天體微中子與宇宙線起源之微中子流量比較

國 立 交 通 大 學

物

理

研

究

所

碩

士

論

文

天體微中子與宇宙線起源之微中子流量比較

The comparison of neutrino flux from astrophysical sources

and that from GZK interactions.

研

究 生

： 謝念潔

指導教授： 林貴林 教授

天體微中子與宇宙線起源之微中子流量比較

The comparison of neutrino flux from astrophysical sources

and that from GZK interactions.

研

究

生 ： 謝念潔

Student: Nien-Chieh Hsieh

指 導 教 授 ： 林貴林

Advisor: Guey-Lin Lin

國 立 交 通 大 學

物

理

研

究

所

碩

士 論 文

A Thesis

Submitted to Institute of Physics

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master

in

Physics

July, 2010

Hsinchu City, Taiwan, Republic of China

天體微中子與宇宙線起源之微中子流量比較

學生： 謝念潔

指 導 教 授 ： 林貴林

國立交通大學物理研究所

摘

要

The comparison of neutrino flux from astrophysical sources and

that from GZK interactions.

Student: Nien-Chieh Hsieh

Advisor: Guey-Lin Lin

Submitted to Institute of Physics

National Chiao Tung University

ABSTRACT

Very high energy neutrinos come from astrophysical sources and GZK interactions. Different neutrino sources produce different neutrino flavor ratios. In this thesis, we compare GRB neutrino flux with GZK neutrino flux in different energy ranges. The ratio between integrated neutrinos fluxes from these two sources are calculated as a function of chosen neutrino threshold energy. We propose a statistical method to reconstruct such a ratio from flavor measurements of neutrino telescopes.

致

致

致

謝

謝

謝

Contents

2 High Energy Neutrino Sources and Neutrino Flavor Ratio 3

2.1 Astrophysical Neutrinos . . . 3

2.1.1 Gamma-Ray Bursts Neutrinos . . . 4

2.1.2 Active Galactic Nuclei Neutrinos . . . 4

2.1.3 Neutrino Flux Upper Bound from Cosmic Ray Observations 5 2.2 GZK Neutrinos . . . 6

2.3 Three Types of Neutrino Sources . . . 6

2.4 Neutrino Oscillation Probability Matrix . . . 7

2.5 Neutrino Flavor Ratios . . . 8

3 The Reconstruction of Neutrino Flavor Ratios at Astrophysical Sources 10 3.1 The Reconstruction of Source Flavor Ratios at Energies Lower Than 1016 eV . . . 11

3.2 The Reconstruction of Source Flavor Ratios at Energies Higher Than 1016 eV . . . 12

4 High Energy Neutrino Spectra and Flux Ratios 16 4.1 Energy Spectra for GRB and GZK Neutrinos . . . 16 4.2 Determining the Flux Ratio of High Energy Neutrinos . . . 21

5 Conclusion 24

List of Figures

3.1 The reconstructed ranges for the neutrino flavor ratios at the source with ∆RI_/RI _{= 10%. The left and right panels are results with}

the muon-damped source and the pion source as the input true source respectively. The numbers on each side of the triangle denote the flux percentage of a specific flavor of neutrino. The red point marks the muon-damped source φ0,µ = (0, 1, 0) and the blue point

marks the pion source φ0,π = (1/3, 2/3, 0). Gray and light gray

areas respectively denote the 1σ and 3σ ranges for the reconstructed neutrino flavor ratios at the source. . . 12 3.2 The reconstructed ranges for the neutrino flavor ratios at the source

with ∆RI/RI = 10% and ∆S/S related to the former by the Pois-son statistics. The left and right panels are results with the muon-damped source and the pion source as the input true source respec-tively. Gray and light gray areas in the left (right) panel denote the reconstructed 1σ and 3σ ranges. . . 13 3.3 Reconstructed ranges for muon-damped source with ∆Ra_/Ra ₌

10% and ∆Sa_/Sa _{= 12%. Gray and light gray areas in the left}

(right) panel denote the reconstructed 1σ and 3σ ranges. The left and right panels correspond to the condition I and II. The pion source can be ruled out at the 3σ level for both conditions. . . 14

3.4 Reconstructed ranges for muon-damped source with ∆Ra_/Ra ₌

10% only. Gray and light gray areas in the left (right) panel de-note the reconstructed 1σ and 3σ ranges. The left and right panels correspond to the condition I and II. The pion source can be ruled out at 3σ level for the condition II but not for the condition I even at 1σ level. . . 14 3.5 Reconstructed ranges for pion source with ∆Ra/Ra = 10% and

