Muon events have been generated with the neutrino simulation package nusim [133, 134]. Neutrinos, sampled from an Eν-1 spectrum, are propagated through the earth, taking full account of the earth's density profile. Neutral current regeneration of the neutrinos is treated exactly using a recursive propagation algorithm. Simulation of a generic Eν-1 spectrum allows for the final events to be re-weighted to any desired neutrino source spectrum, whether it be an atmospheric neutrino spectrum (Eν-3.7), a generic Eν-2 spectrum for AGN neutrinos, or any other spectrum predicted in the literature.
The absorption of neutrinos in the Earth is an important effect, see fig. 14. For energies below a few tens of TeV, the full lower hemisphere is visible, whereas at energies above a few PeV the angular acceptance is reduced to zenith angles smaller than 30 degrees below horizon.
Neutrinos have been generated up to a maximum energy of 100 PeV which was set by technical limitations. Eight million events from downgoing muons which constitute the main source of background have been simulated and are used to demonstrate efficient background rejection. Muons were propagated with full simulation of stochastic energy loss. The detector simulation was performed with the AMANDA simulation package amasim. Ice inhomogenities have been neglected for this study but will be included in the future. We simulated PMTs of 10 in diameter but did not take into account the possible increase in light collection efficiency due to a proposed wavelength shifter coating applied to the outer OM surfaces. The noise of the PMTs has been conservatively assumed to be 500 Hz (i.e. higher than the design goal of 300 Hz). A trigger was defined by the condition of at least 5 hits in a local coincidence. A local coincidence is defined with respect to the four nearest neighbors, and for a time window of 1 μs. Signal Monte Carlo muon events are shown in figs. 15 and 16, where the muon energies are 10 TeV and 6 PeV, respectively.
The reconstruction was done in several steps using the AMANDA reconstruction package recoos. The start values for the full fit are obtained from two simple fast approximations called line-fit and dipole-fit (see [151] for details). As a first reduction step with respect to downgoing muons, relatively weak cuts on the zenith angle obtained from these fast approximations have been applied. These cuts have passing rates of 0.91, 0.78 and 0.04 with respect to the sample from AGN neutrinos, atmospheric neutrinos and downgoing muons, respectively.
Next, a full likelihood fit was performed (see [150] and [151]). This fit yields a series of parameters which can be used for selecting events with high quality reconstruction and for further reduction of background. The cuts chosen are the following:
Note that the standard cuts listed above result in a passing rate of 53% for AGN neutrinos and reduce the background from downgoing muons by 3 · 10-6. As shown in fig. 17, this background is concentrated close to the horizon. It can be easily rejected by a further zenith cut (if lower energies should be accepted), or by cuts on the number of channels (if one focuses to the separation of higher energies). Figure 17 also shows the background from uncorrelated coincident air showers (bottom right) which can produce two hit clusters, and earlier one at the bottom and a later one at the top of the array. Such events are not rejected by a simple angular cut, giving the relatively high passing rate after angular cuts compared to muons from single air showers. However, these events are easily identified by the subsequent cuts, in particular the cut on the smoothness.
Results are shown for the "standard configuration" (triangular pattern, 16 m vertical spacing2, 125 m interstring distance). Numbers at the top corner give the number of events and the passing rates. For AGN neutrinos, a flux of 2 × 10-7 · E-2 GeV-1 cm-2 s-1 sr-1 has been assumed, resulting in 4265 events.3
We have investigated the physics performance of the IceCube detector for different configurations, starting from the default configuration and going into the direction of both smaller and larger spacings [139]. Early results on similar IceCube simulations have been presented in [124, 125]. Whereas in [139] only configurations with about 5000 OMs and equidistant spacing are considered [124, 125] cover also options with twice or half the number of OMs, and nested configurations.
The basic pattern is shown in fig. 18. Apart from the triangular pattern, we also simulated rectangular patterns, which yield similar results. The circle in fig. 18 indicates an area of one square kilometer. The configuration includes 80 strings and 60 OMs per string. For a vertical spacing of 16 m, this results in an instrumented length of 944 m. The top layer of OMs was assumed to be at a depth of 1400 m. Table 2 summarizes the studied configurations.
