Probing the effect of dark matter velocity distributions on neutrino-based dark matter detection

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UPTEC F 19055

Examensarbete 30 hp

Oktober 2019

Probing the effect of dark matter

velocity distributions on neutrino-based

dark matter detection

Martin Ståhl

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Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress:

Box 536 751 21 Uppsala

Telefon:

018 – 471 30 03

Telefax:

018 – 471 30 00

Hemsida:

http://www.teknat.uu.se/student

Abstract

Probing the effect of dark matter velocity distributions

on neutrino-based dark matter detection

Martin Ståhl

Dark matter has a long history, but it was not until modern times that we have a chance of detecting it. This thesis focuses on the velocity distribution and its effect on indirect WIMP detection. Recently a new velocity distribution, based on data from SDSS and GAIA, was proposed. For this reason simulation of capture, annihilation and resulting flux of neutrinos from the Sun and Earth has been made both for the new and Maxwell-Boltzmann velocity distribution. The newly proposed velocity can reduce the annihilation rate in Earth by two thirds. For the Sun the effect depends on the mass of the WIMPs. For 50 GeV WIMPs the newly proposed velocity distribution could increase the annihilation rate by 5%, while for 3 TeV WIMPs it could decrease the annihilation rate by 28%. For Earth and high mass WIMPs the low velocity tail is the important part and the low resolution of this region in the new velocity distribution result in some uncertainties.

Ämnesgranskare: Allan Hallgren Handledare: Carlos Perez de los Heros

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Populärvetenskaplig sammanfattning

’Mörk materia vad är det?’ är en fråga som astronomer länge frågat sig. En theoretisk lösning är WIMP:ar, massiva partiklar som sällan interagerar med vanlig materia. Dessa kan vara sin egen antipartikel vilket betyder att om två WIMP:ar kolliderar så annihileras de. Vid denna annihilation kan det pro- duceras vanlig materia som vi hoppas kunna detektera. När man söker efter mörk materaia genom indirekt detektion så är det dessa produkter man letar efter. Två ställen där annihilationer förväntas ske är Solens och Jordens kärnor.

Genom att dessa fångar upp mörk materia med sin gravitation så kommer den ansamlas i kärnorna och annihileras. Sannolikheten att en WIMP fångas upp beror till stor del på dess hastighet, och antalet annihilationer är proportion- erligt till hur många som fångas upp.

Detta arbete fokuserar på hastighets-distributionen och dess inverkan på in- direkt detektion. Vanligen antar man att den mörka materian i vår galax har nått jämnvikt och hastighets-distributionen är en Maxwell-Boltzmann fördel- ning. Relativt nyligen föreslogs en ny hastighets-distribution baserad på data från Sloan Digital Sky Survey (SDSS) och GAIA. För att undersöka vad effek- ten blir av den nya hastighets-distributionen har simulationer gjorts för olika hastighets-distributioner. Dessa simulerar hur WIMP:ar uppfångas av grav- itationen, annihileras samt vilka partiklar som produceras i Solen respektive Jorden.

Resultatet visade att jämfört med Maxwell-Boltzmann så kan den nya hastighets- distributionen reducera mängden annihilationer i Jorden med två tredjedelar.

För solen har WIMP:ens massa stor betydelse. Den nya hastighets-distributionen innebär att för små 50 GeV WIMP:ar ökar mängden annihilationer med 5%

och för 3000 GeV WIMP:ar minskar den med 28%.

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Contents

1 Introduction . . . .3

1.1 History . . . . 3

1.2 Dark matter candidates . . . . 5

1.2.1 MACHOs . . . . 5

1.2.2 MOND. . . .5

1.2.3 WIMPs . . . . 6

1.3 Particle detection . . . . 6

2 Halo models . . . . 9

2.1 Dark matter density distribution . . . .9

2.2 Velocity distribution . . . . 10

2.2.1 Velocity and capture rate. . . .10

3 Simulation . . . . 11

4 Results . . . . 13

5 Conclusions and discussion . . . . 14

A Appendix - Graphs . . . . 15

A.1 Annihilation in the Sun . . . .15

A.2 Annihilation in Earth . . . . 18

References . . . . 21

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1. Introduction

1.1 History

Dark matter was discussed already in the 19th century, then as invisible matter occupying the galaxy in contrast to light matter that was visible to astronomers of the time i.e. bright stars. Father Angelo Secchi, for example, mentioned dark masses in 1877 when referring to research about nebulae:

