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 Pad\'e approximations. Due to the superior convergence properties of Pad\'e 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 Pad\'e 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 Pad\'e 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 Pad\'e expansion. We obtain fairly good agreement with the Planck results, confirming the $\mathrm{$\Lambda${}}\mathrm{CDM}$ model at small redshifts, although we cannot exclude a dark energy density varying in time with negligible speed of sound.

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 data sets. 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 reparametrizing 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 $\Lambda${}CDM model, although alternative fluid models, with nearly constant pressure and no cosmological constant, match the results accurately as well.

We examine the observational viability of a class of $f(\mathcal{R})$ 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, $\mathcal{R}=\mathcal{R}(z)$, 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 $\mathrm{\Lambda }\mathrm{CDM}$. We then infer late-time observational constraints on $f(z)$ in terms of bounds on the Taylor expansion cosmographic coefficients. In doing so we parametrize possible departures from the standard $\mathrm{\Lambda }\mathrm{CDM}$ 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.