Properties of Interstellar Medium and Cosmic Ray Propagation, Impacts on Dark Matter Indirect Searches

International School for Advanced Studies SISSA/ISAS

Thesis submitted for the degree of Doctor Philosophiae

Candidate: Supervisor: Maryam Tavakoli Prof. Piero Ullio

Trieste, September 2012 2 Abstract

In this thesis the properties of the interstellar medium and cosmic rays propagation are studied in the light of unprecedented data from the Fermi γ-ray telescope. A combined analysis of cosmic rays and γ-rays spectra constrain the thickness of the diffusion zone and the gas distribution in the Galaxy. In- deed, there is a tight correlation between the distribution of gas in the Galaxy and the spectra of diffuse γ-rays. Small scale features in high angular resolution diffuse γ-ray sky maps of the Fermi can be interpreted by detailed models for gas distribution. A new model for three dimensional distribution of atomic hydrogen gas, which is the major ingredient of the interstellar gas, is constructed. Based on established propagation models which account for astrophysical contribution to diffuse γ-rays, limits on possible contributions from dark matter are derived. Inspired by indications for a γ-ray line at energy of about 130 GeV towards the Galactic center, limits on the continuous spectrum which accompanies the γ-ray line are extracted. Extending the window of observation, the bounds on the morphological shape of a dark matter signal associated with the line is discussed. For this purpose, the standard templates for the dark matter profile, such as an Einasto or a NFW profile, and a new more general parametrization are applied. Upper limits on dark matter annihilation cross sections in the Galactic halo are also derived for a broader range of dark matter mass.

3 4 Contents

1 Introduction7

2 Interstellar Medium 11 2.1 Atomic Hydrogen Gas...... 12 2.1.1 Derivation of Atomic Hydrogen Number Density...... 15 2.1.2 Large Scale Features...... 18 2.2 Molecular Hydrogen Gas...... 20 2.3 Ionized Hydrogen Gas...... 24 2.4 Galactic Magnetic Field...... 25

3 Cosmic Rays 27 3.1 Primary Sources...... 28 3.2 Diﬀusion...... 30 3.3 Energy Losses...... 31 3.4 Propagation...... 33

4 Diffuse Gamma Rays 39 4.1 Impact of Cosmic Rays Diffusion...... 42 4.1.1 Diffusion Spectral index...... 42 4.1.2 Diffusion Radial Scale...... 43 4.1.3 Diffusion Scale Height...... 43 4.1.4 Convection...... 48 4.2 Rigidity Break in Injection or Diffusion...... 50 4.3 Influence of Supernova Remnants Distribution...... 53 4.4 γ-ray Yields from pp-Collision Parametrization...... 57 4.5 Significance of the Interstellar Gas Distribution...... 58 4.5.1 2D vs 3D Gas Distribution...... 64 4.6 Reference Model for Diffuse γ-Ray Background...... 68

5 6 CONTENTS

5 Dark Matter Indirect Detection 73 5.1 Galactic Center...... 74 5.1.1 Limits on WIMPs Annihilation Cross Section in the inner 10◦ 10◦ 74 × 5.1.2 Limits on WIMPs Annihilation Proﬁle...... 79 5.1.3 A Speciﬁc Example...... 84 5.2 Galactic Halo...... 85 5.2.1 Dark Gas...... 86 5.2.2 Minimal non-DM Extragalactic Background...... 86 5.2.3 Limits on WIMPs Annihilation Cross Section in the Halo...... 92

6 Conclusion 97 Chapter 1

Introduction

There are convincing pieces of evidence from cosmological and astrophysical observations that non-baryonic dark matter is the dominating matter component in the Universe and the building block for all visible structures [1,2,3,4,5,6,7,8,9, 10]. Among several plausible scenarios for particle physics candidates for dark matter, the case for weakly interacting massive particles (WIMPs) is particularly appealing (for a review on dark matter candidates see, e.g. [11]). WIMPs are generated as thermal relics from the early Universe. A massive stable state with sizable coupling to Standard Model particles would be in thermal equilibrium at high temperatures and tend to decouple from the thermal bath when it becomes non-relativistic. It is easy to show that the relic abundance approximately scales with the inverse of its pair annihilation rate into lighter Standard Model particles [12] and matches the cosmological dark matter abundance for a coupling of weak-interaction strength. The phenomenology of WIMPs is very rich. They can be produced at colliders and detected as missing energy. They can also be directly detected through measuring the recoil energy in their elastic scattering oﬀ nuclei of underground detectors, or indirectly via searching for the products of the WIMPs pair annihilations which happen today in dark matter halos. The traces of WIMPs annihilation might be seen in the spectra of cosmic rays (such as positrons, antiprotons and anti-deuterons), in the spectra of photons in a broad wavelength range (ranging from the radio frequencies, in connection to synchrotron emission from electron/positron WIMPs yields, to the X-ray and γ-ray bands) as well as in WIMP- induced neutrino ﬂuxes. In most of these cases and for most viable observational targets, the WIMPs contribution is in general rather small compared the component induced by more standard astrophysical sources. Therefore, to constrain the WIMPs annihilation properties a deep understanding of astrophysical backgrounds is crucial.

7 8 CHAPTER 1. INTRODUCTION

Among the mentioned detection channels, γ-rays are exceptionally important to study for two reasons. Firstly, γ-rays, unlike the charged particles propagate on straight lines through the Universe almost without losing energy and thus directly point back toward their sources. Secondly and especially timely, the Fermi γ-ray telescope [13] has been providing extremely detailed γ-ray sky maps in the last few years. The Fermi telescope has opened a new era in γ-ray astronomy; its high angular resolution, wide energy coverage with excellent energy resolution together with large effective area and field of view has made it possible to measure the γ-ray radiation with unprecedented quality. Thereby, it gives unique insights into the most energetic processes in our Universe [14]. At energies above few tens of MeV more than 80% of the observed photons are connected to diffuse emission. The bulk of these photons is from the decay of neutral pions which themselves are produced by inelastic collisions of cosmic ray protons and helium nuclei with the interstellar gas. Moreover, cosmic ray electrons and positrons yield γ-rays via bremsstrahlung emission in the interstellar gas or by inverse Compton scattering off the interstellar radiation field. In order to constrain the contribution of astrophysical backgrounds to diffuse γ-rays a thorough understanding of the properties of cosmic rays propagation within the Galaxy is highly demanded. Cosmic rays with energies up to 1015 eV, which are accelerated in our own Galaxy, do not freely propagate in the interstellar medium but rather get deflected by magnetic fields. There are irregularities in magnetic fields which are caused either by fluctuations in the field or the growth of instabilities due to the flow of cosmic rays. The random scattering of cosmic rays by these irregularities give rise to their diffusion through the interstellar medium. The correlation between diffusion and magnetic fields manifests itself as the weaker magnetic field, the larger diffusion length. The energy dependence of diffusion is also correlated to the mechanism that builds up the turbulence in magnetic fields. Furthermore, the stochastic re-acceleration of cosmic rays by scattering on the magnetohydrodynamic waves leads to diffusion in momentum space which is inversely proportional to the diffusion in physical space. In addition, the drift of cosmic rays by the Galactic winds affects the diffusion timescale. Little is known about the absolute value of diffusion and its spatial and spectral properties. In this thesis, a systematic study has been performed on a variety of propagation models in which the spatial and spectral properties of diffusion vary in a fairly large range. The propagation parameters are fitted to the local fluxes of cosmic rays. The local measurements of the spectrum of cosmic ray primaries, the ratio of secondary to primary cosmic ray nuclei and the flux of electrons and positrons are used. To discriminate between the models the evaluated spectra of antiprotons and diffuse γ-rays are compared against the data from PAMELA [15] and 9

Fermi [16]. Another important aspect in estimating the diﬀuse γ-ray background is the distribution of the gas in the interstellar medium. The diﬀuse γ-ray components produced by decays of neutral pions and bremsstrahlung of electrons and positrons are strongly correlated to the gas distribution in the Galaxy. Small scale features of the gas distribution along the line of sights manifest themselves in the sky maps of these components. These structures can be traced in high angular resolution maps of the Fermi γ-ray telescope.

One of the major constituents of the interstellar gas is atomic hydrogen HI . It can be traced by 21cm emission line which is due to the transition between the atomic hydrogen 2S ground state levels split by the hyperﬁne structure. The intensity of 21cm emission line together with rotation curves can be used to derive HI distribution in the Galaxy. Indeed, recent all sky 21cm surveys with high angular resolution, strong sensitivity and large velocity range motivate one to devise a detailed model. In this work, a new model for three dimensional distribution of atomic hydrogen gas in the Milky Way is constructed. To that end, the combined Leiden-Argentine-Bonn (LAB) survey [17] data is used. It is cur- rently the most sensitive 21cm line survey with the most extensive spatial and kinematic coverage. Molecular hydrogen H2 is another important component of the interstellar gas. Its direct observation is diﬃcult since it does not have electric dipole moment. However, collisions with carbon monoxide molecules leads to CO excitation and its 2.6 mm emission line. The CO to H2 conversion factor XCO, which relates the H2 column density

NH2 to velocity-integrated intensity of the CO line, has considerable uncertainties. In an extensive study, the impact of different assumptions about the distributions of XCO and molecular hydrogen on diffuse γ-rays has been investigated. Having performed a thorough analysis on properties of cosmic rays propagation and interstellar medium, models which fit the local spectra of cosmic rays and agree with the spectra of diffuse γ-rays in all sky regions are considered as the models which provide astrophysical backgrounds. The possible contribution of dark matter to the spectra of diffuse γ-rays can then be studied. The Galactic center is a prime target to search for dark matter signal where the rate of WIMPs annihilation peaks. Indeed, there have been indications for a γ-ray line signal towards the Galactic center at energy of about 130 GeV [18, 19]. The observed excess can be interpreted as dark matter signal. There are also alternative explanations [20, 21, 22]. If the excess is of dark matter origin then it is generally expected that such a flux would be correlated to a γ-ray component with continuous spectrum from dark matter pair annihilation into Standard Model particles. In this study, upper limits on the continuum are derived for various dark matter annihilation final states towards the Galactic center. 10 CHAPTER 1. INTRODUCTION

The flux of γ-rays of dark matter origin depends on the properties of dark matter particle itself as well as the distribution of dark matter in the Galaxy. Therefore, by extending the window of observation the bounds on the morphological shape of dark matter signal associated with the line can be extracted. Finally, the Galactic halo can be studied for dark matter searches since the uncertainties in astrophysical backgrounds in the halo are much less than those in the Galactic center region. This thesis is organized as follows. In chapter2, the large scale distribution of hydrogen gas in the Galaxy is discussed and a new model for atomic hydrogen gas is introduced. The structure of the large scale Galactic magnetic field is also described. The journey of cosmic rays from their sources to the Solar System is reviewed in chapter3. Impacts of different assumptions about the properties of the interstellar medium and cosmic rays propagation on diffuse γ-rays are investigated in chapter4. Possible contributions of dark matter to diffuse γ-rays in the Galactic canter region as well as in the Galactic halo are studied in chapter5. Finally, this thesis is summarized in chapter6. Chapter 2

Interstellar Medium

Interstellar matter is mainly made up of gas. The distribution of gas in the Milky Way reveals the global structure and dynamics of the interstellar medium [23]. Indeed, the gravitational potential in which the gas moves is defined by the mass distribution of the stars and of the Galactic dark matter [24]. The kinematics of the interstellar gas therefore act as probes of the gravitational potential field and so provide information about the distribution of mass in the Galaxy. Furthermore, the fragmentation of cosmic rays during their propagation within Galaxy is caused by interactions with interstellar gas. These interactions give rise to production of lighter secondary nuclei, antiparticles and mesons. In addition, the diffuse γ-ray components produced by decays of neutral pions and bremsstrahlung of electrons and positrons are correlated to the gas distribution in the Galaxy. Interstellar gas is composed of hydrogen, helium and small amounts of heavier nuclei.

Hydrogen in our Galaxy is observed in atomic HI , molecular H2 and ionized HII forms. Helium is appeared to follow the hydrogen distribution with a factor of He/H = 0.10 0.08. ± A value of He/H = 0.11, which is widely used in the literature, is adopted [25]. Interstellar dust grains which are composed of heavy elements account for about 1% of the mass in the interstellar matter [26]. They can be found in most parts of our Galaxy except for the hot regions with temperatures of about 103 K where they dissociate [27]. Dust grains may absorb or scatter the star light which passes through them. The absorption dims the light whereas the scattering makes the transmitted light more red. Indeed, the infrared component of the interstellar radiation ﬁeld is a result of processing of star light by dust. Interstellar radiation ﬁeld has two other components, the optical star light and the cosmic microwave background (CMB). Its distribution in the Galaxy plays an important role in energy loss of cosmic ray electrons and emission of γ-rays via inverse Compton

11 12 CHAPTER 2. INTERSTELLAR MEDIUM effect [28]. Magnetic filed is another important ingredient of the interstellar medium. Large scale Galactic magnetic field and its irregularities cause the scattering of cosmic rays and con- duct the diffusion process. Moreover, the energy loss of cosmic ray electrons via synchrotron emission is strongly correlated to the structure of magnetic field in our Galaxy. In section 2.1, a new model for three dimensional distribution of atomic hydrogen gas in our Galaxy, which is derived from the Leiden-Argentine-Bonn (LAB) survey data [17], is discussed in detail. In sections 2.2 and 2.3 it is briefly recapitulated how, respectively, the molecular and ionized gas components are traced. The most recent models of their distribution in the Galaxy are presented too. In section 2.4 the general properties of the Galactic magnetic field is reviewed.

2.1 Atomic Hydrogen Gas

Hydrogen atom emits line radiation with wavelength λ0 = 21.1 cm (ν0 = 1420.4058 MHz) through a hyperﬁne transition when the spins of electron and proton ﬂip from being parallel to antiparallel. Since the transition probability is too small (2.85 10−15 s−1), × collisions have enough time to establish an equilibrium distribution of hydrogen atoms in the upper and lower states labelled 2 and 1 respectively. Thus applying the Boltzmann distribution the ratio of the number of atoms in these states is given by

n2 g2 g2 hν0 = exp ( hν0/kTs) (1 ), (2.1) n1 g1 − ≈ g1 − kTs where g2 and g1 are, respectively, the statistical weights of the upper and lower levels with ratio of g2/g1 = 3. The excitation temperature Ts is called the spin temperature and −2 under most circumstances Ts hν0/k = 7 10 K. × The radiation transfer equation in terms of the radiation intensity Iν, namely the radiant energy per second per unit area per steradian per bandwidth, can be expressed by (see [27, 29, 30, 31] for more details)

dIν kν = χνIν. (2.2) dx 4π −

The increase in intensity, in traversing dx, is kν/4π where kν is the emissivity of the plasma. The decrease in intensity in the same distance increment is χνIν where χν is the absorption per unit path length. The emissivity and absorption per unit frequency interval in terms of the line width of the neutral hydrogen proﬁle δν are 1 kν = n2(x)A21hν0/δν, χν = (n1B12 n2B21)hν0/δν. (2.3) 4π − 2.1. ATOMIC HYDROGEN GAS 13

The Einstein’s coeﬃcients A21, B12 and B21 are the intrinsic properties of atoms and satisfy the following relations

3 B21 g1 A21 2hν0 = , = 2 . (2.4) B12 g2 B21 c Using 2.1 and 2.4, the absorption (2.3) can be rewritten as follows

3hc2 χνδν = A21nHI , (2.5) 32πkTsν0 where we have used the fact that the number density of hydrogen atoms in the lower state is 1/4 of the total hydrogen number density. The optical depth in terms of absorption χν is

τν = χνδr, (2.6) where δr is the path length. The optical depth, itself, is used to deﬁne the brightness temperature

−τν Tb Ts(1 e ). (2.7) ≡ − The line width δν is caused by the motions of hydrogen gas with respect to the observer.

ν0 Thus it can be replaced by c δvr where vr is the gas radial velocity relative to the Sun. Assuming that the motion of gas around the Galactic center is purely circular, the gas located at Galactocentric radius R and altitude z has a radial velocity with respect to | | the Sun which is given by

θ(R, z) θ vr = R sin l cos b. (2.8) R − R In the above, θ(R, z) is the rotation velocity of the gas, l and b are, respectively, the longitude and latitude.

Eq. (2.5) in terms of Tb is then written as

Tb nHI δr = CTs ln (1 )δvr (2.9) − Ts where the constant C is

2 32πkν0 C = 3 − 3hc A21 (2.10) = 1.823 1018cm−2K−1(km/s)−1. − ×

The measurement of the brightness temperature Tb over a large range of radial velocities and directions in the sky is provided by the Leiden-Argentine-Bonn (LAB) survey [17]. This survey merges the Instituto Argentino de Radioastronomia (IAR) southern sky survey [32, 33] with the Leiden/Dwingeloo Survey (LDS) [34]. The data are corrected for nHI_xyz_sigmar4_matrix.fits.gz_0nHI_xyz_sigmar4_matrix.fits.gz_0 14 y(kpc)nHI_xyz_sigmar4_matrix.fits.gz_0y(kpc) CHAPTER 2. INTERSTELLAR MEDIUM 20 20y(kpc)Tb_matrix_lvb2.fits.gz_0 20Tb_matrix_lvb.fits.gz_0Tb_matrix_lvb.fits.gz_0 VelocityVELOVelocityVELO (KM)(KM) (km(KM)Tb_matrix_lvb.fits.gz_0 (km/s)/s) 10 10VelocityVELO (KM) (km/s) 20020010 200 10010000 1000 000 0 -100-10-100-10 -100-10 -200-200 -200-200 -20 -20-20 -2000 -10 100 100 0 200 200 10 300 300 20 -20-200100 -10 -10 100 0 00 200 10-100 300 20 20 LongitudeGLONLongitudeGLON (DEG) (DEG) (deg) (deg) GLONx(kpc)LongitudeGLONx(kpc)x(kpc) (DEG) (deg) 1302 0000 0 13013022 0 K 3 KKK33 cmcmcm

Figure 2.1: The brightness temperature map on the Galactic plane (b = 0◦) in terms of longitude and radial velocity observed by the LAB survey. the stray radiation. This survey has an angular resolution of 0.6◦ and a velocity sampling of 1 km/s, with velocity range of (-450,400) km/s. It is presently the most sensitive 21cm line survey with the most extensive spatial and kinematic coverage. Fig.(2.1) encodes the brightness temperature on the Galactic plane (b = 0◦) at different values of longitude and radial velocity. In the first quadrant (0◦ < l < 90◦) the radial velocities which correspond to the tangent points are extremely positive while in the fourth quadrant ( 90◦ < l < 0◦) they are extremely negative (see section 2.1.1). − The spin temperature is much greater than the brightness temperature for optically thin (τν 1) 21cm line emission. However, toward extragalactic sources Ts varies from 40 to 300 K, depending on the location and velocity [35]. Moreover, outside the solar circle up to Galactocentric radius of 25 kpc the spin temperature is in the range of 250 to 400 K [36]. We assume a globally constant spin temperature equal to 150 K, which is also the maximum observed Tb in the LAB survey data. Although using different values of Ts may change the column density of atomic hydrogen [37], the whole structure of the gas remains unchanged. The column density is obtained by integrating eq.(2.9) over path lengths and radial velocities along a line of sight. Z Z −2 Tb NHI [cm ] = nHI dr = CTs ln 1 dvr. (2.11) l.o.s l.o.s − Ts The sky map of atomic hydrogen column density is shown in fig. (2.2). Most of the gas 2.1. ATOMIC HYDROGEN GAS 15

Figure 2.2: The sky map of atomic hydrogen column density. is distributed in a thin layer around the Galactic plane with more concentration toward the Galactic center.

