Spacetime algebraic skeleton

The cosmological constant Lambda, which has seemingly dominated the primaeval Universe evolution and to which recent data attribute a significant present-time value, is shown to have an algebraic content: it is essentially an eigenvalue of a Casimir invariant of the Lorentz group which acts on every tangent space. This is found in the context of de Sitter spacetimes but, as every spacetime is a 4-manifold with Minkowski tangent spaces, the result suggests the existence of a "skeleton" algebraic structure underlying the geometry of general physical spacetimes. Different spacetimes come from the "fleshening" of that structure by different tetrad fields. Tetrad fields, which provide the interface between spacetime proper and its tangent spaces, exhibit to the most the fundamental role of the Lorentz group in Riemannian spacetimes, a role which is obscured in the more usual metric formalism.Comment: 13 page

Fundamental Plane of Sunyaev-Zeldovich clusters

Sunyaev-Zel'dovich (SZ) cluster surveys are considered among the most promising methods for probing dark energy up to large redshifts. However, their premise is hinged upon an accurate mass-observable relationship, which could be affected by the (rather poorly understood) physics of the intracluster gas. In this letter, using a semi-analytic model of the intracluster gas that accommodates various theoretical uncertainties, I develop a Fundamental Plane relationship between the observed size, thermal energy, and mass of galaxy clusters. In particular, I find that M ~ (Y_{SZ}/R_{SZ,2})^{3/4}, where M is the mass, Y_{SZ} is the total SZ flux or thermal energy, and R_{SZ,2} is the SZ half-light radius of the cluster. I first show that, within this model, using the Fundamental Plane relationship reduces the (systematic+random) errors in mass estimates to 14%, from 22% for a simple mass-flux relationship. Since measurement of the cluster sizes is an inevitable part of observing the SZ clusters, the Fundamental Plane relationship can be used to reduce the error of the cluster mass estimates by ~ 34%, improving the accuracy of the resulting cosmological constraints without any extra cost. I then argue why our Fundamental Plane is distinctly different from the virial relationship that one may naively expect between the cluster parameters. Finally, I argue that while including more details of the observed SZ profile cannot significantly improve the accuracy of mass estimates, a better understanding of the impact of non-gravitational heating/cooling processes on the outskirts of the intracluster medium (apart from external calibrations) might be the best way to reduce these errors.Comment: 5 pages, 1 figure, added an analytic derivation of the Fundametal Plane relation (which is distinctly different from the virial relation), submitted to Ap

A Statistical Strategy for the Sunyaev-Zel'dovich Effect's Cluster Data

We present a statistical strategy for the efficient determination of the cluster luminosity function from the Sunyaev-Zel'dovich (SZ) effects survey. To determine the cluster luminosity function from the noise contaminated SZ map, we first define the zeroth-order cluster luminosity function as a discrepancy between the measured peak number density of the SZ map and the mean number density of noise. Then we demonstrate that the noise contamination effects can be removed by the stabilized deconvolution of the zeroth-order cluster luminosity function with the one-dimensional Gaussian distribution. We test this analysis technique against Monte-Carlo simulations, and find that it works quite well especially in the medium amplitude range where the conventional cluster identification method based on the threshold cut-off usually fails.Comment: final version, accepted by ApJ Letter

Topics in Born-Infeld Electrodynamics

Classical version of Born-Infeld electrodynamics is recalled and its most important properties discussed. Then we analyze possible abelian and non-abelian generalizations of this theory, and show how certain soliton-like configurations can be obtained. The relationship with the Standard Model of electroweak interactions is also mentioned.Comment: (One new reference added). 15 pages, LaTeX. To be published in the Proceedings of XXXVII Karpacz Winter School edited in the Proceedings Series of American Mathematical Society, editors J. Lukierski and J. Rembielinsk

Closed Expressions for Lie Algebra Invariants and Finite Transformations

A simple procedure to obtain complete, closed expressions for Lie algebra invariants is presented. The invariants are ultimately polynomials in the group parameters. The construction of finite group elements require the use of projectors, whose coefficients are invariant polynomials. The detailed general forms of these projectors are given. Closed expressions for finite Lorentz transformations, both homogeneous and inhomogeneous, as well as for Galilei transformations, are found as examples.Comment: 34 pages, ps file, no figure

Kinematics of a Spacetime with an Infinite Cosmological Constant

A solution of the sourceless Einstein's equation with an infinite value for the cosmological constant \Lambda is discussed by using Inonu-Wigner contractions of the de Sitter groups and spaces. When \Lambda --> infinity, spacetime becomes a four-dimensional cone, dual to Minkowski space by a spacetime inversion. This inversion relates the four-cone vertex to the infinity of Minkowski space, and the four-cone infinity to the Minkowski light-cone. The non-relativistic limit c --> infinity is further considered, the kinematical group in this case being a modified Galilei group in which the space and time translations are replaced by the non-relativistic limits of the corresponding proper conformal transformations. This group presents the same abstract Lie algebra as the Galilei group and can be named the conformal Galilei group. The results may be of interest to the early Universe Cosmology.Comment: RevTex, 7 pages, no figures. Presentation changes, including a new Title. Version to appear in Found. Phys. Let

Cosmic microwave background constraints on the epoch of reionization

We use a compilation of cosmic microwave anisotropy data to constrain the epoch of reionization in the Universe, as a function of cosmological parameters. We consider spatially-flat cosmologies, varying the matter density $\Omega_0$ (the flatness being restored by a cosmological constant), the Hubble parameter $h$ and the spectral index $n$ of the primordial power spectrum. Our results are quoted both in terms of the maximum permitted optical depth to the last-scattering surface, and in terms of the highest allowed reionization redshift assuming instantaneous reionization. For critical-density models, significantly-tilted power spectra are excluded as they cannot fit the current data for any amount of reionization, and even scale-invariant models must have an optical depth to last scattering of below 0.3. For the currently-favoured low-density model with $\Omega_0 = 0.3$ and a cosmological constant, the earliest reionization permitted to occur is at around redshift 35, which roughly coincides with the highest estimate in the literature. We provide general fitting functions for the maximum permitted optical depth, as a function of cosmological parameters. We do not consider the inclusion of tensor perturbations, but if present they would strengthen the upper limits we quote.Comment: 9 pages LaTeX file with ten figures incorporated (uses mn.sty and epsf). Corrects some equation typos, superseding published versio

Network conduciveness with application to the graph-coloring and independent-set optimization transitions

We introduce the notion of a network's conduciveness, a probabilistically interpretable measure of how the network's structure allows it to be conducive to roaming agents, in certain conditions, from one portion of the network to another. We exemplify its use through an application to the two problems in combinatorial optimization that, given an undirected graph, ask that its so-called chromatic and independence numbers be found. Though NP-hard, when solved on sequences of expanding random graphs there appear marked transitions at which optimal solutions can be obtained substantially more easily than right before them. We demonstrate that these phenomena can be understood by resorting to the network that represents the solution space of the problems for each graph and examining its conduciveness between the non-optimal solutions and the optimal ones. At the said transitions, this network becomes strikingly more conducive in the direction of the optimal solutions than it was just before them, while at the same time becoming less conducive in the opposite direction. We believe that, besides becoming useful also in other areas in which network theory has a role to play, network conduciveness may become instrumental in helping clarify further issues related to NP-hardness that remain poorly understood