The so-called -gravity has recently attracted a lot of interest since it could be, in principle, able to explain the accelerated expansion of the Universe without adding unknown forms of dark energy/dark matter but, more simply, extending the General Relativity by generic functions of the Ricci scalar. However, apart several phenomenological models, there is no final -theory capable of fitting all the observations and addressing all the issues related to the presence of dark energy and dark matter. An alternative approach could be to β€œreconstruct” the form of starting from data without imposing particular classes of model. Besides, adopting the same philosophy, we take into account the possibility that galaxy cluster masses, estimated at X-ray wavelengths, could be explained, without dark matter, reconstructing the weak-field limit of analytic models. The corrected gravitational potential, obtained in this approximation, is used to estimate the total mass of a sample of 12 well-shaped clusters of galaxies.

1. Introduction

As soon as astrophysicists realized that Type Ia Supernovae (SNeIa) were standard candles, it appeared evident that their high luminosity should make it possible to build a Hubble diagram, that is, a plot of the distance-redshift relation, over cosmologically interesting distance ranges. Motivated by this attractive consideration, two independent teams started SNeIa surveys leading to the unexpected discovery that the Universe expansion is speeding up rather than decelerating as assumed by the Cosmological Standard Model [1–5]. This surprising result has now been strengthened by more recent data coming from SNeIa surveys [6–13], large scale structure [14–18] and cosmic microwave background (CMBR) anisotropy spectrum [19–25]. This large data set coherently points toward the picture of a spatially flat Universe undergoing an accelerated expansion driven by a dominant negative pressure fluid, typically referred to as dark energy [26].

While there is a wide consensus on the above scenario depicted by such good quality data, there is a similarly wide range of contrasting proposals to solve the dark energy puzzle. Surprisingly, the simplest explanation, namely the cosmological constant [27, 28], is also the best one from a statistical point of view [29–31]. Unfortunately, the well known coincidence and 120 orders of magnitude problems render a rather unattractive solution from a theoretical point of view. Inspired by the analogy with inflation, a scalar field , dubbed quintessence [32, 33], has then been proposed to give a dynamical term in order to both fit the data and avoid the above problems. However, such models are still plagued by difficulties on their own, such as the almost complete freedom in the choice of the scalar field potential and the fine tuning of the initial conditions. Needless to say, a plethora of alternative models are now on the market all sharing the main property to be in agreement with observations, but relying on completely different physics.

Notwithstanding their differences, all dark energy models assume that the observed apparent acceleration is the outcome of some unknown ingredient, at fundamental level, to be added to the cosmic pie. In terms of the Einstein equations, , the right hand side should include something more than the usual matter and radiation components in the stress-energy tensor.

As a radically different approach, one can also try to leave unchanged the source side (actually β€œobserved” since composed by radiation and baryonic matter), but rather modifying the left hand side. In a sense, one is therefore interpreting cosmic speed up as a first signal of the breakdown of the laws of physics as described by the standard General Relativity (GR). Since this theory has been experimentally tested only up to the Solar System scale, there is no a priori theoretical motivation to extend its validity to extraordinarily larger scales such as the extragalactic and cosmological ones (up to the last scattering surface!). Extending GR, not giving up to its positive results at local scales, opens the way to a large class of alternative theories of gravity ranging from extra-dimensions [34–38] to nonminimally coupled scalar fields [39–42]. In particular, we are interested here in fourth order theories [43–54] based on replacing the scalar curvature in the Hilbert-Einstein action with a generic analytic function which should be reconstructed starting from data and physically motivated issues. Also referred to as -gravity, some of these models have been shown to be able to both fit the cosmological data and evade the Solar System constraints in several physically interesting cases [55–59].

In this review paper, we will face two of the main problems directly related to the dark energy and dark matter issues: cosmography and clusters of galaxies. These are typical examples where the standard General Relativity and Newtonian potential schemes fail to describe dynamics since data present accelerated expansion and missing matter. Our goal is to address them by -gravity.

