Fractional Differential Equations

Modified Gravity and Cosmology

Superstring theoryをめぐって(放談室)

Progress of theoretical physics

Physical Foundations of Cosmology: Preface

Recently, it has been shown that a four-dimensional (4D) Brans–Dicke (BD) theory with an effective matter field and a self-interacting potential can be achieved from the vacuum 5D BD field equations, where we refer to as a modified Brans–Dicke theory (MBDT). We investigate a generalized Bianchi type I anisotropic cosmology in 5D BD theory, and by employing the obtained formalism, we derive the induced matter on any 4D hypersurface in the context of the MBDT. We illustrate that if the usual spatial scale factors are functions of the time while the scale factor of extra dimension is constant, and the scalar field depends on the time and the fifth coordinate, then, in general, one will encounter inconsistencies in the field equations. Then, we assume that the scale factors and the scalar field depend on the time and the extra coordinate as separated variables in the power-law forms. Hence, we find a few classes of solutions in 5D spacetime through which we probe the one which leads to a generalized Kasner relation among the Kasner parameters. The induced scalar potential is found to be in the power law or in the logarithmic form; however, for a constant scalar field and even when the scalar field only depends on the fifth coordinate, it vanishes. The conservation law is indeed valid in this MBDT approach for the derived induced energy–momentum tensor (EMT). We proceed our investigations for a few cosmological quantities, where for simplicity we assume that the metric and the scalar field are functions of the time. Hence, the EMT satisfies the barotropic equation of state, and the model indicates that the constant mean Hubble parameter is not allowed. Thus, by appealing to the variation of the Hubble parameter, we assume a fixed deceleration parameter, and set the evolution of the quantities with respect to the fixed deceleration, the BD coupling and the state parameters. The WEC allows a shrinking extra dimension for a decelerating expanding universe that, in the constant scalar field, evolves the same way as the flat FRW spacetime in GR. The quantities for the stiff fluid and the radiation-dominated universe indicate an expanding universe commenced with a big bang. There is a horizon for each of the fluids, and the rate of expansion slows down by the time. The allowed ranges of the deceleration and the BD coupling parameters have been obtained, and the model gives an empty universe when the time goes to infinity.

http://iopscience.iop.org/article/10.1088/0264-9381/28/15/155004/pdf

An anisotropic cosmological model in a modified Brans–Dicke theory

Within an algebraic framework, used to construct the induced-matter-theory (IMT) setting, in (D + 1)-dimensional Brans–Dicke (BD) scenario, we obtain a modified BD theory (MBDT) in D dimensions. Being more specific, from the (D + 1)-dimensional field equations, a D-dimensional BD theory, bearing new features, is extracted by means of a suitable dimensional reduction onto a hypersurface orthogonal to the extra dimension. In particular, the BD scalar field in such D-dimensional theory has a self-interacting potential, which can be suitably interpreted as produced by the extra dimension. Subsequently, as an application to cosmology, we consider an extended spatially flat FLRW geometry in a (D + 1)-dimensional space–time. After obtaining the power-law solutions in the bulk, we proceed to construct the corresponding physics, by means of the induced MBDT procedure, on the D-dimensional hypersurface. We then contrast the resulted solutions (for different phases of the universe) with those usually extracted from the conventional GR and BD theories in view of current ranges for cosmological parameters. We show that the induced perfect fluid background and the induced scalar potential can be employed, within some limits, for describing different epochs of the universe. Finally, we comment on the observational viability of such a model.

https://iopscience.iop.org/article/10.1088/0264-9381/31/11/115002/pdf

Modified Brans–Dicke theory in arbitrary dimensions

It has been shown that four dimensional Brans-Dicke theory with effective matter field and self interacting potential can be achieved from vacuum 5D BD field equations, where we refer to as modified Brans-Dicke theory (MBDT). We investigate a generalized Bianchi type I anisotropic cosmology in 5D BD theory, and by employing obtained formalism, we derive induced-matter on any 4D hypersurface in context of the MBDT. We illustrate that if the usual spatial scale factors are functions of time while scale factor of extra dimension is constant, and scalar field depends on time and fifth coordinate, then in general, one will encounter inconsistencies in field equations. Then, we assume the scale factors and scalar field depend on time and extra coordinate as separated variables in power law forms. Hence, we find a few classes of solutions in 5D spacetime through which, we probe the one which leads to a generalized Kasner relations among Kasner parameters. The induced scalar potential is found to be in power law or in logarithmic form, however, for constant scaler field and even when scalar field only depends on fifth coordinate, it vanishes. The conservation law is indeed valid in this MBDT approach for derived induced energy momentum tensor (EMT). We proceed our investigations for a few cosmological quantities, where for simplicity we assume metric and scalar field are functions of time. Hence, the EMT satisfies barotropic equation of state, and the model indicates that constant mean Hubble parameter is not allowed. Thus, by appealing to variation of Hubble parameter, we assume a fixed deceleration parameter, and set evolution of quantities with respect to the fixed deceleration, BD coupling and state parameters. The WEC allows a shrinking extra dimension for decelerating expanding universe that, in the constant scalar field, evolves same as flat FRW spacetime in GR.

Anisotropic Cosmological Model in Modified Brans--Dicke Theory

We study the gravitational collapse of a homogeneous scalar field, minimally coupled to gravity, in the presence of a particular type of dynamical deformation between the canonical momenta of the scale factor and of the scalar field. In the absence of such a deformation, a class of solutions can be found in the literature [R. Goswami and P. S. Joshi], whereby a curvature singularity occurs at the collapse end state, which can be either hidden behind a horizon or be visible to external observers. However, when the phase space is deformed, as implemented herein this paper, we find that the singularity may be either removed or instead, attained faster. More precisely, for negative values of the deformation parameter, we identify the emergence of a negative pressure term, which slows down the collapse so that the singularity is replaced with a bounce. In this respect, the formation of a dynamical horizon can be avoided depending on the suitable choice of the boundary surface of the star. Whereas for positive values, the pressure that originates from the deformation effects assists the collapse toward the singularity formation. In this case, since the collapse speed is unbounded, the condition on the horizon formation is always satisfied and furthermore the dynamical horizon develops earlier than when the phase-space deformations are absent. These results are obtained by means of a thoroughly numerical discussion.

/pdf/gravitational-collapse-of-a-homogeneous-scalar-field-in-1x2c6edhzx.pdf

Gravitational collapse of a homogeneous scalar field in deformed phase space

We study the effects of a generalized uncertainty principle on the classical and quantum cosmology of a closed Friedmann universe whose matter content is either a dust or a radiation fluid. More concretely, assuming the existence of a minimal length, we show that the entropy will constitute a Dirac observable. In addition, 't Hooft conjecture on the cosmological holographic principle is also investigated. We describe how this holographic principle is satisfied for large values of a quantum number, $n$. This occurs when the entropy is computed in terms of the minimal area.

S. M. M. Rasouli

Papers

An anisotropic cosmological model in a modified Brans–Dicke theory

Modified Brans–Dicke theory in arbitrary dimensions

Anisotropic Cosmological Model in Modified Brans--Dicke Theory

Gravitational collapse of a homogeneous scalar field in deformed phase space

Quantum cosmology, minimal length and holography