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Astron. Astrophys. 321, 19-23 (1997)

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1. Introduction

The detection (Smoot et al. 1992; Bennett et al. 1994) and further confirmation (Bennett et al. 1996; Hancock et al. 1994; Ganga et al. 1993) of the presence of anisotropies in the cosmic microwave background (CMB), has endowed cosmologists with a unique tool for a realistic test of cosmological models. The observed large angle [FORMULA] can be produced by a number of sources that go from local, to astrophysical, to cosmological effects, among which the most relevant are: emission from the Galaxy; gravitational fluctuations produced by voids and great attractors; the Doppler effect due to the observer's motion relative to the CMB; mixing of geodesic flow; gravitational waves; cosmic strings; and gravitational potential fluctuations on the last scattering surface - the Sachs-Wolfe effect (for a review see White, Scott & Silk 1994). Clearly the complexity of determining the cause of CMB anisotropies requires several independent analysis techniques. Aside from the standard statistical tests based on the angular auto-correlation function and power spectrum, a number of statistics have been proposed (Sazhin 1985; Vittorio & Juszkiewicz 1987; Bond & Efstathiou 1987; Coles 1988; Gott et al. 1990; Martínez-González & Cayón 1992; Martínez-González & Sanz 1989) as topological descriptors to characterize the observed anisotropies: hot spot number density, length and total curvature or genus of iso-temperature contours, number of upcrossings, area and eccentricity of hot spots, and Euler-Poincaré characteristic. The study of these topological descriptors is based on the well established theory of the geometric properties of the excursion set of random fields (Adler 1981).

The great potential of topological analysis of CMB data can be attested by the large number of results obtained thus far: the genus descriptor has been successfully used to place important restrictions on the shape of the spectrum of primordial perturbations [FORMULA] (Smoot et al. 1994; Torres 1994; Torres 1995a; Torres et al. 1995); further restrictions on [FORMULA] have been obtained from the peak distribution analysis (Fabbri & Torres 1995, 1996); a possible fractal dimension in the universe has been detected with the analysis of the contour characteristics of anisotropy spots (De Gouveia 1995); studies of hot spot number density (Torres 1995b; Cayón & Smoot 1995) support the results given by the genus analysis; tests of Gaussianity of the CMB field have been performed using the genus and the correlation function of local maxima (Kogut et al. 1996); finally, percolation and cluster analysis (Naselsky & Novikov 1995) of CMB maps has been proposed as a powerful tool to test the Gaussian character of CMB anisotropies. The information carried by the CMB properties can even have a direct impact on accelerator physics (Gurzadyan & Margarian 1996).

We propose to use a topological analysis, based on the eccentricity or elongation measure of hot spots, to look for the signature of the mixing of geodesics effect (Gurzadyan & Kocharyan 1991, 1992). Because of the appealing possibility of extracting information about the curvature of the universe from the latter effect, we will deal with it in some detail. A consequence of mixing phenomena in geodesics propagation is the appearance of highly distorted anisotropy spots in cosmic background maps (Gurzadyan & Kocharyan 1993a, 1993b). In the case of an open geometry, anisotropy spots associated with a bundle of photons propagating in free space would appear as highly elongated hot spots, while in a flat space hot spots would show a symmetric circular shape. This visible effect is a direct consequence of the geometry of space and as such it is a unique tool to discriminate between open and closed cosmological models. However, confusion due to galactic contamination at high latitudes and similar elongated patterns produced by instrumental noise and cosmological structure characterized by large coherence angles makes this effect very difficult to observe.

Fortunately, at least 3 different phenomena produced by the effect of mixing geodesics and associated with the observable characteristics of CMB, have been predicted:

(a) Isotropization. The decrease of the amplitude of CMB anisotropy after the decoupling epoch: the degree of isotropization, i.e. the fluctuation damping factor can yield up to [FORMULA] for [FORMULA] = 0.2-0.4 (Gurzadyan & Kocharyan 1991, 1992) 1 ;

(b) Angular dependence. The behaviour of anisotropy as a function of the sky and smoothing (beam) angles, namely, the independence of the autocorrelation function on the sky angle, and its dependence on the smoothing angle (Gurzadyan & Kocharyan 1996);

(c) Maps. The complex topological structure of the CMB sky maps, showing intrinsic threshold independent elongation of the shapes of both hot and cold spots (Gurzadyan & Kocharyan 1993a, 1993b).

In this paper we will deal only with the third phenomenon. Since, in principle, effects of apparent elongation might occur also owing to other physical mechanisms (the same is not excluded also for phenomena (a) and (b)), the unambiguous confirmation of the observational discovery of the effect of mixing geodesics could be reached by means of a comprehensive study of the complete set of predictions. As it is briefly discussed at the end of this paper, at least the available evidences do not contradict to all these predictions. A confirmation of the effect of mixing geodesics would constitute powerful probe of the curvature of the Universe.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998