Cosmography and the redshift drift in Palatini f(R) theories
aa r X i v : . [ g r- q c ] M a r Noname manuscript No. (will be inserted by the editor)
Cosmography and the redshift drift in Palatini f ( R ) theories Florencia A. Teppa Pannia · Santiago E. Perez Bergliaffa · NivaldoManske
Received: date / Accepted: date
Abstract
We present an application to cosmologicalmodels in f ( R ) theories within the Palatini formalismof a method that combines cosmography and the ex-plicit form of the field equations in the calculation ofthe redshift drift. The method yields a sequence of con-straint equations which lead to limits on the parameterspace of a given f ( R )-model. Two particular familiesof f ( R )-cosmologies capable of describing the currentdynamics of the universe are explored here: (i) powerlaw theories of the type f ( R ) = R − β/ R n , and (ii)theories of the form f ( R ) = R + α ln R − β . The con-straints on ( n, β ) and ( α, β ), respectively, limit the val-ues to intervals that are narrower than the ones pre-viously obtained. As a byproduct, we show that whenapplied to General Relativity, the method yields valuesof the kinematic parameters with much smaller errorsthat those obtained directly from observations. Keywords
Modified Gravity · Cosmography · Redshift Drift
PACS · The observational evidence of the accelerated expan-sion of the universe can be described by assuming thatgravity is governed by a theory different from GeneralRelativity (GR) at large scales and late times. One ofthe most studied modifications of GR are the f ( R ) the-ories of gravity, which are formulated by substituting Departamento de F´ısica Te´orica, Instituto de F´ısica, Univer-sidade do Estado de Rio de Janeiro, CEP 20550-013, Rio deJaneiro, Brasil.E-mail: [email protected] the usual Einstein-Hilbert Lagrangian density by an ar-bitrary function of the Ricci curvature scalar R [1,2,3,4,5]. Cosmological models built with them seek to de-scribe current astronomical data without the use of theso-called dark energy , which in the scope of the stan-dard cosmological model amounts up to ∼
70% of thetotal matter content of the observable universe [6].Within f ( R )-theories the dynamics of the gravita-tional degrees of freedom is governed by the field equa-tions derived from the minimisation of the correspond-ing gravitational action. Different variational principles(usually referred as formalisms) can be considered inorder to get such equations, according to the role at-tributed to the connection Γ [1]. Among them, thePalatini formulation is based on the assumption thatthe metric and the connection are independent fields.In such a case, the corresponding Riemann and Riccitensors are constructed with a connection a priori in-dependent of the metric [7]. The Einstein-Hilbert La-grangian is then replaced by a function f ( R ), where R is defined as R ≡ g µν R µν ( Γ ), and R µν is the Riccitensor defined in terms of the independent connection.The dependence of the function f with the scalarcurvature R introduces a set of constants, characteris-ing a particular family of f ( R )-cosmologies. The par-ticular form of f as well as a range of suitable valuesfor the constants must be chosen taking into accountseveral criteria, such as the appropriate sequence of cos-mological eras [8], the correct dynamics of cosmologicaldensity perturbations [9], as well as the correct weak-field limit at both the Newtonian and post-Newtonianlevels [10], the well-posedness of the Cauchy problem Florencia A. Teppa Pannia et al. [11], and the correct fit of cosmological observables [12,13,14]. Among the observable quantities with the poten-tial of discriminating between f ( R )-models and thosebased on GR, particular attention was recently givento the time variation of the cosmological redshift z dueto the variation of the expansion rate of the universe,namely the redshift drift (RD). The possibility of us-ing this observable as a test of cosmological modelswas first proposed by Sandage [16], and later developedby other authors [17,18,19,20,21]. The redshift drift(a.k.a. Sandage-Loeb effect) was considered for manyyears of little use in the task of distinguishing cosmolog-ical models because of the difficulties associated to itsmeasurement. However, this observable may be impor-tant for Cosmology, since it allows the test of the Coper-nican Principle [19], and has the potential to distinguishdifferent cosmological models [20]. As first discussed in[22], the redshift drift can be also used to limit thevalues of the otherwise arbitrary constants of a giventheory of gravity by resorting to its series expansionin powers of z . Such expansion can be computed usingtwo different approaches. The first one is of a cosmo-graphical type, i.e. independent of the dynamics of