Late-Time Cosmology of Scalar-Coupled f(R,G) Gravity
aa r X i v : . [ g r- q c ] F e b Late-Time Cosmology of Scalar-Coupled f ( R, G ) Gravity
S.D. Odintsov, , V.K. Oikonomou, , F.P. Fronimos, ICREA, Passeig Luis Companys, 23, 08010 Barcelona, Spain Institute of Space Sciences (IEEC-CSIC) C. Can Magrans s/n, 08193 Barcelona, Spain Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece Laboratory for Theoretical Cosmology, Tomsk State University ofControl Systems and Radioelectronics, 634050 Tomsk, Russia (TUSUR)
In this work by using a numerical analysis, we investigate in a quantitative way the late-timedynamics of scalar coupled f ( R, G ) gravity. Particularly, we consider a Gauss-Bonnet term coupledto the scalar field coupling function ξ ( φ ), and we study three types of models, one with f ( R ) termsthat are known to provide a viable late-time phenomenology, and two Einstein-Gauss-Bonnet typesof models. Our aim is to write the Friedmann equation in terms of appropriate statefinder quantitiesfrequently used in the literature, and we numerically solve it by using physically motivated initialconditions. In the case that f ( R ) gravity terms are present, the contribution of the Gauss-Bonnetrelated terms is minor, as we actually expected. This result is robust against changes in the initialconditions of the scalar field, and the reason is the dominating parts of the f ( R ) gravity sector atlate times. In the Einstein-Gauss-Bonnet type of models, we examine two distinct scenarios, firstlyby choosing freely the scalar potential and the scalar Gauss-Bonnet coupling ξ ( φ ), in which case theresulting phenomenology is compatible with the latest Planck data and mimics the Λ-Cold-Dark-Matter model. In the second case, since there is no fundamental particle physics reason for thegraviton to change its mass, we assume that primordially the tensor perturbations propagate withthe speed equal to that of light’s, and thus this constraint restricts the functional form of the scalarcoupling function ξ ( φ ), which must satisfy the differential equation ¨ ξ = H ˙ ξ . The latter equationis greatly simplified when late times are considered and can be integrated analytically to yield arelation for ˙ ξ , which depends solely on the Hubble rate, in a model independent way. This leadseventually to an elegant simplification of the Friedmann equation, which when solved numerically,yields a viable late-time phenomenology. A common characteristic of the Einstein-Gauss-Bonnetmodels we considered is that the dark energy era they produce is free from dark energy oscillations. PACS numbers: 04.50.Kd, 95.36.+x, 98.80.-k, 98.80.Cq,11.25.-w
I. INTRODUCTION
The quest for understanding the mysterious late-time acceleration era [1], is still ongoing in modern theoreticalphysics. Many possible theoretical descriptions have been proposed in order to model the dark energy era, amongwhich modified gravity has an elevated role in the successful description of the dark energy era [2–9], since apart frombeing able to describe the late-time era, it is also possible to describe inflation with the same theoretical framework,see for example Refs. [10–16, 73]. Modern modified gravity models are put into stringent test of viability when thedark energy era is considered, since the models must be confronted with the latest Planck 2018 data [18], and also themodels have to be compatible to some inferior extent with the Λ-Cold-Dark-Matter (ΛCDM) model, which is the mostsuccessful model for describing the dark energy era. The ΛCDM model is basically based on the assumptions of thepresence of a cosmological constant, and the presence of particle dark matter [19–24], however both the ingredientsof the model are in question of their existence. Moreover, although the ΛCDM is quite compatible with the CosmicMicrowave Background data, it has several theoretical shortcomings which cannot be harbored by Einstein-Hilbertgravity. At this point, modified gravity offers many possibilities for successful theoretical descriptions. In this lineof research, Einstein-Gauss-Bonnet theory [25–64] could be a potentially correct description of both the early andlate-time era. In this work we shall investigate in a quantitative way the exact effect of the Gauss-Bonnet couplingon the late-time era, in the context of f ( R, φ ) theories of gravity in general. In particular, we shall investigate theeffect of the non-trivial Gauss-Bonnet coupling on a simple canonical scalar field theory, and on f ( R ) gravity in thepresence of a canonical scalar field. We shall perform a thorough numerical analysis of the Friedmann equation andwe shall derive the behavior of several statefinder quantities and of several physical quantities of interest, as functionsof the redshift. Accordingly our findings shall be compared with the ΛCDM and shall be confronted with the latestPlanck constraints on the cosmological parameters [18]. Our findings indicate that when an f ( R ) gravity theory ispresent along with the Gauss-Bonnet coupling, the latter does not significantly affect the late-time phenomenology.Also in the case of a simple Einstein-Gauss-Bonnet theory, we show that it is possible to obtain the phenomenologicalviability of the model under study, but this result could be highly model dependent, and has a minor disadvantage,since it is hard to describe inflation and dark energy with the same Einstein-Gauss-Bonnet model, in general though.Finally, we make a novel assumption