1.1. Cosmography: Why?

It is worth noting that both dark energy models and modified gravity theories seem to be in agreement with data. As a consequence, unless higher precision probes of the expansion rate and the growth of structure will be available, these two rival approaches could not be discriminated. This confusion about the theoretical background suggests that a more conservative approach to the problem of cosmic acceleration, relying on as less model dependent quantities as possible, is welcome. A possible solution could be to come back to the cosmography [60] rather than finding out solutions of the Friedmann equations and testing them. Being only related to the derivatives of the scale factor, the cosmographic parameters make it possible to fit the data on the distance-redshift relation without any a priori assumption on the underlying cosmological model: in this case, the only assumption is that the metric is the Robertson-Walker one (and hence not relying on the solution of cosmological equations). Almost eighty years after Hubble discovery of the expansion of the Universe, we can now extend, in principle, cosmography well beyond the search for the value of the only Hubble constant. The SNeIa Hubble diagram extends up to thus invoking the need for, at least, a fifth order Taylor expansion of the scale factor in order to give a reliable approximation of the distance-redshift relation. As a consequence, it could be, in principle, possible to estimate up to five cosmographic parameters, although the still too small data set available does not allow to get a precise and realistic determination of all of them.

Once these quantities have been determined, one could use them to put constraints on the models. In a sense, we can revert the usual approach, consisting in deriving the cosmographic parameters as a sort of byproduct of an assumed theory. Here, we follow the other way around expressing the model characterizing quantities as a function of the cosmographic parameters. Such a program is particularly suited for the study of fourth order theories of gravity. As it is well known, the mathematical difficulties entering the solution of fourth order field equations make it quite problematic to find out analytical expressions for the scale factor and hence predict the values of the cosmographic parameters. A key role in -gravity is played by the choice of the function. Under quite general hypotheses, we will derive useful relations among the cosmographic parameters and the present day value of , with , whatever is. (As an important remark, we stress that our derivation will rely on the metric formulation of theories, while we refer the reader to [61, 62] for a similar work in the Palatini approach.) Once the cosmographic parameters will be determined, this method will allow us to investigate the cosmography of theories.

It is worth stressing that the definition of the cosmographic parameters only relies on the assumption of the Robertson-Walker metric. As such, it is however difficult to state a priori to what extent the fifth order expansion provides an accurate enough description of the quantities of interest. Actually, the number of cosmographic parameters to be used depends on the problem one is interested in. As we will see later, we are here concerned only with the SNeIa Hubble diagram so that we have to check that the distance modulus obtained using the fifth order expansion of the scale factor is the same (within the errors) as the one of the underlying physical model. Being such a model of course unknown, one can adopt a phenomenological parameterization for the dark energy equation of state (EoS) and look at the percentage deviation as function of the EoS parameters. (Note that one can always use a phenomenological dark energy model to get a reliable estimate of the scale factor evolution even if the correct model is a fourth order one.) We have carried out such exercise using the CPL model, introduced below, and verified that is an increasing function of (as expected), but still remains smaller than up to over a wide range of the CPL parameter space. On the other hand, halting the Taylor expansion to a lower order may introduce significant deviation for that can potentially bias the analysis if the measurement errors are as small as those predicted by future SNeIa surveys. We are therefore confident that our fifth order expansion is both sufficient to get an accurate distance modulus over the redshift range probed by SNeIa and necessary to avoid dangerous biases.

1.2. Clusters of Galaxies: Why?

In the second part of this review we will apply the -gravity approach to cluster of galaxies. In fact, changing the gravity sector has consequences not only at cosmological scales, but also at galactic and cluster scales so that it is mandatory to investigate the low energy limit of such theories. A strong debate is open with different results arguing in favor [63–68] or against [69–71] such models at local scales. It is worth noting that, as a general result, higher order theories of gravity cause the gravitational potential to deviate from its Newtonian scaling [72–77] even if such deviations may be vanishing.