thesubjacent theory and only based on the assumed sym-metries of the space-time. The coefficients of the seriesexpansion depend, in this case, on the so-called cosmo-graphical kinematic parameters defined in terms of thetime derivatives of the scale factor, the values of whichfollow from different observations. The second approachyields a series expansion which explicitly depends onthe dynamics of the scale factor through the gravita-tional field equations of a given theory. The subsequentterm-by-term comparison of the series expansions leadsto a sequence of constraint equations, which explicitlyrelate different orders of derivatives of the scale factor(through the kinematic parameters) and the function f ( R ) and its derivatives (and, consequently, the con-stants of an specific family of f ( R )-models) evaluatedat z = 0. It is worth pointing out that such a compar-ison does not actually depend on the actual measure-ments of the redshift drift.The constraint equations mentioned above can beused in two directions: (i) to get theoretical estima-tions of the kinematic parameters if General Relativityis assumed, and (ii) to constrain the space-parameter ofparticular f ( R )-models if the kinematic parameters arederived from independent observational data. In thisregard, cosmography has been widely used to distin-guish between cosmological models (see for instance [23,24,25,26,27,28,29,30]). In particular, the application The Dolgov-Kawasaki instability does not take place inin f ( R ) theories, see [15]. of the comparative method coming from the redshiftdrift series was used in [22] within the metric formal-ism to constrain the space parameter of f ( R )-models.The main goal of this work is to extend the analysis of[22] to models built with f ( R ) theories in the Palatiniformalism. We shall apply the results to the particularcases of power-law gravity ( f ( R ) = R − β/ R n ) andlogarithmic gravity ( f ( R ) = R + α ln ( R ) − β ).The paper is organised as follows. General consider-ations about the RD and the cosmographical approachfrom which constraints are obtained are presented inSection 2. The application of these ideas to the stan-dard cosmological model are given in Section 3. In Sec-tion 4, we present the features of f ( R )-theories withinthe Palatini formalism and apply the method to set lim-its on the parameter space of the two above-mentioned f ( R ) functions, and we compare our results with previ-ously reported limits. Our final remarks are presentedin Section 5. The redshift of a photon emitted by a source at time t and observed at time t , is defined as follows:1 + z = a ( t ) a ( t ) . (1)Due to the variation of the expansion rate of the uni-verse, the redshift of a source is indeed a function oftime. Then, a second photon emitted at t ′ = t + ∆t willhave a redshift z ( t ′ ). This time variation of the redshiftis the so-called redshift drift , and can be expressed upto first order as [18] ∆z∆t = ˙ a ( t ) − ˙ a ( t ) a ( t ) = (1 + z ) H − H ( z ) , (2)where ∆t is the time delay between the two observedphotons. Note that this observable depends neither onspecific features of the source (such as its absolute lu-minosity) nor on the definition of a standard ruler.We shall present next the two approaches to theredshift drift, each of which involve a series expansionof this observable in terms of z . The first one is basedon geometric and kinematic properties of the metric,namely a cosmographic treatment, whereas the othertakes into account the dynamics imposed by a chosentheory of gravity through the cosmological field equa-tions. A cosmographic approach.
The geometric and kinematicproperties of the metric (14) are characterised by the so-called kinematic parameters, defined as the coefficients osmography and the redshift drift in Palatini f ( R ) theories 3 of the series expansion of the scale factor around t . Inparticular, H ≡ (cid:18) ˙ aa (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t , q ≡ − H (cid:18) ¨ aa (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t , j ≡ H (cid:18) ... aa (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t , where a dot indicates derivatives w.r.t. time, and thesub-index 0 indicates that all quantities are evaluatedat z = 0. The aim of the cosmographic approach in thissetting is then to compute the redshift drift in terms ofthese quantities. By performing a series expansion of H ( z ) in termsof the above defined kinematic parameters, the redshiftdrift can be written as [22] ∆z∆t ( z ) = − H q z + 12 H (cid:0) q − j (cid:1) z + O ( z ) . (3)The coefficients of this equation, which follows from thecosmographic approach, depend purely on the proper-ties of the FLRW metric, in the sense that no dynamicalevolution for the scale factor was assumed. A dynamical approach.