2.1.1 Derivation of Atomic Hydrogen Number Density

The atomic hydrogen number density at a given heliocentric distance r, longitude l and latitude b is derived from (2.9) as T (l, b, v ) δv −3 b r r nHI (r, l, b)[cm ] = CTs ln 1 (2.12) − Ts δr where l = 0, b = 0 corresponds to the Galactic center. The rotation velocity of the gas θ(R, z) away from the Galactic plane becomes smaller than that of the gas at the underlying disk. At small Galactocentric radii the altitude dependence of rotation curves is prominent while it becomes less important at large values of R [38]. Moreover, the vertical extension of hydrogen gas is small inside the solar circle and it increases outward the Galaxy. For these reasons, it is a valid assumption to ignore the lagging rotation.

Inside the solar circle (R < R ) we use the rotation curve of [39] which is ﬁtted by a polynomial of the form 7 n θ(R) = Σn=0AnR . (2.13)

The coeﬃcients An are obtained by assuming R = 8.5kpc and θ = 220 km/s. For

R > R there is a general consensus that it is a fair approximation to assume a ﬂat rotation curve with θ = θ [40, 41]. It is worth noting that the angular velocity

θ(R) vr ω(R) = = + ω , (2.14) R R sin l cos b 16 CHAPTER 2. INTERSTELLAR MEDIUM is always positive and increases toward the Galactic center. Radial velocities giving negative ω are forbidden. They correspond to the peculiar motions of the local gas.

δvr The derivative of radial velocity with respect to heliocentric distance δr is computed by using the chain rule as follows

δvr δvr δω δR 2 1 δω = = R sin l cos b(r cos b R cos l) . (2.15) δr δω δR δr − R δR Therefore, the gas number density at every given (r, l, b) is obtained by inserting eq.(2.15) in eq.(2.12). The heliocentric distance r associated to (l, b, vr) or (l, b, R) is determined by q 2 2 2 R cos l R R sin l r = ± − R R (2.16a) cos b ≤ q 2 2 2 R cos l + R R sin l r = − R > R . (2.16b) cos b In the inner part of the solar orbit there are two kinematically allowed distances (except for the tangent points R = R sin l where they coincide) and for the outer part there | | δvr is only one. Although there is no distance ambiguity at tangent points, because δr is zero eq.(2.12) fails to determine the gas density. We describe the method of obtaining the gas density at tangent points in section 2.1.1. In the case of distance degeneracy the observed intensity must be distributed among the near-far points as explained in

δvr section 2.1.1. Note that δr is also zero toward the Galactic center/anti-center and right above/below the Sun. The number density of points in these directions is calculated by linear interpolation between nH (r, l, b) of nearby points.

Tangent Points

The closest point to the Galactic center at every given direction r cos b = R cos l, has extreme positive (in the first quadrant) or negative (in the fourth quadrant) radial velocity which is called terminal velocity vt. Due to velocity dispersion σv, the velocity profiles do not have a sharp cutoff at tangent points. The emission from the tangent points is ideally a bivariate Gaussian in altitude and velocity. But emission from the nearby radii are not well separated in velocity because of the velocity dispersion. Atomic hydrogen gas in the vicinity of the tangent point has radial velocity with vr vt σv. The number density | | ≥ | | − around the tangent point is obtained by dividing the emission from this velocity range

( vr vt σv) by the corresponding path length [42]. In the ﬁrst quadrant, it is | | ≥ | | − R ∞ Tb(l,b,vr) CTs ln 1 dv vt−σv − Ts nHI (r, l, b) = , (2.17a) r2(vt σv) r1(vt σv) − − − 2.1. ATOMIC HYDROGEN GAS 17

Figure 2.3: Hydrogen number density at tangent points along diﬀerent directions is plotted versus z. The related Galactocentric radius is R = R sin l | | and in the fourth quadrant it is

R vt+σv Tb(l,b,vr) CTs ln 1 dv −∞ − Ts nHI (r, l, b) = . (2.17b) r2(vt + σv) r1(vt + σv) −

In the above, r1 and r2 are, respectively, the near and far heliocentric distances associated to radial velocity of vt σv(vt + σv) in the first (fourth) quadrant. The velocity dispersion − has been estimated to be about 9 km/s for the first quadrant and 9.2 km/s for the fourth quadrant [43], however it is larger close to the Galactic center. The hydrogen number density at tangent points along different directions is shown versus height in Fig.(2.3). The mid-plane, where the gas density is maximum, almost coincides with the Galactic plane. The vertical distribution of the gas around the mid-

z−z0 2 plane can be estimated by a Gaussian function of the form exp[ ( ) ] where z0 and − σz σz are, respectively, the mid-plane displacement and the scale height. The scale height at tangent points varies in a small range between 0.1 to 0.2 kpc.

Inner Galaxy

Inside the solar orbit, for a given radial velocity there are two heliocentric distances. In order to distribute the signal among these points, we assume that the vertical extension of the gas at every given R is a Gaussian function whose scale height and mid-plane displacement are obtained from tangent point with the same R. The point whose height is closer to the mid-plane receives more contribution from the signal and has greater number density. The number density at ri, where i indicates either the near or the far 18 CHAPTER 2. INTERSTELLAR MEDIUM point, is obtained by

T (l, b, v ) δv b r r nHI (ri, l, b) =CTs ln 1 − Ts δr exp [ ( zi−z0 )2] (2.18) − σz zj −z0 2 ×Σj exp [ ( ) ] − σz Local Gas

At every direction the radial velocities associated with negative angular velocities or distances far from the Galactic plane ( z σz) are due to peculiar motions of the | | local gas. We assume that the gas with peculiar radial velocity is locally distributed by a

Gaussian function with radial scale σr of less than about 4 kpc as follows

−( r )2 Z T (l, b, v ) e σr n (r, l, b) = CT ln 1 b r dv (2.19) HI s r R −( r )2 − Ts × e σr dr where the integral is performed over peculiar velocities along the related line of sight. The impact of this assumption is marginal, since the amount of gas with peculiar velocity is only 0.033% of the total amount of atomic hydrogen gas in the Galaxy.

2.1.2 Large Scale Features

Following the method described in section 2.1.1 the number density of atomic hydrogen gas as a function of r, l and b centered at the Sun is derived. It is then transformed into Cartesian coordinates centered at the Galactic center in which the Sun is assumed to be at (x, y, z) = ( 8.5, 0, 0) kpc. The map of number density on the Galactic plane − z = 0 is shown in Fig.(2.4). The distribution of gas on the Galactic plane is north-south asymmetric. The density peaks at the Galactic center however there is a distinct hole right below it. Most of the gas on the Galactic plane is concentrated within the radius of about 10 kpc whereafter it rapidly dilutes away. 9 It is found that the total mass within a radius of 20 kpc is 4.3 10 M and only × 0.033% of that is due to local gas with peculiar velocities. The global properties of the atomic hydrogen gas distribution in the Milky Way are explained in the following sections.

The Warp

There is a large scale warp in the gas disk of the Milky Way (see [40] and references therein). The map of the mid-plane displacement is shown in Fig.(2.5). Inside solar circle the mid-plane and the Galactic plane coincide pretty well however outside this region the 2.1. ATOMIC HYDROGEN GAS 19

nHI_xyz_sigmar4_matrix.fits.gz_0 y(kpc) 20

-10

-20 -20 -10 0 10 20 x(kpc) 0 2 3 cm

Figure 2.4: Hydrogen number density map on the Galactic plane mid-plane is warping. The warp is weak up to radii of about 13 kpc and then it quickly becomes strong. The mid-plane bends up to a height greater than z0 = 2.5 kpc at R=20 kpc in the north and bends down to z0 = 1.5 kpc in the south. − To better illustrate the warping feature of the gas distribution, in fig.2.6 the gas number density maps at different heights are displayed. At negative values of z the dense regions are in the south and at positive values of z they are in the north. In Fig.(2.7) the mid-plane displacement for different values of x is shown against y. The general behavior of increasing the mid-plane distance from the Galactic plane outward the Galaxy is asymmetric with more vertical extension in the north.

The Flare

The balance of gravitational force against the pressure force determines the thickness of the atomic hydrogen disk. The average scale height, which is defined as the distance over which the number density decreases by a factor of e, shows a clear flaring [38, 44, 45, 46]. It increases from about 0.2 kpc in the inner Galaxy up to 0.75 kpc at R=20 kpc as shown in Fig.(2.8, top panel). The mid-plane density is also shown in Fig.(2.8, bottom panel). It peaks at the Galactic center and has fluctuations in the inner part, then falls down in the outer Galaxy. 20 CHAPTER 2. INTERSTELLAR MEDIUM

zc_xy_matrix.fits_0 y(kpc) 20

-10

-20 -20 -10 0 10 20 x(kpc)

1.5 2.8 kpc

-13456 0 1 2 (kpc) Figure 2.5: The warped Galactic plane. The mid (kpc) plane displacement has greater positive (neg- zc_xy_matrix.fits_0 ative) values in darker (lighter) regions. 2

Spiral Structure

The spiral structure can be traced in the surface density distribution as regions with over densities [47, 48]. In Fig.(2.9) the surface density map is shown, Z

Σ(x, y) = dznHI (x, y, z) (2.20) in which several spiral arms are evident. There is one large spiral arm in the north, the so-called Outer arm. In the southern half, the so-called Sagittarius-Carina arm close to the solar circle is prominent. The so-called Perseus arm in the south extends to the north and connects to the Outer arm.

2.2 Molecular Hydrogen Gas

Molecular hydrogen has been detected in large quantities in the interstellar gas. It is found in regions where it is shielded from ultraviolet radiation which causes its photodissociation.

Thus, molecular hydrogen gas can only survive in cold dark clouds. Indeed, H2 is the prime component of molecular clouds. Due to the lack of electric dipole moment and low temperature and density within molecular clouds the excitation of H2 rotational transitions is unlikely [49]. Hence, direct observation of molecular hydrogen gas is diﬃcult.

However, it can be traced by 2.6 mm emission line of CO1−→0, since the collisions between 2.2. MOLECULAR HYDROGEN GAS 21

nHI_xyz_sigmar4_matrix.fits.gz_0 nHI_xyz_sigmar4_matrix.fits.gz_0 nHI_xyz_sigmar4_matrix.fits.gz_0nHI_xyz_sigmar4_matrix.fits.gz_0nHI_xyz_sigmar4_matrix.fits.gz_0nHI_xyz_sigmar4_matrix.fits.gz_0 nHI_xyz_sigmar4_matrix.fits.gz_0 nHI_xyz_sigmar4_matrix.fits.gz_0 y(kpc) y(kpc) y(kpc) y(kpc) y(kpc) y(kpc) y(kpc) y(kpc) 20 20 20 20 20 20 20 20

10 10 10 10 10 10 10 10

0 0 0 0 0 0 0 0

-10 -10 -10 -10 -10 -10 -10 -10

-20 -20 -20 -20 -20 -20 -20 -20 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10-20 0 -10 10 0 20 10 20 -20 -10-20 0 -10 10 020 10 20 -20 -10 0 10 20 -20 -10 0 10 20 x(kpc) x(kpc) x(kpc) x(kpc) x(kpc) x(kpc) x(kpc) x(kpc) 0 0.3 0 0.3 3 3 nHI_xyz_sigmar4_matrix.fits.gz_0nHI_xyz_sigmar4_matrix.fits.gz_0nHI_xyz_sigmar4_matrix.fits.gz_0nHI_xyz_sigmar4_matrix.fits.gz_0cm nHI_xyz_sigmar4_matrix.fits.gz_0nHI_xyz_sigmar4_matrix.fits.gz_0cm nHI_xyz_sigmar4_matrix.fits.gz_0nHI_xyz_sigmar4_matrix.fits.gz_0 y(kpc) y(kpc) y(kpc) y(kpc) y(kpc) y(kpc) y(kpc) y(kpc) 20 20 20 20 20 20 20 20

10 10 10 10 10 10 10 10

0 0 0 0 0 0 0 0

-10 -10 -10 -10 -10 -10 -10 -10

-20 -20 -20 -20 -20 -20 -20 -20 -20 -10-20 0 10 -10 20 0 10 20 -20 -10-20 0 10 -10 20 0 10 20 -20 -10 0-20 10 -10 20 0 10 20 -20 -10 0-20 10 -10 20 0 10 20 x(kpc) x(kpc) x(kpc) x(kpc) x(kpc) x(kpc) x(kpc) x(kpc) 0 0.6 0 0.7 3 3 cm cm

Figure 2.6: Atomic hydrogen density maps at diﬀerent values of z. Plots from left to right are, respectively, due to z=-1,1 kpc on the top panel and due to z=-0.5,0.5 kpc on the bottom panel. 22 CHAPTER 2. INTERSTELLAR MEDIUM

Figure 2.7: The height from the Galactic plane where the number density of the atomic hydrogen gas is maximum.

Figure 2.8: Left: The average scale height versus Galactocentric radius. Right: The mid-plane density versus R. 2.2. MOLECULAR HYDROGENNHI_xy_sigmar4_matrix.fits.gz_0 GAS 23 Y (kpc) NHI_xy_sigmar4_matrix.fits.gz_0NHI_xy_sigmar4_matrix.fits.gz_0 Y Y(kpc) (kpc) 2020

1010

0 0

-10-10

-10-10-10 0 0 0 10 10 20 20 20 XX X(kpc)(kpc) (kpc) 0.01 23 M /pc2

Figure 2.9: The surface density map. The spiral arms can be traced as regions with over densities.

CO and H2 molecules in the clouds are responsible for CO excitation. The observation of

CO emission line therefore provides a measure of H2 density.

The column density of molecular hydrogen NH2 is derived from the velocity integrated intensity of CO line WCO as follows

NH2 = XCOWCO. (2.21)

The conversion factor XCO depends on the metallicity and the ultraviolet radiation background [50, 51]. In clouds with lower metallicity or stronger ultraviolet radiation background, the abundance of CO molecules is smaller. Therefore, the conversion factor must increase to an appreciable extent. However, in clouds with large CO abundance the emission line is fully saturated and XCO is directly proportional to the molecular column density. The measurement of metallicity gradient in our Galaxy can be used to estimate the distribution of XCO. It has been shown that XCO increases with Galactocentric radius although its radial steepness is largely uncertain [52, 53, 54, 55]. The CO surveys performed by CfA 1.2 m telescope and a similar telescope in Chile have been merged to produce a new composite survey of the entire Milky Way [56]. This survey with an angular resolution of 0.6◦ and a velocity sampling of 1.3 km/s is the most sensitive large scale survey to date. The composite maps provide detailed information on individual molecular clouds as seen in Fig.(2.10). A comparison between Fig.(2.2) and 24 CHAPTER 2. INTERSTELLAR MEDIUM

Figure 2.10: The sky map of the velocity integrated brightness temperature of CO.

Fig.(2.10) shows that molecular hydrogen gas is conﬁned to the Galactic plane more than the atomic hydrogen gas.

The most recent model for three dimensional distribution of molecular hydrogen gas in the Milky Way is provided by [57]. This model is obtained by kinematic deconvolution of the composite CO survey of [56] using a new gas ﬂow model [58]. The map of gas distribution on the Galactic plane is shown in Fig.(2.11).

2.3 Ionized Hydrogen Gas

Ionized hydrogen is concentrated in the vicinity of young O and B stars, since the ultraviolet radiation of these stars ionizes the interstellar medium. It is known that the contribution of ionized hydrogen to the total mass in the interstellar medium is negligible

[59]. The distribution of HII in our Galaxy is very similar to that of the free electrons. It is parametrized as [60]

" # " # z R 2 z R R0 2 nHII (R, z) = ne(R, z) = ne 1 exp | | + ne 2 exp | | − h i − z1 − R1 h i − z2 − R2 (2.22) 0 where z1 = 1 kpc, z2 = 0.15 kpc, R1 = 20 kpc, R2 = 2 kpc, R = 4 kpc, ne 1 = h i −3 −3 0.025 cm , and ne 2 = 0.2 cm . h i 2.4. GALACTIC MAGNETIC FIELD 25

Pohl_H2.fits.gz_0 y

-10

-10 0 10 x

Figure 2.11: Molecular hydrogen gas map on the Galactic plane provided by [57]. Sun is at (x, y) = (8, 0) kpc.

2.4 Galactic Magnetic Field

The intensity of magnetic field in our Galaxy is determined by Faraday rotation measurements. The rotation of polarization angle of a linearly polarized radio emission while propagating through a magnetized plasma is known as Faraday rotation. Magnetized plasma, itself, is created by spiraling of ionized interstellar gas around the magnetic field lines. Faraday rotation is proportional to the wavelength squared of the emission λ2 and is calculated as Z 2 θ = 0.812λ neBkds. (2.23) l.o.s −3 In the above, ne is the density of thermal electrons in cm unit, Bk is the strength of magnetic field parallel to the line of sight in µG unit and ds is the path length in parsec 2 2 unit. The rotation measure θ/λ , which is in units of rad/cm , is used to evaluate Bk along the line of sight. Its sign gives information about the weighted mean direction of the magnetic field along the line of sight. If it is positive the field is directed toward the observer and vice versa. The analysis of rotation measures shows that the Galactic magnetic field has a bi- symmetrical spiral structure at large scales and a turbulent part at scales smaller than few hundred parsecs [61]. The strength of the large scale magnetic field is a few µG and it is comparable with that of the turbulent component on the Galactic disk. Because of the small pitch angle of spirals [62], the large scale Galactic magnetic field ~ ˆ can be assumed to be purely azimuthal, B0 = B0φ. It exponentially decreases outward 26 CHAPTER 2. INTERSTELLAR MEDIUM the Galaxy in both radial and vertical directions as

R R z B0 = Bh exp − exp | | . (2.24) − Rh − zh

Based on the analysis of WMAP data on synchrotron intensity and polarization [61] as well as studies of extragalactic rotation measures [62, 63, 64, 65] the local strength of magnetic ﬁeld, its radial and vertical scales are chosen as

Bh = 3 µG,Rh = 11 kpc, zh = 2 kpc. (2.25)

The intensity and polarization of synchrotron emission are proportional to the density of relativistic electrons and the strength of the magnetic field perpendicular to the line of sight. Cosmic ray electrons gyrating around the magnetic field lines lose energy by emitting synchrotron radiation. The strength of magnetic field and its structure has a crucial effect on the evaluating of synchrotron intensity and polarization. However, it has little impact on estimating diffuse γ-rays, since the magnetic field enters only in the electron energy loss, which is anyway dominated by the inverse Compton scattering loss above few GeV. Chapter 3

Cosmic Rays

Cosmic rays are high energy charged particles from outer space that travel at nearly the speed of light and impinge on Earth from all directions. The energy density of cosmic rays with energies greater than 1 GeV is about 1 eV/cm3(1 MeV/m3)[66]. This energy density 2 3 is comparable to that present in the Galactic magnetic field, B /2µ0 0.2 MeV/m . It is ∼ also similar to the local energy density in starlight, 0.3 MeV/m3, and to the energy density of the microwave background radiation about 0.3 MeV/m3. Therefore, cosmic rays are one of the essential factors determining the dynamics and processes in the interstellar medium. As observed at the top of the atmosphere about 89% of the particles are protons, 10% are helium nuclei and the remaining 1% are electrons and heavier nuclei. The chemical abundances of particles in cosmic rays compared with those in Solar System indicate the overall similarities and differences. The abundance peaks at carbon, nitrogen and oxygen and at the iron group are present both in cosmic rays and Solar Sys- tem abundances. However the light elements, lithium, beryllium and boron are grossly overabundant in the cosmic rays relative to their Solar System abundances. These over- abundances imply that cosmic ray primaries, created by primary sources similar to those in Solar System, propagate in the interstellar medium and produce cosmic ray secondaries by spallation processes. The mean amount of matter traversed by cosmic rays must be a few g/cm2 to be compatible with the observed number of secondaries; thus cosmic rays travel in a confined volume with an average residence time of about 107 years. In section 3.1, the primary sources of cosmic rays, their spatial distribution and injection properties are described. The diffusion of cosmic rays in the interstellar medium and its energy and spatial dependence is discussed in section 3.2. The energy loss processes for both nuclei and electrons are presented in section 3.3. Finally, in section 3.4 the methodology of determining relevant parameters which characterize the propagation

27 28 CHAPTER 3. COSMIC RAYS of cosmic rays in the interstellar medium is explained.

