Calculation of Average Flux Limits

The previous section concluded that there are no point sources. The next step is to calculate the neutrino and muon flux limits from a hypothetical point source having an energy spectra given by $E^{-\gamma}$, where $\gamma$ is the spectral index. We present an angular-dependent flux limit and investigate candidate point sources of special interest.

The number of events ($N_{\mu}$) expected from a source with a differential flux ( $\frac{d\Phi_{\nu}}{dE_{\nu}}$) is

$$N_{\mu} = [ \int_{E_{\nu}^{min}}^{E_{\nu}^{max}} A_{eff}^{\nu... ...{\nu} ) \frac{d\Phi_{\nu}}{dE_{\nu}} dE_{\nu} ] \cdot T_{live}$$ (9)

where $A_{eff}^{\nu}$ is the neutrino effective area, determined by

$$A_{eff}^{\nu} = N_{A} \rho_{ice} \sigma_{\nu \mu} (E_{\nu} ) ... ...u} ) \langle R(E_{\nu} ;E_{\mu}^{min} ) \rangle A_{GEN} P_{ev}$$ (10)

where P_ev<> is determined by simulation. The variables, $E_{\nu}^{max}$ and $E_{\nu}^{min}$, are the maximum and minimum neutrino energy, T_live<>, the live time of the experimental data, N_A<>, Avogadro's number, $\rho_{ice}$, the density of ice, $\sigma_{\nu \mu}$, the charge current cross-section for a neutrino-induced muon (ore anti-neutrino where appropriate), $S(E_{\nu})$, the neutrino absorption in the earth, $\langle R(E_{\nu} ;E_{\mu}^{min} ) \rangle$, the average muon range, A_GEN<>, the generation plane area used in the Monte Carlo, and P_ev<>, the fraction of the generated events that in the data set. Hence the neutrino effective area folds in both the detector response and the probability of a neutrino interaction to produce a detectable muon. The neutrino effective area is readily determined because neutrinos originate from a well-defined generation plane. The calculation of the muon effective area is a bit more involved since the muon vertices are distributed in a volume. This is discussed in more detail later in this section.

The flux limit is computed from $N_{\mu}$ according to

$$\phi_{\nu}^{limit} (E_{\nu} > E_{\nu}^{min} , \frac{d\Phi_{\n... ...}^{min}}^{E_{\nu}^{max}} \frac{d\Phi_{\nu}}{dE_{\nu}} dE_{\nu}$$ (11)

where $\mu_{s} (N_{0} , N_{b})$ is the upper limit (90% CL) of the mean of the signal for the number of data events (N₀<>) observed and the given mean of background events (N_b<>) in the source bin, and $\epsilon$ account for the finite angular resolution and the non-central location of a potential source with the bin. The value of $\mu (N_{0} , N_{b})$ is determined by using the number of data events (N₀<>) observed in a bin and assume is equal to the number expected from background (N_b<>). A more detailed discussion of $\mu_{s} (N_{0} , N_{b})$ is in section 3.1. The average neutrino effective area (assuming $\frac{d\Phi_{\nu}}{dE_{\nu}} \propto E^{-\gamma}$) can be use to rewrite equation 11 in a more convenient form:

$$\phi_{\nu}^{limit} (E_{\nu} > E_{\nu}^{min} , \frac{d\Phi_{\n... ...b} ) }{T_{live} \cdot \epsilon \cdot \overline{A}_{eff}^{\nu}}$$ (12)

The average neutrino effective area ( $\overline{A}_{eff}^{\nu}$) is calculated as

$$\overline{A}_{eff}^{\nu} = \frac{A_{GEN}}{N_{GEN}} \cdot \ln ... ...(intwght)_{i} \cdot (E_{\nu})_{i} \cdot (E_{\nu}^{\gamma})_{i}$$ (13)

where A_GEN<> is the generation plane area, N_GEN<> is the number of events generated, $E_{\nu}^{max}$ and $E_{\nu}^{min}$ are the maximum and minimum neutrino energy generated, N_det<> is the number of events survive the analysis, intwght is the probability for a neutrino interaction and a detectable muon, and $E_{\nu}$ is the energy of the neutrino. NUSIM is used to generate the up-going neutrino events. Each vertex (location of neutrino and nucleon interaction) is randomly placed in a volume defined by the maximum range of the induced muon. Each event then can trigger the detector, yet is assigned a probability, intwght, for such an occurrence. This probability, intwght, is defined as $\sigma_{\nu \mu}(E_{\nu}) \cdot S(E_{\nu}) \cdot R(E_{\nu} ;E_{\mu}^{min} )$. The same terms used in equation 10. Since the Monte Carlo distributes the neutrino energies with an E^-1<> spectra, a rescaling is needed in the weighting as shown in equation 13.

The average neutrino effective area was calculated assuming different spectra indices of $\gamma$=2, 2.3, 2.5, 2.7, and 3. Figure 22 shows that there is a strong dependence on the assumed value of the spectra index. The softer spectra ( $\gamma \sim 3$ ) produce smaller average neutrino effective areas. The reason for this will be discussed a little later. These results are then used to calculate the neutrino flux limit above 1 TeV.

  

Figure 22: (Left) The average neutrino effective area for neutrinos above 1 TeV, with the specified power law spectra. (Right) The average muon effective area for neutrino energies above 1 TeV.

