gravitational waves signatures of dark matter

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UNIVERSIDADE FEDERAL DO PAR A

INSTITUTO DE CI ENCIAS EXATAS E NATURAIS

PROGRAMA DE P OS-GRADUAC AO EM F ISICA

QUALIFYING MONOGRAPH

GRAVITATIONAL IMPRINTSOF COMPACT DARK MATTER CONFIGURATIONS

Caio Filipe Bezerra MacedoAdvisor : Prof. Dr. Lus Carlos Bassalo Crispino

Co-Advisor : Prof. Dr. Vtor Cardoso

Belem-Para

July 10, 2014


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Gravitational Imprints of Compact Dark Matter Congurations

Caio Filipe Bezerra Macedo

Monograph presented to the Programa de P os-Graduacao

em Fsica da Universidade Federal do Par a (PPGF-UFPA)

as part of the necessary requirements to qualify for the PhD

Degree in Physics.

Advisor: Prof. Dr. Lus Carlos Bassalo Crispino

Co-Advisor: Prof. Dr. Vtor Manuel dos Santos Cardoso

Examiners

Prof. Dr. Lus Carlos Bassalo Crispino (Advisor)

Prof. Dr. Vtor Manuel dos Santos Cardoso (Co-advisor)

Prof. Dr. Jailson Souza de Alcaniz (Outside member)

Prof. Dr. Ednilton Santos de Oliveira (Inside member)

Prof. Dr. Adalto Rodrigues Gomes dos Santos Filho (Invited member)

Prof. Dr. Jo ao Vital da Cunha Junior (Substitute)

Belem-Para


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July 10, 2014


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Abstract

Dark matter is one of the most intriguing subjects of this century as well as the last

century physics. Its nature is not well understood, but its existence is widely accepted,

and used to explain many observable phenomenology in our Universe, mainly cosmological

and galactic scales. Much less investigated are the effects of dark matter in gravitational

wave physics. Moreover, if dark matter exists throughout space, it can form supermas-

sive compact congurations, like the 106M object in the center of our own galaxy.In this monograph, we describe the phenomenology of a stellar-mass object orbiting a

supermassive compact dark matter object in a perturbative approach. This approach is

also known as extreme mass-ratio inspiral. To analyze of the exterior of the dark matter

supermassive object (outside of an effective radius) we choose to work with stars formed

by solitonic scalar eld congurations, known in the literature as boson stars. Using a

perturbative fully relativistic approach, we describe the gravitational and scalar waves

emitted by the stellar-size object in circular geodesic motion around the supermassive

boson star. The waves emitted excites the spacetime, generically producing resonances,

which appear whenever the frequency of the emitted wave matches a natural frequency

of the background spacetime, also known as quasinormal frequency. These resonances

can provide a signature of these horizonless supermassive compact objects. Inside the

supermassive object, we gain some insight of the phenomelogy modeling the motion of

the particle through quasi-circular orbits around a Newtonian star. The radiation emitted

inside the supermassive object is modeled trough the standard quadrupole radiation. We

also consider the accretion of dark matter particles by the moving stellar-size object and

the dynamical friction due to the gravitational drag of dark matter particles, showing that

they are, in general, dominant over radiation dissipative processes. With these results we

provide a generic prescription to confront black holes against horizonless mimickers.

Keywords: Dark Matter, Boson Stars, Gravitational Waves, Quasinormal

Modes, Extreme Mass-Ratio Inspiral

Programa de Pos-Gradua cao em Fsica - UFPA


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Fields of Knowledge(CNPq): 1.05.01.03-7, 1.05.03.01-3.



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Resumo

Materia escura e um dos objetos de estudo mais intrigantes da fsica deste seculo e

do seculo passado. Sua natureza ainda nao e bem entendida, mas sua existencia e vas-

tamente aceita, e e usada para explicar muitos fen omenos observados no nosso Universo,

principalmente em escala cosmol ogica e galactica. Pouco se tem estudado sobre os efeitos

da materia escura no campo das ondas gravitacionais. Se a materia escura est a espal-

hada por todo nosso Universo, esta pode, em principio, formar conguracoes compactas

supermassivas, como o objeto de massa 106M presente no centro de nossa pr opriagalaxia. Nesta monograa, descrevemos a fsica de um objeto de massa estelar orbitando

uma congura cao de materia escura supermassiva numa aproxima cao perturbativa. Esta

aproxima cao e tambem conhecida como espirais com razao de massa extrema. Na regi ao

exterior a conguracao de materia escura supermassiva (fora de um raio efetivo) escol-

hemos trabalhar com estrelas formadas por congura coes solitonicas de campo escalar,

conhecidas na literatura como estrelas de bosons. Usando uma aproxima cao perturba-

tiva relativstica, descrevemos as ondas gravitacionais e escalares emitidas pelo objeto de

massa estelar em movimento circular geodesico em torno de uma estrela de b osons super-

massiva. As ondas emitidas excitam o espa co-tempo, gerando reson ancias, que aparecem

sempre que a frequencia da onda emitida se iguala `a frequencia natural do espaco-tempo

de fundo, tambem conhecida como frequencia quasinormal. Estas reson ancias podem

fornecer assinaturas destas conguracoes supermassivas sem horizonte. Dentro da con-

guracao supermassiva, modelamos o problema considerando o movimento da partcula

atraves de orbitas circulares em torno de uma estrela newtoniana. A radia cao emitida

pelo movimento dentro da estrela e modelada atraves da aproxima cao padr ao da radia cao

de quadrupolo . Tambem consideramos a acre cao de materia escura pelo objeto de massa

estelar em movimento e a fric cao dinamica devido ao arrasto gravitacional das partculas

de materia escura, mostrando que, em geral, estes s ao dominantes sobre a dissipa cao

devido a emissao de radia cao. Com estes resultados n os obtemos meios genericos para

diferenciar buracos negros em rela cao objetos sem horizontes.



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Palavras-chaves: Materia Escura, Estrelas de B osons, Ondas Gravitacionais,

Modos Quasinormais, Espiral com Razao de Massa Extrema

Areas de Conhecimento (CNPq): 1.05.01.03-7, 1.05.03.01-3.



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Ao meu pai Vilmar Macedo,

que tem sido um pai, irm ao e lho para mim. Te amo Pretinho!



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A man who cant bear to share his habits is a man who needs to quit them.

Stephen King, The Dark Tower



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Acknowledgments

To my parents, Vilmar and Rita Macedo, for the affection, dedication and love theyhave given me throughout my entire life.

To my brothers, Kizz Menezes, Daniela Macedo and Vilmar Macedo Junior, for thecompanionship and friendship.

To Carolina Benone, for the patience, love and affection, that made my life easier(and a bit complicated) during these years.

To my friends from inside the Physics sphere, who have contributed with discussions,ideas and suggestions through my entire course.

To my friends from outside the Physics sphere, who for many times have shown methat there is still life out there.

To the professors of Physics Faculty of UFPA, which that, in the majority, werealways available, answered questions of an aspiring physicist.

To the gravity group of UFPA in Belem-Brazil, for all the meetings and discussions.In particular, to Amanda, Rafael, and Carolina, that agreed to revise part of this

manuscript.

To the CENTRA-IST in Lisbon-Portugal, for the hospitality and for all the thingsI have learned. Also, to the Gravity Group in Lisbon.

To my advisor Lus Crispino, for the excellent guidance and opportunities he pro-vided during the development of my entire career. Thanks for all your work and

dedication.

To my co-advisor Vtor Cardoso, who received me in a unknown land (Portugal),making me feel as I was in my own home. Thanks a lot for all the advices and for

the opportunity to work with you.



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To Paolo Pani (who was also a wonderful co-advisor in Lisbon), whose collaborationwas essential in the development of this work. Thank you Paolo for being my guide

throughout the perturbation theory in stars and black holes, for all illuminatingdiscussions.

To the research nancial agents, in special to the Conselho Nacional de Desenvolvi-mento Cientco e Tecnologico (CNPq), to the Coordenacao e Aperfeicoamento de

Pessoal de Nvel Superior (CAPES), to the Funda cao para a Ciencia e a Tecnologia

(FCT), and to the IRSES-UE project.



