Looking for dark matter near massive black holes with eLISA

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Looking for dark matter near massive

black holes with eLISA

Rob Daniel Hesselink

10587454

Report bachelor project physics and astronomy

15EC

Conducted between April and July 2016

at Nikhef

University of Amsterdam Faculty of science

Supervisor:

Dr. Chris Van Den Broeck Second assessor: Prof. Dr. Patrick Decowski

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Scientific summary

A large part of the matter in the universe has not yet been directly detected. This is because this 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 so far has been inconclusive. Recently, new methods of studying dark matter have become available. 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 study dark 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. This dark matter spike influences the inspiral of objects into the black hole, hereby leaving a footprint in the gravitational waves emitted by the system.

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

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

A wide array of systems was found to be detectable by eLISA, with black hole masses ranging from 103M to 106M being visible, depending on companion mass. A steep dark matter

density severely impacts detectability, due the decreased amount of revolutions around the black hole.

The dark matter density is described by parameters α and β. The first parameter describes the steepness of the density profile, while the second describes the initial conditions of the dark matter density spike at the radius where it is established. Analysis using Fisher matrices has resulted in a measurement accuracy ∆α_α = 2.52 · 10−7 _SNR10 and ∆β

β = 1.04 ·

10−4 _SNR10 , dependent on signal-to-noise ratio (SNR).

Combined with complementary searches into dark matter annihilation, eLISA will provide insight 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 een gewone telescoop lichtgolven kan meten, meet een zwaartekrachtsgolfdetector de zwaartekrachts-golven. Deze techniek opent een nieuw kennisdomein in de sterrenkunde; verre objecten die niet konden worden waargenomen door middel van het licht dat ze uitzenden, kunnen we nu wel waarnemen door naar hun zwaartekrachtsgolven te kijken.

Zwaartekrachtsgolven ontstaan als een massa versneld wordt. Als in het universum twee objecten (bijvoorbeeld twee zwarte gaten) om elkaar heen draaien, dan zenden ze sterke zwaartekrachtsgolven uit. Als deze golven door de ruimte reizen, dan vervormen ze de ruimte: 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 door Albert Einstein voorspeld zijn, heeft het bijna honderd jaar geduurd voordat de technologie bestond om ze te kunnen waarnemen. Nu de technologie er is, kan dit bijdragen aan de oplossing van een van de grootste onopgeloste problemen in de natuurkunde: dat van de donkere materie.

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

De invloed van donkere materie is erg groot wanneer een licht object, zoals een klein zwart gat 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 het zwarte gat wordt veranderd. Deze verandering is zichtbaar in de zwaartekrachtsgolven die uitgezonden 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 via zwaartekrachtsgolven waar te nemen, is de hoop gevestigd op de nieuwe detector eLISA. Deze detector bestaat uit drie satellieten die in een driehoeksvorm in een baan om de aarde bewegen. Elk hoekpunt stu-urt een laser naar een ander hoekpunt, waarmee heel precies de afstand tussen de drie satellieten bepaald kan worden. Wanneer er een zwaartekrachtsgolf door de satellieten beweegt, veranderen de afstanden tussen de hoekpunten, waardoor de zwaartekrachts-golf gemeten kan worden.

In dit bacheloronderzoek is geanalyseerd welke systemen eLISA kan meten en hoe nauwkeurig de dichtheid van donkere materie gemeten kan worden. Wanneer detector eLISA in 2034 gelanceerd wordt, kan deze satelliet bijdragen aan kennis van de donkere materie. Dit brengt ons 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 the universe has not yet been observed. This missing matter does not interact with light, which is why it has been aptly named “dark matter”. It is unclear whether dark matter is a single type of matter, or consists of multiple constituents. While a plethora of candidates has been suggested that might constitute dark matter, research so far has been inconclusive [1]. As both the evidence and potential candidates for dark matter keep increasing, it might be most advantageous to study dark matter via its only mechanism of interaction known so far: gravity.

