Spatial distribution of incident neutrinos

Let's start with analysing the distribution of the neutrino incident angles in the training data. First, what what we might expect to see. The simplest answer is symmetric, that is we might expect that the neutrinos come in equal numbers from all directions. Let's check if the actual distribution fits that model.

Here is the histogram of the azimuth angle ϕ across all batches. It's indeed an almost uniform distribution in [0,2π], except for the small dips - and these dips correspond nicely with the 6 directions that connect the neighboring DOMs. What we see here is that the azimuth distribution bears an imprint of the hexagonal detector geometry, varying DOM efficiency, and also probably effects of the ice in holes having different transparency compared to the bulk of the detector.

Now about the zenith angle. First of all, it is not the azimuth angle that we expect uniformly distributed. Why? Every unit of area of the sphere surrounding the detectors should contain the same number of neutrinos. We know that the area in spherical coordinates is given by $$ dS = \sin\theta d\theta d\phi = -d(\cos\theta)d\phi $$ That is, we need \( \cos\theta \) to be uniformly distributed, so that all directions are equal; let's plot the histogram of the training data. This looks very interesting, with rather prominent maximums at both poles (for \( \theta=0 \) and \( \theta=\pi \)), and a less pronounced one at the equator (for \( \theta=\pi/2 \)). Two factors are mostly in play here: first, due to the DOM specifics the IceCube has very good sensitivity at the horizon (\(\cos\theta=0\)), and second, for the neutrinos coming from below the absorption effect by the Earth becomes noticeable at the high end of the energy spectrum (above a few TeV).

So, we discover that azimuth and zenith angles are distributed close to uniformly, but with some non-negligible deviations. These deviations are the result of accurate modelling of the "physics" of events. The neutrino events are simulated, and the simulation starts with the isotropic distribution and then undergoes several steps of processing, which model various phenomena affecting the neutrino flow (photon propagation through the ice, simulation of photon multipliers angular sensitivity, ice features dependency on depth and direction, simulation of detector trigger logic). The end result is that the acceptance of the detector is not uniform.

Reconstruction anisotropy along the z-axis

As we've just seen, the neutrino events dataset has a fairly isotropic distribution of track directions. Noise is added on isotropically distributed neutrino tracks, for example, in the form of cosmic ray muon tracks. This noise does not seem to correlate with the direction of the target track.

Therefore, it was natural to expect that the model errors would be weakly dependent on the target angles. However, it turns out that this dependence is not negligible. Below are the average error values for various values of the zenith and azimuth angles (the cosine is taken from the zenith angle to obtain a uniform distribution). As you can see, there is a significant monotonic drop in the error when going from zenith=π (particle flies from below) to zenith=0 (particle flies from above). Zero angle roughly coinsides to the total error over all angles (the horizontal line):

This observation forced us to conduct a more detailed analysis of the data in terms of the value of the target zenith angle (its cosine). This analysis shows significant data anisotropy (see the notebook). For example, below are the average values of the number of active strings, triggered sensors and pulses at various values of the zenith angle. Each point has error bars. They are obtained by dividing the standard deviation from the mean by the square root of the number of events with a given angle. The horizontal line is the average value of the value for all angles. The dotted line is the probability density of the number of events as a function of the zenith:

The number of active strings, sensors and pulses is much larger for tracks coming from above. It is clear that the more pulses in the event, the easier it is usually for the model to reconstruct the track. This explains the behavior of the model error, but not the reasons leading to such anisotropy.

Note that "noise" (background) muons enter the IceCube only from above (the Earth screens them completely). It is clear that if background tracks of muons (also coming from above) are superimposed on the track from a neutrino coming from above, then it is easier for the model to reconstruct the track than with counter flows. But it also doesn't explain why there should be more pulses on top tracks.

Most of the events present in the dataset are neutrinos produced by the collision of cosmic rays with the Earth's atmosphere. These atmospheric neutrinos have an isotropic direction. However, in reality the Earth is not completely transparent to neutrinos (and fortunately for the IceCube). Below is the number of events from atmospheric neutrinos depending on the cosine of the zenith angle [source]:

As can be seen, there are significantly fewer atmospheric neutrinos arriving along the Earth's axis than those arriving at an angle. Apparently, it is this effect that leads to the non-uniformity of the zenith angle, noted above (and in this message; see also comment Philipp Eller - @pellerphys). In the diagrams of the distribution density of the cosine of the zenith angle, given above (dashed line), the difference between the minimum and maximum is almost 30%. However, the number of events with a given angle does not affect the average values obtained at a given angle.

The only reasonable explanation that we see so far is as follows. The cross section of the neutrino interaction reaction increases quite strongly with its energy. This means that the Earth absorbs high-energy neutrinos more strongly. Therefore, on average, neutrinos with lower energy enter the IceCube from below, and atmospheric neutrinos arriving from above have, on average, higher energy. Perhaps this is what increases the number of pulses at a small azimuth, since more leptons with high energy are born, which generate more powerful Cherenkov radiation.

Let's note some other factors that make space anisotropic in the vertical direction. They are unlikely to explain the observed effect, but may be useful for understanding the general background of the problem.

  1. The design of the sensors is such that they are much more sensitive to photons coming from below. However, this should rather increase the number of DOMs hits for tracks coming from the bottom, rather than from the top.
  2. The ice inside the IceCube changes its optical properties along the z axis. In particular, at a depth of 2150 m (\(z\sim -100\)m) there is a layer of dust that significantly enhances the scattering and absorption of photons [source]:
  3. Symmetry with respect to \(z\mapsto -z\) is also violated by sensors from DeepCore strings. They are divided into two unequal groups (see the figure above). However, the exclusion of pulses on these strings from events does not qualitatively change the dependence of quantities on z.
  4. The distribution of atmospheric neutrinos towards the horizon has a maximum, because tangential (that is, horizontal) rays tangentially pass through a thicker layer of the atmosphere.
  5. The earth's magnetic field affects cosmic rays and also disrupts the effective isotropy of space.

Below we present a few more graphs for various event characteristics. The logarithm of the total charge, the average time (after the countdown shifted to zero) and the proportion of auxiliary pulses in the event (auxiliary=True):

Average values of event pulse coordinates (in kilometers):
Standard deviations in kilometers (track compactness):