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CONTENTS x

2.3 Greens function method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.4 Computation of the gravitational and scalar wave uxes . . . . . . . . . . . 58

3 Results of the PART I 62

3.1 Quasinormal modes of boson stars . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 Gravitational and scalar uxes from EMRI in boson stars . . . . . . . . . . 66

3.2.1 Axial sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2.2 Emitted polar ux and inspiral resonances . . . . . . . . . . . . . . 67

II Motion in the star interior. Quadrupole and Newtonianapproximation for gravitational waves. 74

4 Gravitational waves from orbiting particles in the Minkowski spacetime 75

4.1 Linearized Einsteins equations . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2 Transverse-traceless gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3 Weak-eld sources with arbitrary velocities . . . . . . . . . . . . . . . . . . 79

4.4 Quadrupole approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4.1 Multipole expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 804.4.2 Radiation by point particles in circular motion . . . . . . . . . . . . 84

4.5 Quasi-circular orbits considering dissipative effects . . . . . . . . . . . . . . 86

5 Dissipative mechanisms: accretion and dynamical friction 91

5.1 Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.1.1 Collisionless accretion . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.1.2 Hydrodynamical accretion . . . . . . . . . . . . . . . . . . . . . . . 96

5.2 Dynamical friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2.1 Perturbations in a uniform steady ideal uid . . . . . . . . . . . . . 102

5.2.2 Gravitational drag force . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Results of PART II 109

6.1 Dissipative effects in the inspiral motion . . . . . . . . . . . . . . . . . . . 110

6.1.1 Accretion: Collisionless versus Bondi-Hoyle . . . . . . . . . . . . . . 110



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CONTENTS xi

6.1.2 Drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.1.3 Gravitational radiation backreaction . . . . . . . . . . . . . . . . . 115

6.2 Numerical evolution of the inspiral in the interior . . . . . . . . . . . . . . 1176.3 Gravitational waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.4 Analytical Fourier waveforms in the stationary-phase approximation . . . . 122

6.4.1 Collisionless accretion, R p . . . . . . . . . . . . . . . . . . . . . 1226.4.2 Bondi-Hoyle accretion, R p . . . . . . . . . . . . . . . . . . . . . 1246.4.3 Gravitational radiation-reaction . . . . . . . . . . . . . . . . . . . . 125

Conclusion and perspectives 127

Appendixes 129

A Expressions for the source terms and wave functions 130

B Massive scalar modes of a constant density star 137

C Radiation-driven inspiral in the exterior 140

D Motion of particles in a spring-like potential 142



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Foreword

This monograph is basically the result of the research developed during the rst two

years of the PhD program of the candidate. The research was carried out in two insti-tutions: The Universidade Federal do Par a (UFPA), in Brazil and the Instituto Superior

Tecnico (IST), in Portugal. The collaboration project between these two institutions was

funded by the CAPES/FCT program, which made possible for the candidate to stay one

year in the IST.

The content of this monograph is based on two published articles, namely:

C. F. B. Macedo, P. Pani, V. Cardoso, and L. C. B. Crispino, Into the lair:gravitational- wave signatures of dark matter, Astrophys. J., vol. 774, p. 48,2013, 1302.2646.

C. F. B. Macedo, P. Pani, V. Cardoso, and L. C. B. Crispino, Astrophysical sig-natures of boson stars: quasinormal modes and inspiral resonances, Phys. Rev. D,

vol. 88, p. 064046, 2013, 1307.4812.

The other publications of the candidate, which are not included in this monograph,

are:

P. Pani, C. F. B. Macedo, V. Cardoso, and L. C. B. Crispino, Slowly rotating blackholes in alternative theories of gravity, Phys. Rev. D, vol. 84, p. 087501, 2011,

1109.3996.

C. F. B. Macedo, L. C. B. Crispino, and V. Cardoso, Semiclassical analysis of thescalar geodesic synchrotron radiation in Kerr spacetime , Phys. Rev. D, vol. 86, p.

024002, 2012.



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Foreword 10

C. F. B. Macedo, L. C. S. Leite, E. S. Oliveira, S. Dolan, and L. C. B. Crispino,Absorption of planar massless scalar waves by Kerr black holes , Phys. Rev. D, vol.

88, p. 064033, 2013, 1308.0018.



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Introduction

General relativity (GR) is the most well accepted theory to describe the nature of the

gravitational interaction [1 3]. Its predictions go beyond the Solar System scale, havingapplications in cosmology, galaxy formation and also in particle physics [ 4 10]. One of

the most intriguing predictions of GR is the existence of black holes [11]: objects with

gravity strong enough to dene a region from which even light cannot escape, hence the

adjective black .

In the last century, black holes (BHs) became a paradigm in physics. It is generally

assumed that there is a BH at the center of most galaxy nuclei. This assumption is

related directly to the fact that the observational data from the region surrounding thecentral objects of galactic nuclei is compatible with the BH hypothesis [ 4]. However, BHs

suffer from one of the most intriguing problem in physics: the presence of singularities.

Generically, true singularities are points in which the theory, and hence the physics, breaks

down. In GR, singularity theorems guarantee that singularities exist [ 3, 12], and the fact

that such a good theory as GR can have points in the spacetime where the physics breaks

down intrigues most physicists.

Although these singularities exist in GR, there is a way to avoid them. The cosmiccensorship [3, 13], conjectured by Penrose in 1969, states that whenever a singularity

occurs in spacetime it is hidden by an event horizon. Since the event horizon is a one

way membrane, one can think that the physics of BHs is important only outside them

and therefore the singularity would not play any special role. The cosmic censorship

is still a conjecture, and its proof is an open problem in physics. Attempts to expose

the singularities by destroying the event horizon of extreme and quasi-extreme BHs with



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Introduction 12

particles have been made lately [14, 15]. In this way, since in general the particle is

charged and has a non-vanishing mass, the self-forces can play a special role, acting as

cosmic censorship agents [16, 17].It is expected that an extension of GR (including quantum descriptions or modica-

tions of GR itself [18]) would heal these singularity problems. Meanwhile there are some

alternatives that avoid singularities. One of them, still within the BHs theory, are the

regular BHs (see, e.g., [19 21] and the references therein): they are called BHs because

they still have an event horizon, and regular by the fact that these objects do not have

singularities. Other alternative which avoid the singularities are the horizonless BH mim-

ickers . These non-standard stars (hence the name horizonless) can be as compact and

as massive as a supermassive BH. Among these objects, lie the gravastars [ 22, 23], which

in the spherically symmetric case have Schwarzschild metric in the exterior and de-Sitter

metric in the interior, separated by a shell.

Another horizonless model, which we will deal extensively in this monograph, are the

boson stars (BSs) models (see [24 26] for reviews). These are stars formed by scalar elds,

which can be fundamental or not. They have been proposed as possible candidates for

the supermassive objects in the center of the galaxies, being indistinguishable from BHs

from the point of view of electromagnetic experimental data from galactic nuclei [27 29].

Bellow we will give a brief motivation for stars formed by light bosonic elds, justifying

their formation by the existence of a kind of matter that has been another paradigm in

physics: dark matter (DM).

Physical motivations for boson stars

In theoretical physics, spin-0 particles have a special place, serving as models or even as

toy models for many phenomena. The results from ATLAS collaboration [ 30,31] released

in 2012 about the discovery of a Higgs-like particle resulted in a Nobel Prize in 2013

(for Fran cois Englert and Peter Higgs), generated a lot of excitement throughout the

physics community, and gave more strength to the belief of the existence of pure spin-

0 particles in nature. Apart from the natural spin-0 particles, fermions can also form



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Introduction 13

congurations that mimic bosonic features. In the formation of a bosonic condensate, the

difference between fermions and boson that plays the main role is the spin, leading to

the Pauli exclusion principle, which tells that fermions have to be in different quantumstates. However, specic combinations of pairs of fermions can behave like bosons. This

makes some non-elementary particles, like the scalar and pseudoscalar mesons, behave

effectively as bosonic spin-0 particles [32]. In this way, one can have effectively a spin-0

particle made of two spin-1/2 particles, for instance.

Apart from the discussion of the existence of spin-0 particles in high-energy particle

physics, there are also motivations from the astrophysical point of view. It is widely

accepted that most of the Universe is populated by a non-standard form of matter, labeled

as DM (see [33] for a review on the subject). This kind of matter can come from many

branches 1 , and it is used to explain the phenomenology of physics in a large range of

astrophysical scales: from the cosmological evolution of the Universe to the size and

mass limitation of neutron stars [ 33 35]. There are many astrophysical experiments that

give indirect evidence for the existence of DM [33], and the results of the Alpha Magnetic

Spectrometer is a major example in that direction [36]. Many models of DM are described

by very light scalar particles [33, 37, 38], and therefore the study of the dynamics of

primordial spin-0 elds in the universe is crucial to understand the inuence of DM in

nowadays Universe.

Given the motivations from both high-energy particle physics and from astrophysics for

the existence of spin-0 elds, including in the DM sector, some authors start to ask whether

or not one could have formation of stars from uctuations of scalar elds in the beginning

of the Universe. Since the seminal works by Kaup [39] and Ruffini and Bonazzolla [40]

in the late 1960s, many works have been done. These BSs2 can be thought as natural

descendents of the Wheeler geons [41], which are stars formed by self-gravitating photonic

congurations, with the difference that they can present stable congurations.

