The Pennsylvania State University The Graduate School HIGH ENERGY NEUTRINOS FROM GAMMA-RAY BURSTS: RECENT OBSERVATIONS AND MODELS

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1 The Pennsylvania State University The Graduate School HIGH ENERGY NEUTRINOS FROM GAMMA-RAY BURSTS: RECENT OBSERVATIONS AND MODELS A Dissertation in Physics by Shan Gao c 2014 Shan Gao Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2014

2 The dissertation of Shan Gao was reviewed and approved by the following: Peter Mészáros Eberly Chair of Astronomy and Astrophysics, Professor of Physics Director, Center for Particle and Gravitational Astrophysics Dissertation Advisor, Chair of Committee Douglas Cowen Professor of Physics Irina Mocioiu Associate Professor of Physics James Kasting Evan Pugh Professor of Geoscience Richard Robinett Professor of Physics Associate Department Head Director of Undergraduate Studies, and Director of Graduate Studies Signatures are on file in the Graduate School.

3 Abstract Neutrino astronomy began with the detection of solar neutrinos, supernova neutrinos (SN1987A) and more recently the 37 events in IceCube which are very likely to be an astrophysical origin. The result from IceCube is perhaps the most exciting discovery of the year 2013, capping a several decades long search. Various astrophysical candidates have been proposed as sources of high energy neutrinos, although the origin of the IceCube neutrinos remains a mystery. Gamma-ray bursts (GRBs), the most energetic explosions in the universe, were considered as the most promising source for high energy cosmic rays and neutrinos (with AGNs). However, a previous search of GRB neutrinos by IceCube surprised the GRB community with negative results, challenging the simple standard picture of GRB prompt emission which is called the internal shock model. In this thesis we give a closer investigation of this model as well as several leading alternative models. With a careful consideration of the particle physics and the model parameters we show that the previous negative result with GRB neutrinos is not surprising, and only those models with extremely optimistic parameters can be ruled out. We predict that GRBs are unlikely to be the sole sources of the IceCube events, but signals of GRB neutrinos may be detected in the near future, with the neutrino telescopes such as IceCube/DeepCore, KM3Net, ARA, ARIANNA, ANITA etc. iii

4 Table of Contents List of Figures List of Tables Acknowledgments vii ix x Chapter 1 Introduction Overview Gamma-ray Burst and Jet Neutrino Production in GRBs Neutrino Detection, Recent Results Chapter 2 GeV-TeV Neutrinos from Collisional GRBs Introduction The Nuclear Collision Scenarios Neutrinos From Longitudinal Nuclear Collisions Neutrinos From Transverse Nuclear Collisions Parameters of the Model GeV Neutrinos and their Detectability in Deep Core and ICECUBE Neutrinos from a Single GRB Diffuse Neutrino Flux Discussion Chapter 3 TeV-PeV Neutrinos from Photospheric GRBs 40 iv

5 3.1 Introduction The Dissipative Photospheric Scenarios Method of Calculation Model Parameters and ν from a Single GRB Diffuse ν Background from GRBs Discussion Chapter 4 PeV-EeV Neutrinos from Pop.III GRBs Introduction Model and Calculation Method Astrophysical Input Photon spectra Proton Acceleration, Cooling and Pion Production The neutrino flux Individual source neutrino fluence Population III GRB rates Diffuse neutrino flux and detectability Discussion Chapter 5 Application : the GRB130427A Case Introduction Model-independent constraints on R dis, Γ bulk and E CR Model dependent constraints Discussion Chapter 6 Discussion and Outlook Summary Possible Future Work Jets Inside a Massive Star Jets Along The Line Of Sight With VHE γ rays Death Of Supermassive Stars Distribution Functions for Secondary Particles 111 π ch & K ch Spectrum in pp Interactions π ch & K ch Spectrum in pγ Interactions ν spectrum from π ch decay v

6 Bibliography 115 vi

7 List of Figures 1.1 Progenitors for Long and Short GRB GRB redshift distribution and luminosity functions Fermi-I acceleration Neutrinos from pn collision pγ and pp interactions Neutrino production sites in GRB jets High energy neutrino events in IceCube Pion spectrum via two types of collisions Comparison of ν spectrum between two methods Muon neutrino spectrum from a nearby GRB by different GRB L and η ν µ + ν µ fluence for different η core and η out ν µ + ν µ fluence for narrow and wide jets Expected event rates in IceCube Diffuse neutrino spectrum using GRB statistics Photon spectrum from a typical photospheric GRB model Proton cooling timescales in a dissipative photosphere Neutrino spectrum from a magnetic dominated jet Neutrino spectrum by magnetic, baryonic and internal shock models Neutrino spectrum with different baryon loading factors Diffuse neutrino spectrum by different model and parameters Diffuse neutrino spectrum by slow and fast jet models Pop III GRB afterglow photon spectrum, case A Pop III GRB afterglow photon spectrum, case B Pop III GRB afterglow photon spectrum, case D Proton cooling timescales, case A Proton cooling timescales, case B Proton cooling timescales, case D vii

8 4.7 Neutrino flux from a pop III GRB Diffuse neutrino flux from Pop III GRB, M h = 30M Diffuse neutrino flux from Pop III GRB, M h = 100M Diffuse neutrino flux from Pop III GRB, M h = 300M Density plot of ν-event in IceCube, model-independent Density plot of ν-event for the internal shock model Density plot of ν-event for baryonic photosphere model Density plot of ν-event for magnetic photosphere model viii

9 List of Tables 3.1 Parameter and neutrino event table Pop. III GRB model parameters Expected number of muon events in IceCube from Pop III GRBs.. 79 ix

10 Acknowledgments I am deeply indebted to my adviser Dr. Peter Mészáros. He has been very supportive and considerate in every aspect. As a student of physics, I had a clean (zero) background in astronomy or astrophysics when I joined his research group. In my own observer frame then, astrophysics appeared very messy and chaotic. However, he lead a way for me to the research in this field, a most efficient one. I was provided with a lot of, both support and freedom when doing my research. He was deeply involved in the discussions of each research work, every paper I wrote, every group meeting, every talk I presented, etc... down to each mis-spelled word due to my poor written English. I also learned a lot just by watching how he did the research, how he taught the classes and even the attitudes he treated tiny things in the daily life. In a word, without his support, none of my achievements today would be possible. I would like to thank the postdoc in our group, Kazumi Kashiyama. He is a knowledgeable and versatile expert in theoretical astrophysics and I benefited greatly from numerous discussions with him. There were a short but very difficult time for me (and for him as well) that I was very puzzled about my future. He gave me lots of encouragements and we both walked out of that eventually. There are many people who helped me significantly with my research and graduate life, but here I probably can give only a partial list to thank: Dr. Asano Katsuaki, Dr. Douglas Cowen, Dr. Irina Mocioiu (especially her course on particle physics, which I would rank as the best course I had during my entire highereducation phase), Dr. Tyce DeYoung, Dr. Richard Robinett, Dr. Kohta Murase, Dr. Kenji Toma, Dr. Xuefeng Wu, Dr. Peter Veres, Dr. Xuewen Liu, Dr. Xiaohong Zhao, Dr. Philipp Baerwald, Mr. Mouyuan Sun and my fellow classmates Xuesong Wang and Enshi Xu (this couple taught me a lot about astronomy and scientific computing). Of course, I am also grateful to my committee Professors: Dr. Douglas Cowen, Dr. Irina Mocioiu and Dr. James Kasting, for their help and thorough review on this dissertation. x

11 Chapter 1 Introduction 1.1 Overview The Neutrino in the Standard Model A neutrino is an electrically neutral, weakly interacting and almost massless elementary particle with spin-1/2. It was first postulated by W. Pauli in 1930 to resolve the problem of energy, momentum and angular momentum conservation associated with beta decay experiments [1]. The name neutrino was later coined by E. Fermi in 1934 in his theory of beta decay where neutr means electrically neutral and ino means tiny. Being free from electromagnetic interactions and subject only to weak interactions, which by its nature is a very short-ranged subatomic force, neutrinos have extremely small cross sections to scatter with normal matter and typically they are able to travel cosmological distances unimpeded. The standard model (SM) of elementary particle physics has three generations of leptons, hence the three flavors of neutrinos: ν e (discovered in 1956 [2]), ν µ (discovered in 1962 [3]) and ν τ (discovered in 2000 [4]), all being massless in this theory. The experiment by C. Wu [5] verified that weak decays are maximally parity violating (theory postulated by CN. Yang and TD. Lee). This is manifested in SM, by saying that neutrinos are always left-handed while anti-neutrinos are always right-handed. Later, the neutrinos from solar nuclear reactions were detected; however, the flux showed discrepancies between the theoretical prediction and actual measurements. Similar problems exist in atmospheric, reactor and ac-

12 2 celerator neutrino experiments. These were resolved by the discovery of neutrino oscillation and mass, which is a departure from SM. In quantum mechanics, each of the neutrino mass eigenstates is a superposition of three flavor quantum states. As the neutrinos propagate, they can oscillate between the three flavors. The probability of observing a neutrino of a specific flavor after oscillation can be calculated by the PMNS matrix [6], [7] and it depends on the path length and energy of the neutrino, besides their intrinsic mixing angles (see e.g. [8], the neutrino section for a complete mathematical description). Today, the neutrino is an important messenger to probe fundamental physics beyond the standard model, such as the issues with its postulated Majorana nature, with neutrino mass, the CP violation and the potential existence of other neutrino species or interactions etc. Astrophysical Neutrinos In the astrophysical context, there are many possible neutrino production mechanisms. At the lower end of the energy spectrum, neutrinos may exist today as the cosmic neutrino background (relic neutrinos, CNB or CνB), a hot big-bang relic of the universe, formed when the primordial protons and neutrons decouple, an event prior to the recombination of electron and H +, which leaves the famous cosmic microwave background (CMB). However, today these primordial cosmic neutrinos have been cooled to a temperature even slightly lower than the CMB photons and are not yet possible to be detected by current means due to their extremely tiny cross sections (although efforts have been proposed [9]). However, the CNB affects the Big Bang Nucleosynthesis and the expansion rate of the universe. Therefore, useful constraints on neutrino mass and number of species can be derived from cosmological observations even before the accelerator experiments (see e.g.[10]). The nuclear reactions in the cores of stars can emit neutrinos roughly at MeV energies, and the solar neutrinos have been detected decades ago (see e.g. the SNO experiment [11]). During the core collapse phase of a massive star, which later triggers the supernova, thermal neutrinos at an energy about 10 MeV are emitted mainly via the electron capture process. SN 1987A was a supernova in the Large Magellanic Cloud and its neutrino emission was detected by the Kamiokande II, IMB and Baksan experiments. Although the total neutrino counts for this

13 3 event were about 20, it was a significant excess over the background [12],[13]. The detection of SN 1987A and solar neutrinos together marks the beginning of neutrino astronomy. Besides these low energy (the CNB) or medium energy (MeV, nuclear energy) neutrinos, there are also high energy (GeV or above) neutrinos in nature. What we know today is that, at least part of those high energy neutrinos must be related to cosmic rays. High energy cosmic rays were discovered more than a century ago. However, their origin remains unknown today. The cosmic ray spectrum as we know today spans from GeV to slightly higher than ev, forming a roughly power-law distribution [14]. A sufficiently high energy cosmic ray proton can interact with another proton (known as pp-process) or nucleon or photon (known as pγ-process) to produce charged mesons (mainly pions and kaons), which ultimately decay to stable leptons as electrons and neutrinos (or their anti-particles). Therefore, two components are guaranteed to exist in the high energy neutrino flux due to cosmic rays: 1) The atmospheric neutrinos. These neutrinos are produced by the cosmic ray particles interacting with the nuclei of the earth atmosphere. 2) The cosmogenic neutrinos ([15],[16]), which are produced by the highest energy (about GeV, or GZK energy [17],[18] ) cosmic ray particles interacting with the CMB photons as they propagate. The former have been discovered while the latter await the future neutrino experiments, due to their low flux. Are there high energy neutrinos besides the atmospheric and cosmogenic neutrinos? Although many theories have been postulating high energy astrophysical neutrinos for long, the question is finally answered today experimentally by the recent IceCube observations. They confirmed the existence of a class of astrophysical neutrinos. The IceCube [19] is a cubic km sized large neutrino telescope located in the South Pole of Antarctica. It uses 86 strings, consisting a total number of about 5000 optical sensors deployed in ice at depths about km. When a high energy neutrino interacts with the target particles of the ice, either a nuclei or very rarely an electron, by weak interaction, secondary charged particles are generated at relativistic speeds, higher than the speed of light in the ice. The characteristic Cheronkov radiation accompanied by these particles is captured by the optical sensors. In the recent all-sky search during 2010 and 2013, three PeV-energy (10 15 ev) and 34 additional sub-pev energy neutrino events were detected [20] and more

14 4 are expected to be published with the completion of data analysis. The excess of the flux above the atmospheric neutrino background was confirmed above the 5σ level. From the data of those events, some flavor and position information on the sky can also be reconstructed. What are the origins of these TeV - PeV neutrinos in IceCube? The recent analysis (e.g. [21]) pointed out that neither the atmospheric neutrinos nor the cosmogenic neutrinos could account for all these events, because the flux of atmospheric neutrino decreases rapidly with energy and the flux is insufficient to produce those especially the two PeV events. For cosmogenic neutrinos, because they are produced by GZK energy cosmic rays, they are expected at much higher energies. Astrophysical sources are plausible at these energies, including active galactic nuclei (AGN), gamma-ray bursts (GRB), supernovae (SNe) and hypernovae (HNe), new-born magnetars, starburst galaxies and galaxy clusters. The detected events in the IceCube are distributed more or less isotropically over the sky and have a flavor ratio that is consistent with ν e : ν µ : ν τ = 1 : 1 : 1, implying an astrophysical origin. The cosmic ray spectrum is highly non-thermal. Non-thermal photon emissions are also a common or even a dominating component in high energy astrophysical phenomena, e.g. where shock waves present. The shock waves are believed to have the capability to accelerate charged particles to very high energies via a diffusive shock acceleration mechanism (known as the Fermi acceleration [22],[23]). The maximum energy of the accelerated particles depends on multiple factors such as the source size, the magnetic field strength and how fast the charged particle loses energy. Electrons suffer energy losses mainly by synchrotron emission when they move in the magnetic fields and also by inverse Compton scattering of photons. The non-thermal photon emission can be derived from the non-thermal electron distribution and compared to observations. The same shocks should also be able to accelerate protons. Those protons can either escape as the high energy cosmic rays and/or produce neutrinos via pp or pγ interactions with other proton or photon targets within the source, or along the path of propagation. Some of the shock waves are guaranteed in certain high energy astrophysical phenomena, e.g. gamma-ray bursts (GRB). The GRB is a sudden flash of gamma-rays with an apparent (isotropic equivalent, see the next section for de-

15 5 tails) luminosity which can be as high as erg/s. GRBs are generally believed to be the result of the collapse of a very massive star or a binary neutron star merger. Relativistic jets should arise from both those scenarios. When the jet is decelerated by sweeping up a sufficient amount of the interstellar medium, a pair of shocks is usually produced (known as a forward and a reverse shock). The shocks are able to accelerate electrons to give the afterglow emission. However, to explain the prompt gamma-ray emissions there are several competing models. Shock waves can be produced as shells of ejecta with different speeds which collide with each other before the afterglow happens. This scenario is referred to as internal shock model [24]. Alternatively there are dissipative photospheric models [25] and magnetic reconnection models [26]. It has been proposed that particle acceleration may be also possible via magnetic reconnection or other mechanisms. The nuclear collision between a proton and a neutron with large relative velocities can also trigger hadronic cascades and give rise to the prompt gamma-rays and neutrinos as well. GRBs were proposed as a leading cosmic ray source (as well as AGNs). Estimating the total energy injection rate of ultra-high energy cosmic rays (UHECRs) and assuming that these protons produce neutrinos efficiently in some modelindependent source, an upper limit for the very high energy neutrino (VHE Nu) flux is obtained, called the Waxman-Bahcall (WB) bound [27]. In this dissertation, the neutrino emission is revisited by means of a more advanced calculation for each major GRB model. It is shown that the total neutrino emission from all-sky GRBs is well below the WB limit. The constraints set by the IceCube observation on correlation with GRBs are not surprising and they only constrain the GRB models with the most optimistic parameters. Although GRBs may not be the sole source for the VHE cosmic rays and neutrinos, with the upgraded version of IceCube and other large neutrino telescopes, we do hope to detect and diagnose the neutrino signals from GRBs in the near future. 1.2 Gamma-ray Burst and Jet A major part of this section is based on the recent review articles by Meszaros,P., Gehrels,N. and Zhang,B.; For details please see [28],[29],[30]

16 6 GRB Progenitors and observations Gamma-ray bursts are observed at a rate about one per day within our Hubble horizon, as an overwhelming flux of gamma-ray photons lasting for a few or tens of seconds that overshadows everything else in the universe. Their X-ray afterglows were discovered by the Beppo-Sax satellite in 1997 and later the optical band, with the redshift measurements marking their origins to be of cosmological distances. The later space missions of Swift and Fermi telescopes obtained data on their redshifts for over 200 bursts, indicating an isotropic distribution on the sky. Observationally, GRBs are classified as long GRBs with a light curve t γ 2s and short GRBs with t γ 2s. The long GRBs are explained by the collapsar scenario where a massive star M 25M undergoes core collapse while the short GRBs are more likely to be of the result of the merger or a compact binary [31], such as a double neutron star (NS-NS) or a neutron star-blackhole (NS-BH), although the nature is less known than long GRBs (See Figure.1.1). For a statistical luminosity function and redshift distribution of GRBs, see Figure.1.2. The core collapse or the binary merger is thought to lead to relativistic jets. For the collapsar case, the centrifugal force leads to the reduced gas density along the rotation axis in the star envelope. The outflow then will be collimated along this direction when it makes its way out. The relativistic beaming effect significantly boosts the observed energy and in this context the most frequently used quantity is the isotropic equivalent energy E iso = E jet (4π/Ω jet ) where Ω jet is the solid angle of the jet. For long GRBs, the averaged value is about < Ω jet /4π > 1/500. For short GRBs the value is less known, but probably higher. A lower limit of the bulk Lorentz factor of the jet can be derived from the highest energy photon observed. These high energy photons (> GeV) exceed the threshold MeV of pair production γγ e + e with lower energy photons which would degrade below the threshold before escaping from the optically thick regions. This can be avoided if the jet is moving relativistically because in the comoving frame of the outflow, where the photon distribution is isotropic, the photon energy is Lorentz translated as ε ε obs /Γ. For two observed photons ε 1 = 30GeV and ε 2 = MeV, it requires that ε 1ε 2 (m e c 2 ) 2. This gives Γ 10 2.

17 7 GRB: standard paradigm Bimodal distribution of t γ duration Short (t g < 2 s) Long (t g >2 s) Figure 1.1: Figure credit: M.Ruffert,H.Janka and P.Meszaros. The left panel shows the possible progenitors for long GRB (collapsar) and short GRB (compact binary mergers). In collapsar scenario it requires that the progenitor has sufficient rotation speed in order to form an accretion disk (red torus) while in the binary merger case the sufficient angular momentum is guaranteed. A relativistic jet is formed (yellow) along the rotation axis. Right panel: The phenomenological classification of long and short GRBs, according to the light curve of the prompt emission t γ > 2s or t γ < 2s. T 90 is the time of t γ which contains 90% of the gamma-ray flux. In the collapsar scenario a blackhole of several solar masses is formed with a Schwarzchild radius r g 3(M/M ) km. The remaining gas from the progenitor, if originally fast rotating, will lead to an accretion disk with a inner radius about r 0 = a few R g 10 7 cm. The typical variability timescale of such accretion can be estimated as t 0 v 0 /r 0 (2GM/r0) 3 1/2 o(1)ms. The gravitational energy is of the order of solar rest mass M c 2, and this is released mostly in the form of thermal neutrinos (E ν o(1) 10MeV) and gravitational waves. A smaller fraction, about erg is converted into a thermal fireball of similar

18 8 comoving temperature E ν 10MeV at r 0. The fireball is initially highly opaque in terms of the pair production process γγ e + e since the optical depth for the pair producing photon E γ > m e c 2 is estimated as τ γγ n γ σ T r 0 r 0 σ T L γ /4πr0cε 2 γ 1. The radiation pressure dominates the jet dynamics initially, and this leads to a jet acceleration as Γ r/r 0 until a saturation radius r sat ηr 0, where η = E jet /N p M p c 2 is the entropy per proton and N p is the total number of protons. On the other hand, if the jet pressure is dominated by magnetic energy, the dynamics of the jet acceleration may be different. The bulk acceleration of the jet can be described as Γ(r) (r/r 0 ) µ where 1/3 µ 2/3. The observed photon spectrum of the prompt emission can be fitted by a broken power law, or Band function [32] dn/de (E/E b ) β where for nominal values β = 1 for E < E b and β = 2 for E > E b. In the function, E b, the energy break, typically falls in the 0.2 1MeV energy bin [33]. This energy can be measured by the Swift Burst Alert Telescope (BAT), which monitors the GRB prompt gammaray emission and sends the alert to the X-ray telescope (XRT) and the UV optical telescope (UVOT) for follow-up observations on their afterglows. The prompt emission is also monitored by the Fermi satellite which has two instruments on board: the Gamma-ray Burst Monitor (GBM) whose scintillation detectors cover an energy range from 8keV to 40MeV, and the Large Area Telescope (LAT), whose pair conversion telescope covers 20MeV to 300GeV. After the launch of the Fermi satellite, several new features in the photon spectrum were observed. The GeV emission in the bright burst GRB C [34] started after the prompt emission as a second pulse, with a time delay of about 4s relative to those in the MeV band. The spectrum can be fitted by a single Band function (no second component needed), with evolving E peak, increasing at first then decreasing. In some other bursts, a second component was clearly detected such as in GRB B [35]. The GeV emission could last in the afterglow for longer than t 1000s. Many of the subsequently detected Fermi bursts show a similar phenomenon. Internal Shock Model and Fermi-I Acceleration The fireball shock model [37],[24] was the first standard picture used to ex-

