Quasi-Equatorial Gravitational Lensing by Spinning Black Holes in the Strong Field Limit

Spherically symmetric black holes produce, by strong field lensing, two infinite series of relativistic images, formed by light rays winding around the black hole at distances comparable to the gravitational radius. In this paper, we address the relevance of the black hole spin for the strong field lensing phenomenology, focusing on trajectories close to the equatorial plane for simplicity. In this approximation, we derive a two-dimensional lens equation and formulae for the position and the magnification of the relativistic images in the strong field limit. The most outstanding effect is the generation of a non trivial caustic structure. Caustics drift away from the optical axis and acquire finite extension. For a high enough black hole spin, depending on the source extension, we can practically observe only one image rather than two infinite series of relativistic images. In this regime, additional non equatorial images may play an important role in the phenomenology.Comment: 13 pages, 9 figures. Improved version with detailed physical discussio

Continuous image distortion by astrophysical thick lenses

Image distortion due to weak gravitational lensing is examined using a non-perturbative method of integrating the geodesic deviation and optical scalar equations along the null geodesics connecting the observer to a distant source. The method we develop continuously changes the shape of the pencil of rays from the source to the observer with no reference to lens planes in astrophysically relevant scenarios. We compare the projected area and the ratio of semi-major to semi-minor axes of the observed elliptical image shape for circular sources from the continuous, thick-lens method with the commonly assumed thin-lens approximation. We find that for truncated singular isothermal sphere and NFW models of realistic galaxy clusters, the commonly used thin-lens approximation is accurate to better than 1 part in 10^4 in predicting the image area and axes ratios. For asymmetric thick lenses consisting of two massive clusters separated along the line of sight in redshift up to \Delta z = 0.2, we find that modeling the image distortion as two clusters in a single lens plane does not produce relative errors in image area or axes ratio more than 0.5%Comment: accepted to GR

Strong Gravitational Lensing in a Charged Squashed Kaluza- Klein Black hole

In this paper we investigate the strong gravitational lensing in a charged squashed Kaluza-Klein black hole. We suppose that the supermassive black hole in the galaxy center can be considered by a charged squashed Kaluza-Klein black hole and then we study the strong gravitational lensing theory and estimate the numerical values for parameters and observables of it. We explore the effects of the scale of extra dimension $\rho_0$ and the charge of black hole $\rho_q$ on these parameters and observables.Comment: 17 pages, 10 figure

Approximate Reachability for Dead Code Elimination in Esterel*

Esterel is an imperative synchronous programming language for the design of reactive systems. Esterel* extends Esterel with a non-instantaneous jump instruction (compatible with concurrency, preemption, etc.) so as to enable powerful source-to-source program transformations, amenable to formal verification. In this work, we propose an approximate reachability algorithm for Esterel* and use its output to remove dead code. We prove the correctness of our techniques

Light's Bending Angle due to Black Holes: From the Photon Sphere to Infinity

The bending angle of light is a central quantity in the theory of gravitational lensing. We develop an analytical perturbation framework for calculating the bending angle of light rays lensed by a Schwarzschild black hole. Using a perturbation parameter given in terms of the gravitational radius of the black hole and the light ray's impact parameter, we determine an invariant series for the strong-deflection bending angle that extends beyond the standard logarithmic deflection term used in the literature. In the process, we discovered an improvement to the standard logarithmic deflection term. Our perturbation framework is also used to derive as a consistency check, the recently found weak deflection bending angle series. We also reformulate the latter series in terms of a more natural invariant perturbation parameter, one that smoothly transitions between the weak and strong deflection series. We then compare our invariant strong deflection bending-angle series with the numerically integrated exact formal bending angle expression, and find less than 1% discrepancy for light rays as far out as twice the critical impact parameter. The paper concludes by showing that the strong and weak deflection bending angle series together provide an approximation that is within 1% of the exact bending angle value for light rays traversing anywhere between the photon sphere and infinity.Comment: 22 pages, 5 figure

