Constraining Magnetic Dipole Dark Matter through

Asteroseismology

Martim Proença da Costa Sousa Branca

Thesis to obtain the Master of Science Degree in

Engineering Physics

Supervisor: Prof. Ilídio Pereira Lopes

Examination Committee

Chairperson: Prof. José Pizarro de Sande e Lemos

Supervisor: Prof. Ilídio Pereira Lopes

Member of the Committee: Prof. Ana Maria Vergueiro Monteiro Cidade Mourão

November 2017

ii

In memory of my grandparents,

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iv

Acknowledgments

I’d like to start by thanking the person without whom this project would have never been possible,

my thesis supervisor, Prof. Ilıdio Lopes, not only for motivating me but also for being a great source of

help throughout the whole process.

I gratefully dedicate this to my family, for being an endless source of support, pushing me along the

years and making me the person I am today.

I also thank Andre Martins for helping me with the numerical codes at the start of the project, as well

as Ana Brito, for helping and clearing some doubts along the way. Last but not least, I thank CENTRA

for providing the necessary tools and support in order to complete this work.

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Resumo

Ao longo das ultimas decadas, a materia escura tem sido um topico de cada vez maior interesse,

especialmente devido a nossa falta de conhecimento sobre o assunto. Na busca de sinais da sua existencia,

varias experiencias tem sido desenvolvidas, apesar de, ate a data, apenas ter sido possıvel obter restricoes

quanto as suas propriedades fundamentais. A identificacao da materia escura poderia potencialmente

significar a confirmacao de certas extensoes ao Modelo Standard.

Nesta dissertacao, conseguimos restringir propriedades da materia escura atraves de estrelas. Em

concreto, analisamos os efeitos de materia escura com dipolo magnetico em duas estrelas mais jovens do

que o Sol, KIC 8379927 e KIC 10454113. E assumido que a nossa galaxia se encontra envolta por um

halo de materia escura, o que faz com que as estrelas capturem as suas partıculas, devido a sua conhecida

interacao gravitacional. Utilizando codigos de evolucao estelar, modelamos estas estrelas e estudamos o

impacto que a materia escura capturada tem nas mesmas. Correndo testes asterosısmicos nas estrelas,

conseguimos limitar as propriedades da materia escura com dipolo magnetico, nomeadamente a sua massa

e momento dipolar magnetico. A nossa analise permitiu obter limites que competem eficazmente com

limites obtidos por experiencias atuais de detecao, conseguindo-se limitar o momento magnetico para

massas na ordem dos 5 GeV, onde os detetores sao menos sensıveis. Atraves deste trabalho, conseguimos

provar que asterosismologia e um metodo valido e eficaz na restricao das propriedades da materia escura,

complementando os contributos de detetores modernos na identificacao deste tipo de materia.

Palavras-chave: Materia Escura, Asterosismologia, Modelacao, Estrelas, Restricoes

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Abstract

Over the last decades, dark matter has been a topic of increasing interest, especially due to our lack

of knowledge about it. Several detection experiments have been developed, attempting to detect signals

of its existence, even though only constraints to its fundamental properties have been obtained to date.

The discovery of the dark matter particle could potentially mean the confirmation of certain Standard

Model extensions.

In this work, we were able to constrain dark matter using stars. In particular, we analysed the effects

of magnetic dipole dark matter in two stars younger than the Sun, KIC 8379927 and KIC 10454113. Our

galaxy is assumed to be surrounded by a dark matter halo, which makes stars capture its particles, due

to their known gravitational interaction. Taking advantage of stellar evolution codes, we were able to

model these stars and study the impact that the captured dark matter particles have on them. Run-

ning asteroseismic tests on the stars, we managed to constrain magnetic dipole dark matter’s properties,

namely its mass and magnetic dipole moment. The analysis performed was successful, obtaining com-

petitive constraints when compared to the ones obtained by current detection experiments. Specifically,

we were able to limit the magnetic moment for masses in the order of 5 GeV, where detectors are less

sensitive. We proved that constraining dark matter through asteroseismology is a valid and effective

method, complementing the efforts by modern detectors to identify the dark matter particle.

Keywords: Dark Matter, Asteroseismology, Modelling, Stars, Constraints

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Dark Matter Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Detection Attempts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Stellar Structure and Evolution 11

2.1 Stars and the Hertzsprung–Russell Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Equations of Stellar Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 CESAM, a Stellar Evolution Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Dark Matter Inside Stars 19

3.1 Magnetic Dipole Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Dark Matter Capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Energy Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Asteroseismic Diagnostics and Dark Matter Constraints 25

4.1 Asteroseismology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Constraining Diagnostic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3.1 KIC 8379927 Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3.2 KIC 10454113 Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.3 Dark Matter Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Conclusions 39

xi

Bibliography 43

xii

List of Tables

4.1 Input observational parameters for KIC 8379927 and KIC 10454113 [67–70], where log g

is the logarithm of the surface gravity, Teff is the effective temperature, Z/X is the

metallicity ratio, and 〈∆ν〉 is the large frequency separation average. . . . . . . . . . . . . 29

4.2 Stellar parameters obtained from the benchmark models of stars KIC 8379927 (A) and

KIC 10454113 (B), where M(M

)is the stellar mass related to the solar mass, Z is the

metallicity, α is the mixing-length parameter, τ is the age of the star, R(R

)is the stellar

radius related to the solar radius, and L(L)

is the stellar luminosity related to the solar

luminosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Parameters used in the computation of DM capture, where ρχ is the DM halo density, v∗

is the stellar velocity, vχ is the DM velocity dispersion, vesc (R∗) is the escape velocity of

the star, and σχχ is the DM self-interaction cross section. . . . . . . . . . . . . . . . . . . 29

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List of Figures

1.1 Superposition of the rotation curves of 21 Sc galaxies. Taken from V. Rubin et al. [3]. . . 3

1.2 Rotation curve of the galaxy NGC 6503 through the fit of a dark matter halo model to

measured data. The dotted, dashed and dash–dotted lines are the contributions of gas,

disk and dark matter, respectively. Edited from K. Begeman et al. [4]. . . . . . . . . . . . 3

1.3 Collision between two clusters inside the Bullet Cluster. In red, we show the hot gas

location according to X-ray imaging, and in blue, the location of the majority of mass in

the cluster, according to gravitational lensing. Credits: X-ray: NASA / CXC / CfA / M.

Markevitch et al. [6]; Lensing map: NASA / STScI; ESO WFI; Magellan / U. Arizona /

D. Clowe et al. [5]; Optical: NASA / STScI; Magellan / U. Arizona / D. Clowe et al. [5]. 4

1.4 Collision between two clusters inside the Bullet Cluster, where the explicit weak gravita-

tional lensing reconstruction of the clusters mass is shown in the green contours (and the

hot gas location through X-rays). White bar represents 200 kpc at the distance of the

cluster. Edited from D. Clowe et al. [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 DM well-motivated candidates along with their limits on the plane of the mass and inter-

action cross section representing the typical strength of interactions with ordinary matter.

The red, pink and blue colors represent hot, warm and cold DM, respectively. Taken from

H. Baer et al. [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Constraints on the WIMP mass and spin-independent (top) or spin-dependent (bottom)

WIMP-nucleon interaction cross section obtained from different detection experiments.

Edited from K. A. Olive et al. [25]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Hertzsprung–Russell diagram showcasing the different groups of stars present in the dia-

gram. Credit: ESO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Observational Hertzsprung–Russell diagram with 22,000 stars plotted from the Hipparcos

Catalogue and 1,000 from the Gliese Catalogue of Nearby Stars. Credit: Richard Powell. . 13

2.3 Evolutionary tracks of stars with masses ranging from 0.5 to 15 solar masses obtained

through CESAM [32]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1 HR diagram showing stars with solar-like oscillation modes detected by Kepler. The solid

black lines represent the evolutionary tracks of stars with different masses. Taken from

Chaplin and Miglio [60]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

xv

4.2 Solar power spectrum measured by BiSON, together with the frequency modes and sepa-

rations illustrated. Taken from Elsworth and Thompson [62]. . . . . . . . . . . . . . . . . 28

4.3 Constraints on MDDM for the stars KIC 8379927 (top) and KIC 10454113 (bottom), with

the χ2r02 diagnostic, showing the 95% (yellow) and 99% (red) confidence level lines (with

11 and 13 degrees of freedom for KIC 8379927 and KIC 10454113, respectively). . . . . . 31

4.4 Constraints on MDDM with no DM self-capture interaction (top) and with a local dark

matter density of ρχ = 0.85 GeV/cm3 (considering DM self-interaction) (bottom), for the

star KIC 8379927, with the χ2r02 diagnostic, showing the 95% (yellow) and 99% (red) CL

lines (with 11 degrees of freedom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.5 Constraints on MDDM with no DM self-capture interaction (top) and with a local dark

matter density of ρχ = 0.85 GeV/cm3 (considering DM self-interaction) (bottom), for the

star KIC 10454113, with the χ2r02 diagnostic, showing the 95% (yellow) and 99% (red) CL

lines (with 13 degrees of freedom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.6 Radius of the DM core related to the stellar radius (top) and DM core’s temperature

(bottom) in function of the MDDM parameter space for KIC 8379927. . . . . . . . . . . . 36

4.7 Radius of the DM core related to the stellar radius (top) and DM core’s temperature

(bottom) in function of the MDDM parameter space for KIC 10454113. . . . . . . . . . . 37

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Glossary

CDM Cold Dark Matter.