19 9 Figure 1.2: Figure credit: Wanderman and Piran 2010 [36]. The left panel (upper) shows the best fit for the GRB luminosity function as a broken power law and (lower) the simulated number of observable GRB in each logarithmic luminosity bin. The right panel shows the best fit of the GRB redshift distribution and the simulated number of bursts for each redshift bin. The two boxes on each bar represent the statistical error range. plain the high energy photons from the GRB. Initially the optical depth of the jet is so high that the plasma is merely thermal. As the jet expands, in the comoving frame the gas cools down rapidly, releasing the thermal photons at the photosphere (where the optical depth drops below unity, at a radius about r ph Lσ T /4πm p c 2 cη 3 ). In order to dissipate the bulk kinetic energy into the random particle energy, diffusive shock acceleration can occur when shocks are formed. This is realized when the jet encounters the interstellar medium, driving a pair of shocks (forward shock and reverse shock). About half of the jet kinetic energy can be dissipated at this external shock when the swept up matter is about M sw M jet /η. By this condition a deceleration radius can be defined as R dec (3E iso /4πn ism m p c 2 η 2 ) 1/3. The shocked interstellar medium is shock heated, with the average energy of a particle < E > ηm p c 2. On the other hand, if shells of different bulk Lorentz factor Γ Γ are ejected by the central engine with a time difference comparable to the variability timescale t var, they collide at a radius r IS Γ 2 ct var. This radius typically lies outside the photospheric radius r ph and inside the external shock radius r dec and the shock is dubbed the name internal shock. For such internal shocks, the shock velocity in the jet comoving frame is

20 10 Γ rel = 1/2(Γ 1 /Γ 2 + Γ 2 /Γ 1 ), being non-relativistic or semi-relativistic. The charged particle bounces across the shock repeatedly, being scattered by the random magnetic turbulence. For non-relativistic or semi-relativistic shocks the particle is assumed to be isotropized in the upstream and downstream of the shock respectively before crossing the shock. Therefore an average energy gain per cycle is f = E/E = (4/3)β rel where β rel is the relative velocity between the upstream and downstream. The accelerated particles form a power-law spectrum dn/de (E/E min ) 1 (Pesc/ f) E 2 α where P esc is the escape probability per cycle. This is the phenomenological point of view of the Fermi-I acceleration process. Due to the difficulty in calculating the plasma instabilities and the feedback effects etc., it has not been fully modeled from first principles and the simulations currently can only mimic a small fraction of particles and the plasma effects. Therefore, to some extent we have to use phenomenological parameters such as the the energy fraction of the total dissipated energy ɛ d, of the random magnetic irregularities ɛ B, of the accelerated electrons ɛ e and protons ɛ p and an acceleration timescale t acc = η acc t gyr where t gyr = γmc/eb is the time for one cycle of gyro-motion. In GRB neutrino calculations, for an even more simplified scenario, an equipartition between those dissipated energy forms can be assumed, namely ɛ e = ɛ B = ɛ p. The minimum electron Lorentz factor can be assumed to equal that of the shock heated thermal electrons γ e,min = ɛ e m p /m e Γ rel in the plasma comoving frame. Above that extends a power law of Fermi-accelerated non-thermal electrons. The maximum energy of the accelerated particle is constrained by both the acceleration region size limit and the energy loss rate. The former is known as the Hillas condition [38], stating that the gyroradius of the particle at E max cannot be larger than the acceleration region size. The latter states that the highest energy particles are also limited by the energy loss rates, e.g. synchrotron, inverse Compton etc., where the acceleration timescale must be shorter than the total energy loss scale t acc t loss. See Figure.1.3 for an illustration of Fermi-I acceleration from non-relativistic shocks and magnetic reconnection. The synchrotron emission from these electrons has a peak energy E peak = α B γe,minm 2 e c 2 1ɛ 1/2 B ɛ3/2 e L 1/2 γ,52/η300t 2.5 var, 2 MeV. Here α B B/B crit is the dimensionless magnetic field strength and B crit = (m e c 2 ) 2 /ce = G is the critical magnetic field. This E peak can be compared with the observed E br break

21 11 Figure 1.3: Figure credit: P.Meszaros (left) and Bosch-Ramen (right). The left panel shows the Fermi-I acceleration from standard diffuse shock scenario. Charged particles bounce across the shock, get isotropized in each comoving frame (upstream and downstream) after being scattered by magnetic irregularities created from the shock. The particles can be accelerated to very high energies by repeatedly crossing the shock, forming a power law distribution in the energy spectrum. Right panel: it shows similar process can happen during magnetic reconnection. Charged particle repeatedly cross the reconnection layer during two converging flows of opposite magnetic field polarization (b). The reconnection can also be triggered when the flow carrying opposite magnetic field lines hits the obstacle (a). energy in the Band spectrum. For the afterglow case, the synchrotron peak energy is lower due to the decrease of the Doppler boost of the jet. The E peak shifts from X-ray to optical and radio band, lasting from minutes to months where the flux also decays as power-laws in time [39]. Alternative Models While the physics of the afterglow scenario is relatively well-understood and the predicted features such as the temporal evolution of E peak and the flux F ν are indeed confirmed by observations, several issues exist with the internal shock model. One is the radiation efficiency being too low to explain the observed flux of gamma-rays, unless the colliding shells have very different Lorentz factors, which should be less common [40],[41]. Another one is the low energy spectral index in the Band spectrum. In the nominal case it has β = 1, but for a fraction of bursts β > 2/3 is observed which is incompatible with the synchrotron emission

22 12 [42]. Besides those, the presence of electron-positron pairs [24] [43], the radiation from the secondaries from potential hadronic interactions and the rapid inverse Compton cooling of the electrons make the picture more complicated. Photospheric models An alternative model is to attribute the E peak to the blackbody spectrum at the fireball photosphere where the pairs annihilate [44]. In this photospheric scenario, the low energy spectral index is due to the Rayleigh- Jeans part of the blackbody spectrum. The high energy power-law part may be realized by the MHD turbulence scattering the thermal photons. Another group of models invoke a dissipative photosphere scenario. Shocks or magnetic reconnection [26] at the photosphere dissipate the kinetic energies to increase the radiative efficiency and the thermal particles can be accelerated, forming a high energy power law extension to reproduce the Band spectrum [45]. The GeV component can be produced when the photospheric MeV photons are up-scattered by the relativistic electrons in the external shock [46]. Magnetic models It is also possible that the jet is dominated by magnetic dynamics, with no baryon load or a sub-dominant baryon load [47],[48]. In such jets, the bulk acceleration of the jet may be different from that in the baryonic jet and therefore having different values of r sat and r pho. The jet still results in an external shock with a similar forward shock wave, but for high magnetization values the reverse shock may be absent, since v s v Alfvén c. The internal shock should also be weaker or absent. But dissipation before the external shock may be viable through magnetic reconnection, if the jet consists of layers of opposite magnetic field polarity. The reconnection rates and particle acceleration from this mechanism are even less clear. The particles may be Fermi-accelerated by the converging flow when it has enough energy to cross the reconnection layer, or it may be accelerated by the electric field in the reconnection layer [49],[50]. In this work, we do not specify the details of this acceleration and treat it in a phenomenological way, with similar parameters as the diffusive shock acceleration mentioned above. A Poynting jet (free of baryons) can be produced via the Blandford-Znajek mechanism, by extracting the energy of a fast rotating blackhole if the blackhole is coupled with the disk by magnetic fields. Hadronic models At least two natural motivations exist for the hadronic models. 1) If there is a non-negligible baryon load, the protons may also be ac-

23 13 celerated. 2) The secondaries from hadronic interactions can easily produce GeV to TeV photons in the observations [51]. Since it takes some time to develop the electromagnetic cascade from the hadronic interactions, even for a one-zone simulation, the delay between the MeV and GeV photons is naturally explained. Proton synchrotron was also proposed to explain the GeV photons. The secondaries may be responsible for an optical flash, which was observed in the naked eye burst GRB B [52] and in many other bursts as well. Besides Fermi acceleration, collisions between protons and neutrons may be important [53]. There are several motivations to speculate that the GRB jet consists of significant fraction of neutrons [54]: the ejection of neutron rich material from the collapsed core, or the free neutrons that come from the photo-disintegrated heavy nucleons. In the bulk acceleration phase of the jet, the neutrons are initially coupled to protons only through the nuclear elastic collisions. After the decoupling happens and the protons continue to be accelerated (by radiation or magnetic pressure), a relative velocity v rel develops between protons and neutrons [55]. As v rel approaches 0.5c, inelastic collisions between a proton and a neutron start to happen (in the longitudinal direction of the jet), leading to a reheating of the jet. If the velocity profile along the transverse direction of the jet is inhomogeneous, (e.g. considering a jet containing a fast jet core and a slow jet sheath), slow neutrons from the jet sheath can drift into the jet core at some radius via thermal motions, leading to inelastic pn collisions as well. The longitudinal collisional dissipation can happen below the photosphere, reproducing the Band spectrum while the transverse collision can give the GeV emission explaining the delay between the MeV and GeV signals. See Fig. 1.4 for a pn-collision scenario in a GRB jet. 1.3 Neutrino Production in GRBs During the core collapse phase, a major fraction of the gravitational energy is carried away by the emission of thermal neutrinos at MeV energies through inverse beta decay. Although the number of emitted neutrinos is huge, e.g for a 10M core, due to the tiny cross section at this energy ( cm 2 ) and different detector techniques (than IceCube), only those close-by core collapses are visible (e.g. SN 1987A).

24 14 neutron proton Slow Jet Sheath Fast Jet Core Longi. Collis. Trans. Collis. R cm R Rph R Rsat Figure 1.4: Multi-GeV neutrino production in a proton neutron collisional model. This includes the longitudinal pn collision at a typical radius R and also transverse pn collision at a radius R when considering a two-component jet model: a fast jet core and a slow jet sheath. R 0, R ph and R sat are the jet base, photospheric and saturation radius, respectively. High energy neutrinos in the GRB jet are mainly produced via pp or pγ interactions. For the pγ process, depending on the photon energy E r in the proton rest frame, the differential cross section is dominated either by the resonances at low energies 0.2 GeV < E r < 1 GeV or multi-pion production at high energies E r > 1 GeV. For the purpose of simplicity, as used in many calculations [56], only the + [1232] resonance [8] - an excited nuclear state of the proton - is considered, dσ pγ (x)/dx 500δ(x 0.3) mb where x = E r /GeV. To satisfy the condition x = 0.3, we have approximately Ep ob Eγ ob = 0.3Γ 2 /(1 + z) 2 GeV 2 in the observer frame. This means a proton with higher energy tends to interact with a photon with lower energy (see Fig.1.5, left panel). The charged pion produced has roughly 20% of the proton energy, and after it decays the energy is roughly evenly distributed between the four leptons. Therefore if the secondary pion and muon do not lose energy until they decay, the neutrino has an energy E ν 0.05E p, tracing the proton energy. However, the neutrino flux F (E ν ) is also affected by

25 15 Log dn de Figure 1.5: Left panel: Due to the [1232] resonance energy condition, a proton with high energy interacts with a photon of a lower energy through this channel, in a pγ interaction. For a Band like spectrum, typical in a GRB prompt emission, the resulting neutrino spectrum also has an energy break E νb 0.05E pb where E pb is energy of the proton that interacts with the photon at energy E γb via [1232] resonance. Right panel (credit: Stanev, High energy cosmic rays springer 2010). It shows the fitted average charged multiplicity of pion and kaon in a single pp collision event as a function of incident proton energy E p with a target proton at rest (lab frame). The grey line is the charged multiplicity for a proton interacting with an air nuclei. the photon spectrum. For multi-pion production and pp interactions, the pion distribution can be fitted by an analytical scaling formula (see Fig.1.5, right panel) and Appendix.6.2.3, although a numerical simulation will be more desirable. For pp interactions, both the neutrino energy and the flux trace those of the parent protons. If the photon density or the magnetic field density is too high in the environment, the secondary pion or muon may suffer significant energy loss via inverse Compton or synchrotron, and the neutrino flux is suppressed accordingly [57]. Various neutrino production sites in the jet are summarized in Figure.1.6. For pn collisional models, the relative drift between proton and neutron leads to multi- GeV neutrinos. For internal shock models, if the same shock accelerates protons, both pp and pγ interactions occur, leading to a neutrino spectrum ranging from multi-gev to multi-tev energies. For dissipative photospheric GRB models, the

26 16 UHE ν from GRB 7 possible (long) GRB ν-sites: e - capt (1) p,n (2) 1) at collapse, make gravitational waves + thermal ν (MeV) 2) If jet outflow is baryonic, have p,n, p,n relative drift, pp/pn collisions pγ, pp (3) (4) (5) pγ (6,7) VHE ν (GeV) 3) Int. shocks while jet is inside star can accel. protons pγ, pp/pn collisions UHE ν (TeV) 4) Photospheric shocks/mag. reconn. pγ collisions UHE ν (~ TeV) 5) Int. shocks outside star, accel. protons pγ collisions UHE ν (100 TeV) 6) Ext. rev. shock EeV ν (10 18 ev) 7) External forward shock? can accelerate CRs, but unless there is a reverse shock, photon field too dilute for effective pγ. May not produce UHENU, nor neutrons which escape Mészáros pan05 Figure 1.6: Figure credit: P.Meszaros. Possible neutrino production sites and the corresponding typical energy in a GRB event. shocks or magnetic reconnections lead to neutrinos spanning roughly a similar energy range as the internal shock models. For internal shocks outside the star and external shocks, the neutrinos peak at higher energies, from multi-tev to multi-eev, since the target photons are usually softer in these environments for pγ interactions. The detailed neutrino predictions from the above sites are discussed in Chapter Neutrino Detection, Recent Results Although neutrinos only participate in weak interactions (besides gravity), we cannot detect weak interactions directly. Instead, we identify the neutrino events by those secondary charged particles and electromagnetic radiations following the weak interaction. The early method of detecting reactor (low energy) neutrinos

27 17 uses the inverse beta decay, ν e + p e + + n, by observing the positron annihilation signal together with a slow neutron. For high energy neutrinos, the popular technique in neutrino telescopes such as IceCube is to observe the Cherenkov radiation, which is produced by the charged secondaries traveling at a speed higher than the phase speed of light in the medium. In IceCube, as the secondary particle travels in the ice, the Cherenkov photons form a cone with opening angle cosθ = 1/nβ where n is the refraction index of the ice (as a function of wavelength λ). The energy de emitted per unit length is given by the Frank-Tamm formula de = (e 2 /4π)µ(ω)sin 2 θdxdω. The intensity is higher as frequency increases (until the refraction index n(ω) becomes less than unity at very high energies to void the Cherenkov condition) and for our naked eye we usually perceive a flash of blue light. In an IceCube detector unit, the photomultiplier tube is sensitive to ultraviolet light around λ 400nm. Neutrinos can undergo charged current (CC) or neutral current (NC) interactions with the electrons or quarks in the nucleon in the atoms of the detector medium. For neutral current interaction, part of the energy is carried away by the outgoing neutrino while the recoiled electron (or nucleon) will induce a leptonic (or hadronic, for the nucleon) cascade inside the detector. If the cascaded shower is fully contained in the detector, the total deposited energy by the incident neutrino can be calculated. If it is a charged current interaction, the corresponding lepton with the same flavor is produced (with a hadronic shower if the nucleon is involved). The shape of the shower event inside the detector can be used to distinguish between those flavors. Electrons lose energy rapidly inside the medium and trigger leptonic showers. Taus have the shortest life time of the three leptons and it may decay inside the detector, producing another shower event after triggering a hadronic shower at its formation. The shape of the two showers are called lollipop or double bang event. The muons lose energy slower than electrons and have a much longer life time than taus. Therefore a long track-shaped shower can be seen inside the detector. This allows a much better angular resolution of the incident neutrino than elliptical shower events, with an accuracy about less than 1 degree and better for higher energy events. However, the tracks of muons may extend well beyond the detector scale and the energy reconstruction is worse than those fully contained showers.

28 18 The background to the high energy neutrino signal in IceCube is mainly due to the cosmic ray muons and atmospheric neutrinos. Solutions to reject those backgrounds rely on a) using the earth to block the cosmic ray particles, by looking at the up-going muons; b) considering only the showers triggered inside the detector; c) using a veto layer at the surface and at depth, to reject those events with an accompanying muon which is typical for the atmospheric neutrinos. d) considering high energy neutrino events, as the atmospheric neutrino flux decreases rapidly with energy (φ E 3.5 ) [58]. In 2013, the IceCube collaboration reported an absence of neutrinos associated with GRBs [59], using the initial 40- and 59-strings operational in the detector between the time from April 2008 to May 2009 and May 2009 to May About 200 GRBs were detected during these periods by electromagnetic means. The onset times for the GRB is marked by the earliest and latest time reported by their gamma-ray emission. No neutrino events were found both on source (within 10 degrees of the GRB) and on time (within the onset times). By taking a simple version of the GRB fireball model, assuming the proton energy to electron energy ratio ɛ p /ɛ e = 10, about 5 events are predicted, or a diffuse flux close to the Waxman-Bahcall level, about 10 8 GeVcm 2 sr 1. This is in tension with the observation, which established an upper limit from GRB diffuse neutrino flux at about GeVcm 2 sr 1 at the energy range about PeV. However, as of 2014, IceCube has detected a total of 37 high energy neutrino events [20], in which a major fraction appear to be astrophysical. Among these, 3 events are within the 1 PeV and 2 PeV energy range while the other ones have the EM deposited energy from 20 TeV to 300 TeV. This 988-day sample of 37 events has a 5.7σ excess over the atmospheric background. Among these events, 8 were seen as muon-track events and the rest were showers. The sky map of those events shows them distributed roughly isotropically. The flavor ratio is consistent with a 1:1:1. This strongly suggests the first evidence of astrophysical neutrinos, as the atmospheric neutrinos or cosmogenic neutrinos (by protons at GZK energy interacting with CMB photons) cannot explain the observed spectrum. However, the origin of these neutrinos remain unknown at this time. See Figure.1.7 for the recently reported events.

29 Figure 1.7: Figure credit: Kopper representing the IceCube Collaboration 2013 [20]. Upper panel: Skymap in equatorial coordinates from the maximum likelihood point source analysis. Those best fit locations for each event are labeled with + for shower events and for muon tracks. Lower panel: best fit values of deposited energy for each event. 19

30 Chapter 2 GeV-TeV Neutrinos from Collisional GRBs This Chapter is reproduced from the paper [60]. 2.1 Introduction The observations of multi-gev photons from GRBs recently accumulated by the Fermi satellite (e.g. [34, 61]) have pointed out the need to re-evaluate the type of models used to explain the prompt photon emission mechanisms and the location of the emission regions in these objects (e.g. [62]). In particular, concerns about the radiative efficiency of usual internal shock models, and the larger radii required to avoid two-photon degradation of the spectra have spurred the investigation of baryonic (non-mhd) jet models where the radiation arises in a jet photosphere [25, 45, 53]. In such baryon-loaded jet models, at a certain radius the timescale of the nuclear elastic collisions (which couple the proton p and neutron n components) becomes longer than the expansion timescale, i.e. the collision optical depth falls below o(1), and the two components decouple from each other [54, 55]. The protons can continue to be accelerated by the radiation, while the neutrons, which have zero electric charge, start to coast with a constant Lorentz factor. Starting at this decoupling radius and for some distance beyond, the longitudinal drift velocity between the n and p becomes v 0.5c, and they start to collide inelastically.

31 21 Such large relative velocities between n and p components also can arise in realistic jets where the bulk Lorentz factor Γ depends on the polar angle θ, which also leads to inelastic collisions [63, 53], as neutrons from the outer parts (sheath) of the jet thermally drift into the jet core. In both pictures, pions are created, which in the case of the dynamics being dominated by the baryons results in multi-gev photons and neutrino production [55, 53]. A different approach towards resolving the radiative efficiency of GRBs involves consideration of magnetically dominated jets [64, 65, 66]. Some of these magnetic models assume an almost baryon-free outflow [67, 47, 66, 68, 69], while in other cases a substantial but dynamically sub-dominant baryon load is assumed [46, 48, 70, 64, 65, 26, 71, 72]. The baryons in such jet models are expected to accelerate at a different rate than in baryonic (non-mhd) jet models, and the different dynamics leads to quantitatively different predictions for the photon spectrum [73]. Similarly, it should lead to quantitatively different neutrino spectra, which we investigate in this chapter. Unlike in the previous investigations cited above, here we consider the neutrino spectra arising from nuclear collision effects in magnetically dominated GRB outflows. In this case, both the radial np decoupling radius as well as the photosphere occur at larger radii from the central engine than in the non-magnetic case, and also the transverse drift of neutrons from the periphery of the jet into the jet core becomes important at different radii, where the physical conditions differ from those previously considered. As a consequence, multi-gev photons are produced at somewhat softer energies and with appreciable time delays [73] respect to the MeV photospheric photons, but the detailed neutrino spectrum for such magnetically dominated jets has not been considered so far. In this chapter we investigate numerically the neutrino spectrum expected in magnetically dominated baryoncollisional GRB models, taking into account both longitudinal and transverse n, p decoupling and inelastic collisions. These neutrinos are in the energy sensitivity range of IceCube and its DeepCore [74, 75, 76] sub-array. In 2.2 we briefly introduce the astrophysical model and present the method of neutrino emission calculation. In 2.3 we present the results for the expected neutrino fluxes and muon events, as well as the detection prospects with Deep Core and IceCube. A discussion and summary of the results is given in 2.4.

