The possibility of obtaining singularity free cosmological solutions in four dimensional effective actions motivated by string theory is investigated. In these effective actions, in addition to the Einstein-Hilbert term, the dilatonic and the axionic fields are also considered as well as terms coming from the Ramond-Ramond sector. A radiation fluid is coupled to the field equations, which appears as a consequence of the Maxwellian terms in the Ramond-Ramond sector. Singularity free bouncing solutions in which the dilaton is finite and strictly positive are obtained for models with flat or negative curvature spatial sections when the dilatonic coupling constant is such that ω < −3/2, which may appear in the so called F theory in 12 dimensions. These bouncing phases are smoothly connected to the radiation dominated expansion phase of the standard cosmological model, and the asymptotic pasts correspond to very large flat spacetimes.

We study cosmological scenarios resulting from effective actions in four dimensions which are, under some assumptions, connected with multidimensional, supergravity and string theories. These effective actions are labeled by the parameters , the dilaton coupling constant, and n which establishes the coupling between the dilaton and a scalar field originating from the gauge field existing in the original theories. There is a large class of bouncing as well as Friedmann-like solutions. We investigate under which conditions bouncing regular solutions can be obtained. In the case of the string effective action, regularity is obtained through the inclusion of contributions from the Ramond-Ramond sector of superstring.

A minimal-length scenario can be considered as an effective description of quantum gravity effects. In quantum mechanics the introduction of a minimal length can be accomplished through a generalization of Heisenberg's uncertainty principle. In this scenario, state eigenvectors of the position operator are no longer physical states and the representation in momentum space or a representation in a quasiposition space must be used. In this work, we solve the Schroedinger equation with a Dirac δ-function potential in quasiposition space. We calculate the bound state energy and the coefficients of reflection and transmission for the scattering states. We show that leading corrections are of order of the minimal length (O(√ β)) and the coefficients of reflection and transmission are no longer the same for the Dirac delta well and barrier as in ordinary quantum mechanics. Furthermore, assuming that the equivalence of the 1s state energy of the hydrogen atom and the bound state energy of the Dirac δ-function potential in the one-dimensional case is kept in a minimal-length scenario, we also find that the leading correction term for the ground state energy of the hydrogen atom is of the order of the minimal length and Δx min ≤ 10 −25 m.

The string effective action at tree level contains, in its bosonic sector, the Einstein-Hilbert term, the dilaton, and the axion, besides scalar and gauge fields coming from the Ramond-Ramond sector. The reduction to four dimensions brings to scene moduli fields. We generalize this effective action by introducing two arbitrary parameters, ω and m, connected with the dilaton and axion couplings. In this way, more general frameworks can be analyzed. Regular solutions with a bounce can be obtained for a range of (negative) values of the parameter ω which, however, exclude the pure string configuration (ω = −1). We study the evolution of scalar perturbations in such cosmological scenarios. The predicted primordial power spectrum decreases with the wavenumber with spectral index ns = −2, in contradiction with the results of the W M AP . Hence, all such effective string motivated cosmological bouncing models seem to be ruled out, at least at the tree level approximation.

We review the Eulerian description of hydrodynamics using Seliger-Whitham's formalism (in classical case) and Schutz's formalism (in relativistic case). In these formalisms, the velocity field of a perfect fluid is described by scalar potentials.With this we can obtain the evolution equations of the fluid and its Hamiltonian.In the scenario of quantum cosmology the Schutz's formalism makes it possible to introduce phenomenologically a time variable in minisuperspace models.PACS number(s): 98.80.Cq, 98.80.Qc