∆Sa_/Sa _{= 12%. Gray and light gray areas in the left (right) panel}

denote the reconstructed 1σ and 3σ ranges. The left and right panels correspond to the condition I and II. The muon-damped source can be ruled out at the 1σ level for both conditions. . . 15 3.6 Reconstructed ranges for pion source with ∆Ra_/Ra _{= 10% only.}

Gray and light gray areas in the left (right) panel denote the recon-structed 1σ and 3σ ranges. The left and right panels correspond to the condition I and II. For condition I, 3σ limit covers all flavor ratio of source. But the muon-damped source can be ruled out at 1σ level for condition II. . . 15 4.1 The GZK neutrino spectra. The dashed curve is the prediction for

an all-proton primary. The solid lines denote the Fe primary models with the highest and lowest predicted neutrino fluxes [27]. . . 17 4.2 The neutrino fluxes in different flavors, 2

νlφνl(normalized to 

2 νeφνe).

φνl stands for the combined flux of νl and νl, and these plots are

valid for neutrinos produced by any combination of π+_{and π}−_decay

[2]. The energy scale 0,µ is about 4 × 1015 eV in GRB. . . 18

4.3 Comparison of muon neutrino fluxes (νµand νµcombined) predicted

by different models with the upper bound implied by cosmic ray observations [3]. We are interested in the GRB neutrino flux which can be matched with results in Fig. 4.2. . . 18

4.4 The neutrino flux from GRB source. Blue curve is νµ flux in Fig.

4.3. Green curve and red curve represent νµ and νe fluxes

respec-tively as a result of combining Fig. 4.2 and 4.3. . . 19 4.5 The comparison of neutrino fluxes from GZK and GRB sources.

Green curve represents GRB νµ flux, red curve represents GRB νe

flux. Blue curve represents total GZK neutrino flux with protons as primary ultrahigh energy cosmic rays. Light-blue curve and purple curve are largest and smallest predicted neutrino fluxes with Fe as primary ultrahigh energy cosmic rays. . . 19 4.6 ∆χ2 _{= 1, ∆R}II_/RII _{increases from left panel to right panel as 5%,}

10% and 15%. . . 23 4.7 ∆χ2 _{= 4, ∆R}II_/RII _{increases from left panel to right panel as 5%,}

10% and 15%. . . 23 4.8 ∆χ2 _{= 9, ∆R}II_/RII _{increases from left panel to right panel as 5%,}

List of Tables

3.1 The definitions of R and S at different energy ranges. . . 10

4.1 Comparison of integrated GRB and GZK neutrino flux . . . 20

4.2 The ratio of νµ flux from GRB to that from GZK . . . 20

4.3 The ratio of νµ flux from GRB to that from GZK . . . 20

4.4 The ratio of νµ flux from GRB to that from GZK . . . 21

4.5 Neutrino flavor ratio measured on Earth (E > 1016_{eV) . . . . 21}

Chapter 1

Introduction

Neutrinos with very high energies (E ∼ 1015 eV or beyond) come from either the interactions between ultrahigh energy cosmic rays and cosmic microwave back-ground photons, the so called GZK neutrinos [1], or the very high energy tails of astrophysical neutrino spectra. Gamma-ray bursts (GRBs) and active galactic nu-clei (AGNs) have been suggested as possible astrophysical sources of high energy neutrinos. Different neutrino sources will induce different neutrino flavor ratios. In this thesis, we are interested in using neutrino flavor ratios as a tool to distinguish GZK neutrino flux from the high energy tails of astrophysical neutrino fluxes. To do this, we propose a statistical method for determining the flavor ratio of very high energy neutrinos and consequently deduce the ratio of GZK neutrino flux to that of astrophysical sources at very high energy.

High energy neutrinos can be generated in astrophysical sources and GZK sources. Neutrino flavor ratio from different sources are different. After the large propagation distance, neutrino flavor composition will change due to neutrino os-cillations. The neutrino flavor composition observed on Earth can be determined by the neutrino oscillation probability matrix. Different neutrino sources can be distinguished by measuring the flavor ratios on Earth. The details of these neutrino sources and flavor ratios are presented in Chapter 2. In Chapter 3, we review the statistical method which has been applied to reconstruct neutrino flavor ratio at

the source. In Chapter 4, we first analyze the energy spectrum of GRB neutrinos using formula derived by Kashti and Waxman [2] as well as the result in Ref. [3]. The neutrino flavor ratio at typical GRB source is also calculated with the same formula. Such a flavor ratio is sensitive to neutrino energies. We then compare GRB neutrino flux with that of GZK neutrinos. The ratio between integrated neu-trinos fluxes from these two sources are calculated as a function of chosen neutrino threshold energy. Finally we discuss the reconstruction of the above ratio from the measurement of neutrino telescopes. Chapter 5 is the conclusion.

