Unitary and non-unitary transitions around a cosmological bounce

In this work, we investigate the notion of time and unitarity in the vicinity of a bounce in quantum cosmology, that is, a turning point for the scale factor. Because WKB methods drastically fail near a turning point, the scale factor cannot play the role of time in such scenarios. We overcome this difficulty by studying the dynamics of matter transitions when using its conjugate momentum as a time. We find precise conditions so as to recover unitarity, and hence, a consistent notion of probability. We then compute transitions in a concrete example, extract the specific feature of a bounce and argue about the necessity of the conjugate momentum representation to go beyond the background field approximation. Our analysis is also equally relevant for a closed Universe undergoing a recollapsing phase.Comment: 25 pages, 1 figure, published versio

Undulations from amplified low frequency surface waves

We study the scattering of gravity waves in longitudinal inhomogeneous stationary flows. When the flow becomes supercritical, counterflow propagating shallow waves are blocked and converted into deep waves. We show that the reflected waves are amplified in such a way that, in the zero-frequency limit, the free surface develops an undulation, i.e., a zero-frequency wave of large amplitude with nodes located at specific places. This amplification involves negative energy waves, and implies that the unperturbed flat surface is unstable against incoming perturbations of arbitrary small amplitude. The relation between this instability and black hole radiation (the Hawking effect) is discussed.Comment: 19 pages, 2 figures - Pulished version. Added clarifications, and relations to fluid mechanics standard treatment

Hawking radiation with dispersion: The broadened horizon paradigm

To identify what replaces the key notion of black hole horizon when working with theories which break Lorentz invariance at high energy, we study the modes responsible for the Hawking effect in the presence of high frequency dispersion. We show that they are regularized across the horizon over a short length which only depends on the scale of dispersion and the surface gravity. Moreover, outside this width, short and long wavelength modes no longer mix. These results can be used to show that the spectrum is hardly modified by dispersion as long as the background geometry does not vary significantly over this length. For relevant frequencies, the regularization differs from the usual WKB resolution of wave singularity near a turning point.Comment: 6 pages, 4 figures, published versio

Dynamical instabilities and quasi-normal modes, a spectral analysis with applications to black-hole physics

Black hole dynamical instabilities have been mostly studied in specific models. We here study the general properties of the complex-frequency modes responsible for such instabilities, guided by the example of a charged scalar field in an electrostatic potential. We show that these modes are square integrable, have a vanishing conserved norm, and appear in mode doublets or quartets. We also study how they appear in the spectrum and how their complex frequencies subsequently evolve when varying some external parameter. When working on an infinite domain, they appear from the reservoir of quasi-normal modes obeying outgoing boundary conditions. This is illustrated by generalizing, in a non-positive definite Krein space, a solvable model (Friedrichs model) which originally describes the appearance of a resonance when coupling an isolated system to a mode continuum. In a finite spatial domain instead, they arise from the fusion of two real frequency modes with opposite norms, through a process that closely resembles avoided crossing.Comment: 31 pages, 13 figures. Small clarifications, title changed, matches published versio

Surface impedance and topologically protected interface modes in one-dimensional phononic crystals

When semi-infinite phononic crystals (PCs) are in contact, localized modes may exist at their boundary. The central question is generally to predict their existence and to determine their stability. With the rapid expansion of the field of topological insulators, powerful tools have been developed to address these questions. In particular, when applied to one-dimensional systems with mirror symmetry, the bulk-boundary correspondence claims that the existence of interface modes is given by a topological invariant computed from the bulk properties of the phononic crystal, which ensures strong stability properties. This one-dimensional bulk-boundary correspondence has been proven in various works. Recent attempts have exploited the notion of surface impedance, relying on analytical calculations of the transfer matrix. In the present work, the monotonic evolution of surface impedance with frequency is proven for all one-dimensional phononic crystals with mirror symmetry. This result allows us to establish a stronger version of the bulk-boundary correspondence that guarantees not only the existence but also the uniqueness of a topologically protected interface state. The method is numerically illustrated in the physically relevant case of PCs with imperfect interfaces, where analytical calculations would be out of reach.Comment: 21+3 pages, 8 figure

Quasi-normal modes and fermionic vacuum decay around a Kerr black hole

We analyze the instability of the non-rotating fermion vacuum in Kerr spacetimes. We describe how the co-rotating Fermi sea is formed as a result of a spontaneous vacuum decay. Most significantly, and drawing upon intuition gained from analogous electrodynamic processes in supercritical fields, we show that this decay process is encoded entirely in the set of quasi-normal fermion modes