that may constrain the functional form of the scalar coupling function ofthe scalar field to the Gauss-Bonnet coupling, and we investigate the late-time phenomenology in this case too.Particularly, we assume that there is a constraint coming from the requirement that the primordial gravitational wavespeed is equal to unity, which imposes a functional constraint on the functional form of the Gauss-Bonnet scalarcoupling function. The reason for demanding that the primordial gravitational wave speed is equal to unity in naturalunits, is coming from the GW170817 event [65], which indicated that the gamma rays and the gravitational wavesarrived almost simultaneously. Thus assuming that there is no fundamental particle physics reason for the graviton tochange its primordial mass, the gravity speed constraint imposed by the GW170817 event, must stretch back to theprimordial inflationary era. For the late-time era, the constraint imposed by requiring that the gravity wave speed ofthe primordial tensor modes is equal to unity in natural units, results to an elegant expression for the time derivativeof the scalar Gauss-Bonnet coupling, which is expressed in terms of the Hubble rate and the redshift, and thus itdepends on statefinder quantities and acquires a model independent description. The late-time viability of such anEinstein-Gauss-Bonnet gravity is examined in detail, and our findings indicate that these can also provide a successfuldescription of the late-time era. Finally, we also conclude that the Einstein-Gauss-Bonnet theories in general, providea dark energy oscillations free late-time era, in contrast to f ( R ) gravity models. However, this result seems to behighly model dependent, at least in the context of f ( R ) gravity and needs to be further discussed in another context. II. ESSENTIAL FEATURES OF f ( R, φ ) EINSTEIN-GAUSS-BONNET GRAVITY
The starting point of our work is the gravitational action, which for the f ( R, φ ) Einstein-Gauss-Bonnet gravity hasthe following form, S = Z d x √− g (cid:18) f ( R, φ )2 κ − g µν ∂ µ φ∂ ν φ − V − ξ ( φ ) G + L ( m ) (cid:19) , (1)where R denotes the Ricci scalar, κ = M P is the gravitational constant with M P being the reduced Planck mass,while V is the scalar potential, while ξ ( φ ) is the Gauss-Bonnet scalar coupling function. Also G = R − R µν R µν + R µνσρ R µνσρ denotes the Gauss-Bonnet invariant with R µν and R µνσρ being the Ricci and Riemann curvature tensorrespectively and finally, L ( m ) specifies the Lagrangian density of both relativistic and non-relativistic perfect matterfluids. For now, the exact form of the function f ( R, φ ) shall remain unspecified, at least for the moment. Concerningthe cosmological geometric background, we shall assume that it corresponds to that of a flat Friedman-Robertson-Walker (FRW) metric with the line element being, ds = − dt + a ( t ) δ ij dx i dx j , (2)where a ( t ) denotes the scale factor. Consequently, the Ricci scalar and Gauss-Bonnet invariant for the FRW back-ground take the forms R = 6(2 H + ˙ H ) and G = 24 H ( H + ˙ H ), where H = ˙ aa signifies the Hubble rate and the“dot” denotes differentiation with respect to the cosmic time t . Furthermore, in order to simplify our work, we shallmake the reasonable assumption that the scalar field is homogeneous and thus it depends solely on the cosmic time.Implementing the variation principle with respect to the metric tensor g µν and the scalar field φ , we obtain thefield equations for gravitational sector and the scalar field equation, which are,3 f R H κ = ρ ( m ) + 12 ˙ φ + V + f R R − f κ − H ˙ f R κ + 24 ˙ ξH , (3) − f R ˙ Hκ = ρ ( m ) + P ( m ) + ˙ φ + ¨ f R − H ˙ f R κ −
16 ˙ ξH ˙ H , (4) V φ + ¨ φ + 3 H ˙ φ − f φ κ + ξ φ G = 0 , (5)where ρ ( m ) and P ( m ) denote the matter density and pressure respectively of any prefect fluid of non-relativistic(baryons, leptons, Cold Dark matter (CDM)) matter and relativistic matter (photons and neutrinos). In particular,for the purposes of this work, for which the focus will be on the dark energy era, we shall assume that the perfectfluids compose of CDM and radiation, so we have, ρ ( m ) = ρ d a (cid:18) χ a (cid:19) , (6) P = X i ω i ρ i , (7)where χ = ρ r ρ d with ρ r being the current density of relativistic matter and ω i the equation of state parameter foreach kind or matter, with i running from relativistic to non-relativistic. As we already mentioned, all matter speciesare described by perfect fluids and a barotropic EoS. Since we have prefect fluids, the continuity equation for each ofthem reads, ˙ ρ i + 3 Hρ i (1 + ω i ) = 0 . (8)In the following, we shall implement two replacements in order to align better our study with the late-time dynamics,with the first being the use of the redshift instead of the cosmic time as a dynamical parameter. From its definition,1 + z = 1 a ( t ) , (9)where we assumed that the scale factor at present time is set to unity, the time derivatives can be expressed in termsof derivatives with respect to the redshift, using the following rule, ddt = − H (1 + z ) ddz , (10)By replacing cosmic time t with the redshift, one can recast the cosmological equations of motion in the followingway, 3 f R H κ = ρ ( m ) + 12 ˙ φ + V + f R R − f κ − H ˙ f R κ − z ) ξ ′ H , (11) V φ + ¨ φ + 3 H ˙ φ − f φ κ + ξ φ G = 0 , (12)where the “prime” denotes differentiation with respect to the redshift. Here, only two equations where rewritten sincethey will be used in our study. In