Chapter 2

High Energy Neutrino Sources

and Neutrino Flavor Ratio

In this chapter, we introduce the probable high energy neutrino sources. These sources produce high energy neutrinos with three types of flavor ratios. They are referred to as pion source, muon-damped source, and beta-decay source respec-tively. Due to neutrino oscillations, the neutrino flavor ratio at the astrophysical source could be quite different from that observed on the Earth. Different neutrino sources can be distinguished by measuring the flavor ratio on Earth.

2.1

Astrophysical Neutrinos

High-energy (> 0.1 TeV) neutrino telescopes are under construction to detect cosmologically distant neutrino sources. The motivation for searching cosmological high-energy neutrino sources is based upon the fact that the cosmic-ray energy spectrum extends to > 1020 eV and is most likely dominated by an extra-galactic source of protons above ∼ 3 × 1018 _eV.

The detection can provide the information of fundamental neutrino properties, and also identify the high energy cosmic ray sources. Gamma-ray bursts (GRBs) [4] and active galactic nuclei (AGNs) jets [5] have been suggested as possible sources

of high energy neutrinos.

High energy neutrinos are likely to be associated with the production of high-energy protons and is produced by the decay of charged pions in astrophysical sources,

π+ → µ+_{+ ν}

µ→ e++ νe+ νµ+ νµ (2.1)

π− → µ−+ νµ→ e−+ νe+ νµ+ νµ (2.2)

These charged pions are produced by photo-meson interaction of the high-energy protons with the radiation field of the source, such as the interaction of protons with photons (pγ) or nucleons (pp, pn). For neutrinos produced in pp or pn collisions, both π+,s and π−,s are produced. In the case of pγ collisions, only π−,s are produced [2].

2.1.1

Gamma-Ray Bursts Neutrinos

In the GRB fireball model, the observed gamma rays are produced by synchrotron emissions of high-energy electrons accelerated in internal shocks of an expanding relativistic wind. In the region where electrons are accelerated, protons are also expected to be shock accelerated, and their photo-meson interaction with observed burst photons will produce a burst of high-energy neutrinos accompanying the GRB [3, 4].

If GRBs are assumed to be the sources of ultra-high-energy cosmic rays [6, 7], then the GRB neutrino flux is expected to be [3, 4]

E2 νΦνµ ≈ E 2 νΦνµ ≈ E 2 νΦνe ≈ 1.5 × 10−9₍fπ 0.2)min{1, Eν/E b ν}GeVcm−2s−1sr−1, E b ν ≈ 1014eV. (2.3) Here, fπ is the fraction of energy lost to pion production by high-energy protons.

2.1.2

Active Galactic Nuclei Neutrinos

AGN have two jets in opposite directions and perpendicular to the accretion disc. The jets accelerate the particles to extremely high energies by Fermi acceleration.

The (Fermi) accelerated ultra high energy protons may collide with other protons or with ambient photons in the vicinity of an AGN or in the associated jets [8]. The interacting chain between high energy protons and gamma rays for generating the pions via ∆+ _{resonance is}

p + γ → ∆+→ p + π0 _{p + γ → ∆}+ _{→ n + π}+ _(2.4)

Neutrinos are produced by the decay of charged pion as Eq.(2.1). Currently, the photohadronically (pγ) produced diffuse flux of high energy neutrinos originating from AGNs dominate over the flux from other sources above the relevant atmo-spheric neutrino background, typically for E ≥ 106 _{GeV [9, 10].}

2.1.3

Neutrino Flux Upper Bound from Cosmic Ray

Ob-servations

Cosmic-ray observations above 1017 eV indicate that an extra-galactic source of protons dominates the cosmic-ray flux above ∼ 3 × 1018 _{eV, while the flux at lower}

energies is dominated by heavy ions of galactic origin [11]. From cosmic ray obser-vations, a model-independent upper bound of E2

νΦν < 2 × 10−8GeV cm−2s−1sr−1

to the flux of neutrinos produced by p − γ interactions for sources optically thin to p − γ reactions can be derived [3, 4].

The neutrino flux predictions of AGN jet models are based on two key as-sumptions. First of all, AGN jets produce the observed gamma-ray background. secondly, high energy photon emission from AGN jets is due to decays of neutral pions produced in photo-meson interactions of protons accelerated in the jet to high energy. Since the neutrino flux predicted by these assumptions is two orders of magnitude higher than the upper bound allowed by cosmic ray observations, at least one of the key assumptions is not valid [3]. The cosmic ray measurements rule out the current version of theories in which the gamma-ray background is due to photo-meson interactions in AGN jets.

Unlike the AGN jet models, the GRB model predicts a neutrino flux satisfy-ing the upper bound from cosmic ray observations [3]. Hence GRB is the more

probable source of high energy neutrinos.

