The geometrical nature of gravity emerges from the universality dictated by the equivalence principle. In the usual formulation of General Relativity, the geometrisation of the gravitational interaction is performed in terms of the spacetime curvature, which is now the standard interpretation of gravity. However, this is not the only possibility. In these notes, we discuss two alternative, though equivalent, formulations of General Relativity in flat spacetimes, in which gravity is fully ascribed either to torsion or to non-metricity, thus putting forward the existence of three seemingly unrelated representations of the same underlying theory. Based on these three alternative formulations of General Relativity, we then discuss some extensions.

The universal character of the gravitational interaction provided by the equivalence principle motivates a geometrical description of gravity. The standard formulation of general relativity à la Einstein attributes gravity to the spacetime curvature, to which we have grown accustomed. However, this perception has masked the fact that two alternative, though equivalent, formulations of general relativity in flat spacetimes exist, where gravity can be fully ascribed either to torsion or to nonmetricity. The latter allows a simpler geometrical formulation of general relativity that is oblivious to the affine spacetime structure. Generalizations along this line permit us to generate teleparallel and symmetric teleparallel theories of gravity with exceptional properties. In this work we explore modified gravity theories based on nonlinear extensions of the nonmetricity scalar. After presenting some general properties and briefly studying some interesting background cosmologies (including accelerating solutions with relevance for inflation and dark energy), we analyze the behavior of the cosmological perturbations. Tensor perturbations feature a rescaling of the corresponding Newton’s constant, while vector perturbations do not contribute in the absence of vector sources. In the scalar sector we find two additional propagating modes, hinting that $f(Q)$ theories introduce, at least, 2 additional degrees of freedom. These scalar modes disappear around maximally symmetric backgrounds because of the appearance of an accidental residual gauge symmetry corresponding to a restricted diffeomorphism. We finally discuss the potential strong coupling problems of these maximally symmetric backgrounds caused by the discontinuity in the number of propagating modes.

Local determinations of the Hubble constant $H_0$ favor a higher value than Planck based on cosmic microwave background and $\mathrm{\Lambda }$ cold dark matter ($\mathrm{\Lambda }\mathrm{CDM}$). Through a model-independent expansion, we show that low redshift ($z\lesssim 0.7$) data comprising baryon acoustic oscillations, cosmic chronometers, and Type Ia supernovae have a preference for quintessence models that lower $H_0$ relative to $\mathrm{\Lambda }\mathrm{CDM}$. In addition, we confirm that an exponential coupling to dark matter cannot alter this conclusion in the same redshift range. Our results leave open the possibility that a coupling in the matter-dominated epoch, potentially even in the dark ages, may yet save $H_0$ from sinking in the string theory swampland.

In this article, we study the implications of string swampland criteria for dark energy in view of ongoing and future cosmological observations. If string theory should be the ultimate quantum gravity theory, there is evidence that exact de Sitter solutions with a positive cosmological constant cannot describe the fate of the late-time universe. Even though cosmological models with dark energy given by a scalar field $\pi$ evolving in time are not in direct tension with string theory, they have to satisfy the swampland criteria $|\mathrm{\Delta }\pi |<d\sim \mathcal{O}(1)$ and $|V^{\prime }|/V>c\sim \mathcal{O}(1)$, where $V$ is the scalar field potential. In view of the restrictive implications that the swampland criteria have on dark energy, we investigate the accuracy needed for future observations to tightly constrain standard dark-energy models. We find that current $3\text{−}\sigma$ constraints with $c\lesssim 1.35$ are still well in agreement with the string swampland criteria. However, stage-4 surveys such as Euclid, LSST, and DESI, tightly constraining the equation of state $w(z)$, will start putting surviving quintessence models into tensions with the string swampland criteria by demanding $c<0.4$. We further investigate whether any idealized futuristic survey will ever be able to give a decisive answer to the question whether the cosmological constant would be preferred over a time-evolving dark-energy model within the swampland criteria. Hypothetical surveys with a reduction in the uncertainties by a factor of $\sim 20$ compared to Euclid would be necessary to reveal strong tension between quintessence models obeying the string swampland criteria and observations by pushing the allowed values down to $c<0.1$. In view of such perspectives, there will be fundamental observational limitations with future surveys.

We systematically study the field equations of $f(\mathbb{Q})$ gravity for spherically symmetric and stationary metric-affine spacetimes. Such spacetimes are described by a metric as well as a flat and torsionless affine connection. In the symmetric teleparallel equivalent of general relativity (STEGR), the connection is pure gauge and hence unphysical. However, in the nonlinear extension $f(\mathbb{Q})$, it is promoted to a dynamical field which changes the physics. Starting from a general metric-affine geometry, we construct the most general static and spherically symmetric forms of the metric and the affine connection. We then use these symmetry reduced geometric objects to prove that the field equations of $f(\mathbb{Q})$ gravity admit general relativity (GR) solutions as well as beyond-GR solutions, contrary to what has been claimed in the literature. We formulate precise criteria, under which conditions it is possible to obtain GR solutions and under which conditions it is possible to obtain beyond-GR solutions. We subsequently construct several perturbative corrections to the Schwarzschild solution for different choices of $f(\mathbb{Q})$, which in particular include a hair stemming from the now dynamical affine connection. We also present an exact beyond-GR vacuum solution. Lastly, we apply this method of constructing spherically symmetric and stationary solutions to $f(\mathbb{T})$ gravity, which reproduces similar solutions but without a dynamical connection.