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

The determination of HOH_O by using the TF relation : About particular selection effects

This paper completes the statistical modeling of the Hubble flow when a Tully-Fisher type relation is used for estimating the absolute magnitude $M\approx a\,p+b$ from a line width distance indicator $p$. Our investigation is performed with the aim of providing us with a full understanding of statistical biases due to selection effects in observation, regardless of peculiar velocities of galaxies. We show that unbiased $H_{\circ}$-statistics can be obtained by means of the maximum likelihood method as long as the statistical model can be defined. We focus on the statistical models related to the Direct, resp. Inverse, Tully-Fisher relation, when selection effects on distance, resp. on $p$, are present. It turns out that the use of the Inverse relation should be preferred, according to robustness criteria. The formal results are ensured by simulations with samples which are randomly generated according to usual characteristics.Comment: 8 pages, Postscript compressed file, to be published in A\&

Closed Spaces in Cosmology

This paper deals with two aspects of relativistic cosmologies with closed (compact and boundless) spatial sections. These spacetimes are based on the theory of General Relativity, and admit a foliation into space sections S(t), which are spacelike hypersurfaces satisfying the postulate of the closure of space: each S(t) is a 3-dimensional, closed Riemannian manifold. The discussed topics are: (1) A comparison, previously obtained, between Thurston's geometries and Bianchi-Kantowski-Sachs metrics for such 3-manifolds is here clarified and developed. (2) Some implications of global inhomogeneity for locally homogeneous 3-spaces of constant curvature are analyzed from an observational viewpoint.Comment: 20 pages, 6 figures, revised version of published paper. In version 2: several misprints corrected, 'redshifting' in figures improved. Version 3: a few style corrections; couple of paragraphs in subsection 2.4 rewritten. Version 4: figures 5 and 6 corrrecte

Topology of the Universe: background and recent observational approaches

Is the Universe (a spatial section thereof) finite or infinite? Knowing the global geometry of a Friedmann-Lema\^{\i}tre (FL) universe requires knowing both its curvature and its topology. A flat or hyperbolic (``open'') FL universe is {\em not} necessarily infinite in volume. Multiply connected flat and hyperbolic models are, in general, as consistent with present observations on scales of 1-20{\hGpc} as are the corresponding simply connected flat and hyperbolic models. The methods of detecting multiply connected models (MCM's) are presently in their pioneering phase of development and the optimal observationally realistic strategy is probably yet to be calculated. Constraints against MCM's on ~1-4 h^{-1} Gpc scales have been claimed, but relate more to inconsistent assumptions on perturbation statistics rather than just to topology. Candidate 3-manifolds based on hypothesised multiply imaged objects are being offered for observational refutation. The theoretical and observational sides of this rapidly developing subject have yet to make any serious contact, but the prospects of a significant detection in the coming decade may well propel the two together.Comment: 5 pages, proceedings of the Workshop ``Cosmology: Observations Confront Theories,'' 11-17 Jan 1999, IIT Kharagpur, West Bengal, to appear in Pramana - Journal of Physic

A measure on the set of compact Friedmann-Lemaitre-Robertson-Walker models

Compact, flat Friedmann-Lemaitre-Robertson-Walker (FLRW) models have recently regained interest as a good fit to the observed cosmic microwave background temperature fluctuations. However, it is generally thought that a globally, exactly-flat FLRW model is theoretically improbable. Here, in order to obtain a probability space on the set F of compact, comoving, 3-spatial sections of FLRW models, a physically motivated hypothesis is proposed, using the density parameter Omega as a derived rather than fundamental parameter. We assume that the processes that select the 3-manifold also select a global mass-energy and a Hubble parameter. The inferred range in Omega consists of a single real value for any 3-manifold. Thus, the obvious measure over F is the discrete measure. Hence, if the global mass-energy and Hubble parameter are a function of 3-manifold choice among compact FLRW models, then probability spaces parametrised by Omega do not, in general, give a zero probability of a flat model. Alternatively, parametrisation by the injectivity radius r_inj ("size") suggests the Lebesgue measure. In this case, the probability space over the injectivity radius implies that flat models occur almost surely (a.s.), in the sense of probability theory, and non-flat models a.s. do not occur.Comment: 19 pages, 4 figures; v2: minor language improvements; v3: generalisation: m, H functions of

Exact Polynomial Eigenmodes for Homogeneous Spherical 3-Manifolds

Observational data hints at a finite universe, with spherical manifolds such as the Poincare dodecahedral space tentatively providing the best fit. Simulating the physics of a model universe requires knowing the eigenmodes of the Laplace operator on the space. The present article provides explicit polynomial eigenmodes for all globally homogeneous 3-manifolds: the Poincare dodecahedral space S3/I*, the binary octahedral space S3/O*, the binary tetrahedral space S3/T*, the prism manifolds S3/D_m* and the lens spaces L(p,1).Comment: v3. Final published version. 27 pages, 1 figur

Cosmic Topology of Polyhedral Double-Action Manifolds

A special class of non-trivial topologies of the spherical space S^3 is investigated with respect to their cosmic microwave background (CMB) anisotropies. The observed correlations of the anisotropies on the CMB sky possess on large separation angles surprising low amplitudes which might be naturally be explained by models of the Universe having a multiconnected spatial space. We analysed in CQG 29(2012)215005 the CMB properties of prism double-action manifolds that are generated by a binary dihedral group D^*_p and a cyclic group Z_n up to a group order of 180. Here we extend the CMB analysis to polyhedral double-action manifolds which are generated by the three binary polyhedral groups (T^*, O^*, I^*) and a cyclic group Z_n up to a group order of 1000. There are 20 such polyhedral double-action manifolds. Some of them turn out to have even lower CMB correlations on large angles than the Poincare dodecahedron

Coherent state quantization of a particle in de Sitter space

We present a coherent state quantization of the dynamics of a relativistic test particle on a one-sheet hyperboloid embedded in a three-dimensional Minkowski space. The group SO_0(1,2) is considered to be the symmetry group of the system. Our procedure relies on the choice of coherent states of the motion on a circle. The coherent state realization of the principal series representation of SO_0(1,2) seems to be a new result.Comment: Journal of Physics A: Mathematical and General, vol. 37, in pres

A natural fuzzyness of de Sitter space-time

A non-commutative structure for de Sitter spacetime is naturally introduced by replacing ("fuzzyfication") the classical variables of the bulk in terms of the dS analogs of the Pauli-Lubanski operators. The dimensionality of the fuzzy variables is determined by a Compton length and the commutative limit is recovered for distances much larger than the Compton distance. The choice of the Compton length determines different scenarios. In scenario I the Compton length is determined by the limiting Minkowski spacetime. A fuzzy dS in scenario I implies a lower bound (of the order of the Hubble mass) for the observed masses of all massive particles (including massive neutrinos) of spin s>0. In scenario II the Compton length is fixed in the de Sitter spacetime itself and grossly determines the number of finite elements ("pixels" or "granularity") of a de Sitter spacetime of a given curvature.Comment: 16 page