2.2

GZK Neutrinos

Ultrahigh energy protons above the “GZK cutoff” (> 5×1019_{eV) [12] interact with}

the cosmic microwave background and infrared background as they propagate over cosmological distances. In this interaction, protons and microwave background photons collide into the resonance state ∆+_{’s, which decay as in Eq. (2.3). Pions}

decay into neutrinos as the decay chain in Eq.(2.1). The expected spectrum of GZK neutrinos can vary considerably, depending on the precise spectrum and chemical composition injected from the cosmic ray sources.

2.3

Three Types of Neutrino Sources

Most of the astrophysical neutrinos are believed to be produced by the decay of charged pion, which leads to the neutrino flux ratio φ0(νe) : φ0(νµ) : φ0(ντ) =

1 : 2 : 0 at the astrophysical source where φ0(να) is the sum of να and να flux.

This flux ratio results from an implicit assumption that the muon decays into neutrinos before losing a significant fraction of its energy. It is possible that muon quickly loses its energy by interacting with strong magnetic fields or with matter in some sources [13, 14]. Such a muon eventually decays into neutrinos with energies much lower than that of νµ(νµ) from π+(π−) decays. This type of source

is referred to as the muon-damped source, which has a neutrino flavor ratio φ0(νe) :

φ0(νµ) : φ0(ντ) = 0 : 1 : 0. Finally, the third type of source emits neutrons

resulting from the photodisassociation of nuclei. As neutrons propagate to the Earth, νe are produced from neutron β decays [15], leading to a neutrino flavor

ratio φ0(νe) : φ0(νµ) : φ0(ντ) = 1 : 0 : 0.

We note that there is no flux of ντ in the above three sources. Actually, ντ may

be produced by the production of charmed mesons. However the higher energy threshold and lower cross section for charmed meson production typically imply a

negligible ντ fraction.

2.4

Neutrino Oscillation Probability Matrix

Neutrinos are generated or detected with a well defined flavor (electron, muon, tau). It has been demonstrated experimentally that neutrinos are able to oscil-late between three flavors while they propagate through space. This quantum mechanical phenomenon was first predicted by Bruno Pontecorvo. Currently, it is understood that oscillations occur due to the fact that the neutrino flavor eigen-states are not identical to the neutrino mass eigeneigen-states (simply called 1, 2, 3). This allows an electron neutrino produced at a given location to be detected as either a muon or tau neutrino with a calculable probability after it has traveled to another location.

In the Standard Model of particle physics, the existence of flavor oscillations implies a nonzero neutrino mass, because the amount of mixing between neutrino flavors depends on the differences in their squared masses.

The neutrino flux at the astrophysical source φ0(να) and that detected on the

Earth φ(να) is related by  φ(νe) φ(νµ) φ(ντ)  = φ0(νe) φ0(νµ) φ0(ντ)      Pee Peµ Peτ Pµe Pµµ Pµτ Pτ e Pτ µ Pτ τ     . (2.5) The matrix element Pαβ is the oscillation probability P (να → νβ). The matrix

element Pαβ is given by [16]

Pαβ = δαβ − 4

P

i>jRe(UαiUβj∗ U ∗ αiUβj) sin2( ∆m2 ijL 4E )+ 4P

i>jIm(UαiUβj∗ U ∗ αiUβj) sin( ∆m2 ijL 4E ) cos( ∆m2 ijL 4E ). (2.6) Here, ∆m2

factors of } and c, one has

∆m2_ij(L/4E) ' 1.27∆m2_ij(eV2) L(km)

E(GeV). (2.7) U is the Pontecorvo-Maki-Nagakawa-Sakata mixing matrix

U =     c12c13 s12c13 s13e−iδ −s12c23− c12s23s13eiδ c12c23− s12s23s13eiδ s23c13 s12s23− c12c23s13eiδ −c12s23− s12c23s13eiδ c23c13     , (2.8) where cij = cosθij, sij = sinθij, and δ is the CP violating phase.

Despite the oscillation, the total neutrino number is conserved such thatP

βPαβ =

1. However, in the presence of a non negligible decay probability or of transitions to additional sterile states, the above sum can be less than unity. In most cases, the neutrino oscillation length λij = 4πE/ | ∆m2ij | is negligibly short compared to

the typical astrophysical distance. Hence for astrophysical neutrinos, it is a good approximation to consider only the averaged oscillation probability which takes the form [17, 18]:

P (να → νβ) =

X

i

| Uαi |2| Uβi |2 (2.9)

The current best-fit values as well as the allowed 1 σ and 3 σ range of the mixing angles are [19]

sin2θ12 = 0.32+0.02,0.08−0.02,0.06, sin2θ23= 0.45+0.09,0.19−0.06,0.13, sin2θ13< 0.019(0.050) (2.10)

2.5

Neutrino Flavor Ratios

The capability of distinguishing between different neutrino sources depends on the knowledge of neutrino mixing parameters and the achievable accuracies in measuring the neutrino flavor ratios R ≡ φ(νµ)/(φ(νe) + φ(ντ)) [20] and S ≡

φ(νe)/φ(ντ) [21] on Earth. R is used to distinguish shower-like events from

track-like ones, while S is used to distinguish between shower-track-like events.

