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.

We study a Friedmann-Lemaître-Robertson-Walker spacetime in the theory of $f(Q)$ gravity, where $Q$ denotes the nonmetricity scalar. It has been previously shown in the literature, that there exist four distinct families of connections, which are compatible with the isometries of the FLRW metric; three for the spatially-flat case and one when the spatial curvature is present. In the spatially-flat case, one connection is dynamically irrelevant and yields the dynamics of the coincident gauge in the Cartesian coordinates. For this, we obtain the general solution of an arbitrary $f(Q)$ theory with a perfect fluid-matter content, and present various examples for specific choices of the $f(Q)$ function. We proceed by studying the effect of the rest of the connections, which are dynamical and affect the equations of the motion. We concentrate in scenarios that depart from the $Q=\mathrm{const}$ case, which just reproduces General Relativity with a cosmological constant, and derive novel vacuum solutions for a power-law $f(Q)$ function.

We present the solution space for the case of a minimally coupled scalar field with arbitrary potential in a Friedmann-Lemaître-Robertson-Walker metric. This is made possible due to the existence of a nonlocal integral of motion corresponding to the conformal Killing field of the two-dimensional minisuperspace metric. Both the spatially flat and nonflat cases are studied first in the presence of only the scalar field and subsequently with the addition of noninteracting perfect fluids. It is verified that this addition does not change the general form of the solution, but only the particular expressions of the scalar field and the potential. The results are applied in the case of parametric dark energy models where we derive the scalar field equivalence solution for some proposed models in the literature.

A scalar-tensor theory of gravity, containing an arbitrary coupling function $F(\varphi )$ and a general potential $V(\varphi )$, is considered in the context of a spatially flat Friedmann-Lemaître-Robertson-Walker model. The use of reparametrization invariance enables a particular lapse parametrization in which the minisuperspace metric completely specifies the dynamics of the system. A requirement of the existence of the maximal possible number of autonomous integrals of motion is imposed. This leads to a flat minisuperspace metric realized by a particular relation between the coupling function and the potential. The space of solutions is completely described in terms of the three autonomous integrals of motion constructed by the Killing fields of the minisupermetric and an additional rheonomous emanating from the homothetic field. The solutions contain the arbitrary function which remains after the imposition of the relation between $F(\varphi )$ and $V(\varphi )$. To exemplify the use of the general results, we select some particular cases and study their physical implications through an effective energy-momentum tensor, which turns out to be that of a perfect fluid.

We investigate the existence of anisotropic self-similar exact solutions in symmetric teleparallel $$f\left( Q\right) $$f Q -theory. For the background geometry we consider the Kantowski–Sachs and the Locally Rotationally Symmetric Bianchi type III geometries. These two anisotropic spacetimes are of special interest because in the limit of isotropy they are related to the closed and open Friedmann–Lemaître–Robertson–Walker cosmologies respectively. For each spacetime there exist two distinct families of flat, symmetric connections, which share the symmetries of the spacetime. We present the field equations, and from them, we determine the functional form of the $$f\left( Q\right) $$f Q Lagrangian which yields self-similar solutions. We initially consider the vacuum case and subsequently we introduce a matter source in terms of a perfect fluid. Last but not least, we report some self-similar solutions corresponding to static spherically symmetric spacetimes.

The existence of self-similar solutions is discussed in symmetric teleparallel $f(Q)$ theory for a Friedmann-Lemaître-Robertson-Walker background geometry with zero and nonzero spatial curvature. For the four distinct families of connections that describe the specific cosmology in symmetric teleparallel gravity, the functional form of $f(Q)$ is reconstructed. Finally, to see if the analogy with GR holds, we discuss the relation of the self-similar solutions with the asymptotic behavior of more general $f(Q)$ functions.