The signature of the mixing of geodesics effect is a clear one: hot spots should have a fixed eccentricity independent of threshold level, and if these hot spots are elongated it is an indication of mixing in open spaces. In principle it is possible to detect eccentricities smaller than 1, the problem however is to disentangle the effect due to mixing from instrumental noise which by its stochastic nature is expected to produce anisotropy spots with certain degree of elongation.
In order to evaluate the statistical significance of a possible detection of the mixing of geodesics effect we have performed Monte Carlo studies of noise maps that take into account instrumental noise, COBE's sky coverage and pixelization. Noise maps were generated by assigning to each pixel on the map a temperature equal to a random number extracted from a Gaussian distribution with dispersion , where is the corresponding sensitivity for one observation and N the number of observations.
Maps for both A and B channels were generated independently. A difference map was formed and Gaussian smoothed just as it is done with the real data. The same algorithm used to obtain the eccentricity parameter of the COBE maps was used for each one of the Monte Carlo noise realizations. Fig. 2 shows the Monte Carlo mean eccentricity and the 1- error bars expected from noise maps. Knowing the eccentricity expected from noise, one can estimate the probability that the observed eccentricity parameter can be produced by noise alone. Table 1 gives for the COBE sum and difference maps, and the deviations from the mean eccentricity of Monte Carlo noise simulations.
Table 1. Eccentricity parameter of hot spots on COBE maps (, ) and comparison with Monte Carlo noise maps (). and denote the difference (in standard deviations) between the measured eccentricities and the mean eccentricity of noise Monte Carlo maps.
The statistic computed with the 9 data points in the range and the corresponding noise Monte Carlo points is 5.6 and 73.0 for the and maps respectively. The low associated with the difference maps was expected and is an indication of the accuracy of the Monte Carlo simulations. On the other hand, the high obtained when the data from the signal maps is compared with noise data is a clear indication of an actual detection of elongated anisotropy spots. The measured eccentricity parameter at threshold levels 1.5 and 2.5 in particular exhibit a - deviation with respect to noise. The average deviation in terms of standard deviations of from the corresponding Monte Carlo result for the 9 bins considered here is . From the mixing of geodesics effect one would expect a constant eccentricity for all threshold levels. Due to the large dispersion of the measured values it is not possible to make a strong statement in favor of the hypothesis of a constant independent of . However, it is seen that the contribution to the is roughly the same independent of . Under the hypothesis of a positive detection of elongated anisotropy spots and using the 9 points in Table 1, the measured eccentricity is .
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998