In [78], the Newtonian limit of power law theories has been investigated, assuming that the metric in the low energy limit () may be taken as Schwarzschild-like. It turns out that a power law term has to be added to the Newtonian term in order to get the correct gravitational potential. While the parameter may be expressed analytically as a function of the slope of the theory, sets the scale where the correction term starts being significant. A particular range of values of has been investigated so that the corrective term is an increasing function of the radius thus causing an increase of the rotation curve with respect to the Newtonian one and offering the possibility to fit the galaxy rotation curves without the need of further dark matter components.

A set of low surface brightness (LSB) galaxies with extended and well measured rotation curves has been considered [79, 80]. These systems are supposed to be dark matter dominated, and successfully fitting data without dark matter is a strong evidence in favor of the approach (see also [81] for an independent analysis using another sample of galaxies). Combined with the hints coming from the cosmological applications, one should have, in principle, the possibility to address both the dark energy and dark matter problems resorting to the same well motivated fundamental theory [82–85]. Nevertheless, the simple power law gravity is nothing else but a toy-model which fail if one tries to achieve a comprehensive model for all the cosmological dynamics, ranging from the early Universe, to the large scale structure up to the late accelerated era [83, 84].

A fundamental issue is related to clusters and superclusters of galaxies. Such structures, essentially, rule the large scale structure, and are the intermediate step between galaxies and cosmology. As the galaxies, they appear dark-matter dominated but the distribution of dark matter component seems clustered and organized in a very different way with respect to galaxies. It seems that dark matter is ruled by the scale and also its fundamental nature could depend on the scale. For a comprehensive review see [86].

In the philosophy of -gravity, the issue is to reconstruct the mass profile of clusters without dark matter, that is, to find out corrections to the Newton potential producing the same dynamics as dark matter but starting from a well motivated theory.

In conclusion, -gravity, as the simplest approach to any extended or alternative gravity scheme, could be the paradigm to interpret dark energy and dark matter as curvature effects acting at scales larger than those where General Relativity has been actually investigated and probed.

Let us discuss now how cosmography and then galaxy clusters could be two main examples to realize this program.

2. The Cosmographic Apparatus

The key rule in cosmography is the Taylor series expansion of the scale factor with respect to the cosmic time. To this aim, it is convenient to introduce the following functions: which are usually referred to as the Hubble, deceleration, jerk, snap and lerk parameters, respectively. It is then a matter of algebra to demonstrate the following useful relations: where a dot denotes derivative with respect to the cosmic time . Equation (2) make it possible to relate the derivative of the Hubble parameter to the other cosmographic parameters. The distance-redshift relation may then be obtained starting from the Taylor expansion of along the lines described in [87–89].

2.1. The Scale-Factor Series

With these definitions the series expansion to the 5th order in time of the scale factor will be It is easy to see that (4) is the inverse of redshift , being the redshift defined by The physical distance travelled by a photon that is emitted at time and absorbed at the current epoch is Assuming and inserting in (4) we have: The inverse of this expression will be Then we reverse the series to have the physical distance expressed as function of redshift : with From this we have with In typical applications, one is not interested in the physical distance , but other definitions: (i)the luminosity distance: (ii)the angular-diameter distance:

where is If we make the expansion for short distances, namely if we insert the series expansion of in , we have To convert from physical distance travelled to r coordinate traversed we have to consider that the Taylor series expansion of - functions is so that (4) with curvature term becomes with Using these one for luminosity distance we have with: While for the angular diameter distance it is with

If we want to use the same notation of [87], we define , which can be considered a purely cosmographic parameter, or if we consider the dynamics of the Universe. With this parameter (12)–(14) become

Previous relations in this section have been derived for any value of the curvature parameter; but since in the following we will assume a flat Universe, we will used the simplified versions for . Now, since we are going to use supernovae data, it will be useful to give as well the Taylor series of the expansion of the luminosity distance at it enters the modulus distance, which is the quantity about which those observational data inform. The final expression for the modulus distance based on the Hubble free luminosity distance, , is with

