Astrophysical spectrum and absence of correlations

In 2013, the IceCube neutrino telescope discovered ultra-high energy neutrinos between 30 TeV and 2 PeV, likely of extragalactic origin, with a significance of more than 5σ [8]. As mentioned above, IceCube is currently the largest operating neutrino telescope with an instrumented vol-ume of 1 km³. It is a Cherenkov detector which uses Antarctic ice as a medium for neutrino interactions, detecting them with photomultipliers buried deep in the ice. These photomultipli-ers are contained in digital optical modules (DOMs), which are in turn mounted onto a string with a spacing of 17 meters. A total of 86 strings with 60 DOMs each are deployed in ∼ 2.5 km deep holes drilled with hot water. In the center of the array, a section of the ice dubbed DeepCore has been more densely instrumented to lower the neutrino energy detection threshold, extending the observable energies below 100 GeV [150].

IceCube can observe different event topologies, of which the main ones are tracks and cascades.

The latter can be observed due to neutral-current (NC) interaction of neutrinos of arbitrary flavor, e.g., ν_x+N → ν_x +N^′, generating hadronic cascade. Cascades can also be created by charged-current (CC) interactions of electron-neutrinos, e.g., ν_e +N → e+N^′, resulting in an electromagnetic cascade, i.e., a cascade with a higher electron and photon content. In either case, the signature is localised, approximately spherical and likely to be fully contained within the detector. Thus, the uncertainty in the energy reconstruction is low (∼15% above 10 TeV), but the reconstruction of the arrival direction is poor (10^◦ to 40^◦). For muon tracks it is the opposite: they are observed following a charged-current interaction of a muon-neutrino,i.e., νµ+N →µ+N^′, where the resulting muon produces a track of photons along its trajectory. Due to the large mean free path of the muon, the track has a much better angular resolution (≲1^◦) than cascade events. On the other hand, it is likely that a part of the track is not contained in the detector, which leads to a poorer energy resolution. Tau-neutrinos at PeV energies can be identified via their expected "double bang" topology, i.e., ντ +N → τ +N^′, which shows a hadronic cascade at the interaction vertex and a second cascade where the τ-lepton decays

Figure 2.6: Diffuse astrophysical neutrino spectrum (single flavor, combined neutrino and anti-neutrino) as a function of the energy. The black data points with the 1σuncertainties attached are from the High Energy Starting Events (HESE) data set, with the blue band showing the 1σ uncertainties of a single power-law best-fit to HESE data.

The pink band represents the best fit of the through-going muon (TGM) data set.

Atmospheric fluxes are also shown (blue: best-fit conventional, green: best-fit upper limit prompt). Taken from [151].

into a tau-neutrino. If only one of the two vertices is contained in the detector, it is called a

"lollipop" signature. See [152] for an overview of the event topologies.

Furthermore, incoming electron-antineutrinos can interact resonantly with electrons to create a W boson, i.e., ̄νe+e → W, if their energy is about 6.3 PeV. The obtained signal can be a hadronic cascade or a leptonic cascade or track, depending on the decay channel of the W boson [153]. This so-called Glashow resonance will become important in Sec. 3.2.3 since for different production mechanisms in astrophysical objects different flavor ratios are obtained for the neutrinos. This results in different event rates for the Glashow resonance, providing a tool for neutrino production diagnosis.

In 6 years of IceCube data, 82 high energy cosmic neutrinos with energies between 100 TeV and 10 PeV have been detected. The flux of astrophysical neutrinos (per flavor, combined neutrino and antineutrino), as it is known today, is shown in Fig. 2.6 as a function of the energy.

There are two main data sets, namely High Energy Starting Events (HESE) and Through-Going Muons (TGM). Fitting the data sets with a single power law yields E^−2.9 (blue band, including 1σ uncertainties) for HESE and E^−2.1 (pink band, including 1σ uncertainties) for TGM as a best fit spectral index. Also shown in the figure are the best fit for the conventional atmospheric

Figure 2.7: Skymap of the arrival direction of detected neutrino events in Galactic coordinates.

Shower-like events are indicated by + while track-like events are marked with ×.

At each location, the color scale represents the test statistics (TS) for point-source clustering. No significant clustering was found. Taken from [151].

neutrino flux (dashed, blue curve) and the best fit for the prompt atmospheric contribution (dashed, green curve) by cosmic ray interactions in the atmosphere [151]. It has been suggested that the TGM are more representative of the astrophysical flux, possibly because the HESE samples are contaminated with atmospheric events [154].