3.1 Primary Sources

The acceleration of primary cosmic rays is assumed to take place in the shocked shells of supernova remnants by first order Fermi acceleration. Cosmic rays get accelerated up to energies of 100 TeV as long as they are confined to the shock region by up-stream ∼ turbulence [67]. Once the supernova remnant enters its radiative phase, all high energy particles are released and diffuse through the interstellar medium. The injection of cosmic ray species i with charge Z into the interstellar medium is a power-law rigidity dependent which is given by

−γi qi(R, z, p) = fs(R, z)q0,i R . (3.1) 0 R

In the above, = pc/Ze is the rigidity, q0,i is the normalization of the injected cos- R mic ray species, and fs(R, z) traces the spatial distribution of supernova remnants. The distribution of supernova remnants is assumed to be cylindrically symmetric. As a reference model the distribution provided by [68], which is derived on the basis of pulsar and progenitor star surveys, is used. Given that protons provide the dominant contribution to diﬀuse γ-ray spectra in the Fermi energy range (see chapter4), a particular attention is paid to the proton spectrum. The local proton spectrum measured by PAMELA [69] and CREAM [70], indicates a hardening at high energies. Motivated by that, cosmic ray proton injection spectrum is a posteriori assumed to be a broken power-law,

p −γj dNp R . (3.2) d ∝ p R R0,j The rigidity break(s) = p and the spectral injection indices γp are determined upon R R0,j j fitting the data. Electrons and positrons which are accelerated between a pulsar and the termination shock of the wind nebula, may also contribute to the high energy e± spectrum [71, 72, 73, 74, 75] and then to the γ-ray flux [76, 77]. In particular, middle aged pulsars are found to be particularly well suited [78, 75]. The injection of e± into the interstellar medium by each pulsar can be described by a power-law E−n with a high energy break ∼ Eb. The value of Eb is estimated at the time that the surrounding pulsar wind nebula is 3.1. PRIMARY SOURCES 29 disrupted and e± escape into the interstellar medium [74] 1 . Furthermore, each pulsar has an initial rotational energy W0 of which only a portion η is injected into the interstellar medium as cosmic ray e±. The ranges of those parameters for different pulsars are very 49−50 broad. Indeed, n can range between 1 and 2, W0 10 erg, η 0.1 [79, 74] and ∼ ∼ ± Eb 10 TeV. Besides, the spectrum of e from a pulsar has a break that is related to ∼ the cooling time (from inverse Compton scattering and synchrotron radiation) of the e± during their propagation in the interstellar medium [71, 74]. To account for these effects, the parametrization of [74] is used to fit the properties of the pulsar distribution. The source term due to distribution of pulsars is described as a power-law with an exponential cut-off given by

−n −E/M Qp(R, z, t, E) = J0E e fp(R, z), (3.3) where M is a “statistical” cut-oﬀ, n the injection index for the distribution of pulsars 2,

ηW0Nb J0 = 2−n , (3.4) Γ(2 n)M Vgal −

(see Eq. 24 of [74]) with Nb the birth rate of pulsars in the Galaxy and

Z zmax Z Rmax Vgal = dz dR 2πR fp(R, z). (3.5) −zmax 0 fp(R, z) describes the spatial distribution of young and middle aged pulsars. Since the kick speeds of pulsars is typically about 200-400 km/s [80, 81], a 105 yr old (middle aged) pulsar would move away from its original position by 30 pc, and thus the spatial ∼ distribution of middle aged pulsars is practically identical to that at their birth time and is given by [82]

a R + R1 R R z fp(R, z) exp b − exp | | , (3.6) ∝ R + R1 − R + R1 − z1

with R1 = 0.55 kpc, z1 = 0.1 kpc, a = 1.64 and b = 4.

1 Although higher energy e± escape earlier into the interstellar medium, the differences in the estimated 4 time scales of escape for e± with energies between 1-10 GeV are negligible compared to their propagation time from the pulsar wind nebula to us [79, 74]. 2 Due to large energy loss, the flux of cosmic ray e± with E TeV probes only the e± sources within ∼ 2 O(10 ) pc around its observed position. Thus, the spectra of these cosmic ray e± are significantly ∼ different within the Galaxy. Since the inverse Compton spectra are studied in wide regions of the sky only the averaged e± flux matters. For such an averaged e± flux from pulsars, the statistical cut-off M and the injection index n refer to the statistically averaged values [74]. 30 CHAPTER 3. COSMIC RAYS

3.2 Diﬀusion

Cosmic rays form a plasma of ionized particles, in which the electric fields can be neglected by virtue of the high conductivity and magnetic fields form Alfvénwaves. If cosmic ray particles have gyro-radii much smaller than the scale of the fluctuations in the magnetic field, the trajectories of the particles follow their guiding centers. In the opposite limit in which the particles have gyroradii much greater than the scale of the fluctuations, the particles do not feel the fine structures in the field but move in orbits determined by the mean magnetic field which is much greater in magnitude than the fluctuating component. The significant scattering occurs only for the case in which the fluctuations have the same scale as the gyro-radii of the particles. The scattering of the particles by the random superposition of these fluctuations leads to stochastic changes in the pitch angles of the particles. Moreover, magnetic fluctuations generated by Alfvénand hydromagnetic waves grow in amplitude under the influence of the streaming motions, so even if there were no magnetic irregularities to begin with, they are generated by the streaming of the high energy particles. In other words, there is a mutual influence between cosmic rays scattering and fluctuations of magnetic fields. Because of the random scattering by irregularities in the magnetic filed either associated with fluctuations in the field or with the growth of instabilities due to the stream of the particles, cosmic rays can be considered to diffuse from their sources through the interstellar medium. The diffusion process makes the distribution of cosmic rays in the Galaxy isotropic and justifies their long residence time. The diffusion of cosmic rays is explained by a tensor [83]

Dij = (Dk D⊥)bibj + D⊥δij + DAijkbk, (3.7) − ~ ~ where b = B0/B0 is a unit vector along the regular magnetic field, Dk and D⊥ are diffusion coefficients, respectively, parallel and perpendicular to the mean field, and DA is the antisymmetric (Hall) diffusion coefficient. In strongly turbulent magnetic fields, which is the case in the Milky Way Galaxy, DA is negligible and D⊥ Dk is given by [84] ∼ c rL D⊥ k= 2π . (3.8) ∝ 3 dEturb/d ln k | rL

Here rL = /B0 is the Larmor radius of a relativistic charged particle of rigidity in the R R regular magnetic field, and dEr/dk is the power spectrum of the random magnetic field −α energy density. The energy density of turbulent component is dEturb/dk k , where the ∝ spectral index α is given by the kind of mechanism that builds up the turbulence. Hence, −α+2 the symmetric components of the diffusion coefficient are proportional to rL , which 3.3. ENERGY LOSSES 31

1/3 1/2 implies D⊥ r for Kolmogorov spectrum [85] and D⊥ r for Kraichnan spectrum ∝ L ∝ L [86, 87]. The diffusion coefficient generally depends on the location because the turbulence is not uniformly distributed in the Galaxy. The diffusion length is expected to increase outward the Galaxy as the Galactic magnetic field decreases. The weaker magnetic field makes the escape of cosmic rays easier and so the diffusion coefficient becomes greater.

In a phenomenological approach D⊥ or simply the diffusion coefficient D is assumed to be described by δ η R R z D(R, z, ) = D0β R exp − exp | | (3.9) R 0 Rd zd R where the radial scale Rd and the scale height zd define the diffusion profile in the Galaxy. The diffusion spectral index δ, as mentioned earlier, is related to the interstellar medium turbulence power-spectrum. The dependence of diffusion on the particle velocity, β = vp/c, is naturally expected to be linear (η = 1), however the analysis by [88] shows an increase in diffusion at low energies. To account for such a possibility, the parameter η has been introduced (see e.g. [89, 75, 90]). The reference rigidity 0 is chosen at 3 GV R and the diffusion normalization D0 defines the local diffusion coefficient at rigidity of 0. R In addition to the spatial diffusion, the stochastic second order Fermi acceleration of cosmic rays is described as diffusion in momentum space. It is due to the scattering of cosmic rays on randomly moving magnetohydrodynamic waves. The corresponding diffusion coefficient in momentum space is inversely proportional to the diffusion coefficient in physical space as follows 2 Dpp p vA/D, (3.10) ∝ where vA is the Alfvénvelocity, associated to the propagation of magnetohydrodynamic waves [91].

3.3 Energy Losses

Cosmic rays lose energy during the propagation in the interstellar medium. The energy loss of cosmic ray electrons and positrons at energies of about 1 GeV is dominated by bremsstrahlung, ionization and Coulomb losses in the interstellar gas. At energies above several GeV cosmic ray e±s lose energy mostly by either inverse Compton scattering oﬀ the interstellar radiation ﬁeld or synchrotron emission. The mean energy loss rate via inverse Compton scattering is expressed by 2 dE 4 v 2 = σT curad γ (3.11) dt ICS 3 c 32 CHAPTER 3. COSMIC RAYS

Figure 3.1: Ratio of the energy loss rate of 1 GeV(top left), 10 GeV (top right) and 100 GeV (bottom) electrons due to synchrotron radiation to the total energy loss rate.

where σT is the Thomson scattering cross section, urad is the energy density of radiation, v is the electron velocity and γ is its Lorentz factor. This expression is valid as long 2 as γ~ω mec whereafter σT must be replaced by the Klein-Nishina cross section. Therefore, the rate of energy loss by inverse Compton scattering off the CMB photons is more significant than that off the infrared and the optical photons. Indeed, the diffuse γ-ray component produced through inverse Compton scattering, away from the Galactic plane, is dominated by the scattering off the CMB photons. Thus, it is almost isotropic all over the sky. The mean energy loss rate via emitting diffuse synchrotron radiation in the Galactic magnetic field is 2 dE 4 v 2 = σT cumag γ (3.12) dt sync 3 c where umag is the energy density of the magnetic filed. It drops fast away from the Galactic plane as the magnetic field decreases (see Eq. 2.24) . The ratio of the rate of synchrotron energy loss to the total rate for electrons with energies of E = 1, 10 and 100 GeV is shown in Fig. 3.1 versus Galactocentric radius and altitude. The synchrotron energy loss is in no part of the Galaxy dominant. It accounts for at most 22% of the total electron energy 3.4. PROPAGATION 33 loss at 100 GeV at the Sun position. Cosmic ray nuclei and protons, lose energy by ionization and Coulomb scattering. In addition, collisions of heavy nuclei with hydrogen or helium of the interstellar gas can lead to inelastic scattering and consequently to the fragmentation of the parent cosmic rays. Cosmic rays can also undergo radioactive decays, with the radioactive isotopes created both by fragmentation of heavier nuclei (e.g. 10Be is created from B, C, N, O) and directly in primary sources (e.g. 26Al). Adiabatic loss can also be important inside expanding supernova remnants but it is negligible when cosmic rays diffusively propagate in the interstellar medium.

3.4 Propagation

The propagation of cosmic rays in the Galaxy at energies below 1017eV can be described by [91] h i ∂ψ(~r, p, t) ~ ~ ∂ 2 ∂ ψ ∂ = q(~r, p, t) + .(Dxx ψ) + p Dpp ( ) (pψ ˙ ) ∂t ∇ ∇ ∂p ∂p p2 − ∂p ~ ∂ hp ~ i ψ ψ .(~vcψ) + ( .~vc)ψ (3.13) − ∇ ∂p 3 ∇ − τfrag − τdecay where ψ(~r, p, t) is the cosmic ray density per unit particle momentum, q(~r, p, t) is the source term including primary, spallation and decay of heavier cosmic ray species, Dxx(~r) is the diffusion tensor in physical space and Dpp(~r) is the diffusion coefficient in momentum space,p ˙ is the momentum loss rate due to interactions with the interstellar gas, the

Galactic magnetic ﬁeld or the interstellar radiation ﬁeld, ~vc is the convection velocity,

τfrag and τdecay are the time scales for, respectively, fragmentation loss and radioactive decay. As explained in section 3.2 the diffusion tensor in strongly turbulent magnetic fields is diagonal and only the component of diffusion which is perpendicular to the mean Galactic magnetic filed has an effect. Therefore, the diffusion tensor Dxx in the Eq. 3.13 can be substituted by the diffusion coefficient D. The Milky Way Galaxy is assumed to be a cylinder with the radius of 20 kpc. Its height L above and below the Galactic plane is taken to be two times the diffusion scale height zd. The density of cosmic rays is supposed to be zero at the boundaries of the diffusion zone where cosmic rays escape into the intergalactic space. The propagation equation is numerically solved in the steady state approximation ∂ψ/∂t = 0. To that end, the DRAGON code [92, 93, 75, 94] is used. Assuming cylindrical symmetry the Eq. 3.13 is solved in 2+1 dimensional grid where each point is specified by 34 CHAPTER 3. COSMIC RAYS the Galactocentric radius R (0, 20) kpc, the altitude from the Galactic plane z ( L, L) ∈ ∈ − and the momentum p. There are many unknowns in the modeling of the propagation of cosmic rays in the interstellar medium. One then is forced to consider a few benchmark scenarios. Among different factors that affect the propagation of cosmic rays within the Galaxy, the diffusion is the most effective process at high energies. The propagation equation at high energies can then be reduced to

∂ψ ~ ~ = q + .(Dxx ψ). (3.14) ∂t ∇ ∇ Cosmic rays diffuse freely in a confined volume with a finite probability to escape. The time scale of escape τesc is inversely proportional to the diffusion coefficient. Thus the diffusion term can be replaced by ψ . At steady state the Eq. 3.14 becomes − τesc ψ = q. (3.15) τesc The primary source of cosmic rays is a power-law energy dependent E−γ. So the density of cosmic ray primaries depends on the energy as

−γ −1 −γ−δ ψpri qpriτesc E D E . (3.16) ∝ ∝ ∝ The energy dependence of the density of the cosmic ray secondaries is given by

−γ−δ −1 −γ−2δ ψsec qsecτsec ψpriPfragτsec E D E (3.17) ∝ ∝ ∝ ∝ where Pfrag is the probability for fragmentation of the primaries. This results in ψ sec E−δ (3.18) ψpri ∝ which implies that the spectral slope of the secondary to primary ratio at high energies is determined by δ. The diffusion spectral index ranges from about δ = 0.3 to about 0.7. The theoretically motivated turbulence spectra namely the Kolmogorov spectrum [85] corresponding to δ = 0.33, the Kraichnan spectrum [86, 87] corresponding to δ = 0.5 and the spectrum with δ = 0.4 are studied in this analysis. The vertical and radial scales of the diffusion are changed in a range of 1 to 10 kpc for zd and in the range of 5 to 20 kpc for Rd. These choices allow one to investigate the effect of highly homogeneous (with zd=10 and Rd=20)/inhomogeneous (with zd=1 and Rd=5) diffusion on the propagation of cosmic rays. The stream of cosmic rays may create convective winds. The Galactic winds affect the diffusion timescale by pushing cosmic rays away from the Galactic plane. In one 3.4. PROPAGATION 35 benchmark model a strong wind is introduced to examine the effect of convection on the propagation of cosmic rays.

The model with δ = 0.5, zd = 4 kpc, Rd = 20 kpc and vc = 0 is chosen as a reference diﬀusion model. Regarding the properties of the interstellar medium the distribution of gas in the Galaxy is of the utmost importance. The reference models which are used for the distribution of atomic and molecular hydrogen gas are provided by, respectively, [42] and [95]. These models are constructed by the same deconvolution method. Concerning the distribution of the supernova remnants in our Galaxy the model which is supplied by [68] is used as a benchmark. In this analysis diﬀerent models for the distribution of gas as well as of cosmic ray primary sources are studied.

For each model with a set of values for δ, zd, Rd and vc the remaining diffusion 2 parameters D0, η and vA are determined by minimizing the χ of the ratio of the secondary to the primary spectrum. To that end, the spectrum of boron to carbon B/C measured by HEAO-3 [96], CRN [97] and CREAM [98] is used. The flux of cosmic rays, which arrive at the outskirt of the Solar System, is modu- lated due to interactions in the heliosphere. The decrease in the magnitude of cosmic rays intensity depends on the level of Solar magnetic activity. Cosmic rays which diffuse through the heliospheric magnetic field are convected and adiabatically decelerated by the outward motions of the Solar wind. In the force field approximation [99] the effect of Solar modulation can be described by a single parameter which is called the modulation potential φ. The local interstellar spectrum (LIS) is related to the locally observed spectrum via

(E + m)2 m2 Z e J(E ,Z,A) = k J (E + φ, Z, A) (3.19) k −2 LIS k | | Z|e| 2 A Ek + m + φ m A − where Ek is the kinetic energy per nucleon, m is the mass of cosmic ray particle with the atomic number of Z and the mass number of A. For electrons and positrons Z/A is taken to be 1. As it is clear the flux of low energy cosmic rays is strongly affected by the Solar modulation whereas the flux of high energy cosmic rays remains almost unchanged. The modulation potential related to different sets of data can be different due to the level of Solar activity at the time of taking the data. So, it is determined upon fitting data. The spectral properties of protons are fitted against the local spectra measured by PAMELA [69] and CREAM [70]. It turns out that two rigidity breaks are needed to fit the protons spectrum. The first one is around p 10 30 GV and the second one is R0,1 ∼ − at p = 300 GV. The spectral index lies in the range of γp (1.85 2.1) at rigidities R0,2 1 ∈ − 36 CHAPTER 3. COSMIC RAYS below p , in the range of γp (2.3 2.5) at intermediate rigidities and in the range of R0,1 2 ∈ − γp (2.18 2.35) at rigidities above p . The protons spectrum is consistent with the 3 ∈ − R0,2 BESS measurements from1997 to1999 [100] with different values of modulation potential. The spectral injection index of helium nuclei is also fitted to the data observed by [69, 70]. Then the evaluated spectrum of antiprotons is compared with the local measurement of PAMELA [15]. Having reproduced the spectra of cosmic ray nuclei and protons the spectral properties of e± are fitted using the flux of e− + e+ measured by Fermi [101]. The spectrum of e± below the energies of about 30 GeV is dominated by the primary electrons which are accelerated at supernova remnants as well as by the secondary e± which are produced through inelastic collisions of cosmic ray nuclei with the interstellar gas. Therefore, the spectral injection index of electrons γe is fitted to the low energy e− + e+ spectrum. Pulsars within the distance of about 3 kpc can contribute to e± spectrum up to O(0.1) at E 50 GeV and up to O(1) at E 500 GeV [74, 72, 102, 73, 103]. The spectral ≈ ≈ properties of pulsars which are parametrized in Eq. 3.3 are determined by assuming that they maximally contribute to e± spectrum. The injection index n and the statistical cutoff M for the distribution of pulsars are found to be, respectively, 1.4 and 1.2 TeV. Then, −1 assuming a constant birth rate of Nb = 30 yr , the averaged energy which is injected ± into the interstellar medium as cosmic ray e per pulsar ηW0 is calculated. Its value is 1049 erg which is well within the allowed range [104, 105, 106, 74, 79]. ' The fitted spectral parameters of electrons are used to evaluate the spectra of e+/(e− + e+) and e−. Then the predicted spectra are compared against the local measurements by [107, 108, 109] for positrons fraction and by [110] for electrons. In Fig. 3.2 the spectra of cosmic rays for the reference model are shown. In the first row the fitted spectra of B/C (left panel) and protons (right panel) are shown. The predicted antiprotons spectrum is shown in the left panel of the second row. The fitted e− + e+ spectrum (second row, right panel), the predicted spectra of e+/(e− + e+) (third row, left panel) and e− (third row, right panel) are shown in Fig. 3.2 as well. Therefore, the reference model is in good agreement with all local cosmic rays spectra. The spectra of e± at high energies are dominated by local sources. Indeed, in this approach it is assumed that the spectral properties of cosmic ray e± sources which are fitted locally are valid in the entire Galaxy. A severe enhancement (suppression) of the local flux of e± above 100 GeV because of the presence (absence) of strong local sources will lead to over-prediction (under-prediction) of diffuse γ-ray component which is produced by inverse Compton scattering. 3.4. PROPAGATION 37

