Hubble tension in a nonminimally coupled curvature-matter gravity model

The presently open problem of the Hubble tension is shown to be removed in the context of a modified theory of gravity with a non-minimal coupling between curvature and matter. By evolving the cosmological parameters that match the cosmic microwave background data until their values from direct late-time measurements, we obtain an agreement between different experimental methods without disrupting their individual validity. These modified gravity models are shown to provide adequate fits for other observational data from recent astrophysical surveys and to reproduce the late-time accelerated expansion of the Universe without the inclusion of a cosmological constant. This compatibility with observations presents further evidence of the versatility of these models in mimicking diverse cosmological phenomena in a unified manner.


I. INTRODUCTION
Cosmology can be a highly volatile field of study, in great part thanks to the rapidly evolving set of experiments and their increasing accuracy.Measurements dating back to the late 1920s have led to the conclusion that we live in an expanding Universe.Naturally, the rate of such an expansion has been a crucial measurement for many decades, starting with Hubble's first discovery in 1929 [1].The rate is typically described through the Hubble parameter H = ȧ/a, where a is the scale factor of the Universe and a dot represents a derivative with respect to cosmic time.As the accuracy and diversity of measurements of the present value of the Hubble parameter (H 0 ) increased, it became clear that there is a significant tension between so-called early and late experimental methods.
Late measurements of the Hubble parameter are those which utilise direct observations of cosmological effects at low redshifts to infer H 0 .The purely empirical method uses the distance-redshift relation, which is implemented through the construction of a "distance ladder".Cepheids and type Ia supernovae are particularly useful for this purpose, as their large and theoretically predictable luminosity allows for accurate observations over several Megaparsecs (Mpc).
The Hubble Space Telescope (HST) was the first experiment to provide the means to measure these objects at sufficiently large distances to yield a value of H 0 = 72 ± 8 km s −1 Mpc −1 [2].Recent improvements in the accuracy of these distance estimates have resulted in values in the range of 73-74 km s −1 Mpc −1 , with the most recent set of 75 Milky Way Cepheids with HST photometry and EDR3 parallaxes [3] yielding H 0 = 73.2± 1.3 km s −1 Mpc −1 [4], which will be taken as the reference value for late measurements throughout this paper.
Early measurements of the Hubble parameter are commonly considered as those relying on observations at redshifts z > 1000 and assuming agreement with the ΛCDM model at those times [5].The same model is then used to evolve the Universe until z = 0, thus obtaining estimates for cosmological parameters in the present.The Planck experiment is widely taken to be the standard for early measurements, predicting H 0 = 67.27± 0.60 km s −1 Mpc −1 in a flat ΛCDM model from observations of the cosmic microwave background (CMB) radiation in their 2018 results [6].Additionally, measurements of Baryon Acoustic Oscillations (BAO) with a CMB prior give a similar value of H 0 = 67.9± 1.1 km s −1 Mpc −1 [7].Late and early methods for the determination of H 0 are thus in a tension of several standard deviations which has yet to be definitively resolved [5].
Several solutions have been proposed to relieve or even remove the Hubble tension, with possibilities ranging from suggestions of possible systematic errors to modifications of the standard cosmological model.The latter option is typically based on altering the evolution from CMB data to the present by adapting the underlying gravity theory in some way.Such modifications include postulating the presence of accelerated expansion in the early Universe as described in early dark energy models [8], late dark energy models like those with a time-varying equation of state parameter [9], among many others (see [5] for an extensive review).It is relevant for the method considered in this paper, different models of modified gravity have been suggested as possible solutions to the current tension.These include f (R) [10,11], f (T ) [12][13][14] and f (Q) [15] models, along with different scalar-tensor theories [16,17].These proposals provide varying degrees of success, with some only alleviating the 4σ tension to roughly (2 − 3)σ and others providing means to fully resolve the gap between different experiments [5].
On top of this tension, there is also very active research on the topic of the accelerated expansion of the Universe, which is not only observed to be expanding but also seems to be doing it increasingly faster as time progresses.While different methods for fitting the observational data have been used to account for this hypothesis [18][19][20], there is still an active debate over what could be causing this acceleration.Clearly, the simplest proposal is the inclusion of a cosmological constant Λ, which raises unanswered questions about its seemingly fine-tuned magnitude and the associated matter/dark energy transition time, while also being in disagreement with predictions made by quantum field theory arguments [21].Alternative proposals on these issues have been presented in Refs.[22][23][24][25].
In this work, we consider a modification of the expansion rate of the Universe between early and late measurements caused by a modified theory of gravity with non-minimal coupling (NMC) of matter and curvature [26].This is similar to the work conducted in Refs.[10,11], where only minimally coupled f (R) theories were considered.Additionally, we seek solutions which simultaneously display accelerated expansion at low redshifts without the inclusion of a cosmological constant, as investigated in Refs.[22,23].Such theories have also been extensively researched in the context of mimicking dark matter profiles [27,28], analysing the modified theory with solar system constraints [29][30][31], sourcing cosmological inflation in the early Universe [32][33][34] and the creation of large-scale structure [35].
The layout of this paper is as follows.We present the nonminimally coupled model, the relevant field equations and their implications for the Friedmann-Lemaître-Robertson-Walker (FLRW) metric in Section II.The methodology for relieving the Hubble tension, the necessary numerical methods and the obtained results are described in Section III.This is followed by a discussion of matching late-time acceleration effects in NMC models with the resolved Hubble tension and their comparison with empirical fitting methods, which is included in Section IV.We conclude the paper in Section V, where we discuss the obtained results along with possible extensions of our work.We use the (−, +, +, +) signature, choose units where c = 1 and define 8πG = κ 2 .

