Quantum cosmology of quadratic f(R) theories with a FRW metric

We study the quantum cosmology of a quadratic $f(R)$ theory with a FRW metric, via one of its equivalent Horndeski type actions, where the dynamics of the scalar field is induced. The classical equations of motion and the Weeler-deWitt equation, in their exact versions, are solved numerically. From the choice of a free parameter in the action follow two cases, inflation + exit and inflation alone. The numerical solution of the Wheeler-DeWitt equation depends strongly on the boundary conditions, which can be chosen so that the resulting wave function of the universe seems to be normalizable and consistent with hermitian operators.Comment: 6 pages, 4 figure

Time in Quantum Cosmology of FRW f(R) Theories

The time problem is a problem of canonical quantum gravity that has long been known about; it is related to the relativistic invariance and the consequent absence of an explicit time variable in the quantum equations. This fact complicates the interpretation of the wave function of the universe. Following proposals to assign the clock function to a scalar field, we look at the scalar degree of freedom contained in f ( R ) theories. For this purpose we consider a quadratic f ( R ) theory in an equivalent formulation with a scalar field, with a FRW metric, and consider its Wheeler-DeWitt equation. The wave function is obtained numerically and is consistent with the interpretation of the scalar field as time by means of a conditional probability, from which an effective time-dependent wave function follows. The evolution the scale factor is obtained by its mean value, and the quantum fluctuations are consistent with the Heisenberg relations and a classical universe today

2-(4-Hydroxyphenyl)-1H-benzimidazol-3-ium chloride monohydrate

The title molecular salt, C13H11N2O+·Cl−·H2O, crystallizes as a monohydrate. In the cation, the phenol and benzimidazole rings are almost coplanar, making a dihedral angle of 3.18 (4)°. The chloride anion and benzimidazole cation are linked by two N+—H...Cl− hydrogen bonds, forming chains propagating along [010]. These chains are linked through O—H...Cl hydrogen bonds involving the water molecule and the chloride anion, which form a diamond core, giving rise to the formation of two-dimensional networks lying parallel to (10-2). Two π–π interactions involving the imidazolium ring with the benzene and phenol rings [centroid–centroid distances = 3.859 (3) and 3.602 (3) Å, respectively], contribute to this second dimension. A strong O—H...O hydrogen bond involving the water molecule and the phenol substituent on the benzimidazole unit links the networks, forming a three-dimensional structure