We shall compute the dynamicalcounterpart of Eq. (3), now considering the dynamicsobeyed by the scale factor a ( t ). Let us consider againthe general expression for the redshift drift given byEq. (2). The dynamics enters through the Hubble pa-rameter function, H ( z ), the evolution of which is com-pletely determined by the gravitational field equationsonce a theory of gravity is chosen. Using the expansionof H ( z ) in a Taylor series of the redshift, together withthe chain rule d H/ d z = d H/ d t · d t/ d z , we obtain ∆z∆t ( z ) = H + ˙ H H ! z ++ ¨ H H − ˙ H H + ˙ H H ! z O ( z ) . (4)Note that Eq. (4) for the redshift drift involves the grav-itational field equations through H and its derivatives. Λ CDM-model
Cosmography is a useful standpoint to study assump-tions of cosmological models which are based entirely onthe Cosmological Principle [33], and provides valuablemodel-independent information about the evolution ofthe scale factor and its derivatives (see also [34,35,36]for an extended discussion). This mathematical frame-work is inherently kinematic in the sense that it relies The same approach has been applied to the luminositydistance in [31,32]. only on geometrical assumptions for the metric and isindependent of the dynamics obeyed by the scale factor.We will focus in this section on the cosmographicinformation provided by the redshift drift in the case ofthe Λ CDM cosmology. The term-by-term comparison ofboth series for different powers of z leads to a sequenceof equations (actually, an infinite number of them) forthe kinematic parameters in terms of the cosmologicalparameters. The first members of the sequence are q = 12 Ω m, − Ω Λ, , (5) j = 52 Ω m, − Ω m, Ω Λ, + Ω Λ, − Ω m, . (6)These relations allow to estimate values for the kine-matic parameters as a function of the value of Ω m, .This estimation has the advantage of being indepen-dent of the restrictions associated to the convergenceand the truncation of the Taylor series usually imple-mented in cosmography (see discussion in [37,25]). Tak-ing for the dimensionless energy densities the values Ω m, = 0 . ± .
007 and Ω Λ, = 0 . ± . q = − . ± . j = − . ± . . (8)The above values are in agreement with other estima-tions of the kinematic parameters coming from differentdata sets [38,39,40], but present significantly smallererrors. This is particularly convenient for the case ofhigher-order parameters, such as j , which have largeerrors when estimated by other methods. f ( R )-cosmologies S [ g, Γ, ψ m ] = Z d x √− g (cid:20) κ f ( R ) + L m ( g, ψ m ) (cid:21) , (9)where f ( R ) is an arbitrary function of the curvaturescalar R ≡ g µν R µν ( Γ ), with R µν ( Γ ) defined as R µν ( Γ ) = − ∂ µ Γ λλν + ∂ λ Γ λµν + Γ λµρ Γ ρνλ − Γ λνρ Γ ρµλ . The matter La-grangian density L m depends on the matter fields ψ m ,the metric g and its first derivatives, but does not de-pend on the affine connection Γ , which appears only inthe gravitational action. The energy-momentum ten-sor is conserved since the total (gravitational plus mat-ter) action is diffeomorphism-invariant, and gravity andmatter are minimally coupled by assumption [41,42].The Palatini f ( R ) gravity is then a metric theory (in Florencia A. Teppa Pannia et al. the sense that the matter is minimally coupled to themetric and not coupled to any other fields), and hencethe energy-momentum tensor T µν and its conservationlaws will remain the ones of GR [1].The variation of the action with respect to the met-ric and the connection yields, respectively, [1] f ′ ( R ) R µν − f ( R ) g µν = κ T µν , (10)¯ ∇ ρ (cid:20) √− g (cid:18) δ ρλ f ′ ( R ) g µν − δ µλ f ′ ( R ) g ρν (11) − δ νλ f ′ ( R ) g µρ (cid:19)(cid:21) = 0 , where f ′ ≡ d f / d R , and ¯ ∇ ρ represents the derivativeoperator associated to the independent connection Γ ρµν ,which is assumed