Gravitational waves were first detected directly on the 14th _{of September 2015 after having}

been predicted by general relativity almost a hundred years prior [2, 3]. These waves offer astronomers a new way to view the cosmos, as all objects that interact gravitationally can emit these waves. Gravitational waves are difficult to detect, since they interact weakly with matter. Measuring displacement due to gravitational waves by conventional laser interferometer detectors requires both extremely high precision equipment and many metres of vacuum tunnels. Existing observatories are most sensitive in the acoustic range (10-10,000 Hz) [4] and unable to detect waves of very low frequencies (0.001-1 Hz), as this would require longer detector arms. Space-based detectors may solve this problem, as space provides a suitable vacuum, which allows for longer detector arms [5]. These space-based detectors may be able to detect primordial gravitational waves - a remnant of the Big Bang [6] - and extreme mass ratio inspirals.

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

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

This thesis aims to provide initial calculations for the eLISA detector that is set to launch in 2034 [5], so that its measurements may be used to expand our knowledge of dark matter and 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 gravitational waves. 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 of a thousand solar masses. Building on the work of Eda et al [8]., we will look at the effects of the dark matter spike on gravitational waveforms in extreme mass-ratio inspirals and in

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

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

Fisher analysis will be used to determine eLISA’s measurement accuracy, by calculating the root-mean-square errors on the parameters that describe the gravitational waveform. The findings will be represented using error ellipses at different confidence levels. The effect of observation 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 and their detection. It will then introduce the TaylorF2 approximant and the systems studied in this thesis: extreme mass-ratio inspirals with dark matter. Section three will present the TaylorF2 dark matter waveform and explain how measurement accuracy of eLISA is determined. Chapter four will present the results that will be used to answer the research questions, which will be discussed in chapters five and six.

Throughout this thesis, in accordance with many works on general relativity, we will use geometric 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 propro-duce electromagnetic waves. A difference between these two types of waves is that gravitational waves couple very weakly to matter, making them incredibly difficult to detect.

Gravitational waves are a direct consequence of the theory of general relativity by Albert Einstein. In this theory masses curve the space around them by virtue of having mass, which is analogous to placing a brick on a trampoline: the brick pulls the fabric downwards, causing surrounding objects to be drawn towards it. The mass of the brick has curved the space-time around it. In real astrophysical systems, these ripples are sent out as gravitational waves that stretch space-time in one of two polarizations h+ and h× [10]. The displacement of a

ring 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 detectors exploit. Ground-based interferometric gravitational wave detectors such as LIGO [11] or VIRGO [12] consist of two long vacuum arms at a 90 degree angle. A laser beam is reflected back and forth in the arms, giving an accurate measure of the length of the arm. When a gravitational wave travels through the detector, the arms are affected differently. This means that the diffraction pattern of the two beams changes as a gravitational wave passes through. We can infer the gravitational waveform by looking at the change in arm length over time. This technique has been successfully used to detect multiple events [2, 13]. Certain astrophysical systems are especially adept at creating gravitational waves with large amplitudes and clear predicted waveform. Binary objects spiral around each other, meaning that they consist of continuously accelerated masses, which continuously create gravitational waves. 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 finally collide and merge [14].

A satellite can orbit a heavier body up until its innermost stable circular orbit (rISCO). At

rISCO, 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 third

law. The radius of the innermost stable circular orbit and the corresponding gravitational wave frequency are given by [10]

rISCO= 6Mtot (1)

fISCO =

1 632πM_tot

. (2)

This process is called an inspiral and it was first proven to be modeled correctly by Einstein’s equations by Hulse and Taylor [15]. Since then, gravitational waves have been directly detected by the LIGO Scientific Collaboration and the Virgo Collaboration [2], offering a new 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 detector arms, which means that the sensitive frequencies of space-based detectors are much lower than earth-based detectors [16]. These detectors consist of separate satellites that send laser signals towards each other, hereby connecting the vertices into a detector arm. Proposed detectors such as LISA and eLISA consist of three satellites in free fall, creating a triangular detector [4, 5].

