Investigating the Influence of Spin-Curvature Coupling on Extreme

Investigating the influence of spin-curvature coupling on extreme mass-ratio inspirals by Alexandra Hanselman Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2020 ○c Massachusetts Institute of Technology 2020. All rights reserved.

Author...... Department of Physics May 8, 2020

Certified by...... Scott A. Hughes Professor of Physics Thesis Supervisor

Accepted by ...... Nergis Mavalvala Associate Department Head, Department of Physics 2 Investigating the influence of spin-curvature coupling on extreme mass-ratio inspirals by Alexandra Hanselman

Submitted to the Department of Physics on May 8, 2020, in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics

Abstract In this report, extreme mass ratio inspiral worldlines and gravitational waveforms were produced considering radiation reaction and spin-curvature effects. Spin-curvature coupling was predicted to modify the inspiral by introducing oscillations to the small body’s worldline. Investigating these effects led to the observation that spin-curvature effects introduce a resonance-like feature in the worldline. This feature is affectedby the mass ratio, spin axis phase angle, and eccentricity of the system. The modifica- tions to the inspiral worldline imprint themselves in the gravitational waves produced. Understanding how spin-curvature coupling modifies these waveforms will help gravitational wave detectors characterize the types of systems they detect.

Thesis Supervisor: Scott A. Hughes Title: Professor of Physics

3 4 Acknowledgments

I would like to thank my thesis advisor, Professor Scott Hughes, for providing guidance and support on my research journey. I would also like to thank Talya Klinger for being a wonderful research partner during the beginning stages of this project. I would also like to thank Lisa Drummond for providing the groundwork for the later stages of my research and for always being available to answer my questions. Finally, a huge thank you to my friends Jay Laone, Yong Hui Lim, and Luke Bordonaro for lending me their computers to assist in generating some data files.

5 6 Contents

1 General Relativity 11 1.1 Advancement of Gravitational Theories ...... 11 1.2 Solutions to Einstein ...... 13 1.3 Orbits in General Relativity ...... 14 1.4 Gravitational Waves ...... 17

2 Extreme Mass Ratio Inspirals 19 2.1 Solution to the Teukolsky Equation ...... 20 2.2 Perturbation Theory ...... 20 2.3 Self-Force Corrections ...... 21 2.3.1 Radiation Reaction ...... 22 2.3.2 Spin-Curvature Coupling ...... 22

3 Implementation 27 3.1 Creating Inspiral Worldlines ...... 27 3.2 Generating Gravitational Waves ...... 29 3.3 Performance ...... 30

4 Results 33 4.1 Accuracy Test ...... 33 4.2 Radiation Reaction and Spin-Curvature Effects on Inspiral Worldlines 34 4.2.1 Effect of Initial Conditions ...... 35 4.2.2 Effect of Inspiral Mass Ratio ...... 36

7 4.2.3 Effect of Spin Axis Phase ...... 39 4.3 Orbital Frequencies During Resonance ...... 41 4.4 Effects Imprinted on Waveforms ...... 42

5 Conclusions and Future Work 47

8 List of Figures

4-1 Comparison of an EMRI inspiral with a mass ratio of 10−3 due to radiation reaction effects starting at an initial point (푝, 푒, 푥) = (8, 0.2, 1). Our Python and Mathematica generators were compared to Scott Hughes’ C code. Hughes’ code uses slightly different stopping criteria and stepping size, which appears to be why the C code trajectory deviates slightly from the other two. Other than this, the three generators agree to excellent precision...... 34 4-2 Comparison of radiation reaction (RR), spin-curvature (SC), and com- bined radiation reaction and spin-curvature (RR+SC) effects on 푝 and 푒 as a function of time 푡 for the case of initial condition (푝, 푒, 푥) = (8, 0.2, 1). Note that adding in spin-curvature effects leads to an oscillation about the radiation reaction inspiral...... 35 4-3 Comparison of radiation reaction and spin-curvature effects for various initial conditions. The first row is for an initial condition of (8, 0.2, 1), the second row for (9, 0.5, 1), and the final row for (10, 0.8, 1). The left column displays the entire inspiral while the right column zooms in on only the beginning of the inspiral. We see that an increase in eccentricity leads to a larger deviation from the radiation reaction only inspiral and changes the spin-curvature oscillations from sine-like to more spread out, peaked waves. Note that the case with 푒 = 0.8 grows to an eccentricity larger than 0.8, which is beyond the domain of our

data set and thus cannot be trusted for 푝 . 8.7. This inspiral is still shown to emphasize the trends we witness as eccentricity increases. . 37

9 4-4 Comparison of spin-curvature coupling for increasing mass ratios of 휇/푀 = 5 × 10−3, 10−3, and 10−4 for the (푝, 푒, 푥) = (8, 0.2, 1) initial condition...... 38 4-5 Comparison of spin-curvature coupling for increasing mass ratios of 휇/푀 = 5 × 10−3 and 10−3 for the (푝, 푒, 푥) = (9, 0.5, 1) initial condition. 39 4-6 Comparison of different small body spin axis orientations for the initial condition of (푝, 푒, 푥) = (8, 0.2, 1) with a mass ratio of 휇/푀 = 10−3. We consider axis phases of 휑 = 휋/2, 휋/4, and 휋/8...... 40

4-7 The evolution of eccentricity 푒 and Mino-time frequency 2Υ푟 − Υ푠 for the case (8, 0.2, 1) with a mass ratio of 10−3 and spin axis phase 휋/2.

