Singular Mean-Field States: A Brief Review of Recent Results

This article provides a focused review of recent findings which demonstrate, in some cases quite counter-intuitively, the existence of bound states with a singularity of the density pattern at the center; the states are physically meaningful because their total norm converges. One model of this type is based on the 2D Gross–Pitaevskii equation (GPE), which combines the attractive potential ∼ r − 2 and the quartic self-repulsive nonlinearity, induced by the Lee–Huang–Yang effect (quantum fluctuations around the mean-field state). The GPE demonstrates suppression of the 2D quantum collapse, driven by the attractive potential, and emergence of a stable ground state (GS), whose density features an integrable singularity ∼ r − 4 / 3 at r → 0 . Modes with embedded angular momentum exist too, but they are unstable. A counter-intuitive peculiarity of the model is that the GS exists even if the sign of the potential is reversed from attraction to repulsion, provided that its strength is small enough. This peculiarity finds a relevant explanation. The other model outlined in the review includes 1D, 2D, and 3D GPEs, with the septimal (seventh-order), quintic, and cubic self-repulsive terms, respectively. These equations give rise to stable singular solitons, which represent the GS for each dimension D, with the density singularity ∼ r − 2 / ( 4 − D ) . Such states may be considered the results of screening a “bare” delta-functional attractive potential by the respective nonlinearities.

On the expanding phase of a singular bounce and intermediate inflation: The modified gravity description

We demonstrate that the intermediate inflation scenario is a singular inflation cosmology, with the singularity at the origin t = 0 being a pressure and energy density singularity and particularly a Type III singularity. Also, we show that the expanding phase of a singular bounce can be identical to the intermediate inflation scenario, if the singular bounce has a Type III singularity at the origin. For the intermediate inflation scenario we examine the cosmological implications on the power spectrum in the context of various forms of modified gravity. Particularly, we calculate the power spectrum in the context of F(R), F(G) Gauss–Bonnet gravity and also for F(T) gravity and we discuss the viability of each scenario by comparing the resulting spectral index with the latest observational data.

COLLECTIVE BARYON DECAY AND GRAVITATIONAL COLLAPSE

While it is widely believed that the gravitational collapse of a sufficiently large mass will lead to a density singularity and an event horizon, we propose that this never happens when quantum effects are taken into account. In particular, we propose that when the conditions become ripe for the formation of a trapped surface, a quantum critical firewall sweeps over the collapsing body, transforming the nucleons in the collapsing matter into a lepton/photon gas together with droplets of a positive vacuum energy. This will happen regardless of the matter density at the time a trapped surface starts to form, and as a result, we predict that at least in all cases of gravitational collapse involving ordinary matter, a large fraction of the rest mass of the collapsing matter will be converted into a burst of neutrinos and γ-rays. We predict that the peak luminosity of these bursts is only weakly dependent on the mass of the collapsing object, and on the order of (ϵq/mPc2)1/4c5/G where ϵq is the mean energy of a nucleon parton and mP is the Planck mass. The duration of the bursts will depend on the mass of the collapsing object; in the case of stellar core collapse, we predict that the duration of both the neutrino and γ-ray bursts will be on the order of 10s.

A Stokesian analysis of a submerged viscous jet impinging on a planar wall

The wall pressure and wall shear stress of a submerged viscous jet impinging on an infinite planar wall are derived. The whole creeping flow of semi-infinite extent is generated via distributions on a cylindrical pipe of tangentially and normally directed Stokeslets which are modified to achieve no-slip at the wall in two stages. First the pressure and vorticity jumps associated with the Poiseuille flow upstream in the pipe are readily forced, and then further distributions, of zero density far upstream but with square-root density singularity at the orifice $z= h$, are added to achieve no-slip on the pipe wall. Thus the adjustment of the interior pipe flow from its upstream parabolic profile to its exit profile is fully included in – and a major feature of – this creeping flow analysis. The maximum plane wall pressure is always located on the axis $r= 0$, and decreases as $h$ increases to alleviate the obstruction effect of the wall. The interaction of the inflow with the ambient fluid in the neighbourhood of $z= 0$ causes the wall stress to rise rapidly to a maximum and then decay with the radial position of this maximum increasing as $h$ increases. This behaviour is discussed in the context of physiological experiments on auditory sensory hair cells that motivated this study.