144 research outputs found
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 form a basis of the vector space
(of dimension ) of the eigenfunctions of the
Laplacian on the sphere, with eigenvalue .
Here we show the existence of a different basis for , where , the power of the scalar product of the current point with a specific null
vector . 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 into thee new basis (and
back) allows to derive new properties for the . In particular, this
leads to a new relation for the , 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
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