CL Confidence Level.

CP Charge-Parity.

DM Dark Matter.

LEP Large Electron-Positron Collider.

LHC Large Hadron Collider.

LTE Local Thermodynamic Equilibrium.

MDDM Magnetic Dipole Dark Matter.

PLATO Planetary Transits and Oscillations of stars.

QCD Quantum Chromodynamics.

SOHO Solar and Heliospheric Observatory.

TESS Transiting Exoplanet Survey Satellite.

WIMP Weakly Interacting Massive Particle.

WMAP Wilkinson Microwave Anisotropy Probe.

ZAMS Zero Age Main Sequence.

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Chapter 1

Introduction

Humanity has always been driven by the search of the unknown. This is why one of the greatest goals

in the past decades and in the next to come is to detect and identify the nature of Dark Matter (DM),

a nonbaryonic type of matter unlike any other seen in our universe. It has never been directly observed,

although there is a good amount of indirect evidence that points towards its existence.

1.1 Motivation

One of the most intriguing questions nowadays is ”What is dark matter?”. In fact, no one really knows

so far. We can only infer through several different tests showing gravitational signatures that there is

some kind of matter present in the universe that we are unable to detect. This is a rather disturbing

statement, especially when considering that this invisible matter makes up to about 84% of the total

matter present in our universe.

This fundamental problem has been gaining traction over the decades, bringing increasingly more

people, teams and institutions into the subject, trying to tackle the problem in different ways. Originated

from particle physics theories attempting to solve problems with the Standard Model, there are several

hypothetical particles which serve as a candidate for dark matter as well, which can help with its detection.

Indeed, along the years, there have been increasingly more experiments with the aim of detecting the

effects of dark matters particles, either directly or indirectly [1].

In this work, instead of using detectors, we will use stars as a laboratory to study DM. Through

asteroseismology, or the study of oscillations in stars, we will be able to set constraints to the properties

of DM, showing that it can be a unique and alternative method to search for dark matter.

1.2 Evidence

There are plenty of arguments that motivate dark matter’s existence. One of them is the problem that

galaxy clusters we see today wouldn’t be able to remain gravitationally bound, having only the luminous

mass we can observe. Fritz Zwicky showed this for the first time in 1933 [2], where he took advantage of

the Virial Theorem, which states that

1

εk = −1

2εp , (1.1)

where εk and εp are the average kinetic and potential energies by unit of mass, respectively. These can

be written as

εk =v2

2(1.2)

and

εp = −3

5GM

R, (1.3)

where v2 is the average done over the different galaxies measured squared Doppler velocity, G is the

gravitational constant, M is the total mass of the cluster, and R is its radius.

By applying this theorem to the Coma galaxy cluster, Zwicky obtained an average velocity that was

far lower than the observed value at the time. This meant that the cluster needed to have a higher mass

than just the luminous matter, some kind of invisible matter, which he named for the first time as dunkle

materie, today known as dark matter.

There is also an anomaly when we analyse the rotation curves of galaxies. Rotation curves are the

circular velocities of the stars and gas in a galaxy as a function of their distance to the center. As Vera

Rubin first observed in her famous 1980 paper [3] and shown in figure 1.1 for a set of 21 galaxies, it

is seen that these rotation curves present a flat behaviour at very large distances from the center, even

further than the edge of the visible luminous disks, instead of decreasing according to the disk velocity,

like one would expect. Therefore, this behaviour cannot be explained only by the luminous matter of

the galaxies. However, models considering the presence of a dark matter halo surrounding the galaxy are

able to produce good fits to the observational data [4], as can be seen in figure 1.2, indicating once more

the presence of DM in our universe.

Gravitational lensing also constitutes good evidence in favor of dark matter. Gravitational lensing

is the deflection of light from background cosmic objects due to the space-time curvature caused by the

presence of matter in-between. Therefore, one can use the observed effects of gravitational lensing to

infer the presence of matter. In figures 1.3 and 1.4, we can observe the collision between two galaxy

clusters inside the Bullet Cluster. In figure 1.3, we are able to see the hot gas of the clusters located

through an X-ray image in red, whereas in blue we have the gravitational lensing reconstruction of where

the majority of the cluster mass should be. In figure 1.4, we show the same situation, with the explicit

weak gravitational lensing reconstruction, as presented in D. Clowe et al. [5].

2

Figure 1.1: Superposition of the rotation curves of 21 Sc galaxies. Taken from V. Rubin et al. [3].

Figure 1.2: Rotation curve of the galaxy NGC 6503 through the fit of a dark matter halo model tomeasured data. The dotted, dashed and dash–dotted lines are the contributions of gas, disk and darkmatter, respectively. Edited from K. Begeman et al. [4].

We see that there is a discrepancy between the hot gas location (that should account for the highest

mass in the cluster) and the zone that contains the most mass, according to gravitational lensing. This

means that the blue zones must contain a much higher mass than the one observed, which is yet another

solid argument proving the existence of dark matter. This separation between the blue and red zones

also disproves alternative gravity theories, as if these were valid, the hot gas location would coincide with

the location that had the highest mass inside the cluster.

3

Figure 1.3: Collision between two clusters inside the Bullet Cluster. In red, we show the hot gas locationaccording to X-ray imaging, and in blue, the location of the majority of mass in the cluster, accordingto gravitational lensing. Credits: X-ray: NASA / CXC / CfA / M. Markevitch et al. [6]; Lensing map:NASA / STScI; ESO WFI; Magellan / U. Arizona / D. Clowe et al. [5]; Optical: NASA / STScI; Magellan/ U. Arizona / D. Clowe et al. [5].

Figure 1.4: Collision between two clusters inside the Bullet Cluster, where the explicit weak gravitationallensing reconstruction of the clusters mass is shown in the green contours (and the hot gas locationthrough X-rays). White bar represents 200 kpc at the distance of the cluster. Edited from D. Clowe etal. [5].

The most accurate determination of the energy density of dark matter in our universe comes from

global fits of cosmological parameters to a wide number of observations. In particular, the most recent

Wilkinson Microwave Anisotropy Probe (WMAP) results [7] show values for the cosmological parame-

ters based on the ”ΛCDM” model: a model accounting for the presence of Cold Dark Matter (CDM)

on a flat universe dominated by a cosmological constant Λ. Different results for the cosmological pa-

4

rameters can be obtained through different combinations of observational data. In this case, we present

the results obtained with the WMAP data combined with observations from the South Pole Telescope,

the Atacama Cosmology Telescope, Supernova Legacy Survey’s three year sample, and measurements

of the baryon acoustic oscillation scale and of the Hubble constant. According to these results, only

a very small part of the universe is composed by baryonic matter, corresponding to a baryon energy

density of Ωb = 0.04601± 0.00091, while the vast majority corresponds to dark energy, having a density

of ΩΛ = 0.7165+0.0093−0.0094. Finally, the result obtained for cold dark matter shows an energy density of

Ωχ = 0.2375± 0.0085.

The amount of research work and importance given around dark matter has been increasing greatly

over the years. With several pieces of evidence motivating its existence plus its high abundance, the

relevance of such a central part of our universe becomes even more apparent.

1.3 Dark Matter Candidates

A crucial part in the study of dark matter is of course to determine which kind of particle actually

constitutes this type of matter. There is a large number of candidates, from which some will be discussed.

In figure 1.5 we show a number of well-motivated candidates for dark matter in the mass vs. interaction

cross section plane, along with their corresponding limits according to the typical strength of interactions

with ordinary matter, with distinction between hot, warm and cold dark matter [8]. Hot dark matter

is a form of dark matter where the DM particles travel at relativistic velocities, such as neutrinos,

while cold dark matter is the opposite, their particles move slowly when compared to the speed of light,

such as WIMPs. Warm dark matter particles have properties that lie in-between those of hot and cold

dark matter, which may include the gravitino hypothetical particle. However, it is expected that DM

particles are non-relativistic, thus cold DM, due to the fact that relativistic particles could not produce

the gravitational wells that allow structure formation.

An interesting candidate that is no longer considered but worth to mention is the neutrino. This

particle was particularly well motivated as a DM candidate since it has actually been detected, while

other candidates are hypothetical particles that have not been observed yet. Following the results obtained

by Planck [9], we see that the masses and relic density of the neutrino are constrained, at 95% Confidence

Level (CL), by:

∑mν < 0.23eV (1.4)

Ωνh2 < 0.0025 , (1.5)

where the sum includes the three neutrino flavours. This value for the neutrino relic density is tremen-

dously smaller than the total dark matter relic density [9], which is Ωχh2 = 0.1198 ± 0.0015, meaning

that neutrinos cannot be the main component of dark matter.

