... coordinates

Throughout this paper we use galactic coordinates with origin at $b=90^\circ, ~ l=0^\circ$, and equator defined by $b=90^\circ$, and $l \in [0^\circ,360^\circ]$, with $b=0^\circ$ in the galactic North-pole.

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... pixels

We shall see below that, subject to certain reasonable constraints, the details of this subdivision will not change our results significantly.

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...$\alpha_i \in (0, 180^\circ]$

In this connection it is instructive to note that, for a homogeneous distribution of points on S², all angular separations $0 < \gamma \leq 180^\circ$ are allowed, and the corresponding probability distribution can be calculated to give ${\cal P}^{{\rm full-sky}}_{{\rm exp}}(\gamma) = \frac{1}{2}~\sin \gamma$. This represents the limit of a statistically isotropic distribution of points in S² as the number of points go to infinity. One can thus quantify departure from statistical isotropy by calculating the departure of the mean observed probability distribution $\langle \Phi_{\rm obs}(\alpha_i) \rangle$ from this quantity, i.e. by evaluating $\langle \Phi_{\rm obs}(\alpha_i) \rangle - \Phi_{\rm exp}(\alpha_i)$.

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... variance

We have checked numerically that the b_lms of the $\sigma$-maps derived from the Monte Carlo CMB maps do follow Gaussian distributions, which justifies the use of cosmic variance bounds and the $\chi^2$ test.

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