Time-covariant Schrödinger equation and invariant decay probability: the $$\Lambda $$-Kantowski–Sachs universe

The system under study is the $$\Lambda $$ Λ -Kantowski–Sachs universe. Its canonical quantization is provided based on a recently developed method: the singular minisuperspace Lagrangian describing the system, is reduced to a regular (by inserting into the dynamical equations the lapse dictated by the quadratic constraint) possessing an explicit (though arbitrary) time dependence; thus a time-covariant Schrödinger equation arises. Additionally, an invariant (under transformations $$t=f({\tilde{t}})$$ t = f ( t ~ ) ) decay probability is defined and thus “observers” which correspond to different gauge choices obtain, by default, the same results. The time of decay for a Gaussian wave packet localized around the point $$a=0$$ a = 0 (where a the radial scale factor) is calculated to be of the order $$\sim 10^{-42}{-}10^{-41}~\text {s}$$ ∼ 10 - 42 - 10 - 41 s . The acquired value is near the end of the Planck era (when comparing to a FLRW universe), during which the quantum effects are most prominent. Some of the results are compared to those obtained by following the well known canonical quantization of cosmological systems, i.e. the solutions of the Wheeler–DeWitt equation.

Conservation of radial actions in time-dependent spherical potentials

ABSTRACT In slowly evolving spherical potentials, Φ(r, t), radial actions are typically assumed to remain constant. Here, we construct dynamical invariants that allow us to derive the evolution of radial actions in spherical central potentials with an arbitrary time dependence. We show that to linear order, radial actions oscillate around a constant value with an amplitude $\propto \dot{\Phi }/\Phi \, P(E,L)$. Using this result, we develop a diffusion theory that describes the evolution of the radial action distributions of ensembles of tracer particles orbiting in generic time-dependent spherical potentials. Tests against restricted N-body simulations in a varying Kepler potential indicate that our linear theory is accurate in regions of phase-space in which the diffusion coefficient $\tilde {D}(J_r) \lt 0.01\, J_r^2$. For illustration, we apply our theory to two astrophysical processes. We show that the median mass accretion rate of a Milky Way (MW) dark matter (DM) halo leads to slow global time-variation of the gravitational potential, in which the evolution of radial actions is linear (i.e. either adiabatic or diffusive) for ∼84 per cent of the DM halo at redshift z = 0. This fraction grows considerably with look-back time, suggesting that diffusion may be relevant to the modelling of several Gyr old tidal streams in action-angle space. As a second application, we show that dynamical tracers in a dwarf-size self-interacting DM halo (with $\sigma /m_\chi = 1\, {\rm cm^2g^{-1}}$) have invariant radial actions during the formation of a cored density profile.

Tomographic Description of a Quantum Wave Packet in an Accelerated Frame

The tomography of a single quantum particle (i.e., a quantum wave packet) in an accelerated frame is studied. We write the Schrödinger equation in a moving reference frame in which acceleration is uniform in space and an arbitrary function of time. Then, we reduce such a problem to the study of spatiotemporal evolution of the wave packet in an inertial frame in the presence of a homogeneous force field but with an arbitrary time dependence. We demonstrate the existence of a Gaussian wave packet solution, for which the position and momentum uncertainties are unaffected by the uniform force field. This implies that, similar to in the case of a force-free motion, the uncertainty product is unaffected by acceleration. In addition, according to the Ehrenfest theorem, the wave packet centroid moves according to classic Newton's law of a particle experiencing the effects of uniform acceleration. Furthermore, as in free motion, the wave packet exhibits a diffraction spread in the configuration space but not in momentum space. Then, using Radon transform, we determine the quantum tomogram of the Gaussian state evolution in the accelerated frame. Finally, we characterize the wave packet evolution in the accelerated frame in terms of optical and simplectic tomogram evolution in the related tomographic space.

Epidemics Forecast From SIR-Modeling, Verification and Calculated Effects of Lockdown and Lifting of Interventions

Due to the current COVID-19 epidemic plague hitting the worldwide population it is of utmost medical, economical and societal interest to gain reliable predictions on the temporal evolution of the spreading of the infectious diseases in human populations. Of particular interest are the daily rates and cumulative number of new infections, as they are monitored in infected societies, and the influence of non-pharmaceutical interventions due to different lockdown measures as well as their subsequent lifting on these infections. Estimating quantitatively the influence of a later lifting of the interventions on the resulting increase in the case numbers is important to discriminate this increase from the onset of a second wave. The recently discovered new analytical solutions of Susceptible-Infectious-Recovered (SIR) model allow for such forecast. In particular, it is possible to test lockdown and lifting interventions because the new solutions hold for arbitrary time dependence of the infection rate. Here we present simple analytical approximations for the rate and cumulative number of new infections.

Epidemics forecast from SIR-modeling, verification and calculated effects of lockdown and lifting of interventions

Due to the current COVID-19 epidemic plague hitting the worldwide population it is of utmost medical, economical and societal interest to gain reliable predictions on the temporal evolution of the spreading of the infectious diseases in human populations. Of particular interest are the daily rates and cumulative number of new infections, as they are monitored in infected societies, and the influence of non-pharmaceutical interventions due to different lockdown measures as well as their subsequent lifting on these infections. Estimating quantitatively the influence of a later lifting of the interventions on the resulting increase in the case numbers is important to discriminate this increase from the onset of a second wave. The recently discovered new analytical solutions of Susceptible-Infectious-Recovered (SIR) model allow for such forecast and the testing of lockdown and lifting interventions as they hold for arbitrary time dependence of the infection rate. Here we present simple analytical approximations for the rate and cumulative number of new infections.

Exact theory of material spike formation in flow separation

We develop a frame-invariant theory of material spike formation during flow separation over a no-slip boundary in two-dimensional flows with arbitrary time dependence. Based on the exact curvature evolution of near-wall material lines, our theory identifies both fixed and moving flow separation, is effective also over short time intervals, and admits a rigorous instantaneous limit. As a byproduct, we derive explicit formulae for the evolution of material line curvature and the curvature rate for general compressible flows. The material backbone that we identify acts first as the precursor and later as the centrepiece of unsteady Lagrangian flow separation. We also discover a previously undetected spiking point where the backbone of separation connects to the boundary, and derive wall-based analytical formulae for its location. Finally, our theory explains the perception of off-wall separation in unsteady flows and provides conditions under which such a perception is justified. We illustrate our results on several analytical and experimental flows.