A sketch of the LISA detector. Source: ESA

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

The LISA gravitational wave detector was originally planned as a collaborative mis-sion between ESA and NASA. However, due to financial considerations, NASA has since decided not to pursue the mission further. The project has been adjusted into eLISA, a space-based detector project designed by

ESA. The LISA Pathfinder satellite, a mission with the goal to test technologies for the eLISA detector, was launched in 2015 and is already giving promising results. In figure 2 the sensitivities of three detectors are shown: eLISA, LISA and Advanced LIGO. Of

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these three, only the Advanced LIGO exists currently, with the LISA project having been cancelled 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 1012 St ra in se nsi tivi ty [H z − 0. 5] eLISA LISA Advanced LIGO

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

2.3 Signal-to-noise ratio

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

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

that gives the accuracy of the detector at specific frequency.

The signal strength is given by the amplitude of the gravitational wave. The amplitude of any 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 " Z ∞ 0 |˜h(f )|2 Sh(f ) df #2 . (3)

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The higher the SNR, the higher the confidence in the detector output. In this work, we shall classify events with an optimal SNR lower than 8 as ‘poorly detectable’. While detector orientation with respect to the source has an influence on the SNR, we will not take this into 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 solve numerically, which is why we resort to approximations when calculating these waveforms. The most commonly used approximation is the TaylorF2 approximation. The TaylorF2 approximation is a seventh order Taylor expansion of the frequency domain waveform using a stationary phase approximation. We will use frequency domain waveforms throughout this thesis. The derivation of the frequency domain waveform is beyond the scope of this work, but it can be found in Arun et al. [9].

The gravitational waveform, denoted ˜h(f ), is given below, along with its amplitude and the TaylorF2 approximation of the phase.

˜ h(f ) = Af−7/6eiΨ(f ) (4) A = 5 24 1/2 ₁ π3/2 1 DMc 5/61 + cos2(ι) 2 (5) Mc= (M1M2)3/5 (M1+ M2)1/5 (6) ˜ Ψ(f ) = 2πf tc− φc− π 4 + ˜Φ(f ) (7) ˜ Φ(f ) = 3 128ηv5 " 1 +20 9 743 336 + 11 4 η v2− 16πv3_{+ 10} 3058673 1016064 + 5429 1008η + 167 144η 2 v4 + π 38645 756 − 65 9 η 1 + 3ln _v vlso v5+ 11583231236531 4694215680 − 640 3 π 2 −6848 21 γ − 6848 21 ln(4v) + 15737765635 3048192 + 2255π2 12 η +76055 1728 η 2 −127825 1296 η 3 v6+ π 77096675 254016 + 378515 1512 η − 74045 756 η 2 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 respect to the plane of the detector. The coefficients tc and φc are the time and phase of the

gravitational wave at coalescence. Lastly, the coefficient η is the symmetric mass ratio and v is the parameter that contains the gravitational wave frequency. Here η and v are defined as:

η = M1M2 (M1+ M2)2

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v = (πM f )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]. Black

holes in the centre of galaxies will attract nearby stars, which may be caught in an orbit around the black hole. Companions with highly eccentric orbits will spend a large amount of revolutions in the eLISA frequency band, making these excellent sources of gravitational waves for space-based detectors [18].

There are several objects that can take on the role of companion. To be properly measurable by eLISA, the source will have to get close to the fISCO of the system. This means that the

companion will have to approach the system’s rISCO without being destroyed by the black

hole’s gravity. Normal solar mass stars are therefore excluded as viable candidates, since their radius is too large and they would not be able to approach the black hole close enough without 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 population of black holes, which is rather poorly known since they emit hardly any electromagnetic radiation. 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 in the milky way [19].

2.6 Dark matter density profile

It has now been established that galaxies contain a lot of matter that does not interact with electromagnetic radiation. The exact structure of dark matter is unknown, but one of the most widely used density profiles is the Navarro-Frenk-White profile [20]. This profile contains two parameters, the scale radius rs and the scale density ρs. These parameters

can be varied to model several different assumptions about the inner profile of dark matter halos, namely cuspy and core profiles [21]. The Navarro-Frenk-White profile is given by

ρN F W =

ρs

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

. (11)