In the following chapter, the evaluated diffuse γ-ray spectra for different propagation models are compared against the Fermi data [16]. The impact of different assumptions about the properties of the astrophysical backgrounds on diffuse γ-ray spectrum is systematically studied. 38 CHAPTER 3. COSMIC RAYS

Figure 3.2: The local spectra of cosmic rays for the reference propagation model. The diffusion parameters are δ = 0.5, zd = 4 kpc, Rd = 20 kpc and vc = 0. The models for distribution of gas and primary sources are provided by [42, 95, 60, 68, 82]. The goodness of fit or of prediction is determined by the value of χ2. Upper left: fitted B/C, upper right: fitted protons, middle left: predicted antiprotons, middle right: fitted e− + e+, lower left: predicted positron fraction, lower right: predicted electrons. Dotted lines refer to unmodulated cosmic ray fluxes. See text for details. Chapter 4

Diﬀuse Gamma Rays

Interactions of cosmic rays with the interstellar medium give rise to a copious γ-ray yield. Inelastic collisions of cosmic ray protons and heavier nuclei with interstellar gas produce neutral and charged pions (π0, π±). Subsequent decay of neutral pions into γγ gives a major contribution to the diffuse Galactic γ-rays. Photons produced in this process trace the spatial distribution of cosmic ray protons as well as of the interstellar gas. Indeed the spectrum of π0 decay component of diffuse γ-rays at energies above a few GeV is mainly determined by that of the cosmic ray protons. In addition, cosmic ray electrons and positrons produce γ-rays via bremsstrahlung in the interstellar gas [111, 112, 113, 114] and by inverse Compton scattering off the interstellar radiation field [114, 115]. At distances far from the Galactic disk the density of the optical and the infrared photons is less than that close to the disk. Therefore, inverse Compton component of diffuse γ-rays at high latitudes is predominantly related to the up-scattered CMB photons. Due to the CMB isotropy, it implies that inverse Compton component sets a good probe to study the spectrum of cosmic ray e± far from the disk. Apart from diffuse Galactic γ-rays the observed flux involves contributions from the Galactic point sources as well as the isotropic extragalactic background [16]. In this analysis diffuse γ-rays spectra are employed to investigate the properties of cosmic rays propagation in our Galaxy. To that end, 1-year Fermi LAT spectral data [16] at intermediate and high latitudes (10◦ 60◦) and all | | | | | | longitudes (0◦ < l < 360◦) up to energy of 100 GeV is used. Cosmic rays which are misidentified as γ-ray events have been subtracted in these spectral data [16]. The remaining cosmic rays contamination is negligible up to Eγ = 100

GeV. However, for Eγ > 100 GeV statistical errors and contamination of cosmic ray electrons and nuclei result in great uncertainty on the spectrum of the γ-rays [16] (see also [116]).

39 40 CHAPTER 4. DIFFUSE GAMMA RAYS

The contribution from the Galactic point sources is modeled by [16]. The model includes the sources which are detected with at least 14σ. Yet, very dim γ-ray sources are not accounted for. Such a class of sources can be millisecond pulsars in the Galactic ridge and halo. Millisecond pulsars may contribute to diffuse γ-ray spectra at low latitudes and at energies of about a few GeV. Since the time scale of their energy loss is large ( 10 ∼ Gyr [80, 117, 118]), the estimate of their population and so their contribution is strongly affected by uncertainties in the evolution of the Galactic halo [119]. It has been suggested that millisecond pulsars can contribute to the isotropic diffuse γ-ray flux [120, 121]. Since millisecond pulsars occur in regions of high stellar densities such as the Galactic ridge and possibly the Galactic halo at its earlier stages [119], it is unlikely that the main part of their diffuse contribution to be isotropic. The spectrum of each individual millisecond pulsar may significantly vary. However, based on the measured spectra of 8 millisecond pulsars [122] the spectrum of their distribution can be described by dN γ E−Γe−E/Ec , (4.1) dE ∼

33.9±0.6 where Γ = 1.5 0.4, Ec = 2.8 1.9 GeV and the luminosity in γ-rays is L = 10 erg/s. ± ± Diffuse γ-rays spectra which are reproduced by reference propagation model (see section 3.4 and Fig. 3.2) are shown in Fig. 4.1. The contributions from two kinds of primary sources namely supernova remnants and pulsars are separately displayed. Pulsars mainly contribute to the inverse Compton component. The spectra agree well with the data at the sky regions of interest. The best fit to γ-ray spectra is achieved at higher latitudes ( b > 20◦) that are less affected by uncertainties | | in the distribution of sources. At intermediate latitudes the prediction is slightly below the observed flux while it still gives a fairly good fit. In section 4.1 the properties of cosmic rays diffusion in the interstellar medium are investigated. Diffuse γ-rays which are evaluated for each diffusion model are compared to those for the reference diffusion model. In section 4.2 the origin of the spectral hardening of cosmic ray protons and helium nuclei at high energies is discussed. The impact of different origins of the observed hardening on diffuse γ-ray spectra is inspected. In section 4.3 different assumptions regarding the distribution of cosmic ray primary sources are examined to explore their effect on γ-rays spectra. In section 4.4 the influence of pp- collision parametrization on the π0 decay flux is reviewed. In section 4.5 a variety of models for distribution of gas in the interstellar medium and their effect on the π0 decay and bremsstrahlung spectra are studied. Finally, in section 4.6 a propagation model which accounts for astrophysical backgrounds is introduced. 41

Figure 4.1: Diﬀuse γ-ray spectra at diﬀerent sky regions for the reference propagation model.

The diﬀusion parameters are δ = 0.5, zd = 4 kpc, Rd = 20 kpc and vc = 0. The models for distribution of gas and primary sources are provided by [42, 95, 60, 68, 82]. The goodness of prediction is determined by the value of χ2. The contributions from two kinds of primary sources are separated. dotted lines: supernova remnants, dashed lines: pulsars. 42 CHAPTER 4. DIFFUSE GAMMA RAYS

Figure 4.2: Diffuse γ-ray spectra at different sky regions for models with different diffusion spectral indices δ. dotted lines: δ = 0.5, dashed lines: δ = 0.4, dashed-dotted lines: δ = 0.33.

For all cases zd = 4 kpc and Rd = 20 kpc. Diﬀusion and spectral parameters are shown in Table 4.1.

4.1 Impact of Cosmic Rays Diﬀusion

4.1.1 Diﬀusion Spectral index

In Fig. 4.2 γ-ray spectra predicted by models with different diffusion spectral indices δ are compared in the three sky regions under study. Changing δ affects the proton spectra whose propagation timescale essentially depends on the diffusion timescale. Lower value of δ makes the propagated spectrum of protons harder for the same injection properties. It then results in the need for a softer proton injection index at high energies to reproduce the data. As shown in Table 4.1 the spectral injection indices of protons at high energies are larger for propagation model with δ = 0.33 (“KOL4-20”) than that for propagation model with δ = 0.5 (“KRA4-20”). This in turn produces differences in the π0 fluxes at the highest energies. 4.1. IMPACT OF COSMIC RAYS DIFFUSION 43

Unlike protons, electron propagation at energies above 5 GeV is significantly affected by the energy loss time scale. Since the interstellar radiation field and magnetic field are kept fixed, the inverse Compton and the higher part of the bremsstrahlung spectrum are not largely affected. In the very high energy part of the inverse Compton spectrum there is a hardening for the models with greater δ. That hardening is due to the fact that for greater δ the higher energy e± diffuse faster out of the Galactic disk compared to lower energy e± reaching the higher latitudes where we observe them through their inverse Compton scattering. In the lower energy part of the spectrum, bremsstrahlung component significantly varies among the models since very different Alfvénvelocities and η values are used in those models in order to fit the cosmic ray data. The overall fit of γ-ray spectra is not affected much due to opposite effects of changing the value of δ on the bremsstrahlung and the π0 spectra below a few GeV. The relative ratio of bremsstrahlung to π0 flux among the models with different δ changes by up to a factor of 0 two at Eγ 0.5 GeV. Since both π and bremsstrahlung components are morphologically ' correlated to the gas distribution, discriminating among them is very difficult. On the other hand, the predictedp ¯ spectrum favors larger values of δ.

4.1.2 Diﬀusion Radial Scale

In Fig. 4.3 the effect of varying the radial scale of the diffusion coefficient Rd is shown.

Decreasing the value of Rd results in lower values for the diffusion coefficient towards the Galactic center relative to the Solar position. It then forces the e± and p produced by sources closer to the Galactic center to spend a greater time close to the disk than those produced by sources close to the Sun. After refitting the diffusion coefficient normalization

D0 (see Table 4.2) to the cosmic ray nuclear data, the net change in the ﬂuxes is negligible. It is also found that the diﬀerence between the radial independent case and the case with

Rd = 20 kpc is negligible. Yet the quality of the ﬁt of the predictedp ¯ to the PAMELA p¯ is aﬀected by changing the radial scale and disfavors the smaller values of Rd as is shown in Table 4.2.

4.1.3 Diﬀusion Scale Height

In Fig. 4.4 the effect of varying the diffusion vertical scale zd, that is correlated to the height of the diffusion zone is shown[92]. Since smaller values of zd yield a greater diffusion coefficient above the Galactic plane, an overall rescaling of the diffusion normalization D0 is necessary to fix the secondary to primary ratio (see Table 4.3). Concerning protons, for which energy losses are less significant, changing zd does not affect much their spectrum 44 CHAPTER 4. DIFFUSE GAMMA RAYS

Name KOL4-20 RUN4-20 KRA4-20 diﬀusion spectral index δ 0.33 0.4 0.5

diﬀusion scale zd/Rd (kpc) 4/20 4/20 4/20 28 2 diﬀusion normalization D0 (10 cm /s) 3.85 3.21 2.49

Alfvénvelocity vA (km/s) 24.8 23.1 19.5 - η 0.765 0.32 -0.363 p p p proton injection index γ1 /γ2 /γ3 2.03/2.49/2.35 2.06/2.44/2.28 2.06/2.35/2.18 proton spectral break p / p (GV) 10.7/300 14/300 14.9/300 R0,1 R0,2 e e electron injection index γ1/γ2 1.7/2.64 1.7/2.64 1.6/2.62 49 pulsar energy injection ηW0 (10 ergs) 0.7 0.76 0.77 2 goodness of fit χB/C 0.5 0.34 0.34 2 goodness of fit χp 0.3 0.41 0.6 2 goodness of fit χ(e−+e+) 0.45 0.29 0.48 2 goodness of prediction χp¯ 2.86 1.36 0.73 2 goodness of prediction χγ 1.33/0.54/0.40 1.11/0.45/0.51 1.02/0.42/0.60 2 goodness of prediction χp¯&γ 1.70 0.99 0.71

Table 4.1: Diffusion and spectral parameters for propagation models with different diffusion spectral indices. Convection is neglected. The electrons spectral break is at e = 5GV. The R0 injection index and the exponential cutoff for the distribution of pulsars are, respectively, 1.4 and 1.2 TeV. The spectra of γ-rays are shown in Fig. 4.2 4.1. IMPACT OF COSMIC RAYS DIFFUSION 45

Figure 4.3: Diffuse γ-ray spectra at different sky regions for models with different diffusion radial scales rd. dotted lines: Rd = 5 kpc, dashed lines: Rd = 10 kpc, dashed-dotted lines:

Rd = 20 kpc. For all cases δ = 0.5 and zd = 4 kpc. Diﬀusion and spectral parameters are shown in Table 4.2 46 CHAPTER 4. DIFFUSE GAMMA RAYS

Name KRA4-5 KRA4-10 KRA4-20 diﬀusion spectral index δ 0.5 0.5 0.5

diﬀusion scale zd/Rd (kpc) 4/5 4/10 4/20 28 2 diﬀusion normalization D0 (10 cm /s) 2.76 2.58 2.49

Alfvénvelocity vA (km/s) 16.9 19.1 19.5 - η 0.0 -0.247 -0.363 p p p proton injection index γ1 /γ2 /γ3 2.07/2.35/2.18 2.05/2.35/2.18 2.06/2.35/2.18 proton spectral break p / p (GV) 27/300 17.5/300 14.9/300 R0,1 R0,2 e e electron injection index γ1/γ2 1.6/2.62 1.6/2.62 1.6/2.62 49 pulsar energy injection ηW0 (10 ergs) 0.71 0.78 0.77 2 goodness of fit χB/C 0.64 0.36 0.34 2 goodness of fit χp 0.54 0.52 0.6 2 goodness of fit χ(e−+e+) 0.4 0.41 0.48 2 goodness of prediction χp¯ 1.45 0.93 0.73 2 goodness of prediction χγ 1.06/0.46/0.45 1.02/0.42/0.55 1.02/0.42/0.60 2 goodness of prediction χp¯&γ 1.01 0.78 0.71

Table 4.2: Diffusion and spectral parameters for propagation models with different diffusion radial scales . Convection is neglected. The electrons spectral break is at e = 5GV. The R0 injection index and the exponential cutoff for the distribution of pulsars are, respectively, 1.4 and 1.2 TeV. The spectra of γ-rays are shown in Fig. 4.3 4.1. IMPACT OF COSMIC RAYS DIFFUSION 47

Figure 4.4: Diffuse γ-ray spectra at different sky regions for models with different diffusion scale heights. dotted lines: zd = 1 kpc, dashed lines: zd = 4 kpc, dashed-dotted lines: zd = 10 kpc.

For all cases δ = 0.5 and Rd = 20 kpc. Diﬀusion and spectral parameters are shown in Table 4.3 48 CHAPTER 4. DIFFUSE GAMMA RAYS

Name KRA1-20 KRA4-20 KRA10-20 diﬀusion spectral index δ 0.5 0.5 0.5

diﬀusion scale zd/Rd (kpc) 1/20 4/20 10/20 28 2 diﬀusion normalization D0 (10 cm /s) 0.55 2.49 4.29

Alfvénvelocity vA (km/s) 16.3 19.5 19.1 - η -0.521 -0.363 -0.373 p p p proton injection index γ1 /γ2 /γ3 2.07/2.34/2.18 2.06/2.35/2.18 2.05/2.35/2.18 proton spectral break p / p (GV) 16.5/300 14.9/300 15.2/300 R0,1 R0,2 e e electron injection index γ1/γ2 1.5/2.58 1.6/2.62 1.6/2.62 49 pulsar energy injection ηW0 (10 ergs) 0.27 0.77 1.01 2 goodness of fit χB/C 0.4 0.34 0.32 2 goodness of fit χp 0.51 0.6 0.48 2 goodness of fit χ(e−+e+) 0.57 0.48 0.33 2 goodness of prediction χp¯ 0.76 0.73 0.70 2 goodness of prediction χγ 3.21/1.67/0.29 1.02/0.42/0.60 0.91/0.32/0.6 2 goodness of prediction χp¯&γ 1.29 0.71 0.65

Table 4.3: Diffusion and spectral parameters for propagation models with different diffusion scale heights. Convection is neglected. The electrons spectral break is at e = 5GV. The R0 injection index and the exponential cutoff for the distribution of pulsars are, respectively, 1.4 and 1.2 TeV. The spectra of γ-rays are shown in Fig. 4.4

and keeps the π0 ﬂux unchanged. Bremsstrahlung emission is also weakly aﬀected by the

changes in zd, because it is morphologically correlated to the gas distribution which is concentrated close to the Galactic disk. Finally, the inverse Compton spectrum is mainly affected by the distribution of electrons being confined within thinner (thicker) diffusion

zones resulting in lower (higher) total inverse Compton flux. For the case of zd = 1 this results in a poor fit to γ-ray spectra. Thus the models with thin diffusion zone such as those that have been suggested by [123], in order to give lowp ¯ flux from Kaluza-Klein dark matter annihilation models, are in tension with the combination of cosmic ray and γ-ray spectra.

4.1.4 Convection

In Fig. 4.5 the eﬀect of convective winds in the Galaxy is shown. The wind is assumed to be perpendicular to the Galactic plane and point outward. The convection velocity is 4.1. IMPACT OF COSMIC RAYS DIFFUSION 49

Figure 4.5: Diffuse γ-ray spectra at different sky regions for models with/without convection. dotted lines: vc = 0 km/s/kpc, dashed lines: vc = 50 z km/s/kpc. Diffusion parameters are | | δ = 0.5, zd = 4 kpc and Rd = 20 kpc. Diffusion and spectral parameters are shown in Table 4.4 50 CHAPTER 4. DIFFUSE GAMMA RAYS then dvc dvc vc(z) = z , = 50 km/s/kpc. (4.2) dz · | | dz Convection introduces a new time scale in the propagation of cosmic rays. It has a strong effect on the propagation of protons. However, its effect on electrons propagation is marginal because of significant energy loss of electrons which dominates the time scale of their propagation. The diffusion properties for propagation model with convection change to a great extent (see Tab. 4.4). This results in a significantly altered inverse Compton and bremsstrahlung components. The effect is most evident at the low energy part of the spectra, where convection is more important than the inverse Compton scattering and synchrotron losses. It is found that high convection models are not favored by γ-ray data in the intermediate latitudes. The model is also in tension with PAMELA positron fraction data at low energies.

4.2 Rigidity Break in Injection or Diﬀusion

The spectra of cosmic ray protons and helium nuclei measured by PAMELA indicate a hardening at high energies [69]. The harder spectral power-law at high rigidities is confirmed by the CREAM data as well [70]. One possible explanation for the observed rigidity break is that it originates at the cosmic ray acceleration sites namely at supernova remnants shocks. Such a scenario has been suggested by study of supernova remnants [124] and semi-analytical calculation of diffusive shock acceleration [125, 126, 127, 128, 129]. The pressure on accelerated particles around the shock leads to the formation of a precursor [124] where the upstream fluid is slowed down and compressed [124]. The energy gain for diffusively accelerated particles which move with respect to the shock, and thus between regions with different pressures, depends on the ”compression ratio”. The high energy particles which have large diffusion lengths probe the entire (or a great part of the) precursor. Thus the highest energy particles ”feel the total compression ratio” [124]. This results in a spectrum harder/softer than E−2 from the first order Fermi acceleration at high/low energies [124, 129, 130, 126]. Another possible explanation is that at energies about the break a population of supernova remnants that accelerate cosmic rays with a harder injection index emerge. If this second population of supernova remnants is common enough in the Galaxy and if its distribution is the same as that with softer cosmic rays injection, both possibilities can be modeled in the same way with the DRAGON code; i.e. with the injection of cosmic rays given by Eq. 3.1-3.2. This scenario is referred to as “scenario A”, under which γ-ray 4.2. RIGIDITY BREAK IN INJECTION OR DIFFUSION 51

Name CON4-20 KRA4-20 diﬀusion spectral index δ 0.6 0.5

diﬀusion scale zd/Rd (kpc) 4/20 4/20

convection gradient dvC /dz (km/s/kpc) 50 0 28 2 diﬀusion normalization D0 (10 cm /s) 0.645 2.49

Alfvénvelocity vA (km/s) 27.2 19.5 - η 0.755 -0.363 p p p proton injection index γ1 /γ2 /γ3 1.85/2.48/2.19 2.06/2.35/2.18 proton spectral break p / p (GV) 12.3/300 14.9/300 R0,1 R0,2 e e electron injection index γ1/γ2 1.6/2.62 1.6/2.62 49 pulsar energy injection ηW0 (10 ergs) 0.44 0.77 2 goodness of fit χB/C 0.61 0.34 2 goodness of fit χp 0.44 0.6 2 goodness of fit χ(e−+e+) 0.82 0.48 2 goodness of prediction χp¯ 1.05 0.73 2 goodness of prediction χγ 2.18/0.98/0.25 1.02/0.42/0.60 2 goodness of prediction χp¯&γ 1.09 0.71

Table 4.4: Diﬀusion and spectral parameters for propagation models with/without convection. The electrons spectral break is at e = 5GV. The injection index and the exponential cutoﬀ for R0 the distribution of pulsars are, respectively, 1.4 and 1.2 TeV. The spectra of γ-rays are shown in Fig. 4.5 52 CHAPTER 4. DIFFUSE GAMMA RAYS

Figure 4.6: Absolute difference between diffuse γ-rays spectra for scenarios A and B normalized to the flux of scenario A (see text). Left panel: dotted lines: π0 decay and dashed-dotted lines: inverse Compton. Red: 10◦ 60◦. | | | | | | The normalized differences for the π0 diffuse component have almost identical spectra in sky regions under study. Right panel: absolute difference between diffuse γ-rays spectra (including bremsstrahlung) for scenarios A and B. Colors are the same as the left panel. The difference 0 between diffuse γ-rays spectra is at most 2% for Eγ < 300 GeV due to cancellation between π and inverse Compton differences.

spectra of Fig. 4.1-4.4 were produced.