32 The Nuclear Collision Scenarios We consider two types of nuclear collision scenarios, one where the collisions occur as a result of longitudinal (radial) velocity drifts, and another where they occur as result of transverse (relative to the jet axis) drifts of neutrons from an outer jet sheath to an inner jet core where the bulk Lorentz factor is different. We consider both of these cases in the context of magnetically dominated jet dynamics, which differs from the usually considered baryonically dominated dynamics. See Figure.1.4 for an illustration of this scenario Neutrinos From Longitudinal Nuclear Collisions In a magnetized outflow the bulk Lorentz factor of the jet increases initially as [77, 78, 79] (r/r 0 ) 1/3 r < r sat Γ = (2.1) η r > r sat where r sat = η 3 r 0 is the saturation radius beyond which the jet material starts to coast. (This is in contrast to the baryonic dominated dynamics, where Γ r up to an r sat ηr 0 ). At the radius where the co-moving baryon collision timescale becomes longer than the adiabatic expansion timescale, the protons decouple from the neutrons, and they continue to accelerate as Γ r 1/3. The condition above is expressed as t exp = r/cγ < t col = 1/n bσ π c (2.2) where n b = L tot /4πr 2 cηγm p c 2 (2.3) is the total comoving baryon density in the flow, L tot is the jet total isotropic equivalent luminosity, and η = L tot /Ṁc2 is the dimensionless entropy or energy to mass outflow ratio. The neutrons thereafter coast with a bulk Lorentz factor Γ n = (r d /r 0 ) 1/3, where r d = η 3 π(η π /η) 3/5 r 0 (2.4)

33 23 is the decoupling radius r d r π defined from the condition (2.2) (see also MR11). Here η π is a dimensionless parameter given by L tot σ π η π = ( ) 1/ L 1/6 4πcm p c 2 54 r 1/6 0,7 (2.5) r 0 where σ π cm 2. Decoupling occurs at r d < r sat if η > η π, otherwise, if the condition (2.2) is met at a radius above r sat, decoupling never happens and the p and n just coast together. Beyond the decoupling radius, the accelerated protons collide longitudinally 1 with the neutrons, which have a smaller Lorentz factor. The collisions becomes mostly inelastic when their relative Lorentz factor Γ rel 1.3; see eq. (2.8). In the star frame, each shell encompassed within (r, r + dr) contributes to the pionproduction optical depth for those protons by an amount dτ(r) = n n (r) βσ π dr (2.6) where β is the relative speed between protons and neutrons in the lab-frame, β = [1 Γ 2 (r)] 1/2 [1 Γ 2 (r d )] 1/2 To estimate the pion spectrum, we can, for simplicity, assume that for each collision, in the center of mass frame the proton and neutron are approximately at rest after the collision, with a maximum number of pions created, which are also approximately at rest (we will use a more detailed treatment in 2.2.2). Therefore, the invariant energy is s = (p µ p + p µ n) 2 = 2(1 + Γ rel ) (2.7) where the proton Lorentz factor viewed in neutron co-moving frame is This relation is valid when both Γ p, Γ n 1. Γ rel = 1 2 (Γ p Γ n + Γ n Γ p ) (2.8) 1 neglecting random thermal motions and transverse collisions, which are discussed in the next section.

34 24 The maximum number of pions that can be created (the pion multiplicity) is λ π = ( 2(1 + Γ rel ) 2)/(m π /m p ) (2.9) with a Lorentz factor Γ π (r) = Γ p(r) + Γ d 2[1 + Γrel (r)] (2.10) The radius-dependent probability for a proton to interact with a neutron is P (r) = e τ(r) (2.11) where τ(r) = r r d dτ and dτ(r) is given by eqn. (2.6) The number of pions created per proton is N π = rmax r d λ π dp (r) (2.12) where the maximum interaction radius is estimated as r max ct duration / β 3r sat (the last equality is for the choice of nominal parameters in MR11. r max ct n decay Γ n ). Otherwise Beyond the decoupling radius r d, the protons continue to be accelerated until reaching r sat. When the energy is above the threshold to create one pion in a collision with a neutron, we define this radius as, e.g. r 1. At a larger radius, protons with greater energies create more pions per collision. The pion Lorentz factor as a function of their production radius is given by eqn. (2.10). The inelastic interactions starts from r 1 and may last to a radius above r sat. Beyond r sat, the protons coast with a Lorentz factor η and (provided η > η π ) create pions with a monochromatic energy (in the first-order approximation used in this section). In fact, the resulting neutrino spectrum, after the pions decay, will be broadened due to various factors, e.g. a) thermal motions of both protons and neutrons; b) energy dispersion of the created charged pions; c) kinematics of the pion and muon decay process; d) pion and muon cooling and re-scattering. In this the above four factors are not considered in the calculation, except in a qualitative way. This is adequate because, as discussed in and in Fig.2.1, it turns out that the dominant process for neutrino production is through transverse nuclear collisions. As an example, for a case with η = 500, r 0 = r 0,7, L tot = L 54, L p = L n = 0.5L b

35 25 (where the sub-index p, n, b refer to proton, neutron, baryon luminosity respectively), the charged pion spectrum from longitudinal nuclear collision is shown in Fig.2.1. For simplicity reasons, a Gaussian distribution is assumed to represent the dispersion. The pion spectrum from transverse collisions is also shown in the figure, anticipating the results of the more detailed treatment of the latter in and after Neutrinos From Transverse Nuclear Collisions In more realistic jet models, the jet properties vary in the transverse (θ) direction. Hydrodynamical simulations indicate a smaller Lorentz factor in the jet outer regions (jet edge) than that in the jet core [80], and qualitatively similar results appear also in some MHD outflows [81]. As a simple ideal model, we consider the transverse structure of the jet in the region outside the star to be represented by a two-step function, consisting of an inner jet core with Γ given by eqn. (2.1) and a slower outer jet, or jet sheath, with a saturation Lorentz factor of η out = 10 2 η out,2. Both inner and outer jets will have been populated with protons and free neutrons already near the black hole, where any nuclei present would have been photodissociated. Due to thermal diffusion, neutrons in the jet sheath can drift sideways into the jet core and interact with baryons in the core. For significant effects, this requires the neutrons in the jet sheath to drift through a substantial transverse distance rθ into the core. This condition can be roughly estimated as r η 6 πθ < Γ rel > r 0 /η, (2.13) where < Γ rel > is the average relative Lorentz factor between the neutron and the baryons encountered along its path 2. We estimate the number of neutrons which have drifted into the core in a timescale t (all measured in the star frame) 2 The path calculation is a well-defined but complicated problem. To derive eqn. (2.13) we set the pionization optical depth τ π = 1 along the neutron path for which its transverse displacement amounts to rθ. The path itself is not transverse to the jet axis, due to relativistic beaming according to different sheath and core bulk velocities. < Γ rel > depends on the details of the jet transverse structure, and the density along the neutron path varies due to the jet dynamics. However, eqn. (2.13) serves as a rough estimate.

36 26 dnπ AdΓ Π cm L1 T L Figure 2.1: Charged pion spectrum at the source from longitudinal and transverse nuclear collisions (analytical approximation), normalized to the Earth observer frame. The charged pions decay in the source, and are observed at Earth only via their neutrino decay products. The component L1 is from longitudinal pn collisions at radii r d < r < r sat, while the component L2 arises at r > r sat. The component T is from transverse drift pn collisions (see 2.2.2). A Gaussian dispersion is assumed for simplicity to represent the broadening of the real spectrum arising from various effects (see 2.2.1). The dominant pion (and neutrino) production comes from the transverse pn collisions, as discussed in We assume a source with L γ = 0.1L tot = erg/s, η = 500, θ jet = 0.01 and z = 0.1 (see 2.2.3). as t N (t ) πr 2 θ φ n (t)dt (2.14) 0 where the diffusive flux φ n (t) is from eqn. (14) in MR11. Meanwhile, the number of baryons (n and p) passing longitudinally through the jet core is

37 27 N (t ) πr 2 θn b ct (2.15) The number of collisions per baryon in the core is roughly N /N, because the pionization optical depth is τ 1 for those neutrons. (The case N /N 1 is less likely for typical jet parameters, and would involve protons in the core undergoing multiple scatterings resulting in cascades, which is beyond the scope of this work). From the equations above, N /N 2η 1 outη 3 π (1 T 2 ) 1/2 (yη/t ) 1/2 r (2.16) where r is expressed in units of r 0 and T is the comoving temperature in units of m p c 2 (T o(1) in this case, see MR11). The parameter y is the ratio of neutron density in the sheath to baryon density in the core, y n n,out /n b, in the star frame Parameters of the Model The neutrino spectrum at Earth depends on a number of parameters. Among these are the total luminosity L tot of the jet, which is initially mostly in magnetic form; the baryon luminosity L b, which is the the dominant energy form beyond the saturation radius r sat = η 3 r 0 (the ratio of these two being defined as ɛ b = L b /L tot ); the photon luminosity produced by dissipation process around the photosphere L ph (whose ratio to the total initial luminosity is defined as ɛ ph = L ph /L tot ). In the inner jet core, the proton and neutron luminosity are assumed, for simplicity, to be L p = L n = (1/2)L b throughout this chapter 3. We take for the ratio y of baryon density in the outer jet to that in the inner jet a nominal value y = The discussions on results from different η core and η out are presented in Fig.2.3,2.4. We adopt a nominal jet opening angle of θ = 0.01, other values being discussed in Fig.2.5. We also adopt as a standard burst duration in the source frame a value of t = 20s, which is a rough average value for long bursts. The redshift-distance 3 The outer jet neutrons can also collide with jet core neutrons and produce neutrinos. However, the relative Lorentz factor with neutrons is smaller than that with the accelerated protons in the jet core, so the neutrino production through nn collisions is less efficient than for pn collisions. Only the latter are discussed in this work.

38 28 relation used is that for a standard ΛCDM cosmology. 2.3 GeV Neutrinos and their Detectability in Deep Core and ICECUBE Neutrinos from a Single GRB The neutrinos are produced by nuclear collisions between protons and neutrons leading to pions, the charged pions subsequently decaying 4 as π ± µ ± + ν µ ( ν µ ) e ± + ν e ( ν e ) + ν µ + ν µ. (2.17) The neutrino flavor mix produced at the source is determined by eqn.[2.17], but as a result of neutrino oscillations, the neutrino flavor received at Earth depends on energy and distance. Since in this work we discuss the general case, not a specific GRB, we approximate the received neutrino flux as having equal numbers in all three flavors. To calculate the neutrino spectrum from nuclear collisions the use of a numerical code is desirable. The commonly used PHYTIA-8 code uses a minimum threshold energy of E cm = 10 GeV, which for the energies considered here leads to inaccuracies. For this reason, we have used two different numerical methods. One method uses the publicly available code of [82]. These authors use a parametrization formula for γ,e ±,ν,and ν which is carried out separately for diffractive, non-diffractive processes and resonance-excitation processes, with a logarithmic rising pp inelastic cross section with T p. The secondary particle spectra are initially extracted out of events generated for mono-energetic protons (0.488GeV < T p < 512T ev ), using several simulation programs. The spectra are then fitted by a common parametrized function, separately for the physical processes listed above. Finally, the parameters determined for mono-energetic protons are fitted as a function of proton energy. The procedure was repeated for all those secondary particle types mentioned above. In order to approximate better the experimental data at lower 4 The charged Kaon leading decay channel is similar and can be approximated as effective pions. However, the number ratio of Kaons to pions produced by hadron collision at these energies is less than 0.1 and can be neglected for the purpose of this work.

39 29 energies, two baryon resonance contributions have been included, one representing the (1232) and the other representing multiple resonances around 1600 MeV/c 2. However, as pointed out also by [82], the pion mean energy in this code is slightly under-estimated at incident proton kinetic energy (in the fixed target lab frame) of T p 2 GeV and above, compared to experimental data. Therefore in this work we have corrected this discrepancy by multiplying the resultant neutrino energy by a factor of 1.3 using an approximate fit to the Fig.5 of their paper. This causes a slightly over-estimated pion mean energy near the threshold energy T p 0.3 GeV; however, this energy range is not of interest for our purposes in this work, since it results in a non-detectable neutrino flux is associated at the associated energy. We refer to this numerical calculation as method A. We have also developed a different code, which is independent of method A, in order to cross-check the validity of method A around energies T p 4 GeV, and in order to gain better transparency on the underlying physical processes where this is not otherwise made explicit in method A. We refer to this second method as method B. The energy T p 4 GeV is of particular importance here for at least three reasons: i) it arises naturally in the astrophysical model considered here; ii) nuclei in this range can produce substantial neutrino fluxes; iii) IceCube and its DeepCore sub-array neutrino detectors are sensitive to the details of the neutrino spectrum in the energy range GeV. In method B, the charged pion spectrum is approximated by a radial scaling [83], based on the apparent Feynman scaling violation at x R E/( s/2) 1. The meson decay kinematics are well established, and in method B we follow the formulation of [84] 5. The method of calculation is summarized in Appendix A comparison of the muon neutrino and anti-neutrino spectra at the source calculated using method A and B is plotted in Fig.2.2, for an incident proton energy T p = 3.8 GeV. It is seen that the two results agree well with each other. We note that a) both methods would result in a very small number of of neutrinos which violate energy-momentum conservation. Although the total amount of energy involved in these neutrinos is well below a fraction 10 3, and thus negligible, we have nonetheless applied a cutoff in the spectrum beyond the energy where this occurs for both method A and B. b) both these methods simulate pp collisions, 5 Noting a typo in this reference, where in his Eq.[14], second line, η ν should be replaced by ξ.

40 Log 10 EdN de o o o o o o o o o o ooooooooooooooooooooooooooooooooooo oo E GeV Figure 2.2: The sum of the muon neutrino and anti-neutrino spectrum from nuclear collisions, calculated in the stellar frame using the numerical method A (curve marked by + ) and the method B (curve marked by o ), as discussed in Both spectra are normalized to one pp collision event. In the outer jet comoving frame (where target protons are at rest) the incident proton kinetic energy is T p = 3.8 GeV. In the case shown here the outer jet Lorentz factor is η out = 70 and the inner jet s is η core = 700. instead of pn collisions. For pp collisions, the π + multiplicity is therefore greater than that of π due to charge conservation near the pion creation threshold energy. At s few GeV or higher, the two multiplicities tend to equal each other. In method A we sum the neutrino and anti-neutrino from both π + and π channels. In method B, Hillas used one fitted formula to describe both π + and π spectra. Thus, for the summed neutrino and anti-neutrino spectrum resulting from the decay products of these mesons, the discrepancy between pp and pn collisions becomes less noticeable. The muon neutrino and anti-neutrino differential number fluence at Earth for a single burst (neutrinos per energy decade, integrated over the duration of the outburst) is shown, after oscillations, in Fig. 2.3,2.4,2.5. In Fig.2.3,2.4, we note a saturation effect in the dependence of L ν on L p : at low L p, L ν grows fast with it

41 31 and then stabilizes above L p erg/s. This effect is mainly due to the change of relative Lorentz factors between η core and η out by choices of different astrophysical parameters. Γ rel grows with L p if we fix other parameters, and saturates at some L p,0 level (may be already saturated at the low end of L p in parts of our parameter space. A higher Γ rel is associated with more secondary leptons (including neutrinos) per pn collision. However when saturated, the received neutrino flux grows at a slower rate with L p, because the increase of fluence is then only due to the fact that we have more protons in the jet. As discussed in 2.2, most of these neutrinos come from transverse drift collisions, rather than from radial drifts. We have used parameters representative of standard long GRB, extending from moderate to high intrinsic luminosity, and the fluence is calculated for a nominal redshift of z = 0.1, corresponding to a luminosity distance of 450 Mpc in a standard Ω m = 0.28, Ω V = 0.72, h = 0.72 cosmology. This is at the lower end of the classical redshift distribution, since for the GeV neutrino energies considered here only rare, very nearby bursts might be expected to be detectable individually, due to the detector effective area decrease with energy. The typical neutrino fluence in Fig.2.3,2.4,2.5 are GeV cm erg cm 2 for z = 0.1. Since pp collisions are expected to produce a comparable amount of energy in photons from π 0 decay as in neutrinos, it is important to check that such a photon luminosity 6 does not violate electromagnetic observation constraints. There are no gamma-ray detections at TeV energies so far, and the smattering of GeV detections involve bursts typically at much higher redshifts than considered here, so the possible constraints are mainly the kev energy range fluence statistics from the BATSE 4B [85] compilation of bursts. We can conservatively assume that at most a fraction 0.3 of the total π 0 photon energy will appear in the kev range, i.e erg cm 2. Such fluences are marginally compatible with the BATSE 4B statistics, which does include some objects with fluences 10 3 erg/cm 2 (BATSE does not provide redshift information, but it is known that most BATSE bursts are at z 1; here we considered our bursts at z < 0.1, and if these were placed at the typical z > 1, they would show a BATSE fluence erg cm 2 ). Thus, our electromagnetic fluences 6 An evaluation of the final photon spectrum would require the use of a detailed electromagnetic cascade code, which is beyond the scope of this work.

42 32 appear compatible with the current observations. The number of muon events arising from the above incident muon neutrino and anti-neutrino fluences is calculated based on the specific detector characteristics. For the IceCube full 86-string operation, the effective area is larger than that of Deep Core at 100 GeV, these effective areas being given by [75]. For a detection relying exclusively on neutrinos one would need to consider GRBs which show at least one muon event. However, bursts able to give an average of > 0.01 muon events are also of interest, because of the expectation of natural fluctuations in the distance or in the luminosity, and because in some cases temporally coincident electromagnetic observations can be expected. The burst values of L γ and redshift z yielding different muon event numbers for different burst parameters are shown by the contours in Fig.2.6. The average rate per year at which GRBs occur producing 1 muon events can be estimated using the GRB luminosity and redshift distributions discussed in 2.3. For a baryon to photon luminosity ratio of 10, i.e. ɛ b = 10ɛ e 1, with inner to outer jet Lorentz factor contrasts of η core = 10η out = 300, 700, 1000, respectively, this average rate is expected to be around /yr, 0.04/yr and 0.06/yr Diffuse Neutrino Flux The diffuse neutrino flux from all GRBs in the sky can be calculated using a GRB luminosity distribution (luminosity function, LF) and a redshift distribution (RD) 7. Here we adopt for these the functions given in [36](see Fig.1.2), (L γ /L ) m 1 L min < L γ < L φ(l γ ) (2.18) (L γ /L ) m 2 L < L γ < L max (1 + z) n 1 z < z 1 R GRB (z). (2.19) (1 + z) n 2 z > z 1 Here equation [2.18] is the luminosity function, and Eqn. [2.19] is the redshift distribution function, L γ is the peak photon luminosity (typically in the MeV range), L min = erg/s, L = erg/s, L max = erg/s, m 1 = 0.2, 7 For the method of calculation, see e.g. Appendix B of [86]

43 33 Log 10 E dn AdE, cm E GeV Figure 2.3: The ν µ + ν µ fluence in the Earth observer frame from a single burst, integrated over the outburst duration, for a nominal redshift of z = 0.1. The other nominal parameters are the same as in Fig.[2.1], except for η core = 10η out = 300, 700, 1000, indicated with dashed, solid and dot-dashed lines respectively. The lines in each style are arranged from top to bottom in the sequence L tot L b = 10 55, , 10 54, , erg/s. m 2 = 1.4, n 1 = 1.0, n 2 = 1.4, z 1 = 3, and we have used these values from the [36] parameter ranges which best reproduce the actual GRBs with measured L γ and z statistics, e.g. Fig.2 of [36], Fig.4 of [33], or [87]. We note that, especially in the range z 0.5, the index of the redshift distribution, i.e. the rate, is very uncertain, due to poor statistics. The differential co-moving rate of GRBs at a redshift z is given by R(z) = R GRB(z) dv (z) (1 + z) dz (2.20) where V (z) is the comoving volume in the ΛCDM cosmology model adopted, with Ω m = 0.28, Ω V GRB per unit redshifts is given by = 0.72 and H 0 = 72 Km/s/Mpc. The differential number of

44 34 Log 10 E dn AdE, cm E GeV Log 10 E dn AdE, cm E GeV Figure 2.4: The ν µ + ν µ fluence in the Earth observer frame from z = 0.1, for different parameters. Top panel: solid lines from bottom to top are for η core = 300, 400, 500, 600, 700; dashed lines from bottom to top are for η core = 2000, 1500, 1000, 900, 800, with a fixed η out = 100. Bottom panel: solid lines, bottom to top are η core = 300, 400, 500, 600, 700; dashed lines, bottom to top are η core = 1500, 1200, 1000, 900, 800 with a running η out = 0.1η core. Both panels are for L p = erg/s, the value of η core yielding the largest neutrino flux being around η 700. When η core approaches 300 the pion multiplicity becomes smaller due to a smaller relative Lorentz factor between outer and inner jet particles (top figure, Eq. [2.9]) or due to a slightly smaller diffusion rate (Eq. [2.16]). For very high η core 10 3, the pion multiplicity similarly decreases.