5

Figure 1.5: DM well-motivated candidates along with their limits on the plane of the mass and interactioncross section representing the typical strength of interactions with ordinary matter. The red, pink andblue colors represent hot, warm and cold DM, respectively. Taken from H. Baer et al. [8].

Axions are also a popular candidate, that were introduced as a solution to the strong CP problem

(where CP stands for charge-parity). The strong CP problem is the question of why CP-symmetry does

not seem capable of being broken according to QCD (Quantum Chromodynamics), despite the presence

of CP violation in the standard model. In 1977, Roberto Peccei and Helen Quinn suggested the Peccei-

Quinn solution, which would introduce a pseudo-Nambu-Goldstone boson, to be called an axion [10,11].

Such a particle would not only naturally solve the strong CP problem, but also be a viable candidate for

dark matter [12].

Perhaps the most widely studied candidate corresponds to Weakly Interacting Massive Particles, also

known as WIMPs, which are hypothetical particles only able to interact through gravity and the weak

force, from which the most well-known and motivated type of particle is the neutralino. Neutralinos are

particles that arise from supersymmetry, a greatly motivated extension to the Standard Model, although

still unverified. Assuming R-parity conservation in the supersymmetric theory, the lightest of the four

types of neutralinos is stable and can account for the dark matter that exists in the universe.

In fact, it can be shown that if a GeV-TeV scale particle was thermally produced and has a similar

abundance to the measured density of dark matter, then the thermally averaged annihilation cross section

6

〈σAv〉 must be on the order of 3× 10−26 cm3/s [13]. On the other hand, the obtained value of 〈σAv〉 for

a generic weak-scale interacting particle (such as WIMPs) is 〈σAv〉 ∼ 3× 10−26 cm3/s. This similarity is

quite remarkable, and has led people to often call it as the ”WIMP miracle”, since this basically means

that if a WIMP particle was thermally produced, it could account for the currently measured density of

dark matter. This leads to a great interest and motivation in the study of WIMPs as the main component

of dark matter. There are still other possibilities for DM candidates, such as sterile neutrinos, gravitinos

and WIMPZILLAs [8].

1.4 Detection Attempts

Over the last decades, there have been several experiments with the objective of detecting in some

way the existence of WIMPs in our universe. Nowadays, we have developed different types of detection

mechanisms with that purpose. We have direct detection, characterized by trying to observe nuclei recoils

caused by their interaction with WIMPs via scintillation detectors. We also have indirect detection, in

which we try to detect the WIMP annihilation products, such as neutrinos or gamma rays. Finally,

particle accelerators can also help us in the search of WIMPs, by trying to produce them through particle

collisions.

Regarding direct detection, in 2010, the cumulative exposure of DAMA/NaI and DAMA/LIBRA

experiments suggested the presence of dark matter based on an annual modulation obtained with 8.9σ

confidence level, which was expected to exist due to the relative motion of the Earth around the Sun [14].

This became an exciting yet controversial positive result, since null results from several experiments

including CDMS [15], PandaX-II [16], LUX [17], and XENON1T [18] seem to exclude the results obtained

by DAMA/LIBRA, as well as other positive results from experiments such as CoGeNT [19]. V. Barger et

al. [20] has suggested that certain DM particle models could explain this inconsistency. One of the DM

models suggested is Magnetic Dipole Dark Matter (MDDM), which will be the main focus of this work.

XENON1T, successor of the previous experiment XENON100 [21], is the most sensitive detector to date

searching for WIMPs and has set the most stringent constraints on the spin-independent WIMP-nucleon

interaction cross section for WIMP masses above 10 GeV.

WIMPs can be captured and trapped by massive bodies, such as planets or stars, which will increase

their density and consequently the probability of annihilation. In an attempt to detect the products

of such annihilations, a number of detectors and space telescopes have been developed. The IceCube

Neutrino Observatory attempts to detect neutrinos originated by WIMP annihilation in the Sun, for

example. It has obtained strong constraints for masses above 35 GeV [22]. Other indirect detection

experiments include the Super-Kamiokande [23] and the Fermi Large Area Telescope [24]. In figure 1.6,

we can see the constraints obtained by several different experiments to the spin-independent (top) and

spin-dependent (bottom) WIMP-nucleon interaction cross section [25]. Additionally, particle accelerators,

like the Large Hadron Collider (LHC), can also help us put constraints to dark matter’s properties [26].

7

Figure 1.6: Constraints on the WIMP mass and spin-independent (top) or spin-dependent (bottom)WIMP-nucleon interaction cross section obtained from different detection experiments. Edited from K.A. Olive et al. [25].

8

1.5 Outline

Following this introductory chapter, we continue with chapter 2, where we describe observational

stellar properties and the HR diagram, as well as the stellar structure equations and the stellar evolution

code used in this work. In chapter 3, a description of the behaviour of dark matter inside the star is made.

We start by explaining the properties of the dark matter model we will be using, which will be magnetic

dipole dark matter. We will then describe DM capture, how the number of captured DM particles inside

the star changes over time, and the effect that these DM particles have on the stellar energy transport.

Furthermore, in chapter 4, we explain the concept of asteroseismology, together with the asteroseismic

diagnostics that we will use. We will follow by providing our results, obtaining constraints for MDDM’s

mass and magnetic dipole moment. We will still analyse the core of dark matter particles that exists

inside the stars and the effect it will have in the stellar core. Finally, in chapter 5, we present some

conclusions on the results obtained.

9

10

Chapter 2

Stellar Structure and Evolution

A crucial part of understanding our universe is the study of stars, incredibly complex celestial objects

that exist throughout the entire cosmos. The evolution of stars may vary depending on their size and

mass. However, all stars share one aspect, which is the gentle balance between their own self-gravity

and the radiated energy supplied by the fusion reactions in their interiors. The battle between these two

opposing forces is what allows stars to maintain their size and structure throughout most of their lifetime.

2.1 Stars and the Hertzsprung–Russell Diagram

In order to measure and understand the properties of stars, observational tools and methods have

been developed and improved over the years, ensuring an unprecedented precision in observational tests.

One of the main intrinsic properties of a star is its luminosity L, the total amount of energy radiated per

unit time. We can determine this quantity through the measurement of the apparent brightness B, the

amount of energy radiated by the star per unit time caught by unit area of a telescope, for example. As

the luminosity is the energy radiated per unit time through a spherical surface area at a certain distance

D from the star, an observer at that distance would measure the apparent brightness as [27]

B =L

4πD2. (2.1)

The distance to the star D may be determined through parallax and the stellar luminosity can

consequently be inferred from the measurement of its apparent brightness.

Stars spectra are dominated by black body radiation, in which a complex web of spectral lines is

superimposed. The properties of these lines are strongly dependent of the specific thermodynamic con-

ditions of the atmosphere in these stars. Thus, we define the effective temperature of a star Teff as

the temperature of a black body that would emit the same radiation per surface area LS , which can be

calculated using the Stefan–Boltzmann law:

LS = σT 4eff , (2.2)

11

where σ is the Stefan–Boltzmann constant. Therefore, the total luminosity of a star of radius R can be

given by

L = 4πR2σT 4eff , (2.3)

which means that the effective temperature of a star can be described through the relation

σT 4eff =

L

4πR2. (2.4)

This effective temperature of the star can be considered the same as the temperature of the star’s

outer shell, the photosphere, since this region originates most of the emitted radiation.

Both the luminosity L and the effective temperature Teff can be measured through observational

methods and are essential in the description of a star. This led two astronomers to independently make

an attempt to find an empirical correlation between these two quantities. Indeed, Ejnar Hertzsprung in

1911 and Henry Norris Russell in 1913 plotted observed stars according to their luminosity and effective

temperature, reason why it has become to be called the Hertzsprung–Russell diagram, or HR diagram,

in short, whose axes usually are the luminosity and the decreasing effective temperature, as can be seen

in figure 2.1. This sparked a great deal of interest, since the relationship between the luminosity and

effective temperature of a star could clearly be seen to fall into distinct groups in the plot, instead of

being random as one would first expect. In figure 2.2, we can see an observational HR diagram with

22,000 stars plotted from the Hipparcos Catalogue and 1,000 from the Gliese Catalogue of Nearby Stars.

Nowadays, we understand the physics behind the different groups observed and the separations be-

tween them, allowing us to track the evolution of a star through the diagram and to infer stellar properties

based on a star’s location in it. For instance, most of the stars in the HR diagram fall into a region called

main sequence, in which stars spend roughly 90% of their lifetime, fusing hydrogen into helium in their

cores. On the other hand, when a star falls into the supergiants region of the diagram, one can conclude

that it has already exhausted its hydrogen and is then fusing helium or heavier elements.

The HR diagram has become a landmark in astrophysics and represents a major step in stellar

evolution, for its ability to understand and track critical events and changes through the lifetime of a

star.