A third possibility is that the hardening originates from a change in the turbulence power spectrum of the interstellar medium. The properties of the interstellar magnetic turbulence can be indirectly inferred from cosmic rays fluxes. Accurate data from PAMELA make it possible to consider the break in the spectral index of diffusion coefficient. Coinci- dentally, the needed change in the diffusion index ∆δ 0.17 is the same as the transition ' from a Kraichnan type turbulence at low to a Kolmogorov type at high . This scenario R R is referred to as “scenario B”. The spectra of diffuse γ-rays are evaluated for scenarios A and B by applying the same propagation parameters as the reference model. It turns out that the fitted diffusion and spectral parameters for both scenarios are exactly the same. Thus, local fluxes of cosmic rays can not discriminate between these two scenarios. The difference between diffuse γ-rays spectra for scenarios A and B is shown in Fig. 4.6 . The maximum difference between inverse Compton and π0 decay spectra for these scenarios is up to O(0.1) at energies of about 100 GeV as can be seen in Fig. 4.6 (left). Because of cancellation between π0 and inverse Compton differences, the difference between total diffuse galactic γ-rays is less than O(10−2) (see Fig. 4.6, right) over the whole considered spectrum and in all three regions of interest. Such differences are too small to be probed 4.3. INFLUENCE OF SUPERNOVA REMNANTS DISTRIBUTION 53

Figure 4.7: Radial profile of supernova remnants. Red: parametrization of [68] given in Eq. 4.3, green: discribed by Eq. 4.4[132] and blue: given by Eq. 4.5. by γ-rays especially because of large uncertainties at energies of about 100 GeV. The best way to discriminate between scenarios A and B is throughp ¯ flux [131]. The scenario A would give a soft break only at 10 GV because of the break in the spectra R ∼ of protons and helium nuclei at 230 GV while scenario B would give also a harder ∼ break at 230 GV (with the same change in the spectral index as in the p and He fluxes) ∼ from the diffusion of protons which produce secondaryp ¯ in the interstellar medium (see discussion in [131]).

4.3 Inﬂuence of Supernova Remnants Distribution

Since in this study γ-rays data above b > 10◦ are used, the results are not very sensitive | | to the distribution of sources. In particular, diffuse γ-rays at higher latitudes are poorly sensitive to the inner few kpc, that are also hardly probed by direct observations of single sources. To study the significance of the source distribution in the inner part three different source profiles are tested as shown in Fig. 4.7. The Ferriere et al. [68] profile (“Source A”) is recovered from Type I and Type II supernovae distribution models and is defined by

−(R−R )/4.5−|z|/0.325 fs(R, z) = 0.138e −(z/0.212)2 −(z/0.636)2 −(R2−R2 )/6.82 + 0.79e + 0.21e 0.943e R > 3.7 kpc × 2 2 2 2 + 0.79e−(z/0.212) + 0.21e−(z/0.636) 3.349e−(R−3.7) /2.1 R < 3.7 kpc(4.3). × 54 CHAPTER 4. DIFFUSE GAMMA RAYS

The relative normalization of two populations is defined based on the averaged occurrence frequencies of supernovae Type I and Type II in other galaxies [133, 134]. The spatial profiles are determined based on the assumption that Type I have the same distribution as that of old disk stars [135] and Type II (which are the most frequent) are tightly correlated to the arms. Especially in the inner 3.7 kpc the profile of the Type II is correlated to the distribution of pulsars in our Galaxy and thus it is sensitive to selection effects. The distribution given in Eq. 4.4 has been used extensively in GALPROP [132, 136] as a conventional distribution for supernova remnants.

2.35 −5.56(R−R )/R |z|/0.2 fs(R, z) = (R/R ) e e , (4.4)

The parameterization of this distribution (“Source B”) comes from observations of Galac- tic supernovae [137]. The detailed values are selected in such a way to agree with diﬀuse ﬂuxes of analyzed EGRET γ-ray data [138, 132]. Therefore, this parametrization has to be taken with some care, in particular with respect to its prediction on the inner parts of the Galaxy. Since both parametrization of [68] and [132] are least predictive towards the inner regions of the Galaxy, a third parametrization is also studied which is denoted by “Source C”:

4 −(R/R ) |z|/0.2 fp(R, z) = 0.078 + 2.57e e . (4.5)

As it is shown in Fig. 4.7 this parametrization gives an almost constant radial distribution in the inner 3 kpc. Its averaged distribution at R > 5 kpc is similar to that of “Source A” and “Source B” and its vertical behavior is the same as that of “Source B”. Since it is expected that Type II supernovae are correlated to the spiral arms, and since the inner few kpc are populated by old stars, “Source C” is studied as a probe of the maximal effect that uncertainties in the distribution of supernova remnants could have on diffuse γ-rays, rather than as an optimal model for supernova remnants. Yet as it can be seen from Table 4.5 and Fig. 4.8 “Source C” provides a better fit to the γ-ray data at low latitudes that probe the inner parts of the Galaxy. That could be an indication that “Source A” and “Source B”, which are connected to observations, either under-predict the density of “recent” supernova remnants towards the inner part of the Galaxy or unresolved point sources are present. Also changing the gas assumption only in the inner few kpc could have an effect. To probe such uncertainties a study including lower latitude regions toward the Galactic center would be very well suited. 4.3. INFLUENCE OF SUPERNOVA REMNANTS DISTRIBUTION 55

Figure 4.8: Diffuse γ-ray spectra at different sky regions for models with different distribution of supernova remnants. Dotted lines: Source A of Eq. 4.3[68], dashed lines: Source B of Eq. 4.4

[132] and dashed-dotted lines: Source C of Eq. 4.5. Diﬀusion parameters are δ = 0.5, zd = 4 kpc and Rd = 20 kpc. Diﬀusion and spectral parameters are shown in Table 4.5. 56 CHAPTER 4. DIFFUSE GAMMA RAYS

Name Source A Source B Source C diﬀusion spectral index δ 0.5 0.5 0.5

diﬀusion scale zd/Rd (kpc) 4/20 4/20 4/20 28 2 diﬀusion normalization D0 (10 cm /s) 2.49 2.49 2.33

Alfv´envelocity vA (km/s) 19.5 19.6 19.2 - η -0.363 -0.355 -0.44 p p p proton injection index γ1 /γ2 /γ3 2.06/2.35/2.18 2.04/2.34/2.18 2.05/2.34/2.18 proton spectral break p / p (GV) 14.9/300 15.9/300 16.6/300 R0,1 R0,2 e e electron injection index γ1/γ2 1.6/2.62 1.6/2.62 1.6/2.62 49 pulsar energy injection ηW0 (10 ergs) 0.77 0.77 0.84 2 Fitted χB/C 0.34 0.3 0.33 2 Fitted χp 0.6 0.36 0.58 2 Fitted χ(e−+e+) 0.48 0.39 0.53 2 Predicted χp¯ 0.73 0.86 0.73 2 Predicted χγ 1.02/0.42/0.60 1.26/0.5/0.43 0.73/0.34/1.44 2 Predicted χp¯&γ 0.71 0.79 0.79

Table 4.5: Diffusion and spectral parameters for propagation models with different distribution of supernova remnants. The electrons spectral break is at e = 5GV. The injection index and R0 the exponential cutoff for the distribution of pulsars are, respectively, 1.4 and 1.2 TeV. The spectra of γ-rays are shown in Fig. 4.8 4.4. γ-RAY YIELDS FROM PP-COLLISION PARAMETRIZATION 57

Figure 4.9: The spectrum of π0 decay component of diffuse γ-rays obtained from the Kamae et al. [139] parametrization for pp collisions versus that from [140] and Pythia runs. Red solid: Kamae et al. parametrization. Blue dotted: parametrization of [140] and blue dashed: its difference from the [139]. Green dashed-dotted: parametrization of Pythia, and green dashed- dotted-dotted its difference from the [139]. The π0 fluxes are normalized at 10 GeV.

4.4 γ-ray Yields from pp-Collision Parametrization

In Fig. 4.9 the π0 decay component of diffuse γ-rays spectrum at 10◦ < b < 20◦ is shown | | using three different parametrization for γ-rays spectra produced by pp collisions. It must be clarified that these spectra are called “π0 spectra” since their contribution to γ-rays spectra is the dominant one for all parameterization. However, γ-rays produced by decay of other mesons such as K±, K0, η, D±, D0 are also taken into account. The parametrization of Kamae et al. [139, 141] is used as a reference. It is derived for the cross-sections of diffractive, non-diffractive and excitation of resonance processes, based on simulation and experimental data on pp collisions. This parametrization is compared to that from Kelner et al. [140] which is based on running SIBYLL [142] simulations of pp collisions. It is also compared against the γ-ray spectrum from Pythia (version 6.4) simulations [143]. For the parametrization of Kamae et al. the updated tables in [141] relevant to those of tables 2 and 3 of [139] are used. For the parametrization of Kelner et al. the information given in eq. 58-61 of [140] is used. For Pythia simulations, pp collisions are calculated with center of mass energy from 2.33 GeV up to 7 TeV and with subsequent decay of all mesons and including final state radiation. The information for 3D momenta of the final stable particles are kept to re-boost to the proper observer frame (where a comic ray proton hits a practically stable interstellar medium proton). The spectra of π0 decay normalized at 10 GeV agree well from 100 GeV down to energies 58 CHAPTER 4. DIFFUSE GAMMA RAYS of 1 GeV where it is expected that the simulations from [140] and Pythia would be no longer reliable. Thus, it is safe to say that uncertainties in γ-ray spectra produced by pp collisions, that could be due to missing processes in the parametrization of [139], are too small to have a strong impact on the constraints imposed on the interstellar medium properties in this analysis.

4.5 Signiﬁcance of the Interstellar Gas Distribution

The π0 decay and the bremsstrahlung components of diﬀuse γ-rays are produced by interactions of cosmic ray protons and e± with the interstellar gas. In this analysis two dimensional azimuthally averaged gas distribution is used. The models which are provided by Nakanishi & Sofue for distribution of atomic [42] and molecular [95] hydrogen gases are used as reference models. This choice is consistent because the same method has been applied to construct both gas distributions. The reference model for distribution of ionized hydrogen gas is given by [60].

The contributions of HI , H2 and HII gases to diﬀuse γ-rays spectra at three sky regions of interest are separately shown in Fig. 4.10. As it is clear HII contribution to each of diﬀuse γ-ray components is at most about 10% at all energies and sky regions.

However, the contributions of HI and H2 gases are determining. Using different models 0 for HI and H2 gas distributions significantly affects the spectra of the π decay and the bremsstrahlung components of diffuse γ-rays. On the contrary, the uncertainties in assumptions on HII (such as [144]) result in at most a few percent change in the γ-ray

ﬂux which is well below the uncertainties in HI and H2 gases. Concerning the atomic hydrogen gas the other model which is examined is given by Gordon & Burton [145] for the radial distribution and by Dickey & Lockman [146] for the vertical distribution. The HI density is shown versus R at z = 0 in Fig. 4.11 (upper left panel) and versus z at R = R in Fig. 4.11 (upper right panel).

The distribution of molecular hydrogen gas is signiﬁcantly uncertain. The H2 mid plane density in the Nakanishi & Sofue model has large error bars. To address the eﬀect of these uncertainties high and low values of the mid plane density, in addition to mean values, are tested. Another model which is considered in this analysis is given by Bronfman et al. [147]. The conversion factor XCO is chosen to exponentially increase with R as suggested by [95]. It is

20 −2 −1 −1 XCO(R) = 1.4 exp(R/RCO) 10 H2cm K (km/s) (4.6) × where RCO = 11 kpc. The radial and vertical distributions of H2 are shown in Fig. 4.11 4.5. SIGNIFICANCE OF THE INTERSTELLAR GAS DISTRIBUTION 59

Figure 4.10: The spectra of the π0 decay (red) and the bremsstrahlung (green) components of diffuse γ-rays from three major interstellar gas ingredients. Contributions to the mentioned diffuse γ-ray components are from HI (dashed lines), H2 (dotted lines), HII (dashed-dotted lines) and total (solid lines). The spectra are the averaged fluxes over all longitudes and latitudes of 10◦ 60◦ (lower left). The ratio | | | | | | of the spectra of diffuse γ-ray components at b > 60◦ to that at 10◦ < b < 20◦ due to different | | | | gas ingredients are shown as well (lower right). 60 CHAPTER 4. DIFFUSE GAMMA RAYS

(the second row). There is a tight correlation between the π0 decay component of diffuse γ-rays and gas distribution. The reason is the following. Cosmic ray protons diffuse in the interstellar medium almost without energy loss since the timescale of their energy loss is large compared with the time they spend in the diffusion zone. Therefore, the change of gas distribution in the Galaxy does not affect much the spectral shape and profile of cosmic ray protons. Indeed, after refitting the diffusion and spectral parameters for different gas distribution models the profiles of cosmic ray protons remain almost unchanged. In Fig. 4.12 the radial and vertical profiles of the steady state differential flux of cosmic ray protons at E = 10 GeV are shown for different models for the gas distribution. The difference among the profiles becomes even smaller at higher energies. This implies that the π0 decay component of diffuse γ-rays essentially probes the gas distribution along the line of sight. The evaluated spectra of diffuse γ-rays for different gas distributions are shown in

Fig. 4.13. In “Bronf” the models given by Bronfman et al. [147] for H2 distribution and given by Gordon & Burton [145] and Dickey & Lockman [146] for HI distribution are used. In “NS high”, “NS mean” and “NS low” the reference gas distributions provided by Nakanishi & Sofue [42, 95] with, respectively, high, mean and low values of the H2 mid plane density are used. The “NS low” model yields small amount of γ-rays at lower latitudes (10◦ 60◦). The “Bronf” model is also in tension with diffuse γ-ray data | | especially at higher latitudes ( b > 60◦). The π0 decay and the bremsstrahlung fluxes for | | the “Bronf” model are about 50% greater than those for the “NS mean” model in the entire range of spectra and at latitudes above 10◦. This is mainly due to the higher density of the local H2 as shown in Fig. 4.11 (lower row). The importance of H2 distribution can also be understood from Fig. 4.13 (lower right panel). The relative differences of the π0 decay fluxes which are produced by H2 in the “Bronf” and the “NS mean” models are of ◦ O(1) at latitudes above 10 . Whereas, the change of HI distribution has much smaller effect on the evaluated π0 decay spectra as can be seen in Fig. 4.13 (lower right panel). This is because the steeper decrease with distance from the Galactic plane in the Dickey & Lockman model [146] (see Fig 4.11 upper right panel) in contrast to the Nakanishi & Sofue model [42] is compensated by higher density on the Galactic plane in the Gordon & Burton model [145] (see Fig. 4.11 upper left panel). As mentioned earlier, the higher gas density yields greater flux of the π0 decay and bremsstrahlung. In order not to overproduce secondaries cosmic ray nuclei need to escape faster from the Galaxy. Thus the diffusion coefficient normalization must be increased 4.5. SIGNIFICANCE OF THE INTERSTELLAR GAS DISTRIBUTION 61

Figure 4.11: Large scale distributions of the atomic (top) and the molecular (bottom) hydrogen gases in the Galaxy versus R at z = 0 (left) and versus z at R = R (right). For HI , “NS” refers to the Nakanishi & Sofue [42], “GB” refers to the Gordon & Burton [145] and “DL” to the Dickey & Lockman [146] models. For H2, “Bronf” refers to the Bronfman et al. [147], “NS high”, “NS mean” and “NS low” refer to the Nakanishi & Sofue [95] model using, respectively, high, mean and low values of the mid plane density. 62 CHAPTER 4. DIFFUSE GAMMA RAYS

Figure 4.12: The profiles of the steady state differential flux of cosmic ray protons at E = 10 GeV versus R for given values of z (left) and versus z for given values of R (right). In “Bronf” for

H2 the Bronfman et al. [147] and for HI the Gordon & Burton [145] and Dickey & Lockman

[146] gas distributions are used. In “NS high”, “NS mean” and “NS low” for H2 the Nakanishi & Sofue [95] distribution with, respectively, high, mean and low values of the mid plane density is used. They share the same HI distribution modeled by Nakanishi & Sofue [42]. 4.5. SIGNIFICANCE OF THE INTERSTELLAR GAS DISTRIBUTION 63

Figure 4.13: Diffuse γ-ray spectra at different sky regions for models with different gas distributions. Dotted lines are due to propagation model with H2 from Bronfman et al. [147] and

HI from Gordon & Burton [145] and Dickey & Lockman [146]. Dashed lines, solid lines and dashed-dotted lines are due to models with the same HI from Nakanishi & Sofue [42] and H2 from Nakanishi & Sofue [95] with, respectively, high, mean and low values of the mid plane density. Diffusion parameters are δ = 0.5, zd = 4 kpc and Rd = 20 kpc. Diffusion and spectral parameters are shown in Table 4.6. The relative differences of the π0 decay fluxes which are produced by different gas components/distributions are shown in the lower right panel. Red: 10◦ 60◦. The dotted lines coincide since the | | | | | | vertical distribution of H2 models are similar. 64 CHAPTER 4. DIFFUSE GAMMA RAYS to keep the flux of B/C ratio the same (see Tables 4.6). It implies that cosmic ray e± propagate to larger distances from the Galactic disk where they produce diffuse γ-rays through inverse Compton scattering off the CMB photons. This leads to a 10% increase in the inverse Compton flux. A possible way to reconcile the high gas density with diffuse γ-rays data is to decrease the thickness of the diffusion halo. The effect of interplaying between the gas distribution and the thickness of the diffusion halo is shown in Table 4.7. The “KRA1-20 Bronf” represents the model with the thinnest diffusion halo and the highest interstellar gas density. This model can not be considered in tension with any data. The reverse case which is labeled by “KRA10-20 NS low” is the model with a thick halo and a low interstellar gas. This model tends to under-predict γ-ray spectra at low latitudes. Finally, the extreme cases of very thin diffusion halo with low gas density “KRA1-20 NS low” or thick diffusion halo with high gas density “KRA10-20 Bronf” systematically under-predict or over-predict the spectra of diffuse γ-rays. While these models predict well the spectrum of antiprotons they are in serious tension with γ-ray fluxes. As a result, a combined analysis of cosmic rays and γ-rays can probe the uncertainties in the large scale distribution of gas.