45 35 Log 10 E dn AdE, cm E GeV Figure 2.5: The ν µ + ν µ number fluence in the Earth observer frame from z = 0.1, calculated for η core = 10η out = 700, L T L b = erg/s for various jet opening angles θ jet = 0.01, 0.02, 0.03, 0.04, 0.05, 0.1 (top to bottom). dn = ρ 0 φ(l γ )R(z)dlogL γ dz (2.21) where ρ 0 is the normalization factor. In this chapter, we normalize the total electromagnetically detected GRB rate to 300/yr in the range given by Eqns. [2.18,2.19]. The neutrino flux depends on the baryon luminosity L b, and the ratio between L b and L γ adopted in is L b /L γ 10. This ratio is characteristic of many hadronic GRB models, the implied photon radiative efficiency of 10% being moderate. The resulting diffuse neutrino fluxes (neutrinos per year) are shown in Fig.2.7, and are discussed in Discussion We have calculated the neutrino emission in the range of a few GeV to a few hundred GeV arising in magnetized collisional GRB models, such as have been recently used for interpreting the electromagnetic properties of these objects. The neutrino

46 36 emission considered here arises partly from longitudinal proton and neutron collisions following their decoupling in the jet, and in larger part from collisions caused by transverse thermal drift of neutrons from an outer jet sheath into the jet core, having different bulk Lorentz factors. This neutrino production model differs from the commonly considered photohadronic pγ models, and it also differs from previous pn model calculations in incorporating explicitly the (dominant) transverse drift collision effects. Also, the emission region characteristics are here determined by the magnetically dominated dynamics, differing from those in previous neutrino calculations, which mostly use baryonic dynamics. Furthermore, the pn neutrino spectra are calculated numerically, using two different codes which are suited for the GeV range energies considered. Our present results indicate that for the burst parameters suitable to explain the photon spectra and the MeV-GeV photon lags [73] indicated by the recent Fermi satellite observations, a low level of neutrino emission is expected at GeV energies. This is in the sensitivity range of the Deep Core sub-array of IceCube, and extends into the lower range of the main IceCube array. For a neutrino detection of an individual burst, unaided by a coincident electromagnetic detection, one would require 1 muon event. For neutrino-induced muon track events in IceCube and Deep Core, considering the angular resolution (e.g. Ω 10 2 deg 2 ) and time search bins o(1) 20s per burst, the atmospheric neutrino background is low. During the total sum of the search bins (e.g. assuming 300 electromagnetically observed long GRBs in a year, from the atmospheric neutrino spectrum the expected number of background muon events is o(1) However, any GRBs which might be expected to yield 1 muon event need to be at the high end of the luminosity function, and located at very low redshifts, L γ,iso erg/s and z 0.1. From the discussion of 2.3, the occurrence of such GRB is estimated to be very rare, 1/17 per year. This estimate is uncertain, because of the poor statistics in the determination of the redshift distribution in this range. Such bursts would in general also be detectable by photon detectors such as Swift and Fermi, except for Earth occultations or possible outages. For the more frequent weaker or more distant bursts, taking into account fluctuations in the average quantities, a neutrino observation correlated with a photon detection can narrow the time bin search, increasing the effective sensitivity of the detection.

47 37 The rate of occurrence of such lower fluence GRBs can be calculated from the luminosity function and redshift distribution, and is shown in Fig.2.6. Because of the low occurrence rate of bursts which can be expected to be detected individually, it is useful to consider also the diffuse neutrino fluxes. The same luminosity function and redshift distribution as above were used for this, as discussed in 2.3, extending the integration to all bursts with z In Fig.2.6 we show the diffuse neutrino fluences over the period of a year, for a set of burst parameters η core = 10η out = 300, 700, 1000 and ɛ b = 10ɛ e 1. For these parameters, the corresponding number of average expected muon events in IceCube and its DeepCore array are estimated to be 0.03, 0.4 and 0.5 per year over the whole sky, respectively. Due to the poor statistics in the rate of low redshift GRBs, the uncertainty in these numbers could be a factor 2. The IceCube IC40+IC59 muon event rates per year being used to set upper limits on the TeV- PeV neutrino flux from putative Waxman-Bahcall GRB models (different from the present model) are larger than the the event rates discussed here, but within one order of magnitude of those for the highest η values. Thus, imposing weak limits may perhaps be possible in the long term. In conclusion, both the individual burst fluences and the expected diffuse flux in the GeV range are significantly low. Fortuitous fluctuations above the mean values could increase this somewhat, but any conclusions based on the current IceCube and Deep Core arrays are likely to require years of data accumulation with the full array. The proposed upgrades to these installations (e.g. PINGU) would help, but a next generation of larger effective volume neutrino detectors could be the best way to accelerate the detection or non-detection of GRB neutrinos in this energy range, and to test GRB models such as discussed here.

48 38 Redshift z Luminosity Log 10 L Γ erg s Redshift z Luminosity Log 10 L Γ erg s Redshift z Luminosity Log 10 L Γ erg s 1 Figure 2.6: Expected number of muon events in the full 86-strings IceCube and its Deep Core sub-array, from a single GRB with η core = 10η out = 300(top panel),700(middle panel), 1000(bottom panel), from various redshifts z and for different photon luminosities L γ, assuming a baryon to photon luminosity ratio of 10, or ɛ b = 2ɛ p 10ɛ γ. The effective detection areas are taken from [75], with the angular position averaged over the northern sky (the effective areas have some dependence on the incident angle of the neutrinos). These contours show the L γ and z ranges that give muon events.

49 39 Log 10 E dn AdE, sr 1 cm E GeV Figure 2.7: The diffuse muon neutrino and anti-neutrino fluence per year (after oscillations) for all GRBs down to z > 0.01, as discussed in 2.4, using the luminosity function and redshift distribution of 2.3. The conventions and parameters are the same as in Fig.2.6, except for using η core = 10η out = 300 (dashed), 700 (solid) and 1000 (dot-dashed).

50 Chapter 3 TeV-PeV Neutrinos from Photospheric GRBs This chapter is reproduced from the paper [88]. 3.1 Introduction The standard GRB internal shock scenario of neutrino production [56, 89] used so far to compare against the IceCube 40 string and string observations [90, 59] predicted an abundant neutrino flux in the TeV to PeV energy range. However, it assumed some simplifications in the neutrino physics. Also, the astrophysical model itself of the GRB prompt gamma-ray emission based on the same internal shocks has been the subject of discussions in the gamma-ray community [91, 42, 92, 28], due to issues with the radiation efficiency and the spectral properties in the standard version of this internal shock scenario. For this reason, modified internal shock models that address these issues (e.g. [93, 94, 95, 96]) as well as alternative models where the prompt gamma-ray emission arises in the jet photosphere have been considered (e.g. [25, 97, 45, 98, 53, 62]. In such photospheric models the high radiative efficiency is due to dissipation processes in it. A separate question that has also been the subject of debate in the astrophysics community is whether the jets in such relativistic sources are dominated

51 41 by the baryons or by magnetic fields, which imply different macroscopic acceleration rates, different proper densities in the jet rest-frame, and implying a major role for magnetic dissipation in the process of particle acceleration. Such magnetically dominated jets in GRBs have been investigated by [77, 65, 71, 26] and others, and the gamma-ray emission is ascribed in such models, again, to dissipative processes mainly in the photosphere, e.g. [64, 99]. It is unclear at present whether the above mentioned modified internal shocks or the photospheric models are best for interpreting the prompt gamma-ray emission, nor whether the jet dynamics is dominated by baryonic or magnetic stresses (e.g. [29]). However, the expected neutrino emission is strongly dependent on the specific overarching dissipation and dynamic model of GRBs. For this reason, here we explore the neutrino features of the three main types of models which are currently under consideration. We illustrate the variety of astrophysical uncertainties involved in these models, and how these can affect the expectations for detection with IceCube or future instruments. 3.2 The Dissipative Photospheric Scenarios In the typical GRB model a high energy-to-mass ratio, jet-like relativistic outflow is launched which initially accelerates with a bulk Lorentz factor Γ, averaged over the jet cross section, whose dependence on distance from the center of the explosion can be parametrized as Γ = (r/r 0 ) µ. (3.1) This behavior is assumed valid up to a saturation radius r sat = r 0 η 1/µ, where the Lorentz factor has reached the asymptotic value Γ sat = η, where η L/Ṁc2 is the dimensionless entropy of the outflow, L and Ṁ being the average energy and mass flux. The index 1/3 µ 1 ranges between the extreme µ = 1/3 magnetically dominated radial outflow and the usual µ = 1 baryonically dominated outflow regimes, e.g. [73]. In the extreme magnetic case and baryonic cases, the saturation radius is given by

52 42 η 3 r η 300 r 0,7 cm, for µ = 1/3 r sat = ηr η 300 r 0,7 cm, for µ = 1. (3.2) In the dissipative photospheric scenario, a fraction of the outflow bulk kinetic energy is converted into radiation energy via some dissipation mechanism in the neighborhood of the photosphere 1, giving rise to the prompt photon luminosity L γ = ɛ e L tot, where L tot = L 53 erg/s is the isotropic equivalent total luminosity of the jet 2. The photospheric radius is estimated by setting the Thomson optical depth τ γe n eσ T R ph /Γ = 1, where n e = n p L tot /4πRph 2 m pc 3 ηγ is the comoving density of electrons if the e + e pairs are absent 3. By using Eqn.3.1 and the condition above, we obtain R ph r 0 = = ( ) Ltot σ T 1 = 4πm p c 3 r 0 ηγ 2 ph { 1/µ η T (η T /η) 3 if η < η T η 1/µ T (η (3.3) T /η) 1/(1+2µ) if η > η T [99], where ( ) µ Ltot σ 1+3µ T η T =. (3.4) 4πm p c 3 r 0 Typically, for a magnetically dominated µ = 1/3 case the photosphere occurs in the acceleration phase r r sat, if η > η T, where η T 150L 1/6 53 r 1/6 0,7. On the other hand the photosphere occurs in the coasting phase r > r sat for η < η T, 1 We do not specify a particular dissipation mechanism; e.g. the sudden drop of photon density might trigger magnetic reconnection as proposed by [26], or it could be due to MHD turbulence [46] or shocks [45], etc. We simply assume that as far as the detectable radiation the dissipation around the photosphere plays the largest role, and we concentrate on the photospheric dissipation region. 2 We also assume that the dissipation gives rise to a prompt photon spectrum of the observed Band function type, without specifying the underlying mechanism, e.g. seed photons scattered by electrons associated with turbulent Alfven waves, synchrotron radiation from Fermi-I accelerated electron or collisional mechanism by decoupled proton and neutron [46, 64, 53, 26] 3 The presence of pairs will increase the radius of the photosphere by a factor of a few in a magnetized photosphere [99, 100] or in a baryonic photosphere where dissipation is via MHD turbulence, or by a factor if baryonic dissipation is via pn collisions [53]. In 3.5 we exemplify the effects of the effective photospheric radius being larger.

53 43 which is typical for baryonic cases, where µ = 1 and η T 1900 L 1/4 53 r 1/4 0,7. The Lorentz factor of the photosphere Γ ph has an r ph dependence for η > η T, being Γ ph = (r ph /r 0 ) L µ/(2µ+1) η µ/(2µ+1) r µ/(2µ+1) 0, while Γ ph η in the case η < η T. In the baryonic photospheres the dissipation may be due to dissipation of MHD turbulence [46] or it may occur in the form of semi-relativistic shocks [45] with Lorentz factor Γ r 1, of different kinematic origin but similar physical properties as internal shocks, with a mechanical dissipation efficiency ɛ d. These result in a proton internal energy, and result also in random magnetic fields with an efficiency ɛ B, relativistic protons with ɛ p, and relativistic electrons with ɛ e 4. In the magnetically dominated jets the total jet luminosity L tot in the acceleration phase before dissipation occurs consists of a toroidal magnetic field component and a proton bulk kinetic energy component. In the dissipation region a fraction ɛ d of L tot is assumed to be dissipated, consuming a fraction from each of the toroidal field and bulk proton energy, and resulting in proton internal energy and in a fraction ɛ B which appears as random magnetic fields, and ɛ p and ɛ e which appear as relativistic protons and relativistic electrons. In both baryonic and magnetically dominated cases we assume ɛ B + ɛ p + ɛ e = 1, and we take ɛ d 0.3 and ɛ B 1/3 as examples in this chapter. In the jet comoving frame the random magnetic field after the dissipation is parametrized by an energy density U B,random = B 2 /8π = κɛ B ɛ d L tot /(4πRphΓ 2 2 phc) (3.5) where for semi-relativistic shocks (Γ rel 1), and a compression ratio of κ 4 is assumed. For the magnetically dominated outflow, during the acceleration phase, the energy remaining in toroidal fields after dissipation is U B,toroid = (1 ɛ d )(1 Γ ph /η)l tot /(4πR 2 phγ 2 phc) (3.6) where Γ ph is the bulk Lorentz factor of the protons at the photosphere. Calculations and simulations of of such baryonic and magnetic dissipative photospheres as well as internal shocks generally result in an escaping photon spectrum similar to the observed characteristic Band spectrum [32], parametrized as 4 An alternative baryonic dissipation involves pn collisions [53] (see also [55]); here for simplicity and for intercomparison with other models we just assume shock dissipation in the photosphere, whose effects are comparable to those of magnetic dissipation.

54 44 dn γ /de (E/E br ) x ph (3.7) in the observer frame. Observationally, for average bursts at redshifts z 2 the mean values are E br 300 kev, x ph = 1 below E br and x ph = 2 above E br. In a photosphere this spectral shape is the product of the modification of a thermal spectrum by the dissipation. For the purposes of this article, we treat this photon spectrum as the input for our calculations, transformed to the rest frame of the outflow. While the bulk Lorentz factors in the photosphere and internal shock models may differ, for the purposes of comparison we adopt here as a test case the same comoving frame photon spectral break energy for the dissipation zones of the various models considered, E br = 0.01 MeV, and x ph = 1 below E br and x ph = 2 above E br. The lower and upper branches can have cut-off energies, e.g. determined by synchrotron self-absorption below and acceleration restrictions or γγ e + e pair production above, the cut-off values depending on the specific model and its parameters. For simplicity, here we adopt the same constant values of a lower limit E min = 1 ev and an upper limit E max = 0.5 MeV, which are adequate for our purposes since the neutrino results are insensitive to these values. The total luminosity of this Band-function spectrum is normalized to ɛ d ɛ e L tot, where L tot represents the total luminosity. (An additional softer thermal spectral component can also be present at the photosphere. However, the temperature of this component is estimated as T o(1) kev at the photosphere [99], corresponding to a thermal luminosity L thermal o(1) erg/s which is low compared to L tot and L γ. Hence we have neglected this component for the purposes of the present neutrino calculation.) radius When the outflow encounters the external medium, it starts to decelerate at a 3L tot t dur R d ( 4πn ISM m p c 2 η 2 )1/3 = L 1/3 t,53(t dur /10s) 1/3 n 1/3 ISM,2 η 2/3 300 (3.8) where an external shock forms which is also able to produce neutrinos. Here a uniform interstellar medium of particle density n ISM = 10 2 n ISM,2 cm 3 is assumed

55 45 and a jet outflow duration time t dur in the central engine frame. The interstellar density value does not affect the photospheric or internal shock neutrinos, but it does affect the external shock neutrinos. Here we have adopted a density which is optimistic for the external shock neutrinos, since even so the external neutrino fluxes predicted are low and more moderate densities such as the typically used n ISM = 1cm 3 would lead to even smaller external shock neutrino fluxes. The corresponding deceleration timescale is estimated as t d R d/cη. At this radius deceleration R d the external shock has fully developed, consisting of a forward shock, and possibly also a reverse shock (if the magnetization parameter σ is or has become low enough at this radius). If present, for our parameters the reverse shock is marginally in the so-called thin-shell regime, the reverse shock having become semi-relativistic as it crosses the ejecta at about the deceleration time t d. The turbulent magnetic fields generated in these external shocks lead to synchrotron radiation, as well as synchrotron self-compton (SSC) and external inverse Compton (EIC) scattering of non-thermal photons from the dissipation region near the photosphere or the internal shocks. The detailed method of calculation of these photon spectra are discussed in [99, 68, 51]. As shown below, however, the neutrino fluence from the external shock region is a few orders of magnitude lower than that from the photospheric or baryonic internal shock regions, due to a much lower photon density leading to a lower interaction rate and lower pion production efficiency. Under the assumptions made here, the input photon spectrum of a GRB with typical parameters is shown in Fig.3.1 as an example 5. For the reverse shock, we include both the self-generated photons from the reverse shock and the prompt emission as target photons for inverse Compton scattering as well as for pγ interaction. For the forward shock, we include the forward shock (FS), reverse shock (RS) and prompt photons. 5 The photospheric spectrum here does not include the effect of the relativistic leptons injected if we had included nuclear collisions [53]; the effect would be to extend the upper branch of the Band photon spectrum into the GeV range; however, it is the photons around the Band peak that affect significantly the photo-pion neutrino production discussed here.

56 46 4 prompt Log 10 E 2 dn de GeV cm SYN,F SYN,R SSC,F EIC,R EIC,F Log 10 E OBS Γ ev Figure 3.1: Photon spectrum in the observer frame for a typical GRB with parameters L tot = erg/s, t dur = 10s, η = 300, z = 1.0, n ISM = 100 cm 3, ɛ d = 0.3, ɛ e = ɛ p = ɛ B = 1/3, where ɛ d is defined as the total dissipated energy from jet total energy, in the forms of ɛ e,ɛ p and ɛ B. The subindex ph refers to photosphere, prompt refers to the prompt emission from the photospheric region, for which a Band-like spectrum is assumed. The SYN,R and SYN,F are the synchrotron from the reverse and forward shock; SSC is the synchrotron-self Compton spectrum, EIC is the inverse Compton scattering of the prompt photons in the external shock region. (At very high energies, a Klein-Nishina break may be expected; however, for the neutrino calculation this contribution can be neglected since it contains very few photons, hence these KN breaks are not shown here). A smaller dissipation value for external shock emission used, ɛ B,FS = ɛ B,RS = 0.02ɛ B, suitable to explain the external shock photon emission [99]. 3.3 Method of Calculation In the baryonic dissipation regions, whether these are in the photosphere or in internal shocks beyond the photosphere, it is usually assumed that protons, as well as electrons, are accelerated through a Fermi-first order (Fermi-I) acceleration mechanism in the disordered magnetic fields created in the region. A process similar to Fermi acceleration is also expected in magnetic reconnection regions where layers of

57 47 magnetic field of opposite polarity meet and drive converging flows. [101, 102, 103]. The particles bounce back and forth in the converging flow between the layers and can reach similar maximum energies as in the usual Fermi mechanism. We assume that the injection of accelerated protons has a spectrum dn p /de E xp (E p,min < E p < E p,max ) (3.9) with x p = 2 as a nominal value. The acceleration timescale is t acc t cycle = ξ p r g/c = ξ p E p/eb c, where r g is the average gyroradius. Here we have adopted a minimum injection energy of protons E p,min = 10 GeV since the neutrino spectrum is insensitive to this value. We have also assumed a high compression ratio and weakly disordered magnetic fields, corresponding to ξ p 10. The accelerated proton spectral energy (eqn.3.9) here is normalized to a fraction of the jet total luminosity ɛ p L tot, the value of ɛ p being discussed in section 3.4. The maximum proton energy is constrained by the gyroradius being smaller than the size of the acceleration region r g < R ph /Γ ph, or by radiative cooling t p,acc < t p,cool where t p,cool is the total cooling timescale for the proton t 1 p,cool = t 1 pγ + t 1 pp +t 1 BH +t 1 sy +t 1 IC +t 1 ad. The terms on the right hand side are the photohadronic, pp collisional, Bethe-Heitler (photopair), proton synchrotron, inverse Compton and adiabatic inverse cooling timescales in the fluid comoving frame (we have dropped the prime superscript here), given respectively by t 1 pγ c de 2γpE 2γp 2 0 E n ph(e) dɛɛσ 2 pγ (ɛ)k pγ (ɛ) (3.10) ɛ TH t 1 pp = cn p σ pp (γ p )K pp (γ p ) (3.11) t 1 BH 7(m ec 2 ) 2 α f σ T c 9 2πm p c 2 γ 2 p t 1 γ p 1 dγ e γ 2 e n ph (γ e m e c 2 ) {(2γ p γ e ) 3/2 [log(2γ p γ e ) 2/3] + 2/3} (3.12) sy = 4σ T m 2 eγ p (B 2 /8π)/3m 3 pc (3.13) t 1 IC = t 1 ad 3(m ec 2 ) 2 σ T c 16γ 2 p(γ p 1)β p 0 de E 2 F (E, γ p)n ph (E) (3.14) Γc/R (3.15)

58 48 where each individual cooling inverse timescale is defined as t 1 (dγ p /dt)γ p and n ph (E) dn/dedv is the photon differential spectral density. Neutrinos result mainly from charged pion and kaon decays, to the first and second leading order of approximation respectively here. These charged mesons come from pγ and pp interactions (eqn.3.10,3.11). The calculation of neutrinos from pp and pγ interactions are summarized in Appendix In eqn.3.10,3.11, K(E) is the inelasticity function (see Appendix for values. In eqn.3.14, the kernal function F E, γ p is derived by [104]. To derive eqn.3.12, we have used a cross section σ φe (7/6π)α f σ T log(ɛ γ /2m e c 2 ) (3.16) where α f = 1/137 is the fine structure constant, σ T = 665mb is the Thomson cross section and ɛ is the photon energy in the proton rest frame. We note that although pγ pe ± has a larger cross section than the photopion process, the effective inelasticity of the proton is smaller and the relative photopion and photopair energy loss rate for protons interacting with the peak of the νf ν target photon spectrum is K φπ σ φπ /K φe σ φe 100. An expression for the function F (E, γ p ) in eqn.3.14 is given by [104]. With the above analytical approximate expressions we can calculate the energy fraction from the parent proton spectrum converted into pions, f pγ t 1 pγ /t 1 p,cool and f pp t 1 pp /t 1 p,cool. With the leading order approximation that the pions are created at the rest frame of the protons, we can obtain the pion spectrum. We also roughly approximate the produced kaon number density as 1 10% of the pions from pγ or pp interactions, motivated by [105, 106] or simulations using PYTHIA-8. Neutrinos from kaon decays are generally subdominant but they become the main component at the high end of the neutrino spectrum. At these energies charged kaons suffer less from radiative cooling than charged pions due to their larger mass and shorter lifetime. The various cooling timescales for a GRB with typical parameters are shown in Fig.3.2. The pion decay kinematics are well established. Neutrinos result mainly from the following channel: π ± µ ± + ν µ ( ν µ ) e ± + ν e ( ν e ) + ν µ + ν µ. (3.17)

59 49 Log 10 t X 1 s 1 2 ac TOTAL pg ad... pp oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo o ic bho oooo o ooo sy o oo o o Log 10 E p GeV Figure 3.2: Proton inverse cooling timescales at the photosphere as a function of proton energy in the magnetic reconnection region comoving frame, defined as t 1 = Edt/dE in s 1. The symbols used are Triangle(pg): p-gamma (photopion) interaction; Circle(bh): Bethe-Heitler (photopair); Square(pp): pp interaction; Dot(sy): proton synchrotron; Cross (ic) : proton inverse Compton; Solid line(ad): adiabatic cooling; Plus(ac) : Fermi acceleration timescale; Minus(TOTAL) : total cooling timescale. The astrophysical parameters for this GRB are L tot = erg/s, η = 300. At lower energies, protons are mainly cooled by adiabatic expansion and pp collisions. At intermediate energies in this figure, protons are mainly cooled by photopion production and at higher energies by proton synchrotron radiation. where the details of the calculation are also summarized in Appendix A high energy pion may lose a significant fraction of its energy through synchrotron radiation before it decays. Therefore we first calculate the pion spectrum after the cooling process has set in (using a method similar to eqns.[3.13,3.14 and 3.15]). Then we calculate the neutrino spectrum from the final pion and muon spectra. The µ ± has a longer mean life-time and smaller mass which makes its synchrotron cooling more severe than that of charged pions. Finally we note that the leading decay channel of the charged kaon is the same as that of the charged pion so in

