arXiv subjects

M. Lachieze-Rey

Publications and source records attributed to M. Lachieze-Rey.

16 recordsLinked to original sources

arXiv2020

Connections and Frame Bundle Reductions

In general relativity, the gravitational potential is represented by the Levi-Civita connection, the only symmetric connection preserving the metric. On a differentiable manifold, a metric identifies with an orthogonal structure, defined as a Lorentz reduction of the frame bundle. The Levi-Civita connection appears as the only symmetric connection preserving the reduction. This paper presents generalization of this process to other aproaches of gravitation: Weyl structure with Weyl connections, teleparallel structures with Weitzenbock connections, unimodular structure, similarly appear as frame bundle reductions, with preserving connections. To each subgroup H of the linear group GL correspond reduced structures, or H-structures. They are subbundles of the frame bundle (with GL as principal group), with H as principal group. A linear connection in a manifold M is a principal connection on the frame bundle. Given a reduction, the corresponding preserving connections on M are the linear connections which preserve it. I also show that the time gauge used in the 3+1 formalism for general relativity similarly appears as the result of a bundle reduction.

math-ph↗

arXiv2005

Laplacian eigenmodes for spherical spaces

The possibility that our space is multi - rather than singly - connected has gained a renewed interest after the discovery of the low power for the first multipoles of the CMB by WMAP. To test the possibility that our space is a multi-connected spherical space, it is necessary to know the eigenmodes of such spaces. Excepted for lens and prism space, and in some extent for dodecahedral space, this remains an open problem. Here we derive the eigenmodes of all spherical spaces. For dodecahedral space, the demonstration is much shorter, and the calculation method much simpler than before. We also apply to tetrahedric, octahedric and icosahedric spaces. This completes the knowledge of eigenmodes for spherical spaces, and opens the door to new observational tests of cosmic topology. The vector space V^k of the eigenfunctions of the Laplacian on the three-sphere S^3, corresponding to the same eigenvalue λ_k = -k (k+2), has dimension (k+1)^2. We show that the Wigner functions provide a basis for such space. Using the properties of the latter, we express the behavior of a general function of V^k under an arbitrary rotation G of SO(4). This offers the possibility to select those functions of V^k which remain invariant under G. Specifying G to be a generator of the holonomy group of a spherical space X, we give the expression of the vector space V_X^k of the eigenfunctions of X. We provide a method to calculate the eigenmodes up to arbitrary order. As an illustration, we give the first modes for the spherical spaces mentioned.

astro-ph↗

arXiv2003

A new basis for eigenmodes on the Sphere

The usual spherical harmonics $Y_{\ell m}$ form a basis of the vector space ${\cal V} ^{\ell}$ (of dimension $2\ell+1$) of the eigenfunctions of the Laplacian on the sphere, with eigenvalue $λ_{\ell} = -\ell ~(\ell +1)$. Here we show the existence of a different basis $Φ^{\ell}_j$ for ${\cal V} ^{\ell}$, where $Φ^{\ell}_j(X) \equiv (X \cdot N_j)^{\ell}$, the $\ell ^{th}$ power of the scalar product of the current point with a specific null vector $N_j$. We give explicitly the transformation properties between the two bases. The simplicity of calculations in the new basis allows easy manipulations of the harmonic functions. In particular, we express the transformation rules for the new basis, under any isometry of the sphere. The development of the usual harmonics $Y_{\ell m}$ into thee new basis (and back) allows to derive new properties for the $Y_{\ell m}$. In particular, this leads to a new relation for the $Y_{\ell m}$, which is a finite version of the well known integral representation formula. It provides also new development formulae for the Legendre polynomials and for the special Legendre functions.

math.SP↗

arXiv2003

Examples of Berezin-Toeplitz Quantization: Finite sets and Unit Interval

We present a quantization scheme of an arbitrary measure space based on overcomplete families of states and generalizing the Klauder and the Berezin-Toeplitz approaches. This scheme could reveal itself as an efficient tool for quantizing physical systems for which more traditional methods like geometric quantization are uneasy to implement. The procedure is illustrated by (mostly two-dimensional) elementary examples in which the measure space is a $N$-element set and the unit interval. Spaces of states for the $N$-element set and the unit interval are the 2-dimensional euclidean $\R^2$ and hermitian $\C^2$ planes.

