceleration pa-rameter at present time.The values found in literature for both q (0) and z t dif-fer widely from one reference to another, see e.g. .In the work , in order to estimate these values, the au-thors make 3 different parameterizations containing only2 free parameters, which are constrained by Supernovaeobservational data. - - - k ‡ k ‡ k ‡ FIG. 3. Deceleration Parameter vs. Redshift¯ k = 0 . k = 0 .
05 ¯ k = 0 . z eq q (0) -0.5619 -0.5754 -0.5798 z t B. Hubble Parameter
The Hubble parameter is plotted in Fig. 4 for the threeanalysed values of ¯ k . The data in red corresponds to ob-servational data from 25 supernovas . The threecurves practically overlap each other for small redshiftsbut evolve differently as it increases. In future times,more accurate data will provide us enough informationto constrain such models. H z L k ‡ k ‡ k ‡ = = FIG. 4. Hubble Parameter vs. Redshift
VII. FINAL REMARKS
In the beginning of this work, we first wrote the La-grangian density on the metric and only then variedthe system with respect to the generalized coordinates,applying the Euler-Lagrange equations. Point-like La-grangian is necessary in order to apply the generator ofsymmetry, but obviously one could derive the same re-sults trading orders, doing the metric variation first, andthen writing the equations of motions on the metric. Oneimportant result is obtained from the energy-momentumtensor’s divergent. The metric variations yields δSδg µν = F R µν − g µν F R + V ∂ µ φ∂ ν φ p − ∂ µ φ∂ µ φ + 12 g µν (cid:18) V p − ∂ µ φ∂ µ φ + ρ m a (cid:19) = 0 , (25)out of which the component µ = ν = 0 implies the Fried-mann equation (5). Note that the energy-momentumtensor is not conserved anymore as its four-divergent doesnot vanish, instead there is an energy flow from the darkenergy density to the gravitational field. If we differenti-ate the Friedmann equation with respect to the time,˙ ρ + 3 H ( ρ + p ) = k ˙ φρ, (26)and consider that the matter field is not coupled, nor in-teracts with other fields, the equation above is equivalentto ˙ ρ φ + 3 H ( ρ φ + p φ ) = − k ˙ φρ φ . (27)The role of the coupling constant becomes clear at thispoint. We see that, as already mentioned before, the big-ger the constant is, the stronger is the coupling, and fromeq. (27) we see that this provokes a more intense energyflow from the dark component to the gravitational field.Moreover, on the right hand side of the above equation,we also have a contribution from the generalized velocityof the field and its energy density. However, the tachyonfield performs a slow roll, and for all times ˙ φ ≪
1, notcontributing significantly for the energy flow.
VIII. CONCLUSIONS
The present model investigated the dynamics of non-minimally coupled tachyon field, constrained to theNoether symmetry and under the Palatini approach. Al-though it was not possible to find an analytical solutionto the system, the fact that we only had two free param-eters (out of which ¯ V did not seem to cause significantdifference when admitting a wide range of values), makesit easy to find plausible numerical solutions. The symme-try imposition showed us naturally which functions wereallowed to the potential and coupling, and they turnedout to be linearly dependent. The coupling is representedby a well behaved function which varies very softly as re-quired, a rapidly varying coupling is not solution to oursystem. As already mentioned in the previous sections,¯ k must be small to avoid instabilities. By inferring thatthe metric and the connection might be independent fromeach other, our point-like Lagrangian acquired one extraterm. Nevertheless, because both the tachyon field andthe coupling function vary smoothly, such term does notcontribute quite much for the dynamics. From the so-lutions, it became clear the importance of the couplingconstant, specially as the redshift increases. The compar-ison with the observational data was very satisfactory asevery calculated parameter lies within the observationalerrors. ACKNOWLEDGMENTS
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