Cosmological effects in the local static frame

What is the influence of cosmology (the expansion law and its acceleration, the cosmological constant...) on the dynamics and optics of a local system like the solar system, a galaxy, a cluster, a supercluster...? The answer requires the solution of Einstein equation with the local source, which tends towards the cosmological model at large distance. There is, in general, no analytic expression for the corresponding metric, but we calculate here an expansion in a small parameter, which allows to answer the question. First, we derive a static expression for the pure cosmological (Friedmann-Lema\^itre) metric, whose validity, although local, extends in a very large neighborhood of the observer. This expression appears as the metric of an osculating de Sitter model. Then we propose an expansion of the cosmological metric with a local source, which is valid in a very large neighborhood of the local system. This allows to calculate exactly the (tiny) influence of cosmology on the dynamics of the solar system: it results that, contrary to some claims, cosmological effects fail to account for the unexplained acceleration of the Pioneer probe by several order of magnitudes. Our expression provide estimations of the cosmological influence in the calculations of rotation or dispersion velocity curves in galaxies, clusters, and any type of cosmic structure, necessary for precise evaluations of dark matter and/or cosmic flows. The same metric can also be used to estimate the influence of cosmology on gravitational optics in the vicinity of such systems.Comment: to appear in Astron. & Astrop

Space and Observers in Cosmology

I provide a prescription to define space, at a given moment, for an arbitrary observer in an arbitrary (sufficiently regular) curved space-time. This prescription, based on synchronicity (simultaneity) arguments, defines a foliation of space-time, which corresponds to a family of canonically associated observers. It provides also a natural global reference frame (with space and time coordinates) for the observer, in space-time (or rather in the part of it which is causally connected to him), which remains Minkowskian along his world-line. This definition intends to provide a basis for the problem of quantization in curved space-time, and/or for non inertial observers. Application to Mikowski space-time illustrates clearly the fact that different observers see different spaces. It allows, for instance, to define space everywhere without ambiguity, for the Langevin observer (involved in the Langevin pseudoparadox of twins). Applied to the Rindler observer (with uniform acceleration) it leads to the Rindler coordinates, whose choice is so justified with a physical basis. This leads to an interpretation of the Unruh effect, as due to the observer's dependence of the definition of space (and time). This prescription is also applied in cosmology, for inertial observers in the Friedmann - Lemaitre models: space for the observer appears to differ from the hypersurfaces of homogeneity, which do not obey the simultaneity requirement. I work out two examples: the Einstein - de Sitter model, in which space, for an inertial observer, is not flat nor homogeneous, and the de Sitter case.Comment: 21 pages, 6 figures. Astronomy & Astrophysics, in pres

Spin and Clifford algebras, an introduction

40 pages ; published version with slight modifications for on-line reading (Ed. UNAM-FQ 2002, México)International audienceIn this short pedagogical presentation, we introduce the spin groups and the spinors from the point of view of group theory. We also present, independently, the construction of the low dimensional Clifford algebras. And we establish the link between the two approaches. Finally, we give some notions of the generalisations to arbitrary spacetimes, by the introduction of the spin and spinor bundles

Wavelet Analysis of Inhomogeneous Data with Application to the Cosmic Velocity Field

In this article we give an account of a method of smoothing spatial inhomogeneous data sets by using wavelet reconstruction on a regular grid in an auxilliary space onto which the original data is mapped. In a previous paper by the present authors, we devised a method for inferring the velocity potential from the radial component of the cosmic velocity field assuming an ideal sampling. Unfortunately the sparseness of the real data as well as errors of measurement require us to first smooth the velocity field as observed on a 3-dimensional support (i.e. the galaxy positions) inhomogeneously distributed throughout the sampled volume. The wavelet formalism permits us to introduce a minimal smoothing procedure that is characterized by the variation in size of the smothing window function. Moreover the output smoothed radial velocity field can be shown to correspond to a well defined theoretical quantity as long as the spatial sampling support satisfies certain criteria. We argue also that one should be very cautious when comparing the velocity potential derived from such a smoothed radial component of the velocity field with related quantities derived from other studies (e.g : of the density field).Comment: 19 pages, Latex file, figures are avaible under requests, published in Inverse Problems, 11 (1995) 76

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 \lambda_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.Comment: 17 pages, no figure, to appear in CQ

Orbifold construction of the modes of the Poincaré dodecahedral space

International audienceWe provide a new construction of the modes of the Poincaré dodecahedral space S3=I. The construction uses the Hopf map, Maxwell's multipole vectors and orbifolds. In particular, the *235-orbifold serves as a parameter space for the modes of S3=I, shedding new light on the geometrical significance of the dimension of each space of k-modes, as well as on the modes themselves

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 $\lambda_{\ell} = -\ell ~(\ell +1)$. Here we show the existence of a different basis $\Phi ^{\ell}_j$ for ${\cal V} ^{\ell}$, where $\Phi ^{\ell}_j(X) \equiv (X \cdot N_j)^{\ell}$, 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.Comment: 6 pages, no figure; new version: shorter demonstrations; new references; as will appear in Journal of Physics A. Journal of Physics A, in pres