12

Figure 2.1: Hertzsprung–Russell diagram showcasing the different groups of stars present in the diagram.Credit: ESO.

Figure 2.2: Observational Hertzsprung–Russell diagram with 22,000 stars plotted from the HipparcosCatalogue and 1,000 from the Gliese Catalogue of Nearby Stars. Credit: Richard Powell.

13

2.2 Equations of Stellar Structure

A description of the interior of stars is possible thanks to stellar structure models, supported by a

set of certain general equations. These equations of stellar structure, as demonstrated in R. Kippenhahn

et al. [28], while following a Lagrangian formalism and considering a non-rotating spherically symmetric

star in hydrostatic equilibrium, are shown to be given by

∂r

∂m=

1

4πr2ρ(2.5)

∂P

∂m= − Gm

4πr4(2.6)

∂l

∂m= εn − εν − cP

∂T

∂t+δ

ρ

∂P

∂t(2.7)

∂T

∂m= − GmT

4πr4P∇ (2.8)

∂Xi

∂t= Ψi −

∂Fi∂m

, 1 6 i 6 nX , (2.9)

where m, t, r, P , l, T and Xi are the mass, time, radius, pressure, luminosity, temperature and the

fraction of a unit mass which consists of nuclei of type i, respectively. We now give a brief description of

each of the equations above.

We take the mass to be the new space-like independent variable, rather than the radius. This means

that the radius and other properties are now dependent on the mass m, which is defined as the total mass

contained within a sphere of radius r. The variation of this radius with the mass is shown in equation

(2.5), where ρ is the density.

Secondly, equation (2.6) accounts for the hydrostatic equilibrium inside the star, which consists on the

equilibrium between the self-gravity of the star and the outward pressure caused by the nuclear reactions

in the stellar core. We assume a quasi-static process, since most stars present evolutionary phases with

a duration long enough such that changes in the gravity or pressure may be considered to be negligible,

allowing the star to maintain hydrostatic equilibrium. It should be noted that this equation would have

the additional term −∂2r/∂t2(4πr2)−1 if the assumption of hydrostatic equilibrium is not satisfied.

In equation (2.7) we have the energy equation, where εn is the energy released per unit mass per

second through thermonuclear reactions, εν is the energy lost per unit mass per second from the stellar

material in the form of neutrinos, cP is the specific heat at constant pressure and δ is defined by

δ = −(∂ ln ρ

∂ lnT

)P

. (2.10)

14

The thermonuclear reactions accounted by the term εn occur in stars mainly through the proton-

proton chain reaction and the CNO cycle, two different types of chain processes that are able to convert

hydrogen into helium, among other products. Also, the last two terms in equation (2.7) account for

internal energy changes over the star’s lifetime.

The energy transport equation is shown in equation (2.8), where the temperature gradient ∇ is defined

by

∇ =d lnT

d lnP. (2.11)

This energy transport can occur through radiative transfer, convection and thermal conduction, al-

though the latter is only relevant in the interior of very dense stars as well as in white dwarfs, being

negligible for the Sun and solar-like stars. Assuming that the energy is transported by radiation only

within a certain region, the temperature gradient ∇ can be shown to be given by

∇ = ∇rad =3

16πacG

κlP

mT 4, (2.12)

where c is the velocity of light, a is the radiation density constant and κ is the Rosseland mean of the

opacity. We can test a layer’s stability and therefore the presence of convection by using the Schwarzschild

criterion for dynamical stability, in which

∇rad < ∇ad , (2.13)

where ∇ad is the adiabatic temperature gradient, given by

∇ad =P δ

Tρ cP. (2.14)

If the Schwarzschild criterion is satisfied, a small spatial upwards perturbation of a mass element in

the star would be quenched, resulting in a stable situation where there are no convection mechanisms.

However, if the criterion is not fulfilled, such a perturbation would result in an adiabatic expansion of

the mass element that would not be stabilized due to the net buoyancy force upwards, resulting in a

dynamically unstable region where there is energy transfer between hotter and cooler layers of the star

through the exchange of macroscopic mass elements, a process which goes by the name of convection.

In the case of convection, the temperature gradient ∇ in equation (2.8) can take the form of ∇ad in the

deep interior of stars, whereas in the outer layers, this gradient is determined through a proper theory of

convection, such as the mixing-length theory [29].

Our final equation of stellar structure represents the variation of the chemical composition of the star

with time, which is shown in equation (2.9) [30]. As already stated, Xi is defined as the fraction of a unit

mass which consists of nuclei of type i, given by

15

Figure 2.3: Evolutionary tracks of stars with masses ranging from 0.5 to 15 solar masses obtained throughCESAM [32].

Xi =miniρ

, (2.15)

where mi and ni are the mass and number density of the nuclei, respectively. Furthermore, nX is the

number of chemical species, Fi is the flow of Xi due to diffusion and Ψi is the rate of change of Xi caused

by thermonuclear reactions, which can be written in the form

Ψi =mi

ρ

∑j

rji −∑k

rik

, 1 6 i 6 nX , (2.16)

where rji are the reaction rates of reactions which create the element i, whereas rik are the reaction rates

of reactions which destroy it, from interactions with the different elements in the star. In purely radiative

regions, ∂Xi/∂t = Ψi, as the only relevant interactions will consist of nuclear reactions.

Taking advantage of measured stellar properties, stellar models can be numerically evolved through

equations (2.5-2.9) and the equation of state, while setting appropriate boundary conditions. The par-

ticular code we will be using in this work for stellar evolution will be CESAM, described next.

2.3 CESAM, a Stellar Evolution Code

Originally created and developed by P. Morel, CESAM [30,31] is a code consisting on a set of programs

and routines that are able to compute one-dimensional quasi-hydrostatic stellar evolution models. The

16

differential equations are solved through the treatment of the physical unknowns as piecewise polynomials

projected on a local linear basis of normalised B-splines. This method allows stable, efficient and robust

calculations, while also being able to find a numerical solution at any location, even for discontinuous

variables such as the stellar density profile, among other advantages. In figure 2.3, we see the evolutionary

tracks of several stars with masses ranging from 0.5 to 15 solar masses, computed using the CESAM

code [32].

The fact that CESAM is structured by modules greatly facilitates the deployment and customization

of the different physical processes. In our work, the nuclear reaction rates are computed according

to NACRE compilations [33] and the Rosseland mean of the opacity according to OPAL 1996 opacity

tables [34]. Additionally, the diffusion coefficients are calculated based on the formalism developed in

the work of Michaud and Proffitt [35], the equation of state is determined according to the OPAL 2001

tables [36], while the mixing-length theory [29] is used to compute the convection mechanisms inside the

star. This type of structure also allows for facilitated future improvements and extensions to the code.

17

18

Chapter 3

Dark Matter Inside Stars

Our whole galaxy and the stars inside it are presumed to be surrounded by a dark matter halo. Due

to gravitational interaction, these DM particles will inevitably be attracted to stars, which will cause

a change in the behaviour and properties of those stars. The analysis of these changes of behaviour

will ultimately allow us to test the impact of DM in the star’s evolution. This reveals the importance

of studying DM effects in stellar interiors, which goes through how the population of DM particles Nχ

changes inside a star and the way those particles will affect the energy transport mechanisms of the star.

The description of capture and transport in this work follows the one taken in I. Lopes et al. [37], based

on the work of Spergel and Press [38]. We will start by describing the properties of the type of dark

matter we will be treating, which is magnetic dipole dark matter, followed by a description of DM capture

and energy transport by the star.

3.1 Magnetic Dipole Dark Matter

In this work, we will consider the existence of dipole moments in dark matter. Motivated by the fact

that many elementary particles have an intrinsic magnetic dipole moment but no observational evidence

of a non-zero electric dipole moment (as the electron, for example), the existence of magnetic dipole dark

matter is very plausible. Additionally, V. Barger et al. [20] has shown that a dark matter particle with a

magnetic dipole moment could explain the discrepancies between direct detection results and successfully

explain the positive results obtained by DAMA/LIBRA and CoGeNT. One should be aware that these

dipole moments, if existent, must be very weak, since the coupling of dark matter with the photon is

considered to be either non-existent or extremely weak. However, it is shown in I. Lopes et al. [37] that

this type of matter can make a significant impact in the evolution of the star and help to effectively

constrain dark matter properties, our ultimate goal.