While the NFW profile in equation 11 is present throughout the whole galaxy, we will look at a local deviation from this profile, close to a black hole [7]. It has been shown that an NFW profile can lead to the formation of a density spike near a black hole. This second density profile, ρspike, is a direct consequence of the NFW profile. When considering an initial NFW

profile which obeys ρ(r) ∼ r−αini _{with 0 < α}

ini≤ 2, depending on the parameters ρs and

rs, the ρspike will be ρ(r) ∼ r−α with α = 9−2α_4−αini

ini. Explicitly, this means

ρspike= ρsp

r_sp r

α

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where the exponent α of the dark matter spike density profile varies between 2.25 and 2.50 [7, 22]. Here rspis the radius at which the dark matter spike is formed and ρsp is the dark

matter 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 3.0 Radius [parsec] 1018 1016 1014 1012 1010 108 106 104 102 100 102 104 ρ( r) [ kg /m 3] ρspike ρNFW

Figure 3: The two density profiles. The density obeys the ρspikeprofile up until rsp(marked

with a black line), after which it switches to the ρN F W profile. The dashed line indicates

the 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 halo profile.

2.7 Dark matter effects on inspiral

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

dEDF

dt = vfDF = 4π

µ2ρDM(r)

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Here ln(Λ) is the Coulomb parameter, a measure for the amount of collisions the dark matter particles 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 emitting gravitational waves. This is a secondary effect of the dynamical friction. The energy radiated away in the form of gravitational waves and the orbital frequency are then given by [8, 10]

dEGW dt = 32 5 µ 2_R4_ω6 s, (14) ωs(R) = Mef f R3 + F Rα 1/2 (15) where Mef f, F and MDM(< rISCO) are defined as:

Mef f = ( MBH− MDM(< rISCO) (rISCO ≤ r ≤ rsp) MBH (r < rISCO) (16) F = ( rα−3_M DM(< 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 secondary effect is less pronounced, because the alteration of orbital frequency ωsis relatively small.

The second dark matter effect is due to the modified Newtonian potential. The companion mass orbits a black hole with a steeper gravitational potential, since the region contained within the orbital radius is inhabited by dark matter particles. The increased gravitational potential 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 orders

of magnitude larger. This means that most of the dark matter effects are due to dynamical friction.

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

2.8 Dark matter waveform

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

Firstly, we need to add a correction to the phase of the gravitational wave. The first order phase 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 Z f fISCO df0f0−11/3L−1+ Z f fISCO df0f0−8/3L−1 # (19) L(f ) = 1 +5 8π 2α−11 3 M− 10+2α 6 BH βf 2α−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 means that the dark matter effects are dependent on just two parameters, which we shall henceforth refer to as α and β. Of these parameters, α is directly connected to the steepness of the dark matter density spike, whereas β provides information on the density and location of the 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 used in section 3.3 to determine the measurement accuracy of the eLISA detector. From the Fisher matrix we can determine error ellipses, give us the confidence of measurement within a specific confidence level. The elements of the Fisher matrix for a system ˜h, dependent on parameters ~θ, are given by [23]:

Fij≡ ∂˜h ∂θi ∂˜h ∂θj ! , (22)

where the inner product is defined as

(h1|h2) ≡ 4Re Z 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 Fisher matrix 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 covariance matrix. Explicitly:

∆θi=p(F−1₎

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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 TaylorF2 expression when all dark matter parameters are set to zero. This requirement is satisfied by choosing specific values for the aforementioned tc and φc. Let us first calculate the

expression for the phase without dark matter.

ΦDM(f ) _α=β=0 = 10 3 (8πMc) −5/3 " − f Z f fISCO df0f0−11/3+ Z f fISCO df0f0−8/3 # =10 3 (8πMc) −5/3 " 3 8f f 0−8/3₋3 5f 0−5/3 #f fISCO =10 3 (8πMc) −5/3 " − 9 40f −5/3₋3 8f f −8/3 ISCO+ 3 5f −5/3 ISCO # (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 (P f aN ) is defined as: P f aN = 3 128ηv = 3 128· M1M2 (M1+ M2)1/3 · (πM f )−5/3 = 3 128(πM f ) −5/3 M1M2 (M1+ M2)2 . (27)

Equations 26 and 27 are equal, which means the two phases are equal, except for two residual terms in the phase expression in equation 25. Noting that one of these terms is linear in f and one is independent of f , we can compensate for these two terms by defining the following: 2πtc≡ 10 3 (8πMc) −5/3 3 5f −5/3 ISCO (28) φc≡ 10 3 (8πMc) −5/3 3 8f −8/3 ISCO (29) This convention assures that when we remove dark matter from our system, we obtain the TaylorF2 results once again. In short, replacing the first order term in TaylorF2 by the dark

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

correct 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 pessimistic values for detector arm length (L) and the low-frequency acceleration (Sn,acc). This means

that the SNR’s calculated here are lower limits to eLISA’s capabilities. The full noise curve can be seen in equation 30.

