There are a few spectral fits of the astrophysical neutrino flux that are based on different event selections in the IceCube detector. An overview of these fits in which the spectrum is modeled as a single power-law dφν/dEν ∼ Eν−γ is given in Table6.4. It illustrates the large uncertainties of the spectral shape, as the spectral index γ varies from 2.1 to 2.9. Not all listed analyses are statistically independent, and some are sensitive to different energy ranges.4 A softer spectrum
is fitted by analyses with a low energy threshold around ∼ 20 TeV and a harder one above ∼ 200 TeV. For different years of the high-energy starting event selection, the spectral index is fitted increasingly soft over time, although the uncertainties are large and measurement results consistent. A spectral cutoff was studied in the global fit, where multiple data samples were combined. It was found that there was a slight (but insignificant) preference to a single power- law flux with cutoff [48]. A dedicated search for a spectral cutoff is under development [246]. More complex spectral shapes such as a dual power-law have been recently investigated without a clear indication of whether the observed data is better explained than by a single power-law flux model [4]. With the currently available data, the shape of the astrophysical neutrino spectrum is not well known. Although there are hints that a single power-law flux model might not be the final answer, it is still the model with the fewest assumptions that is able to consistently explain the observed data within uncertainties.
4
They are also sensitive to different declination bands, e.g. the through-going track analysis is only sensitive to the Northern Hemisphere. This will become important if a Galactic neutrino component is detected.
Diffuse Astrophysicalν Analysis Ethrν [TeV] γ φν [GeV−1cm−2sr−1s−1] Ec [PeV]
2-year HESE [43] ∼ 60 2.2± 0.4 1.2± 0.4 – 3-year HESE [44] ∼ 60 2.3± 0.3 1.5± 0.5 – 4-year HESE [50] ∼ 60 2.6± 0.3 2.2± 0.7 – 6-year HESE [4] ∼ 60 2.9± 0.3 2.5± 0.8 – 2-year starting events [45] ∼ 25 2.5± 0.1 2.1± 0.4 – 4-year cascade [61] ∼ 20 2.5± 0.1 1.6± 0.2 – 6-year through-going tracks [49] ∼ 200 2.1± 0.1 0.9± 0.3 – Global fit without cutoff [48] ∼ 20 2.5± 0.1 2.3± 0.3 – Global fit with cutoff [48] ∼ 20 2.3± 0.1 2.7± 0.4 2.7+7.7−1.4 Table 6.4: Overview of different spectral fits of the diffuse astrophysical neutrino flux with the IceCube
detector. The tabulated values are the lower threshold neutrino energy Ethr
ν of the analysis,
the spectral index γ, the per-flavor normalization φνat Eν= 100 TeV, and the spectral cutoff
energy Ec of the astrophysical neutrino flux.
The sensitivity of this analysis to the astrophysical tau-neutrino flux is strongly dependent on the assumed spectral shape of the neutrino flux.5, which is due to the energy-dependent
identification efficiency. Tau-neutrino interactions at higher energies transfer more energy to the secondary tau, which increases its average propagation distance before it decays. Double cascade events with a longer decay length can be more efficiently identified. A softer spectrum induces more tau-neutrino interactions at lower energies for which they cannot be identified. A spectral cutoff causes a similar effect by reducing the number of identifiable tau-neutrino interactions at high energies. This number can also be significantly lower for a dual power-law flux model if the spectral break (see below) happens to be within the energy range where the sensitivity is maximal (c.f. Section 4.4). In summary, the ability to constrain the astrophysical tau-neutrino flux increases with any spectrum that has a large contribution at high energies. In the previous sections, the number of identifiable tau-neutrino interactions and the corre- sponding flux sensitivity of the likelihood fit were quoted for a benchmark astrophysical neutrino flux that is modeled as a single power-law spectrum. Systematic uncertainties were considered and incorporated, as discussed in Section 6.2.3. It became apparent that the uncertainties of the expected tau-neutrino signal, unlike the background expectation, are not dominated by an imprecise knowledge of the detector properties. Instead, the model uncertainties of the astro- physical neutrino spectrumare much more significant to the estimated tau-neutrino sensitivity. As the spectrum is a quantity to be measured, it is not treated as a systematic uncertainty. However, it is important to be aware of this dependence.
