= Neutrino directional-coincidence model
{title2=$q(\psi)=(1-\cos\psi)/2$}
How much does a small directional uncertainty help? Consider an original idealized model: a source direction fixed in advance, uniform background directions over the full <sphere>, and circular acceptance radius $\psi$. The accepted <solid angle> and single-event background <probability> are
$$
\Omega(\psi)=\int_0^{2\pi}\!d\phi\int_0^\psi\sin\theta\,d\theta=2\pi(1-\cos\psi),\qquad
q(\psi)=\frac{\Omega}{4\pi}=\frac{1-\cos\psi}{2}.
$$
The derivation integrates the area element on a unit <sphere>. For small $\psi$ in <radians>, $q\simeq\psi^2/4$: halving the radius quarters the accidental acceptance, to leading order. At $psi=0$ acceptance vanishes; at $\psi=\pi$ it covers the full <sphere>. The <solid angle> has units of <steradians>; $q$ is dimensionless.
For $0<\psi\leq\pi$, take a deliberately crude <point spread function>: stipulate that every signal direction lies uniformly inside that cap. The signal and background <probability density functions> per <steradian> are $S=1/\Omega$ inside the cap and zero outside, and $B=1/(4\pi)$ everywhere. Both integrate to one. For an event inside the cap their <likelihood ratio> is
$$
\frac{S}{B}=\frac1q.
$$
This ratio compares two specified directional models; it is not the <probability> that the event came from the source. Such a <conditional probability> would additionally require prior source odds and the other selection information. For fixed source fraction $f$ among selected events, a normalized <mixture distribution> has density $(1-f)B+fS$. For independent events the <likelihood> is the product of those densities; its ratio to background alone is
$$
\Lambda(f)=\prod_i\left[1+f\left(\frac{S_i}{B_i}-1\right)\right].
$$
Outside-cap events contribute factors $1-f$ rather than being silently discarded. At $f=0$, $\Lambda=1$; for a full-sky cap, $S=B$ and every $f$ gives $\Lambda=1$. A directional model that cannot distinguish source from background carries no directional evidence.
\Image[/neutrino-angular-background-coincidence.png]
{title=Chosen angular acceptance, directional likelihood and repeated background opportunities}
{description=Original exact spherical-cap calculation and uniform-cap signal model. Radii and event counts are chosen illustrations, not IceCube angular resolutions or discovery data. The fixed source direction and independent uniform background assumptions are essential.}
As an original numerical example, radii $1^\circ$ and $10^\circ$ give $q\simeq7.615\times10^{-5}$ and $7.596\times10^{-3}$, respectively: nearly a hundredfold difference. For a fixed number $N$ of independent background events, the <binomial distribution> gives
$$
P(\text{at least one inside})=1-(1-q)^N.
$$
With chosen $N=100$, those radii give approximately $0.00759$ and $0.5335$. A small single-event acceptance can become a common coincidence after many opportunities. This <probability> concerns a fixed cap, rather than a scan for the most favorable cap anywhere. For $m$ independent searches with common local tail <probability> $p$, the corresponding global tail <probability> is $1-(1-p)^m$; overlapping sky searches generally violate that independence. Actual source tests must include directional response, nonuniform exposure, <energy> information and the <look-elsewhere effect> in their null calibration.
Back to article page