The two following tables give the number of triggered and accepted events per year for the various configurations. Table 3 gives results for atmospheric neutrinos. Numbers in table 4 are for AGN neutrinos, again assuming a flux of 2 × 10-7·E-2 GeV-1 cm-2 s-1 sr-1. The upper numbers refer to triggered events, the lower numbers (in italics) to events after reconstruction and quality cuts.
| vertical/horizontal | 100m | 125m | 150m | 175m |
|---|---|---|---|---|
| 12m | X | X | X | - |
| 16m | X | X | X | - |
| 20m | X | X | X | X |
Table 2: Simulated configurations. Columns indicate different string spacings, rows different spacings of OMs along a string. An "X" indicates that the particular configuration was simulated.
| vertical/horizontal | 100 m | 125 m | 150 m | 175 m |
|---|---|---|---|---|
| 12 m | 2.9 k 1.5 k | 3.6 k 1.8 k | 4.3 k 2.2 k | - - |
| 16 m | 3.4 k 1.9 k | 4.3 k 2.3 k | 5.5 k 2.5 k | - - |
| 20 m | 4.0 k 2.1 k | 4.9 k 2.4 k | 5.7 k 2.6 k | 6.5 k 2.8 k |
Table 4: Number of triggered and accepted neutrino events per year (in thousands) assuming a flux of 2 × 10-7 · E-2 GeV-1 cm-2 s-1 sr-1. The upper numbers refer to triggered events, the lower numbers in italics to the events after quality cuts.
| vertical/horizontal | 100 m | 125 m | 150 m | 175 m |
|---|---|---|---|---|
| 12 m | 600 k 140 k | 620 k 130 k | 640 k 120 k | - - |
| 16 m | 590 k 140 k | 600 k 130 k | 620 k 110 k | - - |
| 20 m | 580 k 140 k | 590 k 110 k | 600 k 90 k | 560 k 60 k |
Table 3: Number of triggered and accepted events per year for atmospheric neutrinos (in thousands). The upper numbers refer to triggered events, the lower numbers in italics to the events after quality cuts.
Clearly, larger spacing is preferable for the AGN case but tends to suppress more atmospheric neutrinos after quality cuts. Going from 125 m string spacing to 150 m gives a 10% increase in accepted AGN events. The rectangular configuration gives a 5% increase in events rate. Decreasing the spacing to 12 m/100 m gives 30% loss in AGN neutrinos, but a slight increase for atmospheric neutrinos. The angular resolution for all configurations is ≤ 1 degree.
We calculate the sensitivity of the detector to diffuse fluxes of neutrinos with a generic E-2 spectrum, expected for sources such as active galactic nuclei (AGN). For the purposes of these calculations we assume a source strength for muon-neutrinos and antineutrinos of a level E2dN/dE = 10-7GeV cm-2 s-1 sr-1. We use the Model Rejection Potential formalism [132, 135], and the associated "model rejection factor" (MRF), to optimize the limit analysis. Figure 19 shows the results of the minimization of the MRF for the hit multiplicity (Nch) cut and an exposure time of three years. The top left figure shows the differential numbers of events expected for both the E-2 spectrum (blue solid line) and for the atmospheric neutrinos (red dashed line). The top right plot shows the same quantities as an integral distribution. Also shown is the Feldman-Cousins 90% confidence level average upper limit. As the background falls toward zero, this classical average upper limit converges toward a value of 2.44. The best limit is obtained where the model rejection factor, (μ(nb)/ns), is minimised (bottom left plot).This occurs for a cut of 168 channels, where 120.9 signal events would be expected upon a background of 17.2. The average upper limit for this background is 8.4, leading to a MRF = 6.9 × 10-2 ,and therefore to a flux limit of E2dN/dE = 6.9 × 10-7GeV cm-2 s-1 sr-1. The bottom right plot shows the parent neutrino energy distribution for those events with multiplicity above the optimal cut of 168 channels. The multiplicity cut corresponds to a minimum energy of about 30 TeV, and to a typical neutrino energy of 1 PeV. Table 5 summarizes the results of the MRF optimization for exposure times of one and three years.
| time (yrs) | Nch cut | S(≥ Nch) | B(≥ Nch) ν(B(≥ Nch)) | limit (= E-2dN/dE×ν/S)) | |
|---|---|---|---|---|---|
| 1 | 175 | 53.1 | 5.4 | 5.3 | 1.0 × 10-8 |
| 221 | 93.1 | 3.2 | 4.5 | 4.8 × 10-9 |
Table 5: IceCube model rejection factor for an E2dN/dE = 10-7cm-2s-1sr-1GeV2 flux as a function of exposure time.
The sensitivity of IceCube for point sources of neutrinos has been assessed. Since the background may be greatly reduced by cutting to a small angular region about the direction of the point source, the multiplicity cut (Nch) can be relaxed. The sensitivity has been checked in two ways, first by averaging the sensitivity over the upward going hemisphere, and second by looking at a point source from a specific zenith angle.
