Looking for dark matter near massiveblack holes with eLISA

Rob Daniel Hesselink10587454

Report bachelor project physics and astronomy15EC

Conducted between April and July 2016at Nikhef

University of AmsterdamFaculty of science

Supervisor:Dr. Chris Van Den Broeck

Second assessor:Prof. Dr. Patrick Decowski

Submitted on 7 July 2016

1

Contents

Scientific summary 4

Popular summary (Dutch) 5

1 Introduction 61.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Theoretical framework 82.1 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Space-based detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 TaylorF2 approximant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Extreme mass-ratio inspiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 Dark matter density profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Dark matter effects on inspiral . . . . . . . . . . . . . . . . . . . . . . . . . . 132.8 Dark matter waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.9 Fisher information matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Method 163.1 Combining waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 eLISA Noise curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Measurement accuracy of eLISA . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Results 194.1 Higher order phase terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.4 eLISA measurement accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Discussion 26

6 Conclusion 27

A Determination of initial observation frequency 28

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My thanks to Laura van der Schaaf, Jeroen Meidam, Chris van den Broeck and the Virgoteam at Nikhef, for their untiring support and good company.

My thanks to my friends and family who proofread this thesis, whose suggestions andcriticisms have helped me greatly.

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Scientific summary

A large part of the matter in the universe has not yet been directly detected. This is becausethis matter does not interact with light, which has resulted in it being named “dark matter”.While there are many possible candidates that might constitute dark matter, research sofar has been inconclusive. Recently, new methods of studying dark matter have becomeavailable. In the past year, gravitational wave detectors have made their first detections.This has opened up a new domain in astrophysics, one which may be well suited to studydark matter, since its only known mechanism of interaction is gravity.

Recent studies have shown that a dark matter density spike can form around a black hole.The shape of this density spike depends on the initial profile of the dark matter halo. Thisdark matter spike influences the inspiral of objects into the black hole, hereby leaving afootprint in the gravitational waves emitted by the system.

The effect of dark matter on gravitational waves is especially pronounced in extreme massratio inspirals (EMRI’s). In these events, a lighter companion falls into a much heavierblack hole, which is heavier by at least a factor thousand. This light companion is affectedby gravitational effects of the dark matter, which causes it to fall into the black hole morequickly. This altered path changes the gravitational waves emitted by the system. Darkmatter will leave a footprint in the gravitational waves emitted, which will be detectable byspace-based gravitational wave detector eLISA.

The aim of this thesis was to investigate which EMRI systems will be detectable by eLISAand how precisely the dark matter parameters can be determined. We classify systemsdetectable when they have a signal-to-noise ratio (SNR) higher than 8. eLISA’s measurementwas determined using Fisher matrices.

A wide array of systems was found to be detectable by eLISA, with black hole masses rangingfrom 103M� to 106M� being visible, depending on companion mass. A steep dark matterdensity severely impacts detectability, due the decreased amount of revolutions around theblack hole.

The dark matter density is described by parameters α and β. The first parameter describesthe steepness of the density profile, while the second describes the initial conditions ofthe dark matter density spike at the radius where it is established. Analysis using Fishermatrices has resulted in a measurement accuracy ∆α

α = 2.52 · 10−7(

10SNR

)and ∆β

β = 1.04 ·10−4

(10

SNR

), dependent on signal-to-noise ratio (SNR).

Combined with complementary searches into dark matter annihilation, eLISA will provideinsight into the nature of dark matter particles, as well as the dark matter density profile,providing valuable insight into this unresolved question in physics.

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Popular summary (Dutch)

Op 14 september 2015 hebben de eerste waarnemingen plaatsgevonden van zwaartekrachts-golven. Die waarnemingen zijn gedaan door een zwaartekrachtsgolfdetector. Zoals eengewone telescoop lichtgolven kan meten, meet een zwaartekrachtsgolfdetector de zwaartekrachts-golven. Deze techniek opent een nieuw kennisdomein in de sterrenkunde; verre objecten dieniet konden worden waargenomen door middel van het licht dat ze uitzenden, kunnen wenu wel waarnemen door naar hun zwaartekrachtsgolven te kijken.

Zwaartekrachtsgolven ontstaan als een massa versneld wordt. Als in het universum tweeobjecten (bijvoorbeeld twee zwarte gaten) om elkaar heen draaien, dan zenden ze sterkezwaartekrachtsgolven uit. Als deze golven door de ruimte reizen, dan vervormen ze deruimte: ze doen de ruimte uitzetten en krimpen, zoals een kiezelsteen die in een meer valt.Het effect van zwaartekrachtsgolven is echter heel erg klein. Hoewel ze al in 1916 doorAlbert Einstein voorspeld zijn, heeft het bijna honderd jaar geduurd voordat de technologiebestond om ze te kunnen waarnemen. Nu de technologie er is, kan dit bijdragen aan deoplossing van een van de grootste onopgeloste problemen in de natuurkunde: dat van dedonkere materie.

Een groot deel van de materie in het universum hebben we tot nu toe nog niet kunnenwaarnemen. Dat komt omdat deze materie geen licht uitzendt of opneemt. Daarom noemenwe het ook wel donkere materie. Hoewel we deze donkere materie dus niet kunnen zien,weten we dat het bestaat. Donkere materie heeft zwaartekracht en beınvloedt daarmeeobjecten die we wel kunnen waarnemen, zoals sterren.

