QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

Black Hole Explosions

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Models where dark matter and dark energy interact with each other have been proposed to solve the coincidence problem. We review the motivations underlying the need to introduce such interaction, its influence on the background dynamics and how it modifies the evolution of linear perturbations. We test models using the most recent observational data and we find that the interaction is compatible with the current astronomical and cosmological data. Finally, we describe the forthcoming data sets from current and future facilities that are being constructed or designed that will allow a clearer understanding of the physics of the dark sector.

https://iopscience.iop.org/article/10.1088/0034-4885/79/9/096901/pdf

Dark matter and dark energy interactions: theoretical challenges, cosmological implications and observational signatures

The Cauchy Problem

[Shortened abstract:] In this thesis we investigate a solution to the `problem of time' in canonical quantum gravity by splitting spacetime into surfaces of constant mean curvature parameterised by York time. We argue that there are reasons to consider York time a viable candidate for a physically meaningful notion of time. We investigate a number York-time Hamiltonian-reduced cosmological models and explore some technical aspects, such as the non-canonical Poisson structure. We develop York-time Hamiltonian-reduced cosmological perturbation theory by solving the Hamiltonian constraint perturbatively around a homogeneous background for the physical (non-vanishing) Hamiltonian that is the momentum conjugate to the York time parameter. We proceed to canonically quantise the cosmological models and the perturbation theory and discuss a number of conceptual and technical points, such as volume eigenfunctions and the absence of a momentum representation due to the non-standard commutator structure. We propose an alternative, wavefunction-free method of quantisation based on an ensemble of trajectories and a dynamical configuration-space geometry and discuss its application to gravity. We conclude by placing the York-time theories explored in this thesis in the wider context of a search for a satisfactory theory of quantum gravity.

Gravitation and cosmology with York time

We consider a solution to the problem of time in quantum gravity by deparameterization of the ADM action in terms of York time, a parameter proportional to the extrinsic curvature of a spatial hypersurface. We study a minisuperspace model together with a homogeneous scalar field, for which we can solve the Hamiltonian constraint exactly and arrive at an explicit expression for the physical (non-vanishing) Hamiltonian. The scale factor and associated momentum cease to be dynamical variables, leaving the scalar field as the only physical degree of freedom. We investigate the resulting classical theory, showing how the dynamics of the scale factor can be recovered via an appropriate interpretation of the Hamiltonian as a volume. We then quantize the system in the Schrodinger picture. In the quantum theory we recover the dynamics of the scale factor by interpreting the spectrum and expectation value of the Hamiltonian as being associated with volume rather than energy. If trajectories in the sense of de Broglie–Bohm are introduced in the quantum theory, these are found to match those of the classical theory. We suggest that these trajectories may provide the basis for a perturbation theory in which both background and perturbations are quantized.

http://inspirehep.net/record/1299641/files/arXiv:1406.2036.pdf

Classical and quantum cosmology with York time

It has been suggested that the York parameter T (effectively the scalar extrinsic curvature of a spatial hypersurface) may play the role of a fundamental time parameter. In a flat, forever expanding cosmology the York parameter remains always negative, taking values $$T=-\infty $$
 at the big bang and approaching some finite non-positive value as $$t\rightarrow \infty $$
 , t being the usual cosmological time coordinate. Based on previous results concerning a simple, spatially flat cosmological model with a scalar field, we provide a temporal extension of this model to include ‘times’ $$T>0$$
 , an epoch not covered by the cosmological time coordinate t, and discuss the dynamics of this ‘other side’ and its significance. We argue that the extension is necessary if a consistent quantisation scheme is to exist. Furthermore, we investigate which types of potentials lead to smooth transitions, paying particular attention to currently favoured inflaton potentials.

An extension of cosmological dynamics with York time

One method to overcome the notorious problem of time in the quantisation of gravity is the identification of a physically preferred time parameter, a promising candidate being so-called `York time'. The dynamical equations for matter and spatial geometry in York time may be obtained via Hamiltonian reduction, that is, by solving the Hamiltonian constraint for the physical, non-vanishing Hamiltonian density identified as the variable conjugate to the chosen time parameter. Yet in general this equation cannot be solved algebraically. Here we show how in a cosmological scenario, where one may treat geometric and matter inhomogeneities as small perturbations, one is able to obtain the physical Hamiltonian density by solving the constraint equation perturbatively. By construction the Hamiltonian density is quadratic in the perturbation variables, which makes it easily quantisable, although subtleties arise due to the non-canonical form of the Poisson brackets and the time-dependent coefficients. The latter are determined by the evolution of the background variables.

Cosmological perturbation theory with York time

For some time the York time parameter has been identified as a candidate for a physically meaningful time in cosmology. An associated Hamiltonian may be found by solving the Hamiltonian constraint for the momentum conjugate to the York time variable, although an explicit solution can only be found in highly symmetric cases. The Poisson structure of the remaining variables is not canonical. Here we quantise this dynamics in an anisotropic minisuperspace model via a natural extension of canonical quantisation. The resulting quantum theory has no momentum representation. Instead the position basis takes a fundamental role. We illustrate how the quantum theory and the modified representation of its momentum operators lead to a consistent theory in the presence of the constraints that arose during the Hamiltonian reduction. The quantised reduced Hamiltonian is Hermitian, although the momentum operators are not, the causes and implications of which we discuss. We are able to solve for the eigenspectrum of the Hamiltonian. Finally we discuss how far the results of this model extend to the general non-homogeneous case, in particular perturbation theory with York time.

https://iopscience.iop.org/article/10.1088/0264-9381/33/6/065001/pdf

Philipp Roser

Papers

Gravitation and cosmology with York time

Classical and quantum cosmology with York time

An extension of cosmological dynamics with York time

Cosmological perturbation theory with York time

Quantum mechanics on York slices