For each exposure time tested, we optimise in both angular bin size and Nch. Figure 20 shows the pointing accuracy of the array. Ninety percent of events reconstruct to within 4° of the true direction, and 60% to within 1°. The MRF results from varying angular bin and exposure time are shown in table 6. These show that a signal strength of the level dN/dE = 10-7cm-2s-1sr-1GeV2 would produce a very significant signal in IceCube, and in its absence, very constrained upper limits would be obtained. For both exposure times, the best choice of angular cut is about 1ffi, but the limit is ultimately not very sensitive to the choice of angular cut, suggesting that a smaller cut, to enhance the significance of a possible observation, may be warranted. Although the cuts have been optimised to minimise the MRF, thereby optimising the limit setting potential of the detector, we can make some basic estimates of the level of a point source flux needed to give a significant detection of a signal in the exposure time. A chance probability of ∼ 10-7 for a background fluctuation corresponds roughly to a "5 sigma" observation. For any quoted background level, we can calculate the number of events needed to get this chance probability. For the case of 1 year exposure with a 1° angular bin, the expected background is 0.53 events. If 8 events were observed, this would give the required chance probability of ∼ 10-7. A signal strength of 8 - 0.5 = 7.5 events is about 1/5 of the number of events expected from the dN/dE = 10-7cm-2s-1sr-1GeV2 flux, therefore a point source of level dN/dE ∼ 2 × 10-8cm-2s-1sr-1GeV2 would be required to produce a "5 sigma" observation in one year of live time. Of course, this analysis assumes that we only look at a single candidate source. If we do a full sky search, we must pay a penalty for statistical trials. There are about 6500 bins of 1° radius in a half sky (upgoing neutrino) search, so a significance of ∼ 10-9 would be needed for an unknown point source to give a ∼ 10-7 result after the trials are accounted for. This requires that about 10 events are observed (the chance probability to observe 10 or more events from a background of 0.53 is 3 × 10-10), increasing the required signal strength by about 30%.
Table 7 shows the MRF results for a point source at a zenith angle of 130° The limit setting potential is slightly less than the average seen in table 6. Figure 21 shows the model rejection potential optimization for a point source.
Recently, Waxman and Bahcall [94] have proposed that gamma-ray bursts (GRBs) might be sources of neutrinos. The search for neutrinos from GRBs is simplified over that of a point source, due to the time stamp available from satellite observations of the gamma rays from the burst. In the Waxman-Bahcall model the neutrinos are expected to arrive within approximately
| t(y) | ΔΨ | Nch cut | S(≥ Nch) | B(≥ Nch) | μ(B(≥ Nch)) | limit (= E-2dN/dE μ/S) |
|---|---|---|---|---|---|---|
| 1 | 0.5 | 25.5 | 43.1 | 0.53 | 2.9 | 6.7 × 10-9 |
| 1 | 1.0 | 32.5 | 60.6 | 0.91 | 3.2 | 5.3 × 10-9 |
| 1 | 2.0 | 42.5 | 55.6 | 1.10 | 3.4 | 6.0 × 10-9 |
| 1 | 5.0 | 55.5 | 43.9 | 2.20 | 4.0 | 9.0 × 10-9 |
| 3 | 0.5 | 32.5 | 110.2 | 0.67 | 3.0 | 2.7 × 10-9 |
| 3 | 1.0 | 42.5 | 137.6 | 0.82 | 3.1 | 2.3 × 10-9 |
| 3 | 2.0 | 53.5 | 128.8 | 1.25 | 3.5 | 2.7 × 10-9 |
Table 6: IceCube model rejection factor for a point source, averaged over zenith angles 90-180° with flux E2dN/dE = 10-7cm-2s-1GeV2. The optimum choice of cuts for both exposure times are highlighted in bold.
| time (yrs) | ΔΨ | Nch cut | S(≥ Nch) | B(≥ Nch) | μ(B(≥ Nch)) | limit (= E-2dN/dE × μ/S) |
|---|---|---|---|---|---|---|
| 1 | 0.5 | 24.5 | 36.6 | 0.71 | 3.0 | 8.2×10-9 |
| 1 | 1.0 | 33.5 | 47.7 | 1.0 | 3.3 | 6.9×10-9 |
| 1 | 2.0 | 41.5 | 45.8 | 2.2 | 4.0 | 8.8×10-9 |
| 1 | 5.0 | 64.5 | 25.6 | 1.6 | 3.7 | 1.4×10-8 |
| 3 | 0.5 | 29.5 | 98.0 | 1.2 | 3.4 | 3.5×10-9 |
| 3 | 1.0 | 33.5 | 143.2 | 3.0 | 4.4 | 3.1×10-9 |
| 3 | 2.0 | 42.5 | 132.8 | 5.8 | 5.5 | 4.1×10-9 |
Table 7: IceCube model rejection factor for an point source at 130° zenith angle, with flux E2dN/dE = 10-7cm-2s-1GeV2.