"Among these studies there is the interesting probable discovery of dark masses scattered in space, whose existence was revealed thanks to the bright background on which they are projected. Until now they were classified as black cavities, but this explanation is highly improbable, especially after the discovery of the gaseous nature of the nebular masses." [27]1

Fritz Zwicky is often referred as a pioneer of dark matter. He analysed the Coma cluster, a big cluster of more than 1000 galaxies, first in 1933 [36]

and then in a refined and extended analysis 1937 [37]. The velocities of the galaxies were derived from the redshift of the light, and from there he derived the mass using the virial theorem. His conclusion was that the Coma cluster contains 500 times more mass than visible light would indicate. It should be noted that the derivation of velocity from redshift was not very accurate at the time. If a modern derivation is used the result would be 60 times more mass than visible. A similar study was conducted by Sinclair Smith on the Virgo cluster. Published in 1936 he estimated that it contained 200 times more mass than visible light would indicate [30]. Smith and Zwicky suggested that the dark matter or missing mass could be in the form of cold stars, clouds of macroscopic and microscopic solid bodies or gases.

World War II halted much of the work in astronomy, but had some unex- pected benefits. Developments in radio telescopes led to its usage in astron- omy, and with measurements of radio waves and x-rays gas clouds could be ruled out as significant contributions of dark matter [3].

Early observations of the rotational curve of galaxies indicated the exis- tence of dark matter but these observations were not taken seriously. In 1970s that changed with more observations of improved quality. Newtons theory of gravity predicts that the rotational speed of galaxies would decline farther out from the center. Instead the observed rotational curves were flat, meaning that rotational speed remained constant even far out, see Figure 1.1.

At the same time several numerical simulations also showed that rotating disk galaxies should be unstable [3]. Simulations in 1973 by Jerry Ostriker

1qtd in [3]

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and Jim Peebles showed that this problem could be solved with a galactic halo of dark matter [25].

In the end of the 1970s suggestions started appearing that neutrinos might account for the missing mass. Neutrinos are stable and do not experience electromagnetic or strong interactions, they interact only through gravity and weak interactions. Such a particle would be able to solve the dark matter problem in galaxy clusters. It could exist in clouds around galaxies asserting gravity and binding them together. The only problem is that neutrinos are very light. As time went on it became apparent that the neutrino mass must be less than 10 eV. 2 There could be another type of particle that had the same properties but is massive, dubbed Weakly Interactive Massive Particle (WIMP).

2In a 2017 limit from cosmological data the sum of all three neutrino masses is less than 0.151 eV [33]

Figure 1.1. Rotational curves from 25 galaxies published by A. Bosma in 1978 [5].

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The aim of this study is to examine how the velocity distribution of WIMPs effects the possibilities of detecting dark matter. For this purpose simulations have been done and chapter 3 describes this in more detail. Chapter 2 goes through the dark matter distribution, and this chapter explain what dark matter is and how one could detect it.

1.2 Dark matter candidates

Dark matter candidates can be categorized as either hot dark matter or cold dark matter. Hot dark matter consists of light particles that travel close to the speed of light, for example the neutrino. In contrast cold dark matter consists of particles that travel slower, at sub-relativistic speed, such as WIMPs or axions. There can also be warm dark matter that is in between, such as the sterile neutrino. Simulations have shown that hot dark matter and cold dark matter forms galaxies very different from each other. If compared to actual observation this strongly disfavor hot dark matter [35]. This leaves warm and cold dark matter as the valid candidates. The latter in the form of WIMPs will be the focus of this study, but first we will briefly go through two alternative theories that tries to refute that dark matter is a new particle.

1.2.1 MACHOs

Could dark matter be normal baryonic matter that we just can not see; ob- jects such as asteroids, planets and non-luminous stars? Instead of WIMPs we would have Massive Astrophysical Compact Halo Objects (MACHOs). These objects would produce gravitational lensing that should be detectable by tele- scopes when they appear between bright stars and Earth. The space survey EROS-2 studied this and found that MACHOs could only constitute 8% of the dark matter mass [32]. Furthermore there are constraints on the universe’s baryon budget. Looking at cosmic microwave background we expect a cer- tain amount of baryonic matter created during big bang, and this would not be enough for MACHOs [12].