In the following, three different astrophysical neutrino flux models are considered in order to estimate their effect on the tau-neutrino sensitivity. These models are motivated by similar properties observed in the cosmic-ray flux (c.f. Chapter 2).
1. a single power-law spectrum is defined by dφν dEν = φν· ( Eν 100 TeV )−γ , (6.13)
with a spectral index γ and a flux normalization φν at Eν = 100 TeV. It is the standard astrophysical neutrino flux model with minimal assumptions and based on the correlation to the cosmic-ray flux. However, an E−2
ν spectrum that had been used as a benchmark model for many years was excluded at a significance of 4.6 σ [48].
5
The sensitivity also depends on the assumed astrophysical flavor composition. However, a variation is simply an energy-independent scaling of the number of identifiable tau-neutrino interactions.
2. a single power-law spectrum with a spectral cutoff is defined by dφν dEν = φν· ( Eν 100 TeV )−γ · exp ( −EEν c ) , (6.14)
with an additional cutoff energy Ec that is modeled with a soft exponential suppression at high energies. Note that this model is in contrast to a hard cutoff, for which exactly zero events would be expected above Ec. A spectral cutoff could be caused by a lack of sources that are capable of producing neutrinos at high energies.
3. a dual power-law spectrum with a spectral break is defined by dφν dEν = dφν1 dEν +dφν2 dEν = [ φν1· ( Eν 100 TeV )−γ1] + [ φν2· ( Eν 100 TeV )−γ2] , (6.15) which is a simple superposition of two independent single power-law spectra with spectral indices γ1 and γ2 and flux normalizations φν1 and φν2 at Eν = 100 TeV, respectively.
The experimentally relevant case in the context of this thesis is a soft component that is dominant in the low-energy region and a hard component that is dominant in the high- energy region. For this case, it follows that φν1 > φν2 and γ1 > γ2. A more convenient
way of parametrizing the dual power-law spectrum under these conditions is to replace the second normalization by a spectral break energy Eb. It is defined by
dφν1 dEν (Eb) = dφν2 dEν (Eb) =⇒ log ( Eb 100 TeV ) = log φν1− log φν2 γ1− γ2 , (6.16)
and denotes the neutrino energy above which the harder spectral component dφν2/dEν
becomes dominant over the softer dφν1/dEν. A dual power-law spectrum could be in-
dicative for different neutrino source populations such as a diffuse neutrino emission from galactic and extra-galactic sources.
Naturally, there are many other possible flux models. However, the focus is on testing the dependence of the tau-neutrino sensitivity on the shape of some representative astrophysical flux models and not on testing the models themselves.
Only one parameter of interest is varied for each model while all other parameters are fixed to a chosen value. For the single power-law model, the parameter of interest is γ, for the cutoff model it is Ecwith fixed γ, and for the dual-power law, the parameter of interest is Ebwith fixed γ1and γ2. Note that for every model, the respective flux normalization φν is rescaled to exactly predict the number of observed events that are expected from the benchmark flux model of φν(Eν) = 1.5· 10−18(Eν/100 TeV)−2.3GeV−1cm−2sr−1s−1. The change in sensitivity is thus only due to a different spectral shape and not due to a statistical variation of the total expected number of events. For the single power-law model, the spectral index is scanned in γ = 2 . . . 3 to cover a wide range of measured values (c.f. Table 6.4). The cutoff model is scanned in Ec = 500 TeV . . . 10 PeV while the spectral index is fixed to the benchmark value of γ = 2.3. The lower limit of the chosen cutoff energies is purely for illustrative purposes, as there is no indication for a sub-PeV cutoff [48]. For the dual power-law model the spectral break energy is scanned in Eb= 200 TeV . . . 5 PeVwhile the spectral indices are fixed to γ1= 2.9and γ2= 2.1. These choices are representative for a lower and upper bound of the measured spectral index as listed in Table 6.4. The value region of the spectral break energy covers a wide range, which is indicative for the lack of knowledge regarding if and where a break in the spectrum could be located. In a recently updated measurement which combines the result of the high-energy starting event sample with the result of a through-going track analysis, there are hints (albeit insignificant) that a spectral break energy could be of the order of a few hundred TeV [4].