With the current statistics, it appears that the signal is isotropic, as it shows no significant clustering in the search for point sources, as shown in Fig. 2.7. A maximum-likelihood clustering method with the test statistic (TS) defined as the logarithm of the ratio between maximum likelihood to include a component of a point source and the likelihood for the isotropic null hypothesis was performed. The analysis by IceCube did not find any clustering, i.e., no point sources or hot spots in the sky [151]. Furthermore, the signals are neither coincident with the occurence of gamma-ray bursts [14] nor active galactic nuclei [155] and there is no definite detection of multiple neutrinos from an individual source [156]. This translates into a number of constraints,e.g., the point source limits, stacking bounds or multiplet constraints, which will be discussed in place when they are relevant for the source classes investigated in this thesis.

Nuclear cascades in combined source-propagation models

As mentioned in Chapter 2, a wide variety of tools including analytical models as well as nu-merical simulations are used in theoretical astrophysics to tackle a great variety of astrophysical problems. While, in general, the analytical approach enhances the understanding of the physical processes which are involved, it does not always lead to precise results for complex systems. In this case, numerical simulations are needed to perform these calculations with high accuracy in order to obtain meaningful predictions and to study the observational consequences. If the observation is inconsistent with the prediction, the model is adjusted by minimal modifications to fit the data. However, if the inconsistencies are too large, a model representing a certain physical scenario can be disfavored or even ruled out. Following this approach, a numerical simulation will be set up to test source scenarios. By comparing with the data presented in Chapter 2, it is possible to study the properties of potential sources and determine whether they can address the observations.

Neutrinos are, for example, produced in cosmic ray interactions with ambient photons. These processes can be described by providing the corresponding particle densities and cross-sections.

While cross-sections are input, the particle densities are determined mainly by the source envi-ronment, i.e., how much energy is available and how the source geometry is defined. The idea in this thesis is to set up a fully flexible, deterministic simulation to calculate nuclear processes, which is then applied to certain types of sources. The key novelty of this work is the nuclear cascade, which represents the interactions of heavy isotopes and how they break up ("cascade down") due to different radiation processes, leading to neutrino production and altering the cosmic ray composition.

In this chapter, the nuclear cascade is introduced in great detail, along with other ingredients needed to set up a generic model. An overview of the different components is given in Fig. 3.1. In the first step, the progenitor scenario determines the main parameters,i.e., the available energy and volume, and with that the energy densities. Further input obtained from the progenitor is, e.g., the duration of the emission or what the spectral shape of accelerated cosmic rays and target photons looks like. Another important input is the composition of cosmic rays which are

Progenitor,

Figure 3.1: Schematic representation of the key components of a generic model. The first in-gredient is the progenitor, which determines the source geometry. The densities of accelerated particles and target photons serve then as an input to calculate the nu-clear cascade in the radiation zone. Particles escape from the source and propagate through the Universe in the next step, before they are eventually detected at Earth.

The colors indicate which components are modeled in this thesis (green) and for which generic assumptions are used (red).

accelerated. Based on this information, the nuclear cascade is calculated in the next step. This yields the particle densities after interactions and energy losses. In this process, neutrinos and gamma-rays are produced and the composition can change because of the gradual disruption of heavy nuclei into lighter fragments. This is the heart of the simulations presented in this thesis, as it is the first fully self-consistent and efficient computation of time-dependent nuclear cascades. A certain fraction of the cosmic rays can escape from the source and undergoes further interactions during propagation through the Universe. These processes can further alter the composition of cosmic rays and produce cosmogenic neutrinos, i.e., neutrinos produced by cosmic ray interactions during propagation. In the last step, the predicted cosmic rays and neutrinos that eventually arrive at Earth are compared to the UHECR spectrum and composition data as well as high energy neutrino data. This will ultimately test the model and with that the source candidates to constrain the origin of UHECRs.

The color of the circles in Fig. 3.1 indicates which components are part of the modeling (green) and for which part certain input assumptions are used (red). In this thesis, the progenitor and acceleration scenarios are not explicitly modeled. Instead, the remaining three components are assembled first and then applied to several source classes which are discussed in the following chapters. Most numerical models in the literature are propagation-only models in contrast to the models developed here, i.e., including only the last two components (see for example [157, 60, 42, 158]). Thus, there are predictions about the composition of cosmic rays at the ejection from the source, after the nuclear cascade developed. However, with including the nuclear cascade as a part of the simulation, the fit to the data can be studied as a function of the injection composition of the source. This is required to uncover the properties of potential

UHECR sources. In addition, different from existing models which take into account the nuclear processes in the source (see for example [23]), the simulations presented in this thesis are efficient, such that parameter scans can be performed.