De invloed van donkere materie is erg groot wanneer een licht object, zoals een klein zwartgat of een neutronenster, door de zwaartekracht in een veel groter zwart gat wordt getrokken.Donkere materie rondom het zwarte gat trekt aan het lichte object, waardoor de baan om hetzwarte gat wordt veranderd. Deze verandering is zichtbaar in de zwaartekrachtsgolven dieuitgezonden worden. De donkere materie laat een duidelijk spoor na in de zwaartekrachts-golven die we waarnemen.

Een hoekpunt van de eLISA detector.Bron: ESA

Nu het mogelijk is om donkere materie viazwaartekrachtsgolven waar te nemen, is de hoopgevestigd op de nieuwe detector eLISA. Deze detectorbestaat uit drie satellieten die in een driehoeksvormin een baan om de aarde bewegen. Elk hoekpunt stu-urt een laser naar een ander hoekpunt, waarmee heelprecies de afstand tussen de drie satellieten bepaaldkan worden. Wanneer er een zwaartekrachtsgolfdoor de satellieten beweegt, veranderen de afstandentussen de hoekpunten, waardoor de zwaartekrachts-golf gemeten kan worden.

In dit bacheloronderzoek is geanalyseerd welke systemen eLISA kan meten en hoe nauwkeurigde dichtheid van donkere materie gemeten kan worden. Wanneer detector eLISA in 2034gelanceerd wordt, kan deze satelliet bijdragen aan kennis van de donkere materie. Dit brengtons een stap dichterbij de oplossing van dit belangrijke vraagstuk in de natuurkunde.

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1 Introduction

1.1 Motivation

The dynamics of stars, clusters and galaxies all suggest that the majority of matter in theuniverse has not yet been observed. This missing matter does not interact with light, whichis why it has been aptly named “dark matter”. It is unclear whether dark matter is a singletype of matter, or consists of multiple constituents. While a plethora of candidates has beensuggested that might constitute dark matter, research so far has been inconclusive [1]. Asboth the evidence and potential candidates for dark matter keep increasing, it might bemost advantageous to study dark matter via its only mechanism of interaction known sofar: gravity.

Gravitational waves were first detected directly on the 14th of September 2015 after havingbeen predicted by general relativity almost a hundred years prior [2, 3]. These waves offerastronomers a new way to view the cosmos, as all objects that interact gravitationally canemit these waves. Gravitational waves are difficult to detect, since they interact weaklywith matter. Measuring displacement due to gravitational waves by conventional laserinterferometer detectors requires both extremely high precision equipment and many metresof vacuum tunnels. Existing observatories are most sensitive in the acoustic range (10-10,000Hz) [4] and unable to detect waves of very low frequencies (0.001-1 Hz), as this would requirelonger detector arms. Space-based detectors may solve this problem, as space provides asuitable vacuum, which allows for longer detector arms [5]. These space-based detectorsmay be able to detect primordial gravitational waves - a remnant of the Big Bang [6] - andextreme mass ratio inspirals.

Gravitational waves may be the key to effectively study dark matter. One specific area ofinterest is the structure of dark matter around massive black holes. Research has suggestedthat dark matter around a black hole will form a density spike [7], due to dark matter beingpulled towards the black hole. This configuration is only plausible for cold dark matter andits structure is influenced by the initial dark matter density profile.

We will investigate (i) which EMRI systems containing a dark matter density spike aredetectable by the eLISA gravitational wave detector and (ii) eLISA’s measurement accuracywhen confronted with EMRI systems containing dark matter.

This thesis aims to provide initial calculations for the eLISA detector that is set to launchin 2034 [5], so that its measurements may be used to expand our knowledge of dark matterand its behaviour.

1.2 Approach

While initially the detection of these dark matter density spikes relied purely on the luminos-ity of dark matter annihilations, dark matter signatures can also be found in gravitationalwaves. Dark matter signatures are especially pronounced at extreme mass-ratio inspirals(EMRI’s), for example a single solar mass star orbiting an intermediate mass black hole ofa thousand solar masses. Building on the work of Eda et al [8]., we will look at the effectsof the dark matter spike on gravitational waveforms in extreme mass-ratio inspirals and in

6

how far space-based detectors such as eLISA will be able to discover the characteristics ofthese mini spikes.

We will use the TaylorF2 [9] approximation to generate gravitational waveforms in thefrequency domain. After adding the dark matter contribution as found in Eda et al. to thestandard waveform, we shall explore in what ways it alters gravitational wave phase andamplitude.

Fisher analysis will be used to determine eLISA’s measurement accuracy, by calculating theroot-mean-square errors on the parameters that describe the gravitational waveform. Thefindings will be represented using error ellipses at different confidence levels. The effect ofobservation time on measurement accuracy will also be investigated.

The thesis is structured as to gradually introduce the components of the investigated astro-physical system. Chapter two will briefly introduce the principle of gravitational waves andtheir detection. It will then introduce the TaylorF2 approximant and the systems studiedin this thesis: extreme mass-ratio inspirals with dark matter. Section three will presentthe TaylorF2 dark matter waveform and explain how measurement accuracy of eLISA isdetermined. Chapter four will present the results that will be used to answer the researchquestions, which will be discussed in chapters five and six.

Throughout this thesis, in accordance with many works on general relativity, we will usegeometric units. This means that in all subsequent formulae G = c = 1.

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2 Theoretical framework

2.1 Gravitational waves

Gravitational waves are ripples in the fabric of space-time[4]. All accelerating masses pro-duce gravitational waves, just like accelerating charged particles produce electromagneticwaves. A difference between these two types of waves is that gravitational waves couplevery weakly to matter, making them incredibly difficult to detect.