addition, every time derivative participating in the equations of motion above shallbe replaced as well. Specifically, we have, ˙ H = − H (1 + z ) H ′ , (13)˙ φ = − H (1 + z ) φ ′ , (14)¨ φ = H (1 + z ) φ ′′ + H (1 + z ) φ ′ + HH ′ (1 + z ) φ ′ , (15)˙ f R = ˙ Rf RR + ˙ φf Rφ , (16)˙ R = 6 H (1 + z ) (cid:18) HH ′′ + ( H ′ ) − HH ′ z (cid:19) , (17)Now more importantly, instead of using the Hubble rate and its derivatives in order to quantify the cosmologicalevolution, we shall use a statefinder quantity defined as follows [66–68, 73], y H = ρ DE ρ d , (18)with ρ DE denoting the dark matter energy density and ρ d the current value of density for non-relativistic matter.Here, we shall assume that the dark energy density is comprised of all the geometric terms in the Friedmann equation.In particular, ρ DE = 12 ˙ φ + V + f R R − f κ − H ˙ f R κ + 24 ˙ ξH + 3 H κ (1 − f R ) , (19)Similarly, from the Raychaudhuri equation, the corresponding pressure for the dark energy fluid is defined as, P DE = − V −
24 ˙ ξH − ξH ˙ H − f R R − f κ − Hκ (1 − f R ) , (20)where, ˙ ρ DE + 3 H ( ρ DE + P DE ) = 0 , (21)Hence, equations (3) and (4) obtain the usual Friedmann equation-like form of Einstein-Hilbert gravity,3 H κ = ρ ( m ) + ρ DE , (22) − Hκ = ρ ( m ) + P ( m ) + ρ DE + P DE , (23)Consequently, the newly defined statefinder parameter y H can be written in terms of the Hubble rate, and vice-versa.Specifically, we have, H = m s (cid:18) y H ( z ) + ρ ( m ) ρ d (cid:19) , (24)where m s = κ ρ d = 1 . · − . This extends to the derivatives of the Hubble rate as well since now, HH ′ = m s y ′ H + ρ ′ ( m ) ρ d ! , (25) H ′ + HH ′′ = m s y ′′ H + ρ ′′ ( m ) ρ d ! , (26)In the following, we shall numerically solve the system of differential equations (11) and (12) with respect to thestatefinder quantity y H and the scalar field φ . Afterwards, we shall compare the theoretical results with the obser-vations. This can be achieved by utilizing further statefinder parameters. Concerning dark energy, we define theequation of state parameter ω DE and the density parameter Ω DE with respect to z and y H as follows [67, 68, 73], ω DE = − z d ln y H dz Ω DE = y H y H + ρ ( m ) ρ d , (27)Furthermore, for the overall evolution, we shall use the following statefinder parameters [68, 73], q = − − ˙ HH j = ¨ HH − q − s = j − (cid:0) q − (cid:1) Om ( z ) = (cid:16) HH (cid:17) − z ) − , (28)which in the order of appearance above, are the deceleration parameter, the jerk, the snap parameter and Om ( z ),which is indicative of the current CDM energy density parameter. III. f ( R ) EINSTEIN-GAUSS-BONNET GRAVITY: UNIFYING EARLY AND LATE TIME
Let us commence our study by introducing the arbitrary functions of the previous models. Hereafter, we shall limitour work to only simple cases for the scalar functions, namely V ( φ ) and ξ ( φ ), for which it is known that the early-timecan be described successfully. Suppose that the scalar functions of the previous section obtain the following forms,which are arbitrary for the moment, meaning that there is no fundamental relation between the scalar potential andthe scalar coupling function, ξ ( φ ) = e φMP , (29) y H × - × - × - × - × - × - × - × - z φ / M P FIG. 1: Solutions y H (left) and φ over reduced Planck mass (right) for the f ( R ) case. The main difference seems to be thescalar field which does not oscillate. In general, the addition of a canonical scalar field and a linear Gauss-Bonnet topologicalinvariant coupled to a scalar function do not suffice to nullify dark energy oscillation at large redshifts. - - q j - - - s FIG. 2: Cosmological parameters q (upper left), j (upper right), s , (bottom left) and s (bottom right) as functions of redshift.Once again, the same results as in the pure f ( R ) case are obtained. and also assuming that there is no scalar potential present, we assume that the there is also an f ( R ) gravity partpresent too, and has the form [68, 73], f ( R ) = R + (cid:18) RM (cid:19) − γ Λ (cid:18) R m s (cid:19) δ , (30)where γ a dimensionless parameter, Λ a constant with mass dimensions [ m ] , M = 1 . · − N M P with N beingthe e-folding number and δ an exponent which satisfies the relation 0 < δ <
1, see Refs. [68, 73] for details. In thisparticular case we assumed for simplicity that the scalar potential is absent, as we already mentioned. In this generalframework however, there is no physical constraint that connects the scalar potential and the scalar coupling function,nevertheless if one takes into account the primordial gravitational wave speed constraints, these two scalar functionsare interconnected fundamentally. For the moment though we assume that these can be freely chosen given that noconstraints on the speed of gravitational waves are imposed, i.e c T does not necessarily coincide with unity. The f ( R )model we chose, was chosen simply because it is capable of uniting early with late time acceleration era of our Universe,due to the fact that for large R , R becomes dominant whereas for R → R δ becomes the dominant term, see Ref.[68, 73] for a detailed analysis on this issue. It is therefore interesting to examine whether the addition of a scalar fieldcan alter the dynamics of such model. Essentially, such a model predicts non negligible dark energy oscillations for z ≥ y H ( z ), which become even more dominant in higher order derivatives. Recently,it was showcased that the addition of a function depending on the Gauss-Bonnet topological invariant G , which alonedescribes an oscillation-free late-time era, cannot nullify such oscillations on the f ( R ) model, implying that the lateris more dominant [69]. It is therefore sensible to try and examine whether the addition of a canonical scalar fieldcoupled to the Gauss-Bonnet topological invariant can achieve such phenomenological behavior. Essentially, by usingthe same parameters for the f ( R ) gravity as in Ref. [73], meaning that γ = 2, Λ = 1 , · − eV , δ = , N = 60with the initial conditions chosen as y H ( z = 10) = Λ3 m s (cid:16) z f (cid:17) , dy H dz (cid:12)(cid:12)(cid:12) z =10 = Λ3 m s , φ ( z = 10) = 10 − M p , dφdz (cid:12)(cid:12)(cid:12) z =10 = − − M p , then by solving numerically equations (3 ) and (5) in the interval [ z i , z f ] = [ − . ,
