Rotation and scale space random fields and the Gaussian kinematic formula

We provide a new approach, along with extensions, to results in two important papers of Worsley, Siegmund and coworkers closely tied to the statistical analysis of fMRI (functional magnetic resonance imaging) brain data. These papers studied approximations for the exceedence probabilities of scale and rotation space random fields, the latter playing an important role in the statistical analysis of fMRI data. The techniques used there came either from the Euler characteristic heuristic or via tube formulae, and to a large extent were carefully attuned to the specific examples of the paper. This paper treats the same problem, but via calculations based on the so-called Gaussian kinematic formula. This allows for extensions of the Worsley-Siegmund results to a wide class of non-Gaussian cases. In addition, it allows one to obtain results for rotation space random fields in any dimension via reasonably straightforward Riemannian geometric calculations. Previously only the two-dimensional case could be covered, and then only via computer algebra. By adopting this more structured approach to this particular problem, a solution path for other, related problems becomes clearer.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1055 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Axially Symmetric Bianchi I Yang-Mills Cosmology as a Dynamical System

We construct the most general form of axially symmetric SU(2)-Yang-Mills fields in Bianchi cosmologies. The dynamical evolution of axially symmetric YM fields in Bianchi I model is compared with the dynamical evolution of the electromagnetic field in Bianchi I and the fully isotropic YM field in Friedmann-Robertson-Walker cosmologies. The stochastic properties of axially symmetric Bianchi I-Einstein-Yang-Mills systems are compared with those of axially symmetric YM fields in flat space. After numerical computation of Liapunov exponents in synchronous (cosmological) time, it is shown that the Bianchi I-EYM system has milder stochastic properties than the corresponding flat YM system. The Liapunov exponent is non-vanishing in conformal time.Comment: 18 pages, 6 Postscript figures, uses amsmath,amssymb,epsfig,verbatim, to appear in CQ

One-loop graviton corrections to Maxwell's equations

We compute the graviton induced corrections to Maxwell's equations in the one-loop and weak field approximations. The corrected equations are analogous to the classical equations in anisotropic and inhomogeneous media. We analyze in particular the corrections to the dispersion relations. When the wavelength of the electromagnetic field is much smaller than a typical length scale of the graviton two-point function, the speed of light depends on the direction of propagation and on the polarisation of the radiation. In the opposite case, the speed of light may also depend on the energy of the electromagnetic radiation. We study in detail wave propagation in two special backgrounds, flat Robertson-Walker and static, spherically symmetric spacetimes. In the case of a flat Robertson-Walker gravitational background we find that the corrected electromagnetic field equations correspond to an isotropic medium with a time-dependent effective refractive index. For a static, spherically symmetric background the graviton fluctuations induce a vacuum structure which causes birefringence in the propagation of light.Comment: 15 pages, revte

A Class of Nonperturbative Configurations in Abelian-Higgs Models: Complexity from Dynamical Symmetry Breaking

We present a numerical investigation of the dynamics of symmetry breaking in both Abelian and non-Abelian $[S U (2)]$ Higgs models in three spatial dimensions. We find a class of time-dependent, long-lived nonperturbative field configurations within the range of parameters corresponding to type-1 superconductors, that is, with vector masses ($m_v$) larger than scalar masses ($m_s$). We argue that these emergent nontopological configurations are related to oscillons found previously in other contexts. For the Abelian-Higgs model, our lattice implementation allows us to map the range of parameter space -- the values of $\beta = (m_s /m_v)^2$ -- where such configurations exist and to follow them for times t \sim \O(10^5) m^{-1}. An investigation of their properties for $\hat z$-symmetric models reveals an enormously rich structure of resonances and mode-mode oscillations reminiscent of excited atomic states. For the SU(2) case, we present preliminary results indicating the presence of similar oscillonic configurations.Comment: 21 pages, 19 figures, prd, revte

Isolated Horizons and Black Hole Entropy in Loop Quantum Gravity

We review the black hole entropy calculation in the framework of Loop Quantum Gravity based on the quasi-local definition of a black hole encoded in the isolated horizon formalism. We show, by means of the covariant phase space framework, the appearance in the conserved symplectic structure of a boundary term corresponding to a Chern-Simons theory on the horizon and present its quantization both in the U(1) gauge fixed version and in the fully SU(2) invariant one. We then describe the boundary degrees of freedom counting techniques developed for an infinite value of the Chern-Simons level case and, less rigorously, for the case of a finite value. This allows us to perform a comparison between the U(1) and SU(2) approaches and provide a state of the art analysis of their common features and different implications for the entropy calculations. In particular, we comment on different points of view regarding the nature of the horizon degrees of freedom and the role played by the Barbero-Immirzi parameter. We conclude by presenting some of the most recent results concerning possible observational tests for theory

Variations on Birkhoff's theorem

The relation between the expanding universe and local vacuum solutions, such as that for the Solar System, is crucially mediated by Birkhoff's theorem. Here we consider how that relation works, and give generalizations of Birkhoff's theorem when there are geometric and matter and perturbations. The issue of to what degree dark matter might influence the solar system emerges as a significant question.Comment: Conference proceeding for ERE 2012, submitted to GRG for ERE2012 special issue, based on arXiv:1005.1809, arXiv:1101.4520 and arXiv:1202.024

Acoustics of early universe. I. Flat versus open universe models

A simple perturbation description unique for all signs of curvature, and based on the gauge-invariant formalisms is proposed to demonstrate that: (1) The density perturbations propagate in the flat radiation-dominated universe in exactly the same way as electromagnetic or gravitational waves propagate in the epoch of the matter domination. (2) In the open universe, sounds are dispersed by curvature. The space curvature defines the minimal frequency $\omega_{\rm c}$ below which the propagation of perturbations is forbidden. Gaussian acoustic fields are considered and the curvature imprint in the perturbations spectrum is discussed.Comment: The new version extended by 2 sections. Changes in notation. Some important comments adde