Gravitational waves are a direct consequence of the theory of general relativity by AlbertEinstein. In this theory masses curve the space around them by virtue of having mass, whichis analogous to placing a brick on a trampoline: the brick pulls the fabric downwards, causingsurrounding objects to be drawn towards it. The mass of the brick has curved the space-timearound it. In real astrophysical systems, these ripples are sent out as gravitational wavesthat stretch space-time in one of two polarizations h+ and h× [10]. The displacement of aring of test particles due to a gravitational wave is shown in figure 1a and 1b.

(a) Plus polarization (b) Cross polarization

It is exactly this expansion and compression of space that gravitational wave detectorsexploit. Ground-based interferometric gravitational wave detectors such as LIGO [11] orVIRGO [12] consist of two long vacuum arms at a 90 degree angle. A laser beam is reflectedback and forth in the arms, giving an accurate measure of the length of the arm. Whena gravitational wave travels through the detector, the arms are affected differently. Thismeans that the diffraction pattern of the two beams changes as a gravitational wave passesthrough. We can infer the gravitational waveform by looking at the change in arm lengthover time. This technique has been successfully used to detect multiple events [2, 13].

Certain astrophysical systems are especially adept at creating gravitational waves with largeamplitudes and clear predicted waveform. Binary objects spiral around each other, meaningthat they consist of continuously accelerated masses, which continuously create gravitationalwaves. As the binary objects circle each other, they lose energy by emitting gravitational

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waves. This loss of energy slowly brings the two objects closer together, until they finallycollide and merge [14].

A satellite can orbit a heavier body up until its innermost stable circular orbit (rISCO). AtrISCO, the gravitational wave frequency (fISCO) is approximately twice its orbital frequency(ωGW = 2ωs) [8]. The orbital frequency is given in good approximation by Kepler’s thirdlaw. The radius of the innermost stable circular orbit and the corresponding gravitationalwave frequency are given by [10]

rISCO = 6Mtot (1)

fISCO =1

632πMtot

. (2)

This process is called an inspiral and it was first proven to be modeled correctly by Einstein’sequations by Hulse and Taylor [15]. Since then, gravitational waves have been directlydetected by the LIGO Scientific Collaboration and the Virgo Collaboration [2], offering anew way of looking at the cosmos.

2.2 Space-based detectors

Space-based detectors are based on the same principle as earth-based detectors. Interfero-metric techniques are used to sense the expansion and compression of space due to gravita-tional waves. The readily available vacuum of space however allows for much larger detectorarms, which means that the sensitive frequencies of space-based detectors are much lowerthan earth-based detectors [16]. These detectors consist of separate satellites that send lasersignals towards each other, hereby connecting the vertices into a detector arm. Proposeddetectors such as LISA and eLISA consist of three satellites in free fall, creating a triangulardetector [4, 5].

A sketch of the LISA detector. Source: ESA

Each of these three vertex points containsa test mass in free fall around the earth.Laser beams are used to determine the dis-tances between the test masses. A passinggravitational wave will change the separa-tion between the masses, which allows thesatellite to detect it.

The LISA gravitational wave detector wasoriginally planned as a collaborative mis-sion between ESA and NASA. However, dueto financial considerations, NASA has sincedecided not to pursue the mission further.The project has been adjusted into eLISA,a space-based detector project designed byESA. The LISA Pathfinder satellite, a mission with the goal to test technologies for theeLISA detector, was launched in 2015 and is already giving promising results. In figure2 the sensitivities of three detectors are shown: eLISA, LISA and Advanced LIGO. Of

9

these three, only the Advanced LIGO exists currently, with the LISA project having beencancelled and eLISA projected to launch in 2034.

10­5 10­4 10­3 10­2 10­1 100 101 102 103 104 105 106

Frequency [Hz]

10­24

10­23

10­22

10­21

10­20

10­19

10­18

10­17

10­16

10­15

10­14

10­13

10­12Strain sensitivity [H

z−0.5]

eLISALISAAdvanced LIGO

Figure 2: Strain sensitivity for three gravitational wave detectors. Advanced LIGO is anearth-based detector, whereas eLISA and LISA are space-based detectors. The ground-based and space-based detectors are sensitive in different regimes due to the difference inarm length.

2.3 Signal-to-noise ratio

We are also concerned with the detectability of the gravitational waves emitted by theextreme mass-ratio inspiral. The detectability depends on two things, the amplitude ofthe gravitational waves and the sensitivity of our detector. The ratio between these twogives us an indication of how sure we can be of our measurements. This ratio is called thesignal-to-noise ratio (SNR).

The detector sensitivity is characterised by the power spectral density (PSD) [4]. Thisfunction can be obtained by measuring the output of a detector with zero input and thentransforming this function to frequency domain. The PSD is then given by Sh(f), a functionthat gives the accuracy of the detector at specific frequency.

The signal strength is given by the amplitude of the gravitational wave. The amplitude ofany complex wave is given by the square root of the dot product with itself, denoted as|h(f)|2. The optimal signal-to-noise ratio for any detector and waveform is then given by[4]

SNRopt = 2

[∫ ∞0

|h(f)|2

Sh(f)df

]2

. (3)

10

The higher the SNR, the higher the confidence in the detector output. In this work, we shallclassify events with an optimal SNR lower than 8 as ‘poorly detectable’. While detectororientation with respect to the source has an influence on the SNR, we will not take thisinto account, in order to determine the systems theoretically detectable by eLISA.