3. -Gravity versus Cosmography

3.1. Preliminaries

As discussed in the introduction, much interest has been recently devoted to the possibility that dark energy could be nothing else but a curvature effect according to which the present Universe is filled by pressureless dust matter only and the acceleration is the result of modified Friedmann equations obtained by replacing the Ricci curvature scalar with a generic function in the gravity action. Under the assumption of a flat Universe, the Hubble parameter is therefore determined by (we use here natural units such that ): where the prime denotes derivative with respect to and is the energy density of an effective curvature fluid (note that the name curvature fluid does not refer to the FRW curvature parameter , but only takes into account that such a term is a geometrical one related to the scalar curvature ): Assuming there is no interaction between the matter and the curvature terms (we are in the so-called Jordan frame), the matter continuity equation gives the usual scaling , with the present day matter density parameter. The continuity equation for then reads: with the barotropic factor of the curvature fluid. It is worth noticing that the curvature fluid quantities and only depends on and its derivatives up to the third order. As a consequence, considering only their present day values (which may be naively obtained by replacing with everywhere), two theories sharing the same values of , , , will be degenerate from this point of view. (One can argue that this is not strictly true since different theories will lead to different expansion rate and hence different present day values of and its derivatives. However, it is likely that two functions that exactly match each other up to the third order derivative today will give rise to the same at least for so that will be almost the same.)

Combining (29) with (27), one finally gets the following master equation for the Hubble parameter: Expressing the scalar curvature as function of the Hubble parameter as: and inserting the result into (31), one ends with a fourth order nonlinear differential equation for the scale factor that cannot be easily solved also for the simplest cases (for instance, ). Moreover, although technically feasible, a numerical solution of (31) is plagued by the large uncertainties on the boundary conditions (i.e., the present day values of the scale factor and its derivatives up to the third order) that have to be set to find out the scale factor.

3.2. -Derivatives and Cosmography

Motivated by these difficulties, we approach now the problem from a different viewpoint. Rather than choosing a parameterized expression for and then numerically solving (31) for given values of the boundary conditions, we try to relate the present day values of its derivatives to the cosmographic parameters so that constraining them in a model independent way gives us a hint for what kind of theory could be able to fit the observed Hubble diagram. (Note that a similar analysis, but in the context of the energy conditions in , has yet been presented in [90]. However, in that work, the author give an expression for and then compute the snap parameter to be compared to the observed one. On the contrary, our analysis does not depend on any assumed functional expression for .)

As a preliminary step, it is worth considering again the constraint equation (32). Differentiating with respect to , we easily get the following relations Evaluating these at the present time and using (2), one finally gets which will turn out to be useful in the following.

Let us now come back to the expansion rate and master equations (27) and (31). Since they have to hold along the full evolutionary history of the Universe, they naively hold also at the present day. As a consequence, we may evaluate them in thus easily obtaining Using (2) and (34), we can rearrange (35) as two relations among the Hubble constant and the cosmographic parameters , on one hand, and the present day values of and its derivatives up to third order. However, two further relations are needed in order to close the system and determine the four unknown quantities , , , . A first one may be easily obtained by noting that, inserting back the physical units, the rate expansion equation reads which clearly shows that, in gravity, the Newtonian gravitational constant is replaced by an effective (time dependent) . On the other hand, it is reasonable to assume that the present day value of is the same as the Newtonian one so that we get the simple constraint: In order to get the fourth relation we need to close the system, we first differentiate both sides of (31) with respect to . We thus get with . Let us now suppose that may be well approximated by its third order Taylor expansion in , that is, we set In such an approximation, it is for so that naively . Evaluating then (38) at the present day, we get We can now schematically proceed as follows. Evaluate (2) at and plug these relations into the left hand sides of (35), (40). Insert (34) into the right hand sides of these same equations so that only the cosmographic parameters and the related quantities enter both sides of these relations. Finally, solve them under the constraint (37) with respect to the present day values of and its derivatives up to the third order. After some algebra, one ends up with the desired result: where we have defined Equations (41)–(51) make it possible to estimate the present day values of and its first three derivatives as function of the Hubble constant and the cosmographic parameters provided a value for the matter density parameter is given. This is a somewhat problematic point. Indeed, while the cosmographic parameters may be estimated in a model independent way, the fiducial value for is usually the outcome of fitting a given dataset in the framework of an assumed dark energy scenario. However, it is worth noting that different models all converge towards the concordance value which is also in agreement with astrophysical (model independent) estimates from the gas mass fraction in galaxy clusters. On the other hand, it has been proposed that theories may avoid the need for dark matter in galaxies and galaxy clusters [74, 76, 78, 81–83, 91]. In such a case, the total matter content of the Universe is essentially equal to the baryonic one. According to the primordial elements abundance and the standard BBN scenario, we therefore get with [92] and the Hubble constant in units of . Setting in agreement with the results of the HST Key project [93], we thus get for a baryons only Universe. We will therefore consider in the following both cases when numerical estimates are needed.