symmetric (torsion-less). We use unitssuch c = 1 and κ = 8 πG .The trace of Eq. (10) yields an algebraic equationfor the Palatini Ricci scalar R in terms of the trace ofthe energy-momentum tensor T , f ′ ( R ) R − f ( R ) = κ T . (12)The trace of Eq. (11), written as¯ ∇ ρ (cid:0) √− gf ′ ( R ) g µν (cid:1) = 0 , (13)is used to define the connection of the metric h µν ≡ f ′ ( R ) g µν , conformal to g µν . Then, ¯ ∇ ρ h µν = 0 and theconnection is actually the Christoffel symbols of theconformal metric h µν . The General Relativity case isrecovered when f ′ ( R ) = 1 (see Eq.13).Let us consider cosmological solutions by assumingthe homogeneous and isotropic flat FLRW metric, de-scribed by the line elementd s = − d t + a ( t )[d r + r d Ω ] . (14)The modified Friedmann equation for a dust-dominateduniverse then becomes [8] H = 16 f ′ ( R ) 2 κ ρ m + R f ′ ( R ) − f ( R ) h − f ′′ ( R )( R f ′ ( R ) − f ( R )) f ′ ( R )( R f ′′ ( R ) − f ′ ( R )) i . (15)4.2 Constraints on particular f ( R )-modelsTwo different approaches to compute a series expan-sion of the redshift drift were presented in the SectionII, namely the cosmographic and dynamical treatments.As exemplified by the Λ CDM model, the term-by-termcomparison of both series expansion leads to a sequenceof constraint equations to be satisfied for a given cosmo-logical model. In the case of f ( R )-models these equa-tions relate kinematic quantities with the parameters of the f ( R ) function. Therefore, in this section we willconsider observational values for the kinematic param-eters (already estimated in the literature from differentdata sets) to constrain the parameter space of the f ( R )functions.From the comparison of the linear terms in z in bothexpansion series (3) and (4), we obtain a relation of theform G ( q , Ω m, ; R , f , f ′ , f ′′ , f ′′′ ) = 0 , (16)with G a lengthy algebraic function of its arguments .This expression, together with the trace Eq. (12), di-rectly implies a constraint on the space-parameter of agiven f ( R ). Note that the absence of H in the con-straint equation 16 reduces the sources of error. Weshall work with the values q = − . +0 . − . , estimatedin [39] using the supernova type Ia JLA compilation[44], and Ω m, = 0 . ± .
007 from the last results ofPlanck Collaboration [6].We present next the analysis of two f ( R )-cosmologicalmodels: (i) power-law type models, characterized by thefunction f ( R ) = R − β/ R n , and (ii) logarithmic typemodels of the form f ( R ) = R + α ln R − β . These par-ticular choices of the function f ( R ) allow the occur-rence of three cosmological phases (radiation-, matter-, and de Sitter-dominated eras) [8], and describe theaccelerated expansion without the introduction of non-standard sources of matter. Moreover, these theoriessatisfy the general criteria for cosmological viability dis-cussed in Sect. 1. It is important to remark that both modelshave been shown to accommodate a phase oflate-time accelerated expansion also by the in-tegration of the corresponding equations of mo-tion, and for a range of parameters compatiblewith our findings. Model (i) presents a transi-tion from non-accelerated to accelerated expan-sion at around z = 1 , as shown in [46, 47] bymeans of the evolution of the effective equationof state parameter w eff in terms of z , and alsoin [48, 47] by studying the behaviour of q ( z ) . Thesame can be said about model (ii), for which thefunction w eff ( z ) was analyzed in [46, 47], and thedeceleration factor q ( z ) was presented in [47]. Additional constraints can be obtained by taking into ac-count higher-order terms of both series, which involve higherorder derivatives of the scale factor (that is, additional kine-matic parameters). See for instance [43]. In addition to these models, the cosmology of the so-called exponential