10] withrespect to y H , φ , it becomes apparent that simply adding a canonical scalar field cannot negate the dark energyoscillations. This result seems to be in agreement with the one obtained in Ref. [69] for the f ( R ) + g ( G ) case giventhat G is quite small in terms of the rest of the parameters. The results of our numerical analysis for the particularmodel at hand can be found in Figs. 1, 2 and 3, while in Table I we compare the values of several statefinder quantitiesat present time with the corresponding values of the ΛCDM model and we confront the values of the dark energydensity parameter Ω DE (0) and the dark energy EoS parameter ω DE (0) with the latest constraints of the Planck 2018collaboration on cosmological parameters [18]. As it can be seen from Table I, the resulting cosmological quantitiesand the statefinder values at present time corresponding to the model at hand are quite close to the ΛCDM values,and both Ω DE (0) and ω DE (0) are compatible with the observational data. In Fig. 1 we present the behavior ofthe statefinder y H (left) and φ (right) as functions of the redshift, for the model at hand. The main difference withthe pure f ( R ) gravity seems to be the scalar field which does not oscillate. In general, the addition of a canonicalscalar field and a linear Gauss-Bonnet topological invariant coupled to a scalar function do not suffice to nullify darkenergy oscillation at large redshifts. Also in Fig. 2 we present the behavior of the cosmological statefinder quantities q (upper left), j (upper right), s , (bottom left) and s (bottom right) as functions of redshift. Once again, the samequalitative behavior as in the pure f ( R ) case are obtained. Finally, in Fig. 3 we present the dark energy variables,namely the EoS (left) and the dark energy density parameter Ω DE (right) as functions of the redshift. Out of thesetwo parameters, only the latter is free of oscillations, however neither the canonical scalar field nor the Gauss-Bonnettopological invariant are responsible for such feature. The f ( R ) contribution is the dominant term as it can also beinferred from the rest results. TABLE I:
Parameter f ( R ) Λ CDM Value q(z=0) -0.520954 -0.535j(z=0) 1.00319 1s(z=0) -0.00104169 0 Om ( z = 0) 0.319364 0.3153 ± DE (0) 0.683948 0.6847 ± ω DE (0) -0.995205 -1.018 ± Thus for this particular class of potential-less models, the f ( R ) gravity part seems to dominate the late-timeevolution. In the following sections we shall also introduce a potential, and in parallel we shall assume an Einstein-Hilbert f ( R ) term. At a later section we shall constrain the functional forms of the scalar potential and the scalarcoupling function, in order to see how the late-time dynamics are affected by these changes.As a final comment, we should mention that even though the value of the scalar field seems to increase with respectto time, the rate of increase is smaller and subdominant when it is compared to the rate of the f ( R ) gravity terms,and in particular from the term ∼ R δ . Subsequently, the f ( R ) part is dominant in comparison to the scalar termsand thus the increasing value of φ does not contradict the overall phenomenology. For this exact reason, one observesdark energy oscillations in the high redshift area, a feature which arises in the pure f ( R ) case. - - - - (cid:6)D(cid:7) (cid:8)(cid:9)(cid:10) FIG. 3: Dark energy variables, the EoS (left) and the Density parameter Ω DE (right). Out of these two parameters, only thelatter is free of oscillations however neither the canonical scalar field nor the Gauss-Bonnet topological invariant are responsiblefor such feature. The f ( R ) contribution is the dominant term as it can also be inferred from the rest results. IV. EINSTEIN-GAUSS-BONNET GRAVITY IN THE PRESENCE OF A SCALAR POTENTIAL
Let us now proceed with a different approach. We shall assume that the f ( R ) case is reduced to a simple f ( R ) = R case and that the scalar potential is now present in the formalism. Let us assume that the potential has the followingarbitrary form, V ( φ ) = (cid:18) φM P (cid:19) , (31)while the scalar coupling function has the following form, y H FIG. 4: Statefinder y H for the R case in the presence of a scalar potential. ξ ( φ ) = (cid:18) φM P (cid:19) , (32)In this case we shall assume simple power-law models for the scalar functions which are normalized with respect tothe reduced Planck mass. Such models are frequently used in the inflationary era where the slow-roll conditions for thescalar field are usually assumed to hold true. In the late time era, there exists no need to apply the slow-roll conditionssince they do not hold true. Let us proceed with the numerical results. In this case, the Einstein-Hilbert form of f ( R ) implies that certain terms in Eq.