It is worth noticing that only plays the role of a scaling parameter giving the correct physical dimensions to and its derivatives. As such, it is not surprising that we need four cosmographic parameters, namely , to fix the four related quantities , , , . It is also worth stressing that (41)–(44) are linear in the quantities so that uniquely determine the former ones. On the contrary, inverting them to get the cosmographic parameters as function of the ones, we do not get linear relations. Indeed, the field equations in theories are nonlinear fourth order differential equations in the scale factor so that fixing the derivatives of up to third order makes it possible to find out a class of solutions, not a single one. Each one of these solutions will be characterized by a different set of cosmographic parameters thus explaining why the inversion of (41)–(51) does not give a unique result for .

As a final comment, we reconsider the underlying assumptions leading to the above derived relations. While (35) are exact relations deriving from a rigorous application of the field equations, (40) heavily relies on having approximated with its third order Taylor expansion (39). If this assumption fails, the system should not be closed since a fifth unknown parameter enters the game, namely . Actually, replacing with its Taylor expansion is not possible for all class of theories. As such, the above results only hold in those cases where such an expansion is possible. Moreover, by truncating the expansion to the third order, we are implicitly assuming that higher order terms are negligible over the redshift range probed by the data. That is to say, we are assuming that over the redshift range probed by the data. Checking the validity of this assumption is not possible without explicitly solving the field equations, but we can guess an order of magnitude estimate considering that, for all viable models, the background dynamics should not differ too much from the CDM one at least up to . Using then the expression of for the CDM model, it is easily to see that is a quickly increasing function of the redshift so that, in order (52) holds, we have to assume that for . This condition is easier to check for many analytical models.

Once such a relation is verified, we have still to worry about (37) relying on the assumption that the cosmological gravitational constant is exactly the same as the local one. Although reasonable, this requirement is not absolutely demonstrated. Actually, the numerical value usually adopted for the Newton constant is obtained from laboratory experiments in settings that can hardly be considered homogenous and isotropic. As such, the spacetime metric in such conditions has nothing to do with the cosmological one so that matching the two values of is strictly speaking an extrapolation. Although commonly accepted and quite reasonable, the condition could (at least, in principle) be violated so that (37) could be reconsidered. Indeed, as we will see, the condition may not be verified for some popular models recently proposed in literature. However, it is reasonable to assume that with . When this be the case, we should repeat the derivation of (41)–(44) now using the condition . Taylor expanding the results in to the first order and comparing with the above derived equations, we can estimate the error induced by our assumption . The resulting expressions are too lengthy to be reported and depend in a complicated way on the values of the matter density parameter , the cosmographic parameters and . However, we have numerically checked that the error induced on , , are much lower than for value of as high as an unrealistic . We are confident that our results are reliable also for these cases.