60 50 this sense they can be viewed as effective pions. While the maximum proton energy at injection is determined by the condition t acc < t cool, the maximum cosmic ray energy of the escaped protons can be smaller, being given by the condition t cool > t dyn. The resultant values are summarized in Table Model Parameters and ν from a Single GRB We numerically compute the neutrino spectrum by using the method described in the previous section. Several parameters are needed: the jet total luminosity L tot and its duration in the source frame t dur, the target photon luminosity L γ = ɛ d ɛ e L tot, the Fermi accelerated proton luminosity and its power-law spectral index x p, the magnetic field calculated from the parameter ɛ B and eqn.3.5, the dissipation radius R, the outflow energy to mass ratio η and finally the source redshift z. We discuss three main representative scenarios: an extreme magnetic photosphere model where Γ r 1/3, a baryonic photosphere model where Γ r, and a modified internal shock (IS) scenario where two ejecta shells of different bulk Lorentz factors collide. For the modified internal shocks we assume that a high mechanical dissipation efficiency ɛ d is achieved, e.g. [93, 94, 95], and that the photon spectral issues raised about traditional internal shocks are avoided in such mechanisms. As far as neutrino production, the location and seed photon spectrum is similar to that in the standard internal shock, but without the simplifications of [89, 59] in the neutrino physics; that is, we treat the pγ interaction with the whole photon spectrum, not just the break region, and include besides the -resonance also multi-pion effects, kaons and detailed secondary particle distributions for the charged meson and muon decay, as in e.g. [107, 108]. We assume for the internal shocks a dissipation efficiency ɛ d = 0.3, which for comparison is taken to be similar to that of the photospheric models. For the two photospheric models, the extreme magnetic one satisfies η > η T, while for the baryonic one η < η T, so the photospheric dissipation radii are R ph η 18/5 T η 3/5 for µ = 1/3; = r 0 ηt 4 η 3 for µ = 1, (3.18)

61 51 Plot Line Fig No. jet comp. dissip. region R R 13 R dis cm Η R B 5 R max E CR,PeV source Μ event z 0.1 IC 86 Μ event z 1.0 IC 86 Ν Ph Fig3 dash 4 M Ph E 3 5 dash 4 M e ± Ph E 5 dot 4 B e ± Ph E 4 dotdash 4 5 B IS E 3 dotdash 4 B IS E 4 dotdash 4 B IS E 5 5 B IS E 3 8.4E 6 5 M Ph E 3 dot 4 5 B Ph E 4 5 B Ph E 4 Table 3.1: Parameter list for different models calculated. The common parameters are: jet total luminosity L tot = erg/s, source frame duration t dur = 10 s. source redshift z = 0.1, and the dissipation partition fractions ɛ e = ɛ B = ɛ p = 1/3 with ɛ d = 0.3. The first column identifies the different curve symbols and the figure used. The M and B in the second column refer to Magnetic dominated and Baryonic dominated. The third column identifies the type of dissipation region: Ph for photosphere with R ph from eqn.3.3, e ± Ph for pair-photosphere, and IS for internal shock of radius R IS from eqn In the next columns R R 13 cm gives the corresponding radii, η 100 is the initial dimensionless entropy, Γ and B 5 = B are the bulk Lorentz factor and comoving magnetic field (in the unit of 10 5 G) in the dissipation region. The max E CR,PeV is the maximum escaping cosmic ray (proton) energy in the source frame, calculated by setting t pγ = t dyn, or t p,syn = t dyn, or t p,cool = t p,acc, whichever gives the smallest E. The number of µ-events are in the last two columns; these are estimated from the neutrino flux and the effective area of the IceCube 86-string configuration, for a source at z = 0.1 or z = 1.0. where η T is defined in eq. (3.4). For the internal shock models, the dissipation radius at which these shocks occur is R IS cη 2 t var. (3.19) Here η is the average Lorentz factor of the two shells and t var 1 ms is the variability timescale which represent the time interval between the ejection of the two shells in the source frame. Note that a range of t var is indicated by observations, extending down to t var = s. This introduces a large uncertainty in the internal shock radius and the corresponding final neutrino spectrum. Here we use optimistic values for the internal shock neutrino production, for comparison

62 52 purposes. Unless specified otherwise, in the following we assume a nominal parameter set of L tot = erg/s, t dur = 10 s (source frame), η = 300, ɛ d ɛ e = 0.1 corresponding to an isotropic equivalent total photon luminosity of L γ erg/s (which is roughly the average luminosity from GRB statistics). The exact energy partition fractions in the jet are not well known, here we have assumed ɛ B = ɛ e = ɛ p = 0.33, ɛ d = 0.3. As an example of the fluence in the observer frame, we consider a GRB at redshift z = 1 corresponding to a luminosity distance of 6.6 Gpc, and at z = 0.1 corresponding to a luminosity distance of 450 Mpc in a standard ΛCDM cosmology with Ω m = 0.28, Ω V = 0.72, H 0 = 72km/s/Mpc. (3.20) The detailed parameters of different models are listed in Table.3.1 for which the neutrino spectra are plotted in figs. 3.4 and 3.5. The neutrino spectral fluence (ν µ + ν µ ) from a magnetically dominated µ = 1/3 GRB is shown in Fig.3.3, including both photospheric and external shock contributions. In principle, the neutrino flavor distribution at the source can be calculated from by Eqn.3.17 and then recomputed at the observer frame after neutrino oscillations. However the large range of distances, source geometry and density distribution introduce a large variability in final result, so here simply approximate the received neutrino flux as having equal numbers in all three flavors. In this type of model the dominant neutrino emission comes from the magnetic photosphere. The neutrino spectrum from the external shock peaks at a higher energy because the magnetic fields and photon densities there are lower than in the photosphere, and the charged mesons suffer much less synchrotron cooling before they decay. A uniformly distributed interstellar density of n ISM = 100 cm 3 is assumed for the external shock calculation. The neutrino fluence is nonetheless low compared to the photospheric fluence, due to the low target photon and proton column density and therefore the low pp or pγ collision rate. A smaller value of n ISM would lead to a more distant shock, and even lower neutrino fluences. Fig.3.3, as well as Fig.3.4,3.5 also illustrate the effect on the neutrino spectra of including additional physical processes besides the pγ production from the + - resonance used in many previous studies, including the recent IceCube GRB data analyses [59]. The processes included in Figs.3.3, 3.4 and the rest are the produc-

63 53 Log 10 E 2 dn de, GeV cm L tot 10^53.5erg s, Η 300 z 1.0, n_ism 100cm o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o.. o o o o. o o o o. o. o o. o. o. o Ν RS o o o o.. o. o o o. o Ν Ph Log 10 E Ν OBS GeV. Ν sπ o Ν mπ Ν K Ν pp Ν FS Log 10 E 2 dn de, erg cm 2 Figure 3.3: The observer frame ν µ + ν µ fluence E 2 dn/de for a µ = 1/3 magnetically dominated burst at z = 1. Solid line(ν Ph ): total muon neutrino spectrum from -resonance (dotted), pγ multi-pion production (circle), pp collision (cross) and kaons(square). Large dashed line (ν RS and ν F S ): total neutrino spectrum from external reverse and forward shock in the early afterglow. For the forward shock neutrino calculation, we included the photons from the reverse shock as well. tion of π ± via + in pp as well as pγ, as well as multi-pion production and K ± in pp and pγ. We note that pp collisions can become important at lower energies (before pγ interactions set in). This is especially true if the dissipation radius R dis is small, where the pp collision optical depth τ pp approaches unity or above. We also note that the neutrino fluence is comparable to the photon fluence in fig.3.1. This may be a relatively conservative value; in some other works a higher acceleration efficiency, pionization efficiency and proton luminosity e.g L p = 10L γ are adopted corresponding to a neutrino luminosity L ν a few L γ. In Fig.3.4, we also show the possible effect of having a dissipative photosphere at a radius larger than that given by Eqn.(3.3), e.g. due to pair effects. Without specifying an underlying mechanism giving rise to the dissipation and Band function spectrum this increase is uncertain. If the photon spectrum is generated by the scattering of thermal electrons associated with turbulent Alfven waves [46],

64 54 Log 10 E 2 dn de, GeV cm M,Ph M,e ± Ph B,Ph B,e ± Ph IS t_var 1ms IS t_var 10ms IS t_var 100ms L_tot 10^53.5 erg s, Η 300,z Log 10 E 2 dn de, erg cm Log 10 E Ν OBS GeV Figure 3.4: The ν µ + ν µ fluence spectra for different models assuming a source at redshift z = 1. L tot = erg/s, η = 300 are used. (External shock neutrino spectrum is not calculated here.) The models shown here are the µ = 1/3 magnetic photosphere (M,Ph), µ = 1 baryonic photosphere (B,Ph), pair-photosphere dissipation (M,e ± Ph or M,e ± Ph) and internal shock shock models (IS) with t var = 1, 10, 100 ms (black, dark gray, light gray) the maximum energy of the comoving photons can hardly exceed the electron rest mass m e c 2 and negligible pairs are produced. On the other hand if the prompt emission is due to synchrotron radiation from Fermi-I accelerated electrons e.g. [49] and the attenuation beyond E max is due to γγ e + e process, we can calculate τ γγ and the amount of pairs. The existence of pairs with such parameters (Fig.3.1) could boost the pair-photosphere radius to 20 times the original R ph. In the collisional scenario where protons and neutrons decouple [53], high energy γs and injected positrons from decaying pions induce leptonic cascades and significant pair production, which gives a similar boost of 20 to a pair-photosphere radius. Without going into specific model details, we just consider for simplicity the dissipative and spectrum formation effects associated with such larger effective pair photospheres, and use this radius for calculating the neutrino spectrum. In Fig.3.4 the curve labelled M, e ± Ph is for a magnetized dynamics photosphere a

65 55 Log 10 E 2 dn de, GeV cm M,Ph,Η 1000 M,Ph,Η 300 B,Ph,Η 1000 B,Ph,Η 300 IS,Ph,Η 1000 IS,Ph,Η Log 10 E Ν OBS GeV Log 10 E 2 dn de, erg cm 2 Figure 3.5: The neutrino spectrum from a magnetic photosphere (red upstanding triangle), a baryonic photosphere (blue inverted triangle) and an internal shock model (black square), for η = 300 (filled) η = 1000 (empty). Both the two lines for internal shock models have used t var = 1 ms. The other parameters are the same as in fig.3.4. Note that a slower jet produces a higher neutrino flux in all these cases. The more detailed quantities for those models (and in fig.3.4) are listed in Table.3.1. factor 20 larger, and the curve labelled B, e ± Ph is for a baryonic photosphere a factor 3.6 larger than the value of Eqn.(3.3). The case of an internal shock is also shown, for a dissipation radius estimated by Eqn.3.19 and variability time t var = 1 ms. Qualitatively, the harder spectra from larger radius dissipation regions arise because the magnetic field and photon density is lower at larger radii, reducing the pion and muon electromagnetic cooling. The effect of a magnetic versus a baryonic jet bulk dynamics on the neutrino signatures are compared in Fig.3.5, where we plot the neutrino spectra from a magnetic photosphere where Γ r 1/3, a baryonic photosphere where Γ r and an internal shock model. Each case is calculated for two different terminal Lorentz factors, η = 300 and η = The neutrino fluence in general decreases for increasing η in the baryonic photosphere model. The analytical expression for the photospheric radius is R η 3

66 56 in this model (Eqn.3.3), and having assumed a fixed Band function in the comoving frame, the number density of target photons is n ph u ph /E E L/R 2 η 2 η 4. Therefore, the pγ inverse cooling timescale t 1 pγ n ph η4 We also have t 1 dyn ηc/r η 4 t 1 ad. The pionization efficiency f π = t 1 pγ /t 1 ad has no dependence on η in the leading order approximation, or on R, which is treated as a function of η here. But, the synchrotron cooling for π ±, µ ± (t sync /t dec η 4 for given π, µ energy) diminishes the neutrino flux for larger η. 6. From Fig.3.5 we see that the peak energy of η = 1000 is smaller than η = 300 case in the comoving frame. This indicates strong pion cooling taking place. Therefore a higher η is associated with a smaller ν flux. For the magnetic photosphere, a larger η gives smaller R ph and Γ ph. Both are advantageous for neutrino production (although a higher magnetic field leads to stronger cooling of pions and muons, it is less severe than the baryonic case, e.g. see B column in Table.3.1). In Fig.3.6, upper panel, we compare the luminosity dependence of the single source spectra for a standard η = 300 of an optimistic internal shock model (with a high dissipation rate and t var = 1, 10, 100 ms), a magnetic photosphere model and a baryonic photosphere model. The optimistic internal shock model with t var = 1 ms gives the highest flux at energies 100 TeV, while the magnetic photosphere models have a lower flux at these energies, with their spectrum also peaking towards higher energies. In Fig.3.7 upper panel we show those cases with η = We see that the magnetic photosphere case is the least affected by η. An interesting possibility in the case of magnetic dissipation regions, e.g. due to reconnection, is that they may produce an an accelerated proton spectrum which is harder than the typical Fermi case of dn/de E 2. As discussed by [49, 50] for an extreme case all the protons entering the acceleration process are essentially confined by the magnetic field and the zero escape probability leads to a spectrum dn/de E 1. If this scenario is valid, the bulk of the energy for the accelerated protons is concentrated in the high energy end of their injection spectrum. The neutrino energy associated with these protons lies in the PeV-EeV energy range. This energy is in the sensitivity range of IceCube and proposed 6 the proton radiative cooling is also relevant; however, it is relatively weak due to their heavy mass compared to pion and muons

67 57 ARIANNA neutrino detector. However this scenario requires significant magnetic reconnection process where the toroidal magnetic field is still present. Charged pions and muons at this energy suffer strong synchrotron cooling which suppresses the neutrino spectrum significantly. Therefore, no significant neutrino emission is expected from this scenario (since at lower energies there are too few protons due to the nature of a dn/de E 1 spectrum.) 3.5 Diffuse ν Background from GRBs Since the individual source fluxes are very low, except for the unlikely event of an extremely nearby occurrence, it is useful to consider the cumulative diffuse neutrino flux from all GRBs in the sky. We calculate this diffuse flux based on two methods. Method-I uses a GRB luminosity distribution (luminosity function) and a redshift distribution, which is identical to the one we used in An alternative way, which we label Method-II, is to use the Wisconsin GRB catalog 7. We consider only long GRBs and use the parameters such as the Band photon index, fluence, redshift, photon peak energy etc. to compute the neutrino spectrum from each individual burst and stack them together. Finally we normalize the resultant flux to an all sky rate of 700 GRBs/yr 8. These two methods are then used to calculate the diffuse neutrino flux assuming that the GRB neutrinos are due to a magnetic photosphere, a baryonic photosphere, and an internal shock model (Fig.3.6,3.7). In Fig.3.6 we plot those cases with η = 300 while in Fig.3.7 η = An inspection of this Fig.3.6,3.7 (lower panel) shows that the above models, with the parameters used in this work, predict a diffuse neutrino flux which is likely to be within the current constraints set by IceCube string observations, as suggested by the fact that they lie below the two IceCube constraint lines labeled WB for IC and IC A caveat is that these constraints are upper limits on the flux as a function of the break energy assuming a Band function with α = 1, β = 2, or assuming a slope -2. They are not directly applicable to other types of spectra; thus, these constraint lines are here intended only for a Since these instruments have a limited sky coverage and operation time bin, a smaller number of GRBs per year are actually recorded

68 58 3 M,Ph B,Ph B,IS L Γ 10^53 top,52 mid,51 bottom erg s Η 300,z Log 10 E 2 dn de, GeV cm Log 10 E 2 dn de, erg cm Log 10 E OBS Ν GeV Log 10 E 2 dn de, GeVcm 2 s 1 sr 1 Diffuse Ν Μ Ν Μ Dash:M,Ph Dotted:B,Ph DotDash:IS Thick:Method I Thin:Method II WB for IC IC40 59 limit 95 CL Log 10 E OBS Ν GeV ATM Log 10 E 2 dn de, ergcm 2 s 1 sr 1 Figure 3.6: Upper panel: Neutrino fluence from a single GRB from different dissipation regions. Red, dashed: magnetic photosphere; blue, dotted: baryonic photosphere; Dot-dash: baryonic internal shock for t var = 1, 10, 100 ms. (black, dark gray, light gray). These are computed for different luminosities (in each model, L γ = 10 53, 10 52, erg/s (top,middle,bottom). Lower panel: Diffuse ν µ + ν µ neutrino spectral flux from the three models above (same line style), calculated for an all-sky GRB rate of 700/yr using statistical Method I (thick lines) and Method II (thin lines; see 3.5). Also shown is the IceCube collaboration s representation of the diffuse flux from a standard Waxman-Bahcall internal shock model, and the IC observational upper limit (see Fig.3 of [59] for description). The gray zone labeled ATM is the atmospheric neutrino spectrum. The plots from both panels suggest that the occasional electromagnetically bright GRBs can contribute significantly to the total diffuse flux.

69 59 3 M,Ph B,Ph B,IS L Γ 10^53 top,52 mid,51 bottom erg s Η 1000,z Log 10 E 2 dn de, GeV cm Log 10 E 2 dn de, erg cm Log 10 E Ν OBS GeV Log 10 E 2 dn de, GeVcm 2 s 1 sr 1 Diffuse Ν Μ Ν Μ WB for IC Dash:M,Ph Dotted:B,Ph DotDash:IS Thick:Method I Thin:Method II IC40 59 limit 95 CL ATM Log 10 E Ν OBS GeV Log 10 E 2 dn de, ergcm 2 s 1 sr 1 Figure 3.7: The parameters and conventions are the same as those in fig.3.6 except we have used η = 1000 (in fig.3.6,η = 300 is used).

70 60 rough comparison; they would have to be re-evaluated for the spectra shown here. Nonetheless, they do provide some guidance, and it appears that the only models which are close to being constrained at present are those using the most optimistic parameters; e.g. for internal shock models with a t var = 1 ms one would expect a small dissipation radius leading to a high τ pp and a high neutrino fluence in the lower energy range from pp collisions (where τ pγ has saturated to unity with τ pp increasing, making pp collisions more important.) Also, if we were to assume a higher value of ɛ p /ɛ e, such as 10 (in this work we used a value of 1, see also section. 3.4) and/or if we were to adopt a lower magnetic field fraction (ɛ B 10 2, rather than the ɛ d ɛ B = 0.1 used here), the diffuse neutrino fluence would be likely to violate the above constraints, especially for the internal shock model. However, this is for the optimistic internal shock case where one uses t var = 1 ms, whereas there is a larger uncertainty in this quantity, and in most bursts t var can often be several orders or magnitude larger (see comments in the third paragraph of 3.4). For such larger (and more reasonable) values of t var the model would appear to be still compatible with the constraints, even if a high ɛ p /ɛ e ratio were assumed. We have also included the approximate atmospheric neutrino background in Fig.3.6,3.7. It is worth noting that GRBs are transient sources whose prompt emission duration in the observer frame is within t obs dur 100 s. The angular resolution for TeV neutrino and above is within 1 deg 2. Therefore, considering the search time bin and the small solid angle set by optical observations, the effective atmospheric neutrino background is well below these GRB diffuse fluxes. In other words, even one or two muon events in IceCube correlated with photon detections would give a high signal to noise ratio. A caveat for such calculations of the diffuse background is that the statistical description of both the source physics parameters and the spatial-temporal distribution of the sources has large uncertainties; this applies to both the luminosity function (Method I) and the observed burst catalog (Method II). In Method-I, the parameters have large uncertainties; especially in the low redshift region (e.g. z < 0.3), very few GRBs are observed. However, in order to expect more than one muon event in IceCube, we need a GRB of moderate or high luminosity located at low redshift (such as the example GRB in fig.3.4 with L γ = erg/s and z = 0.1, which results in about 0.2 muon events in the IceCube 86-string configu-

71 61 ration.) The estimated rate of GRBs which satisfy N µ,icecube 1 is about o(1) per ten years. In Method-II, we actually have a very limited number of GRBs with well measured redshifts in the catalog. Those without a redshift are assigned a default redshift value of z = 2.0. Thus, improvements in the photon-based statistics of bursts, as well as neutrino observations over multi-year periods appear required. 3.6 Discussion We have calculated the ν µ + ν µ signatures expected from photospheric GRB models where the dynamics is either magnetically or baryonically dominated, including also the effects of external shocks, and have compared these signatures with those expected from the baryonic internal shock models. This comparison is timely in view of recent developments, the most pressing being the recently published Ice- Cube constraints [90, 59] on the standard internal shock GRB models. exploration of alternative models to the internal shocks has its own separate motivation, independently of the IceCube observations. One reason is the increased realization that magnetic fields may play a dominant role in the GRB phenomenon, and the dynamics of magnetically dominated GRB jet models 9 are being considered in detail [48, 65, 26]. Also, issues related to the efficiency and spectrum of standard internal shock models of the prompt γ-ray emission [42, 92, 28] have led, on the one hand, to considering modified internal shock models that address these issues, e.g. [110, 94, 95], etc., and, on the other hand, to considering the photospheric emission as the source of the prompt γ-rays, including both baryonic photospheres [25, 97, 45, 98, 53, 62] and magnetic photospheres [64, 73, 99]. Some previous calculations of neutrino spectra from baryonically dominated photospheres have been carried out [111, 112], but so far none from dissipative photospheres 10 or magnetic photospheres, which we treat here. The One of the notable features about the neutrino emission from dissipative photosphere models, both in magnetized of baryonic dynamics, is that the peak energy of the spectrum (which is around PeV, as seen in Fig. 3.3) differs from the 9 Magnetically dominated jets are naturally expected if the black hole energy is extracted via a Blandford-Znajek mechanism [109], or if the source is a temporary magnetar [71]. 10 As we were ready to submit we received a preprint on this subject [113], with results compatible with ours.