2.4 TaylorF2 approximant

Einstein’s exact field equations that describe gravitational waves are very difficult to solvenumerically, which is why we resort to approximations when calculating these waveforms.The most commonly used approximation is the TaylorF2 approximation. The TaylorF2approximation is a seventh order Taylor expansion of the frequency domain waveform usinga stationary phase approximation. We will use frequency domain waveforms throughoutthis thesis. The derivation of the frequency domain waveform is beyond the scope of thiswork, but it can be found in Arun et al. [9].

The gravitational waveform, denoted h(f), is given below, along with its amplitude and theTaylorF2 approximation of the phase.

h(f) = Af−7/6eiΨ(f) (4)

A =5

24

1/2 1

π3/2

1

DMc

5/6 1 + cos2(ι)

2(5)

Mc =(M1M2)3/5

(M1 +M2)1/5(6)

Ψ(f) = 2πftc − φc −π

4+ Φ(f) (7)

Φ(f) =3

128ηv5

[1 +

20

9

(743

336+

11

4η

)v2 − 16πv3 + 10

(3058673

1016064+

5429

1008η +

167

144η2

)v4

+ π

(38645

756− 65

9η

){1 + 3ln

(v

vlso

)}v5 +

{11583231236531

4694215680− 640

3π2

− 6848

21γ − 6848

21ln(4v) +

(15737765635

3048192+

2255π2

12

)η +

76055

1728η2

− 127825

1296η3

}v6 + π

(77096675

254016+

378515

1512η − 74045

756η2

)v7

](8)

Here Mc is called the chirp mass, D is the distance to the barycenter of the inspiral, γ =0.57721... is the Euler-Mascheroni constant and ι is the angle of inclination with respectto the plane of the detector. The coefficients tc and φc are the time and phase of thegravitational wave at coalescence. Lastly, the coefficient η is the symmetric mass ratio andv is the parameter that contains the gravitational wave frequency. Here η and v are definedas:

η =M1M2

(M1 +M2)2(9)

11

v = (πMf)1/3, (10)

where M is the total mass of the binary system.

2.5 Extreme mass-ratio inspiral

An extreme mass-ratio inspiral (EMRI) is defined as an inspiral event where a lighter com-panion (∼ 1−102M�) is gravitationally pulled into a black hole (∼ 103−107M�) [17]. Blackholes in the centre of galaxies will attract nearby stars, which may be caught in an orbitaround the black hole. Companions with highly eccentric orbits will spend a large amountof revolutions in the eLISA frequency band, making these excellent sources of gravitationalwaves for space-based detectors [18].

There are several objects that can take on the role of companion. To be properly measurableby eLISA, the source will have to get close to the fISCO of the system. This means that thecompanion will have to approach the system’s rISCO without being destroyed by the blackhole’s gravity. Normal solar mass stars are therefore excluded as viable candidates, sincetheir radius is too large and they would not be able to approach the black hole close enoughwithout being deformed by tidal forces. Possible candidates include solar mass black holes,white dwarfs and neutron stars [8].

It is difficult to estimate the rate of EMRI’s. The rate of EMRI’s depends on the populationof black holes, which is rather poorly known since they emit hardly any electromagneticradiation. Calculations have however put the rate near 25 ∼ 50 events every two years [5].Conversely, the detection rate of eLISA can be used to study the black hole population inthe milky way [19].

2.6 Dark matter density profile

It has now been established that galaxies contain a lot of matter that does not interactwith electromagnetic radiation. The exact structure of dark matter is unknown, but one ofthe most widely used density profiles is the Navarro-Frenk-White profile [20]. This profilecontains two parameters, the scale radius rs and the scale density ρs. These parameterscan be varied to model several different assumptions about the inner profile of dark matterhalos, namely cuspy and core profiles [21]. The Navarro-Frenk-White profile is given by

ρNFW =ρs

(r/rs)(1 + r/rs)2. (11)

While the NFW profile in equation 11 is present throughout the whole galaxy, we will look ata local deviation from this profile, close to a black hole [7]. It has been shown that an NFWprofile can lead to the formation of a density spike near a black hole. This second densityprofile, ρspike, is a direct consequence of the NFW profile. When considering an initial NFWprofile which obeys ρ(r) ∼ r−αini with 0 < αini ≤ 2, depending on the parameters ρs andrs, the ρspike will be ρ(r) ∼ r−α with α = 9−2αini

4−αini . Explicitly, this means

ρspike = ρsp

(rspr

)α. (12)

12

where the exponent α of the dark matter spike density profile varies between 2.25 and 2.50[7, 22]. Here rsp is the radius at which the dark matter spike is formed and ρsp is the darkmatter density at rsp. An example of these two profiles combined can be found in figure 3.

0.0 0.5 1.0 1.5 2.0 2.5 3.0Radius [parsec]

10­18

10­16

10­14

10­12

10­10

10­8

10­6

10­4

10­2

100

102

104

ρ(r)   [kg

/m3]

ρspike ρNFW

Figure 3: The two density profiles. The density obeys the ρspike profile up until rsp (markedwith a black line), after which it switches to the ρNFW profile. The dashed line indicatesthe course of the NFW profile in absence of the dark matter spike

Since the dark matter density spike profile is a direct result of the initial halo profile,knowledge about the density spike may be used to construct the entire dark matter haloprofile.