Since we will be treating magnetic dipole dark matter, it is important to understand the properties

of this type of DM. We have that the interaction of a dark matter particle χ with a magnetic dipole

moment µχ, with an electromagnetic field Fµν is given by [37]

19

L =1

2µχχσµνF

µνχ , (3.1)

where σµν = i2 (γµγν − γνγµ), with γµ being the Dirac matrices. Additionally, this interaction vanishes

for Majorana fermions, which are particles that are their own anti-particles. Therefore, MDDM particles

can only consist of Dirac fermions. We are also able to determine the differential cross section for the

scattering between the dark matter particle and a nucleus, via an electromagnetic interaction, which can

be written as

dσ

dER=e2µ2

χ

4πv2r

Sχ + 1

3Sχ

[Z2

(v2

ER− 1

2mN− 1

mχ

)|GE(ER)|2 +

SN + 1

3SN

(µN

e/(2mp)

)2mN

m2p

|GM (ER)|2],

(3.2)

where ER = q2

2mNis the recoil energy for momentum transfer q, with mN being the nucleus mass, Sχ is

the DM particle spin, vr is the incoming velocity of the DM particle in the lab. frame, Ze is the nucleus

charge, mχ is the dark matter particle mass, SN is the nucleus spin, mp is the proton mass, µN is the

nucleus magnetic dipole moment, and GE(ER) and GM (ER), normalized to GE,M (0) = 1, are the electric

and magnetic form factors, respectively.

The relation above describes the interactions between the MDDM particles and the nuclei inside the

star and will be dependent on Z, which will differ according to the different elements in the star. As we

are treating solar-like stars, we know that even though there are many different elements inside the star,

metals will mostly account for a very small part of the star’s mass, the majority being hydrogen and

helium. The amount of helium can increase to considerable values, but mostly in the core of the star,

and since it has an atomic number of Z = 2, we can consider only the interaction of the dark matter with

hydrogen, to a good approximation, as it always remains at a higher abundance than the other elements.

The total interaction cross section of MDDM with hydrogen can then be written as a sum of the

spin-dependent and spin-independent cross sections [39]:

σχp = σSDχp + σSIχ p , (3.3)

where

σSDχp =µ2χe

2

2π

(µp

e/(2mp)

)2m2r

m2p

(3.4)

and

σSIχ p =µ2χe

2

2π

(1− m2

r

2m2p

− m2r

mpmχ

), (3.5)

with µp being the magnetic dipole moment of the proton.

20

3.2 Dark Matter Capture

In order to describe DM accumulation in stars, we must calculate the number of DM particles inside

a star, which, as reviewed by Petraki and Volkas [40], changes according to the following differential

equation:

dNχ(t)

dt= CC + (CS − CE − CCA)Nχ(t)− CSAN2

χ(t) , (3.6)

where CC is the capture rate due to DM-nucleon collisions, while CS Nχ(t) is the capture rate due to DM

self-interaction. This happens when a DM particle enters the star and scatters with a nucleus inside the

star, or with another already captured DM particle in the case of self-interaction, losing enough energy

for its velocity to become smaller than the star’s escape velocity, effectively becoming gravitationally

bound to the star.

There can also be DM self-annihilation inside the star, which is described by the term CSAN2χ(t).

In this work we will assume that dark matter possesses an asymmetry, which will have a significant

impact on this term. There are several models that suggest explanations for the existence of asymmetric

dark matter, such as mirror dark matter [41], darkogenesis [42], and pangenesis [43]. Furthermore, by

assuming a dark matter asymmetry, we can neglect CSA, since the DM particles inside the star will

be much more abundant than the anti-particles, hence making a bigger impact in the star’s evolution.

CCANχ(t) accounts for the potential co-annihilation of the DM particles with nucleons, which we will

neglect in this work.

Finally, CE is the evaporation rate. DM evaporation happens when collisions between light DM

particles and nuclei inside the star cause those light DM particles to escape the star’s gravitational field,

significantly reducing the DM impact on the stellar core. However, this effect is only relevant when

mχ 6 4 GeV [44], and can therefore be considered negligible for higher masses.

As such, for our work, the only non-negligible terms that will remain on the right-hand side of equation

(3.6) will be CC and CS Nχ(t), which will indeed be the terms responsible for capture of DM by the star

and will therefore maximize its impact on the star’s evolution. The number of DM particles inside the

star will then change according to the relation

Nχ(t) =CCCS

(eCSt − 1

), (3.7)

where the capture rate due to DM-nucleon collisions CC can be described by [40,45]

CC =√

6πρχ

mχvχ

(2GM∗R∗

1− 2GM∗R∗

)min

[1,σχnσsat

], (3.8)

in whichG is the gravitational constant, M∗ and R∗ are the star’s mass and radius, mχ is the DM particle’s

mass, ρχ and vχ are the density and average velocity of the DM particles in the star’s surrounding DM

halo, σχn is the cross section between the DM particles and nucleons inside the star, and σsat is the

saturation cross section. The last factor of equation (3.8) incorporates the effect of saturation, which

21

happens when all the incident DM particles are captured, if the saturation cross section is reached. This

cross section is given by

σsat =R2∗

0.45Nn min[1,

mχ0.2 GeV

] , (3.9)

where the denominator accounts for the possibility of nucleon degeneracy leading to a lower capture rate,

with Nn being the number of nucleons inside the star.

On the other hand, the capture rate due to DM self-interaction CSNχ(t) can be written as

CS Nχ(t) =

√3

2

ρχmχvχ

⟨v2esc(r)

v2esc(R∗)

⟩2GM∗R∗

(erf η

η

)min

[σχχNχ, πr

2χ

], (3.10)

with σχχ being the DM self-interaction cross section. The last factor in equation (3.10) represents the

geometric limit that is reached when the DM self-capture saturates. This happens when all DM particles

are captured when incident to the DM core with radius rχ, which is the region inside the star where

the DM distribution is concentrated. When this point is reached, the DM self-capture rate will become

constant and the number of DM particles in the star will start growing linearly, instead of exponentially,

as can be verified by equation (3.6). In this work we will assume σχχ = 10−24 cm2, based on the fact

that the most stringent constraints of this cross section indicate that σχχ < 2×10−24 cm2 through Bullet

Cluster observations [46,47]. On the other hand, η is given by

η =3 v2∗

2 v2χ

, (3.11)

where v∗ is the stellar velocity. The average in equation (3.10) is done over the star’s DM distribution,

which is concentrated within the radius rχ R∗. This radius [40,48] is described by

rχ =

√9Tc

4πGρcmχ, (3.12)

with Tc and ρc being the temperature and density of the stellar core, respectively.

3.3 Energy Transport

We now need to analyse the energy transport inside the star, which will be done through the calculation

of the dark matter density and luminosity as a function of the stellar radius. There are two distinct regimes

where the properties of energy transport differ from each other. These two regimes can be differentiated

through the Knudsen number, K, defined by K ≡ lχ/rχ, where lχ is the DM particle’s mean free path

and rχ is the dark matter distribution’s radius, as defined in equation (3.12).

In stellar regions where lχ rχ, or equivalently K 1, the dark matter scattering cross sections

with the star’s nuclei will be large, which will make collisions much more frequent, and therefore, the

mean free path will be much smaller than the radius of the DM core in the star. This will result in a local

22

energy transport predominated by conduction, causing the DM particles to be in Local Thermodynamic

Equilibrium (LTE) with the star’s nuclei, reason why this type of transport is often called LTE transport.

In this regime, the dark matter number density can be shown to be described by [49,50]

nχ,LTE(r) = nχ(0)

[T (r)

T (0)

]− 32

exp

[−∫ r

0

dr′kBα(r′)dT (r′)

dr′ +mχdφ(r′)dr′

kBT (r′)

], (3.13)

where T (r) is the stellar temperature, α is the thermal diffusivity coefficient and φ(r) is the gravitational

potential. Additionally, the energy carried by DM scattering can be written as

Lχ,LTE(r) = 4πr2κ(r)nχ(r)lχ(r)

[kBT (r)

mχ

] 12

kBdT

dr, (3.14)

where κ is the dimensionless thermal conductivity and lχ is the DM mean free path, described by

lχ(r) =

[∑i

σini(r)

]−1

, (3.15)

with ni(r) being the local number density of the ith species of nuclei and σi its total interaction cross

section with the elementary particles.

The quantities α and κ can be determined for a single nuclear species by the tabulated values presented

in Gould and Raffelt [50]. The total value of those quantities for a mixture of various nuclear species can

then be generalized to be

α(r, t) = lχ(r, t)∑i

σini(r, t)αi(µ) (3.16)

and

κ(r, t) =

[lχ(r, t)

∑i

σini(r, t)

κi(µ)

]−1

. (3.17)

We also have the regime where lχ rχ, or K 1, often called Knudsen regime, where the scattering

cross sections will be smaller and the mean free path larger, making the energy transport non-local, since

two consecutive collisions will be farther apart from each other. We also know that in the star’s core, the

nuclei temperature is higher than the DM particles temperature, which means that collisions will cause

a transfer of energy from the nuclei to the DM particles. Since this regime causes non-local transport,

many of the DM particles will only give this energy back far from the star’s core, which becomes an

effective method of energy evacuation from the center of the star.