Another form of MACHOs are primordial black holes, theorized small black holes created shortly after big bang. From these we would expect hawk- ing radiation and gravitational lensing. Lack of this limits their mass to 1014− 1023 kg. Also only a negligible amount of black holes are expected to have formed in the early universe making them an unlikely candidate [3].

1.2.2 MOND

Modified Newtonian dynamics (MOND) is an alternative to dark matter. It assumes that dark matter does not exist and that it is our understanding of

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gravity that is incomplete. To be able to solve everything related to gravity it gets very complicated, and seems (to the author) like an ad hoc solution.

It is good at predicting the rotation of galaxies, yet cannot fully explain the movement of galaxy clusters [3].

1.2.3 WIMPs

The most common explanation to dark matter is WIMPs. Heavy particles that hardly interact with normal matter, only weakly and through gravity. There are several theories in physics that introduce such candidates.

Theories of supersymmetry, such as Minimal Supersymmetric Standard Model (MSSM), try to solve problems related to the standard model such as the hierarchy problem [31, 34]. They predict that each particle have a su- persymmetric partner. A boson’s supersymmetric partner is a fermion and a fermion’s supersymmetric partner is a boson, i.e. the supersymmetric partner to any particle have a spin difference of 1/2. Most promising as dark matter is the neutralino, supersymmetric partner to the bosons: Z, neutral higgs and photon. The neutralino is also its own antiparticle. The sneutrino (partner to neutrinos) and gravitino (partner to graviton in supergravity extension) have also been suggested [16].

Theories of universal extra dimensions introduce extra dimensions in ad- dition to the known 4 dimensions of space and time. They predict a WIMP in the form of the lightest Kaluza-Klein particle. With results from LHC the mass must be in the TeV range [28].

1.3 Particle detection

There are three main strategies to detect WIMPs and other weakly interacting dark matter: direct detection, indirect detection and production (see Figure 1.2). Production involves creating dark matter in collider based experiments, such as ATLAS and CMS at the LHC [6]. Direct detection involves looking for WIMPs scattering off nuclei. Indirect detection is mainly about detecting the products from WIMP annihilation. It is possible that dark matter is its own antiparticle, in such case one WIMP could collide with another and create matter we can detect. Annihilation would occur where there are large amounts of dark matter such as the galactic halo, the galactic center or in large celestial bodies (Earth, Sun and others). Large celestial bodies would gravitationally capture dark matter and accumulate it in the core, thus we expect annihilation to occur there [26]. Almost all standard matter created in the annihilation will be absorbed before reaching the surface, the exception being neutrinos. They rarely interact with other matter and would be able to reach the surface and beyond.

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Neutrinos are hard to detect, they can only be detected by interactions with other particles. The neutrino-nucleon cross section increases with neutrino energy. Very high energy neutrinos of 1 PeV have a mean free path equal to the Earth diameter [11], so most neutrinos of lower energy will go straight through Earth. When they do interact they produce charged particles, that in turn can be detected through Cherenkov radiation. Cherenkov radiation is photons that are produced when a charged particle travels faster than light in the medium. The neutrino energy we expect from WIMP annihilation depends on the WIMP mass and what particles are initially produced. The maximum neutrino energy is the same as the WIMP’s mass. So from two 1 TeV WIMPs annihilating we might get neutrinos with hundreds of GeV in energy.

As an example of a neutrino detector we have IceCube, located about 1 km from the geographical South Pole. It consists of optical modules attached to strings that have been deployed in the ice. The modules are completely submerged in ice and located 1.5-2.5 km beneath the surface, see Figure 1.3.

The ice serves as a transparent medium that allows Cherenkov radiation to propagate. Each optical module contains a photo multiplier tube (PMT) that can detect the radiation [11].

Figure 1.2. Diagram of dark matter (DM) interactions with standard matter (SM).

The three main strategies to search for dark matter are shown with arrows: ← indirect detection, ↓ direct detection and → production in colliders. Credit: M. Deliyergiyev, originally intended for [9]

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Figure 1.3. Overview of IceCube Neutrino Observatory. Credit: IceCube collabora- tion

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2. Halo models

2.1 Dark matter density distribution

The galaxy is embedded in a halo of dark matter. There are different models on how exactly the distribution looks, and we will briefly look at three common models.

