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

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3. Analysis of DMR maps

COBE's Differential Microwave Radiometers (DMR) have mapped the microwave sky at three frequencies: 31, 53, and 90 GHz (Smoot et al. 1990). The analysis presented here is based on DMR's 53 GHz maps (which have the best sensitivity) of the four year data set (Bennett et al. 1996), and is a continuation of the study based on the one year data-set (Gurzadyan & Torres 1993). Signal and noise maps were prepared by adding and subtracting the two independent DMR channels (ie. [FORMULA] and [FORMULA]) and Gaussian smoothing ([FORMULA]) the resulting maps. A Galaxy cut [FORMULA] has been applied to the maps. The geometric characteristics of hot spots are quite sensitive to galactic cuts below [FORMULA] to [FORMULA] but beyond [FORMULA] our results are stable.

DMR maps are digitized on a grid of 6144 approximately equal area pixels of size [FORMULA] (Torres et al. 1989). A hot spot is defined as a continuous region of the map formed by pixels whose temperature is higher than a preset threshold [FORMULA], where [FORMULA] is the standard deviation of the sky temperature (monopole and dipole subtracted). The algorithm to identify hot spots (Torres 1994, Torres 1995a) relies on the construction of binary tree structures out of the set of connected 'hot' pixels (for a given temperature threshold). The number of pixels in a tree gives the area of the spot. The eccentricity parameter [FORMULA] of the i -th hot spot at threshold [FORMULA] is found as follows: first, the center of the spot is estimated as the point with [FORMULA] coordinates equal to the mean of all the x and y coordinates of the pixels that form the spot respectively; second, the distance from this center point to all the pixels that lay on the boundary of the spot are calculated; [FORMULA] and [FORMULA] are the largest and shortest of these distances. The above mentioned procedure is repeated for several threshold levels ([FORMULA] in steps of [FORMULA]). The geometric descriptor [FORMULA] used to measure the elongation of anisotropy spots at threshold [FORMULA] is the average of all the [FORMULA] at a fixed [FORMULA].

At this point one can question whether the defined [FORMULA] really corresponds to the theoretical parameter of elongation in Fig. 1. The correspondence of the Lyapunov numbers or KS-entropy with parameters observed either with experiments or computer simulations is an essential problem. However, it has been empirically established that a correspondence with macroscopic parameters does exist (at least qualitative) and just this fact determines the role of those parameters as important tools for the study of nonlinear phenomena (Hilborn 1994 and references therein). Another question is whether the specific procedure of estimation of the mean elongation of spots can be crucial or not. Obviously other algorithms in principle can lead to numerically not absolutely equivalent results, but a drastic change of the situation seems unlikely, especially at high thresholds (Sommerer & Ott 1993 and references therein).

Fig. 2 shows [FORMULA] for the sum and difference DMR maps for threshold levels in the range [FORMULA]. Data for [FORMULA] is dominated by noise and for [FORMULA] is limited by the large statistical error due to the small number of spots at high thresholds.

[FIGURE] Fig. 2. Measured eccentricity parameter [FORMULA] for the COBE sum maps (solid) and difference maps (dash) compared with the expected eccentricity for noise Monte Carlo realizations (points with error bars).
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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998