4. -Gravity and the CPL Model

A determination of and its derivatives in terms of the cosmographic parameters need for an estimate of these latter from the data in a model independent way. Unfortunately, even in the nowadays era of precision cosmology, such a program is still too ambitious to give useful constraints on the derivatives, as we will see later. On the other hand, the cosmographic parameters may also be expressed in terms of the dark energy density and EoS parameters so that we can work out what are the present day values of and its derivatives giving the same of the given dark energy model. To this aim, it is convenient to adopt a parameterized expression for the dark energy EoS in order to reduce the dependence of the results on any underlying theoretical scenario. Following the prescription of the Dark Energy Task Force [94], we will use the Chevallier-Polarski-Linder (CPL) parameterization for the EoS setting [95, 96]: so that, in a flat Universe filled by dust matter and dark energy, the dimensionless Hubble parameter reads with because of the flatness assumption. In order to determine the cosmographic parameters for such a model, we avoid integrating to get by noting that . We can use such a relation to evaluate and then solve (2), evaluated in , with respect to the parameters of interest. Some algebra finally gives Inserting (55)–(58) into (41)–(51), we get lengthy expressions (which we do not report here) giving the present day values of and its first three derivatives as function of . It is worth noting that the model thus obtained is not dynamically equivalent to the starting CPL one. Indeed, the two models have the same cosmographic parameters only today. As such, for instance, the scale factor is the same between the two theories only over the time period during which the fifth order Taylor expansion is a good approximation of the actual . It is also worth stressing that such a procedure does not select a unique model, but rather a class of fourth order theories all sharing the same third order Taylor expansion of .

4.1. The CDM Case

With these caveats in mind, it is worth considering first the CDM model which is obtained by setting in the above expressions thus giving When inserted into the expressions for the quantities, these relations give the remarkable result: so that we obviously conclude that the only theory having exactly the same cosmographic parameters as the CDM model is just , that is, GR. It is worth noticing that such a result comes out as a consequence of the values of in the CDM model. Indeed, should we have left undetermined and only fixed to the values in (59), we should have got the same result in (60). Since the CDM model fits well a large set of different data, we do expect that the actual values of do not differ too much from the CDM ones. Therefore, we plug into (41)–(51) the following expressions: with given by (59) and quantifying the deviations from the CDM values allowed by the data. A numerical estimate of these quantities may be obtained, for example, from a Marko chain analysis, but this is outside our aims. Since we are here interested in a theoretical examination, we prefer to consider an idealized situation where the four quantities above all share the same value . In such a case, we can easily investigate how much the corresponding deviates from the GR one considering the two ratios and . Inserting the above expressions for the cosmographic parameters into the exact (not reported) formulae for , and , taking their ratios and then expanding to first order in , we finally get having defined and which, being dimensionless quantities, are more suited to estimate the order of magnitudes of the different terms. Inserting our fiducial values for , we get: For values of up to 0.1, the above relations show that the second and third derivatives are at most two orders of magnitude smaller than the zeroth order term . Actually, the values of for a baryon only model (first row) seems to argue in favor of a larger importance of the third order term. However, we have numerically checked that the above relations approximates very well the exact expressions up to with an accuracy depending on the value of , being smaller for smaller matter density parameters. Using the exact expressions for and , our conclusion on the negligible effect of the second and third order derivatives are significantly strengthened.

Such a result holds under the hypotheses that the narrower are the constraints on the validity of the CDM model, the smaller are the deviations of the cosmographic parameters from the CDM ones. It is possible to show that this indeed the case for the CPL parametrization we are considering. On the other hand, we have also assumed that the deviations take the same values. Although such hypothesis is somewhat ad hoc, we argue that the main results are not affected by giving it away. Indeed, although different from each other, we can still assume that all of them are very small so that Taylor expanding to the first order should lead to additional terms into (62) which are likely of the same order of magnitude. We may therefore conclude that, if the observations confirm that the values of the cosmographic parameters agree within with those predicted for the CDM model, we must conclude that the deviations of from the GR case, , should be vanishingly small.

It is worth stressing, however, that such a conclusion only holds for those models satisfying the constraint (52). It is indeed possible to work out a model having , , but for some . For such a (somewhat ad hoc) model, (52) is clearly not satisfied so that the cosmographic parameters have to be evaluated from the solution of the field equations. For such a model, the conclusion above does not hold so that one cannot exclude that the resulting are within of the CDM ones.

4.2. The Constant EoS Model

Let us now take into account the condition , but still retains thus obtaining the so called quiessence models. In such a case, some problems arise because both the terms and may vanish for some combinations of the two model parameters . For instance, we find that for with: On the other hand, the equation may have different real roots for depending on the adopted value of . Denoting collectively with the values of that, for a given , make taking the null value, we individuate a set of quiessence models whose cosmographic parameters give rise to divergent values of , and . For such models, is clearly not defined so that we have to exclude these cases from further consideration. We only note that it is still possible to work out a theory reproducing the same background dynamics of such models, but a different route has to be used.

