The present work studies one of Einstein’s alternative formulations based on the nonmetricity scalar $Q$ generalized as $f(Q)$ theory. More specifically, we consider the power-law form of $f(Q)$ gravity, i.e., $f(Q)=Q+\alpha Q^n$. Here, we analyze the behavior of the cosmological model at the background and perturbation level. Using the dynamical system analysis, at the background level, we find the effective evolution of the model is the same as that of the $\mathrm{\Lambda }\mathrm{CDM}$ for $|n|<1$. Interestingly, the geometric component of the theory solely determined the late-time acceleration of the Universe. We also examine the integrability of the model by employing the method of singularity analysis. In particular, we find the conditions under which field equations pass the Painlevé test and hence possess the Painlevé property. While the equations pass the Painlevé test in the presence of dust for any value of $n$, the test is valid after the addition of radiation fluid only for $n<1$. Finally, at the perturbation level, the behavior of matter growth index signifies a deviation of the model from the $\mathrm{\Lambda }\mathrm{CDM}$ even for $|n|<1$.

In the cosmological scenario in $f(T)$ gravity, we find analytical solutions for an isotropic and homogeneous universe containing a dust fluid and radiation and for an empty anisotropic Bianchi I universe. The method that we apply is that of movable singularities of differential equations. For the isotropic universe, the solutions are expressed in terms of a Laurent expansion, while for the anisotropic universe we find a family of exact Kasner-like solutions in vacuum. Finally, we discuss when a nonlinear $f(T)$-gravity theory provides solutions for the teleparallel equivalence of general relativity and derive conditions for exact solutions of general relativity to solve the field equations of an $f(T)$ theory.

We investigate the main features of the flat Friedmann-Lema\^\itre-Robertson-Walker cosmological models in the $f(T)$ modified gravity regime. In particular, a general approach to find out exact cosmological solutions in $f(T)$ gravity is discussed. Instead of taking into account phenomenological models, we consider as a selection criterion, the existence of Noether symmetries in the cosmological $f(T)$ pointlike Lagrangian. We find that only the $f(T)={f}_{0}{T}^{n}$ model admits extra Noether symmetries. The existence of extra Noether integrals can be used in order to simplify the system of differential equations (equations of motion) as well as to determine the integrability of the $f(T)={f}_{0}{T}^{n}$ cosmological model. Within this context, we can solve the problem analytically and thus we provide the evolution of the main cosmological functions such as the scale factor of the Universe, the Hubble expansion rate, the deceleration parameter, and the linear matter perturbations. We show that the $f(T)={f}_{0}{T}^{n}$ cosmological model suffers from two basic problems. The first problem is related to the fact that the deceleration parameter is constant which means that it never changes sign, and therefore the Universe always accelerates or always decelerates depending on the value of $n$. Second, we find that the clustering growth rate remains always equal to unity implying that the recent growth data disfavor the $f(T)={f}_{0}{T}^{n}$ gravity. Finally, we prove that the $f(T)={f}_{0}{T}^{n}$ gravity can be cosmologically equivalent with the $f(R)={R}^{n}$ gravity model and the time varying vacuum model $$\Lambda${}(H)=3$\gamma${}{H}^{2}$ (for ${n}^{-1}=1-$\gamma${}$) because the above cosmological scenarios share exactly the same Hubble expansion, despite the fact that the three models have a different geometrical origin. Finally, some important differences with power-law $f(R)$ gravity are pointed out.

Spherically symmetric solutions for $f(T)$ gravity models are derived by the so-called Noether symmetry approach. First, we present a full set of Noether symmetries for some minisuperspace models. Then, we compute analytical solutions and find that spherically symmetric solutions in $f(T)$ gravity can be recast in terms of Schwarzschild-like solutions modified by a distortion function depending on a characteristic radius. The obtained solutions are more general than those obtained by the usual solution methods.

We propose to use a model-independent criterion based on first integrals of motion, due to Noether symmetries of the equations of motion, in order to classify the dark energy models in the context of scalar field (quintessence or phantom) Friedmann-Lema\^\itre-Robertson-Walker cosmologies. In general, the Noether symmetries play an important role in physics because they can be used to simplify a given system of differential equations as well as to determine the integrability of the system. The Noether symmetries are computed for nine distinct accelerating cosmological scenarios that contain a homogeneous scalar field associated with different types of potentials. We verify that all the scalar field potentials, presented here, admit the trivial first integral, namely, energy conservation, as they should. We also find that the exponential potential inspired from scalar field cosmology, as well as some types of hyperbolic potentials, include extra Noether symmetries. This feature suggests that these potentials should be preferred along the hierarchy of scalar field potentials. Finally, using the latter potentials, in the framework of either quintessence or phantom scalar field cosmologies that contain also a nonrelativistic matter (dark matter) component, we find that the main cosmological functions, such as the scale factor of the Universe, the scalar field, the Hubble expansion rate, and the metric of the Friedmann-Lema\^\itre-Robertson-Walker space-time, are computed analytically. Interestingly, under specific circumstances the predictions of the exponential and hyperbolic scalar field models are equivalent to those of the $\Lambda${}CDM model, as far as the global dynamics and the evolution of the scalar field are concerned. The present analysis suggests that our technique appears to be very competitive to other independent tests used to probe the functional form of a given potential and thus the associated nature of dark energy.

We use Dirac’s method for the quantization of constrained systems in order to quantize a spatially flat Friedmann–Lemaître–Robertson–Walker spacetime in the context of f ( Q ) cosmology. When the coincident gauge is considered, the resulting minisuperspace system possesses second class constraints. This distinguishes the quantization process from the typical Wheeler–DeWitt quantization, which is applied for cosmological models where only first class constraints are present (e.g. for models in general relativity or in f ( R ) gravity). We introduce the Dirac brackets, find appropriate canonical coordinates and then apply the canonical quantization procedure. We perform this method both in vacuum and in the presence of matter: a minimally coupled scalar field and a perfect fluid with a linear equation of state. We demonstrate that the matter content changes significantly the quantization procedure, with the perfect fluid even requiring to put in use the theory of fractional quantum mechanics in which the power of the momentum in the Hamiltonian is associated with the fractal dimension of a Lévy flight. The results of this analysis can be applied in f ( T ) teleparallel cosmology, since f ( Q ) and f ( T ) theories have the same degrees of freedom and same dynamical constraints in cosmological studies.