Formation of BSs has been studied extensively in the literature [ 24,42 48]. In particu-

lar, the most plausible explanation for a cooling-down mechanism to generate stellar-like1In fact, there are many claims that the DM present in the Universe is formed by the combination of

a bunch of components, and not just one component.2We shall understand BSs as stars formed by DM or by others scalar elds as well.



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Introduction 14

Property BSConstituents Scalars.Pressure support Self-interacting scalar potential

Size Its compactness can be vary small (modelinggalaxy halos [34]) to very big (mimicking BHs.See section 1.3).

Appearance Transparent. Only interacts gravitationally withordinary baryonic matter.

Surface Atmosphere. Does not have a hard surface.Structure Einstein-Klein-Gordon system (see section 1.1)Circular orbits Can present unstable orbits, depending on the

model (see section 1.3).Rotation Yes. Limited due stability analysis [ 51].Star disruption (tidal radius) Yes.Distinctive observational features Gravitational radiation, gravitational lensing [ 52],

K line prole from accretion disks [53, 54].

Table 1: Generic properties of BSs. Adapted from Ref. [ 24].

bosonic structures in a reasonable time is to start with a highly massive bosonic cloud

(bigger than the stability limit [see also section 1.3.1]), making the star cool-down by

ejecting its own matter (having then a phase transition). Although baryonic matter form

stellar structures in a more efficient way, this does not exclude the possibility of the

existence of DM stellar structures in the Universe.

Generic boson stars

In the literature, there is a plethora of BS models. These models depend intrinsically

on how the boson particles interact with each other inside the bosonic cloud, i.e., they

depend on the scalar eld potential in the Einstein-Klein-Gordon action (see section 1.1).

The simplest model considers the scalar eld only with the mass term 3. Even this simplest

case can generate stars with mass 106M , if the scalar particle is light enough ( 1026GeV) [24,29], and therefore could mimic some properties of supermassive BHs. Also,since the DM component can be light-scalars, one can think that BSs are actually stars

formed by DM constituents [47, 49,50].3Non-interacting massless scalars cannot generate star, ending as BHs or dispersing to innity, de-

pending on the physical conditions [24].



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Introduction 15

In Table 1 we describe some generic properties of BSs [24]. Some of these properties will

be explained with more detail later in this monograph, but here we want to emphasize the

main aspects that make the BS candidates as supermassive astrophysical objects. Onenotable characteristic of BSs appealing in this direction is the non-existence of a hard

surface: the scalar eld spreads in all space, going to zero exponentially at innity (see

section 1.2), and interacts only gravitationally with baryonic matter. This characteristics

makes the spacetime interior to the star to be an allowed region for the baryonic matter

surrounding the star. In fact, tidal disruptions of less compact stars can happen even

inside a supermassive BS [27]. Also, the atmosphere characteristic of the BSs surface

make a distinct feature when compared with neutron stars, for example. Imprints obtained

by replacing the event horizon by a proper hard surface would not be applicable to BSs

as well.

The question of whether or not one can have electromagnetic imprints of the event

horizon is still debatable. While some argue that this is almost impossible [ 55], others say

that the the replacement of an event horizon by a surface would lead to a characteristic

change in the electromagnetic signal, making the event horizon visible [56, 57].

Rotating BS solutions were found in [58], where it is shown that the rotation of the

star is proportional to the angular momentum of the scalar eld, being discrete. Also, it

should be noted that the BS rotation is limited due to the ergoregion instability that it

develops when rapidly rotating [ 51].

Newtonian stars and general relativistic stars

Although BSs only interact gravitationally with normal stars and baryonic matter, in

general some gravitational effects could deviate the geodesic motion of an orbiting star,

when considering the motion inside the BS. The study of gravitational dissipation mech-

anisms like accretion [59 61] (see section 5.1) and dynamical friction [ 62 65] (see section

5.2) and their effects on the motion of a particle in a curved spacetime is still elusive.

Meanwhile, the Newtonian description of such phenomena is well-posed, and, in some

cases, some of the results can be very close to the GR counterpart [61]. With this in



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Introduction 16

mind, one can estimate the motion of stars considering gravitational dissipative effects in

Newtonian regime.

Although almost a hundred of years have passed since GR was created by AlbertEinstein, Newtonian gravity is still used to describe many phenomena in the Universe. In

the weak-eld limit, it is sufficient to say that GR reduces to Newtonian gravity, so that it

is born ready for this regime. In stronger regimes, some modications of it, like, e.g., the

post-Newtonian approximation [ 66,67] are successfully used to describe the gravitational

elds of compact objects.

Regular uid stars are compatible with both Einsteinian and Newtonian gravity. In

fact, even BSs can be modeled in Newtonian gravity [ 43] (see also section 2.7 of [24],

and [34] for a direct application of it in galaxies). In this sense, one can model the

motion of particles inside a DM-like object by using Newtonian effects. Depending on the

compactness of the central object, GR effects should be included, although the qualitative

discussion of the motion should basically be the same (see chapter 5). By using the

Newtonian approximation we reduce drastically the computational cost of the problem

while keeping its main characteristics. Since the motion of point particles does not depend

on the star structure, one could even use the motion of point particles inside uniform

density stars in Newtonian gravity as a toy model to more complicated phenomena. Our

main message here is: Newtonian stars are important even for general (relativistic) motion

discussions.

Extreme mass-ratio inspiral

Test particles in GR follow geodesics. This kind of motion only depends on how the

spacetime locally is, not on its dynamics or the presence of matter in it (as long as the

backreaction in the curvature is small). This makes different spacetimes look the same

from the geodesic point of view. This is also shared with some observable quantities which

consider the geodesic structure of the spacetime. For instance, the electromagnetic emis-

sion spectrum of accretion disks around spherically symmetric BS can be indistinguishable

from a Schwarzschild BH [28, 68].



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Introduction 18

of BSs [85, 86], showing that for an appropriate description of the waveforms, resonances

must be taken into account. Phase measurements through time-delay interferometry [ 89]

is a potential source for the detection with future space-based detectors [ 74].Most astrophysical objects have a hard surface or a horizon. Due to this, the EMRI

systems are generally studied only outside the supermassive object radius/horizon. When

considering the particle motion inside a medium that interacts only gravitationally with

the moving particle, dissipative effects may appear (and dominate), changing the particle

motion. The main gravitational dissipative effects are: accretion and dynamical friction.

As said earlier in this introduction, accretion accounts for the star matter absorbed by

the moving object and dynamical friction accounts for the loss of momentum due to

the gravitational drag caused by the uid wake generated by the moving object (see

Chapter 5). Some works in which the dissipative effects due to the presence of matter

were considered are [90,91], of which [91] has a closer relation to the work presented in

this monograph.

Outline of this monograph

The main goal of this monograph is clear: we want to understand the inuence an object

formed by DM-like components in GW physics. To do so, we used as GW probes the

EMRI phenomenology. We divided the monograph in two parts:

Part I

In part I we analyze the motion of a test particle around a supermassive BS. This motion

can be used to discuss the gravitational wave emission by the particle outside the DM-likeobject, but we also discuss the motion in a region inside the star, without dissipative

effects. Although there is no proper BS radius, we can dene an effective radius, which

we choose to be the one in which englobes 99% of the total mass of the BS.

In chapter 1 we analyze the Einstein-Klein-Gordon system in spherically symmetric

spacetimes, and the generic boundary conditions that a BS must obey in order to form

stellar-like structures. Moreover, we introduce the BS models used in this this monograph:



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Introduction 19

mini BSs, massive BSs and solitonic BSs. Then, we study circular geodesic motion around

these BS spacetimes.

In chapter 2 we study perturbation theory in BS spacetimes in a quite generic way.We use the Regge-Wheeler gauge to simplify the analysis of the perturbed equation, and

illustrate a way to expand the energy-momentum tensor of the particle in terms of the

spherical tensor harmonics. These procedures are then used to nd the fundamental per-

turbed equations. In order to better visualize the equations, we separate the perturbation

into two sectors: axial and polar. Then, we apply different methods to compute the free

oscillations of BSs spacetime, namely the quasinormal modes. The quasinormal modes

found can be compared to the frequency of the emitted wave in order to analyze the

existence of resonant modes. We then discuss the procedure to compute the gravitational

and scalar energy uxes at innity, in both axial and polar sectors.

The numerical calculations and the results of part I are presented in chapter 3. We

show that the quasinormal modes in BS spacetimes present a richer structure, when

compared to the BH case, mainly due to the presence of the scalar eld. The modes

are computed for both axial and polar perturbations, and, in particular, the polar sector

presents a multitude of modes which can be excited due to the motion of the particle

around a BS. We show the energy ux due to the emission of GWs by the orbiting particle.

