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 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 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 class of generalized Galileon cosmological models, which can be described by a pointlike Lagrangian, is considered in order to utilize Noether’s theorem to determine conservation laws for the field equations. In the Friedmann-Lemaître-Robertson-Walker universe, the existence of a nontrivial conservation law indicates the integrability of the field equations. Because of the complexity of the latter, we apply the differential invariants approach in order to construct special power-law solutions and study their stability.

We construct exact, regular and topologically nontrivial configurations of the coupled Einstein-nonlinear sigma model in ($3+1$) dimensions. The ansatz for the nonlinear $SU(2)$ field is regular everywhere and circumvents Derrick’s theorem because it depends explicitly on time, but in such a way that its energy-momentum tensor is compatible with a stationary metric. Moreover, the $SU(2)$ configuration cannot be continuously deformed to the trivial Pion vacuum as it possesses a nontrivial winding number. We reduce the full coupled four-dimensional Einstein nonlinear sigma model system to a single second order ordinary differential equation. When the cosmological constant vanishes, such a master equation can be further reduced to an Abel equation. Two interesting regular solutions correspond to a stationary traversable wormhole (whose only “exotic matter” is a negative cosmological constant) and a ($3+1$)-dimensional cylinder whose ($2+1$)-dimensional section is a Lorentzian squashed sphere. The Klein-Gordon equation in these two families of spacetimes can be solved in terms of special functions. The angular equation gives rise to the Jacobi polynomials while the radial equation belongs to the Poschl-Teller family. The solvability of the Poschl-Teller problem implies nontrivial quantization conditions on the parameters of the theory.