In 1996 Navarro, Frenk and White made numerical simulations to study the dark matter distributions in a wide array of different galaxies and galaxy clus- ters [21]. The density profiles were fitted giving an ’universal’ average, this became known as Navarro-Frenk-White (NFW) density profile. The density of dark matterρ as a function of distance to the galactic center r is given by

ρ(r) = ρ0 rrs

�1 +rrs2 (2.1)

whereρ0and rsare free parameters that vary between galaxies.

Burkert on the other hand looked at dwarf spiral galaxies, which are domi- nated by dark matter, and derived an empirical density profile [8]. The Burkert density profile is as follows

ρ(r) = ρ0

�1 +rrs��

1 +rr22

s

� (2.2)

whereρ0and rsare free parameters. It has shown to fit better to actual obser- vations of the Milky Way [23].

It has been argued that the dark matter distribution from simulations could be better described by the Einasto density profile instead of the NFW [15, 20, 19]. The Einasto density profile is named after Einasto’s extensive use of the same equation to describe properties of galaxies, such as luminosity and density distribution [10]. It is as follows:

d lnρ(r)

d ln r =−2� r r0

α

i.e. ρ(r) ∝ exp(−Arα) (2.3) with free parameters r0andα.

There is a clear difference between density profiles derived from simula- tions (NFW, Einasto) and those derived from observations (Burkert). Numer- ical simulations yield cusps, where the dark matter density spikes in the cen- ter of a galaxy. Meanwhile observations yield cores, where the density are constant in the inner parts of a galaxy forming a core. This is known as the core-cusp problem [4].

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2.2 Velocity distribution

Compared to the density distribution there has not been much focus on the velocity distribution. The dark matter velocity is important for the capture rate in celestial bodies and direct detection. The most commonly used model is the Maxwell-Boltzmann distribution, that assumes a halo in equilibrium [16, 22].

This might not be a valid assumption since our galaxy is not in equilibrium.

We have a lot of accreted matter from mergers with smaller satellite galax- ies. A new study used observational data from the GAIA mission [13, 14]

cross-matched to Sloan Digital Sky Survey (SDSS) [1]. They divided the dark matter in two groups: old dark matter and new accreted dark matter. Old dark matter that was with our galaxy since the beginning is assumed to have reached equilibrium and has a Maxwell-Boltzmann distribution. The larger group of new accreted dark matter can not be observed itself, but the stars that where accreted at the same time can be. It is assumed that the accreted dark matter follows the velocity of the accreted stars reasonably well [22].

In our solar system the planets and the sun will absorb and scatter dark mat- ter. This will result in a different distribution than in free space. In numerical simulations made by Lundberg and Edsjö they used the Maxwell-Boltzmann distribution as a starting point and looked at the effects of Earth, Jupiter, Sun and Venus. The result was that low velocity (below 70 km/s) dark matter is significantly reduced in our solar system, mostly due to capture by the Sun [18]. However a later study found that the effect was not as big and concluded that the local velocity distribution could be approximated as the one in free space [29].

2.2.1 Velocity and capture rate

For capture by earth, and other planets like it, only the WIMPs of low velocity are important, specifically those below 10-20 km/s [18]. They are otherwise too fast to be captured. Earth’s escape velocity is 11.2 km/s at the surface and 15.0 km/s in the center [18]. Incoming WIMPs also need to interact with standard matter in order to be slowed down and captured, otherwise they will move away with the same velocity. Note that heavier WIMPs means more momentum, thus they need more interactions to be slowed down. Resulting in that WIMPs with high mass need to have lower velocities to be captured (assuming constant WIMP-nucleus cross section).

For the Sun how low the velocity needs to be for capture is strongly depen- dent on the WIMP mass. The Sun’s escape velocity is 617 km/s at its surface, but it is less dense than Earth. Low mass WIMPs are easily captured and can have high velocities. High mass WIMPs on the other hand are harder to slow down and therefore only those of low velocity can be captured. For example a 1 TeV WIMP need to have an initial velocity below 20 km/s to have a chance to get captured, while a 50 GeV WIMP need to be below 400 km/s [24].

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3. Simulation

To see how the velocity distribution affects the flux of neutrinos at a detec- tor on Earth simulations have to be made of the capture and annihilation of WIMPs, but also the resulting propagation and decay of the created particles.

