In this work, we consider an extension of symmetric teleparallel gravity, namely, $f(Q)$ gravity, where the fundamental block to describe spacetime is the nonmetricity, $Q$. Within this formulation of gravitation, we perform an observational analysis of several modified $f(Q)$ models using the redshift approach, where the $f(Q)$ Lagrangian is reformulated as an explicit function of the redshift, $f(z)$. Various different polynomial parametrizations of $f(z)$ are proposed, including new terms which would allow for deviations from the $\mathrm{\Lambda }$ Cold Dark Matter model. Given a variety of observational probes, such as the expansion rate data from early type galaxies, type Ia supernovae, quasars, gamma ray bursts, baryon acoustic oscillations data, and cosmic microwave background distance priors, we have checked the validity of these models at the background level in order to verify if this new formalism provides us with plausible alternative models to explain the late time acceleration of the Universe. Indeed, this novel approach provides a different perspective on the formulation of observationally reliable alternative models of gravity.

It is nowadays accepted that the universe is undergoing a phase of accelerated expansion as tested by the Hubble diagram of type Ia supernovae (SNeIa) and several large scale structure observations. Future SNeIa surveys and other probes will make it possible to better characterize the dynamical state of the universe, renewing the interest in cosmography which allows a model independent analysis of the distance-redshift relation. On the other hand, fourth order theories of gravity, also referred to as $f(R)$ gravity, have attracted a lot of interest since they could be able to explain the accelerated expansion without any dark energy. We show here how it is possible to relate the cosmographic parameters (namely, the deceleration $q_0$, the jerk $j_0$, the snap $s_0$, and the lerk $l_0$ parameters) to the present-day values of $f(R)$ and its derivatives $f^{(n)}(R)=d^{n}f/d{R}^n$ (with $n=1$, 2, 3), thus offering a new tool to constrain such higher order models. Our analysis thus offers the possibility to relate the model independent results coming from cosmography to the theoretically motivated assumptions of $f(R)$ cosmology.

Over the last years some interest has been gathered by $f(Q)$ theories, which are new candidates to replace Einstein’s prescription for gravity. The nonmetricity tensor $Q$ allows to put forward the assumption of a free torsionless connection and, consequently, new degrees of freedom in the action are taken into account. This work focuses on a class of $f(Q)$ theories, characterized by the presence of a general power-law term which adds up to the standard (linear in) $Q$ term in the action, and on new cosmological scenarios arising from them. Using the Markov chain Monte Carlo method, we carry out statistical tests relying upon background data such as Type Ia supernovae luminosities and direct Hubble data (from cosmic clocks), along with cosmic microwave background shift and baryon acoustic oscillations data. This allows us to perform a multifaceted comparison between these new cosmologies and the (concordance) $\mathrm{\Lambda }\mathrm{CDM}$ setup. We conclude that, at the current precision level, the best fits of our $f(Q)$ models correspond to values of their specific parameters which make them hardly distinguishable from our general relativity “échantillon,” that is, $\mathrm{\Lambda }\mathrm{CDM}$.

We study the possibility of extracting model independent information about the dynamics of the Universe by using cosmography. We intend to explore it systematically, to learn about its limitations and its real possibilities. Here we are sticking to the series expansion approach on which cosmography is based. We apply it to different data sets: Supernovae type~Ia (SNeIa), Hubble parameter extracted from differential galaxy ages, gamma ray bursts, and the baryon acoustic oscillations data. We go beyond past results in the literature extending the series expansion up to the fourth order in the scale factor, which implies the analysis of the deceleration q0, the jerk j0, and the snap s0. We use the Markov chain Monte~Carlo method (MCMC) to analyze the data statistically. We also try to relate direct results from cosmography to dark energy (DE) dynamical models parametrized by the Chevallier-Polarski-Linder model, extracting clues about the matter content and the dark energy parameters. The main results are: (a)~even if relying on a mathematical approximate assumption such as the scale factor series expansion in terms of time, cosmography can be extremely useful in assessing dynamical properties of the Universe; (b)~the deceleration parameter clearly confirms the present acceleration phase; (c)~the MCMC method can help giving narrower constraints in parameter estimation, in particular for higher order cosmographic parameters (the jerk and the snap), with respect to the literature; and (d)~both the estimation of the jerk and the DE parameters reflect the possibility of a deviation from the $\Lambda${}CDM cosmological model.