Constraints on a scalar-tensor model with Gauss-Bonnet coupling from SN Ia and BAO observations
aa r X i v : . [ g r- q c ] J u l Constraints on a scalar-tensor model with Gauss-Bonnet couplingfrom SN Ia and BAO observations
S. Bellucci a , A. Banijamali b , B. Fazlpour c and M. Solbi ba INFN - Laboratori Nazionali di Frascati, 1-00044, Frascati (Rome), Italy b Department of Basic Sciences, Babol Noshirvani University of Technology, Babol, Iran c Department of Physics, Babol Branch, Islamic Azad University, Babol, Iran
Abstract
In the present work, the observational consequences of a subclass of of the Horndeski theory have beeninvestigated. In this theory a scalar field (tachyon field) is non-minimally coupled to the Gauss-Bonnetinvariant through an arbitrary function of the scalar field. By considering a spatially flat FRW universe,the free parameters of the model have been constrained using a joint analysis from observational data ofthe Type Ia supernovae and Baryon Acoustic Oscillations measurements. The best fit values obtainedfrom these datasets are then used to reconstruct the equation of state parameter of the scalar field. Theresults show the phantom, quintessence and phantom divide line crossing behavior of the equation ofstate and also cosmological viability of the model.
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Keywords:
Dark energy; Gauss-Bonnet coupling; Observational cosmology [email protected] [email protected] [email protected] Introduction
The current accelerated expansion of the universe is one of the great problems of modern cosmology. Thisacceleration was first suggested by Type Ia supernovae (SN Ia) surveys [1, 2] and then by measurementsof the cosmic microwave background (CMB) [3, 4], the Hubble constant [5], Baryon Acoustic Oscillations(BAO) [6] and more measurements of Type Ia supernovae [7]. Although observational cosmology confirmsthe acceleration of the universe, explaining this issue from theoretical point of view is a big challenge. Thesimplest way to obtain an accelerated universe is adding a cosmological constant to the standard cosmologicalmodel. However, a cosmological constant suffers from the fine-tuning problem, that is due to its extremelysmall observed value compared to predictions from theoretical considerations [8]. As a result one can followtwo ways to explain the late-time behavior of the universe: modifying general relativity at large scale [9] orintroducing a new content in the universe such as canonical scalar field, phantom scalar, both scalars, vectorfields etc., that is introducing the concept of dark energy [10, 11, 12].Furthermore, dynamical dark energy models can be extended in a huge class of models. Among them,non-minimally coupled dark energy models in which scalar fields coupled to the curvature terms dubbedscalar-tensor theories have been extensively studied in the literature. The most famous example of suchtheories is known as the Brans-Dicke [13] theory in which the gravitational constant is replaced by a scalarfield φ entering into the action as φ R , R is the Ricci scalar. Another well-known example of non-minimallycoupled system is provided by (1 − ξφ ) R coupling in which ξ is a constant measuring the strength of non-minimal coupling [14].Moreover, due to the novel features of non-minimally coupled scalar field system, such as allowing thephantom divide crossing and having the cosmological scaling solutions, these models are of great interest tothe community [15, 16, 17, 18, 19, 20]. On the other hand, in 1974 Horndeski [21] found the most general classof scalar-tensor theories which lead to the second order differential equations similar to the Einstein generalrelativity. The Horndeski gravity has been considered in many papers in the context of the inflationarycosmology [22, 23]. An interesting subclass of the Horndeski theory is given by the non-minimal couplingof the scalar field to the Gauss-Bonnet invariant in four dimensions [34, 35, 36, 37, 38, 39, 40]. Such anon-minimal coupling originates from the string theory and the trace anomaly and may play an importantrole in cosmological context. For example, this coupling has been proposed to address the dark energyproblem in [39] and various aspects of accelerating cosmologies with Gauss-Bonnet correction have beendiscussed in [41, 42, 43]. Indeed, these studies yield the result that the scalar-curvature coupling predictedby fundamental theories may become important at current, low-curvature universe. It deserves mention thatthe modifications of gravity from the Gauss-Bonnet invariant have been often considered as the result ofquantum gravity effects [44, 45, 46].In the present work, we will consider a model in which the scalar field playing the role of dark energyis coupled to the Gauss-Bonnet invariant. Here we derive constraints on the model parameters from acombination of available SN Ia data, as well as available BAO data and χ minimization technique.The outline of the paper is as follows: In the next section we present the basic formalism of our model in aflat FRW background, along with the definition of different cosmological parameters. We then discuss theobservational dataset and methodology in section III. Our main results in data analysis are summarized insection IV. Finally, section V is devoted to our conclusions. The model we examine in this paper is described by the following action: S = Z d x √− g (cid:20) R κ − V ( ϕ ) p − ∂ µ ϕ ∂ µ ϕ − η ( ϕ ) G + L m (cid:21) , (1)where g is the determinant of the metric tensor, κ = 8 πG , G is the gravitational constant and L m isthe matter Lagrangian density. The second term in the brackets is the Lagrangian of tachyon field withthe potential V ( φ ), while the third term represents a non-minimal coupling between the scalar field and2urvature through a general function η ( φ ). G is the Gauss-Bonnet invariant which is given by: G = R − R µν R µν + R µνλρ R µνλρ , (2)where R , R µν , R µνλρ are the Ricci scalar, the Ricci tensor and the Riemann tensor, respectively.Notice that not only tachyon field originates from the string theory but also the term proportional to theGauss-Bonnet invariant G is considered as a stringy correction in the action. These are our main motivationsto study the model.To analyse the model it is more convenient to use the following redefinition, as proposed in [47] for studyingthe tachyon dynamics, ϕ → φ = Z dϕ p V ( ϕ ) ⇐⇒ ∂ϕ = ∂φ p V ( φ ) . (3)Applying (3) in (1) yields to our starting action as follows: S = Z d x √− g " R κ − V ( φ ) s − ∂ µ φ ∂ µ φV ( φ ) − η ( φ ) G + L m . (4)The variation of the action (4) with respect to the metric leads to the following gravitational equations: R µν − g µν R = κ (cid:0) T φµν + T GBµν + T mµν (cid:1) , (5)where T mµν is the usual energy-momentum tensor for the matter, T φµν corresponds to the energy-momentumtensor of minimally coupled tachyon scalar field and T GBµν is the contribution of the non-minimal Gauss-Bonnet coupling. These last two components are given by T φµν = − u ∇ µ φ ∇ ν φ − g µν u − V ( φ ) (6)and T GBµν = 4 (cid:16)(cid:2) ∇ µ ∇ ν η ( φ ) (cid:3) R − g µν [ ∇ ρ ∇ ρ η ( φ ) (cid:3) R − ∇ ρ ∇ µ η ( φ ) (cid:3) R νρ − ∇ ρ ∇ ν η ( φ ) (cid:3) R µρ +2[ ∇ ρ ∇ ρ η ( φ ) (cid:3) R µν + 2 g µν [ ∇ ρ ∇ σ η ( φ ) (cid:3) R ρσ − ∇ ρ ∇ σ η ( φ ) (cid:3) R µρνσ (cid:17) , (7)where u = q − ∂ µ φ ∂ µ φV ( φ ) .In the derivation of T GBµν , the properties of the 4-dimensional Gauss-Bonnet invariant have been used(see [39, 48] for details). The energy density and pressure derived from these energy-momentum tensors willbe considered as effective ones and we represent them by ρ DE and p DE , respectively.Now, we assume the spatially flat Friedmann-Robertson-Walker (FRW) metric, ds = − dt + a ( t )( dr + r d Ω ) , (8)where a ( t ) is the scale factor. Considering this metric in equations (5)-(7) we obtain the following Friedmannequations: H = κ (cid:0) ρ DE + ρ m (cid:1) , (9)˙ H = − κ (cid:0) ρ DE + p DE + ρ m + p m (cid:1) , (10)where ρ m and p m are the energy density and pressure of the matter, ρ DE and p DE are given by ρ DE = u V ( φ ) + 24 H f ( φ ) ˙ φ, (11)3nd p DE = − u − V ( φ ) − H (cid:16) f ,φ ˙ φ + f ( φ ) ¨ φ (cid:17) − H f ( φ ) ˙ φ ( ˙ H + H ) , (12)where H = ˙ aa is the Hubble parameter, and we have also defined f ( φ ) = dηdφ , f ,φ = df ( φ ) dφ .Further, by varying the action (4) over φ and assuming that φ only depends on time, we obtain the equationof motion for φ , which in FRW background takes the following form¨ φ + 3 u − H ˙ φ + − φ V ! V ,φ + 24 H (cid:16) ˙ H + H (cid:17) f ( φ ) = 0 . (13)Note that in deriving equation (13), we have used the following expression for the Gauss-Bonnet invariantin FRW background G = 24 H ( ˙ H + H ) . (14)In addition, the energy conservation equations for dark energy and the matter are expressed in the followingforms, respectively ˙ ρ DE + 3 H (1 + ω DE ) ρ DE = 0 , (15)and ˙ ρ m + 3 H (1 + ω m ) ρ m = 0 , (16)where ω DE = p DE ρ DE and ω m = p m ρ m are the equation of state parameters of dark energy and matter respectively.Here, we just focus on the late-time eras, so that we can neglect the radiation contribution and assume apressureless fluid for the matter content ω m = p m ρ m = 0. Then, the continuity equation (16) can be easilyintegrated to yield ρ m = ρ m (cid:0) a a (cid:1) − = ρ m (1 + z ) . (17)where ρ m denotes the present value of the matter energy density and z is the redshift parameter z + 1 = a a .In addition, we define the density parameters of dark energy and the matter by Ω DE = ( κ ρ DE ) / (3 H ) andΩ m = ( κ ρ m ) / (3 H ) and here after a subscript ”0” for a parameter stands for the present value of thatparameter.Before closing this section, it is worthwhile to mention that since we are going to constrain the model usingobservational data, which are expressed in terms of the redshift, it is convenient to rewrite the Friedmannequations in terms of the latter, instead of the cosmic time. This can be done straightforwardly by thefollowing replacements in equations (11) and (12),˙ H = − HH ′ (1 + z ) , ˙ φ = − H (1 + z ) φ ′ , ¨ φ = H (1 + z ) φ ′ + HH ′ (1 + z ) φ ′ + H (1 + z ) φ ′′ , (18)where prime denotes the derivative with respect to the redshift. Here, we explain the methodology that we use to constrain the model by using the recent observationaldatasets from Type Ia Supernova (SNe Ia) and Baryon Acoustic Oscillations (BAO).We use the Markov-chain Monte Carlo (MCMC) method for the minimization of χ to perform the statisticalanalysis. We have tested the model using the publicly available codes by S. Nesseris et a.l. (see for example[49, 50, ? ]) and making the necessary changes in the case of our model. Now, we briefly explain the methodfor elaborating the observational data.Our study follows the likelihood L ∝ exp( − χ / χ for combined datasets reads: χ = χ + χ . (19)In the following subsections, the way by which, one can calculate each of χ is described.4 .1 Type Ia Supernova (SN Ia) The χ function for the SNe Ia is given by [52], χ = A − µ B + µ C , (20)where A , B and C are defined by A = X i [ µ obs ( z i ) − µ th ( z i ; µ = 0)] σ i ,B = X i µ obs ( z i ) − µ th ( z i ; µ = 0) σ i ,C = X i σ i . (21)The definition of the distance modulus is µ th ( z ) ≡ D L ( z ) + µ , (22)where µ ≡ . − h , with h ≡ H / / [km sec − Mpc − ] [3] and the subscripts ”th” and ”obs”stand for theoretical and the observed distance modulus. Also, the quantity σ i represents the statisticaluncertainly in the distance modulus.The dimensionless luminosity distance D L for the flat universe is given by D L ( z ) = (1 + z ) Z z dz ′ E ( z ′ ) , (23)where E ( z ) = H ( z ) H = q Ω (0)m (1 + z ) + Ω (0)r (1 + z ) + Ω (0)DE (1 + z ) w DE ) . (24) Here, Ω r is the radiation density parameter and Ω (0)r = Ω (0) γ (1 + 0 . N eff ), where Ω (0) γ is the presentfractional photon energy density and N eff = 3 .