Apart from the difference with the BH counterpart in the ux for resonant frequencies, the

resonances can generate a dephasing in the signal, which could, in principle, be observed

by future GW detectors [79, 86, 89]. This is a distinctive feature of BS spacetimes and

the resonant frequencies are model-dependent, so that, once the observational data is

available, one can conrm the models, or rule them out. Most of our original results

obtained in Part I have been published in Ref. [ 86].

Part II

In part II we analyze the motion of the particle and the emission of GWs for the case

in which the particle is moving inside the central supermassive object. This analysis is

performed using a Newtonian approximation for the motion and the standard quadrupole

approximation for the emission of GWs.



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Introduction 20

In chapter 4 we review the standard quadrupole approximation [92] to compute the

GW emission by a point particle in circular orbits. This is also presented in some intro-

ductory books on GR (see [2], for instance), so that we do not extend our discussion, onlypresenting the main expressions to be used. We also present how quasi-circular orbits can

be treated assuming that the particle loses energy through gravitational wave emission

only.

In chapter 5 we show the dissipative effects considered in the motion inside the star.

These dissipative effects are considered within the context of the Newtonian approxima-

tion, and are due to gravitational forces only . This has to be emphasized because we

want to model DM-like objects, which interact gravitationally only. The orbiting particle

(stellar-mass BH or neutron star) accretes mass [ 59,61,93,94] from the central supermas-

sive object, its mass grows, therefore inuencing its motion. We describe the accretion

models used, discriminating their validity regimes. Moreover, we describe another dissi-

pative effect called dynamical friction [63, 65]. Basically, dynamical friction generates a

drag force due to the wake left by the moving particle in the medium. Since we are work-

ing within the EMRI regime, we use the dynamical friction formulae due to the particle

moving in straight lines [63], as modied to t the case of the particle moving in circular

motion [65].

The numerical computations and results of part II are presented in chapter 6. We

assume a radius-dependent density prole to model the Newtonian star. The prole

can be adjusted according to the central star model. We compute the results for the

dissipative mechanisms for the moving particle, showing that accretion and dynamical

friction can dominate over the GW backreaction in the particle motion. We also show

that, when dynamical friction is negligible (i.e., accretion dominates), the GW signal is

nearly constant, due to angular momentum conservation. Most of our original results

obtained in Part II have been published in Ref. [85].



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Part I

Motion in the star exterior. Boson

stars, quasinormal modes and

resonances of gravitational waves



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Chapter 1

Spherically symmetric boson stars

In this chapter we shall present the procedure to generate boson star (BS) spacetimes. We

present the Einstein-Klein-Gordon system of equations and discuss the proper boundary

conditions to nd the BS congurations, as well the shooting method used to nd the

solutions. We also discuss circular geodesic motion around BS spacetimes, showing that

there exist stable circular orbits deep inside the star and, if the star is compact enough,

circular light-like orbits (light-rings) too.

1.1 Einstein-Klein-Gordon Equations

Here we will discuss some technical details on BS. BSs are generated by self gravitat-

ing (complex) scalar eld. By this we mean that they are solutions of Einstein general

relativity (GR) plus a scalar eld 1. The Einstein-Klein-Gordon action ( S ) is given by

S = S EH + S KG + S MA . (1.1)1Although BSs are also studied in other theories of gravity, we shall be concerned here only with BS

congurations within GR.



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1.1 Einstein-Klein-Gordon Equations 23

In Eq. (1.1), S EH is the Einstein-Hilbert action and S KG is the Klein-Gordon action,

explicitly given by

S EH = d4x g R2 , and (1.2)S KG = d4x g gab a b + V s(||2) , (1.3)

respectively. Here gab are the covariant metric components, g its determinant, (= 8 ) is

the coupling constant, R the Ricci scalar, the scalar eld, and V s (||2) the scalar po-tential. The action S MA represents an additional form of matter, which in this study shall

be considered as a perturbation term (rst order correction), therefore not contributingto the background BS conguration. In this chapter we shall consider S MA = 0, and we

shall be back to it in Chap. 2.

Einstein-Klein-Gordon system of equations can be obtained by extremizing the action

(1.1)(1.3) with respect to the metric and to the scalar eld 2, using the functional deriva-

tive of the action as function of the elds ( gab, ). The functional derivatives with respect

to the metric gives 3

S EH gab

= g2 Rab 12

gabR , (1.4)

S KGgab

= g

2 T ab , (1.5)

where T ab is the energy-momentum tensor of the scalar eld, given by

T ab = a b + b a gab c c + V s (||2) . (1.6)

Therefore, from the equation of motion of the theory, namely

S gab

= 0 (1.7)

2For a discussion on the variational formulation of GR we direct the readers to Refs. [ 2,3].3One simple way to obtain the functional derivative is to expand the action integral assuming that the

eld varies innitesimally. For example, in the case of the metric function we have that gab gab + gab ,leading to S = d4 x S g ab gab , where

S g ab is the functional derivative of S with respect to gab . See, for

instance, [ 2,3].



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we get the Einsteins equations

Rab

1

2gabR = T ab . (1.8)

Extremizing the action ( 1.1) with respect to the complex conjugated of the scalar eld

(S/ = S KG / = 0), we get the Klein-Gordon equation

= 1 g

a ggab b = dV sd||2

, (1.9)

together with its complex conjugated, obtained from S KG / = 0.

In order to obtain self-gravitating scalar eld congurations, Eqs. ( 1.8) and (1.9) mustbe solved. Here we want to nd spherically symmetric solutions, hence the background

line element and the scalar eld can be written as [1 3]

ds2 ds20 = ev(r ) dt2 + eu(r ) dr 2 + r2(d2 + sin 2 d 2) and (1.10) 0 = 0(r )eit , (1.11)

respectively, where v(r ), u(r ) and 0(r ) are real functions to be found by solving thedifferential equations. The frequency > 0 of the background scalar eld is an eigenvalue

that can be found by imposing the proper boundary conditions (see Sec. 1.2). The metric

associated with ( 1.10) is in a generic form to describe spherically symmetric spacetimes [ 2].

Although the scalar eld is time-dependent, the Einstein-Klein-Gordon system admits

static, spherically symmetric metrics [24, 39, 40, 95 97]. Dening

V 0

V s (

|0

|2), and (1.12)

U 0 dV sd|0|2

, (1.13)



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we have that the Einstein-Klein-Gordon system ( 1.8) and (1.9) reduces to

1

r 2r eu

1

r 2 =

, (1.14)

eu vr

+ 1r 2

1r 2

= prad , (1.15)

0 +2r

+ v u

20 = eu U 0 2ev 0 , (1.16)

where

T tt = 2ev20 + eu (0)2 + V 0 , (1.17)

prad T r r = 2ev20 + eu (0)2 V 0 , (1.18) ptan T

= 2ev20 eu (0)2 V 0 . (1.19)

The quantities ( 1.17)(1.19) are dened analogously to the regular uid quantities (see,

e.g., Refs. [1, 2]). We can see that the boson star can be seen as an anisotropic uid star,

which density ( ) and pressures [radial ( prad ) and tangential ( ptan )] are described by the

set (1.17)(1.19). In fact, Eqs. ( 1.14)(1.16) are quite similar to the Oppenheimer-Volkoff

system, which describes uid stars in GR [ 2,61], and can be written in the exact form foranisotropic spherical stars in GR [ 98]. Dening

eu = 1 2m(r )

r (1.20)

one can show from equation (1.14) that

m(r ) =

r

0

4x 2(x)dx. (1.21)

The function m(r ) can be interpreted as a mass within a radius r , the total mass being

M m(r ). Equation ( 1.21) in fact considers the whole energy forming the starconguration, including the gravitational one, so that this is not only the mass formed by

the bosonic particles [61]. One can see this by noting that the proper volume at a xed

time is given by 4(1 2m/r )1/ 2r 2dr , and therefore Eq. ( 1.21) can not be considered as



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1.2 Boundary conditions for boson stars 26

just the mass-energy of the bosonic eld contained inside a radius r .

The background BS congurations can be found by solving equations ( 1.14)(1.16),

once the scalar potential is chosen, and with the proper boundary conditions. We ob-tained these equations directly from the Einstein-Klein-Gordon system, using the ansatz

described above. An alternative way to obtain the background equations is using a 3+1

decomposition of spacetime [26, 99, 100]. This kind of decomposition is suitable to study

numerical evolution of dynamical systems (see, e.g., Refs. [ 101, 102]).