A. Action and field equations
The nonminimally coupled f (R) model can be written in action form as [26] where f 1,2 (R) are arbitrary functions of the scalar curvature R, g is the metric determinant and L m is the Lagrangian density for matter fields [26].The General Relativistic action is recovered by setting f 1 = R and f 2 = 0.The inclusion of a cosmological constant can be considered by choosing f 1 = R − 2κ 2 Λ.By varying the action with respect to the metric g µν we obtain the field equations [26] ( where we have defined ∆ µν ≡ ∇ µ ∇ ν − g µν □ and F i ≡ df i /dR.Taking the covariant derivative of both sides of Eq. (2) and using the Bianchi identities ∇ µ G µν = 0 leads to the non-conservation law [26] which reduces to the usual stress-energy tensor conservation for f 2 = 0.This non-conservation follows directly from the non-minimal coupling of matter and curvature.In this work, as we aim to determine the effects of the NMC model independently of the minimally coupled f (R) model, we set f 1 = R and consider f 2 ̸ = 0.

B. Cosmology in nonminimally coupled gravity
Next, we use the flat FLRW metric given by the line element and the usual perfect fluid stress-energy tensor components T 00 = ρ and T rr = a 2 p, where ρ and p are the energy density and the isotropic pressure of the fluid, respectively.As can be seen from the field and non-conservation equations, the choice of L m is non-trivial, as it explicitly enters these equations via the nonmiminal coupling, while only appearing in the form of the related stress-energy tensor in General Relativity [36,37].With this in mind, in the remainder of this work, we follow the arguments given in Refs.[37,38] and take the Lagrangian density to be L m = −ρ.
We can then use the previously discussed field equations ( 2) to arrive at the modified Friedmann equation where we have defined ρ ≡ κ 2 ρ and for simplicity [39].Additional components of the field equations yield the modified Raychaudhuri equation and the non-conservation equation leads to the usual result where the choice of L m = −ρ causes the modifications in Eq. ( 3) to vanish.This greatly simplifies the cosmological evolution of our system, as we can consider the usual evolution of the energy density of all kinds of matter with respect to the scale factor ρ ∝ a −3(1+ω) , which depends only on the equation of state parameter ω = p/ρ (ω = 0 for non-relativistic matter, ω = 1/3 for radiation).
For the analysis considered here, it is convenient to explicitly write the deceleration parameter and use Eqs.( 5) and ( 6) to write where given our choice f 1 = R, we have F = 1 − 2F 2 ρ.This gives q = 1 2 + κ 2 p 2H 2 when setting f 2 = 0 and assuming a barytropic equation of state-dominated Universe, as expected from General Relativity.More importantly, it correctly implies that q = 1/2 when f 2 = 0 and p = 0, which is the expected result in a matter-dominated Universe, such as the one that we consider as the starting point of the modified regime.This second-order differential equation is useful when evolving the system numerically, as seen in Section III.