101 102 103 104 105
Deposited Energy [TeV] 10−4 10−3 10−2 10−1 100 101 Events in 2078 Da ys γ = 2.0 γ = 2.3 γ = 2.6 γ = 3.0 101 102 103 Length [m] 10−4 10−3 10−2 10−1 100 101 Events in 2078 Da ys γ = 2.0 γ = 2.3 γ = 2.6 γ = 3.0
Spectral-Index Dependence of the Double Cascade Observable Distributions
101 102 103 104 105
Deposited Energy [TeV] 10−4 10−3 10−2 10−1 100 101 Events in 2078 Da ys Ec= 500 TeV Ec= 1 PeV Ec= 5 PeV Ec= 10 PeV Ec=∞ 101 102 103 Length [m] 10−4 10−3 10−2 10−1 100 101 Events in 2078 Da ys Ec= 500 TeV Ec= 1 PeV Ec= 5 PeV Ec= 10 PeV Ec=∞
Spectral-Cutoff Dependence of the Double Cascade Observable Distributions
101 102 103 104 105
Deposited Energy [TeV] 10−4 10−3 10−2 10−1 100 101 Events in 2078 Da ys Eb= 200 TeV Eb= 500 TeV Eb= 1 PeV Eb= 5 PeV Eb=∞ 101 102 103 Length [m] 10−4 10−3 10−2 10−1 100 101 Events in 2078 Da ys Eb= 200 TeV Eb= 500 TeV Eb= 1 PeV Eb= 5 PeV Eb=∞
Spectral-Break Dependence of the Double Cascade Observable Distributions
Figure 6.14: Observable distributions of the double cascade sample for different spectral shapes of the astrophysical tau-neutrino flux. The expected number of signal events in the double cascade sample of six years is depicted as a function of the total deposited energy (left) and the double cascade length (right). The distributions are shown for different realizations of a spectral index γ (top), spectral cutoff energy Ec (center), and spectral break energy Eb
In Figure6.14, the double cascade observable distributions of astrophysical tau-neutrino in- teractions in the respective topology sample are shown for the aforementioned astrophysical neutrino flux models. Note that these are the distributions of identifiable tau-neutrino interac- tions, i.e. only a fraction of all tau-neutrino interactions. It can clearly be seen how it decreases with a softer spectrum. A harder spectral index shifts the maximum of the energy distribution towards higher energies, and the slope becomes more gradual. More importantly, the length distribution strongly depends on the spectral index due to the correlation Eντ ∼ Eτ ∼ λτ.
Consequently, a soft spectrum suppresses the number of efficiently identifiable tau-neutrino interactions for which the tau has a long decay length. For example, a soft ∼ E−3
ν spectrum de- creases the number of identifiable double cascade events at L ≃ 100 m by a factor ∼ 5 compared to a hard ∼ E−2
ν spectrum. A similar effect is caused by a spectral cutoff: the lower in energy the spectral cutoff, the lower the number of high-energy neutrino interactions. Note that the change for a cutoff at Ec= 500 TeVis both drastic and unlikely because it is incompatible with the observed PeV neutrinos [4]. A more realistic cutoff of a few PeV still has a significant impact on the number of identifiable tau-neutrino interactions. However, the relative change weakens with larger cutoff energies due to the decreasing identification efficiency at energies above a few PeV (c.f. Section4.4). The impact of a break in the astrophysical tau-neutrino spectrum is smaller compared to the variations of the other flux models. In effect, the representation of a varying spectral break energy is equivalent to changing the spectral index. For a very low break energy at Eb= 200 TeV, the flux is mostly dominated by the hard ∼ Eν−2component, and for a very high break energy Eb= 5 PeVit is mostly dominated by the soft ∼ Eν−3 component. This parametrization, however, is helpful for studying a potentially more realistic case where the flux is neither described well by a soft spectrum nor by a hard spectrum. It can be seen that the mixture of both, as defined by the spectral break energy, has a significant impact of the number of identifiable tau-neutrino interactions.