quant-ph↗

arXiv2003

Cosmic Topology

General relativity does not allow one to specify the topology of space, leaving the possibility that space is multi-- rather than simply--connected. We review the main mathematical properties of multi--connected spaces, and the different tools to classify them and to analyse their properties. Following the mathematical classification, we describe the different possible muticonnected spaces which may be used to construct universe models. We briefly discuss some implications of multi--connectedness for quantum cosmology, and its consequences concerning quantum field theory in the early universe. We consider in details the properties of the cosmological models where space is multi--connected, with emphasis towards observable effects. We then review the analyses of observational results obtained in this context, to search for a possible signature of multi--connectedness, or to constrain the models. They may concern the distribution of images of cosmic objects like galaxies, clusters, quasars,..., or more global effects, mainly those concerning the Cosmic Microwave Background, and the present limits resulting from them.

gr-qc↗

arXiv2000

Theoretical Necessity of an External Scalar Field in the Kaluza-Klein Theory (I)

We show that the principle of least action is generally inconsistent with the usual Kaluza-Klein program, the higher dimensional Einstein-Hilbert action being unbounded from below. This inconsistency is also present in other theories with higher dimensions like supergravity. Hence, we conclude to the necessity of an external scalar field to compensate this flaw.

gr-qc↗

arXiv2000

The Friedmann-Lemaitre models in perspective

I show that all FRW models (four dimensional pseudo-Riemannian manifolds with maximally symmetric space) can be embedded in a flat Minkowski manifold with 5 dimensions. The pseudo Riemannian metric of space-time is induced by the flat metric. This generalizes the usual embedding widely used for the \dS ~models. I give the coordinate transformations for the embedding. Taking into account the spatial isotropy, one can reduce space-time to a two-dimensional surface, embedded in a three-dimensional Minkowski space. This allows to give exact graphic representations of the FRW models, and in particular of their curvature.

astro-ph↗

arXiv1997

Observational Limits on Machos in the Galactic Halo

We present final results from the first phase of the EROS search for gravitational microlensing of stars in the Magellanic Clouds by unseen deflectors (machos: MAssive Compact Halo Objects). The search is sensitive to events with time scales between 15 minutes and 200 days corresponding to deflector masses in the range 1.e-7 to a few solar masses. Two events were observed that are compatible with microlensing by objects of mass of about 0.1 Mo. By comparing the results with the expected number of events for various models of the Galaxy, we conclude that machos in the mass range [1.e-7, 0.02] Mo make up less than 20% (95% C.L.) of the Halo dark matter.

astro-ph↗

arXiv1996

Cosmic Crystallography

We assume that the Universe has a non trivial topology whose compact spatial sections have a volume significantly smaller than the horizon volume. By a topological lens effect, such a "folded" space configuration generates multiple images of cosmic sources, e.g. clusters of galaxies. We present a simple and powerful method to unveil non ambiguous observational effects, independently of the sign of the curvature and of the topological type.

gr-qc↗

arXiv1995

A symplectic approach to gravitational instability

We present a global approach of non-dissipative physics. Based on symplectic mechanics this technique allows us to obtain the solution of a very large class of problems in terms of a Taylor expand. We apply this method to the problem of gravitational instability and we obtain a general expression of the gravitational potential, solution of the Vlasov- Poisson gravitational potential, solution of the Vlasov-Poisson system, as a function of time in the context of Newtonian dust cosmology.

astro-ph↗

arXiv1995

V/Vmax statistic and Neo-Classic Cosmological Tests

A new cosmological test is derived, based on the distribution of individual V/Vmax in a complete redshift-limited sample of distant objects. The fundamental assumption is that, in any bin of absolute luminosity, individual V/Vmax are required to be uniformly spread over the [0,1] range. This condition of uniformity allows a likelihood function to be computed from the Kolmogorov- Smirnov probabilities of each bin. Monte-Carlo simulations show that the test is mostly sensitive to the density parameter, but under certain conditions, it also sets constraints on the space curvature and, to a lower extent, on the cosmological constant. The test is then applied to the UVX sample of Boyle et al. (1990). A low matter density, and a flat Universe without cosmological constant, are rejected: 0.2<Omega<0.8 within the 95% confidence level.

astro-ph↗

arXiv1995

Pure Luminosity Evolution Hypothesis for QSOs: From Luminosity Functions to Synthetic Catalogues

This paper describes the simulation of realistic Monte-Carlo extragalactic catalogues, aimed at comparing the behaviour of cosmological tests versus input parameters. QSO catalogues are built on a Luminosity Function derived from data through suitable computation of individual maximum volumes in complete (but magnitude- and redshift-limited) samples requiring neither of redshift nor of apparent magnitude histogram. The values of the evolution parameter are derived for various cosmologies, corresponding to =1/2 in the sample of 400 Ultra-Violet Excess (UVX) QSOs (Boyle et al 1990). The various luminosity functions are compared, both for the whole sample and in redshift bins. An evolution characteristic time is defined and computed, depending strongly on the cosmology, but practically constant when expressed in terms of the age of the Universe. Algorithms are given for producing unbiased or biased catalogues based on the null hypothesis that the objects are uniformly distributed in volume but suffer Pure Luminosity Evolution.