04 is the effective number of neutrino species [3].Now, the minimizing of χ with respect to µ yields to˜ χ = A − B C . (25)In our statistical analysis we use (25) for SNe Ia dataset and the Union 2.1 compilation data [51] of 580 datapoints have been used to constrain the model parameters.
Next, we have used BAO measurement dataset to put the BAO constraints on the model parameters.The BAO observable is the distance ratio d z ≡ r s ( z d ) /D V ( z ), where r s is the comoving sound horizon, z d is the redshift at the drag epoch [53] and D V is the volume-averaged distance which is defined as follows [54], D V ( z ) ≡ (cid:20) (1 + z ) D A ( z ) zH ( z ) (cid:21) / . (26)In equation (26) D A ( z ) is the proper angular diameter distance for the flat universe.Here we have considered six BAO data points (see Table 1). The WiggleZ collaboration [55] has measuredthe baryon acoustic scale at three different redshifts, while SDSS and 6DFGS surveys provide data at lower5edshift [53]. 6dF SDSS WiggleZ z d z d z χ function of the BAO data is defined as, χ = (cid:0) x th i, BAO − x obs i, BAO (cid:1) (cid:0) C − (cid:1) ij (cid:0) x th j, BAO − x obs j, BAO (cid:1) , (27)where the indices i, j are in growing order in z , as in Table 1 and C − can be obtained by the covariancedata [55] in terms of d z as follows: C − = − − − − − − . (28)One can now obtain the best fit values of the model parameters by minimizing χ in equation (19). Following the χ analysis ( as presented in the previous section), in this section, we obtain the constraintson the free parameters of the model.Two important functions in our analysis are the function f ( φ ) and the scalar field potential V ( φ ). We willconsider power-law and exponential forms for f ( φ ) and V ( φ ) and thus our study is categorized into thefollowing four different cases:Case I: Exponential f ( φ ) and Power-law V ( φ ) f ( φ ) ∝ e αφ , V ( φ ) ∝ φ β Case II: Power-law f ( φ ) and Exponential V ( φ ) f ( φ ) ∝ φ α , V ( φ ) ∝ e βφ Case III: Power-law f ( φ ) and V ( φ ) f ( φ ) ∝ φ α , V ( φ ) ∝ φ β Case IV: Exponential f ( φ ) and V ( φ ) f ( φ ) ∝ e αφ , V ( φ ) ∝ e βφ , where α and β are the free parameters which will be constrained using the data.In the present work, we identify the parameters of the model as the parameters α and β , the present matterdensity parameter Ω m and dark energy equation of state parameter ω DE . Thus, the model parameters are( α, β, Ω m , ω DE ).Now, performing the combined ( SN Ia + BAO ) analysis using MCMC method for the cases I-IV, yields tothe constraint results as what summarized in Table 2.6 odel χ min Ω m ω DE α β Case I 594 .
216 0 . ± . − . ± . − . ± .
012 0 . ± . .
505 0 . ± . − . ± . − . ± .
025 0 . ± . .
833 0 . ± . − . ± . − . ± .
025 0 . ± . .
747 0 . ± . − . ± . − . ± .