gravity, described by the function f ( R ) = R − αnH (cid:16) − e −R / ( αH ) (cid:17) , has been widely studied withinthe Palatini formalism [45]. However we have verified thatthis type of models does not satisfy the necessary conditionimposed by the correct sequence of cosmological eras.osmography and the redshift drift in Palatini f ( R ) theories 5 f ( R ) = R − β/ R n Theoretical and observational studies determining theviability of power-law type gravity within the Palatiniformalism have been presented by many authors [49,50,51,52,53]. We restrict our analysis here to the function f ( R ) = R − β/ R n , where n is a dimensionless parame-ter and β is reported in units of H n − . Using dynam-ical analysis, it was shown that this type of theories canreproduce the sequence of radiation-dominated, matter-dominated and de-Sitter eras for n > − β > Λ CDM cosmology ( β ≈ − ) [42,13],other analyses combining CMB and BAO data yield( n, β ) = (0 . +0 . − . , . +2 . − . ) [8]. These values werelater improved with combinations of CMB, SNIa andBAO data, along with Hubble parameter estimations[55,56,57], as well as data coming from strong lens-ing analysis [58,14]. Cosmographic constraints were ob-tained from the examination of the deceleration param-eter [48] and the so-called statefinder diagnostic [46].Recent constraints coming from cosmological standardrulers in radio quasars have been reported with consid-erably smaller ranges: ( n, β ) = (0 . +0 . − . , . +0 . − . )[59].In this work we present the constraint relation forthe ( n, β ) space, which must be satisfied by the param-eters to be consistent with the cosmographic and dy-namical approaches of the redshift drift. The restrictivecondition coming from Eq. (16) is shown in Fig. 1, to-gether with the best-fit values previously reported andtheir associated error bars. The shadowed regions indi-cate the error propagation associated to the Ω m, and q values. Since the reported error for the q estima-tion is large, we also include a forecast correspondingto a measurement of q with an error of 10% of its value.The plot shows that mildly improved estimations of thedeceleration parameter would narrow even more the al-lowed region on the parameter space. Our constraintrelation is consistent with the Λ CDM model, which isrecovered for ( n, β ) = (0 , . ± . β nf(R)=R- β /R n RD constraint Λ CDM[8][56][55][54][48]
Fig. 1
Constraint on the parameter space ( n, β ) for theoriesof the type f ( R ) = R − β/ R n coming form the cosmographicapproach to the redshift drift, together with the best-fit val-ues previously reported and the Λ CDM case. The shadowedregions indicate the error propagation associated to the Ω m, and q values, and a forecast corresponding to a measurementof q with an error of 10% of its value. f ( R ) = R + α ln R − β This type of theories within the Palatini formalism wasfirstly studied in [60], where it was shown that logarith-mic terms in the action may be of use in the descriptionof the accelerated expansion and reduce to the standardFriedmann evolution for high redshifts. Furthermore, itwas shown that logarithmic gravity is one of the sim-plest forms capable of reproducing a suitable sequenceof cosmological eras [8]. Limits on the parameter space( α, β ), where α and β are computed in units of H , havebeen less studied than that in the previous case. Esti-mations using different sets of data coming from SN,BAO and CMB observations were obtained in [8], withbest-fit values ( α, β ) = (0 . +1 . − . , . +3 . − . ). A morerecent work has reported ( α, β ) = ( − . +1 . − . , . Ω m, and q parameters, to-gether with a forecast corresponding to a measurementof q with an error of 10% of its value. In this case ourconstraint significantly improves the limits in the pa-rameter space and is also consistent with the Λ CDMmodel.