The flux ratio of the pion source is φ0(νe) : φ0(νµ) : φ0(ντ) = 1 : 2 : 0 at the

source, while the flux ratio observed on Earth is φ(νe) : φ(νµ) : φ(ντ) = 1 : 1 : 1.

Muon-damped source is φ0(νe) : φ0(νµ) : φ0(ντ) = 0 : 1 : 0 at the source, and the

flux ratio observed on Earth is φ(νe) : φ(νµ) : φ(ντ) = 1.8 : 1.8 : 1. Hence we

have R = 0.5 and S = 1 for the pion source, while R = 0.64 and S = 0.56 for the muon-damped source.

The ratio parameters R and S are suitable for Eν < 1016eV. At energies higher

than 1016 _{eV, R ≡ e/(µ + τ ) and S ≡ µ/t is a more suitable set of parameters}

Chapter 3

The Reconstruction of Neutrino

Flavor Ratios at Astrophysical

Sources

One can infer the neutrino flavor ratio at the astrophysical source from the flavor ratio we measured on Earth [23, 24, 25, 22]. In this chapter we review results obtained in Refs. [23, 22]. The suitable ratio parameters for flavor reconstruction is summarized in Table 3.1.

Condition I : Eν < 33 PeV Condition II : Eν > 33 PeV

RI _{≡ φ(ν}

µ)/(φ(νe) + φ(ντ)) RII ≡ φ(νe)/(φ(νµ) + φ(ντ))

SI _{≡ φ(ν}

e)/φ(ντ) SII ≡ φ(νµ)/φ(ντ)

Table 3.1: The definitions of R and S at different energy ranges.

3.1

The Reconstruction of Source Flavor Ratios

at Energies Lower Than 10

eV

To do the reconstruction with a statistical analysis, we use the following best-fit val-ues and 1σ ranges of neutrino mixing parameters set1 sin2θ12= 0.32+0.02−0.02, sin2θ23=

0.45+0.09_−0.06, sin2θ13< 0.019 in Ref. [19].

The fitting to the neutrino flavor ratios at the source is facilitated through χ2 = (R I th− RIexp σ_RI exp )2+ (S I th− SexpI σ_SI exp )2+ X jk=12,23,13 (S 2 jk − (Sjk)2bestf it σ_S2 jk )2, (3.1) with σ_RI exp = (∆R I_/RI_)RI exp, σSI exp = (∆S I_/SI_)SI exp, Sjk2 ≡ sin 2_θ jk and σS2 jk the

1σ range for S_jk2 . Here RI_th and S_thI are theoretical predicted values for RI and SI _{respectively while R}I

exp and SexpI are experimentally measured values. In our

analysis, we scan all possible neutrino flavor ratios at the source that give rise to a specific χ2 _{value. Since we have taken R}I

exp and SexpI as those generated by

input true values of initial neutrino flavor ratios and neutrino mixing parameters, we have (χ2₎

min = 0 occurring at these input true values of parameters. Hence

the boundaries for 1σ and 3σ ranges of initial neutrino flavor ratios are given by ∆χ2 _{= 2.3 and ∆χ}2 _{= 11.8 respectively where ∆χ}2 _{≡ χ}2 _{− (χ}2₎

min = χ2 in our

analysis [23].

By measuring R alone from either an input pion source or an input moun-damped source with a precision ∆RI_/RI _{= 10%, the reconstructed 3σ range for}

the initial neutrino flavor ratio is as large as the entire physical range as shown in Fig. 3.1. By measuring both R and S from an input muon-damped source, we can see from Fig. 3.2 that the pion source can be ruled out at the 3σ level for the parameter sets 1 and 2 with ∆RI_/RI _{= 10% and ∆S}I_/SI _{related to the former}

Figure 3.1: The reconstructed ranges for the neutrino flavor ratios at the source with ∆RI/RI = 10%. The left and right panels are results with the muon-damped source and the pion source as the input true source respectively. The numbers on each side of the triangle denote the flux percentage of a specific flavor of neutrino. The red point marks the muon-damped source φ0,µ = (0, 1, 0) and the blue point

marks the pion source φ0,π = (1/3, 2/3, 0). Gray and light gray areas respectively

denote the 1σ and 3σ ranges for the reconstructed neutrino flavor ratios at the source.