Since both and now deviate from the CDM values, it is not surprising that both and take finite non null values. However, it is more interesting to study the two quantities and defined above to investigate the deviations of from the GR case. These are plotted in Figures 1 and 2 for the two fiducial values. Note that the range of in these plots have been chosen in order to avoid divergences, but the lessons we will draw also hold for the other values.

As a general comment, it is clear that, even in this case, and are from two to three orders of magnitude smaller that the zeroth order term . Such a result could be yet guessed from the previous discussion for the CDM case. Actually, relaxing the hypothesis is the same as allowing the cosmographic parameters to deviate from the CDM values. Although a direct mapping between the two cases cannot be established, it is nonetheless evident that such a relation can be argued thus making the outcome of the above plots not fully surprising. It is nevertheless worth noting that, while in the CDM case, and always have opposite signs, this is not the case for quiessence models with . Indeed, depending on the value of , we can have theories with both and positive. Moreover, the lower is , the higher are the ratios and for a given value of . This can be explained qualitatively noticing that, for a lower , the density parameter of the curvature fluid (playing the role of an effective dark energy) must be larger thus claiming for higher values of the second and third derivatives (see also [97] for a different approach to the problem).

4.3. The General Case

Finally, we consider evolving dark energy models with . Needless to say, varying three parameters allows to get a wide range of models that cannot be discussed in detail. Therefore, we only concentrate on evolving dark energy models with in agreement with some most recent analysis. The results on and are plotted in Figures 3 and 4 where these quantities as functions of . Note that we are considering models with positive so that tends to for so that the EoS dark energy can eventually approach the dust value . Actually, this is also the range favored by the data. We have, however, excluded values where or diverge. Considering how they are defined, it is clear that these two quantities diverge when so that the values of making to diverge may be found solving where and are obtained by inserting (55)–(58) into the definitions (45)-(46). For such CPL models, there is no any model having the same cosmographic parameters and, at the same time, satisfying all the criteria needed for the validity of our procedure. Actually, if , the condition (52) is likely to be violated so that higher than third order must be included in the Taylor expansion of thus invalidating the derivation of (41)–(44).

Under these caveats, Figures 3 and 4 demonstrate that allowing the dark energy EoS to evolve does not change significantly our conclusions. Indeed, the second and third derivatives, although being not null, are nevertheless negligible with respect to the zeroth order term thus arguing in favour of a GR-like with only very small corrections. Such a result is, however, not fully unexpected. From (55) and (56), we see that, having set , the parameter is the same as for the CDM model, while reads . As we have stressed above, the Hilbert-Einstein Lagrangian is recovered when whatever the values of are. Introducing a makes to differ from the CDM values, but the first two cosmographic parameters are only mildly affected. Such deviations are then partially washed out by the complicated way they enter in the determination of the present day values of and its first three derivatives.

5. Constraining Parameters

In the previous section, we have worked an alternative method to estimate , , resorting to a model independent parameterization of the dark energy EoS. However, in the ideal case, the cosmographic parameters are directly estimated from the data so that (41)–(51) can be used to infer the values of the related quantities. These latter can then be used to put constraints on the parameters entering an assumed fourth order theory assigned by a function characterized by a set of parameters provided that the hypotheses underlying the derivation of (41)–(51) are indeed satisfied. We show below two interesting cases which clearly highlight the potentiality and the limitations of such an analysis.

5.1. Double Power Law Lagrangian

As a first interesting example, we set with and two positive real numbers (see, e.g., [98] for some physical motivations). The following expressions are immediately obtained: Denoting by (with ) the values of determined through (41)–(51), we can solve which is a system of four equations in the four unknowns that can be analytically solved proceeding as follows. First, we solve the first and second equation with respect to obtaining while, solving the third and fourth equations, we get Equating the two solutions, we get a systems of two equations in the two unknowns , namely,