In the following, the different astrophysical neutrino flux models defined in Equations6.13to6.15 are incorporated by an extension of the likelihood fit. Only one additional flux parameter is allowed to be free in the fit, while the others are fixed. The single power-law model corresponds to the default hypothesis as described in Section 6.2.1 and has γ as a free fit parameter. For a fit of the single power-law model with cutoff, γ is fixed to its true value and Ec is a free parameter in the likelihood. The dual power-law hypothesis is fitted by fixing γ1 and γ2to their true values and adding Ebas a free parameter.6 Two-dimensional likelihood scans are performed which depict the dependence of the astrophysical tau-neutrino fraction on the spectral index, the spectral cutoff energy, and the spectral break energy. Confidence intervals are obtained using Wilks’ theorem for a likelihood ratio test with two degrees of freedom.
In Figure 6.15, the likelihood scan of the tau-neutrino fraction and spectral index is shown for four different injected values. As expected, the contours are broader for a soft spectrum. Note that the constraints on both the tau-neutrino fraction and the spectral index are stronger for a hard spectrum. In this case, the fit is more sensitive to the tau-neutrino fraction due to an increase of identifiable tau-neutrino interactions, as explained above. The spectral index itself can be better constrained for a harder spectrum due to the increase of high-energy events that make a distinction between astrophysical and atmospheric neutrino flux easier. Note that the two-dimensional contours are approximately parallel to the axis of the tau-neutrino fraction, which means that they can be measured independently by the likelihood fit. As discussed in Section6.3.2, the spectral index is mostly constrained by the single cascade and track samples due to the larger number of events, whereas the tau-neutrino fraction is almost exclusively constrained by the much smaller double cascade sample.
6
Note that the likelihood fit is only changed to study the dependence of the sensitivity on the spectral shape of the astrophysical tau-neutrino flux. These models are not tested in this analysis, because the high-energy starting event sample is not well suited for such a fit due to the low statistical power of the data sample.
1σ 2σ 3σ − 2∆log L
Likelihood Scan of the ν
τFraction and the Spectral Index
1.5 2.0 2.5 3.0 3.5 Spectral Index 0.0 0.2 0.4 0.6 0.8 1.0 F raction of ντ γ = 2.0 γ = 2.3 γ = 2.6 γ = 3.0 1σ 2σ 3σ −2∆log L
Figure 6.15: Likelihood scan of the astrophysical tau-neutrino fraction and the spectral index. In the central panel, the 68% contours are depicted for different injections of γ around their best- fit values marked with ‘×’. The one-dimensional likelihood scans are shown in the top panel for γ and in the right panel for fντ. The test statistic −2∆log L is given in units of Gaussian
standard deviations using Wilks’ theorem.
In Figure 6.16, the likelihood scan of the tau-neutrino fraction and spectral cutoff energy is shown for four different injected values. As expected, the constraints on the tau-neutrino fraction are increasingly stronger for a higher spectral cutoff energy. This is due to the weaker suppression of identifiable tau-neutrino interactions at higher energies. Note that the constraints on the cutoff energy itself become weaker with increasing values. It can be seen that the contours do not close in the depicted energy range for a 10 PeV cutoff. At these energies, the flux is so low that the statistical power of the six-year high-energy starting event sample are too weak to be sensitive to a spectral cutoff. The contours show no correlation between the tau-neutrino fraction and the cutoff energy for the same reason as discussed above for the likelihood scan of the spectral index.
In Figure 6.17, the likelihood scan of the tau-neutrino fraction and spectral break energy is shown for four different injected values. As expected, the contours increase with a higher spectral break energy. The constraints on the tau-neutrino fraction are increasingly weaker due to the stronger contribution of the soft spectral component. As discussed above, a soft spectrum reduces the number of identifiable tau-neutrino interactions significantly. Similar to the spectral cutoff energy, the constraints on the spectral break energy itself become weaker with increasing values. The explanation is the same as above, because the neutrino flux at a few PeV and thereby the number of expected events are too small to be sensitive to a spectral break in this energy range. The contours show no strong correlation between the tau-neutrino fraction and the spectral break energy, as observed for the other flux models.