72 62 peak energies of a typical Waxman-Bahcall (WB97) [56] simplified standard internal shock model such as used in the recent IceCube studies (and in [114, 115] ). The peak energy is also higher than those for the internal shock model with t var = 1ms and η = 300, but lower than those with higher t var and η values. In part, this is due to the inclusion, in addition to resonance production, also of multi-pion effects, kaon decay, etc., which also gives a steeper spectrum above the peak, compared to the flat slope of E 2 dn/de above the peak expected in the usual WB97 internal shock model. This steepening of the spectrum above the peak is not unique to the photospheric dissipation model, it occurs also in internal shocks when we include the above additional physics beyond the -resonance, as can be seen in Fig (Such a steepening for internal shocks is also found by [107]). The energy at which the spectral peak appears depends generically on the radius of the dissipative and photon escape zones, which here we have assumed to be collocated, whether it be a photosphere or an internal shock. The effect of different radii is illustrated in Fig. 3.4, which compares the spectra and peak energies of different photospheric dissipation zones at different radii and also two internal shocks at two different radii (or variability times). The harder spectra from larger radii are expected because of the decrease of the magnetic fields and and photon densities, which allows the higher energy secondary pions and muons to decay before significant electromagnetic cooling has taken place. The difference between a magnetically dominated photosphere (M,Ph), a baryonically dominated photosphere (B,Ph) and internal shock (IS) models are compared in Fig.3.4,3.5, Fig.3.6,3.7(upper panel) and Table.3.1. The dissipation region for the M,Ph typically lies in the acceleration phase of the jet where there is a smaller Γ ph than B,Ph and IS model. The neutrino spectrum in the observer frame is affected by the efficiency of pp and pγ interaction, the cooling of the secondary charged particles (which is mainly synchrotron, determined by the comoving magnetic field) and finally Lorentz boost and redshift. The differences in neutrino spectra from these models are not quite significant for η = 300 where the dissipation region, magnetic field and Lorentz factor are similar. For η = 1000, the B,Ph and IS have harder spectra than the M,Ph model. The B,Ph radius here is rather small where pp, pγ and coolings are all efficient. The large Lorentz boost factor finally pushes the peak neutrino energy in the observer frame over the

73 63 M,Ph model. For IS model with η = 1000 the dissipation radii is much larger than M,Ph and B,Ph case. The inefficient cooling results in the highest peak energy of the three models. However, the flux is also the lowest because pp and pγ are the least efficient of these models. The maximum source-frame energy of the accelerated protons which are able to escape the acceleration region, for the models considered here, are shown in the third from the last column of Table.3.1. Leaving out any consideration of the diffuse flux, it is seen that only the internal shock models approach the highest energies associated with the GZK limit, as originally suggested by [56], while all photospheric models fall several orders of magnitude below this energy. We have not done an exhaustive parameter search 11, since our emphasis has been the neutrino emission, but it is apparent that photospheric models would not be competitive GZK sources. The detection of neutrinos from individual single sources with the 86 string IceCube, as seen from Table.3.1 in the last two columns, is rather difficult, the number of expected muon events at best being 0.2 for a magnetic photosphere model of average luminosity at z = 0.1. The only hope may be the statistically rare observation of a nearby bright GRB, or through observations of the diffuse flux. The diffuse flux offers higher prospects for an eventual detection, and a comparison of the diffuse flux expected from the three different types of models discussed, Figs. 3.6 and 3.7 lower panels, shows that an internal shock scenario with optimistic parameters such as t var = 1ms and η = 300 comes closest to being constrained by the current IceCube limits [59]. This confirms the recent calculations of [119, 120, 107]. On the other hand, a magnetic photosphere model is far from being constrained by the current limits (although the limits will have to be re-evaluated for the specific spectral shapes). A baryonic dissipative photosphere model has an even lower flux, and would take the longest to be detected. Fig.3.7 shows that a higher Lorentz factor, in this case η = 1000, decreases the constraints. The diffuse flux from a magnetic photosphere is the least affected, and both baryonic photosphere and internal shock models have higher peak energy but lower flux. 11 For standard (unmodified) internal shock models, such searches have been done by e.g.[116, 117, 118]

74 64 In conclusion, the calculations discussed in the previous sections indicate that the current IceCube limits are not yet sufficient to distinguish between the dissipative photospheric models based on magnetic or baryonic dynamics, although they are approaching meaningful limits for baryonic internal shocks outside the photospheres. Also, as of now it is not yet possible to set stringent limits on the value of ɛ p /ɛ e, i.e. the putative ratio of accelerated protons to accelerated electrons. One of the reasons, for the photospheric models, is that the magnetic reconnection or dissipation scenario is less straightforward than the simple internal shock scenario, having both more physical model uncertainties and parameters associated with them, e.g. the geometry of the plasma layers or the striped field structures, resistivity, instabilities, reconnection rate, etc. Both for photospheric and internal shock models, the fraction of protons that are injected into the reconnection or Fermi acceleration process is uncertain. The acceleration timescale and escape probability depend on the geometry and dynamics of magnetic reconnection or acceleration region. These factors are also crucial for determining the injection of a cosmic ray proton spectrum and its effect on the neutrino fluence, maximum neutrino energy and spectrum. For these reasons, and in preparation for future improved limits from longer observation times and from possible future higher sensitivity detectors, we have here investigated the typical neutrino spectral features and flux levels expected from three of the basic GRB models currently being considered. There are reasonable prospects that the detection or non-detection of these neutrino fluxes in the next decade with IceCube or next generation of large neutrino detectors will shed light on the underlying physics of the GRB jets and on the cosmic ray acceleration process in them.

75 Chapter 4 PeV-EeV Neutrinos from Pop.III GRBs This chapter is reproduced from the paper [69]. 4.1 Introduction The first generation of stars in the Universe (known as population III stars) are the earliest objects to form from the collapse of pristine gas. There is so far no direct observational evidence for their existence or properties, but simulations [121, 122, 123, 124] suggest that they are likely to form with a top-heavy initial mass distribution, with a much heavier average mass than current stars, in the range M, although models with smaller masses are also possible, e.g. [125]. Such hypermassive stars have very short life times, and except for a limited intermediate mass range, they undergo a core collapse leading to a black hole whose mass is some fraction of the initial stellar mass, depending on how much mass loss occurred during the brief stellar evolution. For sufficiently fast rotating cores, accretion of remnant gas onto the black hole can lead to the formation of a powerful jet, resulting in a gamma-ray bursts (GRB), e.g. [126, 127, 128]. In this chapter, we discuss a specific scenario where the GRB jet is initially dominated by magnetic energy (an MHD jet) [66, 68]. A very large energy output can be realized in the jet, leading to shocks which can accelerate protons to highly

76 66 relativistic energies, leading to significant neutrino emission via the photomeson process in the presence of the accompanying large photon luminosity. Although difficult to observe, these neutrinos propagate almost absorption-free across cosmological distances, and can provide valuable information about cosmic conditions during the reionization epoch, when the first structures formed. We discuss the very high energy neutrino flux expected from Pop. III GRBs and the potential for its detectability with current or future large neutrino detectors such as IceCube [129], ANITA [130], ARIANNA [131] and ARA [132] etc. In section 4.2 we discuss the method of calculation. The main features for our Pop III GRB model and its astrophysical setting are described in subsection The photon spectrum serving as the target for the photomeson process is discussed in subsection The method used to calculate the photomeson process and other competing channels for proton and secondary particle energy losses is detailed in subsection In section 4.3 we discuss the possible Pop III GRB source rates, and the potential for the detection of the corresponding neutrino fluxes. A summary and discussion of the implications is given in section Model and Calculation Method Astrophysical Input In the very massive M Population III star scenario, aside from the approximate M range leading to pair instability supernovae which leave no remnant [127], the core collapse results in a massive black hole (BH) of mass M h encompassing a substantial portion of the original mass. A fraction of these are expected to be fast rotating [133, 134], one of the requirements for GRB progenitors. For a star of initial radius R, infall of the remnant stellar gas onto the black hole leads to an accretion disk of typical outer radius R d R /4 which for a typical disk magnetization parameter α = 10 1 α 1 gets swallowed on a timescale [128] t d 7 3α ( R3 d GM h ) 1/ α 1 1R 3/2,12M 1/2 h,2.5 s, (4.1)

77 67 where M h,2.5 = M is the largest mass of the central black hole considered here, and henceforth we use the convention A X A/10 X. Extraction of the rotational energy from high mass fast-rotating black holes is expected predominantly through MHD effects via the Blandford-Znajek mechanism [109]. The corresponding BZ luminosity of the resulting jet is estimated as L BZ a2 h 128 B2 hr 2 hc. (4.2) Here a h is the dimensionless spin parameter of a Kerr BH, R h GM h /c 2 and B h is the disk magnetic field strength threading the black hole horizon, which should scale with the disk gas pressure P in the advection-dominated disk (ADAF) regime [135], Bh 2 = 8πP/β, where β = 10 β 1. Thus, B h (4 14Ṁc/3αβR2 h )1/ (α 1 β 1 ) 1 M 1 h,2.5 M 1/2 d,2.5 t 1/2 d,4 G, where M d is the accretion disk mass. Assuming a simple scaling M d = δm h, where δ 1, the total energy output of the jet is E j L BZ t d (a 2 hδ/α 1 β 1 )M h,2.5 erg, (4.3) and the isotropic equivalent energy is E iso E j (1 cos θ j ) 1 E j (2/θ 2 j ) (4.4) where (1 cos θ j ) θj 2 /2 is the beaming factor for a jet of collimation half-angle θ j 1. The ratio of the photon scattering mean free path and the jet radial dimension in the comoving frame (the scattering optical depth ) is very large near the black hole and drops outward, until it becomes unity at a photosphere located typically beyond the original stellar radius. As the jet expands beyond this photospheric radius, if the jet is strongly magnetically dominated, internal shocks are unlikely to occur [66, 68], but an an external forward shock (FS) forms as the jet sweeps up the external gaseous matter in its surrounding. For an approximately uniform external density of n/cm 3, at the time t d when the accretion process stops feeding the jet, the jet head has reached a distance r d from the central explosion,

78 68 r d ( E isoct d 4πnm p c 2 )1/ E 1/4 57.6t 1/4 d,4 n 1/4 0 cm (4.5) and the bulk Lorentz factor of the jet head is E iso Γ d ( ) 1/8 97E 4πnm p c 5 t 57.6t 1/8 3/8 3 d,4 n 1/8 0 (4.6) d e.g. [66], where from now on we will write E E iso, and m p is the proton mass. In the standard GRB shock description, the random magnetic field energy density in the comoving frame of the shocked external gas is amplified to some fraction ɛ B of the internal energy, B (32πɛ B nm p c 2 ) 1/2 Γ d 3.8ɛ 1/2 B, 2 E1/8 57.6t 3/8 d,4 n 3/8 0 G. (4.7) In the shock the electrons are Fermi-accelerated into a power law distribution, resulting in synchrotron emission in the above field, and these synchrotron photons are further subjected to scattering by the same electrons leading to a synchrotronself-compton (SSC) radiation field [68]. If by the time t d the original jet magnetization parameter has decreased, which can also be promoted by baryon entrainment from its surroundings, besides the forward shock (FS) also a reverse shock (RS) will develop, which moves into the ejecta [96], and a hydrodynamical approximation can be used for the description. Let the Lorentz factor of the unshocked jet in the source frame be Γ j, the jet head Lorentz factor be Γ d (also measured in the source frame) and the Lorentz factor of the unshocked ejecta measured in the jet head frame be Γ. Since in our case Γ j, Γ d 1, we have The comoving number density in the unshocked ejecta is Γ 1 2 (Γ d Γ j + Γ j Γ d ). (4.8) n j E iso 4πm p c 2 Γ 2 j ct. (4.9) drd 2 The Lorentz factor of the jet as inferred from observed GRB afterglows is , and in these calculations we adopt Γ j 500 as a nominal value. The majority of

79 69 the reverse shock emission is produced when the reverse shock finishes crossing the ejecta and injection of fresh electrons ceases [86]. This shock crossing time is essentially t d, at which time the ejecta thickness is approximately ct d and the shock radius is approximately r d. The details of the photon spectrum vary in time and depend on the energy of the GRB jet, the shock physics parameters and the details of the environment. The fraction of the shock energy that goes into relativistic electrons ɛ e (the electron equipartition parameter) is assumed to be the same for the forward and reverse shocks; a similar assumption is made for the forward and reverse magnetic field and the accelerated proton energy equipartition parameters ɛ B and ɛ p (relevant for neutrino emission). The jet total energy E j is reasonably well defined as a function of the disk mass and the BH spin parameter. However, the corresponding isotropic-equivalent energy E iso depends on the uncertain jet opening angle. Finally, the radiation produced depends, via the shock radius, on the external medium density n, which is largely unknown, depending on details of the star formation process. For simplicity we treat this density as a constant free parameter, our choices being guided by existing numerical simulations of early star formation and the results of analyses and modeling of lower redshift GRBs. The modeling of observed GRB afterglows suggests that for the z 8, i.e. later generation, so-called Pop. I/II GRBs, the densities of the medium in their environment typically range over 0.1 < n < 100 cm 3 [136]. The environments of the first stars prior to their collapse has so far only been inferred from model numerical simulations, which differ significantly among each other. For example, the typical early galactic gas environment could evolve as n (1 + z) 4 [137], or it might be approximately independent of redshift, n 0.1 cm 3, as a result of stellar radiation feedback [138, 139]. The small number of analyses for what are currently the most distant GRBs imply ambient densities for these high redshift GRBs which could be n cm 3 for GRB at z 6.3 [140], and n 1cm 3 for GRB at z 8.2 [141]. As nominal cases, we will discuss mainly the models in Table.4.1, the most energetic examples having M h = 300M, (a 2 h δ/α 1β 1 ) 1, E j = erg, ɛ B =

80 70 case M h /M E j /erg θ 2 ɛ e, 1 n/cm 3 A B B B C D D D Table 4.1: Pop. III GRB model parameters 0.01, ɛ p = 0.1, jet opening angles θ j = 10 1, 10 2 and an external density n = 1, 10 2, 10 4 cm 3, the corresponding isotropic equivalent energies being given by eq. (4.4). We also consider smaller black hole masses M h = 100M and 30M with correspondingly lower isotropic energies. These parameter values are reasonable, given the various uncertainties, but are by no means unique. Different energy equipartition parameters ɛ X are possible, and also lower Pop. III stellar masses [125], which would lead to even lower M h and E j than those in the table. Such lower values, however, will produce dimmer neutrino fluxes (see below) which are not favorable for detection with current or future neutrino detectors. Higher M h (e.g. [142] ) and E j than those in Table.4.1 are also possible but more speculative Photon spectra The GRB photons provide the most abundant targets for the photomeson process (nuclear collisions, despite larger cross sections, are much rarer). For times t t d, a common feature of all jet models is a photospheric spectral component, arising at a radius r ph much smaller than r d, where the photon scattering timescale equals the expansion timescale. This component has a quasi-blackbody spectrum [68], and its fluence is taken to be a fraction ɛ a = 0.1 of the jet total energy. In addition to this quasi-thermal component, the shocks contribute various nonthermal photon spectral components, which typically dominate at energies both

81 71 above and below the photospheric component. Electrons are accelerated in the shocks to a power law spectrum dn/dγ e γ p e for γ e > γ m, where γ m ɛ e m p m e f(p)(γ i 1) (4.10) is the minimum injected electron Lorentz factor in the shock comoving frame (where i =, d is for reverse and forward shock, respectively), and f(p) (p 2)/(p 1) since observations suggest an index 2 p 2.5. Accelerated electrons produce a (comoving frame) synchrotron spectrum peaking at E m 3heB 4πm e c γ2 m, (4.11) and the synchrotron photons are inverse Compton (IC) scattered by the same electrons to produce a synchro-self-compton (SSC) spectrum peaking at E SC m 2γ 2 me m. (4.12) The different parameters of the cases A-D in Table.4.1 lead to non-thermal photon spectra which can be quite different. In case A, the non-thermal photon spectrum consists of the sum of the synchrotron emission and the SSC component. In the cases B-D the energy density of the photons is significantly higher than in case A. The peak of the original SSC spectrum exceeds the threshold energy for two-photon pair formation, Em SC E γγ. Here E γγ is the γγ self-absorption energy above which high energy photons produce cascades of electron-positron pairs. The secondary pairs in turn also produce synchrotron emission and IC-scatter photons. Part of the SSC spectrum, however, can be suppressed by the Klein-Nishina effect. In cases B and C, the effects from one generation of cascade pairs are calculated. The results suggest that the second generation of pairs affects the spectrum less significantly than the first and is neglected for simplicity. In case D the photon energy density is so high in the higher mass sub-cases that the copious e ± created by γγ e ± can modify the original photon spectrum by multiple Compton scatterings. The compactness parameter (defined as the comoving optical depth to γγ effects) can be estimated as

82 72 l = σ Tɛ e E iso t 1 d 60 (4.13) 8πm e c 3 Γ 3 r d for the case D 300, and about l 34 for D 100, so the spectrum is partly thermalized (it would be completely thermalized if l 10 2, e.g. [143] ). In addition, hadronic cascade photons from neutral pion decay could also affect the spectrum, an exact evaluation of such cascades requiring a numerical calculation [143]. In general, however, the effects of hadronic cascades is sub-dominant relative to electromagnetic cascades from γγ effects [144]. To simplify things, in this work we calculate the photon spectrum of case D under two limits: D(1), where the nonthermal spectrum is approximated as a summation of the synchrotron and SSC spectrum, and D(2), where it is assumed that cascades lead to a completely thermalized (black body) spectrum. Due to the uncertainties in case D, only the forward shock emission is considered. (As it turns out, the neutrino spectra calculated for cases D(1) and D(2) give similar predictions for the flux in the ARIANNA energy band of observational interest, since in this energy range the pionization efficiency approaches unity; see 4.3). The details of these various photon spectra are calculated using the methods described in the appendix of [68]. The resulting photon spectra for all cases in Table.4.1 were calculated. In Fig we show only some of the representative cases A, B and D, expressed in the observer frame, for the mass sub-case M h = 300M. The smaller mass cases are roughly similar, downscaled versions of these Proton Acceleration, Cooling and Pion Production Here we assume that protons will be Fermi accelerated at the shocks to form a spectrum N E 2 p, e.g. [57], although a spectrum softer than E 2 is also possible. The various timescales, including the acceleration and cooling of protons, are discussed in 3.3. The calculation method for the charged pion spectrum, which eventually leads to the neutrino spectrum, is discussed in Appendix Charged pions and muons are also subject to energy loss due to radiative cooling before they decay to neutrinos and leptons. In our case, the dominant channel is

83 73 8 Log 10 EF E erg cm 2 s Log 10 E ph ev 2 Figure 4.1: Photon E 2 (dn/de) spectrum in the observer frame from a Pop. III GRB at z = 20, case A 300 (Fig are for M h = 300M ). Solid lines- left: synchrotron component, right: SSC component, both from the forward shock. Dotted: photospheric emission. Solid envelope: total of forward shock plus photospheric emission. Dashed lines- left: synchrotron component, right: SSC component, both from the reverse shock. Dashed envelope: total of the reverse shock emission. via the synchrotron cooling. Due to this effect the logarithmic slope of the neutrino spectrum steepens by 2 units [145, 57] above the muon critical energy E µb at which t sy,µ (E µb ) = t dec,µ (E µb ), (4.14) where t sy,µ is the muon synchrotron cooling time scale and t dec,µ is the muon decay time scale in the comoving frame. The neutrino spectrum is further suppressed above the pion critical energy E πb where E πb is given by t sy,π (E πb ) = t decay,π (E πb ). Relativistic neutrons produced through the channel p + γ n + π + have a much longer decay timescale [146] in the comoving frame and their effect is neglected in this calculation. The re-acceleration timescale for pions is longer than the pion decay timescale for all cases of interest, while for muons t µ,acc < t µ,decay when the magnetic field is

84 74 Figure 4.2: Photon E 2 (dn/de) spectrum in the observer frame from a Pop. III GRB at z = 20, case B 300. Solid lines- left, syn(fs): synchrotron component, right, SSC(FS): SSC component, both from forward shock. Dotted: photospheric emission. Other solid lines labeled p : emission from electron-positron pair cascades. Overall solid envelope: total forward shock emission, included photospheric emission and cascade. Dashed lines- left: synchrotron component, right: SSC component, both from reverse shock. Dashed envelope: total reverse shock emission. Thick vertical dashed line: photon-photon absorption energy. SSC spectrum (FS) above E KN (Vertical dot-dashed line) is suppressed by the Klein-Nishina effect. above 5 G. The latter can be realized in the early phase of the afterglow in our B and D case. However, for simplicity, we shall assume here that re-acceleration is inefficient for all leptons when calculating the photon and neutrino spectrum. A more detailed spectrum including the re-acceleration can be evaluated numerically, as in [86]. The total energy loss rate of the protons is given by t 1 p = t 1 i, (i = all channels) and the photo-pion cooling efficiency is defined through f π (γ p ) = t 1 pγ /t 1 p. This gives the average fraction of energy lost to pions from the injected protons at energy γ p. On average each charged pion carries a fraction 0.2 of the energy of its parent proton and each neutrino carries a fraction of 0.05 (either