1.2 Boundary conditions for boson stars

The boundary conditions used to construct BS congurations are similar to the regular

uid star case [61]. We integrate Eqs. ( 1.14)(1.16) from the origin, where we require

regularity. We have the following boundary conditions

u(r 0) = 0 , [grr (r 0) = 1] (1.22)

v(r 0) = vc, [gtt (r 0) = evc ] (1.23)

0(r

0) = c, (1.24)

0(r 0) = 0 . (1.25)

For the metric function u(r ), the boundary condition ( 1.22) essentially means that there

is no mass-energy inside the sphere of zero radius, cf. equation (1.21). Like in the regular

uid star cases, in the boundary condition ( 1.23) vc can have arbitrary values through a

time reparametrization. Here we set it to zero ( vc = 0), and at the end we multiply gtt

by a constant to achieve the required condition gtt (r ) = grr (r )1 at the innity 4. Theresult becomes a one-parameter family of solutions, with the parameter being the central

value of the scalar eld c. This is a crucial point, because for each value of c there is a

correspondent (zero-node) value of the frequency for which the star conguration can

be formed (see below).4In fact, gtt = 1/g rr is the real boundary condition for the problem. One can think also this condition

as being a shooting problem for the value vc , which can be readily solved through a time reparametrization.Also, it is more suitable from the numerical point of view to work with all the boundary conditions atthe same point.



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1.2 Boundary conditions for boson stars 27

In order to completely determine the BS spacetime, one has to nd the eigenvalues

for which the star congurations exist. These values are found such that the condition

0(r 1) 1r exp ( 2 2 r ) (1.26)is satised. We denote for the scalar eld mass and r for the tortoise coordinate of the

BS spacetime, dened through

dr = e(uv)/ 2dr. (1.27)

The problem becomes a two-point boundary value problem [ 103]. The condition ( 1.26) is

satised only for some values of the frequency , which are the eigenvalues of the problem.However, a single value of c in fact generates a countable innity set of . For a xed c,

solutions with different goes exponentially to zero at the innity in different ways. In

Fig. 1.1 we show three different congurations for the background scalar eld 0, with the

same value of the central eld for the mini BS case (see Sec. 1.3). Different congurations

with the same value of the central eld are characterized by the number of the nodes, i.e.,

the number of times that the scalar eld crosses the r -axis. In Fig. 1.1 we have zero, one

and two nodes congurations. However, it is known that solutions that are not on thezero-node state (excited states) are unstable, and would or decay to the ground state, or

disperse, or collapse into a black hole [43,96, 97, 104]. For this reason, we will consider in

this monograph only zero-node solutions.

Another important denition is the star radius. For BSs, due to the exponential

damping of the eld 0 [cf. Eq. (1.26)], the background scalar eld is zero at innity.

This makes the denition of the star radius, usually denoted as the point at which the

star pressure is zero, more complicated. There is a number of radius denitions for BSs

in the literature [24]. Here we shall dene the star radius R as being the point at which

m(R) = 0 .99M . In this way we guarantee that within the star radius we have basically

all star mass.

In the following section we will specify the BS models used in this monograph. We

shall also discuss the numerical procedures to nd them.



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1.3 Boson stars models used in this monograph 28

0 10 20 30 40 50

0.000

0.005

0.010

0.015

0.020

r

0 r

zero nodeone nodetwo nodes

Figure 1.1: First nodes solutions for mini BS congurations. Here we show zero, one andtwo nodes solutions for the same central value of the scalar eld c = 0.02. The behaviorfor the other models, namely massive and solitonic BSs (see Sec. 1.3), are qualitativelythe same.

1.3 Boson stars models used in this monograph

The stability study of BSs can be analyzed analogously to the uid star case [61], but

also from other mechanisms, like catastrophe theory [ 24,105]. Generically, stable congu-

rations exist for the central eld values in the range 0 < c < critc , where critc represents

the value of the central eld which generates de maximum mass conguration. The con-

guration generated with critc is generally called marginally stable, because it denes a

transition point between stable and unstable congurations. The congurations c > critcstudied in this monograph are unstable 5 .

We have chosen two BS congurations for each BS model presented in this mono-

graph. One of the two congurations is the marginally stable conguration, presenting

the maximum asymptotic mass M max as function of the central eld c. The other is the

one that presents the maximum compactness, dened as being 2 M/R . The maximum

compactness conguration is at the unstable branch of the solutions, i.e., for c > critcand dM/d c < 0 (unstable branch). We choose this unstable conguration in order to

test extreme relativistic effects with BS star congurations, because the GR effects are5We should point out that not all congurations with c > critc are unstable. It was shown in

Ref. [106] that BSs may exhibit two stable regions, in a similar fashion of white dwarfs and neutron stars.



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0.0 0.2 0.4 0.6 0.8 1.0 1.20.2

0.3

0.4

0.5

0.6

c

M

0.0 0.2 0.4 0.6 0.8 1.0 1.20

5

10

15

20

25

30

c

R

M

Figure 1.2: Mass prole and radius, plotted as function of the central eld c in the miniBS model. The dots indicate the two congurations chosen here (maximum mass andmaximum compactness).

1.3.2 Massive boson stars

For this model the potential has, besides the mass term, a quartic interaction, being

V s (||2) = 2||2 + 2 ||

4 , (1.29)

where is a constant. This model is usually called in the literature simply boson star,

but to avoid confusion with the other models used in this monograph, we shall refer to it

as massive boson stars . The name is justied due to the fact that the total mass can easily

be at the same order of the Chandrasekhar limit for fermions, and it differs markedly from

the non-interacting case ( = 0) even when 1 [95]. Here, to be in agreement withRefs. [107, 108], we shall rescale the quantities as

r r

, m(r ) m(r )

, , 0(r )

0(r ). 4 .

The metric and the scalar eld for the two massive BS congurations listed in Table 1.1 are

shown in the middle panels of Figure 1.3, and are compared with the Schwarzschild case.

The mass prole and the radius behave similarly as in the mini BS case (cf. Figure 1.2)

as it can be seen in Ref. [95]. Moreover, the compactness of the congurations increases



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numerical scheme and a extremely ne-tuned shooting method, as shown by the number

of relevant digits presented in Table 1.1 for this model. In the small 0 limit, our results

agree remarkably well with the approximate solutions presented in Refs. [ 84,110] and theyextend those results to generic values of the parameters in the scalar potential ( 1.30).

Unlike the other cases explored in this monograph, solitonic BSs can be very compact,

with the radius of the star comparable to or smaller than the Schwarzschild light-ring [ 84,

110]. In the right panels of Figure 1.3 we show the steep prole of the scalar eld and

we compare the metric components to those of a Schwarzschild spacetime and those of a

uniform density stars with radius R = 3M . The scalar eld approximates a step function,

in agreement with the approximate solution of Refs. [ 84, 110]. In their approximated

solution, grr is discontinuous at the star surface. In our case there is no actual radius,

and the metric is continuous, although it has a sharp peak close to the effective radius of

the star.



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Table 1.1: BS models used in this monograph. For massive BS congurations we used = 100, whereas bhave 0 = 0.05. The signicant digits of do not represent the numerical precision, but they show theachieve the solutions.

c M R M mini BS I 0.1916 0.853087 0.63300 7.86149 0.5400mini BS II 0.4101 0.773453 0.53421 4.52825 0.4131

massive BS I 0.094 0.82629992558783 2.25721 15.6565 1.865massive BS II 0.155 0.79545061700675 1.92839 11.3739 1.533solitonic BS I 1.05 0.4868397896964082036868178070 1.847287 5.72982 0.899solitonic BS II 1.10 0.4931624243761699601334882568 1.698627 5.08654 0.837


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1.4 Geodesics around boson stars 34

1.4 Geodesics around boson stars

Stellar-sized objects gravitating around supermassive BSs have a small back-reaction on

the geometry and, to rst order in the objects mass, move along geodesics of the BS back-

ground. Accordingly, gravitational-wave emission by such binaries requires a knowledge of

geodesic motion together with the consequent perturbative expansion of the gravitational

eld. Many features of the gravitational radiation can be understood from the geodesic

motion, in which we now focus. We will also focus exclusively on circular geodesic motion.

The reasoning behind this is that it makes the calculations much simpler, while retaining

the main features of the physics. Furthermore, it can be shown that generic eccentric

orbits get circularized by gravitational-wave emission [112], on a time scale that depends

on the mass ratio. Eccentric orbits around BS were studied in Ref. [113].

Some empirical results considering geodesic motions around BSs already exist in the

literature. For instance, in Refs. [28, 68, 84] accretion disks were considered, in which the

disk itself follows circular geodesics.

We follow the formalism presented by Chandrasekhar [ 11], and the procedure for a

generic background is presented also in Ref. [114]. Following previous studies, we assume

that the point-particle is not directly coupled to the background scalar eld [28, 68,84].

