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

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5. Discussion

An excess elongation as possible genuine feature of the hot spots on COBE maps is well established at least at threshold levels [FORMULA] between 1.0 and 3.0. The detected excess elongation shows a statistically strong independence on the threshold, contrary to the case for the difference [FORMULA] maps where [FORMULA] shows a clear correlation with threshold. This characteristic dependence of the eccentricity parameter with threshold for noise maps is clearly manifest in Monte Carlo noise realizations. If the detection of high eccentricity of hot spots is attributed to the effect of mixing of geodesics flow, our analysis implies [FORMULA]. However one must also consider the fact that the detected effect can be produced in part by galactic contamination of the maps.

The numerical value of the obtained elongation parameter might have been obviously affected by a number of effects of cosmological and non-cosmological nature. Primordial fluctuations on the surface of last scattering, for example, would also produce non-zero ellipticity even in flat universe (Bond & Efstathiou 1987). However, due to the stochastic nature of this effect, the dependence of the ellipticity parameter with threshold would be just like that of noise, contrary to what we have observed (i.e. a threshold independent eccentricity). More statistics and higher accuracy is necessary for a reliable numerical evaluation of [FORMULA]. A question thus arises as to how can one distinguish whether the observed elongation is really due to the effect of geodesics mixing? Indeed, although the "geodesics mixing" has put forward the idea of looking for the threshold independent elongation of hot spots on CMB sky maps as a signature for [FORMULA], there are other mentioned effects that can be considered as well. Thus, to answer the question posed above it is important to have a thorough consideration of other observable consequences of the effect of mixing geodesics. As it is mentioned above, among those consequences is the decrease of the amplitude of initial fluctuations during the expansion of the Universe; the expected decrease depends on the curvature, the value of [FORMULA] and the time of last scattering (Gurzadyan & Kocharyan 1992).

More interesting, however, can be the property concerning the dependence of the temperature autocorrelation function [FORMULA] on the sky angle [FORMULA] with a beam size [FORMULA]. It was shown (Gurzadyan & Kocharyan 1996), that in negatively curved spaces [FORMULA] tends to become constant with respect to [FORMULA] (i.e. independent on the sky angle) independent of the initial perturbation spectrum, but should depend on the beam size [FORMULA] by the asymptotic law


In other words, the observation with different beam sizes should lead to different values for the amplitude of the CMB anisotropy, namely the measured anisotropy decreases with increasing beam size.

Regarding the latter effect, note that some measurements by smaller beam sizes (Netterfield et al 1995; Ruhl et al 1995) seem to indicate higher values for the anisotropy amplitude as compared with the COBE data, even though one has to distinguish the contributions of other effects such as the Doppler peak for example.

The peculiar feature of the effect of geodesic is its crucial dependence on the curvature of the Universe, i.e. its disappearance in flat and positively curved spaces, and independence on various models of dark matter or the spectrum of initial perturbations, etc.

Obviously nothing is happening to an individual photon during the free propagation after the last scattering epoch, and these effects are entirely statistical and determined by the principal limitations of obtaining information during the measurements, i.e. by the impossibility of the reconstruction of the trajectory of the photon while observing within finite smoothing angle and time period, due to the overlapping of exponentially deviating geodesics in any given cut of phase space (for detailed discussion of this and similar problems see Gurzadyan & Kocharyan 1994). Note that these effects differ from those of classical chaotic Mixmaster models, and can be alternatives to inflationary scenarios (Gurzadyan & Kocharyan 1994; Cornish et al. 1996). The measured effect of elongation of anisotropy spots in CMB sky maps can be the direct footprint of the negative curvature of the Universe and therefore has a considerable impact on cosmology.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998