These simulations were made in DarkSUSY, a program written in Gfortran [7].A model with generic WIMPs that are self conjugated (they are their own anti-particle) was used. The annihilation cross section was set to 3 · 10−26 cm3/s and WIMP-nucleus cross section to 1 ·10−7pb. The values of the cross sections were arbitrary chosen, but are below or close to current limits [2, 17].

Five different annihilation channels were simulated, i.e. the particles produced when two WIMPs annihilate. The channels were W, b, νeµ, and ντ with their respective anti-particles. Each channel was simulated with five WIMP masses: 50, 100, 500, 1000 and 3000 GeV.

Furthermore three different velocity distributions were used. Two based on the function:

f (u)

u =

� 3 2π

e−1.5

(u−vsun)2

v23d − e−1.5

(u+vsun)2 v23d

v3dvsun (3.1)

where u is the WIMP velocity and f the velocity distribution.

First the Maxwell-Boltzmann distribution (in DarkSUSY referred to as stan- dard Gaussian distribution), same as (3.1) but with v3d=270 km/s and vsun= 220 km/s.

Second the new velocity distribution based on data from SDSS and Gaia [22], same as (3.1) but with v3d =185 km/s and vsun=248 km/s. To get this the original function was fitted using Scipy curve_fit. This uses a least square method: the Levenberg–Marquardt algorithm.

The third velocity distribution is based on numerical simulation of the solar system made by Johan Lundberg [18]. This is the velocity distribution used by default in DarkSUSY for Earth. It is in the form of a table of data and will henceforth be called DS table. Note that it is only relevant for Earth and will not be used in simulations of the Sun.

See Figure 3.1 and 3.2 for a comparison between the three velocity distri- butions.

Each of the Einasto, Burkert and NWF halo models were also tested, but they gave the same results.

Summing it up we have three different velocity distributions, 5 annihilation channels and 5 different WIMP masses, in total 75 different runs.

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Figure 3.1. Velocity distribution. Black curve is the new velocity distribution de- rived from SDSS-GAIA. Red curve is the fit used for simulations. Blue curve is the Maxwell-Boltzmann distribution. Green curve, DS table, is local velocity distribution for Earth.

Figure 3.2. Velocity distribution for velocities below 25 km/s. Red curve is the new ve- locity distribution, a fit made from data from SDSS-GAIA. Blue curve is the Maxwell- Boltzmann distribution. Green curve, DS table, is local velocity distribution for Earth.

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4. Results

The simulations gave the result in form of muon neutrino flux, sum of both par- ticles and anti-particles, for a detector located at Earth. For graphs on muon neutrino flux, see the Appendix. The difference between Maxwell-Boltzmann and the new velocity distribution (proposed by [22]) can be seen in table 4.1.

For the Sun the difference in annihilation rate and subsequent neutrino flux goes from an increase of 5% for 50 GeV WIMPs to a decrease of 28% for 3 TeV. For Earth the difference is even more apparent. The new velocity dis- tribution (compared to Maxwell-Boltzmann) gives an increased flux of 11%

for 50 GeV, but a decrease of flux by two thirds for higher mass WIMPs. The velocity distribution DS table give an even lower annihilation rate and subse- quent neutrino flux, and reduce it by more than 90% for 500 GeV or heavier WIMPs (compared to Maxwell-Boltzmann velocity distribution).

Mass (GeV) 50 100 500 1000 3000

Sun 5% 6% -5% -10% -28%

Earth 11% -66% -68% -68% -68%

Earth (DS table) -1% -72% -96% -97% -98%

Table 4.1. The difference between Maxwell-Boltzmann and the new velocity distri- bution in annihilation rates and resulting flux. Last is between Maxwell-Boltzmann (free space) and DS table (solar system). A positive value indicate an increase of annihilations compared to the Maxwell-Boltzmann and a negative value a decrease.

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5. Conclusions and discussion

We can conclude that the different velocity distributions tested affect the cap- ture rate of WIMPs and the subsequent neutrino flux at a detector on Earth, which would impact the prospects of detection.

For annihilation rate in the Sun the new velocity distribution could reduce the rate by 28%, but only for the high energy WIMPs of 3 TeV. Otherwise the difference was 5-10%.