2.7 Dark matter effects on inspiral

The dark matter halo affects the inspiral of the companion in two ways: dynamical frictionand gravitational wave back-reaction. Dynamical friction is an effect of the stellar massobject moving through the dense dark matter region. Its movement through the dark matterhalo accelerates the dark matter particles, who will follow in the wake of the satellite. Allmomentum that the particles gain, has to be lost by the satellite due to conservation ofmomentum. This means that the dark matter following the stellar mass object slows theobject down. This causes the object to spiral in more quickly, since it does not have thevelocity to escape the gravitational pull of the black hole. The loss of energy due to thisdynamical friction, with companion mass µ, is given by [8]

dEDFdt

= vfDF = 4πµ2ρDM (r)

vln(Λ) (13)

13

Here ln(Λ) is the Coulomb parameter, a measure for the amount of collisions the dark matterparticles undergo. We will set ln(Λ) = 3, assuming weakly interacting dark matter particles.

Since the object is now spiraling in faster, the object also loses more energy by emittinggravitational waves. This is a secondary effect of the dynamical friction. The energy radiatedaway in the form of gravitational waves and the orbital frequency are then given by [8, 10]

dEGWdt

=32

5µ2R4ω6

s , (14)

ωs(R) =

[Meff

R3+

F

Rα

]1/2

(15)

where Meff , F and MDM (< rISCO) are defined as:

Meff =

{MBH −MDM (< rISCO) (rISCO ≤ r ≤ rsp)MBH (r < rISCO)

(16)

F =

{rα−3MDM (< rISCO) (rISCO ≤ r ≤ rsp)0 (r < rISCO)

(17)

MDM (< rISCO) = 4πrαspρsprα−3ISCO/(3− α). (18)

It can be seen that energy loss speeds up due to the influence of dark matter. This secondaryeffect is less pronounced, because the alteration of orbital frequency ωs is relatively small.

The second dark matter effect is due to the modified Newtonian potential. The companionmass orbits a black hole with a steeper gravitational potential, since the region containedwithin the orbital radius is inhabited by dark matter particles. The increased gravitationalpotential also increases the energy radiated away by means of gravitational waves.

The coefficients describing the dynamical friction (cDF ) and the modified Newtonian po-tential (cGW ) can be found in Appendix A. Of the two, dynamical friction is many ordersof magnitude larger. This means that most of the dark matter effects are due to dynamicalfriction.

The presence of dark matter causes the inspiral of the companion to occur more quicklythan it would have in a system without dark matter. The effects of this altered motion willbecome evident in the next section.

2.8 Dark matter waveform

Continuing from the TaylorF2 waveform displayed in an earlier section, we need to addseveral terms to come to the waveform that is produced by our EMRI system with darkmatter.

Firstly, we need to add a correction to the phase of the gravitational wave. The first orderphase for the dark matter waveform is taken from Eda et al. and is given by [8]

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ΦDM (f) =10

3(8πMc)

−5/3

[−f∫ f

fISCO

df ′f ′−11/3L−1 +

∫ f

fISCO

df ′f ′−8/3L−1

](19)

L(f) = 1 +5

8π

2α−113 M

− 10+2α6

BH βf2α−11

6 , (20)

where β ≡ ρsprαspln(Λ). Since these parameters are only ever present as a product, we can

only measure their combined effect on the gravitational waveform. This convention meansthat the dark matter effects are dependent on just two parameters, which we shall henceforthrefer to as α and β. Of these parameters, α is directly connected to the steepness of thedark matter density spike, whereas β provides information on the density and location ofthe dark matter spike, since it is a combination of the spike’s radius and density.

The correct expression for the amplitude is the final alteration to our original waveform.The resulting waveform is then described by

h(f) = Af−7/6eiΨDM (f)L(f)−1/2 (21)

2.9 Fisher information matrix

Fisher matrices are used to determine the variance of parameters in a system. This is usedin section 3.3 to determine the measurement accuracy of the eLISA detector. From theFisher matrix we can determine error ellipses, give us the confidence of measurement withina specific confidence level. The elements of the Fisher matrix for a system h, dependent onparameters ~θ, are given by [23]:

Fij ≡

(∂h

∂θi

∣∣∣ ∂h∂θj

), (22)

where the inner product is defined as

(h1|h2) ≡ 4Re

∫ fISCO

fini

h1(f)h∗2(f)

Sn(f)df, (23)

which is a noise-weighted inner product. This is where eLISA’s sensitivity is used to deter-mine the measurement accuracy on gravitational wave parameters. The inverse of the Fishermatrix is the covariance matrix, which contains the root-mean-square errors on the param-eters that describe the system. These errors are found on the diagonal of the covariancematrix. Explicitly:

∆θi =√

(F−1)ii. (24)

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3 Method

3.1 Combining waveforms

It is important that the expression for phase in equation 19 returns to the first order TaylorF2expression when all dark matter parameters are set to zero. This requirement is satisfiedby choosing specific values for the aforementioned tc and φc. Let us first calculate theexpression for the phase without dark matter.

ΦDM (f)∣∣∣α=β=0

=10

3(8πMc)

−5/3

[− f

∫ f

fISCO

df ′f ′−11/3 +

∫ f

fISCO

df ′f ′−8/3

]

=10

3(8πMc)

−5/3

[3

8ff ′−8/3 − 3

5f ′−5/3

]ffISCO

=10

3(8πMc)

−5/3

[− 9

40f−5/3 − 3

8ff−8/3ISCO +

3

5f−5/3ISCO

](25)

And looking specifically at the first term in equation 25:

10

3(8πMc)

−5/3

[− 9

40f−5/3

]=

3

128(πf)−5/3

((M1M2)3/5

(M1 +M2)1/5

)−5/3

=3

128(πf)−5/3 M1M2

(M1 +M2)2.