We start by dening the Lagrangian of the particle motion on the = / 2 plane:

2L p = s2 = ev t2 + eu r 2 + r2 2 . (1.31)

The conserved energy per unit of rest mass E and angular momentum parameter per unit

rest mass L can be obtained via the relations

E = L p t

= ev t , (1.32)

L = L p

= r 2 . (1.33)

From these equations, we get the following equation for the motion around BSs

eu+ v r 2 = E 2 V eff (r ) , (1.34)



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0.2

0.40.60.81.0

e v ( r )

0.2

0.40.60.81.0

c=1.05

c=1.10

BH

Star

2.0

3.0

e u ( r )

c=0.1916

c=0.4101

BH

0 5 10r/M

0.0

0.2

0.4

~ 0 ( r )

1 2 3 4 5r/M

0.0

0.5

1.0

2.0

3.0

c=0.094

c=0.155

BH

5 10r/M

mini BS massive BS solitonic BS

Figure 1.3: Rescaled background proles for different BS models and congurations (cf.Table 1.1). In the top, middle and lower row we show the metric elements ev, eu andthe scalar prole 0, respectively. Each column refers to a different BS model. From leftto right: mini BS, massive BS and solitonic BS. For each model, we compare the metricproles with those of a Schwarzschild BH and for the solitonic BS model we also compare

with the metric elements of a uniform density star with radius R = 3M .



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where

V eff (r ) = ev 1 + L2

r 2, (1.35)

and we used s = 1, which holds for time-like geodesics. Eq. (1.34) can be seen as anequation of energy balance, with the particle mass dependent of its position [ 84].

The energy and angular momentum of the particle in circular orbits follow from

Eq. (1.34), by imposing r |r = r p = 0 and r |r = r p = 0, resulting in

E c = ev 2(r 2m)

2r r 3 prad 6m1/ 2

r = r p, (1.36)

Lc = r2 ( r 3 prad + 2 m)2r r 3 prad 6m

1/ 2

r = r p, (1.37)

where the background Einsteins equations were used to eliminate metric derivatives.

Circular null geodesics correspond to the point at which ( 1.36) and (1.37) diverge, i.e.,

2r r 3 prad 6m = 0 [11]. Finally, the orbital frequency of circular geodesics, using(1.32), (1.33), (1.36) and (1.37), reads

= t =

ev ( r 3 prad + 2 m)2r 2(r 2m)

1/ 2

r = r p . (1.38)

When (E c, Lc, ) are real at some point r p > 0, r p is a circular orbit, which can be stable,

marginally stable or unstable. The stability condition can be seen through the analysis of

the second derivative of the effective potential V eff (r ) given by Eq. (1.35). We have that

d2

dr 2V ef f (r )

r = r p

< 0, Unstable orbits

= 0 , Marginally stable orbits

> 0, Stable orbits .

(1.39)

In the case of BHs, marginally stable orbits are called, in general, innermost stable cir-

cular orbits (ISCO). In Figure 1.4 we plot the logarithm of the second derivative of the

potential as a function of the orbital radii for the BS spacetime congurations I and for

the Schwarzschild BH case. We can see that for mini and massive BS congurations stable

circular geodesics exist all the way down to the center of the star, while for the solitonic



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Figure 1.4: Second derivative of the effective potential ( V eff ) computed at r p for geodesicmotion as function of the orbital radii for different BS congurations compared with theBH case. The logarithm scale helps to visualize the curves, and because of that negativevalues for the second derivative, which correspond to unstable orbits, are not showed inthe plot.

case there is a discontinuity of stable orbits. This transition will be addressed in the

following paragraphs.

The main characteristics of circular geodesics in BS spacetimes are summarized in Fig-

ure 1.5. Up to the innermost stable circular orbit of a Schwarzschild spacetime, r = 6M ,

geodesic quantities are very close to their Schwarzschild value (as dened by the total

mass) as might be expected since these are very compact congurations. For geodesics atr < 6M the structure can be very different. A striking difference is that stable, circular

timelike geodesics exist for BSs even well deep into the star [28, 68].

Solitonic BSs can become truly relativistic gravitating objects. For these objects, an

outer last stable circular orbit (marginally stable) exists at r 6M with M 0.068.This is expected, as the spacetime is very close to Schwarzschild outside the solitonic

BS effective radius. We also nd a rst (unstable) light-ring at roughly r l+ 3M . Theunexpected feature is the presence of a second stable light-ring at r l < r l+ , together witha family of stable timelike circular geodesics all the way to the center of the star. These

light-rings are genuine relativistic features, which was reported only recently [ 85], as far

as we are aware. Uniform density stars, depending on their compactness, also present

two light-ring and stable circular timelike orbits in their interior. In the right panel of

Fig. 1.5 it is also shown the case of a uniform density star with radius R = 3M . In

that case, the two light-rings degenerate in the star surface. What makes solitonic BSs



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Chapter 2

Perturbation of boson stars and the

EMRI approximation

In this chapter we study general perturbations in spherically symmetric boson star space-

times. Here we use the Regge-Wheeler gauge [115], because of its simplicity. We divided

the perturbations in two sectors: axial and polar. The axial sector is described in an

analogous way to the uid star case, but due to the coupling with the scalar eld the

polar sector is more involved. We show two distinct methods to compute the naturaloscillation of the spacetime, namely the quasinormal modes (QNMs). We also study the

perturbations induced by a massive particle orbiting the equatorial plane of a boson star,

solving the linearized differential equations by the Greens function techniques. Using

the solutions, we show the expressions for the gravitational and scalar energy uxes at

innity.

2.1 Perturbation analysisPerturbation analysis in spacetimes is an important issue in any geometrical eld theory.

It allows us to understand how gravitational waves propagate in a curved background.

If the object is a black hole, then we may nd out, for instance, how a wave can be

absorbed or scattered by the black hole [ 116 118]. Moreover, perturbation theory may

tell about the true nature of the spacetime itself [ 11, 78]. Among the great things that



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2.1 Perturbation analysis 41

eld) perturbations. Source terms, like small-mass particles orbiting the spacetime,

should also be included in the form of the action S MA of Eq. (1.1).

Second order : we obtain the effective energy-momentum tensor which we use tocompute the gravitational radiation at innity (Isaacson tensor [ 131,132]). We also

obtain, as a second order effect, the scalar part of the radiation, which exists due

to the gravito-scalar coupling [133].

Dealing with perturbations by directly using the Einsteins equations can lead to cum-

bersome calculations [11]. One way to facilitate the calculations, is spliting the angular

dependence of the perturbations into known functions. In spherically symmetric space-

times, we can do this with the aid of the tensorial spherical harmonics [129,134]1. Within

this formalism, the metric perturbations naturally split into two sectors, named axial and

polar, with dened angular parity 2. Parity properties can be analyzed by applying the

parity transformation, given by ( , ) ( , + ). Because of this, and the fact thatthe background metric is invariant under parity transformations, the two sectors can be

viewed as independent to each other. Therefore, the metric perturbations can be written

as

hab = haxialab + h polarab , (2.3)

where [115, 137 139]

haxialab =l|m |

0 0 h0 S lm h0 S lm

0 h1 S lm h1 S lm

1sin h2 X lm sin h2W lm sin h2X lm

, (2.4)

1Recently this formalism was extended to compute perturbations in slowly rotating BHs [135].2Axial and polar are the names we shall use to distinguish perturbations with different parities.

Under parity transformation, axial perturbations transform as ( 1) l+1 while polar perturbations as ( 1)l .One can also nd them with the names odd and even, or magnetic and electric [136].



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and

h polarab = l|

m

|ev H 0Y lm H 1Y lm 0Y lm, 0Y lm, e u H 2Y lm 1Y lm, 1Y lm, r 2( K Y lm + G W lm ) r2 GX lm

r 2 sin2 ( K Y lm G W lm )

. (2.5)

The symbol denotes symmetric component, such that hab = hba . Here (hi , i , H i , K, G)3

are functions of (t, r ) only, Y lm = Y lm (, ) are the standard scalar spherical harmonics

and

S lm , S lm

Y lm,sin

, sin Y lm, , (2.6)

X lm , W lm 2(Y lm, cot Y lm, ), Y lm, cot Y lm, Y lm,sin2

. (2.7)

Even with the aid of this angular decomposition, the problem is still complicated.

We see that the metric perturbations reduce to ten unknown functions of ( t, r ) . Theproblem is greatly simplied noting that different waves can represent the same physical

phenomena viewed in different systems of coordinates [115]. In other words, we can exploit

the gauge freedom of the theory 4. In their paper, Regge and Wheeler found a gauge in

which the gravitational perturbations are simpler. Basically they exploit the fact that

innitesimal coordinate transformations do not change the background metric, such that

we can choose the innitesimal transformation in order to simplify only the perturbation

functions. With this, we can readily eliminate one function in the axial sector, namelyh2, and three in the polar part, namely ( G, i). We direct the reader to Ref. [ 138] for a

more detailed calculation.3The index i here should not be confused with the spatial components. We just used in order to

collectively denote the perturbation with a compact notation.4GR is invariant under any coordinate transformation, and we can consider this as being the gauge

symmetry of GR [ 92].