III. HUBBLE TENSION
In order to ensure the compatibility of late and early measurements, we assume that the CMB data is accurate and that it is the evolution from early to late times that causes the discrepancy between the different values for H 0 in the present.As will be discussed in the following section, we assume a decoupling of matter and curvature at high redshifts, allowing us to consider starting conditions compatible with General Relativity where we have kept only the non-relativistic matter energy density, as the radiation density is negligible at points where the curvature is small enough to ensure a noticeable effect from f 2 .We also assume no cosmological constant, as the accelerated expansion will be generated by the non-minimal coupling modification.The values H * 0 and ρ(z i ) are taken from the Planck data [6], as this would be the measured value in the present if there were no modifications to gravity and the Universe evolved with a cosmological constant, which is not relevant for the CMB observations due to the small contribution of dark energy at these redshifts.This agreement is also ensured by the standard ρ ∝ a −3 ∝ (1 + z) 3 dependence remaining unaltered with our choice of the Lagrangian density.By evolving the Universe from an initial point between early and late measurements, we can obtain a modified present value H 0 that is in agreement with direct observations conducted at smaller redshifts.This resolves the tension between experiments with no major implications for the independent measurements.
A. Choice of f2(R) A reasonable assumption for cosmology at early times is that f 2 (R) → 0 as R → ∞, as this ensures a decoupling of matter and curvature at high redshifts, such as the ones considered for the CMB measurements and any radiation dominated era.This was already considered in the context of mimicking the effects of dark matter [27] and the accelerated expansion of the Universe [22].A natural choice is then any inverse power of R such as which we can think of as an isolated term in a more complex expansion of integer powers of n.The constant R 0 serves as a characteristic curvature for the modification, which can be interpreted as a characteristic length scale with units of distance.Naturally, such a choice of f 2 leads to the possibility of diverging behaviour when the scalar curvature vanishes [40].If one thinks of systems with R = 0, such as the vicinity of an uncharged black hole, the issue of a diverging f 2 term seems evident.However, this argument fails to capture the transition of the locally sourced values of the curvature onto its cosmologically sourced behaviour, which becomes relevant when considering large enough scales.Additionally, as discussed in Ref. [41], such NMC models are not to be thought of as being composed of functions with unique powers of the curvature scalar, but as a sum of terms that would be relevant at different scales.In the example of nuclear physics given in Ref. [40], this would simply mean that at these scales the NMC effects are suppressed.Details on the effects of non-analytic forms of f 2 in the context of the Solar System were presented in Refs.[31,41], where a screening mechanism that allows for planetary constraints to be calculated was introduced.Further discussions of the consistency of these types of inverse R dependence can be found in Ref. [42], where stability and causality tests of these theories were addressed.

B. Numerical Method
Due to the degree of complexity of the resulting equations, the evolution of the scale factor and other cosmological parameters is conducted via a numerical analysis.While the modified Friedmann equation is an obvious candidate for describing the dynamics of the system, it is found to be prone to numerical instabilities.We thus evolve our system using the second-order Eq. ( 9), which can be usefully rewritten using the relation d(log a) = Hdt as where we have combined several terms such that we evolve the quantity F 2 ρ as a proxy for R throughout the simulation.Due to our simple choice of f 2 , this function can be easily inverted to obtain R at each point of the numerical evolution.We evolve H using the unaltered equation which, together with the previous equation, sets up the dynamics of our system with log a as our "time" parameter.The evolution of the matter density is directly determined by its relation with a = e log a .We then take the usual ΛCDM initial conditions where ρi = ρ(z i ) is the matter energy density evaluated at the initial redshift, as seen in Eq. (10).The remaining condition is where we have taken F 2 and its derivative F ′ 2 to be evaluated at the initial point with R i = ρi .Thus, the 3 quantities we evolve are H, F 2 ρ and its derivative with respect to log a.
The starting point is determined by ensuring an adequately small value of the quantity F 2 ρ that allows a smooth transition between the standard and modified regime.For a given n and some arbitrary small parameter ϵ this is where ρ0 is the value of the energy density at z = 0 taken from the Planck data [6].We can safely consider this value due to the conservation Eq. ( 7) being the same as in General Relativity for the NMC theory with the chosen form of the matter Lagrangian density.
The determination of the required R 0 values to solve the tension is based on defining starting values for H direct 0 and H indirect 0 , with the former being taken from Ref. [4] and serving as a target for the simulation, while the latter is taken from Planck's data [6] and is used to generate the initial conditions of the numerical method.For fixed n, each R 0 then corresponds to some final H value at z = 0, which can be used as the input function in a root-finding method, together with the directly observed value of H 0 and initial choices of R 0 taken from Ref. [22].We find that the simulated H 0 is monotonic with respect to R 0 , thus allowing us to uniquely determine the necessary form of NMC that eliminates the tension.