Sn(f ) = 20 3 4Sn,acc(f ) + Sn,sn(f ) + Sn,omn(f ) L2 × " 1 + _f 0.41c 2L 2# (30) Sn,acc= 9 × 10−28 1 (2πf )4 1 + 10 −4_Hz 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 described

by these parameters, resulting in a 6x6 dimensional Fisher matrix. The derivatives of ˜h with respect to these parameters are given by the following:

∂˜h ∂lnA = ˜h (32a) ∂˜h ∂tc = 2πif ˜h (32b) ∂˜h ∂φc = −i˜h (32c) ∂˜h ∂lnMc = 5 3i˜h ˜Φ (32d) ∂˜h ∂lnα = α˜h i∂Ψ ∂α − 1 2 1 L ∂L ∂α (32e) ∂˜h ∂lnβ = β˜h 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 Fisher

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of observation by eLISA. This initial frequency is dependent on the total observation time until coalescence of the binary objects. It is therefore necessary to determine the frequency evolution of the system, f (t). We use the same method as Eda et al.[8], who have shown that the initial observation frequency fini is strongly dependent on α. The initial observation

frequency can be found using the equations in appendix A.

Error ellipses were made using the variance computed from the fisher matrix. When creating an error ellipse in the plane of parameters x and y with corresponding σx, σy and σxy, and

with probability ρ, the following equations describe an ellipse with major and semi-minor axes a and b and orientation θ [25, 26]:

a2= k2· σ 2 x+ σy2 2 + h(σ_x2− σ2_y)2 4 + σ 2 xy i1/2 ! (33) b2= k2· σ 2 x+ σ 2 y 2 − h(σ2_x− σ_y2)2 4 + σ 2 xy i1/2 ! (34) tan2θ = 2σxy σ2 x− σy2 (35) k =p−2ln(1 − ρ) (36)

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4 Results

To start the results, we will explicitly state the parameters of our standard system. These parameters 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 F2 Only 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 the evolution of phases. Figure 4b shows the absolute value of the phase difference relative to the TaylorF2 phase.

It can be seen that while the two phases are quite coherent at first, they begin to deviate strongly as they approach the fISCO of the system. The qualitative behaviour of the phase

starts to deviate so strongly, that the leading-order approximation breaks down past a frequency close to 1 Hz. This means that the higher order terms are significant at higher frequencies and should not be left out.

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4.2 Signal-to-noise ratio

We wish to investigate which systems will be detectable by the eLISA detector. Firstly, we will keep the dark matter spike fixed, while varying black hole mass MBH and companion

mass µ. This will give an indication which EMRI systems with dark matter can be detected by eLISA. Secondly, we will look at the dark matter effect within a system with masses described in table 1. We will again look at the separate effect of the two dark matter parameters α and β. 102 ₁₀3 ₁₀4 ₁₀5 ₁₀6 ₁₀7 Black hole mass in solar masses 0 10 20 30 40 50 60 SN R

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 frequency

band and are therefore poorly detectable. Companion mass µ is kept fixed at 1M

0 20 40 60 80 100 Companion mass in solar masses 0 10 20 30 40 50 60 70 80 90 SN R

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

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Judging from figure 6, companion mass µ has a different effect on detectability than the black hole mass MBH does. As the ratio of the two masses approaches one, the signal-to-noise

ratio goes up. While this is in principle a preferable situation, this causes problems when attempting to detect the presence of dark matter. As stated before, the most important dark matter effect on the inspiral is mostly due to dynamical friction. This effect is dependent on 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 SN R

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

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

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0.0 0.5 1.0 1.5 2.0 β 1e−6 0 2 4 6 8 10 12 SN R