4.5.1 2D vs 3D Gas Distribution

In general, the use of two dimensional spatially smoothed gas distribution is not accurate enough to interpret γ-ray sky maps with the angular resolution of the Fermi. Nevertheless, the goal of this analysis is not to trace small scale features of γ-ray sky maps but rather to study the large scale properties of the Galaxy. The fine structures in three dimensional gas distribution are washed out by averaging γ-ray spectra over wide regions of the sky. To investigate the effect, three dimensional distributions of the Nakanishi & Sofue models for HI [42] and H2 [95] are implemented to evaluate diffuse γ-rays spectra. The spectra of γ-rays at sky regions of 10◦ 60◦ for the reference | | | | | | propagation model are shown in Fig. 4.14. Comparing these spectra with those in Fig. 4.1 clarifies the minimal influence of using 3D or 2D gas distributions on diffuse γ-rays spectra in the sky regions under study. The effect is much smaller than the effect of using different gas distributions as can be seen in Fig. 4.13. The impact of using 3D rather than 2D gas distribution on the evaluated γ-ray spectra at three sky regions under study is also illustrated in Fig. 4.15. The relative differences of the π0 decay and the bremsstrahlung spectra are at 10% level. As can be seen in Fig. 4.15 the changes of these components are in the opposite direction. So, the relative difference 4.5. SIGNIFICANCE OF THE INTERSTELLAR GAS DISTRIBUTION 65

Name NS low NS high Bronf diﬀusion spectral index δ 0.5 0.5 0.5

diﬀusion scale zd/Rd (kpc) 4/20 4/20 4/20 28 2 diﬀusion normalization D0 (10 cm /s) 1.94 3.04 3.39

Alfvénvelocity vA (km/s) 14.6 24.4 26.5 - η -0.324 -0.411 -0.526 p p p proton injection index γ1 /γ2 /γ3 2.07/2.35/2.18 2.06/2.35/2.18 2.08/2.35/2.18 proton spectral break p / p (GV) 15.4/300 17/300 17.6/300 R0,1 R0,2 e e electron injection index γ1/γ2 1.6/2.62 1.6/2.62 1.6/2.62 49 pulsar energy injection ηW0 (10 ergs) 0.68 0.74 0.79 2 goodness of fit χB/C 0.32 0.31 0.38 2 goodness of fit χp 0.44 0.57 0.58 2 goodness of fit χ(e−+e+) 0.42 0.55 0.52 2 goodness of prediction χp¯ 0.67 0.85 0.69 2 goodness of prediction χγ 3.69/1.77/0.21 0.52/0.55/1.96 1.45/1.42/3.44 2 goodness of prediction χp¯&γ 1.34 0.94 1.47

Table 4.6: Diﬀusion and spectral parameters for propagation models with diﬀerent distributions

of hydrogen gas. “NS low” and “NS high” are due to models with the same HI from Nakanishi

& Sofue [42] and H2 from Nakanishi & Sofue [95] with, respectively, low and high values of the

mid plane density. “Bronf” is due to propagation model with H2 from Bronfman et al. [147]

and HI from Gordon & Burton [145] and Dickey & Lockman [146]. The electrons spectral break is at e = 5GV. The injection index and the exponential cutoﬀ for the distribution of pulsars R0 are, respectively, 1.4 and 1.2 TeV. The spectra of γ-rays are shown in Fig. 4.13 66 CHAPTER 4. DIFFUSE GAMMA RAYS

Name KRA1-20 NS low KRA10-20 NS low KRA1-20 Bronf KRA10-20 Bronf δ 0.5 0.5 0.5 0.5

zd/Rd (kpc) 1/20 10/20 1/20 10/20 28 2 D0 (10 cm /s) 0.414 3.35 0.831 5.6

vA (km/s) 11.9 14.4 25.4 24.6 η -0.454 -0.331 -0.563 -0.576 p p p γ1 /γ2 /γ3 2.07/2.35/2.18 2.06/2.35/2.18 2.07/2.35/2.18 2.09/2.36/2.18 p p R0,1/R0,2 (GV) 16.6/300 15.2/300 18.3/300 17.6 e e γ1/γ2 1.6/2.62 1.6/2.62 1.6/2.62 1.6/2.62 49 ηW0 (10 ergs) 0.31 0.95 0.33 1.15 2 χB/C 0.44 0.36 0.29 0.35 2 χp 0.56 0.85 0.9 0.75 2 χ(e−+e+) 0.67 0.31 0.47 0.58 2 χp¯ 0.74 0.68 0.9 0.68 2 χγ 6.77/3.89/0.77 2.84/1.28/0.14 0.66/0.47/1.11 1.43/1.29/3.23 2 χp¯&γ 2.43 1.09 0.81 1.4

Table 4.7: Diffusion and spectral parameters for propagation models with different distributions of hydrogen gas. See text for details. The electrons spectral break is at e = 5GV. The injection R0 index and the exponential cutoff for the distribution of pulsars are, respectively, 1.4 and 1.2 TeV. 4.5. SIGNIFICANCE OF THE INTERSTELLAR GAS DISTRIBUTION 67

Figure 4.14: Diffuse γ-ray spectra at different sky regions for the reference propagation model with 3D gas distributions of Nakanishi & Sofue [42, 95]. The diffusion parameters are δ = 0.5, zd = 4 kpc, Rd = 20 kpc and vc = 0. 68 CHAPTER 4. DIFFUSE GAMMA RAYS

Figure 4.15: The relative differences of diffuse γ-ray components using 3D and 2D gas distributions at three sky regions of interest. Red: π0 decay, green: bremsstrahlung and black: diffuse Galactic (π0 decay + bremsstrahlung + inverse Compton). of diffuse Galactic γ-rays spectra is smaller at the level of 5%. The total flux of diffuse γ-rays includes the contributions from point sources and extragalactic background as well. Taking all components into account, the goodness of prediction which is determined by the value of χ2 does not get affected much.

4.6 Reference Model for Diﬀuse γ-Ray Background

The Fermi γ-ray space telescope has an angular resolution of about 1◦ for energies above a few GeV. It allows one to trace in detail the distribution of γ-ray emissivity associated to gas in the Galaxy. Although, the 3D gas models of Nakanishi & Sofue predict well γ-rays data at large sky regions they fail to account for ﬁne features observed by Fermi at small windows of the sky. Recent large scale surveys with high angular resolution, strong sensitivity and large velocity range make it possible to devise gas models with detailed features. Therefore, the model constructed in section 2.1 is used for HI distribution and the model which is constructed in [57] is used for H2 distribution. In [57] a constant con- 20 −2 −1 −1 version factor XCO equal to 2.3 10 H2cm K (km/s) is assumed. However, γ-ray × data favor a conversion factor which increases with R [148]. Testing diﬀerent assumptions on radial variation of XCO the conversion factor which is given by [149] has better agreement with the Fermi γ-ray data. It exponentially increases outward the Galaxy with a 4.6. REFERENCE MODEL FOR DIFFUSE γ-RAY BACKGROUND 69

radial scale of RCO = 6.3 kpc as follows

20 −2 −1 −1 XCO(R) = 0.35 exp(R/RCO) 10 H2cm K (km/s) . (4.7) ×

In the following, the propagation model with diﬀusion parameters of δ = 0.5, zd =

4 kpc, Rd = 20 kpc, vc = 0 and with accurate 3-dimensional gas distributions is taken as the reference model for astrophysical backgrounds. The sky maps of diffuse γ-ray components at E = 10 GeV for the reference model are shown in Fig. 4.16. In sky maps of the π0 decay and bremsstrahlung components small scale features of HI and H2 distributions can be recognized (see Fig. 2.2 and Fig. 2.10 for comparison). The sky map of inverse Compton component is remarkably isotropic. The reason is that it is mainly produced by scattering off the CMB photons which are isotropic. In the following, the updated γ-ray data which are taken by the Fermi-LAT in three years from August 2008 to August 2011 is used. The Pass 7 (v9r23p1) “ULTRACLEAN” event class ensures minimal cosmic ray contamination 1. γ-ray events with energies of 200 MeV up to 500 GeV are binned in 30 logarithmically spaced energy bins. The expo- sures and fluxes of front and back-converted events are separately calculated. Then their contributions are summed up to obtain the total flux within the windows of interest. To account for contributions from known point and extended sources within the windows of interest 2-year catalogue of the Fermi-LAT is used (see [150] and references therein). The contributions of the Fermi bubbles [151] and of the Fermi haze [152, 153] are ignored, given that the uncertainties on the exact morphology of these structures are large especially at lower latitudes [153]. Diffuse γ-ray spectra evaluated for the reference propagation model have a good agreement with the updated Fermi data as can be seen in Fig. 4.17.

1http://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/ 70 CHAPTER 4. DIFFUSE GAMMA RAYS

Figure 4.16: Sky maps of diﬀuse γ-ray components at 10 GeV for the reference propagation model. The diﬀusion parameters are δ = 0.5, zd = 4 kpc, Rd = 20 kpc and vc = 0. The models 0 for HI and H2 distributions are provided, respectively, in section 2.1 and in [57]. Top: π decay, middle: bremsstrahlung and bottom: inverse Compton. 4.6. REFERENCE MODEL FOR DIFFUSE γ-RAY BACKGROUND 71

Figure 4.17: Diﬀuse γ-ray spectra for the reference propagation model. The diﬀusion parameters are δ = 0.5, zd = 4 kpc, Rd = 20 kpc and vc = 0. The models for HI and H2 distributions are provided, respectively, in section 2.1 and in [57]. The sky regions are due to 0 < l < 360 and upper left: b < 5, upper right: 5 < b < 10, middle left: 10 < b < 20, middle right: | | | | | | 20 < b < 60 and lower: 60 < b < 90. The goodness of the prediction is determined by the | | | | value of χ2. 72 CHAPTER 4. DIFFUSE GAMMA RAYS Chapter 5

Dark Matter Indirect Detection

Indirect detection of dark matter with γ-rays has a bonus that, unlike charged particles, γ-rays freely stream through the Galaxy and so they directly point back to their sources. Thus, one can search for the regions in the sky where the ratio of dark matter signal to astrophysical background is maximized. Moreover, the Galaxy is transparent to γ-rays up to energies of hundreds of TeV. At very high energies about PeV the mean free path of γ-rays sharply drops due to e−e+ pair production on the CMB photons. It implies that γ-rays essentially reach us without attenuation or energy loss. γ-rays can be produced by WIMPs annihilation in a number of ways. One is the prompt emission which includes final state radiation, virtual internal bremsstrahlung as well as decay of π0s which are, in turn, produced by hadronization or decay of WIMPs annihilation products. The inverse Compton scattering and bremsstrahlung of leptonic final products can also contribute to γ-rays spectrum. These processes give continuous spectra. Direct annihilation into γ’s provides a monochromatic line emission with a very clean signature. However, since dark matter is electrically neutral this process is loop suppressed. The differential flux of γ-rays which are produced by WIMPs annihilation coming to us from angular window of dΩ is described by f Z dφγ 1 dNγ 2 = 2 Σf σv f dsρχ(s, Ω). (5.1) dΩdE 8πmχ h i dE l.o.s

In the above σv f is the velocity averaged annihilation cross section for ﬁnal state of f, h i f dNγ /dE is the γ-ray spectrum generated per WIMP annihilation through channel f, mχ is the mass of the WIMP, ρχ is the dark matter density and the integral is performed along the related line of sight. In Eq. 5.1 the integral depends only on the astrophysical distribution of dark matter whereas the ﬁrst part is determined by the nature of dark matter namely its mass and

73 74 CHAPTER 5. DARK MATTER INDIRECT DETECTION annihilation cross section as well as the spectrum of γ-rays produced by WIMPs annihilation. These aspects are discussed in section 5.1 in the inner region of the Galaxy. In section 5.2 limits on the WIMPs annihilation cross section in wide regions of the Galaxy are derived.

5.1 Galactic Center

The Galactic center is expected to be the brightest dark matter source in the sky since it has the highest dark matter density. Recently, [18] has suggested detection of a line at 129.8 2.4+7 GeV with 3.3σ significance in a wide window towards the Galactic center. ± 13 In addition, [19] has revealed an excess of γ-rays concentrated around the Galactic center. It can be interpreted as a single line at 127.0 2.0 GeV with 5.0σ significance. A pair of ± lines at energies of 110.8 4.4 and 128.8 2.7 GeV can also explain the excess of γ-rays ± ± with 5.4σ significance. Apart from the morphological differences these results agree on the energy of the line. Both signals are also in agreement with the constraints on line searches from the Fermi Collaboration [154]. The morphology of the line signal from both [18] and especially from [19] is so confined that it favors profiles of cuspy annihilating dark matter halos. Thus one leads to concentrate on WIMPs annihilation rather than decay. Most dark matter models that can give a line(s) signal do so with a production rate which is loop suppressed. Typically, the monochromatic γ-ray yield comes together with a γ-ray yield with continuous spectrum from tree-level annihilation into Standard Model final states which in turn hadronize and/or decay into the stable species, p,p ¯ e±, νs and γ-rays. Assuming that the γ-ray line is of dark matter origin, limits on the continuous component that in general accompanies the line signal are derived. The inner few degrees around the Galactic center and the mass range that matches γ-ray line are, respectively, the preferred direction and mass to be looked at.

5.1.1 Limits on WIMPs Annihilation Cross Section in the inner

10◦ 10◦ × In [19] the angular extension of the γ-rays excess at 110-130 GeV is described by a gaussian distribution with FWHM of 3◦ 4◦. If that excess comes from WIMPs annihilation, the − same region must be searched for the room for a γ-ray yield with continuous spectrum due to WIMPs annihilation into other ﬁnal states. For this reason the region of the inner 10◦ 10◦ box namely ( b < 5◦, l < 5◦) is studied. × | | | | 5.1. GALACTIC CENTER 75

γ-rays in this window originate from a combination of sources. There are 29 detected point sources centered in this window [150], 2 close by extended sources with negligible contribution [155, 156] as well as diffuse γ-rays background which includes π0 decay, bremsstrahlung and inverse Compton fluxes. Furthermore, many unknown dim point sources are expected to be located within that window. Finally, the possible contribution from WIMPs annihilations in the halo is expected to peak towards the Galactic center. These γ-rays can be directly produced by WIMPs annihilations into γγ, Zγ or hγ final states and can possibly match the lines detected by [18, 19]. Virtual internal bremsstrahlung (VIB) and final state radiation (FSR) in WIMPs annihilation can also give a very hard spectrum that can be confused as a line over an otherwise featureless power law spectrum [157, 21]. The decay of mesons (predominantly π0s), produced in the decay or hadronization processes of the products of WIMPs annihilation, can also lead to a significant contribution to the γ-ray spectrum. This component is typically harder than the background γ-ray spectra. Nevertheless, it is significantly softer than the VIB/FSR and can not be confused as a dark matter line in γ-rays. Moreover, inverse Compton and bremsstrahlung γ-rays from the leptonic final products (e±) of WIMPs annihilation will also contribute in that window since both the interstellar radiation field energy density and the interstellar gas density peak in the inner part of the Galaxy. The contribution form dark gas is ignored since its uncertainties are significant in the inner 5◦ in latitude [158, 159]. Dark gas can only contribute to diffuse γ-rays background and so results in less room for WIMPs annihilation originated γ-rays. Five individual annihilation channels are studied; χχ W +W −, χχ b¯b, χχ −→ −→ −→ τ +τ −, χχ µ+µ− and χχ e+e−. Typically, dark matter models have sizable −→ −→ branching ratios into more than one of these channels. The exact limits in such models can be recovered by linearly combining the limits from these channels. Annihilation to Z gauge bosons (top quarks) give very similar γ-ray spectra to those of W +W − (b quarks) and thus the constraints to those channels can be taken to be the same (within 10%) ' as those of the χχ W +W − (χχ b¯b). −→ −→ Following [19] it is assumed that a line at energy of 127 2 GeV has been detected. ± The morphology of the excess is described by a bi-gaussian distribution with FWHM of 4◦ in both l and b. That line can come from χχ γγ or χχ Zγ or χχ hγ. −→ −→ −→ In addition, the case where there are 2 lines centered at 128.8 2.7 and 110.8 4.4 GeV ± ± has been indicated by [19]. In that case the lines come from the combination of either γγ&Zγ lines or Zγ&hγ lines. In this analysis, both the case of a single line centered at 127 GeV and the case of 2 lines centered at 129 and 111 GeV are studied. The choice of mass depends on the exact origin 76 CHAPTER 5. DARK MATTER INDIRECT DETECTION

◦ ◦ Figure 5.1: The spectrum of γ-rays in b < 5 , l < 5 . WIMPs with mχ = 130 GeV annihilate | | | | into a W +W − pair with a cross-section of 1.05 10−25 cm3s−1 and to a γγ line with a cross- × section of 1.25 10−27 cm3s−1. ×

of the line(s). For a single line from χχ γγ the mass range of 122 < mχ < 132 GeV, −→ from χχ Zγ the mass range of 137 < mχ < 145 GeV and from χχ hγ the mass −→ −→ range of 149 < mχ < 157 GeV are studied. For 2 lines which originate from γγ&Zγ the mass range of 127 < mχ < 130 is studied while for lines from Zγ&hγ the mass range of

138 < mχ < 143 GeV is considered. The relevant ratio in the luminosity of two lines is taken to be 0.7/1 for the 111/129 GeV lines. For every choice of main annihilation channel (which gives the continuum), subdominant annihilation channel(s) (which give line(s)) and dark matter mass, both annihilation cross sections are fitted to γ-ray data within 10◦ 10◦ box. Thus there are two degrees × of freedom. In Fig. 5.1, γ-ray spectrum within the window of interest for the case of WIMPs with + − mχ = 130 GeV which annihilate into W W and γγ is shown. The fact that the best fit value for the cross section is positive validates the claim of deriving conservative limits on dark matter annihilation. From the best fit values the 3σ upper limits on the annihilation cross section of the main channels are derived keeping the line annihilation cross section fixed at its best fit value. In Fig. 5.2 these 3σ upper limits for the five annihilation channels of W +W −, b¯b, τ +τ −, µ+µ− and e+e− are shown. The limits are due to both the case of a single line (Fig. 5.2, top left) and of the double line (Fig. 5.2, top right). The exact choice of the origin of the line(s) and its energy(ies) has a subdominant effect on the limits in all 5.1. GALACTIC CENTER 77

50 50

L 40 L 40 1 1 - - s s 3 3 cm cm 26 26 - 30 - 30 10 10 ´ ´ H H Σ Σ 3 chan. 3 chan. \ \ v v Σ Σ

X 20 X 20

10 10 120 130 140 150 160 120 130 140 150 160

mΧHGeVL mΧHGeVL

1.0 1.0 L L 1 1

- 0.8 - 0.8 s s 3 3 cm cm

LH 0.6 LH 0.6 26 26 - -

10 0.4 10 0.4 ´ ´ H H BF line BF line \ \

v 0.2 v 0.2 Σ Σ X X

0.0 0.0 120 130 140 150 160 120 130 140 150 160

mΧHGeVL mΧHGeVL

Figure 5.2: Top: The 3σ upper limits for annihilation into “channel” from region of b < 5◦, | | l < 5◦. Channels are solid lines: χχ W +W −, dotted lines: χχ b¯b, dashed lines: | | −→ −→ χχ τ +τ −, dashed dotted lines: χχ µ+µ− and long dashed lines: χχ e+e−. Left: −→ −→ −→ assuming a single line from blue: χχ γγ or red: χχ Zγ or green: χχ hγ. Right: −→ −→ −→ assuming double lines from blue: χχ γγ and χχ Zγ or from red: χχ Zγ and −→ −→ −→ χχ hγ. Bottom: best fit values for the annihilation into line(s). For the double line case −→ the annihilation best fit value refers to the cross-section for the highest energy line; the 111/129 GeV luminosity ratio is taken to be 0.7/1. The Einasto dark matter profile of eq. 5.2 which gives J/∆Ω = 1.21 1024 GeV2 cm−5 is used (see text for more details). × channels except for the case of e+e−. That happens since the contribution of the main annihilation channel to γ-rays spectrum for all channels except for e+e− is at energies bellow 100 GeV where the lines do not contribute. The case of the e+e− channel is an exception due to its significant final state radiation component which peaks at mχ. Thus the final state radiation component competes with the line(s) in the fit and makes its limits sensitive to the exact assumptions on the line(s). In Fig. 5.2 (bottom panels) the best fit values for annihilation cross sections due to line(s) are shown for the relevant combinations of dark matter mass and channel. The assumptions on distributions of the interstellar radiation field and of the interstellar gas affect, respectively, the inverse Compton and bremsstrahlung components. One can derive even more conservative limits on the WIMPs annihilation cross section by con- sidering only the prompt γ-ray contribution. In Fig. 5.3, the 3σ upper limits are shown where only the prompt γ-rays are taken into account. For the W +W −, b¯b and τ +τ − chan- 78 CHAPTER 5. DARK MATTER INDIRECT DETECTION