(26)

Noting then that the first order TaylorF2 term (PfaN) is defined as:

PfaN =3

128ηv

=3

128· M1M2

(M1 +M2)1/3· (πMf)−5/3

=3

128(πMf)−5/3 M1M2

(M1 +M2)2.

(27)

Equations 26 and 27 are equal, which means the two phases are equal, except for two residualterms in the phase expression in equation 25. Noting that one of these terms is linear inf and one is independent of f , we can compensate for these two terms by defining thefollowing:

2πtc ≡10

3(8πMc)

−5/3

[3

5f−5/3ISCO

](28)

φc ≡10

3(8πMc)

−5/3

[3

8f−8/3ISCO

](29)

This convention assures that when we remove dark matter from our system, we obtain theTaylorF2 results once again. In short, replacing the first order term in TaylorF2 by the dark

16

matter phase in equation 19 and choosing the appropriate values for φc and tc, we get thecorrect phase for the dark matter system.

3.2 eLISA Noise curve

In this work, the eLISA noise curve as used as given in [24], using the most pessimisticvalues for detector arm length (L) and the low-frequency acceleration (Sn,acc). This meansthat the SNR’s calculated here are lower limits to eLISA’s capabilities. The full noise curvecan be seen in equation 30.

Sn(f) =20

3

4Sn,acc(f) + Sn,sn(f) + Sn,omn(f)

L2×

[1 +

(f

0.41 c2L

)2]

(30)

Sn,acc = 9× 10−28 1

(2πf)4

(1 +

10−4Hz

f

)Sn,sn = 1.98× 10−23m2Hz−1

Sn,omn = 2.65× 10−23m2Hz−1

(31)

3.3 Measurement accuracy of eLISA

To determine eLISA’s measurement accuracy, we calculate the Fisher matrix for the follow-ing parameters: A, tc, φc, Mc, α and β. The gravitational waveform is completely describedby these parameters, resulting in a 6x6 dimensional Fisher matrix. The derivatives of h withrespect to these parameters are given by the following:

∂h

∂lnA= h (32a)

∂h

∂tc= 2πifh (32b)

∂h

∂φc= −ih (32c)

∂h

∂lnMc=

5

3ihΦ (32d)

∂h

∂lnα= αh

(i∂Ψ

∂α− 1

2

1

L

∂L

∂α

)(32e)

∂h

∂lnβ= βh

(i∂Ψ

∂β− 1

2

1

L

∂L

∂β

). (32f)

The derivatives on Ψ and L are calculated numerically.

The initial observation frequency, fini, is necessary to calculate the elements of the Fishermatrix. The initial observation frequency is the frequency of gravitational waves at the start

17

of observation by eLISA. This initial frequency is dependent on the total observation timeuntil coalescence of the binary objects. It is therefore necessary to determine the frequencyevolution of the system, f(t). We use the same method as Eda et al.[8], who have shown thatthe initial observation frequency fini is strongly dependent on α. The initial observationfrequency can be found using the equations in appendix A.

Error ellipses were made using the variance computed from the fisher matrix. When creatingan error ellipse in the plane of parameters x and y with corresponding σx, σy and σxy, andwith probability ρ, the following equations describe an ellipse with semi-major and semi-minor axes a and b and orientation θ [25, 26]:

a2 = k2 ·

(σ2x + σ2

y

2+[ (σ2

x − σ2y)2

4+ σ2

xy

]1/2)(33)

b2 = k2 ·

(σ2x + σ2

y

2−[ (σ2

x − σ2y)2

4+ σ2

xy

]1/2)(34)

tan2θ =2σxy

σ2x − σ2

y

(35)

k =√−2ln(1− ρ) (36)

18

4 Results

To start the results, we will explicitly state the parameters of our standard system. Theseparameters will be the ones used in all subsequent figures, unless when stated otherwise.

Table 1: Standard inspiral parameters

MBH µ Distance ρsp rsp α ln(Λ)1000 M� 1M� 100 Mpc 226M�/pc3 0.54 pc 7/3 3

4.1 Higher order phase terms

First, we will investigate the effect of the higher-order terms on the dark matter waveform.Using the standard parameters in table 1, we have plotted the two phases.

10­2 10­1 100

Frequency [Hz]

104

105

106

107

Φ

Taylor F2Only leading order

(a)

10­2 10­1 100

Frequency [Hz]

10­4

10­3

10­2

10­1

|∆Φ(f)|/ΦF2(f)

(b)

Figure 4: The difference between TaylorF2 and leading order phase. Figure 4a shows theevolution of phases. Figure 4b shows the absolute value of the phase difference relative tothe TaylorF2 phase.

It can be seen that while the two phases are quite coherent at first, they begin to deviatestrongly as they approach the fISCO of the system. The qualitative behaviour of the phasestarts to deviate so strongly, that the leading-order approximation breaks down past afrequency close to 1 Hz. This means that the higher order terms are significant at higherfrequencies and should not be left out.

19

4.2 Signal-to-noise ratio

We wish to investigate which systems will be detectable by the eLISA detector. Firstly, wewill keep the dark matter spike fixed, while varying black hole mass MBH and companionmass µ. This will give an indication which EMRI systems with dark matter can be detectedby eLISA. Secondly, we will look at the dark matter effect within a system with massesdescribed in table 1. We will again look at the separate effect of the two dark matterparameters α and β.