For annihilations occurring in Earth the rate can be reduced to a third of the rate from standard Maxwell-Boltzmann distribution. In the simulations this occurs when WIMP mass is 100 GeV or more. Since the local distribution relevant for Earth is different from what we observe in free space it is unclear what the actual effect will be. To find out new numerical simulations of so- lar system could be carried out using the new velocity distribution [22] as a base. The numerical simulations that DS table1is based on [18] show that the total effect of gravitational capture and scattering by Jupiter and the Sun can strongly reduce WIMP annihilations in Earth and the new velocity distribution should reduce it even more; but this is not necessary the case since scattering by the Sun or Jupiter of higher velocity WIMP could do the opposite. A later study found that the total effect of capture and scattering was not that signif- icant but stated that changing the starting velocity distribution could possibly change this [29].

Because of the low resolution of the original distribution, that I based my new velocity fit on, there will be some uncertainties. This is especially true for the low velocities below 25 km/s. Thus the result is not entirely reliable for annihilations in Earth and for high mass WIMPs in the Sun (500 GeV or more).

Suggested first step for future studies is to redo the simulation with better data. One can also do simulations on the effect of the new velocity distribution on the local distribution in our solar system. Finally one could look at what the change of velocity distribution would mean for current search for dark matter.

The result indicate that it would be harder to detect annihilations of heavy WIMPs occurring in the Sun and Earth, since they would be rarer. If one were to compare with current data, to for example reevaluate current limits on crosssection, it would be useful to also get the energy and angle of incoming neutrinos, as well as flux of muons and other neutrino flavors.

1the default velocity distribution for Earth in DarkSUSY [7]

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A. Appendix - Graphs

A.1 Annihilation in the Sun

Figure A.1. Annihilation into b quark and b anti-quark. Resulting flux from the Sun for a detector at Earth’s surface. Flux ofνµ. Blue curve is with the Maxwell-Boltzmann velocity distribution. Red curve is the new velocity distribution.

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Figure A.2. Annihilation into W+ boson and W boson. Resulting flux from the Sun for a detector at Earth’s surface. Flux of νµ. Blue curve is with the Maxwell- Boltzmann velocity distribution. Red curve is the new velocity distribution.

Figure A.3. Annihilation intoνeand anti-νe. Resulting flux from the Sun for a detector at Earth’s surface. Flux ofνµ. Blue curve is with the Maxwell-Boltzmann velocity distribution. Red curve is the new velocity distribution.

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Figure A.4. Annihilation into νµ and anti-νµ. Resulting flux from the Sun for a detector at Earth’s surface. Flux ofνµ. Blue curve is with the Maxwell-Boltzmann velocity distribution. Red curve is the new velocity distribution.

Figure A.5. Annihilation intoντand anti-ντ. Resulting flux from the Sun for a detector at Earth’s surface. Flux ofνµ. Blue curve is with the Maxwell-Boltzmann velocity distribution. Red curve is the new velocity distribution.

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A.2 Annihilation in Earth

Figure A.6. Annihilation into b quark and b anti-quark in Earth. Resulting flux ofνµ

for a detector at Earth’s surface. Blue curve is with the Maxwell-Boltzmann velocity distribution. Red curve is the new velocity distribution. Green curve, DS table, is the local velocity distribution in the solar system.

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Figure A.7. Annihilation into W+boson and Wboson in Earth. Resulting flux ofνµ

for a detector at Earth’s surface. Blue curve is with the Maxwell-Boltzmann velocity distribution. Red curve is the new velocity distribution. Green curve, DS table, is the local velocity distribution in the solar system.

Figure A.8. Annihilation intoνeand anti-νein Earth. Resulting flux ofνµ for a detec- tor at Earth’s surface. Blue curve is with the Maxwell-Boltzmann velocity distribution.

Red curve is the new velocity distribution. Green curve, DS table, is the local velocity distribution in the solar system.

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Figure A.9. Annihilation into νµ and anti-νµ in Earth. Resulting flux ofνµ for a detector at Earth’s surface. Blue curve is with the Maxwell-Boltzmann velocity distri- bution. Red curve is the new velocity distribution. Green curve, DS table, is the local velocity distribution in the solar system.

Figure A.10. Annihilation intoντ and anti-ντ in Earth. Resulting flux ofνµ for a detector at Earth’s surface. Blue curve is with the Maxwell-Boltzmann velocity distri- bution. Red curve is the new velocity distribution. Green curve, DS table, is the local velocity distribution in the solar system.

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