200 200

L 150 L 150 1 1 - - s s 3 3 cm cm 26 26 - 100 - 100 10 10 ´ ´ H H Σ Σ 3 chan. 3 chan. \ \ v v Σ Σ

X 50 X 50

0 0 120 130 140 150 160 120 130 140 150 160

mΧHGeVL mΧHGeVL

1.0 1.0 L L 1 1

- 0.8 - 0.8 s s 3 3 cm cm

LH 0.6 LH 0.6 26 26 - -

10 0.4 10 0.4 ´ ´ H H BF line BF line \ \

v 0.2 v 0.2 Σ Σ X X

0.0 0.0 120 130 140 150 160 120 130 140 150 160

mΧHGeVL mΧHGeVL

Figure 5.3: Top: The 3σ upper limits for annihilation into ”channel” using only the prompt γ-rays from the WIMPs annihilation from the region of b < 5◦, l < 5◦. Left: assuming a | | | | single line. Right: assuming double lines. Lines and colors are the same as in Fig. 5.2. Bottom: best fit values for the annihilation into line(s). The Einasto dark matter profile of eq. 5.2 is used. nels, in which the prompt γ-rays are the dominant component, the limits become weaker only by 10 20%. For the µ+µ− and e+e− modes since hard cosmic ray electrons are ' − injected by WIMPs annihilation, the inverse Compton and bremsstrahlung components are significant. Thus ignoring these diffuse components and keeping only the prompt component, the 3σ limits become weaker by a factor of 4-5 in both channels. The limits shown in Figs. 5.2 and 5.3 depend on the dark matter profile assumptions. Here an Einasto dark matter profile is used 2 rα ρχ(r) = ρEin exp α 1 , (5.2) −Rc ∗ Rc − where α = 0.22, Rc = 15.7 kpc and ρEin is set such that the local dark matter density is equal to 0.4 GeVcm−3 [160, 161]. That results in a J-factor of J/∆Ω = 1.21 × 1024GeV2cm−5 in that window. The J-factor is defined by Z Z ∞ 2 J = ρχ(s, Ω)dsdΩ, (5.3) ∆Ω 0 where s is the distance along the line of sight and ∆Ω is the angle of observation. A more cuspy dark matter profile would lead to stronger limits while a more cored (flat) dark matter profile would lead to weaker limits. All the limits shown in Fig. 5.3 and 5.1. GALACTIC CENTER 79 the limits for W +W −, b¯b and τ +τ − in Fig. 5.2 will change inverse proportionally (exactly or approximately) to the value of the J-factors within that window, since the prompt component is dominant in these channels. The same applies for the best fit values to the line(s). Thus these limits can be used for other dark matter profile assumptions once one properly takes into account the different J-factor from that window. For the annihilation channels into µ+µ− and e+e− the limits in Fig. 5.2 have a dependence on the dark matter profile that is more involved. The 3σ limits presented in Figs. 5.2 and 5.3 were derived with the cross section to the line(s) to be the best fit value from the fit to the γ-ray data within l < 5◦, b < 5◦. | | | | Alternatively, the 3σ limits for the same channels are calculated by using the value for the cross section to the line(s) which gives the luminosity stated in [19] for the 4◦ FWHM cusp, that is (1.7 0.4) 1036ph/s or (3.2 0.6) 1035erg/s. The difference in the values ± × ± × of the cross sections to the line(s) between the two methods is 30% (at the same level ' with the stated uncertainty of [19]). In Table 5.1 the 3σ upper limits on the continuous components for these two alternative methods of evaluating the cross section to the line(s) are presented. These limits are due to the case in which the cross section to the line(s) comes from the fit to l < 5◦, b < 5◦ | | | | region, denoted as “free”, and due to the case where the cross section comes form the luminosity stated by [19]. These limits are derived for values of WIMP mass that can give a single line at 127 GeV. The difference in the limits for all five channels and all masses between the two methods is at the 1% level. The same results apply for the case of 2 lines (111 and 129 GeV). ' Thus the exact luminosity assumptions for the line(s) can not influence the results on the continuous component. It is also found that changing the window of observation from ( l , b ) < 5◦, to | | | | ( l , b ) < 3◦, 4◦ or 8◦ the limits for the continuous component and for the best fit value for | | | | the line(s) change by up to 10% and 20% respectively, with the limits from ( l , b ) < 5◦ | | | | being the strongest (see also [162, 163]).

5.1.2 Limits on WIMPs Annihilation Proﬁle

In [19] it has been suggested that the line(s) signal can be morphologically fitted by a 4◦ FWHM gaussian distribution or 3◦ FWHM if one uses only the events and avoids making diffuse maps or masking out any part of the Galactic center. The author of [18] has suggested instead a wider region of best significance for the 130 GeV line. Here, the morphology of dark matter profile is addressed by assuming that the line 80 CHAPTER 5. DARK MATTER INDIRECT DETECTION

Chan. Line 127 GeV (γγ) 140 GeV (Zγ) 150 GeV (hγ) W +W − Free 34.2(40.8) 35.1(42.6) 36.6(44.1) W +W − Fixed 34.5(41.4) 35.4(43.2) 37.2(44.7) b¯b Free 30.0(31.5) 31.5(33.3) 32.7(34.5) b¯b Fixed 30.3(31.8) 31.8(33.6) 33.0(34.8) τ +τ − Free 20.4(21.9) 21.6(23.4) 24.1(24.9) τ +τ − Fixed 20.7(21.9) 21.9(23.7) 23.4(25.2) µ+µ− Free 39.0(155.7) 39.9(169.8) 42.0(185.4) µ+µ− Fixed 41.1(156.3) 40.2(167.7) 42.3(184.5) e+e− Free 18.3(91.8) 13.5(100.8) 18.9(111.0) e+e− Fixed 18.3(92.1) 13.5(99.3) 19.2(110.4)

Table 5.1: 3σ upper limits on dark matter annihilation σv BR to the continuous part h i × (channel)( 10−26 cm3s−1) using full(only prompt) dark matter γ-ray spectra within l < 5◦, × | | b < 5◦. The line signal is taken to be either from its best ﬁt value (free) or ﬁxed using the | | luminosity of [19] (see text for more details). The J-factor/∆Ω from this window is 1.21 1024 × GeV2cm−5.

signal is of dark matter origin and that an associated continuous spectrum exists. The spectra of γ-rays within a wider region of the sky can be used to derive limits on the annihilation cross section for a specific assumption on the dark matter halo profile or vice versa to study the properties of the dark matter halo profile for specific assumptions on the annihilation cross section. Motivated by the γ-ray line signal at energy of 130 GeV, the case of WIMPs with ' + − mχ = 130 GeV annihilating into W W and γγ is considered. The region of b < 25◦, l < 25◦ is chosen since the annihilation from the halo in | | | | this region is dominant. This region is broken into 20 smaller windows, 8 of which in the region of b < 10◦, l < 10◦ with the size of 5◦ in l and 10◦ in b. For b > 10◦, | | | | | | l > 10◦, there are 12 windows which are symmetrically placed with respect to b = 0◦, | | l = 0◦ and each of them is composed of 4 boxes with 5◦ 5◦ size. These windows are × shown in Fig. 5.4. In Fig. 5.4, the dark matter profile is assumed to be either an Einasto as described in eq. 5.2, (Fig. 5.4, left panel) or an NFW profile (Fig. 5.4, right panel):

2 0 ! Rc 1 ρχ(r) = ρNFW r , (5.4) r 1 + 0 Rc 5.1. GALACTIC CENTER 81

3.58 3.60 4.05 4.05 20 20 2.26 2.65 2.77 2.60 3.03 3.15

1.35 1.57 2.15 2.56 1.58 1.84 2.49 2.91 10 10 0.90 0.82 0.94 1.24 1.61 2.66 5.33 1.0 0.97 1.12 1.48 1.89 3.07 6.03

2.23 1.0 0.94 1.98 2.56 1.0 0.94 2.26

b 0 * b 0 *

-10 -10

-20 -20

-20 -10 0 10 20 -20 -10 0 10 20 l l

Figure 5.4: Strength of limits on the annihilation cross-section to W +W −. Left: Einasto proﬁle. Right: NFW proﬁle. Values are normalized to the 3 σ limit on the annihilation cross-section of + − ◦ ◦ ◦ 3σ −25 3 −1 χχ W W from 5 < l < 0 , b < 5 ( σv ◦ ◦ ◦ = 3.72 10 cm s ). Values −→ − | | h i−5

0 −3 with Rc = 14.8 kpc, ρNFW = 0.569 GeV cm . For both cases the local dark matter density is 0.4 GeV cm−3 [160]. That results in a specific J-factor for each window. Similar to section 5.1.1 the 3σ limits on χχ W +W − from each angular window −→ are calculated. These limits include γ-rays background contribution that fits the lower energies. In Fig. 5.4 the ratios of the 3σ limit from each window divided by the 3σ limit from the window of ( 5◦ < l < 0◦, b < 5◦) − | | σv 3σ h iwindow (5.5) σv 3σ h i−5◦ 5◦ and b < 5◦ the strength of the 3σ limits drops. For the Einasto | | | | profile the limits become stronger at ( 10◦ < l < 5◦, 5◦ < b < 10◦) and weaker in all − | | other windows. For the more centrally peaked NFW profile the only region where slightly stronger limits are recovered is that of ( 5◦ < l < 0◦, 5◦ < b < 10◦). − | | Turning the perspective around, the information that can be extracted on the dark matter profile is discussed by fixing the annihilation cross section at a reference value. Rather than introducing some functional form for the dark matter density which is specified in terms of few parameters, as most usually done in the literature, here a much more general approach is considered. This generic spherically symmetric profile is set via: i)

Specifying its value ρi at the seven Galactocentric distances ri = R sin(αi), where R is ◦ ◦ ◦ ◦ ◦ ◦ ◦ the Solar Galactocentric distance and with αi = 5 , 10 , 15 , 20 , 25 , 45 & 65 , namely at the spherical shell corresponding to the angular windows already introduced above, plus three higher latitude patches; ii) Fixing the dark matter density at the local Galactocen- −3 tric distance r8 R to ρ8 0.4 GeVcm and implementing a linear interpolation in ≡ ≡ a double logarithmic scale to retrieve the density profile between any two of these radii, i.e. allowing for an arbitrary power-law scaling between any two ri, with the only extra assumption of imposing that the profile is monotonically increasing for decreasing radius; iii) Assuming that the profile follows an Einasto model for r > R , a choice that has no impact on the analysis that follows. + − The dark matter case with mχ = 130 GeV which annihilate into W W , introduced in Fig. 5.1, is chosen. While the ratio between the cross section into γγ to the one into W +W − is fixed by the best fit value (= 0.012), the absolute value of the cross section which scales with the inverse of the J-factor in the angular window b < 5◦, l < 5◦, in | | | | turn depends on the density profile within all the angular shells introduced above. After choosing a reference value for σv , the maximum possible contribution to J-factor from h i each of the shells are derived with the constraint of not overshooting the data in other angular windows. This gives an indication on how centrally concentrated the dark matter profile should be to provide a signal in the Galactic center direction and, at the same time, to be consistent with data away from it. For this purpose the factors Ji are introduced 5.1. GALACTIC CENTER 83 which are analogous to the J-factor introduced in Eq. 5.3 except for imposing that the density profile is constant below the radius ri, namely ρ(r < ri) = ρi.

The analysis is performed scanning the parameter space defined by the values ρi (with i [1, 7]). For each model the Ji factors for the angular regions displayed in Fig. 5.4 ∈ as well as for the regions at 25◦ < b < 45◦, 45◦ < b < 65◦ and 65◦ < b < 85◦, | | | | | | and 0◦ < l < 20◦ (for all three latitude intervals) are calculated. For all regions we can | | compare against the 3σ upper bound on the flux due to a dark matter candidate with mass 130 GeV annihilating to W +W − (conservatively including prompt emission only, while ignoring the radiative emission from the associated lepton yields). As upper bound to the monochromatic signal we consider instead the mean flux integrated in the energy bin [104.5, 135.7] GeV, obtained using the ULTRACLEAN data sample, and under the very conservative hypothesis of zero background from diffuse emission. Among models passing constraints, we search for configurations giving the maximum for the individual terms Ji

(we also implement the additional limit J1 J, given that any additional contribution to ≤ the line of sight integral at radii r < r1 is always neglected in our setup). Although we are dealing with a very large parameter space, finding the upper bounds to Ji, which we max label Ji , is not exceeding expensive since one can show that, for each radial shell, they mostly correspond to the models with largest changes in profile slope between neighboring shells. max Ratios between Ji and J are shown in Fig. 5.5, where we display separately the max Ji found when applying the limit on the monochromatic flux and when implementing that from the component with continuous spectrum; limits are shown as (very narrow) bands since they were derived for three different values for σv : the “thermal” value h i 3 10−26 cm3 s−1, the best fit value in case of our reference Einasto profile 1.05 10−25 × × cm3 s−1, and ten times the thermal value (this shows that dependence of our analysis on

σv is really very mild). For comparison, we plot also values of Ji/J for our reference h i 2 2 Einasto profile and for a Burkert profile, namely ρ 1/(r + Rc)/(r + R ), with local ∝ c −3 dark matter density 0.4 GeV cm and core radius Rc = 10 kpc [160]. As one can see the Burkert profile is excluded from both line and continuous components, while the line limits are giving stronger evidence towards the need for a more centrally concentrated dark matter profile. This is most probably related to the fact that the limits are derived in part from regions of the sky where the Fermi Bubbles/haze, has been claimed to be needed; we do not try to include such component in our background model and most probably this translates into an extra room (or a less severe constraint) on the continuous emission from dark matter annihilations. On the other hand, the Fermi Bubbles/haze are expected to play a marginal role at high energy, hence the sharper constraint from 84 CHAPTER 5. DARK MATTER INDIRECT DETECTION

à 1.0 à à à 0.8 à æ J

0.6 max i à J 0.4 à æ

0.2 æ æ æ æ 0.0 æ 0 2 4 6 8 RHkpcL

Figure 5.5: Upper bounds on the partial Ji factor, normalized to the total Galactic center line of sight integration factor J, for the parametric dark matter density proﬁle introduced in the text.

The Ji factor are computed assuming a constant dark matter density within the corresponding max radial shell ri. The Ji values displays are derived implementing separately the limits on the monochromatic flux (lower green bands) and the continuous spectrum (higher orange bands) derived from the other angular windows considered in the analysis. Limits are shown as narrow bands since they refer to three different values of the annihilation cross section, see the text for details. Also shown are the values for Ji/J for our reference Einasto profile (blue dots) and for a cored Burkert profile (red squares); as it can be seen the Einasto profile is allowed, while the Burkert shape is excluded.

the line emissivity.

5.1.3 A Speciﬁc Example

The limits that are shown in Figs. 5.2 and 5.3 can be linearly combined to give the limits for a wide class of models. As a speciﬁc example the model presented in [164] to explain a 130 GeV line is considered here. This model suggests that dark matter is composed of Winos and Axions with almost equal amounts of mass density towards the Galactic center. Following that, the mass density of the Winos is taken to be 49% of the total in the Galactic center and in the entire Galaxy. The proﬁle of dark matter is chosen to be the Einasto in Eq. 5.2. The mass of Wino is assumed to be mχ = 145 GeV. The Winos annihilate to Zγ line with cross section of 1.26 10−26. The total annihilation cross × section of the Winos is 3.2 10−24 cm3s−1 and it is dominantly to W +W −. Assuming a × BR=0.96 for annihilation to W +W − it is shown that such a model is excluded as can be 5.2. GALACTIC HALO 85

−26 3 −1 Figure 5.6: Wino/Axion model of [164]. mχ = 145 GeV σv χχ−→Zγ = 1.26 10 cm s , h i × σv tot = 3.2 10−24 cm3s−1. h iχχ × seen in Fig. 5.6. In fact such a cross section is O(10) larger than the relevant 3σ limit for that mass and channel shown in Fig. 5.2. It is worth noting that even ignoring the inverse Compton and bremsstrahlung components and all the background contribution, the prompt component which also includes the line signal overshoots the total γ-ray spectrum between 10 and 40 GeV. This result can not depend on the assumptions for the dark matter proﬁle or on the ratio of Wino to Axion mass density within the Galaxy since by changing any of these assumptions the γ-ray line will decrease/increase by the same amount.

5.2 Galactic Halo

Although dark matter contribution to γ-rays spectra from the Galactic halo is lower than that from the Galactic center, the astrophysical uncertainties are also less in the Galactic halo. Therefore, the Galactic halo can serve as a target for dark matter searches. The contribution of dark gas to the π0 decay and bremsstrahlung components of diffuse γ-rays can be modeled in the halo with reasonable accuracy. Including that results in an overall very good agreement to the Fermi data from 200 MeV up to 300 GeV. The method for deriving dark gas component is described in section 5.2.1. Furthermore, to extract conservative upper limits the minimal contribution of non dark matter extragalactic background radiation to diﬀuse γ-rays is applied. This is explained in detail in section 5.2.2. Finally, in section 5.2.3 the best angular windows to search for dark matter signal 86 CHAPTER 5. DARK MATTER INDIRECT DETECTION are probed and the upper limits on dark matter annihilation cross section are presented.

5.2.1 Dark Gas

As discussed in sections 2.1 and 2.2, atomic and molecular hydrogen gasses are traced by, respectively, 21cm and 2.6mm emission lines. However, both lines may suffer from absorption by interstellar gasses. Dust is known to mix with both of these phases and so can provide an alternative tracer for the interstellar gas distribution. On the other hand, γ-rays which are produced by interactions of cosmic ray protons and electrons with the interstellar gas typically do not suffer any attenuation up to energies of several TeV. Thus, there is a potential mis-match between the Fermi γ-ray data and the predictions of diffuse Galactic gamma-rays associated to hydrogen gas distribution. This has already been shown for the EGRET γ-ray sky map in [158] and has been quoted as the ”dark gas” (see also [159] for a dark gas evaluation related to the Fermi γ-ray data). Here the same method as [158] is applied to derive the dark gas component using the 3-year Fermi γ-ray data and gas distributions of section 2.1 and [57]. As has been discussed both in [158] and [159] the part of the Galactic sky that is closer to the disk suffers from great uncertainties in the evaluation of the dark gas. Thus, the part of the Galaxy which is close to the disk is ignored. This is done by masking out either the b < 1◦ or the b < 5◦. In the final model the b < 1◦ masked maps are used. | | | | | | A maximization of likelihood fit of a linear combination of the HI and CO column density maps to the 100 µm SFD dust map [165] is performed. A 2◦ FWHM smoothing in all maps is applied. For the maximization of likelihood fit the method described in [152] is followed. It is worth noting that there is always a freedom in the normalization of the dark gas. It is chosen in such a way to avoid overshooting γ-ray spectral data in any of the windows.

