Obstruction flat rigidity of the CR 3-sphere

We consider the obstruction flatness problem for small deformations of the standard CR 3-sphere. That rigidity holds for the CR sphere was previously known (in all dimensions) for the case of embeddable CR structures, where it also holds at the infinitesimal level. In the 3-dimensional case, however, a CR structure need not be embeddable. Unlike in the embeddable case, it turns out that in the nonembeddable case there is an infinite-dimensional space of solutions to the linearized obstruction flatness equation on the standard CR 3-sphere and this space defines a natural complement to the tangent space of the embeddable deformations. In spite of this, we show that the CR 3-sphere does not admit nontrivial obstruction flat deformations, embeddable or nonembeddable.

Holographic cosmology solutions of problems with pre-inflationary cosmology

In this paper we describe in detail how to solve the problems of pre-inflationary cosmology within the holographic cosmology model of McFadden and Skenderis [1]. The solutions of the smoothness and horizon problems, the flatness problem, the entropy and perturbation problems and the baryon asymmetry problem are shown, and the mechanisms for them complement the inflationary solutions. Most of the paper is devoted to the solution of the monopole relic problem, through a detailed calculation of 2-loop correlators of currents in a toy model which we perform in d dimensions, in order to extract its leading dependence on $$ {g}_{\mathrm{eff}}^2={g}_{\mathrm{YM}}^2N/q $$ g eff 2 = g YM 2 N / q and find a dilution effect. Taken together with the fact that holographic cosmology gives as good a fit to CMBR as the standard paradigm of Λ CDM with inflation, it means holographic cosmology extends the inflationary paradigm into new corners, explorable only through a dual perturbative field theory in 3 dimensions.

The flatness problem and the age of the Universe

ABSTRACT Several authors have made claims, none of which has been rebutted, that the flatness problem, as formulated by Dicke and Peebles, is not really a problem but rather a misunderstanding. Nevertheless, the flatness problem is still widely perceived to be real. Most of the arguments against the idea of a flatness problem are based on the change with time of the density parameter Ω and normalized cosmological constant λ and, since the Hubble constant H is not considered, are independent of time-scale. An independent claim is that fine-tuning is required in order to produce a Universe which neither collapsed after a short time nor expanded so quickly that no structure formation could take place. I show that this claim does not imply that fine-tuning of the basic cosmological parameters is necessary, in part for similar reasons as in the more restricted flatness problem and in part due to an incorrect application of the idea of perturbing the early Universe in a gedankenexperiment; I discuss some typical pitfalls of the latter.

From Fractional Quantum Mechanics to Quantum Cosmology: An Overture

Fractional calculus is a couple of centuries old, but its development has been less embraced and it was only within the last century that a program of applications for physics started. Regarding quantum physics, it has been only in the previous decade or so that the corresponding literature resulted in a set of defying papers. In such a context, this manuscript constitutes a cordial invitation, whose purpose is simply to suggest, mostly through a heuristic and unpretentious presentation, the extension of fractional quantum mechanics to cosmological settings. Being more specific, we start by outlining a historical summary of fractional calculus. Then, following this motivation, a (very) brief appraisal of fractional quantum mechanics is presented, but where details (namely those of a mathematical nature) are left for literature perusing. Subsequently, the application of fractional calculus in quantum cosmology is introduced, advocating it as worthy to consider: if the progress of fractional calculus serves as argument, indeed useful consequences will also be drawn (to cite from Leibnitz). In particular, we discuss different difficulties that may affect the operational framework to employ, namely the issues of minisuperspace covariance and fractional derivatives, for instance. An example of investigation is provided by means of a very simple model. Concretely, we restrict ourselves to speculate that with minimal fractional calculus elements, we may have a peculiar tool to inspect the flatness problem of standard cosmology. In summary, the subject of fractional quantum cosmology is herewith proposed, merely realised in terms of an open program constituted by several challenges.

The quasi-steady-state cosmology in a radiation-dominated phase

Analytical solutions for radiation-dominated phase of Quasi-Steady-State Cosmology (QSSC) in Friedmann–Robertson–Walker models are obtained. We find that matter density is positive in all the cases [Formula: see text]. The nature of Hubble parameter (H) in [Formula: see text] is discussed. The deceleration parameter [Formula: see text] is marginally less than zero indicating accelerating universe. The scale factor [Formula: see text] is graphically shown with time. The model represents oscillating universe between the above-mentioned limits. Because of the bounce in QSSC, the maximum density phase is still matter-dominated. The models represent singularity-free model. We also find that the models have event horizon i.e. no observer beyond the proper distance [Formula: see text] can communicate each other in FRW models for radiation-dominated phase in the frame work of QSSC. The FRW models are special classes of Bianchi type I, V, IX spacetimes with zero, negative and positive curvatures, respectively. Initially i.e. at [Formula: see text], the models represent steady model. We have tried to show how a good fit can be obtained to the observations in the framework of QSSC during radiation-dominated phase. The present model is free from singularity, particle horizon and provides a natural explanation for the flatness problem. Therefore, our model is superior to other models.

The Flatness Problem and the Variable Physical Constants

We have used the varying physical constant approach to resolve the flatness problem in cosmology. Friedmann equations are modified to include the variability of speed of light, gravitational constant, cosmological constant, and the curvature constant. The continuity equation obtained with such modifications includes the scale factor-dependent cosmological term as well as the curvature term, along with the standard energy-momentum term. The result is that as the scale factor tends to zero (i.e., as the Big Bang is approached), the universe becomes strongly curved rather than flatter and flatter in the standard cosmology. We have used the supernovae 1a redshift versus distance modulus data to determine the curvature variation parameter of the new model, which yields a better fit to the data than the standard ΛCDM model. The universe is found to be an open type with a radius of curvature R c = 1.64 ( 1 + z ) − 3.3 c 0 / H 0 , where z is the redshift, c 0 is the current speed of light, and H 0 is the Hubble constant.