102 103 104 105 106 107

Black hole mass in solar masses

0

10

20

30

40

50

60

SNR

Figure 5: Signal-to-noise ratio varying black hole mass MBH . The peak SNR lies at approx-imately 105 M�. Inspirals with higher masses occur outside eLISA’s sensitive frequencyband and are therefore poorly detectable. Companion mass µ is kept fixed at 1M�

0 20 40 60 80 100Companion mass in solar masses

0

10

20

30

40

50

60

70

80

90

SNR

Figure 6: Signal-to-noise ratio varying black hole mass µ. Signal-to-noise ratio increases asthe mass ratio approaches 1. MBH is kept fixed at 1000 M�.

20

Judging from figure 6, companion mass µ has a different effect on detectability than the blackhole mass MBH does. As the ratio of the two masses approaches one, the signal-to-noiseratio goes up. While this is in principle a preferable situation, this causes problems whenattempting to detect the presence of dark matter. As stated before, the most important darkmatter effect on the inspiral is mostly due to dynamical friction. This effect is dependenton the ratio of the dark matter mass and the companion mass. The heavier the companion,the smaller the dark matter effect on the inspiral path.

1.5 2.0 2.5 3.0 3.5

α

0

2

4

6

8

10

12

14

SNR

Figure 7: Signal-to-noise ratio varying α. The two vertical lines indicate the expected rangeof 2.25 ≤ α ≤ 2.5, assuming an initial Navarro-Frenk-White density profile. All other valuesare in accordance with table 1

As can be seen in figure 7, the dark matter spike severely impacts detectability. Since darkmatter causes the inspiral to occur more quickly, it decreases the amount of revolutionsaround the black hole, hereby decreasing the total signal. While the dark matter effectsare more pronounced at very massive dark matter halos, the probability of detecting thesesystems decreases as the SNR decreases.

21

0.0 0.5 1.0 1.5 2.0

β 1e−6

0

2

4

6

8

10

12

SNR

Figure 8: Signal-to-noise ratio varying β. The vertical line indicates the β value found intable 1. All other values are in accordance with table 1

4.3 Phase

Next, to quantify dark matter effect on phase, we have analysed the difference betweena waveform emitted by a system containing a dark matter spike and a system without adark matter spike. The phase difference is denoted as Φ(f)− Φ0(f) and it is crucial to thedetection of dark matter, as the dark matter effects are most pronounced in the phase ofthe gravitational waveform. Firstly, we will look at the effect of the steepness of the darkmatter spike, characterised by α. After this, we will see the effect of our second dark matterparameter, β on the phase of the gravitational waves.

Judging from figures 9 and 10 we can see that of the two parameters the effect of α, thesteepness of the dark matter spike, is much more pronounced than the phase change due tothe second parameter, β. Since α describes the density profile of the complete dark matterdensity spike, increased α increases total dark matter mass greatly. The parameter β on theother hand describes the radius where the spike is established and its density at that point,which affects the total dark matter mass less strongly than α does.

22

10­2 10­1 100

Frequency [Hz]

10­1

100

101

102

103

104

105

106

107

108

Φ−Φ

0

2.52.452.42.352.32.25

Figure 9: The accumulated phase difference between a system with a dark matter spike anda system without a dark matter spike, for varying degrees of α. All other parameters are inaccordance with table 1

10­2 10­1 100

Frequency [Hz]

10­1

100

101

102

103

104

105

106

107

108

Φ−Φ

0

3e­072e­071e­07

Figure 10: The accumulated phase difference between a system with a dark matter spikeand a system without a dark matter spike, for varying degrees of β. Note that β is expressedin geometric units. All other parameters are in accordance with table 1

23

4.4 eLISA measurement accuracy

Fisher matrices were used to determine the effect of observation time on measurementaccuracy and to create confidence ellipses representing a full five year observation. Theerror on α and β was determined for observation times of 1, 2 and 5 years. The results canbe found in table 2.

Table 2: Measurement error on the dark matter parameters as a function of observationtime τ . These values correspond to a signal-to-noise ratio of 10 by varying the distance tothe object. Measurement accuracy declines greatly at shorter observation times.

τ [years] ∆α/α ∆β/β1 2.41 · 10−5 1.03 · 10−3

2 1.40 · 10−5 5.92 · 10−4

5 2.52 · 10−7 1.04 · 10−4

Secondly, we investigated the measurement accuracy for a five year observation until coa-lescence. The values depend inversely on signal-to-noise ratio.

∆AA

= 0.1

(10

SNR

)(37a)

∆tc = 1.12

(10

SNR

)(37b)

∆φc = 1.17

(10

SNR

)(37c)

∆Mc

Mc= 2.55 · 10−7

(10

SNR

)(37d)

∆α

α= 2.52 · 10−7

(10

SNR

)(37e)

∆β

β= 1.04 · 10−4

(10

SNR

)(37f)

These values are used to generate confidence ellipses as shown in the following figures. Allsubsequent ellipses will show the 1σ, 2σ and 3σ confidence regions, which correspond to68.3%, 95.4% and 99.7% confidence. In a 95.4% confidence region, we can be 95.4% certainthat the region contains the true mean of the sampled data.

The confidence ellipses in figures 11a, 11b and 12 have a small semi-minor axis in comparisonto the semi-major axis. This indicates that Mc, α and β are strongly correlated. This isespecially true for the dark matter parameters α and β. This is due to the fact that β is inpart dependent on α as β ≡ ρsprsplnΛ. The correlation can be further explained by noting

that the two parameters are both featured in the phase ΦDM and are together responsiblefor the change in gravitational wave amplitude due to L(f).