Now that the signals which are measured at Earth were introduced in the previous chapter, the parts of the model will be presented in this chapter, following the first part in our paper [159]. In Sec. 3.1, it will be described how cosmic rays are accelerated in the source. The second component, namely the nuclear cascade and the radiation modeling will be reviewed in detail.

Lastly, it will be discussed how particles can escape from the sources. After that, in Sec. 3.2, the interface to cosmic ray propagation is discussed together with energy budget considerations and cosmological evolution scenarios. Neutrinos do not interact on their way to Earth, but they do oscillate. Flavor mixing and its consequences will be reviewed in detail in the last section of this chapter. The flavor composition can be used to infer on source properties via the Glashow resonance, which will be discusses at the end, based on our paper [160].

3.1 Radiation modeling in collisionless internal shocks

There are two different regimes for which radiation of astrophysical transients is observed: the prompt emission and the afterglow. The principle is depicted in Fig. 3.2 in the case of a black hole engine [161]. In general, it is assumed that the prompt emission is related to the actual event, e.g., a star collapsing to a black hole. The stellar material is ejected with high but different velocities depending on the dynamics of the source. This can happen in the form of a collimated jet, but that is not necessarily the case, as the jet could also be choked or the emission could originate from a cocoon [162, 163]. In a simple picture, spatial over- and under-fluctuations of the matter density with different speeds are represented by shells. These shells collide eventually due to the non-vanishing relative Lorentz factor and so-called internal shocks can form. Particles can be further accelerated by scattering off the shock front (see Sec. 3.1.1 for details) and nuclear interactions are triggered if the shock is dense enough.

On the other hand, the afterglow emission happens when the ejected material eventually collides with the ambient medium. External shock waves in the forward (forward shock) and backward (reverse shock) direction are formed which can lead to acceleration and nuclear inter-actions as well. The afterglow emits radiation over much longer time scales than the prompt phase. In practice, it is therefore more challenging to observe the prompt emission of an as-trophysical source, if this part of the sky is not monitored by chance. It is believed that the highest energies are reached during the prompt emission phase, since much more energy is re-leased therein [164, 165]. For that reason, this thesis focuses on modeling the prompt emission

Figure 3.2: Schematic scenario for prompt emission and afterglow of an astrophysical transient.

An engine (here: black hole) emits material (not necessarily in a collimated jet) with high velocities. Spatial fluctuations of the matter density, here represented as shells, may collide and internal shocks are formed, leading to particle acceleration, gamma-ray emission and nuclear interactions. Later, further particle acceleration and interactions can happen in external shocks in the afterglow. Taken from [161].

phase in the internal shock model. The main relation of the internal shock model, R = 2Γ²c t_v

1 +z , (3.1)

connects the radius at which the shells of plasma collide with the observed variability tv of the light curve. The variability time scale is also indicative for the shell width ∆d^′ = Γctv/(1 +z), with the shock Lorentz factor Γ and redshift z.

Besides the already mentioned internal shock model, there are two more types of models for the prompt emission phase. One of them are photospheric models, in which the radiation is assumed to originate from thermal electrons inside the plasma. The thermal photons emitted by the electrons can interact again via inverse Compton scattering, pushing them to very high energies. Once the plasma density decreased sufficiently such that the optical depth to Thomson scattering falls below unity, the photons can escape from the plasma. This decoupling can depend on the wavelength of the corresponding photons. Indeed, multi-wavelength observations indicate

a delayed emission of the high energy radiation. In this model, neutrino production is believed to take place near the photosphere, i.e., the transition of the plasma being opaque to transparent.

The photospheric radius is typically smaller than the radius at which internal shocks occur, which leads to higher radiation densities. See for example [166, 167] for a review.

The second alternative are jets powered by magnetic reconnection, as for example the Internal Collision Induced Magnetic Reconnection and Turbulence (ICMART) model [168, 169]. In this model, particles are accelerated due to turbulent reconnection of magnetic fields inside the plasma. In contrast to photospheric models, the radius at which photo-hadronic interactions happen, i.e., interactions between photons and nuclei, are expected to be larger than in the internal shock scenario. The advantage of this model is that it describes pulses and the fast time variability of the light curve at the same time by introducing two instead of one characteristic time scale. However, this also means that the interaction rate is suppressed by the ratio of these time scales. Because of this and due to the different model geometry in all three cases the predicted neutrino fluxes can be different.