Alternatives to General Relativity may be the key todescribe the accelerated expansion of the universe with-out the use of the so-called dark energy. In particular,
Florencia A. Teppa Pannia et al. β α f(R)=R- α lnR- β RD constraint Λ CDM[8][46][47]
Fig. 2
Constraint on the parameter space ( α, β ) for theoriesof the type f ( R ) = R + α ln R − β coming from the cos-mographic approach to the redshift drift, together with thebest-fit values previously reported and the Λ CDM case. Theshadowed region indicates the error propagation associatedto the Ω m, and q values, and a forecast corresponding to ameasurement of q with an error of 10% of its value. cosmological models based on f ( R )-theories of gravitywithin the Palatini formalism have been constructedto deal with observational data sets and solar-systemconstraints, yielding a satisfactory description of thelate time dynamics of the universe. The investigationof cosmological observables capable to distinguish be-tween different models can also be used as a tool toset constraints on the parameters of such theories. Inthis direction, a novel method for f ( R )-theories withinthe metric formalism was recently proposed in [22]. Itsessence is the comparison of two Taylor expansions ofa given observable. While the first one is of the cos-mographic type, namely based on the assumed symme-tries of the space-time and independent of the dynam-ics obeyed by the scale factor, the second takes intoaccount the specific dependence of H with z , deter-mined by the field equations of a given gravity theory.The order-by-order comparison of these two series leadsto a sequence of constraints relating the parameters ofthe theory. Furthermore, the method does not rely onthe actual measurement of the observable, but on thecondition that both series coincide term-by-term. Thislast feature can be taken as an advantage for explor-ing limits imposed by yet to-be-measured quantities asis the case of the redshift drift. Using values for thecosmological parameters Ω m, and Ω Λ, obtained fromobservations, we have shown that a direct application ofthis method to GR yields theoretical estimations of thekinematic parameters with an error much smaller thanthe one that follows straight from the cosmographic ap-proach.The main goal of this work was to apply the methodto f ( R )-cosmological models within the Palatini for- malism. Using two series expansions of the redshift driftwe obtained a constraint relation, given by Eq.(16), onthe parameter space of such models in terms of theso-called kinematic parameters. Two particular f ( R )-models were studied: (i) those based on power-law typetheories, characterised by a Lagrangian function of theform f ( R ) = R − β/ R n , and (ii) logarithmic-gravitytype models of the form f ( R ) = R + α ln R − β . Thesetwo particular models allow the occurrence of the rightsequence of cosmological eras, and have been previouslystudied in the literature. We have used here the red-shift drift expansions to impose an independent con-straint relation on their parameters in terms of q and Ω m, . Our results, presented in Figs. 1 and 2, are ingood agreement with previously reported values, butwith considerably smaller errors. These could be re-duced even more taking into account future improvedestimations of the kinematic parameters, as shown bythe forecast obtained using an error of ten percent ofthe value of q .To close, we would like to emphasise that the boundsobtained by the method presented here do not dependon the actual measurement of the redshift drift. Suchbounds can also be considered together with those com-ing from theoretical considerations (such as the energyconditions [43]), in order to decide whether a given f ( R ) theory furnishes an appropriate description of thecurrent state of the universe. Acknowledgements
FATP acknowledges support from PNPD/CAPESand UERJ. SEPB and NM would like to acknowledge supportfrom FAPERJ and UERJ.
References
