We give a short survey on the concept of contact Hamiltonian dynamics and its use in several areas of physics, namely reversible and irreversible thermodynamics, statistical physics and classical mechanics. Some relevant examples are provided along the way. We conclude by giving insights into possible future directions.

We propose a novel approach for parametrizing 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 data, 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 $\mathrm{\Lambda }\mathrm{CDM}$ 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’s. equation of state, constraining the dark energy term in a model-independent way.

A cosmographic reconstruction of $f(\mathcal{T})$ models is here revised in a model independent way by fixing observational bounds on the most relevant terms of the $f(\mathcal{T})$ Taylor expansion. We relate the $f(\mathcal{T})$ models and their derivatives to the cosmographic parameters and then adopt a Monte Carlo analysis. The experimental bounds are thus independent of the choice of a particular $f(\mathcal{T})$ model. The advantage of such an analysis lies on constraining the dynamics of the Universe by reconstructing the form of $f(\mathcal{T})$, without any further assumptions apart from the validity of the cosmological principle and the analyticity of the $f(\mathcal{T})$ function. The main result is to fix model independent cosmographic constraints on the functional form of $f(\mathcal{T})$ which is compatible with the theoretical predictions. Furthermore, we infer a phenomenological expression for $f(\mathcal{T})$, compatible with the current cosmographic bounds and show that small deviations are expected from a constant $f(\mathcal{T})$ term, indicating that the equation of state of dark energy could slightly evolve from the one of the Lambda cold dark model.