3.2

The Reconstruction of Source Flavor Ratios

at Energies Higher Than 10

eV

Let us take the muon-damped source as the input true source and consider its reconstruction. The reconstructed regions of neutrino flavor ratio are comparable for a = I and II. For an input muon-damped source, the pion source can be ruled out at the 3σ level as shown in Fig. 3.3.

It is well known that measuring Ra _{is easier than measuring S}a _{for neutrino}

telescopes. If the measurements on Saare not available, the results for flavor-ratio reconstruction are quite different. Fig. 3.4 shows the reconstructed flavor ratios with ∆Ra_/Ra _{= 10%. The reconstructed region of neutrino flavor ratio for a =I}

Figure 3.2: The reconstructed ranges for the neutrino flavor ratios at the source with ∆RI_/RI _{= 10% and ∆S/S related to the former by the Poisson statistics.}

The left and right panels are results with the muon-damped source and the pion source as the input true source respectively. Gray and light gray areas in the left (right) panel denote the reconstructed 1σ and 3σ ranges.

is much larger than that for a =II. The pion source can only be ruled out at 3σ level for the condition II but not for the condition I.

For an input pion source, the muon-damped source can be ruled out at the 1σ level as shown in Fig. 3.5 for both energy conditions. If one only measures Ra with ∆Ra/Ra = 10%, it is shown in Fig. 3.6 that the reconstructed region for a =I covers all physical parameter space while the reconstructed region for a =II remains comparable to that in Fig. 3.5. Once again, the muon-damped source can be ruled out for condition II at 1σ level, but not for condition I at the same confidence level. From the reconstructions of pion source and muon-damped source, it is evident that this new parameter RII _{is more efficient than R}I _for

Figure 3.3: Reconstructed ranges for muon-damped source with ∆Ra/Ra = 10% and ∆Sa_/Sa _{= 12%. Gray and light gray areas in the left (right) panel denote}

the reconstructed 1σ and 3σ ranges. The left and right panels correspond to the condition I and II. The pion source can be ruled out at the 3σ level for both conditions.

Figure 3.4: Reconstructed ranges for muon-damped source with ∆Ra/Ra = 10% only. Gray and light gray areas in the left (right) panel denote the reconstructed 1σ and 3σ ranges. The left and right panels correspond to the condition I and II. The pion source can be ruled out at 3σ level for the condition II but not for the condition I even at 1σ level.

Figure 3.5: Reconstructed ranges for pion source with ∆Ra_/Ra _{= 10% and}

∆Sa/Sa = 12%. Gray and light gray areas in the left (right) panel denote the reconstructed 1σ and 3σ ranges. The left and right panels correspond to the con-dition I and II. The muon-damped source can be ruled out at the 1σ level for both conditions.

Figure 3.6: Reconstructed ranges for pion source with ∆Ra_/Ra _{= 10% only. Gray}

and light gray areas in the left (right) panel denote the reconstructed 1σ and 3σ ranges. The left and right panels correspond to the condition I and II. For condition I, 3σ limit covers all flavor ratio of source. But the muon-damped source

Chapter 4

High Energy Neutrino Spectra

and Flux Ratios

High energy neutrinos can be generated by astrophysical sources and GZK inter-actions. These sources of neutrinos have distinct flavor ratios. For astrophysical sources, the neutrino flavor ratios depend on energies [2, 26]. In this chapter, we propose a statistical method for determining the flux ratio of the above two sources of very high energy neutrinos.

4.1

Energy Spectra for GRB and GZK

Neutri-nos

In this section, we compare the GZK neutrino spectra (Fig. 4.1) [27] with the astrophysical neutrino spectra (Figs. 4.2,4.3) [2] for determining the relative con-tributions of these sources to the total high-energy neutrino flux.

We note that Fig. 4.2 gives the energy dependencies of GRB νe and νµ fluxes

while Fig. 4.3 gives the normalization of GRB νµ flux at 1014 eV. Combining

both figures, we obtain GRB neutrino flux as presented in Fig. 4.4. The blue curve is the flux in Fig. 4.3, red curve and green curve are νe and νµ fluxes

respectively. The neutrino spectrum is steep at high energy (> 1016eV), where 16

Figure 4.1: The GZK neutrino spectra. The dashed curve is the prediction for an all-proton primary. The solid lines denote the Fe primary models with the highest and lowest predicted neutrino fluxes [27].

neutrinos are produced by the decay of muons and pions whose lifetime τµ,π exceeds

the characteristic time for energy loss due to adiabatic expasion and synchrotron emission.