(3) are discarded which facilitates our study. Furthermore, since ˙ R is nowabsent, there exists no second derivative of statefinder y H . In fact, if it was not for the scalar field which has a term ¨ φ proportional to y ′ H in the continuity equation, the aforementioned statefinder function would need no initial conditionsto be specified. Here, we shall only assume that y H ( z = 10) is once again equal to y H ( z = 10) = Λ3 m s (cid:16) z f (cid:17) - - q (cid:11) - - - s (cid:12)(cid:13)(cid:14) FIG. 5: Deceleration q (upper left), jerk j (upper right), snap s (bottom left) and Om (bottom right) as functions of redshift.In this case, no dark energy oscillations are present since such feature is generated from additional R terms. All variables arein agreement with the ΛCDM however what is fascinating is the snap parameter which decreases with time. and in addition, φ ( z = 10) = M P , dφdz (cid:12)(cid:12)(cid:12) z =10 = M P then the results of our analysis can be found in Figs. 4, 5 and6. Also in Table II, as in the model of the previous section, we compare the values of several statefinder quantitiesat present time with the corresponding values of the ΛCDM model and we confront the values of the dark energydensity parameter Ω DE (0) and the dark energy EoS parameter ω DE (0) with the latest constraints of the Planck 2018collaboration on cosmological parameters [18]. As it can be seen from Table II, the resulting cosmological quantitiesand the statefinder values at present time corresponding to the model at hand are quite close to the ΛCDM values,and both Ω DE (0) and ω DE (0) are compatible with the observational data. It can easily be inferred from the plots TABLE II:
Parameter R Λ CDM Value q(z=0) -0.522521 -0.535j(z=0) 1.0002 1s(z=0) -0.00006431 0 Om ( z = 0) 0.318319 0.3153 ± DE (0) 0.681713 0.6847 ± ω DE (0) -1 -1.018 ± that the qualitative behavior of the model under study is quite close to the ΛCDM model. One striking feature is thatthe dark energy density ρ DE is nearly constant throughout the interval [-0.9,10], as indicated by y H , and thereforethe EoS parameter ω DE is also nearly equal to −
1. This result is robust towards changing the free parameters forthis particular model. The rest of the cosmological parameters however seem to have an infinitesimal evolution, forinstance the jerk parameter is quite close to j = 1 but not exactly equal to unity as ω DE is −
1. As a final comment, - - - - (cid:15)(cid:16)(cid:17) (cid:18)(cid:19)(cid:20) FIG. 6: Equation of state parameter ω DE and density parameter Ω DE . Due to the fact that statefinder y H is numerically foundto be slowly evolving or nearly constant during the stage of our Universe’s evolution, the EoS is subsequently exactly nearlyequal to ω DE = − it should be noted that both models studied so far do not have a fixed value for the velocity of the primordialgravitational waves these models produce. Since [25, 54–58], c T = 1 − Q f Q t , (33)with Q f = 16( ¨ ξ − H ˙ ξ ) and Q t = M P − ξH , then it stands to reason that the velocity obtains arbitrary values. cT FIG. 7: Gravitational wave velocity as function of redshift. The main result seems to be a constant value which in turn impliesthat since Q f ≪
1, the scalar coupling function ξ ( φ ) satisfies the differential equation ¨ ξ = H ˙ ξ . Despite being arbitrary, its value is essentially equal to unity due to the fact that Q f ≪ Q t , and thus an infinitesimalvalue is subtracted from unity in Eq. (33). In particular, Q f ∼ O (10 − ) and Q t ∼ O (10 ) hence the reason whythe velocity is equal to unity, and this can also be seen in Fig. 7, but the fact that Q f ≪
1, even in Planck unitswhere κ = 1, implies that a relation of the form Q f = 0 or ¨ ξ = H ˙ ξ for the Gauss-Bonnet scalar coupling functionis not a random choice. In the next section we shall examine the phenomenological implications of constraining theaforementioned velocity be letting Q f = 0. V. PHENOMENOLOGY WITH THE CONSTRAINT ¨ ξ = H ˙ ξ The above formulation seems to be in general in good agreement with not only the observational data, but alsowith the ΛCDM model itself. As stressed in the last model however, the Einstein-Gauss-Bonnet models have aflaw, having to do with a production of a primordial tensor power spectrum, with propagation speed different fromunity. The primordial gravitational wave speed however, must be equal to that of light’s in order to comply with the0GW170817 event [65]. The deviation from unity for the above models is perhaps small in magnitude, implying thatthe effective value is c T ≃ ξ = H ˙ ξ . During the inflationary era, where the slow-roll conditions are assumed to hold true,the previous differential equation can define the time evolution of the scalar field, ˙ φ . This study though was performedusing the slow-roll assumptions, which of course do not hold true in the present late-time context. Although one caneasily imply that the scalar functions of the model continue to have the same primordial relation they had during theinflationary era, and also that the gravitational wave speed remains unity after the horizon crossing of the primordialtensor modes, here we shall adopt a different approach, and since the redshift is used as a variable, we shall solveanalytically the equation. Since ¨ ξ = H ˙ ξ , the solution reads,˙ ξ = λe R Hdt , (34)where λ is an