5.2.2 Minimal non-DM Extragalactic Background

The Fermi collaboration has measured in [16, 166] the total Extra-Galactic Background −2.41 Radiation (EGBR) described by a single power-law of dNγ/dE E in the energy ∼ range of 100 MeV to 100 GeV with the total flux of γ-rays in that range being 1.09 10−5 × ph cm−2s−1sr−1. Dark matter annihilation in the main halo and in the substructures can contribute to diffuse γ-ray spectrum even at the highest latitudes. Thus it has to be included to the isotropic γ-ray spectrum. Additionally, dark matter annihilation in prototype halos 5.2. GALACTIC HALO 87 at high redshifts can contribute in both the EGBR flux and its power-spectrum recently measured in [167]. In this work, we aim to derive conservative upper limits on the annihilation rate in the dark matter halo. Thus the minimum contribution to the EGBR and its power-spectrum from known non-dark matter γ-rays sources is considered. Among the known sources that can contribute to the EGBR we will consider the contribution of BL Lacertae-like objects (BL Lacs), Flat Spectrum Radio Quasars (FS- RQs), Milli-Second Pulsars (MSPs), unresolved Star-Forming Galaxies (SFG), Ultra High Energy Cosmic Rays producing γ-rays from cascades after interactions with the Cosmic Microwave Background and the interstellar radiation field (UHECR CMB) as well as the contribution of Fanarov Riley I and II sources (FRI, FRII). We will also include the possible contribution to the EGBR from undetected/unidentified Gamma-Ray Bursts (GRBs), from Starburst and luminous IR galaxies (SBG), the contribution of isotropic γ-rays from cascades produced by UHECR protons interacting with the Inter-cluster medium (UHEp ICM) and finally from gravitationally induced shock-waves in the intergalactic medium accelerating electrons that then up-scatter low energy photons into γ-rays (IGS). BL Lacs and FSRQs are two populations of Blazar point sources (Active Galactic Nuclei with a relativistic jet pointing towards our line of sight) that are common among the known detected point sources and thus their unresolved members are expected to contribute to the EGBR. From identified point sources belonging in those categories we can probe their spectral properties and extrapolate their contribution to the EGBR. BL Lacs, having taken their name form the BL Lacertae AGN, are objects with rapid and strong variability in their optical luminosity and a non-thermal optical spectrum and strong radio emission. Following the assumptions of [166], the differential spectrum of unresolved BL Lacs is described by a power-law of 2.23 with a normalization which is given by dNγ −8 −2.23 −1 −2 −1 −1 = 3.9 10 Eγ GeV cm s sr , (5.6) dE BLLac × −7 −2 −1 −1 where Eγ is in GeV. Thus, their contribution to the EGBR flux is 5.4 10 ph cm s sr × at 10 MeV < Eγ <10 GeV which represents the 1σ low value as it was estimated by [166]. FSRQs are energetic flat spectrum quasi-stellar radio sources. Their spectrum is slightly softer than that of the BL Lacs and is described by

dNγ −8 −2.45 −1 −2 −1 −1 = 3.1 10 Eγ GeV cm s sr . (5.7) dE FSRQ × Their contribution to the EGBR is almost the same as BL Lacs. In terms of photon numbers it is (6.1 10−7 ph cm−2s−1sr−1 at 1 σ low), being more important below 0.5 × GeV. 88 CHAPTER 5. DARK MATTER INDIRECT DETECTION

The millisecond pulsars are the faint Galactic γ-ray sources. However, they are expected to give significant contribution at high latitudes. In [77] their minimum contribution to the γ-ray flux at latitudes b > 40◦ was estimated to be 8.0 10−7 ph cm−2s−1sr−1 | | × for 100 MeV < Eγ < 10 GeV. Their spectral properties have been measured by [168]. Thus, their contribution is modeled as dNγ −7 −1.5 Eγ −1 −2 −1 −1 = 1.8 10 Eγ exp GeV cm s sr , (5.8) dE MSP × −1.9 for b > 40◦. | | Unresolved star-forming Galaxies contribute to EGBR by at least 5% of the EGBR ∼ [169]. Though, their contribution to EGBR has been evaluated with great uncertainty. Thus SFGs can even account for 50% of the flux [170]. Assuming that the main part of their γ-ray spectrum is of hadronic origin (from π0 decays that have been produced in inelastic pp collisions in Galaxies) the spectrum has a power-law above 1 GeV with the π0 decay peak at 0.5 GeV. For the minimal assumptions on SFGs we take '

dNγ −7 −2.75 −1 −2 −1 −1 = 1.3 10 Eγ GeV cm s sr for Eγ > 1GeV, (5.9) dE SFG ×

−7 −2 −1 −1 which gives a total contribution of 5.5 10 ph cm s sr for 100 MeV < Eγ < 100 × GeV. Ultra High Energy CRs interaction with the CMB also produce cascades which give hard γ-rays spectrum. We adopt the minimum estimate of [171] (see also [172]) which is obtained by assuming that the UHECR is proton dominated with a hard power law 21 spectrum ( 2) up to Ep = 10 eV and also an evolution is done up to redshifts of z = 2. ' Their spectrum comes from monte carlo simulations for the development of the cascades p + γ π0, p + γ π± and can be approximated by −→ −→ " 0.35# dNγ −9 −1.8 Eγ −1 −2 −1 −1 = 4.8 10 Eγ exp GeV cm s sr for Eγ > 1GeV. dE UHECRCMB × − 100 (5.10) Their contribution is minimal to the total number of extragalactic γ-rays, (3.3 10−8 ph × cm−2s−1sr−1 between 100 MeV and 100 GeV) and becomes important only at the highest energies of the observed spectrum. Fanarov Riley objects are misaligned AGNs with type I, which are 2-sided radio jet subsonic structures, and type II, which are edge darkened objects with lobes with indications for supersonic jets. Both classes suﬀer from small statistics in detection at γ-rays, leading to large uncertainties. Since the jets of the AGNs are not along our line of sight, the mechanism that produces the detectable γ-rays for us is expected to be synchrotron 5.2. GALACTIC HALO 89 self Compton scattering. Using the assumptions of [173], the diﬀerential spectrum at

Eγ > 5 MeV is given by a power law of 2.39. Including attenuation of γ-rays leads to dNγ −6 −2.39 Eγ −1 −2 −1 −1 = 5.7 10 Eγ exp GeV cm s sr . (5.11) dE FR × − 50

Even at this lower/conservative estimate these sources are expected to give 1.0 10−6 ph × cm−2s−1sr−1 between 100 MeV and 100 GeV. Thus they are responsible for at least 10 % of the EGBR. Gamma Ray Bursts that are not detected as transient events due to small number of photons and also a duration longer than typical can also contribute to the EGBR. Based on the previous estimates on their contribution to the EGRET extragalactic background [174], their contribution is still expected to be at the level of 1 % of the total EGBR (1.0 10−7 ph cm−2s−1sr−1 above 100 MeV). We take them to be described by ×

dNγ −9 −2.1 −1 −2 −1 −1 = 8.9 10 Eγ GeV cm s sr . (5.12) dE GRB × where attenuation has to be added for γ-rays above 50 GeV. ∼ Starburst Galaxies have enhanced star formation rates and interstellar medium densities compared to the Milky Way. Their γ-ray contribution is given by a spectrum from inverse Compton scattering and so it is more important at high energies. Following [175], they are given by

dNγ −7 −2.4 −1 −2 −1 −1 = 0.3 10 Eγ GeV cm s sr , (5.13) dE SBG × which leads to a total of 5.4 10−7 ph cm−2s−1sr−1 or 5% of the EGBR above 100 × ' MeV. Ultra High Energy CR protons interacting with the inter-cluster medium give from the cascades π0 γ-rays and secondary e± which then up-scatter photons. In our conservative approach we will ignore the inverse Compton component which depends on the assumptions of the inter-cluster radiation ﬁeld. The remaining π0 component above 1 ∼ GeV can be approximated by

dNγ −9 −2.75 −1 −2 −1 −1 = 3.1 10 Eγ GeV cm s sr . (5.14) dE UHEp ICM ×

The normalization results in 1.0 10−7 ph cm−2s−1sr−1 above 100 MeV [176, 177]. × Finally gravitationally induced shock waves in the intergalactic medium can accelerate electrons and protons. Since protons do not loose energy fast enough the major contribution to γ-rays is due to inverse Compton of electrons with very hard spectrum E−2. ∼ 90 CHAPTER 5. DARK MATTER INDIRECT DETECTION

" 10 1

! EGB Fermi

sr BL Lac 1

! !7 s 10 FSRQ 2

! MSP SFG 10!8 UHECR CMB FRI & FRII GeV cm ! GRB

dE !9 SBG

# 10

Γ IGS

dN UHEpr ICM 2

Γ !10 Total

E 10

0.1 0.5 1.0 5.0 10.0 50.0 100.0

EΓ GeV

Figure 5.7: The measured Extra-Galactic! γ-ray" Background (EGB Fermi) (black solid) and the minimum contribution of various known sources. BL Lacs (blue solid), FSRQs (red solid), millisecond pulsars (MSP) (purple solid), star-forming galaxies (SFG) (orange solid), from interaction of Ultra High Energy CRs with CMB and galactic radiation ﬁeld (UHECR CMB) (green solid), Fanarov Riley I and II galaxies (FRI & FRII) (blue dashed), Gamma Ray Bursts (GRB) (red dashed), Starburst galaxies (purple dashed), from shock waves in the intergalactic medium (IGS) (orange dashed) and from Ultra High Energy CR protons in the inter-cluster medium (UHEpr ICM) (green dashed). The minimum combined EGB (Total) (thick solid green).

While this is not an extra-galactic component it can give a contribution at high latitudes and thus mimic the EGBR. Following [178] (see also [179]) we take

 −2.04  Eγ −1 −2 −1 −1 GeV cm s sr for Eγ < 10GeV dNγ −10  10  = 0.87 10 −2.13 (5.15) dE IGS × × Eγ  10 for Eγ > 10GeV. 

We also include attenuation of γ-rays for those sources that are expected to be important at high redshifts.

The spectra of all of the mentioned sources are shown in Fig. 5.7. The combination of these spectra gives a total ﬂux of 4.3 10−6 ph cm−2s−1sr−1 between 100 MeV and 100 × GeV or 40 % of the total EGBR measured by [16]. ' The spectra of γ-rays which are obtained by including dark gas and by using the minimal contribution from extragalactic background radiation are shown in Fig. 5.8. These spectra are used as background for extracting limits on dark matter annihilation cross section in the halo. 5.2. GALACTIC HALO 91

Figure 5.8: The spectra of diﬀuse γ-ray background. The diﬀusion parameters are δ = 0.5, zd = 4 kpc, Rd = 20 kpc and vc = 0. The models for HI and H2 distributions are provided, respectively, in section 2.1 and in [57]. Dark gas is also included. The minimal contribution from extragalactic background radiation is used. The sky regions are due to 0 < l < 360 and upper left: b < 5, upper right: 5 < b < 10, middle left: 10 < b < 20, middle right: 20 < b < 60 | | | | | | | | and lower: 60 < b < 90. | | 92 CHAPTER 5. DARK MATTER INDIRECT DETECTION

5.2.3 Limits on WIMPs Annihilation Cross Section in the Halo

In order to constrain dark matter contribution to diffuse γ-rays in the Galactic halo the individual annihilation channels of χχ W +W −, χχ µ+µ−, χχ τ +τ − −→ −→ −→ and χχ tt¯ are studied. As mentioned earlier in section 5.1.1 the limits for specific −→ dark matter models can be derived by linear combination of limits from these individual channels. The mass range for each of these channels is chosen in such a way that their γ- rays spectrum lies in the energy range of Fermi. The sky is broken into 30 windows which cover the entire sky. These windows are defined by the limits b 5◦, 5◦ b 10◦, | | ≤ ≤ | | ≤ 10◦ b 20◦, 20◦ b 60◦ and 60◦ b 90◦ in latitude and 0◦ l 30◦, ≤ | | ≤ ≤ | | ≤ ≤ | | ≤ ≤ ≤ 30◦ l 60◦, 60◦ l 180◦, 180◦ l 300◦, 300◦ l 330◦ and 330◦ l 360◦ in ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ longitude. To extract limits on wide regions in the sky low latitude region ( b < 5◦) is excluded. | | The best angular windows with the most stringent limits on WIMPs annihilation are then searched for. It is found that the regions of l < 30◦, 5◦ 60◦ are the best regions to search for dark matter. | | | | The WIMPs annihilation cross sections which best fit diffuse γ-rays in these regions are derived. In Fig. 5.9 the best fit spectra for WIMPs with mχ = 400 GeV which annihilate into W +W − are shown for the regions under study. As can be seen lower latitude regions give tighter constraints. The best fit spectra for four annihilation channels in the region of l < 30◦, 5◦ < | | ◦ b < 10 which gives the most tight limits are shown in Fig. 5.10 for mχ = 1 TeV. The | | best annihilation cross sections for all of these channels are very similar. However, τ +τ − channel gives slightly stronger limits. From the best fit values for annihilation cross sections, 1 4σ upper limits are derived − for all annihilation channels and regions under study. These limits are shown in Fig. 5.11 versus dark matter mass. The most constraining annihilation channels are respectively W +W − and τ +τ −. 5.2. GALACTIC HALO 93

Figure 5.9: The spectra of diﬀuse γ-rays for dark matter model with mχ = 400 GeV annihilating + − into W W . The best ﬁt value for annihilation cross section is 111 σv th. The sky regions × h i are upper left: l < 30◦, 5◦ 60◦ which give the most stringent limits on WIMPs annihilation. | | 94 CHAPTER 5. DARK MATTER INDIRECT DETECTION

Figure 5.10: The spectra of diﬀuse γ-rays in l < 30◦, 5◦ < b < 10◦ sky region for dark matter | | | | + − models with mχ = 1 TeV. The annihilation channels are upper left: W W , upper right: µ+µ−, lower left: tt¯ and lower right: τ +τ −. The best ﬁt values for annihilation cross sections are shown in terms of thermal annihilation cross section 3 10−26cm3/s. × 5.2. GALACTIC HALO 95

Figure 5.11: Upper limits on WIMP annihilation cross sections versus mχ. The annihilation channels are upper left: W +W −, upper right: µ+µ−, lower left: tt¯ and lower right: τ +τ −. The lines represent solid lines: the best ﬁt and dotted lines: 1σ, dashed lines: 2σ, dashed-dotted lines: 3σ, dotted-dotted-dashed lines: 4σ upper limits. The sky regions are red: l < 30◦, 5◦ < | | b < 10◦, blue: 30◦ < l < 60◦, 10◦ 60◦. | | | | | | | | 96 CHAPTER 5. DARK MATTER INDIRECT DETECTION Chapter 6

Conclusion

The wealth of high accuracy data on cosmic rays and diffuse γ-rays is the key to un- derstand the properties of the interstellar medium and cosmic rays propagation. The measurement of cosmic rays fluxes is a tool to locally constrain these properties. Com- bining that with the measurement of diffuse γ-rays enables us to acquire much deeper understanding of those properties on the Galactic scale. In this study, a large subset of the benchmark models which have rather diverse settings for modeling the interstellar medium and cosmic rays propagation are tested. While many of these models provide a good fit to both the cosmic rays and the γ-rays data, still there are a few scenarios that this analysis can disfavor. In particular, rather extreme limits on cosmic rays diffusion profiles ranging from essentially constant diffusion coefficient everywhere in the Galaxy to exponentially suppressed vertical and radial profiles with 1 kpc scale height and 5 kpc radial scale are studied . The combined fit of cosmic rays and γ-rays suggests a slight preference for thicker diffusion zones, while there is a weak dependence on the variation of the diffusion coefficient in the radial direction, which is however better probed and constrained by antiprotons spectrum. While this result is not conclusive, it suggests a trend that better statistics and smaller systematics on γ-rays spectra will lead to constrain the position dependence of the diffusion coefficient D(R, z) of cosmic rays in the Galaxy. The high statistics measurements of the local flux of radioactive isotopes by AMS-02 [180], placed on the International Space Station, will add further information on the vertical thickness of the diffusion region, possibly allowing to break the degeneracy between thicker regions of emissivity populated by cosmic rays diffusing out of the Galactic disk and exotic sources with an intrinsically thicker scale height, such as from dark matter. On the other hand, it is found that cosmic rays and γ-rays data do not constrain strongly the diffusion spectral index δ within the considered range.

97 98 CHAPTER 6. CONCLUSION

Furthermore, the implications from the rigidity break in the spectra of cosmic ray protons and helium nuclei which are indicated by [69] and confirmed by [70] are discussed. The possibility of discriminating whether the break is in the injection spectrum (connected to either acceleration effects in the sources or to the presence of an extra population of primary sources injecting cosmic rays with harder spectra) or in the energy dependence of the diffusion coefficient are addressed. It is found that the Galactic diffuse γ-rays can not be used to this aim, neither with the current nor with the near future projected accuracy of the spectra, leaving this task to other observables, such as antiprotons as suggested by [131]. Moreover, diffuse γ-ray spectra can discriminate (and even constrain) profiles for the interstellar gasses. Before Fermi-LAT γ-ray data, in order to place constraints on the interstellar medium properties, the secondary to primary cosmic rays spectra were used, with the best measured data sets coming from B/C andp/p ¯ . Yet, with only the cosmic rays data as a handle, a variation of the large scale gasses distributions could almost always be compensated by changing the diffusion properties (mainly the normalization of the diffusion coefficient). It is shown that by exploiting diffuse γ-ray fluxes at latitudes above 10◦ and combining them with the cosmic rays data the degeneracy between diffusion and interstellar gas distribution can be broken. Indeed, detailed gas distributions are needed to interpret the fine structures in high angular resolution maps of the Fermi gamma ray telescope. For this reason, a new model for three dimensional distribution of atomic hydrogen gas in the Milky Way is constructed. Atomic hydrogen gas is the major component of the interstellar gas whose total mass 9 within the radius of 20 kpc is found to be 4.3 10 M . Using the derived model for × distribution of atomic hydrogen gas together with the most recent model for distribution of molecular hydrogen gas [57] results in a very good agreement between the predicted diffuse γ-rays and the Fermi data in all regions of the sky. Having established the propagation models which account for astrophysical contribution to diffuse γ-rays, a natural application will be to place limits on a possible contribution from dark matter. Inspired by indications for a monochromatic γ-ray signal towards the Galactic center, possibly connected to WIMPs annihilation, limits on the continuous component that in general accompanies such a signal are derived. The 3σ upper limits for WIMPs annihilation into W +W −, b¯b, τ +τ −, µ+µ− and e+e− channels are extracted in the l < 5◦, b < 5◦ | | | | region. The limits are extracted from the total dark matter γ-ray emission, including the prompt, the inverse Compton and the bremsstrahlung components (as given in Fig. 5.2), 99 a well as from only prompt γ-rays (as given in Fig. 5.3). These limits do not depend on the exact normalization of the line(s) since they are dominated by the γ-ray data below 100 GeV, where the lines do not contribute (see Table 5.1). While these limits depend on the choice of the dark matter halo profile, they can be easily rescaled to a different halo configuration. This happens since, apart from the µ+µ− and e+e− channels, the prompt γ-rays are the dominant component. Studying the γ-ray data from other angular windows of the Galaxy, it is found that for cuspy dark matter profiles such as the NFW profile, the most stringent constraints come from l < 5◦, b < 5◦ window while for the Einasto profile slightly stronger limits | | | | can come from 5◦ < b < 10◦ (see Fig. 5.4). A new general parametrization for the | | dark matter profile is also introduced to discuss how centrally concentrated the profile should be to give a flux compatible with the suggested Galactic center line signal without violating bounds from other angular windows. The results for upper bounds on partial line-of-sight integration factors, which are readily applicable to any dark matter profile, show e.g. that a Burkert dark matter halo can not be compatible with those bounds (see Fig. 5.5). The limits on specific dark matter annihilation channels can be linearly combined and readily applied to most dark matter models in this mass range. Applying our limits to the model of [164] we conclude that it is excluded given that its prompt component exceeds the total γ-ray flux (Fig. 5.6). Finally, conservative bounds on WIMPs annihilation cross sections are derived by using the minimal contribution from non-DM extragalactic background radiation to diffuse γ- rays in the Galactic halo. It is found that the best angular windows for dark matter searches in the Galactic halo are l < 30◦&5◦ 60◦. The lower latitude region gives the most tight constraints. | | Upper limits are extracted for individual annihilation channels of W +W −, µ+µ−, tt¯ and τ +τ − (see Fig. 5.11). 100 CHAPTER 6. CONCLUSION Bibliography

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