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After simplifying the perturbations with the Regge-Wheeler gauge, we can expand the

metric perturbation in a Fourier decomposition in time. Hence, the metric perturbations

in the frequency domain are given by:

haxialab =l|m | d

0 0 1sin h0(r ) sin h0(r ) 0 1sin h1(r ) sin h1(r ) 0 0

0

Y lm eit (2.8)

and polar sector

h polarab =l|m | d

evH 0(r ) iH 1(r ) 0 0

e u H 2(r ) 0 0

r 2K (r ) 0

r 2 sin2 K (r )

Y lm eit . (2.9)

We should note that each metric and scalar eld perturbation, e.g. h0(r ), explicitly

depends on the frequency ( R ) and on the wave numbers l and m.The scalar eld perturbations can be expanded in terms of the standard scalar spherical

harmonics, namely

=l|m | d

+ (r )r

Y lm ei(+ )t , (2.10)

=l|m | d(r )r Y

lm ei()t , (2.11)

where > 0 is the frequency of the background scalar eld (see Chapter 1)5. Note that

the ansatz above differs from that used in Refs. [ 107,108]. The scalar eld potential can5We should note that the frequency shift in Eqs. (2.10) and ( 2.11) is to simplify thecalculations.



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be written as

V = V 0 +l|m | dV (r ) Y

lm eit . (2.12)

Likewise,6dV

d||2 = U 0(r ) +

l|m | dU (r ) Y lm eit . (2.13)

In the presence of matter elds other than the complex scalar, the energy-momentum

tensor of the matter T matterab has to be expanded in tensorial harmonics [ 126, 137]. In the

time domain, the matter stress-energy tensor of a particle in the = / 2 plane reads

T matterab = p

xa (t)xb(t)r p(t)2xt (t) e

1

2(v+ u)

(r r p(t)) (cos ) ( p(t)), (2.14)

where xa (t p, r p, 0, p) and p are the particles four-velocity and mass, respectively.The overdot denotes derivative with respect to the proper time of the particle.

In the case of BS spacetimes, the harmonic decomposition of the metric perturbations

was rst studied by Kojima et al. [108] and used to analyze the QNMs of mini BS con-

gurations by Yoshida et al. [107]. Note, however, that their ansatz for the scalar eld

perturbations is different from ours (although there is a direct correspondence betweenthem). The axial part of the perturbations does not couple with the scalar perturba-

tions, since the latter are expanded with angular functions of even parity, and because

of it Ref. [107] analyzes only the polar sector. Despite of that, we also analyze here the

axial part, obtaining quasinormal modes that are similar to the w-modes of regular uid

stars [136], which are purely spacetime modes 7. Moreover, the gravitational wave emitted

by the moving particle around a BS comes from both axial and polar sectors [128, 140].

Moving particles around BSs also emit scalar waves, depending on the value of the particleangular velocity.

In the following, we shall study the two sectors separately. As mentioned above, only

the polar sector of the gravitational perturbations couples with the scalar eld perturba-6The expansion of the potential ( 2.12) and of its derivative ( 2.13) are generic. Each scalar potential has

a specic form for V and U , which should be found through the explicit expression of the potential andthe eld perturbations ( 2.10) and ( 2.11). The expansion validity relies on the fact that these quantitiesare scalars in S 2 .

7To a classication of QNMs of star, we direct the reader to Ref. [136].



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tions, and this makes its structure richer than the axial sector.

2.1.2 Reduction and simplication of the perturbed equationsThe Einstein-Klein-Gordon system of equations for a point particle orbiting a BS is given

by

Rab 12

gabR = (T ab + T matterab ), (2.15)

= 1 g

a ggab b = dV sd||2

, (2.16)

together with the complex conjugate of Eq. ( 2.16). T matterab is the energy-momentum tensorof the point particle expanded in terms of the tensorial harmonics (see Appendix A). The

linearized Einsteins equations are obtained by considering the rst order corrections of

the above system, using the perturbation scheme presented in Sec.( 2.1.1). In the following

we shall treat the axial and polar sector separately.

Axial sector

As in the regular uid star case [136, 141, 142], the axial modes do not couple with thematter terms. Therefore, the simplications of the axial sector follow a similar fashion

of the regular star case. From the linearized Einsteins equations, we have that the rst

order perturbation terms are:

eu h1 + ievh0 + 12

( prad )h1 + 2r 2

m(r )h1 = P lm ,

(2.17)

ih 0 + 2ir h0 2 ev

r 2 (l(l + 1) 2) h1 = P rlm ,(2.18)

ih 1 + h0 12

re u ( + prad )(h0 + ih 1) + 2i

r h1

+ h0eu ( prad + ) l(l + 1)

r 2 +

4m(r )r 3

= P tlm ,

(2.19)



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As explained in Sec. 2.1.1, the metric perturbations depend on the values of ( , l ,m). In

Eqs. (2.17)(2.18), P lm , P rlm and P tlm are source terms which depend on the particles

stress-energy tensor [81] (see also Appendix A). We can also dene h1(r ) in terms of theRegge-Wheeler function RW (r ),

h1(r ) = e12 (uv) r RW (r ) . (2.20)

Substituting the relation above into Eq. ( 2.17), the function h0(r ) can be written in terms

of RW as

h0(r ) =

i

e12 (vu ) d

dr [r RW (r )]

i

P lm (r ) . (2.21)

Equations ( 2.17)-(2.19) are not all independent, due to the Bianchi identities 8 . Indeed,

they are equivalent to a single Regge-Wheeler equation for RW , namely:

d2

dr 2+ 2 V RW (r ) RW (r ) = S RW (r ) , (2.22)

where r is the Regge-Wheeler coordinate, dened through dr = e(uv)/ 2dr , V RW (r ) isthe Regge-Wheeler potential

V RW (r ) = evl(l + 1)

r 2 6m(r )

r 3 2

( prad ) , (2.23)

and S RW (r ) is the source term

S RW = e

12 (vu)

r2ev

r1

rv2

P lm evP lm + P rlm .

Note that the homogeneous Regge-Wheeler equation ( 2.22) with the potential ( 2.23) isequivalent to that of an isotropic, perfect-uid star with pressure equal to prad [120, 141,

142].8The Bianchi identities read [a Rbc ]de = 0, where Rabcd is the Riemann tensor and [ ] denotesantisymmetric part of the tensor.



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Polar sector

Due to the coupling between the metric and scalar eld perturbations, the polar sector

is much more evolved than the axial one. Here we shall follow a procedure similar tothe one presented in Ref. [ 126], although choosing different fundamental functions 9. The

linearized Einsteins equations read

K + K 2r

3 eu 1 + r2prad + H 12r 2

l(l + 1) 2r 2( ptan + ) H 0

r+

r 2

r (( + )+ + ( ))0 + 0 + r + + r =

1

A(1) (, r ) 2rF (, r ) , (2.24)H 0 +

K 2r

3 eu 1 + r2prad H 0

r2 eu (1 + r2prad )

+12

H 1l(l + 1)

r 2 2ev2 2( ptan + )

+ r 2

r (( )+ ( + ))0 + 0 + r + + r =

1

A(1) (, r ) + B(, r ) rF (, r ) 1 eu 1 + r2prad , (2.25)H 1 + ( H 0 + K )eu +

H 1r (r 2m)

2m r 3V 0 2r

eu 0(+ )= e

u

B (0) (, r ) + 2 r 2eu F (, r ) , (2.26)

where the source terms A(1) , F, B and B (0) read

A(1) (, r ) = 2 2 dt A(1)lm (r, t )eit , (2.27)

F (, r ) = 2 2(l 2)!(l + 2)! dt F lm (r, t )eit , (2.28)

B(, r ) = r

2 l(l + 1) dt Blm (r, t )eit , (2.29)

B (0) (, r ) = r

2 l(l + 1) dt B (0)lm (r, t )e

it , (2.30)

9Zerilli [126] used (K, H 1 , H 2) to describe the gravitational perturbations and we shall use ( K, H 0 , H 1).The reason behind this is to facilitate the comparison with the equations presented in Refs. [107,108].



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and the functions A(1)lm (r, t ), F lm (r, t ), Blm (r, t ) and B(0)lm (r, t ) for the Schwarzschild back-

ground are explicitly given in Ref. [137]. In the background ( 1.10), these functions can be

computed in a similar fashion and they reduce to those in Ref. [ 137] in the vacuum case(see Appendix A). We have also used that

H 2 = H 0 2r 2F (, r ), (2.31)

which is obtained from the linearized Einsteins equations. The scalar eld perturbations

are governed by the following inhomogeneous equations:

d2

dr 2 + ( )2 V (r ) = S (2.32)

where

V = evl(l + 1)

r 2 +

2mr 3

+ U 0 V 0 ,S =

eu 2

2r ( 2)0 4 eu r 2( prad + ) 0 H 1

r0 (

2)H 0 + K + eu H 1

+ evr eu (K H 0) 0 (U 0H 0 + U )0+ r3F 0 (2evU 0 (2 )) + evu 0

2r

+ F F

.