C. Results
The obtained values of R 0 are shown in Figure 1.Results for n = 1, 2 are not presented, as these powers do not enforce a significant modification of the evolution of the Hubble parameter, thus being unable to resolve the tension.Non-integer values of n have similar effects to those shown in the remainder of this work, with the relatively smooth form of the relation between R 0 and n allowing for a simple extrapolation for non-integer exponents.The error on the determined values of R 0 for each n has been estimated by considering the effect of varying both the late and early measurements of the Hubble parameter and corresponding matter density.A larger value of a late-time determination of H 0 leads to the largest possible required R 0 to remove the tension for a fixed early-time value.Conversely, larger values of indirect measurements of H 0 reduce the required R 0 for fixed late-time values.This allows us to easily combine the errors of both ends of the data range and obtain an overall error estimate for each R 0 .The associated characteristic lengths r c = R −1/2 0 are all around 1450 Mpc, which is comparable to the Hubble length r H ≈ 4000 Mpc.This illustrates the scale at which the effects of these modifications to the theory would become significant.
For the presented values, it is interesting to note that n = 4 seems to be at a relative minimum, while the necessary values of R 0 seem to increase for larger values of n.This follows from the late impact of these terms on the evolution of the Universe, which then requires a larger value of R 0 to reconcile the evolution of H(z) with the late-time observational value of H 0 .All of these values would be smaller if we had included a cosmological constant, as it would then be a matter of patching the approximately 10% difference between values of H 0 .However, as will be discussed in Section IV, the absence of a cosmological constant can be accounted for by the effects of the NMC model.With this in mind, we have taken the more encompassing approach, which not only attempts to provide possible solutions to the Hubble tension but also the accelerated expansion problem.

D. Comparison with observational data
Besides analysing its present value, we can also compare the modified evolution of the Hubble parameter with recent observational data from Cosmic Chronometers (CCs) and Baryon Acoustic Oscillations (BAO).Data for the former were taken from Refs.[43][44][45][46][47][48][49], while the latter were taken from Refs.[50][51][52][53][54][55][56][57][58][59][60].For this comparison, we consider the n = 4 and n = 10 model simulations, as these were found to encapsulate the general behaviour of smaller and larger values of n while being within computationally reasonable boundaries [22].We compare our models with theoretical expectations from the ΛCDM model assumed by the Planck experiment [6].The results are shown in Figure 2. The smaller z data points seem to be in better agreement with the ΛCDM model, while some of the larger z BAO data are surprisingly more in line with the n = 10 modified model, which has a noticeably better fit than the n = 4 model.This is confirmed by the χ 2 test of the quality of each fit -χ 2 4 = 26, χ 2 10 = 18 and χ 2 ΛCDM = 15.However, as discussed at the start of this paper, the BAO values are obtained using a prior from ΛCDM and are therefore naturally inclined to agree with its predictions.Additionally, as noted in Ref. [5], the CC values have large uncertainties and therefore provide relatively poor insight into late values of the Hubble parameter.Nevertheless, the similarity between ΛCDM evolution and the considered NMC models with no cosmological constant is already a relevant result, as the standard matter-dominated Universe would fall significantly short of data points at small values of z.Additionally, we can use the simulated H(z) data to calculate the theoretical distance moduli of astronomical objects at redshift z as where the luminosity distance is given by allowing us to add yet another comparison with observational data.The Pantheon+SH0ES dataset contains 1701 light curves of 1550 supernovae in the range 0.001 ≤ z ≤ 2.2613 from 18 surveys [61,62].These measurements are model-independent, thus providing a solid comparison for our results.These are shown in Figure 3. Due to the large number of data points, it becomes difficult to assess the quality of the fit of each model, leading us to the comparison of their respective χ 2 values.These are χ 2 4 = 875, χ 2 10 = 990 and χ 2 ΛCDM = 2361.This expresses quantitatively the better fit provided by modifying the evolution of CMB data with the NMC theory.Even when considering a ΛCDM model with late values of the Hubble constant, we obtain χ 2 ΛCDM = 902, which is at the quality level of the NMC theory fit.However, this value of H 0 would fail to reproduce the CMB data observed by Planck [6], while the NMC model can accurately match both ends of the observations.Therefore, we conclude that these modified models go beyond the initially proposed task of matching present values of the Hubble parameter by providing an alternative method to fit existing observational data.