integration constant. Since the definition of redshift is dzdt = − H (1 + z ), the above integral can be solvedanalytically, and thus the final solution is written as,˙ ξ = a ( t ) λ = λ z . (35)This is a quite useful result, to say the least, since by simply imposing constraints on the velocity of gravitationalwaves in the late-time era, the degrees of freedom of the model are decreased by one, similar to the inflationary era,and obtain a functional constraint on ˙ ξ which seems to be model independent. In contrast to the previous sections, nodefinition for ξ ( φ ) is needed since essentially a transformation was performed which replaced ˙ ξ ( φ ) with ˙ ξ ( z ), and giventhat in the equations (3) and (5) which we aim to solve numerically, only ˙ ξ is present, the overall phenomenology isnow significantly altered. Now, the equations of motion are altered as shown below,3 f R H κ = ρ m + 12 ˙ φ + V + f R R − f κ − H ˙ f R κ + 24 λ z H , (36) − f R ˙ Hκ = ρ m + P m + ˙ φ + ¨ f R − H ˙ f R κ − λ z H ˙ H , (37) V φ + ¨ φ + 3 H ˙ φ − f φ κ + λ z G ˙ φ = 0 . (38)It should be noted that all the previous equations acquired in section II are still valid even when the constraint isapplied. Furthermore, given that G is small from its nature, for certain values of λ , the phenomenology for such choiceis indistinguishable from the one obtained without the constraint. For instance, if we recall the results for the f ( R )model studied previously, it was mentioned that the scalar field cannot alter the results. The same can be said aboutthe case of f ( R ) with ˙ ξ = λ z for a plethora of values for λ . By altering λ and giving it a quite large value, say λ = 10 , then the solution diverges and compatibility cannot be achieved. Let us proceed with a specific model andexamine the impact the constraint has on the late-time evolution.Consider a non-minimally coupled model of the form, f ( R, φ ) = h ( φ ) R , (39) h ( φ ) = φ φ , (40) V ( φ ) = V φ , (41)and obviously ˙ ξ = λ z , (42)1 - - q (cid:21) - - - s (cid:22) O m FIG. 8: Cosmological parameters q ( upper left), j (upper right), s (bottom left) and Om (bottom right) with respect to z .Once again, no dramatic change is present in the non minimally coupled constrained case with the R unconstrained. where φ and V are auxiliary parameters with mass dimensions [ m ] and [ m ] respectively. In this case as well,since there exists only a linear R term, only a single initial condition for y H is needed. As was the case with theprevious two models, we shall use the same value, meaning y H ( z = 10) = Λ3 m s (cid:16) z f (cid:17) . In consequence, letting φ = 1, V = 1, λ = 1, φ ( z = 10) = 10 − M P , dφdz (cid:12)(cid:12)(cid:12) z =10 = − − M P then the results obtained are compatible withthe ΛCDM model as shown in Fig. 9, while in Fig. 10 we present the behavior of the dark energy EoS parameter andthe dark energy density parameter as functions of the redshift. The results of our numerical analysis correspondingto the values of the statefinders and of the dark energy EoS parameter and the dark energy density parameter atpresent time, can be found in Table III. As it can be seen in Table III our model is in good qualitative agreement withthe ΛCDM model and also is compatible with the 2018 Planck constraints on the cosmological parameters, when thedark energy EoS parameter and the dark energy density parameter are considered. TABLE III:
Parameter h ( φ ) R Λ CDM Value q(z=0) -0.520794 -0.535j(z=0) 0.99987 1s(z=0) -0.00004423 0 Om ( z = 0) 0.319364 0.3153 ± DE (0) 0.680671 0.6847 ± ω DE (0) -0.99984 -1.018 ± Genuinely speaking, the linear R case we studied in the previous section, without constraints and the h ( φ ) R casewith constraints studied in this section, do not differ so much. The main difference lies in the evolution of statefinder y H and subsequently the EoS on the latter case, however even when the EoS is dynamically evolving, it does so with2 - - (cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28) - - (cid:29)(cid:30)(cid:31) !" ’ D E FIG. 9: Dark energy parameters ω DE (left) and Ω DE for the non minimally coupled model. Since y H is depending on z , ω DE evolves as well however infinitesimally near the expected value − DE increases with time as it is expected. an infinitesimal rate that its value essentially cannot be distinguished from unity.As a comment, it should be noted that in this case no difference between the constrained and the unconstrainedGauss-Bonnet phenomenology is found, however the latter seems quite arbitrary from one perspective, meaning thatthe velocity of gravitational waves is dynamically evolving in various cosmological eras and c T just happens to beunity since Q f ≪ Q t . However, even uncontrolled, the fact that Q f ≪ ξ = H ˙ ξ is satisfied one way or another, hence instead of coming to such numerical conclusion at the end, itis beneficial to begin with such statement as the degrees of freedom seem to decrease in the first place. The mainidea was to examine a model with the constraint c T = 1 and prove that compatibility can be achieved by taking intoconsideration that the constraint imposed from the velocity of gravitational waves decreases the degrees of freedomsuch that the Gauss-Bonnet scalar coupling function can be replaced by a single parameter. In the literature [70]such a question has been addressed, and the results were quite different quantitatively in comparison to the presentmodel. In principle