25 0 . ± . χ min and the best fit values of the model parameters Ω m , ω DE , α and β for the casesI-IV.In this table, the reader may see a compact presentation of the best fit values of the model parametersas well as χ min for each case, separately.Furthermore, in Figures 1 − σ, σ and 3 σ confidence level contour plots for several combina-tions of the model parameters as well as their likelihood analysis for the cases I-IV, respectively. Additionally,in figure 5, using the same combined analysis SN Ia + BAO, we have shown the qualitative evolution of thedark energy equation of state parameter. Figure 6 shows the Hubble diagram for 580 SN Ia from (Union2.1) sample. The curves represent the distance modulus predicted by the four cases I-IV in our model.Further, the case with the lowest value of χ min is the case I and as it is clear from Table 2 the cases withthe exponential coupling function f ( φ ) (cases I & IV) have a lower χ min than the cases with power-law f ( φ )(cases II & III).The joint analysis on cases I-IV shows that the best fit values of the dark energy equation of state parame-ter, exhibit phantom behavior, although very close to the cosmological constant boundary. As one see fromTable 2, in case I, we have the nearest value of the equation of state parameter to the cosmological constant( ω Λ = −
1) and in case IV, the phantom character of the current dark energy equation of state is the clearestone. Notice that the quintessence behavior of ω DE is excluded in all cases of Table 2. Further, from Figure5, it is clear that the transition from quintessence phase ( ω DE > −
1) to the phantom phase ( ω DE < −
1) orthe so-called phantom divide line crossing, occurs in all four cases, which is in agreement with observationalresults [56, 57, 58].From the constraints on α and β as shown in Table 2, it is clear that the combination of SN Ia + BAO datafavors negative values for α and positive values for β in cases I-IV.Between the best-fit values of α , the case II i.e. when the coupling function is in power-law form, has theminimum value, while the case I in which f ( φ ) is exponential has the maximum value of α . On the otherhand, the best-fit value of the free parameter β , has its minimum and maximum values for the cases I andIV, that is for power-law and exponential potentials, respectively.It deserves mention here that the values of the dark matter density parameter at present Ω m for all fourcases are very close to the desired value in cosmology.7igure 1: 1 σ (68.3%), 2 σ (95.4%) ans 3 σ (99.7%) confidence level contour plots for different combinationsof the model parameters with also 1-dimensional posterior distributions in the case I for combined observa-tional dataset from SN Ia + BAO. The black dot in each contour plot represents the best fit values of thecorresponding pair. 8igure 2: 1 σ (68.3%), 2 σ (95.4%) ans 3 σ (99.7%) confidence level contour plots for different combinationsof the model parameters with also 1-dimensional posterior distributions in the case II for combined obser-vational dataset from SN Ia + BAO. The black dot in each contour plot represents the best fit values of thecorresponding pair. 9igure 3: 1 σ (68.3%), 2 σ (95.4%) ans 3 σ (99.7%) confidence level contour plots for different combinationsof the model parameters with also 1-dimensional posterior distributions in the case III for combined obser-vational dataset from SN Ia + BAO. The black dot in each contour plot represents the best fit values of thecorresponding pair. 10igure 4: 1 σ (68.3%), 2 σ (95.4%) ans 3 σ (99.7%) confidence level contour plots for different combinationsof the model parameters with also 1-dimensional posterior distributions in the case IV for combined obser-vational dataset from SN Ia + BAO. The black dot in each contour plot represents the best fit values of thecorresponding pair. 11igure 5: The evolution of the dark energy equation of state parameter, for the best fit values of ( α, β )that arises from the analysis of SN Ia + BAO datasets, for the cases I (blue), II(green), III (orange) and IV(pink).Figure 6: The Hubble diagram for 580 data of SN Ia from Union 2.1 sample [51].The curves correspond tothe distance modulus predicted by the four cases I-IV with the best-fit values coming from the joint analysisof SN Ia + BAO as presented in Table 2. In this paper, we have focused on the analysis of a non-minimally coupled scalar field theory in which thescalar field is considered as a candidate of dark energy. In this model a tachyon field is non-minimally cou-pled to the Gauss-Bonnet invariant, via a general coupling function f ( φ ), as in action (4). The cosmologicalevolution of the model is studied by assuming a flat FRW universe. Then, we placed constraints on the freeparameters of the model by performing a joint statistical analysis using the recent cosmological data fromSN Ia and BAO measurements. We have considered the exponential and power-law forms for the scalarfield potential, as well as the non-minimal coupling function. Then, we have obtained the best fit values ofthe free parameters α and β in the potential and the coupling function, respectively. The equation of state12arameter of dark energy, ω DE and the present value of the matter density Ω m have also been fitted.According to the contents of Table 2, where our results are summarized, the joint analysis of SN Ia + BAO,favors the negative values for α and positive values for β . In addition, by constraining with the datasets ofSN Ia and BAO, we found that ω DE < −
1, for all cases, which means our universe slightly biases towardsphantom behavior while the values of Ω m , i.e. the present-day dark matter density, are very close to thedesired value Ω m ≃ .
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