The DM number density and the energy carried by DM scattering in this regime can be calculated

through an empirical correction to the correspondent LTE transport expressions. Indeed, these can be

written as [49,51]

nχ(r) = f(K)nχ,LTE + [1− f(K)]nχ,iso (3.18)

23

and

Lχ(r, t) = f(K)h(r, t)Lχ,LTE , (3.19)

where nχ,LTE and Lχ,LTE correspond to equations (3.13) and (3.14), respectively. The parameter K is

the already discussed Knudsen number and nχ,iso is given by

nχ,iso(r) = N(t)e− r2

r2χ

π32 r3χ

, (3.20)

with N(t) being the total number of DM particles. Finally, f(K) and h(r, t) are described by

f(K) =1

1 +(KK0

) 1τ

(3.21)

and

h(r, t) =

(r − rχrχ

)3

+ 1 , (3.22)

where K0 = 0.4 and τ = 0.5 through empirical determination [52]. Furthermore, the transported energy

rate per unit mass is given by

εχ(r) =1

4πr2ρ(r)

dLχ(r)

dr, (3.23)

with ρ(r) being the stellar radial density profile.

24

Chapter 4

Asteroseismic Diagnostics and Dark

Matter Constraints

Analysing oscillating frequencies in stars throughout the universe plays a major role in this work, as

we will see next. As such, we will start this chapter by describing stellar oscillations, followed by the

explanation of asteroseismic diagnostics used in our analysis, including a brief description of the numerical

codes used. Finally, we present the results obtained, effectively constraining magnetic dipole dark matter

through asteroseismology using two different stars, while also analysing the behaviour of the dark matter

cores inside the stars.

4.1 Asteroseismology

Over the last years, we have detected stellar oscillations in many stars, which have been incredibly

useful to probe stellar interiors. These oscillations primarily consist on two types of standing waves:

p-modes, which are acoustic waves driven by pressure as a restoring force for a star perturbed from

equilibrium; and g-modes, or gravity modes, where buoyancy is the restoring force instead. Stellar

oscillations occur due to one of two different driving mechanisms that will feed energy into the pulsation.

We have the heat-engine mechanism, when a radial layer of the star is able to drive the pulsation through

heat gained during the oscillation’s compression, converting thermal energy into mechanical energy. This

driving mechanism is directly related with the opacity, since when the opacity of a stellar layer increases

sufficiently high, the layer will be able to trap radiation, resulting in an increase of its temperature and

expansion. This is the reason why it is also called κ mechanism, being responsible for oscillations in most

pulsating variable stars, such as classical Cepheids, showing changes in their luminosity much higher than

in the Sun, for example. However, when this heat-engine mechanism cannot drive the oscillations, as

for the Sun and solar-like stars, we have stochastic driving, where the stochastic noise from near-surface

convection zones turbulence is able to excite some of the star’s natural oscillation frequencies [53].

The study of the propagation of these wave oscillations existent in stars, called asteroseismology, is

of critical importance, since it can give valuable information about the star’s internal layers, such as the

25

Figure 4.1: HR diagram showing stars with solar-like oscillation modes detected by Kepler. The solidblack lines represent the evolutionary tracks of stars with different masses. Taken from Chaplin andMiglio [60].

sound speed profile inside the star, or the determination of stars properties, like their mass and radii [54],

being able to improve previous values. Furthermore, it can also be extremely helpful in the study of dark

matter [37,55,56]. The study of these oscillations specifically in the Sun is called helioseismology, and has

led to a description of its internal layers with remarkable precision, with thousands of oscillation modes

detected by the SOHO (Solar and Heliospheric Observatory) spacecraft [57]. Additionally, the CoRoT

spacecraft [58] by ESA and NASA’s spacecraft Kepler [59] have detected oscillation modes in several

stars, enabling a much better description of these stars. In particular, Kepler has detected oscillation

modes in over 500 different main-sequence solar-like stars, among a rough total of 20,000 oscillating stars,

which are plotted in figure 4.1 in an HR diagram [60].

The detectable modes in the Sun and in solar-like stars consist mainly in p-modes, which have high

radial order n, making an asymptotic analysis possible in the limit l n, where l is the degree of the

mode. Indeed, an approximate expression for the oscillation frequencies νn,l can be written to second

26

order as [53,60,61]

νn,l ' ∆ν

(n+

l

2+ ε

)− ∆ν2

νn,l

(A [l (l + 1)]− δ

), (4.1)

where

∆ν =

(2

∫ R

0

dr

c

)−1

(4.2)

is the inverse of twice the time sound would take to travel from the centre to the surface of the star,

R is the stellar radius, c is the sound speed, ε and δ are terms that depend on the surface boundary

conditions, and the term A is given by

A =(4π2∆ν

)−1

[c(R)

c−∫ R

0

dc

dr

dr

r

]. (4.3)

We now define the large frequency separation ∆νn,l as the separation between two adjacent oscillation

modes with the same l, written as

∆νn,l = νn,l − νn−1,l ' ∆ν . (4.4)

The last term on the right-hand side of equation (4.1) lifts the degeneracy caused by the first term,

which by itself would imply νn,l ' νn−1,l+2. We then define the small frequency separation as

δνn,l = νn,l − νn−1,l+2 ' −(4l + 6)∆ν

4π2νn,l

∫ R

0

dc

dr

dr

r. (4.5)

Satellites have the ability to detect periodic oscillations in stars through the measurement of small

changes in their luminosity over an extended period of time. Using Fourier transforms, one is then

able to resolve these periodic changes of luminosity over time into a power spectrum of the individual

frequencies of each mode. With these measurements, it becomes possible to determine the oscillation

modes frequencies of a star and, consequently, different frequency separations as the ones discussed

above. This can be seen in figure 4.2, where we show the solar power spectrum measured by BiSON

(Birmingham Solar Oscillations Network), with the frequency separations illustrated [62].

Due to gravitational interaction, dark matter particles will accumulate more heavily in the star’s core

rather than in the outer layers. Therefore, to search for dark matter signatures it’s better to use methods

that are more sensitive to the star’s core. Even though the large and small separation are two very useful

diagnostics in asteroseismology, Roxburgh and Vorontsov [63] concluded that the ratio of small to large

separations rij would be a diagnostic much more sensitive to the star’s core, since it is independent of

the structure of the star’s outer layers. In our work, we will be using the ratio r02, given by

r02(n) =d02(n)

∆1(n), (4.6)

27

Figure 4.2: Solar power spectrum measured by BiSON, together with the frequency modes and separationsillustrated. Taken from Elsworth and Thompson [62].

where

d02(n) = νn,0 − νn−1,2 (4.7)

and

∆1(n) = νn,1 − νn−1,1 , (4.8)

which will help us in the asteroseismic tests done next.

4.2 Constraining Diagnostic

The main objective of this work is to be able to constrain dark matter properties. In order to do this,

we start by taking advantage of the stellar evolution code dmp2k [64], an adaptation of CESAM [31] that

accounts for dark matter effects, which will compute a grid of several stellar models with varying mixing-

length parameters, stellar masses and metallicities, while maintaining all measured quantities within

their observational error. The mixing-length parameter α is used to describe the effects of convection

in the star through the mixing-length theory [29]. CESAM will evolve a star from the Zero Age Main

Sequence (ZAMS), the moment when the star reaches a sufficiently high enough temperature to start

28

Star log g(

cm/s2)

Teff (K) Z/X 〈∆ν〉 (µHz)

KIC 8379927 (A) 4.5± 0.4 6050± 250 0.020± 0.010 120± 5

KIC 10454113 (B) 4.3± 0.2 6200± 200 0.026± 0.006 105± 4

Table 4.1: Input observational parameters for KIC 8379927 and KIC 10454113 [67–70], where log g is thelogarithm of the surface gravity, Teff is the effective temperature, Z/X is the metallicity ratio, and 〈∆ν〉is the large frequency separation average.

Star M(M

)Z α τ (Gyr) R

(R

)Teff (K) L

(L)

Z/X log g(

cm/s2)〈∆ν〉 (µHz)

A 1.12 0.0180 1.80 1.77 1.12 6157 1.62 0.0236 4.39 120.7

B 1.19 0.0180 2.00 1.96 1.23 6399 2.29 0.0221 4.33 107.7

Table 4.2: Stellar parameters obtained from the benchmark models of stars KIC 8379927 (A) and KIC10454113 (B), where M

(M

)is the stellar mass related to the solar mass, Z is the metallicity, α is the

mixing-length parameter, τ is the age of the star, R(R

)is the stellar radius related to the solar radius,

and L(L)

is the stellar luminosity related to the solar luminosity.

ρχ

(GeV/cm

3)

v∗ (km/s) vχ (km/s) vesc (R∗) (km/s) σχχ(cm2

)0.38 220 270 618 10−24

Table 4.3: Parameters used in the computation of DM capture, where ρχ is the DM halo density, v∗ isthe stellar velocity, vχ is the DM velocity dispersion, vesc (R∗) is the escape velocity of the star, and σχχis the DM self-interaction cross section.

having fusion reactions in its core, burning hydrogen into helium, until its present age. This evolution

will be mediated by observational parameter constraints, namely the logarithm of the surface gravity

log g, the effective temperature Teff , the metallicity ratio Z/X, where Z is the metals content and X the

hydrogen content inside the star, and also the large frequency separation average 〈∆ν〉. These models

do not consider dark matter yet. Additionally, we also use ADIPLS [65], an oscillation frequencies and

separations computing code, which will compute the normal oscillation frequencies for each of the stellar

models obtained, allowing us to determine which stellar model is the closest to reality, making it our

benchmark model.