Actions infinit\'esimales dans la correspondance de Langlands locale p-adique

Let V be a two-dimensional absolutely irreducible p-adic Galois representation and let Pi be the p-adic Banach space representation associated to V via Colmez's p-adic Langlands correspondence. We establish a link between the infinitesimal action of GL_2(Q_p) on the locally analytic vectors of Pi, the differential equation associated to V via the theory of Fontaine and Berger, and the Sen polynomial of V. This answers a question of Harris and gives a new proof of a theorem of Colmez: Pi has nonzero locally algebraic vectors if and only if V is potentially semi-stable with distinct Hodge-Tate weights.Comment: Completely revised version, to appear in Math. Annale

From Contracts in Structured English to CL Specifications

In this paper we present a framework to analyze conflicts of contracts written in structured English. A contract that has manually been rewritten in a structured English is automatically translated into a formal language using the Grammatical Framework (GF). In particular we use the contract language CL as a target formal language for this translation. In our framework CL specifications could then be input into the tool CLAN to detect the presence of conflicts (whether there are contradictory obligations, permissions, and prohibitions. We also use GF to get a version in (restricted) English of CL formulae. We discuss the implementation of such a framework.Comment: In Proceedings FLACOS 2011, arXiv:1109.239

Gravitational Lensing and f(R) theories in the Palatini approach

We investigate gravitational lensing in the Palatini approach to the f(R) extended theories of gravity. Starting from an exact solution of the f(R) field equations, which corresponds to the Schwarzschild-de Sitter metric and, on the basis of recent studies on this metric, we focus on some lensing observables, in order to evaluate the effects of the non linearity of the gravity Lagrangian. We give estimates for some astrophysical events, and show that these effects are tiny for galactic lenses, but become interesting for extragalactic ones.Comment: 7 Pages, RevTex, 1 eps figure; references added; revised to match the version accepted for publication in General Relativity and Gravitatio

Cosmic histories of star formation and reionization: An analysis with a power-law approximation

With a simple power-law approximation of high-redshift ($\gtrsim3.5$) star formation history, i.e., $\dot{\rho}_*(z)\propto [(1+z)/4.5]^{-\alpha}$, we investigate the reionization of intergalactic medium (IGM) and the consequent Thomson scattering optical depth for cosmic microwave background (CMB) photons. A constraint on the evolution index $\alpha$ is derived from the CMB optical depth measured by the {\it Wilkinson Microwave Anisotropy Probe} (WMAP) experiment, which reads $\alpha\approx2.18\lg{\mathscr{N}_{\gamma}}-3.89$, where the free parameter $\mathscr{N}_\gamma$ is the number of the escaped ionizing ultraviolet photons per baryon. Moreover, the redshift for full reionization, $z_f$, can also be expressed as a function of $\alpha$ as well as $\mathscr{N}_{\gamma}$. By further taking into account the implication of the Gunn-Peterson trough observations to quasars for the full reionization redshift, i.e., $6\lesssim z_f \lesssim7$, we obtain $0.3\lesssim\alpha\lesssim1.3$ and $80\lesssim\mathscr{N}_{\gamma}\lesssim230$. For a typical number of $\sim4000$ of ionizing photons released per baryon of normal stars, the fraction of these photons escaping from the stars, $f_{\rm esc}$, can be constrained to within the range of $(2.0-5.8)$.Comment: 10 pages, 4 figures, accepted for publication in JCA

The Inflammasome Drives GSDMD-Independent Secondary Pyroptosis and IL-1 Release in the Absence of Caspase-1 Protease Activity.

Inflammasomes activate the protease caspase-1, which cleaves interleukin-1β and interleukin-18 to generate the mature cytokines and controls their secretion and a form of inflammatory cell death called pyroptosis. By generating mice expressing enzymatically inactive caspase-1 <sup>C284A</sup> , we provide genetic evidence that caspase-1 protease activity is required for canonical IL-1 secretion, pyroptosis, and inflammasome-mediated immunity. In caspase-1-deficient cells, caspase-8 can be activated at the inflammasome. Using mice either lacking the pyroptosis effector gasdermin D (GSDMD) or expressing caspase-1 <sup>C284A</sup> , we found that GSDMD-dependent pyroptosis prevented caspase-8 activation at the inflammasome. In the absence of GSDMD-dependent pyroptosis, the inflammasome engaged a delayed, alternative form of lytic cell death that was accompanied by the release of large amounts of mature IL-1 and contributed to host protection. Features of this cell death modality distinguished it from apoptosis, suggesting it may represent a distinct form of pro-inflammatory regulated necrosis