1σ 2σ 3σ − 2∆log L
Likelihood Scan of the ν
τFraction and the Spectral Cutoff
102 103 104 Ecutoff [TeV] 0.0 0.2 0.4 0.6 0.8 1.0 F raction of ντ
Ec= 500 TeV Ec= 1 PeV Ec= 5 PeV Ec= 10 PeV
1σ 2σ
−2∆log L
Figure 6.16: Same as in Figure 6.15, but for the spectral cutoff energy.
1σ 2σ 3σ − 2∆log L
Likelihood Scan of the ν
τFraction and the Spectral Break
102 103 104 Ebreak[TeV] 0.0 0.2 0.4 0.6 0.8 1.0 F raction of ντ
Eb= 200 TeV Eb= 500 TeV Eb= 1 PeV Eb= 5 PeV
1σ 2σ
−2∆log L
2.0 2.2 2.4 2.6 2.8 3.0 Spectral Index 1σ 2σ 3σ Median Detection Sensitivit y Spectral-Index Dependence 102 103 104 Ecutoff [TeV] 1σ 2σ 3σ Median Detection Sensitivit y Spectral-Cutoff Dependence 102 103 104 Ebreak[TeV] 1σ 2σ 3σ Median Detection Sensitivit y Spectral-Break Dependence 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Events in 2078 Da ys Detection Sensitivity Signal Double Cascades Background Double Cascades
0.0 0.5 1.0 1.5 2.0 2.5 3.0 Events in 2078 Da ys Detection Sensitivity Signal Double Cascades Background Double Cascades
0.0 0.5 1.0 1.5 2.0 2.5 3.0 Events in 2078 Da ys Detection Sensitivity Signal Double Cascades Background Double Cascades
Figure 6.18: Median detection sensitivity and ex- pected number of signal and back- ground events as a function of the spec- tral index (top), cutoff energy (center) and break energy (bottom). The black line corresponds to the detection sen- sitivity as defined in Equation 6.9 (left axis). The colored lines correspond to the number of signal and background events in the double cascade sample as presented in Section6.3.1(right axis).
It can already be deduced from the two- dimensional likelihood scans that the ability to constrain the tau-neutrino fraction strongly varies with the spectral shape of the assumed astrophysical neutrino flux. In Figure 6.18, this is quantified more precisely in terms of the median detection sensitivity as defined in Equation 6.9 and the expected number of signal and background events as presented in Section6.3.1. A strong correlation between the median detection sensitivity and the spec- tral shape is visible for all three flux models. A hard spectrum with ∼ E−2.1
ν as observed us- ing the six-year through-going track sample (c.f. Table6.4) would correspond to a detec- tion sensitivity of ∼ 2.9 σ. However, for the much softer spectrum ∼ E−2.9
ν as observed using the six-year high-energy starting event sample, the detection sensitivity would dras- tically decrease to ∼ 1.8 σ. While the num- ber of background events remains approxi- mately constant, the number of identifiable tau-neutrino interactions is reduced by ∼ 40% in comparison. In the intermediate range for an ∼ E−2.3
ν spectrum, the median detection sensitivity ∼ 2.6 σ of the benchmark model is recovered (c.f. Section 6.3.2). A similar variation of the sensitivity can be seen for a spectral cutoff. An unrealistic cutoff energy of Ec = 100 TeV would imply that the tau- neutrino fraction cannot be constrained. A more realistic scenario of Ec = 2.7 PeV, as obtained in the global fit, would correspond to a detection sensitivity of ∼ 2.4 σ. This does not deviate much from the sensitivity for the assumed benchmark flux model. As discussed above, the variation due to the spectral break energy is slightly smaller. A break energy at a few PeV only corresponds to a superposed flux that is increasingly dominated by the soft spectral component but does not reduce the number of high-energy neutrino interactions as abruptly as a spectral cutoff. As expected, the detection sensitivity decreases with an in- creasing break energy. In the potentially in- teresting region of a few hundred TeV, the detection sensitivity varies between ∼ 2.4 σ to ∼ 2.6 σ. Also, the occurrence of a potential spectral break does not cause a large deviation from the sensitivity expected for the bench- mark flux model. Note that the number of
expected background events in the double cascade sample is approximately constant for all spectral shapes. This is due to the unique features of the double cascade topology, for which