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Kinematic reconstructions of extended theories of gravity at small and intermediate redshifts

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Abstract

In the last few decades, extensions of general relativity have reached always more attention especially in view of possible breakdowns of the standard \(\varLambda \)CDM paradigm at intermediate and high redshift regimes. If general relativity would not be the ultimate theory of gravity, modifying Einstein’s gravity in the homogeneous and isotropic universe may likely represent a viable path toward the description of current universe acceleration. We here focus our attention on two classes of extended theories, i.e., the f(R) and f(RG)-gravity. We parameterize the so-obtained Hubble function by means of effective barotropic fluids, by calibrating the shapes of our curves through some of the most suitable dark energy parameterizations, XCDM, CPL, WP. Afterwards, by virtue of the correspondence between the Ricci scalar and the Gauss–Bonnet topological invariant with the redshift z, we rewrite f(RG) in terms of corresponding f(z) auxiliary functions. This scheme enables one to get numerical shapes for f(RG) and f(R) models, through a coarse-grained inverse scattering procedure. Although our procedure agrees with the simplest extensions of general relativity, it leaves open the possibility that the most suitable forms of f(R) and f(RG) are rational Padé polynomials of first orders. These approximations seem to be compatible with numerical reconstructions within intermediate redshift domains and match fairly well small redshift tests.

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Notes

  1. 1.

    The hypothesis of flatness is today debated [35,36,37,38,39,40]. In this work, however, we assume \(k=0\) for simplicity, without entering the issue of a non-flat universe. Our results will not be significantly influenced by this choice.

  2. 2.

    We here take pressure-less matter and we neglect neutrinos and radiations. For a different perspective over the form of standard matter, see Ref. [49].

  3. 3.

    Here, we take \(H_0\) as present value of H evaluated at our time, i.e., \(t_0\). Although a severe tension occurs [54], this leaves unaltered our final outcomes, since all quantities of interest are re-written accordingly to our choice.

  4. 4.

    The differential equation (26) is a third-order differential equation for H in the z variable. Therefore, the Cauchy problem requires three initial conditions for \(H(z=0) = H_0\), \(\dot{H}(z=0) = \dot{H}_0\), and \(\ddot{H}(z=0) = \ddot{H}_0\). The value of these initial conditions is in Eq. (13).

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Acknowledgements

OL acknowledges INFN, Frascati National Laboratories, for Iniziative Specifiche MOONLIGHT2 for support. This article is also supported in part by the Ministry of Education and Science of the Republic of Kazakhstan and OL acknowledges Program ’Fundamental and applied studies in related fields of physics of terrestrial, near-earth and atmospheric processes and their practical application’ IRN: BR05236494.

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Correspondence to Lorenzo Sebastiani.

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Calzá, M., Casalino, A., Luongo, O. et al. Kinematic reconstructions of extended theories of gravity at small and intermediate redshifts. Eur. Phys. J. Plus 135, 1 (2020). https://doi.org/10.1140/epjp/s13360-019-00059-2

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