The mathematical research of William Parry FRS

In this article we survey the mathematical research of the late William (Bill) Parry, FRS

Pattern Formation in the Early Universe

Systems that exhibit pattern formation are typically driven and dissipative. In the early universe, parametric resonance can drive explosive particle production called preheating. The fields that are populated then decay quantum mechanically if their particles are unstable. Thus, during preheating, a driven-dissipative system exists. We have shown previously that pattern formation can occur in two dimensions in a self-coupled inflaton system undergoing parametric resonance. In this paper, we provide evidence of pattern formation for more realistic initial conditions in both two and three dimensions. In the one-field case, we have the novel interpretation that these patterns can be thought of as a network of domain walls. We also show that the patterns are spatio-temporal, leading to a distinctive, but probably low-amplitude peak in the gravitational wave spectrum. In the context of a two-field model, we discuss putting power from resonance into patterns on cosmological scales, in particular to explain the observed excess power at 100 h^{-1}Mpc, but why this seems unlikely in the absence of a period of post-preheating inflation. Finally we note our model is similar to that of the decay of DCCs and therefore pattern formation may also occur at RHIC and LHC.Comment: 9 pages, 11 figure

Droplet shapes on structured substrates and conformal invariance

We consider the finite-size scaling of equilibrium droplet shapes for fluid adsorption (at bulk two-phase co-existence) on heterogeneous substrates and also in wedge geometries in which only a finite domain $\Lambda_{A}$ of the substrate is completely wet. For three-dimensional systems with short-ranged forces we use renormalization group ideas to establish that both the shape of the droplet height and the height-height correlations can be understood from the conformal invariance of an appropriate operator. This allows us to predict the explicit scaling form of the droplet height for a number of different domain shapes. For systems with long-ranged forces, conformal invariance is not obeyed but the droplet shape is still shown to exhibit strong scaling behaviour. We argue that droplet formation in heterogeneous wedge geometries also shows a number of different scaling regimes depending on the range of the forces. The conformal invariance of the wedge droplet shape for short-ranged forces is shown explicitly.Comment: 20 pages, 7 figures. (Submitted to J.Phys.:Cond.Mat.

Tricritical wedge filling transitions with short-ranged forces

We show that the 3D wedge filling transition in the presence of short-ranged interactions can be first-order or second order depending on the strength of the line tension associated with to the wedge bottom. This fact implies the existence of a tricritical point characterized by a short-distance expansion which differs from the usual continuous filling transition. Our analysis is based on an effective one-dimensional model for the 3D wedge filling which arises from the identification of the breather modes as the only relevant interfacial fluctuations. From such analysis we find a correspondence between continuous 3D filling at bulk coexistence and 2D wetting transitions with random-bond disorder.Comment: 7 pages, 3 figures, 6th Liquid Matter Conference Proceedings (to be published in J. Phys.: Condens. Matter

The large CP phase in B(s)-anti-B(s) mixing from primary scalar unparticles

In this letter we consider the case of primary scalar unparticle contributions to B(d,s) mixing. With particular emphasis on the impact of the recent hint of new physics in the measurement of the B(s) mixing phase, phi(s), we determine the allowed parameter space and impose bounds on the unparticle couplings.Comment: 8 pages, 8 jpeg figures, using pdflatex. Typo corrected, reference adde

Interfacial Structural Changes and Singularities in Non-Planar Geometries

We consider phase coexistence and criticality in a thin-film Ising magnet with opposing surface fields and non-planar (corrugated) walls. We show that the loss of translational invariance has a strong and unexpected non-linear influence on the interface structure and phase diagram. We identify 4 non-thermodynamic singularities where there is a qualitative change in the interface shape. In addition, we establish that at the finite-size critical point, the singularity in the interface shape is characterized by two distint critical exponents in contrast to the planar case (which is characterised by one). Similar effects should be observed for prewetting at a corrugated substrate. Analogy is made with the behaviour of a non-linear forced oscillator showing chaotic dynamics.Comment: 13 pages, 3 figure

Brane Gas Inflation

We consider the brane gas picture of the early universe. At later stages, when there are no winding modes and the background is free to expand, we show that a moving 3-brane, which we identify with our universe, can inflate even though it is radiation-dominated. The crucial ingredients for successful inflation are the coupling to the dilaton and the equation of state of the bulk. If we suppose the brane initially forms in a collision of higher-dimensional branes, then the spectrum of primordial density fluctuations naturally has a thermal origin.Comment: 4 pages, 1 figur