however, the compatibility is a model dependent feature.Before closing we need to discuss an interesting scenario, in view of the unified description of inflation with the darkenergy era that Einstein-Gauss-Bonnet theory, combined with the fact that it is a string theory originating theory. Inthe present paper we assumed that the dark matter perfect fluid consists of an unknown particle, but string theoryhas also offered the possibility of having axion like particles present even in the pre-inflationary era. In fact, in theaxion like particle phenomenology with a primordial pre-inflationary era broken U (1) symmetry. The axion due tothe breaking of this symmetry is frozen in its vacuum expectation value, but as the Universe expands, the axionbehaves as a condensate and evolves as a dark matter perfect fluid, which makes it a perfect candidate for a low-massweakly interactive massive dark matter particle. Such scenarios in the context of modified gravity have been studiedin the literature [71–74], so one interesting scenario is to have the combined presence of the axion coupled to theGauss-Bonnet scalar. This would utterly change the symmetry breaking patter of the primordial U (1) symmetry, dueto the presence of the coupling ξ ( φ ) in the axion equation of motion, even pre-inflationary. The calculation might geteasier if it is assumed that the Gauss-Bonnet corrections and the flat four dimensional spacetime, FRW-like, are theresulting outcomes of the quantum era. Also, the presence of the Gauss-Bonnet non-minimal coupling would alterthe post-inflationary evolution, and in addition, in this scenario, the axion U (1) symmetry might be unbroken duringthe inflationary era. These issues are interesting material for a focused future work. VI. CONCLUSIONS
In this work we investigated the late-time phenomenology aspects of scalar-coupled f ( R, G ) gravity. We focused ontheories of Einstein-Gauss-Bonnet form, and we examined three types of models, f ( R ) gravity Einstein-Gauss-Bonnetmodels, and pure Einstein-Gauss-Bonnet models, with arbitrary choice of the scalar functions of the models and withconstrained functions of the models. Our numerical analysis indicated that for the models containing the f ( R ) gravity,the late-time dynamics is very much affected by the f ( R ) gravity part, and thus in those cases, the Einstein-Gauss-Bonnet coupling does not affect the dynamics. For the pure Einstein-Gauss-Bonnet, we made a novel assumptionrelated to the requirement that the primordial gravitational wave speed is equal to unity, which in turn imposed afunctional constraint on the functional form of the Gauss-Bonnet scalar coupling function. The exiting feature inthe late-time study by taking into account the gravitational wave speed constraint, is the fact that the functional3form of ˙ ξ is model independent, and has a specific form given in terms of the Hubble rate and the redshift. Thissimplification is rather interesting to think that there is a strong motivation to assume that the primordial gravitywave speed should be set equal to unity for all the cosmic times after the first horizon crossing of the primordialtensor modes. This could in fact constrain the functional forms of the scalar potential and of the scalar couplingfunction, directly from the inflationary era and thereafter, but we did not go to deep to this study, since it seemsthat the Gauss-Bonnet coupling as we studied it in section II does not play a significant role during the late-timeera. It should actually, it is of the order ∼ H , but we aimed to formally investigate the phenomenology of thesemodels. The positive outcome we keep is that the primordial gravitational wave speed constraint has some effect onthe late-time cosmological dynamics, and this is a motivation for us to go investigate astrophysical scenarios relatedto Einstein-Gauss-Bonnet models, with the potential and the scalar coupling functions being related in the way wedemonstrated in [54–58]. Such task we aim to address in a future work.Let us discuss at this point the issue of choosing the scalar functions of the models we studied in this paper. In orderto address this comment we need to elaborate further on the concept of gravitational waves. Theories containing aGauss-Bonnet term coupled to an arbitrary scalar function are notorious for producing primordial gravitational waveswhich propagate with a velocity which is different from the speed of light. Our initial work on Gauss-Bonnet startedby confronting this nasty feature during the inflationary era and coming up with ways in order to remedy the theory,see Ref. [58]. The reason that the theory needs to be rectified, from our point of view, it that there exists no knownmechanism which could produce massive primordial gravitons which were later turned into massless according to therecent GW170817 event. Therefore, a reasonable assumption is to impose that the theory is described by masslessgravitons throughout the evolution of the Universe. The main result extracted from this simple statement is that theGauss-Bonnet scalar coupling function must satisfy the differential equation ¨ ξ = H ˙ ξ and since Hubble’s parameteris given from the Friedmann equation, it becomes apparent that in this scenario both scalar coupling functions areconnected, i.e. V ( φ ) and ξ ( φ ) are chosen in such a way so that the aforementioned differential equation is satisfiedproperly. This can be seen easily in the slow-roll regime where