Our asteroseismic analysis will be based on the statistical quantity χ2r02 , defined by

χ2r02 =

∑n

(robs02 (n)− rmod02 (n)

σrobs02(n)

)2

, (4.9)

where σrobs02is the observational error of robs02 , already described in equation (4.6), which, as already

explained, is a particularly useful diagnostic for the presence of DM in a star, since it is especially

29

sensitive to the stellar core, where DM effects are more significant. This quantity represents the sum

over the different modes of the error between the observed r02 ratio values and the modelled ones. By

running this diagnostic through a certain DM parameter space within our stellar benchmark model, we

will be able to exclude DM models that cause an incompatibility between our models and reality to a

certain confidence level, effectively obtaining constraints on DM properties, as we will see next.

4.3 Results

We were then able to run the codes mentioned before and model two different solar-like stars, as well

as constrain the properties of magnetic dipole dark matter.

In this work, two stars were analysed: KIC 8379927 and KIC 10454113, both observed by the Kepler

spacecraft (KIC meaning Kepler Input Catalog). Each of these stars is slightly more massive than

the Sun, showing a high precision in the determination of both the observational parameters and the

detected oscillation modes. A great number of oscillation frequency modes has been observed in either

star, presented in T. Appourchaux et al. [66], which is important in order to obtain realistic values of

the asteroseismic diagnostics. In table 4.1 we show the observational constraints chosen for the models

of both stars analysed in this work [67–70]. The benchmark models are shown in table 4.2. Additionally,

based on other works studying the DM impact on the Sun or solar-like stars [37, 55, 64], we show the

numerical values for parameters used in the computation of DM capture in table 4.3.

4.3.1 KIC 8379927 Star

From table 4.2, we can see that the optimal model obtained for the star KIC 8379927 presents a mass

of M = 1.12M, an age of τ = 1.77 Gyr and a radius of R = 1.12R, being more massive and bigger

in size than the Sun, while also being a much younger star, since the Sun has an age of τ = 4.57 Gyr.

Furthermore, this star is a good subject to run asteroseismic tests, considering that it possesses 11 distinct

measured values of the r02 ratio, compared to the Sun’s 17 observable ones.

We can now run the χ2r02 diagnostic through the MDDM-influenced stellar models with DM masses

between 5 GeV 6 mχ 6 15 GeV and DM magnetic dipole moments between 10−17 e · cm 6 µχ 6

2×10−15 e · cm. Calculating χ2r02 for each of the models, we were able to obtain the contour plot for KIC

8379927 shown in figure 4.3 (top).

Our χ2r02 analysis points to the exclusion of MDDM particles with low masses and magnetic dipole

moment surrounding µχ ' 10−16 e · cm, since there is a large disagreement between our modelled results

and measured data in this region. This incompatibility between models and observational data allows

certain MDDM parameters to be excluded up to a determined certainty.

In particular, for an MDDM particle mass of mχ ' 5 GeV, we can exclude candidates with a magnetic

dipole moment within 3.5×10−17 e ·cm 6 µχ 6 3.9×10−16 e ·cm with a 95% confidence level and within

3.9× 10−17 e · cm 6 µχ 6 3.2× 10−16 e · cm with a 99% confidence level.

Additionally, we can also perform the same χ2r02 tests as the ones done in figure 4.3 but without

considering the effect of DM self-interaction in the capture of DM by the star. This effect contributes to

30

Figure 4.3: Constraints on MDDM for the stars KIC 8379927 (top) and KIC 10454113 (bottom), with theχ2r02 diagnostic, showing the 95% (yellow) and 99% (red) confidence level lines (with 11 and 13 degrees

of freedom for KIC 8379927 and KIC 10454113, respectively).

31

Figure 4.4: Constraints on MDDM with no DM self-capture interaction (top) and with a local dark matterdensity of ρχ = 0.85 GeV/cm3 (considering DM self-interaction) (bottom), for the star KIC 8379927, withthe χ2

r02 diagnostic, showing the 95% (yellow) and 99% (red) CL lines (with 11 degrees of freedom).

32

Figure 4.5: Constraints on MDDM with no DM self-capture interaction (top) and with a local dark matterdensity of ρχ = 0.85 GeV/cm3 (considering DM self-interaction) (bottom), for the star KIC 10454113,with the χ2

r02 diagnostic, showing the 95% (yellow) and 99% (red) CL lines (with 13 degrees of freedom).

33

the capture of DM particles as can be seen by equation (3.6), and as such, neglecting this term is expected

to result in a lower DM impact in the star. However, for our case, by observing figure 4.4 (top), we can

verify that no relevant difference is seen in the constraints when compared to the case considering DM

self-interaction, shown in figure 4.3 (top). This may be explained by the fact that this is a rather young

star, with less than half the age of the Sun, and hasn’t been able to capture a large enough amount of dark

matter particles in its short lifespan. Since the capture rate due to DM self-interaction is proportional

to the number of dark matter particles inside the star, we can see the possible reason why it seems to be

negligible at this age.

Although the currently standard value for the local dark matter density used in the literature is

ρχ = 0.38 GeV/cm3, S. Garbari et al. [71] suggests a value of ρχ = 0.85+0.57−0.50 GeV/cm3, based on the

kinematics of K dwarf stars. Therefore, we tested our models (considering DM self-interaction) setting

the local dark matter density value to ρχ = 0.85 GeV/cm3 instead of the value used in the previous tests

ρχ = 0.38 GeV/cm3. Increasing the dark matter density will cause stars to capture a higher number of

DM particles, increasing their impact on the star. We present the obtained results in figure 4.4 (bottom),

showing the exclusion of mχ ' 5 MDDM particles with magnetic moment between 2.7× 10−17 e · cm 6

µχ 6 5.8× 10−16 e · cm with 95% CL and between 3.0× 10−17 e · cm 6 µχ 6 4.9× 10−16 e · cm with 99%

CL. As expected, when compared to figure 4.3 (top), we verify that stronger constraints were obtained

with this higher value of local dark matter density.

4.3.2 KIC 10454113 Star

On the other hand, the obtained benchmark model for KIC 10454113 shows a mass of M = 1.19M,

an age of τ = 1.96 Gyr and a radius of R = 1.23R. We see that this star is slightly more massive,

bigger in size and older than KIC 8379927, while also having a substantial amount of 13 measured r02

ratios.

Performing the χ2r02 test for the star KIC 10454113, we can observe by figure 4.3 (bottom) that for

MDDM masses of mχ ' 5 GeV, it allows the exclusion of models with magnetic dipole moment between

2.9× 10−17 e · cm 6 µχ 6 5.9× 10−16 e · cm with a 95% confidence level and within 3.9× 10−17 e · cm 6

µχ 6 4.1× 10−16 e · cm with a 99% confidence level.

Even though KIC 10454113 yields lower values of χ2r02 below the exclusion lines, we see that not only

does it produce stronger constraints on MDDM but it also has higher values of χ2r02 in the rest of the

parameter space, which evidences an overall greater DM impact in the evolution of this star. This effect

can be explained due to the fact that this star is older than KIC 8379927, and since DM capture will

produce a cumulative effect over time, older stars will experience a stronger DM impact. Indeed, this

star does provide stronger constraints than KIC 8379927, both in terms of magnetic dipole moment as

in DM mass. We see that at 95% CL, constraints can be obtained up to nearly mχ ' 8.5 GeV, whereas

in KIC 8379927 these can only be observed up until mχ ' 7 GeV.

Similarly to what was done with KIC 8379927, we tested our χ2r02 analysis without the effect of DM

self-interaction for KIC 10454113, presented in figure 4.5 (top). When comparing to 4.3 (bottom), we

can see that, although a small change in the 99% CL constraint can be observed, there are still no major

34

differences. Again, this may be caused by the fact that, while being a slightly older star than KIC

8379927, it still is a young star, making the self-interaction term nearly negligible.

Increasing the local DM density to ρχ = 0.85 GeV/cm3, we also obtained stronger constraints for

KIC 10454113, comparing to the previous case shown in figure 4.3 (bottom), where ρχ = 0.38 GeV/cm3.

Indeed, by observing the χ2r02 diagnostic test in figure 4.5 (bottom), we see that we can exclude MDDM

particles with mχ ' 5 GeV and magnetic moments within 2.2 × 10−17 e · cm 6 µχ 6 8.3 × 10−16 e · cm

with 95% CL and within 3.1× 10−17 e · cm 6 µχ 6 5.9× 10−16 e · cm with 99% CL.

4.3.3 Dark Matter Core

As we have seen, the DM particles trapped by a star will be concentrated inside the stellar core.