IV. LATE-TIME ACCELERATION
Besides providing a natural agreement between direct and indirect measurements of the Hubble constant, we also consider the resulting late-time acceleration of the Universe, as revealed by recent data [18].Indeed, as we have postulated a model with no cosmological constant, any deviation from the standard matter-dominated value of q = 1/2, which describes a constantly decelerating expansion, must follow from the modifications introduced by the NMC model.However, as the main purpose of this work was to resolve the Hubble tension, we fix the corresponding value of R 0 for each n and analyse the compatibility of these very models with observations of late-time acceleration.
Throughout this section, we will continue to consider the values n = 4, 10 as references for the behaviour of the spectrum of values for n.As will become clear, both of these models have a smooth transition from the matterdominated regime (q = 1/2) at large z to different asymptotic regimes as z → −1.These asymptotic values seem to be in agreement with the results of Ref. [22], where a power-law form of the scale factor was considered in the limit of weak/strong matter-curvature coupling.As expected from the rapid increase of the magnitude of F 2 ρ as the curvature becomes increasingly smaller, our values agree with the strong coupling regime.This provides further motivation for the analysis of a power-law form for the expansion of the Universe in the future of the NMC gravity model, as conducted in the same paper [22].
We consider two different empirical fittings for q(z).The first, proposed in Ref. [19] as the best of 3 parametrizations, is given by where a = 0.855 ± 0.034 and b = 3.85 ± 0.19, and is shown in Figure 4.The fit for this model was obtained using the previously mentioned CC and BAO data [7,63], together with the also mentioned distance moduli [61,62].Note that this function approaches q = −1 as we move towards the future (z = −1), similarly to the behaviour of standard ΛCDM.Interestingly, while the n = 4 model tends to stray from the empirical fitting region near z = 0, the n = 10 model seems to agree with this fit, being within 1σ of the fitted model up until the same point.All of these agree on a transition redshift z T ≈ 0.9, while distinctly disagreeing with the transition redshift of the standard model, this being typically around z = 0.65.However, the modified models predict a weaker asymptotic acceleration phase in the far future, while the asymptotic value of the deceleration for the empirical fit follows ΛCDM and is thus fixed at -1.
In fact, even though our agreement between the empirical and NMC models could be expected from their agreement with the same observational data, we should note that each NMC model was merely chosen to remove the Hubble tension, with their respective consistency with other observational data being a consequence and not a requirement.
FIG. 4. Comparison of observationally fitted parametrisation of q(z) [19] with the results for the Hubble tension-free NMC model.The shaded regions show the 1σ, 2σ and 3σ errors of the empirical fit.Also presented is the prediction given by the ΛCDM model.
The second parametrisation, presented in Ref. [64], is given by where q 1 = 1.47 +1.89 −1.82 and q 2 = −1.46 ± 0.43, and is shown in Figure 5.This fit was obtained from the data for 182 gold type Ia supernovae [65], which determined the distance of these astrophysical events for several redshifts around z = 1, analogously to the Pantheon+SH0ES data taken in the formerly discussed parametrisation and in this work in Section III.In this case, the n = 4 model is the one which appears to be within an error of (1 − 2)σ of the empirical form of q(z), with its n = 10 counterpart providing better agreement at smaller redshifts.Additionally, this parametrisation exhibits a much lower acceleration transition of z T ≈ 0.35, with the 1σ region ranging from z = 0.3 to the approximate ΛCDM value of z = 0.65, placing the transition of the modified models only within 2σ.However, the parametrisation presents a divergence at z = −1, leading to an increasingly accelerated expansion of the Universe as we move past the present stage.It is thus possible to consider this as a pathological solution for q(z), which relatively lowers its importance when compared with the formerly considered example.FIG. 5. Comparison of observationally fitted parametrisation of q(z) [64] with the results for the Hubble tension-free NMC model.The shaded regions show the 1σ, 2σ and 3σ errors of the empirical fit.Also presented is the prediction given by the ΛCDM model.