Figure 8: Signal-to-noise ratio varying β. The vertical line indicates the β value found in table 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 between a waveform emitted by a system containing a dark matter spike and a system without a dark matter spike. The phase difference is denoted as Φ(f ) − Φ0(f ) and it is crucial to the

detection of dark matter, as the dark matter effects are most pronounced in the phase of the gravitational waveform. Firstly, we will look at the effect of the steepness of the dark matter spike, characterised by α. After this, we will see the effect of our second dark matter parameter, 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, the steepness of the dark matter spike, is much more pronounced than the phase change due to the second parameter, β. Since α describes the density profile of the complete dark matter density spike, increased α increases total dark matter mass greatly. The parameter β on the other 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.

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102 ₁₀1 100 Frequency [Hz] 101 100 101 102 103 104 105 106 107 108 Φ − Φ0 2.5 2.45 2.4 2.35 2.3 2.25

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

102 ₁₀1 ₁₀0 Frequency [Hz] 101 100 101 102 103 104 105 106 107 108 Φ − Φ0 3e07 2e07 1e07

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

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4.4 eLISA measurement accuracy

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

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

τ [years] ∆α/α ∆β/β 1 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.

∆A A = 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. All subsequent ellipses will show the 1σ, 2σ and 3σ confidence regions, which correspond to 68.3%, 95.4% and 99.7% confidence. In a 95.4% confidence region, we can be 95.4% certain that 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 comparison to the semi-major axis. This indicates that Mc, α and β are strongly correlated. This is

especially true for the dark matter parameters α and β. This is due to the fact that β is in part 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 responsible

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−4 −3 −2 −1 0 1 2 3 4 ∆Mc/Mc 1e−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/Mc 1e−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

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5 Discussion

Our analysis has placed the detectable black hole masses MBH surrounded by a dark matter

density spike between 103_M

and 106M. Companions as light as ∼ 0.1M can still be

detectable. It must be noted however that we have used a simplified model of black holes so that black hole spin has not been taken into account. Other studies have reported on detectability in narrower ranges [5], but the definition of ‘detectable’ is not a clearly defined one, so the detectable ranges are bound to vary between definitions.

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

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

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6 Conclusion

The inspiral of a compact object into a black hole is influenced nearby by dark matter due to dynamical friction and modified Newtonian potential. These two effects cause the companion 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 the system, which allows space-based gravitational wave detectors such as eLISA to determine the dark matter density profile.

We have found that extreme mass-ratio inspirals in a system with a dark matter density spike with black hole mass 103M≤ MBH ≤ 106M and companion mass µ ≥ 0.1M may

be detectable by eLISA. The detectability depends heavily on dark matter density. High amounts 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 measurement accuracy decreases greatly with decreased observation time. It has also shown that although highly correlated, Mc and dark matter parameters α and β can be determined to a high

accuracy 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 _SNR10 and ∆β_β = 1.04 · 10−4 _SNR10 .

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

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A

Determination of initial observation frequency

The method for determining the initial observation frequency is due to Eda et al. To find the initial frequency fini we need to solve a differential equation with the boundary condition

that fISCO corresponds with t = 0. The time corresponding to fini is therefore defined to

be 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 Mef f to be MBH. The terms on the right hand side of

equation 39 are given by:

J (x) = 4x 11/2−α 1 + xα−3 (40a) K(x) =(1 + x 3−α₎5/2_{(1 + αx}3−α_/3 1 + (4 − α)x3−α (40b) X = (δ)1/(α−3)x (40c) f0= c3 8πGMc (40d) x = 1/(3−α)R (40e) ≈ F GMBH (40f) δ ≈ GMBH π2_f2 (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 of the dynamical friction and the modified Newtonian potential cDF/cGW, which are given by

cDF ≡ 256 5 Gµ c3 GMef f c 2 4/(3−α) (41) cGW ≡ 8πG2µβ (GMef f)−3/2(2α−3)/[2(3−α)] (42)

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

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1 2 3 4 5 6 7 8 9 10 time [years] 0.010 0.015 0.020 0.025 0.030 0.035 0.040 fini [H z]

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 is directly 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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Looking for dark matter near massive black holes with eLISA