2 On the existence of the intrinsic anisotropy

In order to test the isotropy of 2025 GRBs we test the three dipole and five quadrupole terms. One obtains that, except for the terms defined by $\omega_{2,-1}$ and $\omega_{2,-2}$, the remaining six terms may still be taken to be zero. This means that there is a clear anisotropy defined by term $\sim \sin 2b \sin l$. The probability that this term is zero is $0.1\%$. In addition, the second quadrupole term being proportional to $\cos^2b \sin 2l$ is non-zero, too, with the probability $0.6\%$.

A straightforward counting of GRBs in those regions of equal areas, where $\sin 2b \sin l$ has either positive or negative signs, respectively, shows that 930 GRBs are in the first area and 1095 are in the second one. Taking p=0.5 probability for the binomial (Bernoulli) test, one obtains a $0.03\%$ probability that this detected distribution is caused by a chance. The observed distribution of all GRBs on sky is anisotropic with a certainty.

Instrumental effects of BATSE instrument should also play a role, since the sky exposure of BATSE instrument is non-uniform. The known dependence of the detection probability of this instrument predicts a similar kind of anisotropy. Hence, the question is the following: is this anisotropy caused either exclusively by the non-uniform sky-exposure function of BATSE instrument, or is there also an intrinsic anisotropy in the distribution of GRBs?

To clarify the situation we divided the GRBs into two groups according to their durations T₉₀. We excluded the dimmest GRBs. Then, 932 GRBs were separeted into the "short" ones (251 GRBs; T₉₀<2 s), and "long" ones (681 GRBs; T₉₀>2 s).

Dividing the sky into the two equal areas, as described above, we obtain a different behaviour for the short and long GRBs, respectively (see Table 1).

  

Table 1: Results of the binomial test of subsamples of GRBs with different durations. N is the number of GRBs at the given subsample, $k_{\rm obs}$ is the observed number GRBs at the first area for this subsample, and % is the probability in percentages that the assumption of isotropy still holds

Table 1 shows that the short GRBs are further distributed anisotropically; there is a smaller than $1\%$ probability of isotropy. On the other hand, the long GRBs can still be distributed isotropically. Up to this point all results mentioned in this section, were also written down in [Balázs et al. 1998] together with the relevant references.

The application of the 2-sample Kolmogorov-Smirnov test ([Press et al. 1992]) on $\omega_{2,-1}$ shows that the significance of the difference among the samples of short and long GRBs is $98.7\%$. The short and long ones are obviously distributed differently with a probability $98.7\%$.Note here that this important quantification of the different behaviour of two subclasses is a new result not presented in [Balázs et al. 1998].

We mean that these values confirm the expectation that there must exist some intrinsic anisotropy in the distribution of GRBs. Once there were an exclusive instrumental origin of the anisotropy all GRBs, the character of anisotropy should be the same for both types of GRBs; there should exist no difference among the short and long samples. Of course, the character of anisotropy is quite different than expected for the Galactical origin. For this one would need a clear non-zero $\omega_{2,0}$ spherical harmonic. Hence, there is no doubt concerning the cosmological origin of GRBs.