astro-ph↗

arXiv1995

EROS VARIABLE STARS : FUNDAMENTAL-MODE AND FIRST OVERTONE CEPHEIDS IN THE BAR OF THE LARGE MAGELLANIC CLOUD

We present CCD phase-binned light curves at 490 nm for 97 Cepheid variable stars in the bar of the LMC. The photometry was obtained as part of the French EROS project and has excellent phase coverage, permitting accurate decomposition into Fourier components. We identify as `sinusoidal' or s-Cepheids those stars with periods less than 5.5 d and small second-harmonic components. These stars comprise $\sim$30% of our sample and most form a sequence $\sim$1 mag brighter than the LMC classical Cepheids in the period-luminosity diagram. They are also generally bluer and have lower-amplitude light curves. We infer that the s-Cepheids are first-overtone pulsators because, when their periods are converted to expected fundamental-mode values, they obey a common period-luminosity-colour relation with classical Cepheids. This also confirms the reality of the colour term in the Cepheid period-luminosity-colour relation. Further, the blue edge of the classical Cepheid instability strip agrees well with the theoretical calculations for the fundamental mode made by Chiosi et al. (1993) for the Hertzsprung-Russell and period-luminosity diagrams, but we find that our observed s-Cepheids are $>0.2$ mag brighter and bluer than the Chiosi et al.\ predictions for the first-overtone. We identify a number of features in plots of our stars' Fourier-component amplitude ratios and phase differences. These features have been identified with resonances between different pulsation modes. In the LMC we find these features seem to occur at periods very similar to Galactic ones for classical Cepheids, but at different periods for s-Cepheids. We discover a double-mode Cepheid in the LMC, for which $P({\rm first overtone})/P({\rm fundamental}) = 0.710 \pm 0.001$, very similar to observed ratios for Galactic double-mode Cepheids.

astro-ph↗

arXiv1995

SEARCH FOR VERY LOW MASS OBJECTS IN THE GALACTIC HALO

We present results from a search for gravitational microlensing of stars in the Large Magellanic Cloud by low mass objects in the Galactic Halo. The search uses the CCD light curves of about 82,000 stars with up to 46 measurements per night over a period of 10 months. No light curve exhibits a form that is consistent with a microlensing event of maximum amplification greater than 1.2. This null result makes it unlikely that the Halo is dominated by objects in the mass range $5 10^{-8}M_{\odot} < M <5 10^{-4}M_{\odot} $. keywords{Galaxy : Halo, kinematics and dynamics, stellar content -- Cosmology : dark matter, gravitational lensing}

astro-ph↗

arXiv1994

Spectroscopic Studies of the Two Eros Candidate Microlensed Stars

Low resolution spectroscopy, and UBVRI photometry, have been obtained for the two EROS microlensing candidates. Radial velocities indicate that both stars are members of the Large Magellanic Cloud. The spectrum and the colours of EROS1, the first candidate, reveal that it is a moderately reddened main-sequence B star with H emission lines. The presence of H absorption lines seems to be more the signature of a normal star than that of a cataclysmic variable. As to EROS2, the second candidate, its spectrum and photometry are those of an unreddened normal main-sequence A star, but it cannot be totally excluded that they represent those of a nova in the pre-outburst phase. Although it is not yet possible to exclude intrinsic stellar variations, the interpretation in terms of microlensing effects remain the most natural one.

astro-ph↗

arXiv1994

About the Malmquist bias in the determination of H0 and of distances of galaxies

We provide the mathematical framework which elucidates the way of using a Tully-Fisher (TF) like relation in the determination of the Hubble constant $H_0$, as well as for distances of galaxies. The methods related to the so-called Direct and Inverse TF Relations (herein DTF and ITF) are interpreted as maximum likelihood statistics. We show that, as long as the same model is used for the calibration of the TF relation and for the determination of $H_0$, we obtain a coherent Hubble's constant. The choice of the model is motivated by reasons of robustness of statistics, it depends on selection effects in observation which are present in the sample. The difference on the distance estimates when using either the ITF or the DTF model is only due to random fluctuations. It is interesting to point out that the DTF estimate does not depend on the luminosity distribution of sources. Both statistics show a correction for a bias, inadequately believed to be of Malmquist type. The repercussion of measurement errors, and additional selection effects are also analyzed

astro-ph↗