24

−4 −3 −2 −1 0 1 2 3 4

∆Mc/Mc1e−7

−4

−3

−2

−1

0

1

2

3

4

∆α/α

1e−6

(a) Confidence contours in the Mc-α plane

−4 −3 −2 −1 0 1 2 3 4

∆Mc/Mc1e−7

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

∆β/β

1e−4

(b) Confidence contours in the Mc-β plane

−4 −3 −2 −1 0 1 2 3 4

∆α/α 1e−6

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

∆β/β

1e−4

Figure 12: Confidence contours in the α-β plane

25

5 Discussion

Our analysis has placed the detectable black hole masses MBH surrounded by a dark matterdensity spike between 103M� and 106M�. Companions as light as ∼ 0.1M� can still bedetectable. It must be noted however that we have used a simplified model of black holesso that black hole spin has not been taken into account. Other studies have reported ondetectability in narrower ranges [5], but the definition of ‘detectable’ is not a clearly definedone, so the detectable ranges are bound to vary between definitions.

The cumulative phase difference between a dark matter and a non-dark matter systemconcerning parameter α is in close agreement with previous results by Eda et al. The effectof parameter β has so far never been reported and while its effects are weaker, they aremost certainly measurable by eLISA. While Eda et al. have found that the relative erroron α decreases with higher values of α, we add to this that the signal-to-noise ratio of theinspiral decreases significantly with high α. This means that there is a trade off for highvalues of α: while measurability increases, signal strength decreases.

The Fisher information matrices have shown that measurement accuracy increases withobservation time. The covariance matrix yielded values which bear close resemblance toEda et al, except for measurement accuracy of α, which is an order of magnitude moreprecise due to the higher-order approximant and the eLISA noise curve used here.

26

6 Conclusion

The inspiral of a compact object into a black hole is influenced nearby by dark matterdue to dynamical friction and modified Newtonian potential. These two effects cause thecompanion to spiral into the black hole more quickly than in a system without dark matter.Dark matter leaves a clearly detectable imprint on the gravitational waves emitted by thesystem, which allows space-based gravitational wave detectors such as eLISA to determinethe dark matter density profile.

We have found that extreme mass-ratio inspirals in a system with a dark matter densityspike with black hole mass 103M� ≤MBH ≤ 106M� and companion mass µ ≥ 0.1M� maybe detectable by eLISA. The detectability depends heavily on dark matter density. Highamounts of dark matter strongly decrease the amount of revolutions around the black hole,resulting in a lower signal-to-noise ratio.

Determination of measurement accuracy using Fisher matrices has shown that measurementaccuracy decreases greatly with decreased observation time. It has also shown that althoughhighly correlated, Mc and dark matter parameters α and β can be determined to a highaccuracy for a favourable signal-to-noise ratio of 10. For a five year observation until coa-lescence, the measurement error on the dark matter parameters are ∆α

α = 2.52 · 10−7(

10SNR

)and ∆β

β = 1.04 · 10−4(

10SNR

).

Combined with complementary research into the emission spectrum of annihilating particlesin the dark matter halo, the techniques studied in this thesis can provide insight into thenature of dark matter. Weakly annihilating dark matter will form a clear dark matterdensity spike, detectable by eLISA. Strongly annihilating dark matter will be detectable bydetection of electromagnetic emission. This allows us to study the nature of dark matterparticles. Furthermore, by probing the structure of the dark matter density spike aroundthe black hole, eLISA can be used to determine the initial dark matter halo profile of thesurrounding galaxy, since the dark matter spike is a direct consequence of the initial densityprofile.

27

A Determination of initial observation frequency

The method for determining the initial observation frequency is due to Eda et al. To find theinitial frequency fini we need to solve a differential equation with the boundary conditionthat fISCO corresponds with t = 0. The time corresponding to fini is therefore defined tobe negative. Defining τ ≡ −t leads to the following set of equations:

fini = f(τ = observation time, α) (38)

df

dτ= −3

5π

(f

f0

)5/3

f2X−11/2 [K(1 + cJ ] (39)

For simplicity, we approximate Meff to be MBH . The terms on the right hand side ofequation 39 are given by:

J(x) =4x11/2−α

1 + xα−3(40a)

K(x) =(1 + x3−α)5/2(1 + αx3−α/3

1 + (4− α)x3−α (40b)

X = (δε)1/(α−3)x (40c)

f0 =c3

8πGMc(40d)

x = ε1/(3−α)R (40e)

ε ≈ F

GMBH(40f)

δ ≈(GMBH

π2f2

)(3−α)/3

(40g)

R ≈(GMBH

πf

)(40h)

where F is given by equation 17. Lastly, c is defined as the ratio between the coefficient ofthe dynamical friction and the modified Newtonian potential cDF /cGW , which are given by

cDF ≡256

5

(Gµ

c3

)(GMeff

c

)2

ε4/(3−α) (41)

cGW ≡(8πG2µβ

)(GMeff )

−3/2ε(2α−3)/[2(3−α)] (42)

Where as before, β ≡ ρsprαsplnΛ. This results in a initial observation frequency depending

on total observation time as seen in figure 13.

28

1 2 3 4 5 6 7 8 9 10time [years]

0.010

0.015

0.020

0.025

0.030

0.035

0.040

f ini [Hz]

Figure 13: The initial observation frequency depending on total observation time. All grav-itational wave parameters are in accordance with table 1 (α = 7/3). The initial frequency isdirectly related to the amount of revolutions the companion makes around the black hole.As observation time increases, so does measurement accuracy.

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