A. More complex combinations
Considering the different compatibilities between lower and higher values of n and observational constraints, we are led to consider the possibility of using linear combinations of inverse powers of R. Indeed, functions of the form have already been examined in the literature and provide a more detailed description of phenomena such as galaxy rotation curves [27,28].A more general power-law expansion with various integer powers of n is not just able to capture the behaviour of NMC models at different scales, but also provides an alternative template to consider more complex functions of R.
Naturally, one should not expect that the presence of two linearly combined terms in f 2 translates into a linear effect in the evolution of H(z) due to the non-linearity of the theory.Nonetheless, the possibility of better understanding the behaviour of various sources of cosmological data observed at different epochs of the late Universe strongly motivates such considerations.Unfortunately, this leads to forms of F 2 that cannot be directly inverted to give the value of R at each point in the simulation, forcing us to resort to root-finding functions at each step of our method.This increases the error in the already sensitive numerical evolution of the system, which then leads to severe instabilities and meaningless results.The analysis of these linear combinations in this context is thus left as the topic of future research.

V. CONCLUSIONS
In this work, we have applied a modified theory of gravity which nonminimally couples curvature and matter to the Hubble tension problem.In order to do that, we started by taking data from the CMB as accurate and hypothesising that the differences in measurements arise from the subsequent model-dependent evolution of the Universe.To ensure concordance between our model and the CMB data, we take the simple form of f 2 = (R 0 /R) n , where n > 0 is some positive exponent, leading to a decoupling of matter and curvature at high redshifts.By numerically evolving the modified field equations, we find the necessary values of R 0 for each n that evolve the CMB observables to their measured late-time correspondents, therefore eliminating the Hubble tension.
The functional form of H(z) was also tested against observational data from Baryon Acoustic Oscillations, Cosmic Chronometers and the observed distance moduli of supernovae.We considered n = 4, 10 as specific examples of the behaviour of the NMC model and found that n = 10 is in better agreement with some of the BAO data, while n = 4 seems to better fit the supernovae distance data [22].
Additionally, we have assumed no cosmological constant in our model, allowing us to investigate if the same form of f 2 can generate late accelerated expansion, similarly to what was done in Ref. [22].By comparing with empirically motivated parametrisations of the deceleration parameter, we have confirmed that late-time acceleration can be recreated by the presence of a non-minimal coupling between matter and curvature.Both the n = 4 and n = 10 models yield a present value q 0 < 0, along with similar acceleration transition redshifts z T ≈ 0.9, leading to the mimicking of the effects of a cosmological constant.This result is non-trivial and provides further proof of the versatility of the NMC model, which adequately competes with current models in matching cosmological behaviour at low redshifts even when just burdened by the initially proposed challenge of removing the Hubble tension.
Due to the different characteristics of the various values of n considered here, the possibility of more complex combinations of exponents in f 2 was raised, with numerical stability issues impeding the development of testable results.Nevertheless, this possibility of capturing cosmological behaviours at different scales is promising in providing a better match between theory and data.We should note that patching the Hubble tension and reproducing the accelerated expansion of the Universe at late times are a small subset of the phenomena that can be explained through the nonminimally coupled theory of gravity, as shown in many examples in the literature [22,27,29,32,35,39].The NMC theory considered in this paper is therefore incomplete without all the terms in the power expansion of f 2 , which would dominate at different scales.Such an expansion would have manifestly different behaviours in different scenarios, allowing for the simultaneous explanation of effects such as dark matter (f 2 ∝ R −1 , R −1/3 ) [27,28], dark energy (f 2 ∝ R −4 , R −10 ) [22] and inflation (f 2 ∝ R, R 3 ) [33,34], among others.This combination of terms can be interpreted as the effective form of a putative full expansion of a more general nonminimal coupling that provides a more consistent description of gravity.Naturally, this interpretation further depends on arguments about the origin and the emergence of this non-minimal coupling, which remains a promising topic for future research in the field.

FIG. 1 .
FIG. 1. Values of R0 required to remove the Hubble tension for different powers n of the curvature scalar.The y-axis is in arbitrary units and simply serves to provide a comparison between magnitudes of values.

FIG. 2 .
FIG. 2. Comparison of modified model with various Baryon Acoustic Oscillation and Cosmic Chronometer data.The flat ΛCDM model with 2018 Planck data [6] is shown for the standard theory.

FIG. 3 .
FIG. 3. Comparison of modified model with Pantheon+SH0ES distance moduli data [61, 62].To allow for better visualisation, points in dense data regions have been removed for this figure.The flat ΛCDM model with 2018 Planck data [6] is shown for comparison.