85 Log 10 EF E erg cm 2 s D2 D Figure 4.3: Photon E 2 (dn/de) spectrum in the observer frame of a Pop. III GRB at z = 20, case D 300. Two extreme limits are shown: Case D(2) assumes complete thermalization of the original spectrum due to copious pair formation and Compton scatterings; Case D(1) solid lines show the original synchrotron and SSC from the forward shock, dashed is the photospheric emission, and the solid envelope is the sum of these, neglecting spectral changes due to pair formation. from pion decay or muon decay). Thus we have J ν (E ν ) = (1/4)f π (E p )f πb,µb (E p )J p (E p ) (4.15) where E ν 0.05E p and the flux J X is defined by J X EX 2 dn(e X)/dE X dt. The function f πb,µb (E p ) = Min[1, ((E π /E πb ) 2 )]{[(1/2)Min[1, (E µ /E µb ) 2 ] + (1/2)Min[1, ((E π /E πb ) 2 )]} approximates the effects of pion and muon cooling discussed above. The use of the factor 1/2 assumes that half the neutrinos come from charged pion decay and the other half from muon decay. Neutrinos with different flavors approximately contribute equally when they oscillate [146] over cosmological distances although muons are more cooled than pions before they decay and may induce a different flavor ratio [147]. In Fig we show the comoving frame cooling timescales for the various

86 76 Inverse Time Scale Log 10 t 1 s TOTAL pγ XΠ pγ p e ± Synchrotron IC Adiabatic Figure 4.4: Proton cooling, case A 300 inverse timescale in the forward shock region, plotted in the jet comoving frame. The thickness of the acceleration line shows the uncertainty in acceleration efficiency. proton interaction channels, corresponding to photon spectra of Fig , for the forward shock regions in the cases A 300, B 300 and D(2) The neutrino flux Individual source neutrino fluence Pop. III stars are expected in the re-ionization epoch, at redshifts z 7 [148, 149], and the majority could arise at redshifts z 20, with the very first objects at redshifts possibly as high as z 70 [150]. Here we calculate the neutrino spectrum for an individual Pop. III GRB using the model and approximations described in section 4.1 and 4.2, for the various cases listed in Table.4.1 of section 4.2. The total neutrino fluence spectra (the time integrated energy spectral flux) are evaluated at the deceleration time t d, when the GRB emissivity is the largest. As an example, the source fluences for the highest mass cases M h = 300M are plotted

87 77 Inverse Time Scale Log 10 t 1 s TOTAL pγ XΠ pγ p e ± Synchrotron IC Adiabatic Figure 4.5: Proton cooling, case B 300 inverse timescale in the forward shock region plotted in the jet comoving frame. in Fig.4.7 in the observer frame, for the four nominal cases A 300 through D(1, 2) 300, assuming a redshift z = 20. At low energies the spectrum is dominated by the - resonance where the proton energy reaches the threshold energy. At higher energies multi-pion processes become significant. The total emission is dominated by the interaction with the external shock photons, especially synchrotron photons (due to the original electrons and, depending on the case, secondary pairs), for which the photo-pion efficiency approaches order of unity o(1). The IC and SSC photons contribute less significantly due to their low number densities. This is the reason why the shape of the total neutrino spectrum is similar between the cases A C. However in case D(2), where photons are assumed to be complete thermalized, there is a dearth of high energy photons above the Wien tail so only very high energy protons meet the photo-pion threshold, and the neutrino spectrum is different from those in cases A C. As seen in Fig.4.7, the neutrino emission from such Population III MHDdominated GRB models is characterized by a very high peak flux energy, as well

88 Inverse Time Scale Log 10 t 1 s TOTAL pγ XΠ pγ p e ± Synchrotron IC Adiabatic Figure 4.6: Proton cooling, case D(2) 300 inverse timescale in the forward shock region, in the jet comoving frame. The cutoff in the dashed line is due to the photomeson threshold and the cutoff in the dotted line are due to the photo-pair threshold, where the photon energy in the proton frame are m π 140MeV and m e ± 1MeV respectively. as a high fluence. The latter is due to the high black hole mass, which implies a high intrinsic luminosity as well as a long duration of the external shock peak emissivity phase (which is further lengthened in the observer frame by the high redshift). The magnetic field strength at the external shock is weak compared to that in internal shocks, so the cooling effect for charged pions becomes much less significant than when internal shocks may be important, such as in lower redshift GRBs. An internal shock component is not included here, since internal shocks are unlikely in strongly MHD dominated GRBs, e.g. [66] (for internal shock neutrino emission in hydrodynamical models see, e.g. [56, 151, 152, 86, 118]). We can obtain an initial quick estimate of the number of muon events expected in a km 2 detector from one burst such as, e.g., cases B or D(1), D(2) in Fig.4.7. Taking the fluence to be E 2 (dn/de) 1 GeV cm 2 in all flavors in a band

89 79 Figure 4.7: Neutrino fluence (time-integrated energy flux) from one Pop. III GRB of M h = 300M located at z = 20. Model cases A 300 through C 300 are the labeled solid lines, model D is shown as dashed lines for the two extreme cases D(1) and D(2) corresponding to the photon spectra of Fig.4.3. The neutrino emission from the forward shocks is shown by thick lines and that from the reverse shocks as thin lines. The thick dashed line is the neutrino emission in case A (forward shock) if photospheric photon emission were absent, to show the effect of the latter. Cases D(1) and D(2) result in very similar fluence levels in the multi-pev range, because the efficiency for photo-meson pion production is nearly unity at these energies. B D(2) D(1) M h E T ev P ev P ev EeV T ev P ev P ev EeV T ev P ev P ev EeV Table 4.2: Number of muon events from an individual burst, in models B and D for M h = 300, 100, 30M from redshifts z = 10, 20, 70.

90 80 E GeV around E GeV, there are N ν cm 2 neutrinos received over a time t d,obs s, or N ν km 2 per burst. Complete mixing occurs over cosmological distances, so 1/3 of those are muon neutrinos, and for an approximate conversion probability P ν µ (E ν /TeV) (valid in the range TeV E ν PeV) this translates into N µ 4 km 2 muon events per burst from a redshift z = 20. A more accurate calculation takes into account that the conversion probability changes to P ν µ 10 2 (E ν /EeV) 0.47 in the range PeV E ν EeV, where EeV= ev. Table.4.2 shows the result of such a calculation performed numerically using the detailed spectra of Fig.4.7, which gives the number of muon events separately in the TeV-PeV and PeV-EeV energy ranges. The above are only the muon events, in addition to which there would be tau events, which can increase the signal (up to at most a factor 2 at the highest energies). These Pop. III GRB neutrino signals need to be considered against at least three different sources of background. (a) One is the diffuse atmospheric neutrino background, which has a steep spectrum, and at energies E ν GeV has an upper limit of [153] E 2 Φ E 10 9 GeV cm 2 s 1 sr 1. Taking an angular resolution circle of 0.7 degree, this gives over the duration of the burst N ν,atm km 2 or N µ,atm km 2 muons per burst, a negligible background. (b) Another background is the GZK cosmogenic neutrino background, due to the photo-meson interactions of the observed ultra-high energy cosmic rays with the cosmic microwave background photons [154, 155, 156, 157, 158, 159]. While the exact value depends on the assumed evolution with redshift of the cosmic ray sources, it is generally important only at energies higher than those where the signals predicted here could be important. (c) A third background is that which may be expected from lower redshift (Pop. I/II) GRBs, the typical model for which [160, 159] is currently close to being constrained by IceCube measurements [118, 161]. From Table.4.2 one sees that for masses M h 300M at z = 20 the signal could be a doublet or a triplet of events within a θ 0.7 degree error circle within a day, for models B 300 and D 300. In model D 300 the spectral sub-cases D(1) (no pairs) and D(2) (full thermalization due to pairs) bracket the range of possibilities, the answer being probably closer to the latter. For z 10 the signals would be larger, while for M h = 100M the signal is a factor 3 4 times smaller, and even

91 81 smaller for M h = 30M. For M h = 30M, a model B 30 would be very hard or impossible to detect, especially against a background of assumed Pop. I/II GRBs; but even for this mass, the signal might be a doublet from a redshift z 10 for D(1) 30 (note that D(2) 30 is not appropriate since no pair formation is expected for this mass) Population III GRB rates For calculating the diffuse flux we need to know the burst rate as a function of redshift, which is very uncertain due to the lack of observations of confirmed Pop. III objects of any kind. Population III GRBs are both much rarer and located at higher redshifts than the usually considered GRBs from the second and subsequent generations of stars (i.e. Pop. I/II GRBs [157, 162]). If we assume that the Pop. III GRB rate traces the Pop. III star formation rate (SFR), the observed all-sky GRB rate can be parametrized as (1 + z)dṅ GRB obs /dz φobs GRB (z) = φco SFR (z)ɛ GRBP ph (z)dv/dz [133, 68], where φ co SFR (z) is the Pop. III SFR per unit comoving volume (in units of M yr 1 Mpc 1 ), ɛ GRB is the efficiency of the GRB formation (in units of M 1 ), P ph (z) is the detection efficiency of photons for a specific instrument such as the Swift BAT, and dv/dz is the comoving volume element of the observed area per unit redshift. Adopting a specific result of the extended Press-Schechter simulation for φ co SFR (z) and assuming that ɛ GRB 10 8 M 1 (similar to that for the ordinary Pop. I/II cases) and P ph (z) 0.3 for z (which corresponds to the case that the luminosity function of the Pop. III GRBs is similar to the ordinary Pop. I/II GRBs), one obtains φ obs GRB (z) 0.5 yr 1 [133] (note that Swift BAT only covers 2π/3 of the sky, and then we use ɛ GRB and φ obs GRB (z) 6 times larger than those in [133] to obtain the all-sky GRB rate). However, for such bright bursts as we consider, the photon emission can be generally above the threshold of Swift BAT, i.e., P ph (z) 1 [66, 68]. Furthermore, the Pop. III stars can have a higher GRB formation efficiency ɛ GRB. This may be written as ɛ GRB = η beam ɛ BH η env, where η beam θ 2 j /2 is the jet beaming factor. In cases A and C, η beam = 1/200 and in the cases B and D, η beam = 1/ The efficiency for the collapse to lead to a central black hole leading to a GRB is parametrized as ɛ BH (in units of M 1 ). Several simulations, e.g., [121, 122, 123]

92 82 show that Pop. III stars are likely to have a top-heavy initial mass function (IMF, i.e. mass distribution), instead of a traditional negative index power-law (Salpeter) mass function. Stars in the mass range M undergo a disruptive pairinstability supernova explosion and leave no black hole remnant at all. Given the major uncertainties, we take here as a simple approximative example a delta function IMF leading to black holes of M h = 300M, M h = 100M or 30M, so that the black hole formation efficiency is assumed to be as high as ɛ BH o(1), o(3) or o(10) per 10 3 solar masses. The remaining parameter η env denotes an efficiency factor related to the environment under which the GRB jet can be formed. According to [133], a requisite is that the envelope of the star be removed by a binary stellar companion in order to let the jet break out, which would suppress the GRB efficiency by a further order of magnitude. This condition implicitly assumes the jet durations of order s in the GRB frame known from low redshift observations. In our model, however, the jets last t d 10 4 s, which is enough to break through even rather large stars, such as Pop. III without significant envelope mass loss. As an effective upper limit, we can assume η env o(1). The above represents an optimal theoretical case, which can be used as an upper limit, ɛ GRB M 1 for cases A 300, C 300, and M 1 for cases B 300, D 300. Taking into account the larger P ph (z), we have a theoretical (and highly uncertain) upper limit of φ GRB 10 yr 1 for cases B, D, and in principle two orders of magnitude higher for cases A, C. We consider now some possible indirect observational constraints on Pop. III GRB rates, since direct observational constraints are so far not available. afterglow of the Pop. III GRB as well as its prompt emission is bright enough to be detected by Swift and by ground based optical/near-ir telescopes [66, 68]. On the other hand, the current observed GRB rate at z > 6 is 0.6 yr 1, i.e., 3 GRBs (GRB ,GRB , and GRB ) during the 5-yr operation of Swift. This may be partly because the optical and near-ir observations are more difficult at redshifts z > 6 due to the Lyα absorption in the intergalactic medium. According to the current statistical data [163], only a small fraction 25% of GRBs detected by BAT have redshift determinations, because of bad conditions for optical and near-ir observations (not only the Lyα drop-off effect but also conditions such as weather as well as dust extinction in the host and our galaxy). The

93 83 Although the GRB redshift determination rate may be a function of redshift (i.e., higher redshift GRBs suffer stronger Lyα drop-off), we may crudely estimate that the intrinsic GRB rate at z > 6 detected by BAT is 0.6/0.25 = 2.4 yr 1. Since the BAT covers only 2π/3 of the sky, the estimate of the GRB rate at z > 6 from the isotropic sky can be yr 1. Another indirect constraint on the rate of GRBs from the Pop. III very massive stars may be their production of very long and bright radio afterglows [68]. The predicted durations at 1 GHz can be typically as long as 200 yr 1 for z > 10 radio afterglows arising from such GRBs. A comparison of the NVSS catalog (spanning over ) and the FIRST catalog (spanning over ), which effectively cover 1/17 of the sky, did not find any radio transient sources which could be due to GRB afterglows with significant flux changes over timescales of 5 yr [164, 165]. This indicates that the isotropic rate of Pop. III GRBs is constrained to be < 17/5 = 3.4 yr 1. From the above two rough estimates, we adopt in this work a conservative observational constraint on the massive Pop. III GRB rate of n b 3 yr Diffuse neutrino flux and detectability With the above estimates of the rate of Pop. III GRBs we can now evaluate the diffuse neutrino flux expected over a given integration time, for comparison with the capabilities of some current or planned neutrino telescopes such as IceCube [129], ANITA [130] and ARIANNA [131]. In Fig.4.8 we plot the diffuse neutrino flux from M h = 30M Pop. III GRBs, the lowest mass case considered, averaged over a period of a year of observation, in units of GeV cm 2 s 1 sr 1. This figure assumes, for illustrative purposes, a conservative GRB rate of n b = 1 observed event per year (taking into account beaming effects), and assumes that the bursts occur predominantly at a given redshift (or narrow range of redshifts) indicated in the figure, up to an upper limit of z = 70 for Pop. III formation [150]. The nominal case of one burst per year implies of course an anisotropic flux, but for a multi-year integration time, an averaged diffuse flux would be approximated. Also, if the rate were larger (e.g. in accord with the above observational indirect limit, say n b 3yr 1 ), the diffuse

94 84 fluxes would be higher by a factor n b. The neutrino spectra shown in Fig.4.8 are for the cases B 30 and D(1) 30 discussed above, since these are the preferred candidates for detection. However, for this M h = 30M sub-case, even from z = 10 the predicted fluxes just approach the IceCube 5 year sensitivity [130] or the ARIANNA 5 year sensitivity [131] (if in the latter we extrapolate the 6-month values to 5 years using a t 1/2 scaling, which could be too conservative if the sensitivity is signal limited). For this M h = 30M mass we have used the spectral case D(1) instead of D(2) because the lower luminosity leads to a γγ compactness too low to lead to significant thermalization. diffuse fluxes for the cases A and C are not shown, since they are too low to be of observational interest. The relatively larger chance of detection in cases B and D is attributable to the smaller jet opening angle and/or a higher external medium density. The In Fig.4.9 we plot the diffuse neutrino flux from M h = 100M Pop. III GRBs, averaged over a period of a year of observation, assuming again a GRB rate of n b = 1 event per year (including beaming effects) and assuming they arise from various redshifts. For this and the higher mass case we show the B and D(2) spectral models, the latter assuming that γγ effects have thermalized the photon spectrum. In this mass case, 5 years of observations would appear to make detection feasible if they arise predominantly from redshifts z 10, or from z 20 for a larger n b 3 yr 1. Fig.4.10 shows the diffuse neutrino fluxes predicted for the highest mass case M h = 300M, averaged over a year of observation for a GRB rate of n b = 1 events per year (beaming effects included), from various redshifts. The spectral cases shown are again the B and D(2) models, which give the highest fluxes (cases A and C giving observationally negligible fluxes, even at this high mass 1 ). For these high mass models, 5 years of observations would make detection feasible even if they arise predominantly from redshifts z 20 and the rate is as low as n b = 1 event per year. These Pop. III GRB diffuse neutrino fluxes of Figs , for all three mass values and for n b = 1 yr 1, do not exceed the Waxman-Bahcall cosmic ray limit 1 In order to reach the minimum detectability with IceCube or ARIANNA, even for M h = 300M one would need for cases A and C a rate of at least 100 yr 1 if the redshift is as low as z = 7, and an even higher rate if z is higher.

95 85 Figure 4.8: Diffuse neutrino flux in units of GeV cm 2 s 1 sr 1 for GRBs of M h = 30M, averaged over a year in the observer frame, based on the nominal assumption of a rate n b = 1 yr 1, for the cases B 30 and D(1) 30 (the latter because the compactness parameter is not large enough as to thermalize the photons to produce a D(2) 30 case). We have assumed different typical source redshifts. Also shown are the atmospheric neutrino background, the IceCube 5 year limits, the ARIANNA 6 month limits and the ANITA II 45 day limits. (including evolution in redshift [27], line marked WB bound ; see also [166]) or the nominal GZK cosmogenic neutrino flux [154, 156, 159] (line labeled GZK ), and they are compatible with the observational constraints set by the ANITA- 2 mission [130, 167]. The atmospheric neutrino background decreases extremely steep and is only relevant here at energies E ν GeV, where its value is GeV cm 2 s 1 sr 1, e.g. [129]. 4.4 Discussion The earliest macroscopic objects to arise out the primeval Universe plasma are the first generation Population III stars. These are potentially invaluable probes

96 86 Figure 4.9: Diffuse neutrino flux in the same units as Fig.4.8, for GRBs of M h = 100M, averaged over a year in the observer frame, based on the nominal assumption of a rate n b = 1 yr 1, averaged over a year in the observer frame, for the cases B 100 and D(2) 100, at different typical source redshifts, with the atmospheric neutrino background, the IceCube 5 year limits, the ARIANNA 6 month limits and the ANITA II 45 day limits. of the cosmology in the epochs z leading to the currently detected Universe. The lack of heavy elements in the primordial metal-free gas makes the cooling process less efficient than that of the metal-polluted gas as it collapses under its own gravity. The Jeans mass the critical mass above which the gas begins self-contraction spontaneously increases with temperature. Therefore, the primordial gas is expected to have higher temperature than metal-rich gas when it collapses and more massive fragments are expected, e.g. in excess of hundreds of solar masses. This is confirmed by detailed numerical simulations such as [121, 122, 123], although there is debate about the possibility of much lower masses [125]. The almost complete lack of observational data about these earliest objects is a major problem. However, objects more massive than about 30M may collapse to become gamma-ray bursts (GRBs) based on what is observationally known at

97 87 Figure 4.10: Diffuse neutrino flux in units of GeV cm 2 s 1 sr 1, for GRBs of M h = 300M, averaged over a year in the observer frame, based on the nominal assumption of a rate n b = 1 yr 1, for the cases B 300 and D(2) 300, at different typical source redshifts, with the atmospheric neutrino background, the IceCube 5 year limits, the ARIANNA 6 month limits and the ANITA II 45 day limits. lower redshifts, and these could serve as invaluable tracers for this first generation of star formation. These GRBs are expected to be bright γ-ray, X-ray and optical/infrared sources, but optical/ir detections providing redshift determinations are largely prevented by the Ly-α absorption from the intervening intergalactic gas, while gamma- or X-ray observations would be hard to disentangle from those of lower luminosity and lower redshift GRBs. For this reason, predictions about the expected high energy neutrino spectrum from Pop. III GRBs, providing a new channel completely free of absorption by the intervening medium, could provide an invaluable handle about the rate and characteristics of these objects, and through them, about the formation of the earliest stars and structures. One of the largest uncertainties is the initial stellar mass function. If the Pop. III star masses are as large as often assumed [121, 122, 123, 125] the largest black hole masses M h M used here are probably a conservative upper limit to

98 88 the stellar core collapse remnants. The corresponding Pop. III GRB luminosities (eq. (4.2)) used here are based on the electromagnetic extraction of the rotational energy of a fast rotating black hole, and the average beaming angle θ j 10 2 assumed is comparable to the values inferred for high luminosity Fermi bursts [34]. There is no agreement on the value of such parameters; for instance, ours differ from those of [168] who assumed a lower jet luminosity (i.e., a 2 h /αβ ) and larger jet angle θ j 10 1 calibrated on those of Pop. I/II low redshift GRBs, implying a larger energy input into the stellar envelope and less into the emergent jet. On the other hand, the higher jet luminosity and narrower jets adopted here lead to a higher ratio of emergent jet to envelope energy. Other uncertainties concern the rate of occurrence with redshift of Pop. III stars, the resulting Pop. III GRB rate, and the external medium density into which the GRB relativistic jet expands. For the formation rates we have based ourselves on recent numerical simulations and theoretical estimates ( 4.3.2), moderated by current upper limits on the number of possible unidentified high redshift bursts in the currently observed sample, as well as the possible contribution of their afterglows to existing transient radio source observations. A conservative upper limit that we have considered is one Pop. III burst, on average, per year. The generally considered range of redshifts of occurrence of Pop. III stars is 10 z 30, with a likely typical redshift of z 20. As far as the external gas densities, we have used 1 n 10 4 cm 3, spanning a range of plausible values. Among the cases considered, those with high mass, high luminosity and narrow beam angles lead to high isotropic equivalent energies, which together with moderate to high external densities result in relatively high fluxes. In view of the considerable uncertainty concerning the appropriate parameter values, the predicted fluxes must be considered as nominal values, to be tested via observations in order to narrow down the range of possibilities. Our calculation of the proton acceleration and the production of high energy neutrinos via photo-meson interactions in the GRBs own photon field involve new features not present in previous calculations. Typical calculations of the neutrino emission from GRBs, e.g. [56, 105, 86, 169], considered lower redshift Pop. I/II objects (c.f. [170], and generally assumed acceleration in purely hydrodynamic internal shocks (c.f. [171]). This is also the kind of bursts assumed for the recent