The distribution of such DM particles will have a certain radius rχ, given by equation (3.12). This DM

core radius is inversely proportional to the square root of the DM particle’s mass, and as such, we are

expecting the size of the DM core to increase as the MDDM particle’s mass decreases. Additionally, a

dark matter core inside the stellar centre provides an additional mechanism for transferring energy from

the stellar core to the outer layers of the star, effectively being able to decrease the temperature in the

DM core’s region. In order to observe the changes that the DM particle’s properties cause in the DM

core’s behaviour, we calculated the DM distribution’s radius rχ and the DM core’s temperature Tχ for

our MDDM parameter space, obtaining the contour plots shown in figure 4.6 for KIC 8379927 and figure

4.7 for KIC 10454113.

From these plots, one can verify that, as expected, the radius of the DM core does increase with

decreasing DM masses. As seen by figure 4.6, for KIC 8379927, with MDDM particles of mχ ' 6.5

GeV and µχ ' 10−16 e · cm, the radius of the DM core would reach ∼ 4.4% of the total stellar radius.

On the other hand, as for the temperature of the DM core, a similar behaviour to the χ2r02 analysis

plot presented in figure 4.3 can be seen. This shows the link between DM impact and a lower core

temperature, as one would expect. In particular, for KIC 10454113, with MDDM particles of mχ ' 5.5

GeV and µχ ' 10−16 e · cm, the DM core can reach temperatures of Tχ ' 1.59× 107 K, down from the

benchmark’s model (no DM influence) stellar core temperature of Tχ = 1.81 × 107 K. We can therefore

see how effective DM can be in lowering the core’s temperature and the impact it can have in the stellar

structure.

35

Figure 4.6: Radius of the DM core related to the stellar radius (top) and DM core’s temperature (bottom)in function of the MDDM parameter space for KIC 8379927.

36

Figure 4.7: Radius of the DM core related to the stellar radius (top) and DM core’s temperature (bottom)in function of the MDDM parameter space for KIC 10454113.

37

38

Chapter 5

Conclusions

In this work we analysed the effect that magnetic dipole dark matter can have in the evolution and

behaviour of solar-like stars. We started by describing the stellar structure equations along with the

stellar evolution code used in this work, CESAM. We continued by providing the theoretical background

related to MDDM and its capture and energy transport inside the star. Afterwards, we modelled two stars

up to their current ages, based on measured observational parameters. We then were able to calculate

the statistical quantity χ2r02 , defined by equation (4.9), which allowed us to constrain MDDM properties.

Our asteroseismic analysis based on this χ2r02 parameter, shows that MDDM particles with lower

masses and magnetic dipole moments surrounding µχ ' 10−16 e·cm are more likely to be excluded, as can

be shown in figure 4.3. For KIC 8379927, our analysis showed that we can exclude MDDM particles with

a mass mχ ' 5 GeV and with a magnetic dipole moment within 3.5×10−17 e·cm 6 µχ 6 3.9×10−16 e·cm

with a 95% CL. On the other hand, with KIC 10454113, we were able to exclude mχ ' 5 GeV particles

with 2.9× 10−17 e · cm 6 µχ 6 5.9× 10−16 e · cm also with a 95% CL. This shows an improvement on the

95% CL constraints of the magnetic dipole moment of ∼ 17% on the lower boundary and ∼ 51% on the

upper boundary from KIC 8379927 to KIC 10454113 atmχ ' 5 GeV. The fact that KIC 10454113 provides

stronger constraints can be explained by the fact that it is an older star, allowing DM to accumulate in

its core for a longer period of time and leaving a larger signature. Additionally, for KIC 10454113, with

mχ ' 10 GeV, we can exclude MDDM particles with a lower boundary of µχ > 6.0× 10−17 e · cm with

a 90% CL, improving the 90% CL constraints put by the LHC of µχ & 10−15 e · cm, using the mono-

jet events plus missing transverse energy, and also the constraints put by the Large Electron-Positron

Collider (LEP) of µχ & 10−16 e · cm, using the mono-photon plus missing transverse energy events [72].

E. Del Nobile et al. [73] shows that all the experimental results are compatible with an MDDM particle

of mχ ' 10 GeV and µχ ' 1.5 × 10−18 e · cm. We see that these results do not get conflicted by the

constraints provided in this work. This procedure provided us with a weaker constraint than the one

obtained in the helioseismologic analysis carried out by I. Lopes et al. [37], in which we can exclude

particles with mχ ' 10 GeV and µχ > 1.6 × 10−17 e · cm. This is, however, somewhat expected at the

moment, since not only does the Sun have a significant number of measured r02 ratios, but it also has

observational data measured with a remarkably high precision. Our analysis provides competitive and

39

independent results using the studied stars, in different evolutionary stages than the Sun, showcasing its

utility and potential growth in precision within the next decades.

Afterwards, we performed the same χ2r02 analysis as before on both stars, while neglecting the DM self-

interaction capture rate. When comparing the correspondent plots in figure 4.3 (with DM self-interaction)

and in figure 4.4 and 4.5 (top) (neglecting DM self-interaction), we can see that neither star shows relevant

differences in the constraints obtained, which is unexpected, since neglecting the DM self-interaction term

results in a lower number of DM particles in the star, and should therefore decrease its impact, giving

less competitive constraints. This may be explained by the fact that both stars are quite young, not

having had enough time to accumulate sufficient dark matter particles in their interiors to make the DM

self-interaction capture rate significant, since it is proportional to the number of DM particles present

inside the stars, which grows over time according to equation (3.7). It is therefore expected that this

effect becomes noticeable in older stars.

The effect of the local dark matter density on the DM impact inside the star was also analysed. While

previous models shown in figure 4.3 were computed with a local DM density of ρχ = 0.38 GeV/cm3,

we verified that by increasing this density to ρχ = 0.85 GeV/cm3, based on the work of S. Garbari et

al. [71], much stronger constraints could be obtained, as can be seen in figure 4.4 and 4.5 (bottom). As

an example, for KIC 10454113, at a 95% CL, we can exclude MDDM particles with mχ ' 5 GeV and

2.2 × 10−17 e · cm 6 µχ 6 8.3 × 10−16 e · cm. This corresponds to an improvement on the 95% CL

constraints of the magnetic dipole moment of ∼ 24% on the lower boundary and ∼ 41% on the upper

boundary from models with ρχ = 0.38 GeV/cm3 to models with ρχ = 0.85 GeV/cm3, at mχ ' 5 GeV.

We also see that, for KIC 10454113, we can only obtain 95% CL constraints up to DM masses of nearly

mχ ' 8.5 GeV with the lower DM density, while models with the increased density allow for constraints

up to almost mχ ' 10 GeV. From these results we can verify that, as expected, the local DM density can

have significant effects on the DM impact in a star.

Finally, we analysed the DM core, a region inside the star where the captured DM particles will be

concentrated. We observed that, as expected from equation (3.12), the radius of the DM core increases

with decreasing DM particle masses, shown in figures 4.6 and 4.7 (top). In particular, for KIC 8379927,

with MDDM particles of µχ ' 10−16 e · cm, the DM core radius to stellar radius ratio goes from ∼ 3.1%

with DM masses of mχ = 15 GeV to ∼ 4.7% with mχ = 5 GeV. On the other hand, analysing the

temperature of the DM core, we can see that it decreases with decreasing DM particle masses, presented

in figures 4.6 and 4.7 (bottom). This is to be expected since, as already explained, dark matter will

be able to transfer energy from the stellar core to different regions of the star, and as such, a higher

DM impact will be linked to a lower core temperature, which is what we can observe in the lower mass

regions. In fact, for KIC 10454113, with MDDM particles of mχ ' 5.5 GeV and µχ ' 10−16 e · cm, we

see that the DM core reaches a temperature of Tχ ' 1.59 × 107 K, while the benchmark model for this

star with no DM influence would present a core temperature of Tχ = 1.81× 107 K. This corresponds to

a core temperature reduction of ∼ 12% due to the presence of dark matter, highlighting the potentially

significant effects that DM can have in stellar cores.

In the future, asteroseismology will continue evolving as more complex stellar models regarding the

40

evolution and dark matter effects inside the star are developed and as more precise observational data

is collected through increasingly better spacecrafts. A possible improvement to the work developed here

could be to account for the evaporation effects that become relevant for masses mχ 6 4 GeV, enabling the

study and constraining of lower masses. Another improvement would be accounting for the interaction

of dark matter with other elements in the star, instead of only hydrogen, in order to obtain more realistic

and precise results. There are also two spacecrafts to be deployed in the upcoming years, which represent

an important step in asteroseismology. These are the Transiting Exoplanet Survey Satellite (TESS ),

from NASA and planned to be launched in March 2018, and Planetary Transits and Oscillations of stars

(PLATO), from ESA and planned to be launched in 2026. Both these spacecrafts will be able to measure

numerous stellar oscillation mode frequencies with an impressive level of precision, in our case allowing

us to study and constrain dark matter inside stars much more precisely.

As a final note, we have shown that asteroseismology can be an alternative and unique method to

study dark matter and provide strong and competitive constraints to it, representing one step forward in

the the search for the elusive dark matter particle.

41

42

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