GRB and GZK neutrino spectra are compared in Fig. 4.5. We calculate the integrated neutrino flux with threshold energy varied from 1014 eV to 1018 eV. The results are shown in Table 4.1. We calculate the ratio of GRB muon neutrino flux to GZK muon neutrino flux as shown in Table 4.2. GRB neutrino sources dominate when threshold energy is below 1016 _{eV, while GZK neutrino sources}

dominate as threshold energy is higher than 1017 _{eV. For a 10}16 _{eV threshold}

energy, the contribution of GRB and GZK to the total flux is almost the same. In Table 4.3 (Table 4.4), we calculate the muon-neutrino flux ratio of GRB source to GZK source with Fe as primary ultrahigh energy cosmic rays. GZK neutrino source dominates as the neutrino energy is higher than 1017 _{eV (Table 4.3) and}

1018 eV (Table 4.4).

If we take the ratio of total neutrino flux from muon-damped source to that from pion source as c/(1 − c), the neutrino flux ratio r of GRB to GZK in Table

Figure 4.2: The neutrino fluxes in different flavors, 2

νlφνl (normalized to 

2 νeφνe).

φνlstands for the combined flux of νl and νl, and these plots are valid for neutrinos

produced by any combination of π+and π−decay [2]. The energy scale 0,µis about

4 × 1015 _{eV in GRB.}

Figure 4.3: Comparison of muon neutrino fluxes (νµ and νµ combined) predicted

by different models with the upper bound implied by cosmic ray observations [3]. We are interested in the GRB neutrino flux which can be matched with results in Fig. 4.2.

Figure 4.4: The neutrino flux from GRB source. Blue curve is νµ flux in Fig. 4.3.

Green curve and red curve represent νµ and νe fluxes respectively as a result of

combining Fig. 4.2 and 4.3.

Figure 4.5: The comparison of neutrino fluxes from GZK and GRB sources. Green curve represents GRB νµ flux, red curve represents GRB νe flux. Blue curve

rep-resents total GZK neutrino flux with protons as primary ultrahigh energy cosmic rays. Light-blue curve and purple curve are largest and smallest predicted neutrino fluxes with Fe as primary ultrahigh energy cosmic rays.

N(cm−2s−1sr−1)

log[E(eV)] GRB νe GRB νµ GZK proton GZK Fe-max GZK Fe-min

> 14 3.60 × 10−14 6.76 × 10−14 1.17 × 10−16 2.13 × 10−17 4.76 × 10−18 > 15 4.84 × 10−16 1.78 × 10−15 1.06 × 10−16 1.51 × 10−17 1.14 × 10−18 > 16 2.58 × 10−19 7.72 × 10−17 8.57 × 10−17 1.17 × 10−17 6.53 × 10−19 > 17 4.14 × 10−24 2.20 × 10−19 6.52 × 10−17 8.59 × 10−18 3.84 × 10−19 > 18 2.29 × 10−22 1.02 × 10−17 1.72 × 10−18 1.29 × 10−20

Table 4.1: Comparison of integrated GRB and GZK neutrino flux

N(cm−2s−1sr−1) Flux Ratio log[E(eV)] GRB νµ GZK νµ(proton) r=GRB/GZK > 14 6.76 × 10−14 0.78 × 10−16 870 > 15 1.78 × 10−15 0.71 × 10−16 25 > 16 7.72 × 10−17 5.71 × 10−17 1.4 > 17 2.20 × 10−19 4.35 × 10−17 5.1 × 10−3 > 18 2.29 × 10−22 0.68 × 10−17 3.4 × 10−5 Table 4.2: The ratio of νµ flux from GRB to that from GZK

N(cm−2s−1sr−1) Flux Ratio log[E(eV)] GRB νµ GZK νµ(Fe-max) r=GRB/GZK > 14 6.8 × 10−14 1.4 × 10−17 4.8 × 104 > 15 1.8 × 10−15 1.0 × 10−17 176 > 16 7.7 × 10−17 0.8 × 10−17 9.9 > 17 2.2 × 10−19 5.7 × 10−18 0.04 > 18 2.3 × 10−22 1.2 × 10−18 2.0 × 10−4 Table 4.3: The ratio of νµ flux from GRB to that from GZK

4.2-4.4 is equal to 3c/2(1 − c). For example, the neutrino flux ratio of GRB to GZK proton is r ≡ 3c/2(1 − c) = 1.4 when the threshold energy is 1016 _{eV. Hence}

the fraction of contribution of GRB neutrino source to the total neutrino flux is c=0.5. We also determine the neutrino flavor ratio measured on Earth in Table 4.5. With these neutrino flavor ratios, we can distinguish different sources.

