Cosmography can be considered as a sort of a model-independent approach to tackle the dark energy/modified gravity problem. In this review, the success and the shortcomings of the ΛCDM model, based on General Relativity and standard model of particles, are discussed in view of the most recent observational constraints. The motivations for considering extensions and modifications of General Relativity are taken into account, with particular attention to f (R) and f (T ) theories of gravity where dynamics is represented by curvature or torsion field respectively. The features of f (R) models are explored in metric and Palatini formalisms. We discuss the connection between f (R) gravity and scalar-tensor theories highlighting the role of conformal transformations in the Einstein and Jordan frames. Cosmological dynamics of f (R) models is investigated through the corresponding viability criteria. Afterwards, the equivalent formulation of General Relativity (Teleparallel Equivalent General Relativity) in terms of torsion and its extension to f (T ) gravity is considered. Finally, the cosmographic method is adopted to break the degeneracy among dark energy models. A novel approach, built upon rational Padé and Chebyshev polynomials, is proposed to overcome limits of standard cosmography based on Taylor expansion. The approach provides accurate model-independent approximations of the Hubble flow. Numerical analyses, based on Monte Carlo Markov Chain integration of cosmic data, are presented to bound coefficients of the cosmographic series. These techniques are thus applied to reconstruct f (R) and f (T ) functions and to frame the late-time expansion history of the universe with no a priori assumptions on April 3, 2019 0:31 WSPC/INSTRUCTION FILE Review 2 S. Capozziello, R. D'Agostino, O. Luongo its equation of state. A comparison between the ΛCDM cosmological model with f (R) and f (T ) models is reported.

We use cosmography to present constraints on the kinematics of the Universe, without postulating any underlying theoretical model. To this end, we use a Monte Carlo Markov Chain analysis to perform comparisons to the supernova Ia Union 2 compilation, combined with the Hubble Space Telescope measurements of the Hubble constant, and the Hubble parameter datasets. We introduce a sixth order cosmographic parameter and show that it does not enlarge considerably the posterior distribution when comparing to the fifth order results. We also propose a way to construct viable parameter variables to be used as alternatives of the redshift z. These can overcome both the problems of divergence and lack of accuracy associated with the use of z. Moreover, we show that it is possible to improve the numerical fits by re-parameterizing the cosmological distances. In addition, we constrain the equation of state of the Universe as a whole by the use of cosmography. Thus, we derive expressions which can be directly used to fit the equation of state and the pressure derivatives up to fourth order. To this end, it is necessary to depart from a pure cosmographic analysis and to assume the Friedmann equations as valid. All our results are consistent with the ΛCDM model, although alternative fluid models, withnearly constant pressure and no cosmological constant, match the results accurately as well.PACS numbers: 98.80.Jk, 98.80.Es

Cosmography is used in cosmological data processing in order to constrain the kinematics of the universe in a model-independent way, providing an objective means to evaluate the agreement of a model with observations. In this paper, we extend the conventional methodology of cosmography employing Taylor expansions of observables by an alternative approach using Padé approximations. Due to the superior convergence properties of Padé expansions, it is possible to improve the fitting analysis to obtain numerical values for the parameters of the cosmographic series. From the results, we can derive the equation of state parameter of the universe and its first derivative and thus acquire information about the thermodynamic state of the universe. We carry out statistical analyses using observations of the distance modulus of type 1a supernovae, provided by the union 2.1 compilation of the supernova cosmology project, employing a Markov chain Monte Carlo approach with an implemented Metropolis algorithm. We compare the results of the original Taylor approach to the newly introduced Padé formalism. The analyses show that experimental data constrain the observable universe well, finding an accelerating universe and a positive jerk parameter. We demonstrate that the Padé convergence radii are greater than standard Taylor convergence radii, and infer a lower limit on the acceleration of the universe solely by requiring the positivity of the Padé expansion. We obtain fairly good agreement with the Planck results, confirming the ΛCDM model at small redshifts, although we cannot exclude a dark energy density varying in time with negligible speed of sound.

We here propose a new model-independent technique to overcome the circularity problem affecting the use of gamma-ray bursts (GRBs) as distance indicators through the use of Ep−Eiso correlation. We calibrate the Ep−Eiso correlation and find the GRB distance moduli that can be used to constrain dark energy models. We use observational Hubble data to approximate the cosmic evolution through Bézier parametric curve obtained through the linear combination of Bernstein basis polynomials. In doing so, we build up a new data set consisting of 193 GRB distance moduli. We combine this sample with the supernova JLA data set to test the standard ΛCDM model and its wCDM extension. We place observational constraints on the cosmological parameters through Markov Chain Monte Carlo numerical technique. Moreover, we compare the theoretical scenarios by performing the Akaike and Deviance Information statistical criteria.the 2σ level, while for the wCDM model we obtain $\Omega _m=0.34^{+0.13}_{-0.15}$ and $w=-0.86^{+0.36}_{-0.38}$ at the 2σ level. Our analysis suggests that ΛCDM model is statistically favoured over the wCDM scenario. No evidence for extension of the ΛCDM model is found.

