We study the John term of Fab Four cosmology in the presence of a scalar potential. We show here how this theory can describe a wide range of cosmological solutions. This theory has two general functions of the scalar field: the potential V (φ) and the John coefficient function V j (φ). We show that for very simple choices of those functions, we can describe an accelerated expansion, a radiation-dominated era, and a matter-dominated era. By means of simple modifications, it is also possible to describe nonsingular bouncing versions of those solutions and cyclic universes. We also address some quantum issues of that theory, showing that, for the most significant singular cases, the theory admits a classically well behaved quantization, even though the Hamiltonian has fractional powers in the momenta.

We study a quantization via fractional derivative of a nonminimal derivative coupling cosmological theory, namely, the Fab Four John theory. Its Hamiltonian version presents the issue of fractional powers in the momenta. That problem is solved here by the application of the so-called conformable fractional derivative. This leads to a Wheeler-DeWitt equation of second order, showing that a Bohm-de Broglie interpretation can be constructed. That combination of fractional quantization and Bohmian interpretation provides us a new quantization method, in which the quantum potential is the criterion to say if a quantum solution is acceptable or not to be further studied. We show that a wide range of solutions for the scale factor is possible. Among all of those, a bouncing solution analogous to the perfect fluid cosmology seems to deserve special attention. *

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In a Bohmian quantum cosmology scenario, we investigate some quantum effects on the evolution of the primordial universe arising from the adoption of an alternative non-trivial ordering to the quantization of the constrained Hamiltonian of a minimally coupled scalar field. The Wheeler-DeWitt equation has a contribution from the change in factor ordering, hence there are new quantum effects. We compare the results between the non-trivial and the trivial ordering cases, showing that the classical limit is valid for both orderings, but new bouncing and cyclic solutions are present in the non-trivial case. Additionally, we show that the non-singular solutions already present in the trivial ordering formalism keep valid. * Bohmian) quantum cosmology. In Refs. [7-9], it is described how such an alternative interpretation can be generalized from quantum mechanics to cosmological models such as those of Eq. (1). In particular, even when V = 0 one is able to solve the singularity problem thanks to quantum corrections, which induce a bounce. The case of an exponential potential, related with a matter-dominated universe, was recently presented in Ref. [10].The Lagrangian (1) can also be seen as one of the simplest particular cases of Horndeski modified gravity theory [11], thus it is free of Ostrogradsky instability [12,13]. It is also in agreement with the recent constraints imposed by GW170817 and GRB170817A [14-16] on the velocity of gravitational waves. See e.g. Ref.[13]. Lagrangian (1) is also related with effective string theory [7] and is also the Einstein frame version of several other scalar-tensor theories of gravity [5].The motivation for adopting an alternative interpretation of quantum mechanics in a cosmological setting comes from the fact that the exterior domain hypothesis [17], tacitly present in most standard interpretations of quantum mechanics, is considered by some authors as being a conceptual problem when the system under investigation is the universe [18][19][20]. Because of that, it has been proposed [20-23] to adopt in cosmology the Bohmian interpretation of quantum mechanics [24,25]. Besides that, the Bohmian interpretation also solves the problem of time ambiguity in quantum cosmology and quantum gravity [26,27], thanks to the guidance equations. More about Bohm-de Broglie quantum mechanics can be found in Refs. [28][29][30][31][32][33].Another conceptual problem faced in the quantisation of a gravitational theory like (1) is the factor ordering ambiguity, a direct consequence of Dirac's quantisation rule. In Ref. [23], it is shown that the basic features of Bohmian quantum gravity do not really depend on the factor ordering, although some ordering must be chosen to actually apply quantisation. Basead on that argument, it is common to apply only the trivial ordering in the quantization of the constrained Hamiltonian, like it is done in Ref. [23]. But the factor ordering ambiguity remains, so that we can ask ourselves: if the general features of that theory are invariant under a change of ordering...