Therefore, the polar sector is described by three rst-order Einstein equations coupled

to two second-order scalar equations. There exists an algebraic relation between K , H 0

and H 1 that can be used to eliminate one of these gravitational perturbations. Finally, the

system (2.24)(2.26) and (2.32) can be reduced to three coupled second order differential

equations, consisting, e.g., in the Eqs. ( 2.32) and a third equation. This is in contrast

with the case of perfect-uid stars, where the polar sector is described by a system of two

second-order equations [120,142 144]. Here, rather than working with three second order

equations, we shall use the system of equations given by Eqs. ( 2.24)(2.26) and (2.32).



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2.2 Quasinormal modes of boson stars 50

2.2.1 Boundary conditions

Axial modes

The source-free (S RW = 0) axial perturbations can be reduced to the homogeneous Regge-

Wheeler equation d2

dr 2+ 2 V RW (r ) RW (r ) = 0 , (2.33)

where V RW is dened in Eq. (2.23) and it is shown in Fig. 2.1 for some BS models and

for the case of a Schwarzschild BH. Note that Eq. ( 2.33) does not involve the scalar eld

perturbations, in analogy to the uid perturbations of an ordinary star, which are only

coupled to the polar sector. This decoupling led Yoshida et al. [107], based on Ref. [87], toassume that the axial sector of BSs is not coupled to gravitational waves and therefore

not interesting. However, we show here that BS models generically admit axial QNMs,

in analogy to the w-modes of ordinary stars, which are in fact curvature modes similar

to those of a BH [142, 145]. Moreover, for ultracompact stars ( R < 3M ), a potential

well appears in the Regge-Wheeler potential, cf. the right plot in Fig 2.1, generating the

possibility of having trapped QNMs, which are long-living modes [136, 146]. In chapter

3, we shall also show that axial perturbations with odd values of l + m are emitted bypoint particles orbiting the BS and therefore they contribute to the gravitational-wave

signal emitted during the inspiral.

At the center of the star, we require regularity of the Regge-Wheeler function,

RW (r 0) r l+1N

i=0

a(i)0 ri , (2.34)

where the coefficients a(i)0 can be obtained solving the Regge-Wheeler equation order by

order near the origin. At innity, the solution of Eq. ( 2.33) is a superposition of ingoing

and outgoing waves. The QNMs are dened by requiring purely outgoing waves at innity,

i.e.

RW (r ) eirN

i=0

a(i)r i

, (2.35)

10 The homogeneous equations are obtained by neglecting the source terms in Eq. ( 2.22) for the axialperturbations and in Eqs. ( 2.24)(2.26) and (2.32) for the polar perturbations.



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where again the coefficients a(i) can be obtained perturbatively. These boundary condi-tions are satised only by some characteristic frequencies, which are the QNM frequencies.

Polar modes

As in the axial case, at the origin we require regularity of the perturbations and we can

expand them in powers of r as

X (r 0) r lN

i=0

x(i)0 r i , (2.36)

where X collectively denotes H 2 = H 0, K , H 1 and . By substituting the expansion(2.36) in the homogeneous polar system of equations and solving iteratively the equations

in a power series in r near r 0, one can show that the expansion ( 2.36) depends onthree free parameters.

At innity, the background scalar eld vanishes and gravitational and scalar pertur-

bations decouple [107]. Therefore, the metric and the scalar eld perturbations at innity

can be analyzed separately.

Let us now discuss the asymptotic behavior of the gravitational perturbations. In the

vacuum, all polar metric perturbations can be written in terms of one single function Z

which obeys the Zerilli equation [126, 137],

d2

dr 2+ 2 V Z (r ) Z = 0 , (2.37)

where dr/dr = 1 2M/r ,

V Z (r ) = drdr 2

2

r2

(3M + ( + 1)r ) + 18M

2

(r + M )r 3(r + 3 M )2 , (2.38)

and = (l 1)(l + 2) / 2. Once again, the QNMs are dened by requiring purely outgoingwaves at innity [119], i.e.

Z (r ) eirN

i=0

x(i)r i

, (2.39)



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where the coefficients x(i) can be obtained perturbatively through differential equation

(2.37). The metric perturbations can be written in terms of the Zerilli function through

the following equations [126, 137]:

H 1 = r 2 3Mr 3M 2(r 2M )(r + 3M )

Z r 2dZ /drr 2M

,

K =( + 1)r 2 + 3 M r + 6 M 2

r 2(r + 3 M )Z +

dZ dr

,

H 0 = H 2 =r (r 2M ) 2r 4 + M (r 3M )

(r 2M )(r + 3 M )K

+M ( + 1) 2r 3

r (r + 3 M )H 1 .

Due to the mass term, the asymptotic behavior of the scalar eld perturbations is more

complicated than that for the gravitational perturbations. In the vacuum, the homoge-

neous equations for the scalar perturbations ( 2.32) reduce to

d2

dr 2+ ( )2 V (r ) = 0 , (2.40)

whereV (r ) = 1

2M r

2 + l(l + 1)

r 2 +

2M r 3

. (2.41)

The asymptotic solution for the scalar perturbations reads

(r ) Bek r r + C ek r r , (2.42)

where we have dened M2/k and

k = 2 ( )2 . (2.43)Without loss of generality, we choose the root such that Re[ k] > 0. Different physicallymotivated boundary conditions are possible for the scalar eld, depending on the sign

of the imaginary part of k, Im[k] (R ) I . As usual, a purely outgoing-waveboundary condition at innity, i.e. ei|Im[k ]|r , denes the QNMs. On the other hand,



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Table 2.1: Possible boundary conditions at innity for the scalar eld perturbations with eigenfrequency = R + iI .

I R Im[k ] QNM condition Bound-state condition

I stable, I < 0 R > Im[k+ ] > 0, Im[k ] > 0 B+ = 0, B = 0 C + = 0, C = 0II stable, I < 0 R < Im[k+ ] > 0, Im[k ] < 0 B+ = 0, C = 0 C + = 0, C = 0III unstable, I > 0 R > Im[k+ ] < 0, Im[k ] < 0 C + = 0, C = 0 C + = 0, C = 0IV unstable, I > 0 R < Im[k+ ] < 0, Im[k ] > 0 C + = 0, B = 0 C + = 0, C = 0

due to the presence of the mass term it is possible to have quasi bound-state modes, i.e.

states that are spatially localized within the vicinity of the compact object and decay

exponentially at spatial innity [135,147, 148]. Therefore, quasi bound-states are simply

dened by C = 0. In BS cases, the QNM conditions depend on R and on I , as shownin Table 2.1, where all cases are listed. In the following, we detail the QNM condition for

stable and unstable modes.

Let us start discussing the boundary conditions for stable modes ( I < 0). When

R > the QNM condition is the same for both scalar perturbations, B = 0. However,if R < , the QNM condition for the scalar elds perturbations is different, being B+ = 0

and C = 0. Note that in this case the stable QNMs of decay exponentially anddegenerate in the bound-state modes condition.

For unstable modes ( I > 0) the situation is different. In this case when R > ,

the QNM condition is the same for both scalar perturbations, C = 0 and coincide withthe bound-state conditions. However, when R < the QNM conditions read C + = 0

and B = 0, so that only the unstable QNM of + degenerates with the bound-statecondition.

This peculiar behavior is due to the presence of a mass term in the scalar eld (which

allows for bound states) and of a complex background scalar eld, = 0, which essentiallyshifts the real part of the frequency of the scalar perturbations. Note that in the case

of probe complex scalars around a Schwarzschild BH, the terms introduced by can be

eliminated by a simple shift of the wave frequency, but in the BS case this shift is physical

because of the coupling with the gravitational perturbations.



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2.2.2 Methods to compute quasinormal modes

Here we shall discuss two distinct methods used in this monograph to compute the quasi-

normal modes of stars: the direct integration method and the continued fraction method.The direct integration method here discussed is slightly different from the one presented in

Ref. [139]. The continued fraction method is similar to the one discussed in Refs. [ 138,149],

which we adapted to our case in order to suit the scalar eld as a star matter distribution.

The methods shall be discussed using a system of N second order differential equations,

and can be applied for both axial and polar perturbations 11 .

Direct integration methodLike the eigenvalue problem to nd the background frequencies , we can also nd QNMs

frequencies through a direct integration shooting method. The method consists of in-

tegrating the differential equations from two different regions: near the origin and far

from the star with the proper QNM boundary conditions. The problem of nding the

QNM frequencies then consists in nding the proper values of for which the solutions

obtained integrating fr

gravitational waves signatures of dark matter

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