10 seconds of the gamma rays. The IceCube detector will reconstruct neutrino-induced muons from GRBs to less than ten degrees of the true direction (fig. 22).
The Waxman-Bahcall flux corresponds to the expected neutrino flux from 1000 GRBs. Within a 10° angular search bin and 10 second time window, we would expect a total of 15 upgoing muon events from 1000 GRBs after quality cuts were applied to remove the most obvious mis-reconstructed downgoing muons. The atmospheric neutrino background is negligible (total 0.32 events) but about 500 misreconstructed downgoing muons would remain. Tightening the cuts to the final cut set discussed previously removes the last of the muon background, reduces the atmospheric neutrino background to 0.23 events, and retains 12 GRB induced neutrino events. It should be noted that more Monte-Carlo simulations are needed to further check the belief that the background of mis-reconstructed downgoing muons is indeed zero.
Optimising the Nch cut leads to a cut of Nch = 12.5 and model rejection factor of 0.22, a severe constraint on the Waxman-Bahcall model. Since the GRB search is essentially background free, the improvement in the limit goes nearly linearly with the increasing number of GRBs searched. This behaviour is shown in table 8. It is interesting that the observation of only 210 GRBs is enough to rule out the Waxman-Bahcall model at 90% classical confidence.
Since the angular cut and time window search can be made so tight, the GRB searches are close to background-free. Then the observation of even a few events would be very significant. For 300 GRBs searched, and a 2°, window, the Waxman-Bahcall model predicts about 3 events, on an atmospheric neutrino background of 0.003. The chance probability of observing 3 or greater events given a background of 0.003 is about 5 × 10-9.
Table 8 shows that the MRF is insensitive to the choice of angular bin size. In this case, a choice of a smaller search window would be preferred, in order to reduce the background (for limited signal loss) and give a more significant observation of a potential signal. Ultimately however, the uncertainty in the knowledge of the GRB direction (from the satellite observation) will limit how small the angular search bin can be made. Figure 23 shows the model rejection potential optimization for GRBs.
| nbursts | ΔΨ | Nch cut | S(≥ Nch) | B(≥ Nch) | μ(B(≥ Nch)) | limit (= E-2dN/dE × μ/S) |
|---|---|---|---|---|---|---|
| 100 | 2.0 | 1.5 | 0.98 | 0.001 | 2.4 | 2.49 |
| 5.0 | 1.5 | 1.13 | 0.005 | 2.4 | 2.16 | |
| 10.0 | 1.5 | 1.19 | 0.02 | 2.46 | 2.06 | |
| 20.0 | 1.5 | 1.20 | 0.086 | 2.51 | 2.07 | |
| 30.0 | 12.5 | 1.22 | 0.19 | 2.60 | 2.12 | |
| 210 | 10 | 1.5 | 2.50 | 0.045 | 2.49 | 0.99 |
| 300 | 2.0 | 1.5 | 2.94 | 0.003 | 2.44 | 0.829 |
| 5.0 | 1.5 | 3.40 | 0.016 | 2.45 | 0.721 | |
| 10.0 | 1.5 | 3.51 | 0.065 | 2.46 | 0.70 | |
| 20.0 | 1.5 | 3.66 | 0.26 | 2.66 | 0.72 | |
| 1000 | 2.0 | 1.5 | 9.81 | 0.009 | 2.45 | 0.249 |
| 5.0 | 1.5 | 11.34 | 0.05 | 2.48 | 0.219 | |
| 10.0 | 12.5 | 11.9 | 0.215 | 2.62 | 0.22 | |
| 20.0 | 20.5 | 11.3 | 0.471 | 2.83 | 0.25 |
Table 8: IceCube model rejection factor for a Gamma-ray burst search.
The sensitivities given above could change in reality. We estimate that uncertainties in ice modelling might change them by up to 50%–in either direction. Use of an energy estimator
better than Nch may result in an improvement of up to 30%. Use of wavelength shifter could enhance the sensitivity of the OMs by up to 35% and result in higher signal efficiency and improved background rejection. The application of reconstruction methods which make use of the full waveform instead of using only the time of the first arriving photon will lead to a better angular reconstruction, in particular for high-energy events (see Section 5.6). We hope to reach an accuracy of about 0.5 degrees for moderate cuts, resulting in a slightly better point source sensitivity. Neutrinos with >100 PeV have not been taken into account in the present simulation. Their inclusion will also result in a better limit. Other lines of improvement are selection of cuts with a better passing rate for for AGN neutrinos (giving possibly a 10–15% effect), and a change of the configuration to larger spacing, which is discussed in section 5.8.