N(cm−2s−1sr−1) Flux Ratio log[E(eV)] GRB νµ GZK νµ(Fe-min) r=GRB/GZK > 14 6.8 × 10−14 3.2 × 10−18 2.1 × 104 > 15 1.8 × 10−15 0.8 × 10−18 2.3 × 103 > 16 7.7 × 10−17 4.4 × 10−19 177 > 17 2.2 × 10−19 2.6 × 10−19 0.9 > 18 2.3 × 10−22 0.9 × 10−20 0.03 Table 4.4: The ratio of νµ flux from GRB to that from GZK

Sources c φ0,c = ((1 − c)/3, (2 + c)/3, 0) φc= P φ0,c RII

GRB+GZK proton 0.47 (0.18, 0.82, 0) (0.26, 0.38, 0.36) 0.35 GRB+GZK Fe-max 0.86 (0.05, 0.95, 0) (0.21, 0.41, 0.38) 0.27 GRB+GZK Fe-min 0.99 (0.003, 0.997, 0) (0.19, 0.42, 0.39) 0.24

Table 4.5: Neutrino flavor ratio measured on Earth (E > 1016_eV)

4.2

Determining the Flux Ratio of High Energy

Neutrinos

We determine the flux ratio of very high energy neutrinos which come from GRB and GZK sources. The former is a muon-damped source with φ0(νe) : φ0(νµ) :

φ0(ντ) = 0 : 1 : 0 and the latter is a pion source with φ0(νe) : φ0(νµ) : φ0(ντ) = 1/3 :

neutrino flavor ratio is then φ0(νe) : φ0(νµ) : φ0(ντ) = (1 − c)/3 : (2 + c)/3 : 0. Due

to the neutrino oscillation (Eq. 4.1), the fluxes φ(νe), φ(νµ) and φ(ντ) measured

on the Earth are different from those at the source. They are given by

 φ(νe) φ(νµ) φ(ντ)  =  (1 − c)/3 (2 + c)/3 0      Pee Peµ Peτ Pµe Pµµ Pµτ Pτ e Pτ µ Pτ τ     (4.1) We use the statistical method mentioned in Chapter 3 to reconstruct the neutrino flavor ratio at the source. In this high energy limit, one measures RII = e/µ + τ . Hence χ2 = (R II th − RIIexp σ_RII exp )2+ X jk=12,23,13 (S 2 jk− (Sjk)2bestf it σ_S2 jk )2. (4.2) We determine whether the reconstructed ratio coutput is consistent with the

input ratio cinput. In χ2 fitting, we also take ∆RII/RII as 5%, 10% and 15%

respectively. The probable ranges of the flux ratio “c” are presented in Fig. 4.6-4.8. The color regions in Fig. 4.6, 4.7 and 4.8 correspond to 1σ, 2σ and 3σ reconstructed ranges for the ratio c. We find that the reconstructed ratio coutput

is better constrained when the input ratio cinput is larger. A larger value of cinput

implies the dominance of GRB neutrino source. In this situation, we can clearly reconstruct the very high energy neutrino source. However, for a small cinput(GZK

dominant case), one can not rule out the GRB source in the flavor reconstruction since the allowed range for coutput is large.

Figure 4.6: ∆χ2 = 1, ∆RII/RII increases from left panel to right panel as 5%, 10% and 15%.

Figure 4.7: ∆χ2 = 4, ∆RII/RII increases from left panel to right panel as 5%, 10% and 15%.

Figure 4.8: ∆χ2 = 9, ∆RII/RII increases from left panel to right panel as 5%, 10% and 15%.

Chapter 5

Conclusion

Flavor ratio of astrophysical neutrinos varies with the neutrino energy and the neutrino spectra. It has been argued in Ref. [2, 27] that this ratio evolves from that of a pion source to that of a muon-damped source as neutrino energy increases. However, the flavor ratio of the GZK neutrino is fixed to be that of a pion source. The source composition of high energy neutrinos in different energy range has been calculated in Table 4.2. The contribution of integrated GRB and GZK (pro-ton dominant) neutrino fluxes to the total integrated flux is almost the same for 1016 _{eV threshold energy. GRB neutrino source dominates when the threshold}

energy is below 1016 eV, while GZK neutrino source dominates as the energy threshold is above 1017 _eV.

We have presented the results for the reconstruction of source composition of very high energy neutrino flux. The reconstructed composition coutputis better

con-strained when the input composition cinput is larger. A larger value of cinput means

that GRB neutrino source dominates. In contrast, coutput is poorly constrained for

a small cinput.

In summary, the flavor ratio of astrophysical neutrinos in different energy range is worth studying. We can infer the composition of neutrino sources from such a study.

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