many terms are assumed to be subleading but during thelate-time era, we are interested in all of them. The main idea was to start without imposing the constraint derived fromthe velocity of gravitational waves and afterwards compare the results between a constrained and an unconstrainedmodel. As it turns out, both assumptions lead to relative similar results given that in the set of equations, other termsare dominant during the late-time era. Moreover, in order to perform a fully self-consistent study, further auxiliaryparameters such as the jerk, snap and Om ( z ) where used in order to see the pros and cons of the models studied inthis case and to see explicitly where and if some cosmological parameters deviate from observations, thus renderingthe model unsuitable. Concerning the initial conditions at redshift z = 10, the first variable is designated in order tocoincide with observations while the initial value of the scalar field, since it cannot be observed, is chosen arbitrarily.The previous value seems to produce a smooth evolution for the scalar field with respect to cosmic time and moreoverall the physical observables values of the model are compatible with the observational data.Moreover, let us comment that in the present text, the early-time phenomenology was not considered, howeverin order have an inferior Gauss-Bonnet term in the late-time era, one would expect that the scalar field relatedterms are subleading from the first horizon crossing and thereafter. No comment can be made about the Planck eraunfortunately, but at least in the inflationary era, the R part of the equations of motion should at best receive mildcorrections so that to obtain acceptable values for the scalar spectral index and the tensor-to-scalar ratio. Assumingthat the scalar-field evolves with either a slow-roll or a constant rate of roll, then by implementing either similar slow-roll conditions for the Gauss-Bonnet coupling in the unconstrained case, or working in the constrained case and in thepresence of an arbitrary scalar potential, then compatibility with the observational data can be achieved relativelyeasily, while simultaneously the order of magnitude of several scalar functions and their derivatives can be negligible.To summarize, a smooth early-time description which ensures that the scalar components of the model act as aneffective dark energy density in the Friedmann equation can be achieved by working with the classical prescription ofinflation and assume a potential driven inflation with either slow or constant-roll evolution. In either case, the R partof the equations of motion becomes dominant in comparison to the scalar one as time flows and especially during thelate-time era.Before closing we need to discuss an importance issue, related with the choice of the arbitrary functions of ourmodels. As we demonstrated, the Gauss-Bonnet coupling contributes slightly to the late-time era. The fact thatthe Gauss-Bonnet coupling does not affect so significantly the late-time phenomenology of the models studied in theunconstrained case, is a direct consequence of their relevance. It turns out that the Gauss-Bonnet term G is notdominant in the dark energy era (it scales as G ∼ H for a flat FRW spacetime). In consequence, all the scalarterms manage to produce what is perceived as a a nearly constant term in the Friedmann equation, which in turn isinterpreted as an effective late-time cosmological constant. In principle, in order to see different results, one wouldneed a really strong contribution from the scalar components of the model, however no such case was observed in therespective computer program which was developed for this purpose. In fact as we evinced, in the first case, the f ( R )contribution, and mainly from the third term R δ , is the dominant driving force. Similarly, in the second case, a simple4 R term even in the presence of a scalar potential is the driving force. Finally, in the constrained model where thenon-minimally coupled case was examined, the realization that the scalar field itself is subleading is the factor whichensures that the R contribution is once again dominant. The reason behind such designation for the scalar couplingwas mainly the simplicity of the resulting expressions and also the production of a viable inflationary era, which canbe ascertained from recent observations, hence the reason why power-laws and also dilatonic couplings were assumed.Finally, it is interesting to compare the results obtained in this paper, with the ones obtained in Ref. [75] using againa dilatonic Einstein-Gauss-Bonnet gravity, but in the context of an emergent gravity scenario. The results producedin our paper are different from the ones obtained in Ref. [75]. This can be attributed to the specific form of thescale factor that the authors have chosen and/or to the non canonical kinetic term of the scalar field. In the presentarticle, we did not assume a specific form for the scale factor and in fact the scale factor is directly derivable fromthe equations of motion once V ( φ ), ξ ( φ ) and φ ( t ) are chosen. Moreover, the kinetic term is assumed to be canonical,meaning that ˙ φ is not coupled to φ . These two different assumptions may be the reason why the results do not agree,since Ref. [75] deals with the emergent Universe scenario, which has its own attributes however, so this issue wasworthy of mentioning. Acknowledgments
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