The average muon effective area was also calculated, which is more involved because the muon vertices are distributed within a volume near the detector. Since the signal Monte Carlo assumes that the muon direction is the same as the neutrino direction, the area of the neutrino generation plane can also apply for the generated muons. It is important to include only the events in which the muons reach the detector (ie. are detectable). A fortran routine was written to select only muons that reach the geometric volume (assumed to be a cylinder 60m in radius and 480m in height centered on the detector). The calculation for the muon effective area is given by

$$\overline{A}_{eff}^{\mu}= \frac{A_{GEN} \cdot \sum _{i=1}^{N_... ...ntwght)_{i} \cdot (E_{\nu})_{i} \cdot (E_{\nu}^{-\gamma})_{i}}$$ (14)

Equation 14 is normalized by summing over all detectable generated events (i.e., muons which reach the geometric volume are detectable) weighted by the interaction probability times the assumed power law spectra of the source. The right plot in figure 22 show the effective area assuming different power law spectra. The effective area has been averaged over the azimuth angle.

The right plot in figure 22 illustrates that the average muon effective area decreases for softer energy spectra, yet not nearly as dramatically as the neutrino effective area. To understand why, a simulation which included lower energy neutrinos (10 GeV to 10^5.5<> GeV) was also processed. Initially, this analysis only needed to study high energy neutrino signal ( $E_{\nu}> 1$ TeV) because our interest was in hard energy spectra (E^-2<>). Even though the analysis was optimized for E^-2<>, there is significant sensitivity to softer spectra (figure 22). Hence, simulations were initiated that included lower energy neutrinos to investigate this behavior.

  

Figure 23: neutrino and muon energy distribution for the high energy MC signal ($E_{\nu}> 1$ TeV) and the low energy MC signal ( $E_{\nu}>10$ GeV). Each plot show distributions for a different power law spectra (ie $\gamma=$ 2, 2.3, 2.7, and 3).

Figure 23 shows the neutrino and muon energy distributions of the lower threshold ( $E_{\nu}>10$ GeV) and higher threshold ($E_{\nu}> 1$ TeV) energy signal. For softer spectra two observations are made. The peak of the neutrino and muon energy distributions shift to lower energies. Since the neutrino and muon effective area decrease with energy (figure 35), the average neutrino and muon effective area will decrease for softer spectra. Also for softer energy spectra, the muon energy distribution is truncated at lower energies (<<> 1 TeV) for the 1 TeV neutrino energy threshold. Hence the average effective area calculated from the high energy signal applies only for muon energies above 1 TeV, while the low threshold simulation would apply for muon energies down to threshold of the detector ($\sim$ 10 to 50 GeV). Note the average effective area for the energy spectra of E^-2<> appears to be unaffected by the 1 TeV cut off.

  

Figure 24: (Left) The average neutrino effective area for neutrinos above 10 GeV, with the specified power law spectra. (Right) The average muon effective area for neutrino energies above 10 GeV.

 

Table 4: Efficiency ($\epsilon$) for different energy spectra and angular bin size.

Energy Spectra Index	18x36 Binning	24x48 Binning	36x72 Binning
2	.589	.499	.378
2.3	.610	.518	.398
2.5	.621	.528	.408
2.7	.632	.536	.417
3	.645	.543	.428

For softer energy spectra, the previous conclusion is no longer valid, so the average neutrino effective area was computed for energy threshold of 10 GeV (figure 24). The average neutrino effective area decreases by a factor of 100 relative to the results for $E_{\nu}> 1$ TeV. This is the result of the interaction cross-section decreasing for smaller neutrino energies combined with the finite muon range from ionization losses. Also, the lower energy muons produce the lowest number of Cherenkov photons per unit length The average muon effective area significantly decreases with a softer energy spectra, as expected.

This efficiency ($\epsilon$) term accounts for the possibility that the center of a fixed tiling may not coincide with the source location. It also accounts for signal events which fall outside the bin due to finite angular resolution. The bin size is selected to optimize signal to noise. The binning that has been tried are 18 by 36 (i.e.. 18 zenith bins by a maximum of 36 azimuth bins for the entire sky), 24 by 48, and 36 by 72. With these different binnings, the efficiency was calculated using the Monte Carlo by determining the fraction of reconstructed tracks that fall within the same bin as the true track. The results for $\epsilon$ are shown in table 4 for each of the power law spectra. The differences for each energy spectra are due to the energy dependence of the space angle error (Figure 25). Somewhat unexpectedly, error in the space angle increases with neutrino and muon energy. The higher neutrino energy will on average produce higher energy muons, which will in turn have more cascade like interactions (such as bremsstrahlung, and electron-positron production, etc) per path length of the muon. These cascade-like processes produce bright spherically symmetric bursts of light, which are not accounted for in the reconstruction. Thus at higher energies the space angle error is larger. Of course, harder spectra produce a larger fraction of higher energy neutrinos, thus the efficiency ($\epsilon$) is lower.

The live-time used for the calculation is 138.2 days. The details are shown in appendix B.

    

Figure 25: (Left) The space angle error versus muon energy and neutrino energy.
Figure 26: (Right) Comparison of flux limits from the entire 1997 experimental data set shown in this report, to the results presented by John Kim at the ICRC (August, 1999) [7].