1. T.P. Sotiriou, V. Faraoni, Reviews of Modern Physics ,451 (2010)2. A. de Felice, S. Tsujikawa, Living Reviews in Relativity , 3 (2010)3. S. Capozziello, V. Faraoni, Beyond Einstein Gravity. Asurvey (...) (Springer Science, 2011)4. S. Nojiri, S.D. Odintsov, Phys. Rept. , 59 (2011)5. S. Nojiri, S.D. Odintsov, V.K. Oikonomou, Phys. Rept. , 1 (2017)6. N. Aghanim, et al., (2018)7. G.J. Olmo, International Journal of Modern Physics D , 413 (2011)8. S. Fay, R. Tavakol, S. Tsujikawa, PRD , 063509 (2007)9. K. Uddin, J.E. Lidsey, R. Tavakol, Classical and Quan-tum Gravity , 3951 (2007)10. G.J. Olmo, Phys. Rev. D (8), 083505 (2005)11. M. Salgado, Classical and Quantum Gravity , 4719(2006)12. T. Koivisto, Phys. Rev. D , 083517 (2006)13. B. Li, K.C. Chan, M.C. Chu, Phys. Rev. D (2), 024002(2007)14. K. Liao, Z.H. Zhu, Physics Letters B , 1 (2012)osmography and the redshift drift in Palatini f ( R ) theories 715. T.P. Sotiriou, Physics Letters B , 389 (2007)16. A. Sandage, The Astrophysical Journal , 319 (1962)17. G.C. McVittie, The Astrophysical Journal , 334(1962)18. A. Loeb, Astrophysical Journal Letters , L111 (1998)19. J.P. Uzan, C. Clarkson, G.F.R. Ellis, Physical ReviewLetters (19), 191303 (2008)20. C. Quercellini, L. Amendola, A. Balbi, P. Cabella,M. Quartin, Phys. Rept. , 95 (2012)21. L. Amendola et al., JCAP , 042 (2013)22. F.A. Teppa Pannia, S.E. Perez Bergliaffa, JCAP , 030(2013)23. F.Y. Wang, Z.G. Dai, S. Qi, A&A , 53 (2009)24. S. Capozziello, V. Cardone, V. Salzano, Phys. Rev. D ,063504 (2008)25. S. Capozziello, R. Lazkoz, V. Salzano, Phys. Rev.D (12), 124061 (2011)26. A. Aviles, A. Bravetti, S. Capozziello, O. Luongo, ArXive-prints (2012)27. A. Shafieloo, A.G. Kim, E.V. Linder, Phys. Rev.D (12), 123530 (2012)28. S. Capozziello, O. Luongo, ArXiv e-prints (2014)29. S. Capozziello, O. Farooq, O. Luongo, B. Ratra, Phys.Rev. D (4), 044016 (2014)30. L. Pizza, Phys. Rev. D (12), 124048 (2015)31. T. Chiba, T. Nakamura, Progress of Theoretical Physics , 1077 (1998)32. M. Visser, General Relativity and Gravitation , 1541(2005)33. S. Weinberg, Gravitation and Cosmology: Principles andApplications (...) (John Wiley & Sons, 1972)34. V.C. Busti, ´A. de la Cruz-Dombriz, P.K.S. Dunsby,D. S´aez-G´omez, Phys. Rev. D (12), 123512 (2015)35. P.K.S. Dunsby, O. Luongo, International Journal of Ge-ometric Methods in Modern Physics , 1630002-606(2016)36. A. de la Cruz-Dombriz, PoS DSU2015 , 007 (2016)37. C. Catto¨en, M. Visser, Classical and Quantum Gravity , 5985 (2007)38. Y.N. Zhou, D.Z. Liu, X.B. Zou, H. Wei, The EuropeanPhysical Journal C (5), 281 (2016)39. A. Aviles, J. Klapp, O. Luongo, Physics of the Dark Uni-verse , 25 (2017)40. S. Capozziello, R. D’Agostino, O. Luongo, MNRAS (3), 3924 (2018)41. R.M. Wald, General Relativity (University of ChicagoPress, 1984)42. T. Koivisto, Classical and Quantum Gravity , 4289(2006)43. S.E. Perez Bergliaffa, Physics Letters B , 311 (2006)44. S. collaboration; M. Betoule et al., A&A , A22 (2014)45. M. Campista, B. Santos, J. Santos, J.S. Alcaniz, Phys.Lett. B699 , 320 (2011)46. S.L. Cao, S. Li, H.R. Yu, T.J. Zhang, Research in As-tronomy and Astrophysics , 026 (2018)47. Z.X. Zhai, W.B. Liu, Research in Astronomy and Astro-physics , 1257 (2011)48. N. Pires, J. Santos, J.S. Alcaniz, Phys. Rev. D (6),067302 (2010)49. D.N. Vollick, Phys. Rev. D (6), 063510 (2003)50. ´E.´E. Flanagan, PRL (7), 071101 (2004)51. A. Dom´ınguez, D. Barraco, Phys. Rev. D , 043505(2004)52. G.J. Olmo, W. Komp, ArXiv General Relativity andQuantum Cosmology e-prints (2004)53. S. Capozziello, V.F. Cardone, M. Francaviglia, GeneralRelativity and Gravitation (5), 711 (2006) 54. D. M¨uller, V.C. de Andrade, C. Maia, M.J. Rebou¸cas,A.F.F. Teixeira, European Physical Journal C , 13(2015)55. M. Amarzguioui, Ø. Elgarøy, D.F. Mota, T. Multam¨aki,A&A , 707 (2006)56. F.C. Carvalho, E.M. Santos, J.S. Alcaniz, J. Santos,JCAP , 008 (2008)57. J. Santos, J.S. Alcaniz, F.C. Carvalho, N. Pires, PhysicsLetters B , 14 (2008)58. X.J. Yang, D.M. Chen, MNRAS , 1449 (2009)59. T. Xu, S. Cao, J. Qi, M. Biesiada, X. Zheng, Z.H. Zhu,ArXiv e-prints (2017)60. X.H. Meng, P. Wang, Physics Letters B584