We examine the observational viability of a class of f (R) gravity cosmological models. Particular attention is devoted to constraints from the recent observational determination of the redshift of the cosmological deceleration-acceleration transition. Making use of the fact that the Ricci scalar is a function of redshift z in these models, R = R(z), and so is f (z), we use cosmography to relate a f (z) test function evaluated at higher z to late-time cosmographic bounds. First, we consider a model independent procedure to build up a numerical f (z) by requiring that at z = 0 the corresponding cosmological model reduces to standard ΛCDM. We then infer late-time observational constraints on f (z) in terms of bounds on the Taylor expansion cosmographic coefficients. In doing so we parameterize possible departures from the standard ΛCDM model in terms of a two-parameter logarithmic correction. The physical meaning of the two parameters is also discussed in terms of the post Newtonian approximation. Second, we provide numerical estimates of the cosmographic series terms by using Type Ia supernova apparent magnitude data and Hubble parameter measurements. Finally, we use these estimates to bound the two parameters of the logarithmic correction. We find that the deceleration parameter in our model changes sign at a redshift consistent with what is observed.

Cosmography represents an important branch of cosmology which aims to describe the universe without the need of postulating a priori any particular cosmological model. All quantities of interest are expanded as a Taylor series around here and now, providing in principle, a way of directly matching with cosmological data. In this way, cosmography can be regarded a model-independent technique, able to fix cosmic bounds, although several issues limit its use in various model reconstructions. The main purpose of this review is to focus on the key features of cosmography, emphasising both the strategy for obtaining the observable cosmographic series and pointing out any drawbacks which might plague the standard cosmographic treatment. In doing so, we relate cosmography to the most relevant cosmological quantities and to several dark energy models. We also investigate whether cosmography is able to provide information about the form of the cosmological expansion history, discussing how to reproduce the dark fluid from the cosmographic sound speed. Following this, we discuss limits on cosmographic priors and focus on how to experimentally treat cosmographic expansions. Finally, we present some of the latest developments of the cosmographic method, reviewing the use of rational approximations, based on cosmographic Padé polynomials. Future prospects leading to more accurate cosmographic results, able to better reproduce the expansion history of the universe are also discussed in detail.

We propose a novel approach for parameterizing the luminosity distance, based on the use of rational "Padé" approximations. This new technique extends standard Taylor treatments, overcoming possible convergence issues at high redshifts plaguing standard cosmography. Indeed, we show that Padé expansions enable us to confidently use data over a larger interval with respect to the usual Taylor series. To show this property in detail, we propose several Padé expansions and we compare these approximations with cosmic data, thus obtaining cosmographic bounds from the observable universe for all cases. In particular, we fit Padé luminosity distances with observational data from different uncorrelated surveys. We employ union 2.1 supernova data, baryonic acoustic oscillation, Hubble space telescope measurements and differential age data. In so doing, we also demonstrate that the use of Padé approximants can improve the analyses carried out by introducing cosmographic auxiliary variables, i.e. a standard technique usually employed in cosmography in order to overcome the divergence problem. Moreover, for any drawback related to standard cosmography, we emphasize possible resolutions in the framework of Padé approximants. In particular, we investigate how to reduce systematics, how to overcome the degeneracy between cosmological coefficients, how to treat divergences and so forth. As a result, we show that cosmic bounds are actually refined through the use of Padé treatments and the thus derived best values of the cosmographic parameters show slight departures from the standard cosmological paradigm. Although all our results are perfectly consistent with the ΛCDM model, evolving dark energy components different from a pure cosmological constant are not definitively ruled out. Finally, we use our outcomes to reconstruct the effective universe equation of state, constraining the dark energy term in a model independent way.PACS numbers: 98.80.Jk, 98.80.Es

We revise the cosmological standard model presuming that matter, i.e. baryons and cold dark matter, exhibits a non-vanishing pressure mimicking the cosmological constant effects. In particular, we propose a scalar field Lagrangian L1 for matter with the introduction of a Lagrange multiplier as constraint. We also add a symmetry breaking effective potential accounting for the classical cosmological constant problem, by adding a second Lagrangian L2. Investigating the Noether current due to the shift symmetry on the scalar field, ϕ → ϕ + c 0 , we show that L1 turns out to be independent from the scalar field ϕ. Further we find that a positive Helmotz free-energy naturally leads to a negative pressure without introducing by hand any dark energy term. To face out the fine-tuning problem, we investigate two phases: before and after transition due to the symmetry breaking. We propose that during transition dark matter cancels out the quantum field vacuum energy effects. This process leads to a negative and constant pressure whose magnitude is determined by baryons only. The numerical bounds over the pressure and matter densities are in agreement with current observations, alleviating the coincidence problem. Finally assuming a thermal equilibrium between the bath and our effective fluid, we estimate the mass of the dark matter candidate. Our numerical outcomes seem to be compatible with recent predictions on WIMP masses, for fixed spin and temperature. In particular, we predict possible candidates whose masses span in the range 0.5 − 1.7 TeV.