99 89 IceCube observational upper limits [90]. A basic difference in our case is that Pop. III GRBs are thought to involve MHD jets leading to prominent photospheric and external shock emission [128, 66, 68], where the target photon field is different. Also, due to the higher luminosity the photon density and pair formation effects are more important. This, together with a more detailed treatment of the photomeson cross section and multiplicity results in different neutrino spectra. The prospects for detecting neutrinos from single Pop. III GRBs with IceCube [129] (and KM3NeT [172] or ARIANNA [131] ) appear realistic, provided they have large masses compared to their low redshift counterparts, and provided they are efficient proton accelerators. They are also limited the case where the external medium density encountered by the jets is high, n 10 2 cm 3. Such values, although highly uncertain, are within the range of what is expected from numerical simulations. As discussed in 4.3.1, up to once a year, at PeV energies a 300M GRB at z 20 could yield a doublet or a triplet of events over a time of a day in IceCube, and in some cases even a 30M GRB at z 10 could produce a doublet over a day. In the range PeV these signals would be above a diffuse background from low-redshift Pop. I/II GRBs, and also of GZK cosmogenic neutrinos of a different origin (while atmospheric neutrinos are not important at these energies). An accurate evaluation of the signal to noise ratio would however depend on model considerations for these backgrounds, which is beyond the purpose of this work. While doublet and triplet searches have been done by IceCube over shorter timescales, 100 s, recent extensions of such searches to multiplets over day windows are of great interest for the signals discussed here. The detection of single sources would be aided if there were a simultaneous electromagnetic detection. The gamma-ray and X-ray flux would in principle be detectable [66, 68], but for the long observer frame durations t d,obs 10 5 s 1 day the rise-time is very gradual and poses difficulties for normal triggering algorithms [68]. Optical detections are out of the question, due to the blocking by the intergalactic Ly-α absorption [173, 149]. However, infrared L-band (3.4µm) or even K-band (2.2µm) detections may be possible for some models and redshifts. For instance, using the photon fluxes of Figs.4.2 and 4.3, a rough estimate indicates that for M h = 300M at z = 20, neither models B 300 nor D(2) 300 are detectable in K but are detectable in L at the level of m L 3 and 5, respectively, which is

100 90 very bright. Scaling for an M h = 30M burst, one would expect at z = 20 neither B 30 nor D(1) 30 to be detectable in K, but to be detectable in L at m L 5 and 6 respectively. The L-band at z = 20 corresponds to source-frame UV frequencies, so there could be some intra-source absorption making these dimmer. Observations of z 8 GRBs [174] and galaxy candidates [175] do not show evidence for dust, although atomic or molecular resonant scattering could conceivably have some effect. Observations in the L-band, while not common, are being done with a few telescopes, but these generally have a small field of view, so one-day transients once a year such as described above could very easily have gone unnoticed. A detection at these wavebands would be dependent on having alerts based on automated triggers from gamma-ray, X-ray or neutrino signals. The prospects for detection of the diffuse neutrino flux are also encouraging, under the above caveats of large masses, efficient proton acceleration and high external densities. Using a conservative Pop. III burst rate of n b 1 yr 1 and assuming black hole mass M h 300M or even M h 100M at z 20 would lead to a diffuse flux, averaged over five years, which at energies 1 PeV is within the reach of IceCube and ARIANNA, even in the presence of a diffuse neutrino background from lower-redshift Pop. I/II GRBs and GZK cosmogenic neutrinos of a different origin. The Pop. III GRB diffuse neutrino flux signals have a spectrum which differs significantly from that of the backgrounds mentioned above. Thus, if a sufficiently large number of events is accumulated, the spectrum should help to distinguish between these signals and the backgrounds. We note that the cosmic rays from these Pop. III GRB sources, after pγ losses within the sources and also in the CMB once outside of them, do not provide a significant contribution to the diffuse cosmic ray background. Similarly, the neutrinos they produce do not contribute significantly to the GZK cosmogenic neutrino fraction. Thus, in five years or maybe less, IceCube would be able to rule out massive GRBs whose formation rate is n b 1 yr 1 and M h 300M at redshifts z 20, or n b 3 yr 1 and M h 100M at z 20, based on diffuse PeV neutrino measurements. The same measurements which are currently setting constraints on the previously considered Pop. I/II GRB diffuse background [118, 161] will also be able to set constraints - or perhaps confirm - such models as considered

101 91 here for the Pop. III GRBs and their environment. Since the neutrino detection of either individual Pop. III GRBs or their diffuse neutrino background is only possible for large black hole masses, implying progenitor stars of M 3M h 100M, large area neutrino experiments at energies TeV would be able to address the currently unresolved question of whether Pop. III stars have very large masses or perhaps more modest masses approaching solar values. In the latter case, core collapse black holes and GRBs may not follow from the demise of this first generation of stars; the first black holes may arise from subsequent stellar generations, with smaller black hole masses, which would be much less luminous than those discussed here. (An ancillary implication would be that a faster growth or coalescence rate would be needed to go from such later low mass black holes to the supermassive ones inferred in early quasars). In the former case, the presence already in the Pop. III era of large mass M black holes resulting in anomalously luminous GRB would provide a head-start for the growth into supermassive black holes, as well as provide information about the GRB physics in an extreme regime, testing questions of jet physics and probing the star forming medium composition and density. Perhaps even more interestingly, PeV neutrino measurements could provide the first positive detections of the earliest massive structures to form in the Universe, at the so far unexplored 10 z 20 range of redshifts. Such measurements would be invaluable for a better understanding of the earliest generation of stars, tracing the cosmic structure formation and the environment conditions at the dawn of the Universe.

102 Chapter 5 Application : the GRB130427A Case This chapter is reproduced from the paper [176]. 5.1 Introduction It is expected that if a major fraction of the GRB energy is converted into ultrahigh energy cosmic rays, a detectable neutrino fluence should appear in IceCube [118]. However, the two-year data gathered by the IceCube string configuration has challenged this scenario by a null result in the search for correlation with hundreds of electromagnetically detected GRBs [59]. Constraints on the conventional internal shock fireball models have been derived [120] and several alternative models have been discussed [177, 96, 88]. Recently a super-luminous burst, GRB A, was detected simultaneously by five different satellites, with an isotropic equivalent energy of E iso ergs in gamma-rays at a low redshift of z 0.34 [178]. Disappointingly, a neutrino search for this GRB reported by the IceCube collaboration yielded a null result 1. Here we show that this null detection is not surprising, and show that it provides interesting information about the properties of this GRB, some of which are otherwise difficult to obtain through conventional electromagnetic channels. We discuss the constraints on the physical parameters of this GRB, both (a) using a modelindependent procedure patterned after that of [56] and [113] (section 5.2), and (b) 1

103 93 for three specific GRB models (IS, BPH and MPH, section 5.3), summarizing our results in section 5.4). 5.2 Model-independent constraints on R dis, Γ bulk and E CR We assume a simple GRB jet model, whose Lorentz factor averaged over the jet cross has a value Γ at the dissipation radius R d. Generally R d is model dependent and is a function of Γ. However, in the interest of generality, in this section we do not specify the underlying models, leaving this for section 5.3. Here, the parameters Γ and R d are treated as two independent variables. At R d, a fraction of jet total energy E tot (in the form of kinetic energy and a possible toroidal magnetic field energy if the jet is highly magnetized) is dissipated and converted into energy carried by accelerated cosmic rays protons ɛ p E tot, turbulent magnetic fields ɛ B E tot and high energy non-thermal electrons ɛ e E tot (the latter promptly converting into photons). For GRB A, the photon spectrum is well fitted by a Band-function spectrum with dn/de (E/E γb ) s in the observer frame, where s = 0.79 for E < E γb and s = 3.06 for E γb < E < 10 MeV, with a spectral break energy E γb = 1.25 MeV and a total isotropic equivalent E γ ergs [178]. For simplicity, in this chapter we assume ɛ e = 0.1, corresponding to a jet total energy, E tot = 10E γ. The value of ɛ B is uncertain see e.g. [179]; here we use a value ɛ B = A high magnetic field would suppress the neutrino spectrum at very high energies, since the π ± and µ ± would have time to cool by synchrotron emission before they decay, see e.g. [57]. However for GRB A, as we show below, the neutrino flux decreases rapidly as the energy increases above the peak, the final expected event rate in IceCube being insensitive to ɛ B. For the neutrino calculation, we follow the outlines in [56], and [113]; see also [113, 119, 120, 107] for detailed treatments. For this specific GRB, the analytical approximations in this section lead to an error of < 30% compared to the results obtained with the methods in section 5.3, for most of the realistic parameters. The cosmic rays are accelerated to a dn p /de E 2 spectrum up to a maximum energy E p,max

104 94 determined either by the Hillas condition or by t dyn = t acc (in the jet frame and converted to the observer frame), whichever is smaller. Here t dyn R d /Γc is the dynamical time scale and t acc o(1) E p /ebc is the acceleration time scale in the jet frame. High energy protons lose energy in the jet frame due to pγ interaction, at a rate t 1 π dγ p γ p dt = c 2γ 2 p E T H deσ pγ κe E/2γ p x 2 n(x)dx (5.1) The second integral can be solved analytically for the broken-power-law photon energy distribution n(e) = dn ph /de, while for the first integral is approximated using the 1232 resonance, in which, σ pγ = δ(e E pk ) cm 2 where E pk = 0.3 GeV is the peak of the pγ cross section, E T H = 0.2 GeV is the threshold and E = 0.3 GeV is the width of the resonance measured in the proton rest frame. κ = 0.2 is the averaged inelasticity of the pγ interaction, or E p = 5E π. It is convenient to define the pionization efficiency f π min(1, t 1 π /t 1 dyn ) (5.2) which describes the fraction of energy flowing from parent protons to pions within the dynamical time scale. Of f π, about 1/2 goes to π + and 1/2 to π 0, and neutrinos are produced by the charged pion decay π ± µ ± + ν µ ( ν µ ) e ± + ν e ( ν e ) + ν µ + ν µ. (5.3) The energy of the charged pion is approximately equally divided among the four leptons, E π = 4E ν. Due to neutrino oscillations and the large uncertainties in the exact energy and distance of the source, we assume that they arrive at the earth in equal numbers per flavor. Thus, we can express the muon + anti-muon neutrino flux (φ dn/de) in the observer frame as φ fπ<1(e) = L 2 γ,53(ɛ p /ɛ e )(ɛ γb/mev )(1 + z) 2 Γ 6 300D 2 27 R 1 13 (E/E νb ) s 3 GeV 1 cm 2 (5.4)

105 95 φ fπ 1(E) = L γ,53 (ɛ p /ɛ e )(1 + z) 3 Γ 4 300D 2 27 (E/E νb ) 2 GeV 1 cm 2 (5.5) where s = 3.06 (higher Band index) for E < E νb and s = 0.79 (lower Band index) for E > E νb. The pionization efficiency in the above equation is f π = 6.13L γ,53 R 1 13 (E γb /MeV)Γ 2 300(1 + z) 1 (E/E νb ) α 1 (5.6) with the first neutrino break energy (due to those protons interacting with the E = E γb photons) at E νb = (E γb /MeV)Γ 2 300(1 + z) 2 GeV (5.7) and a second neutrino break energy (due to synchrotron and inverse Compton cooling by charged pions, assuming Thomson regime for simplicity) 2 at E νb,2 = L 1/2 γ,53 R 13 Γ 2 300(1 + z) 1 (ɛ B, 1 + ɛ e, 1 ) 1/2 GeV (5.8) The model-independent expected number of neutrino events N tot, including all types (track and cascade events) and flavors, are shown for this GRB in Fig.[5.1]. The neutrino flux φ in the observer frame is computed using as input the two free parameters R d and Γ on a grid, and this is then integrated with the IceCube effective area profile at the source incident angle dec = 27 degrees to obtain N tot, from an energy range 10 2 E 10 9 GeV which is sufficiently broad to cover the entire neutrino spectra discussed in this chapter. The value of N tot is represented by color. Since N tot ɛ p by eqn. 5.4 and 5.5, we show five contours where N tot = 1 for different ɛ p /ɛ e values. The allowed region in the R d Γ parameter space favors a moderately high value of R d and Γ for small ɛ p ; for high proton to lepton energy ratio, e.g. ɛ p /ɛ e = 10, only large Γ and R d 2 For energy above this, it is good approximation to treat it as a cutoff for this GRB

106 96 values are admitted (blue, or top right region of the figure). This is consistent with the nature of the pγ interactions and the photon spectrum of this GRB, for two reasons: a) The wind comoving frame photon energy density u γ = L/4πR 2 d Γ 2 c decreases rapidly with R d and Γ; a low u γ suppresses f π and thus the final neutrino flux. b) The neutrino break energy, where the neutrino flux contributes most to the final N tot in IceCube, is proportional to Γ 2. A higher E νb is associated with a smaller contribution (due to the effective area function and the photon indices for this GRB). 5.3 Model dependent constraints In this section we discuss these three scenarios, labeled IS, BPH and MPH. After specifying any one of them, R d is derivable from Γ and the other parameters, and hence one degree of freedom is eliminated from the parameter space. the IS model, semi-relativistic shocks are formed by the collision of two shells of different velocities. The dissipation radius is estimated as R d,is Γ 2 c t = Γ t ms cm, where t = 10 3 t ms s is the variability time scale in milliseconds and Γ is the averaged bulk Lorentz factor of the two shells. For the two photospheric models, the dissipation is assumed to take place at the photosphere where the optical depth for a photon to scatter off an electron is unity τ γe = 1. Depending on whether the jet is dominated by the kinetic energy of the baryons or by the toroidal magnetic energy, the photospheric scenario is either a baryonic photospheric (BPH) or a magnetic photospheric (MPH) type. For the magnetic type, the fast rotating central engine (a black hole or magnetar) can lead to a highly magnetized outflow which is initially Poynting dominated with a sub-dominated baryonic load. If the magnetic field is striped, carrying alternating polarity and the jet is roughly one dimensional, the bulk acceleration of the jet is approximately Γ(r) = (r/r 0 ) 1/3 until a saturation radius r sat = r 0 η 3, where r 0 = 10 7 R 7 cm is the base radius of the jet and η = L tot /Ṁc2 is the baryon load portion. Around the photosphere region 3, we assume a major fraction of the jet energy (whether baryonic or magnetic) is dissipated, leading to proton acceleration, resulting in a 3 While in the ICMART model by [96], the magnetic dissipation is determined by the variability timescale, which resembles the internal shock model. For

107 97 proton spectrum dn p /de E 2 similar to that expected from a Fermi process, as is the case with the internal shock model, e.g. see [111, 112, 49]. For most reasonable parameters, in the magnetic MPH model we have Γ η where the jet is still in the acceleration phase at the dissipation radius. On the other hand, for the BPH and IS models, the dissipation radius almost always occurs outside the saturation radius, namely Γ = η. The determination of R d is more complicated for photosphere models than it is for the IS models, due to many factors (e.g. [177, 180].) For example, the jet may be contaminated by the electron positron pairs which will substantially increase τ γe and R phot. A more realistic consideration should also include the fact that the dissipation can start from the sub-photosphere all the way out, until the jet is saturated. The magnetic field configuration is also complex (e.g. the geometry of the layers, the reconnection rates etc) which will eventually affect the proton and neutrino spectrum. Although a multi-zone simulation is beyond the scope of this work, it would be desirable in order to increase the precision of the neutrino spectrum. In this section, we have used a calculation scheme similar to that in [88] 4. We make the same assumptions suitable for a one-zone calculation, but the consideration on the micro-processes is here more complete, compared to that of II, and is is based on a numerical code. The pionization efficiency is obtained by calculating all the leading order processes, e.g. pγ (Delta-resonance and multi-pion productions), Bethe-Heitler, pp, synchrotron, inverse Compton and adiabatic losses. The cooling of the secondary charged particles via synchrotron and inverse Compton, and the energy distribution functions for the neutrinos from pion and muon decay are also included. The expected neutrino events detectable in IceCube are plotted in Fig.[5.2,5.3,5.4], for the three models separately. The asymptotic Lorentz factor η (instead of Γ at the dissipation radius) and the ratio of accelerated protons to electrons ɛ p /ɛ e are the two input free parameters. The calculation is performed on a grid for this parameter space for each model. The resultant neutrino event number, N µ (tracks only) and N tot (track+cascade), are represented by colors. We also show the contours in each figure where N µ = 1 and N tot = 1. Constraints on IS models (Fig.[5.2]): Here we have used a variability timescale 4 The code is updated to a parallel version to allow the fast computation of a large region parameter space, for this work.

108 98 t = 1 ms. This corresponds to a minimal dissipation radius R d Γ cm which is optimally advantageous for neutrino production. A higher value of t would increase R d and decrease f π and φ from those shown in Fig.[5.1]. Thus, the constraints should be considered looser when using a higher value of t. The results suggest that ɛ p /ɛ e values from 1 to 10 can all be admitted, but for ɛ p /ɛ e > 5, Γ > 600 is required in order not to violate the IceCube null result. We note also that N tot and N µ increase when η = Γ is lowered, but their values saturate at about η = 300 and then decrease when η < 300. For η 300, f π reaches unity and protons lose almost all their energy to pions. A smaller radius is associated with a higher energy density u B and u γ, which causes the charged pions and muons to cool faster before they decay to produce neutrinos. A smaller Γ value gives a smaller E νb value, for which IceCube has a smaller effective area. Constraints on the BPH model (Fig.[5.3]): The photospheric radius is estimated as R phot σ T L tot /4πΓ 3 m p c 3. The result coincidentally resembles the IS model with t = 1 ms. However, at low Γ values, R phot rapidly increases, which is different from the R d Γ 2 behavior of the IS model. On the contrary, only at high η values does the magnetic field begin to cool the charged secondaries significantly, leading to a suppressed neutrino spectrum. Constraints on the MPH model (Fig.[5.4]): The most interesting constraints are obtained for the MPH model. Due to the nature of the magnetic jet, the photosphere, if one neglects the effects of e ± formation [99], should occur in the acceleration phase for the likely parameter values. Even if the jet is initially loaded with a small amount of baryons (a high η value), Γ at R phot is roughly a constant value 150 Γ 200 for most η choices. This fact is also revealed in Fig.[5.4] by the contours being almost parallel to the Γ-axis. The magnetic photospheric radius is larger than the R d computed for the IS model with t = 1 ms, but it is not large enough to suppress f π much below unity. The secondaries suffer somewhat less cooling from synchrotron and IC than in the IS case considered. A somewhat lower bulk Lorentz factor 5 is advantageous for neutrino production. Therefore, the MPH model generally has an equal or higher neutrino efficiency than the IS and BPH model. Although the result is insensitive to η, a relatively stringent 5 However, E νb is also lower, which decreases the event number in IceCube.

109 99 constraint on ɛ p /ɛ e 1 2 is obtained for this burst, independent of η or Γ. For very low values of η, there is also a saturation effect for reasons similar as in the IS case. 5.4 Discussion We have discussed the implications of the non-detection by IceCube in the gammaray burst GRB130427A. Using the results of the electromagnetic observations of this burst, we have analyzed the implications of this neutrino null-result for constraining the physical parameters of this burst. Using first a simplified analysis which is independent of specific GRB models, we find that the null-result implies a simple inverse relationship between the bulk Lorentz factor Γ at the dissipation radius R d and this radius, as a function of the relativistic proton to electron ratio ɛ p /ɛ e (Fig.[5.1]), which suggests values of Γ 500 and R d cm. We then performed more detailed numerical calculations for three different specific GRB models, the internal shock (IS), baryonic photosphere (BPH) and magnetized photosphere (MPH) models. We find that the IS model (Fig.[5.2]) with the shortest variability time and the highest neutrino luminosity is able to comply with the null-result constraint if its bulk Lorentz factor Γ = η , depending on ɛ p /ɛ e, a fairly modest constraint. Longer variability times only weaken the constraint. For the baryonic photosphere BPH model (Fig.[5.3]) the constraint for compliance is comparable, Γ = η depending on ɛ p /ɛ e. The most restrictive constraint is for the magnetic photosphere MPH model of Fig.[5.4]. Here it is found that, for this burst GRB130427A, the null result implies an allowed value of ɛ p /ɛ e 1 2, almost independently of the asymptotic bulk Lorentz factor η. More careful calculations of all three types of models will clearly be required for reaching firm conclusions, but based on the above considerations, the generic internal shock and baryonic photosphere models are not significantly constrained by the lack of observed neutrinos.

110 100 Figure 5.1: (See e-print for colored version) Density plot of the expected number of neutrino events (track+cascade) in IceCube for GRB A on the 2D parameter space of the dissipation radius R 13 = R d /10 13 cm and the bulk Lorentz factor Γ of the jet at this radius. This calculation uses the semi-analytical method similar to [56, 113] but assuming no specific scenario (e.g. neither an internal shock, nor other model, see II for details). The blue color (top-right region) denotes fewer events while the red (lower regions) denotes more events. The five dashed lines from top to bottom show contours where one event is expected, for different proton to electron energy ratios ɛ p /ɛ e = 10, 5, 3, 2, 1. The other two energy partition parameters are taken to be constants, ɛ e = 0.1 and ɛ B = Based on the null result in the IceCube neutrino search reported in [59], the parameter space below each contours is more likely to be ruled out for the corresponding ɛ p /ɛ e.

111 101 Figure 5.2: Density plot of expected neutrino event number in IceCube based on the internal shock model for GRB A. The total number N tot (muons + cascades) and N µ (muons only) are calculated in the 2D parameter space η and ɛ p. Here R d is the internal shock radius for a 1 ms variability timescale (see III). The value of N is represented the by color; however, for N tot and N µ the value of the color coding is different - see the upper and lower legend. The contours where N tot = 1 and N µ = 1 are also shown by solid and dashed lines, respectively. The other energy fractions are taken as constants, ɛ e = 0.1 and ɛ B = The IceCube null result favors the yellow and blue regions, e.g. a high η.

112 102 Figure 5.3: The expected neutrino event number in IceCube based on the baryonic photospheric model for GRB A. The conventions are the same as those in Fig.[5.2]. A high η (or Γ) is favored by the null result in IceCube.

113 103 Figure 5.4: The expected neutrino event number in IceCube based on the magnetic photospheric model for GRB A. The conventions are the same as those in Fig.[5.2]. The result is insensitive to η (see III for an explanation.). The null result in IceCube favors the region where ɛ p < , roughly independent of η or Γ.