We see that the intersection of 2Υ푟 − Υ푠 = 0 corresponds to the time right before the resonance feature begins...... 41 4-8 Gravitational waves produced at the beginning of the inspiral for initial condition (8, 0.2, 1) comparing radiation reaction (top) and spin- curvature effects at spin axis phases 휋/2 (middle) and 휋/4 (bottom). We see that both spin-curvature waveforms are similar at beginning times, but are different from the radiation reaction only waveform. .43 4-9 Gravitational waves produced during the resonance feature of the inspiral for initial condition (8, 0.2, 1) comparing radiation reaction (top) and spin-curvature effects at spin axis phases 휋/2 (middle) and 휋/4 (bottom). We see that all three waveforms have different behavior during resonance...... 44 4-10 Gravitational waves produced at the end of the inspiral for initial condition (8, 0.2, 1) comparing radiation reaction (top) and spin-curvature effects at spin axis phases 휋/2 (middle) and 휋/4 (bottom). We see that each waveform ends at a different coordinate time and have different forms at the end of the inspiral...... 45

10 Chapter 1

General Relativity

In 1915, Albert Einstein published his theory of general relativity, which is currently the best description of gravity. General relativity has been used to describe many astrophysical phenomena and has led to major insights in fields such as cosmology and astrophysics.

1.1 Advancement of Gravitational Theories

Before Einstein, the best description of gravity was Newtonian gravity, developed by Isaac Newton in the late 1600’s. Newtonian gravity is described by the famous inverse-square law, given by 퐺푀푚 F = − ^r (1.1) 푟2 where 퐺 is the gravitational constant, 푀 and 푚 are masses of two particles, and 푟 is the distance between the particles. Newton’s second law then describes the resulting motion of a particle due to the gravitational force acting on it [1]. There are some obvious limitations to Newton’s theory of gravity. The first is that the force acting between two particles would be propagated instantaneously across any distance. The second is the force’s dealing with light. Light does not have mass, but it does have energy. Knowing that gravity does in fact act on light, we use light’s energy to describe the force gravity acts on it. However, light’s energy changes in

11 different reference frames, leading to different actions on light for different observers. Because Newton’s theory implies that actions should be described the same way by different observers, these limitations suggest that Newton’s theory of gravity is incomplete.

In 1905, Einstein proposed his theory of special relativity, which postulated that there was no universal reference frame and that the speed of light was the same in every reference frame. These postulates also suggested that the Newtonian theory of gravity was not complete. In special relativity, it would take time for information to travel, meaning events that could appear simultaneous in one reference frame could appear to happen at different times in another reference frame. Furthermore, relative times between events could be different in different reference frames. This implies that a force acting from one body to another over a distance would lead to different actions occurring in different reference frames.

Einstein spent the next ten years following his proposal of special relativity re- alizing that all of the challenges of making gravity consistent with the principles of relativity could be fixed by recognizing that the gravitational force arises from the motion of bodies in curved spacetime. Spacetime is governed by what are now called the Einstein field equations, given by

퐺휇휈 = 8휋퐺푇휇휈 (1.2)

where 퐺휇휈 is the Einstein tensor (a second order nonlinear differential operator acting on the spacetime metric), 퐺 is the same gravitational constant as in (1.1), and 푇휇휈 is the stress-energy tensor [1]. In this and all future equations, the speed of light 푐 is taken to be equal to one. The Einstein Field Equations describe ten second order nonlinear differential equations describing how systems interact with the geometry of spacetime.

12 1.2 Solutions to Einstein

The Einstein Field Equations are notoriously difficult to solve analytically given that they are ten coupled, nonlinear partial differential equations. It is easiest to solve these equations by imposing certain simplifications. There are two main ways to sim- plify these equations: assuming the spacetime is sufficiently close to that of special relativity to linearize the equations, or secondly to assume some symmetries in the system. One category of solutions that arises from invoking symmetries are black holes. The first (and simplest) black hole solution was discovered by Karl Schwarzschild and describes a static, spherically symmetric mass in an empty universe. The arising Schwarzschild metric, given by

(︂ 2퐺푀 )︂ (︂ 2퐺푀 )︂−1 푑푠2 = − 1 − 푑푡2 + 1 − 푑푟2 + 푟2(푑휃2 + sin2 휃푑휑2) (1.3) 푟 푟

is solely characterized by the black hole mass, 푀 [1]. The interval 푑푠 describes the proper distance between two events given coordinate separations 푑푡, 푑푟, 푑휃, and 푑휑. Right away, a well-known feature of black holes can be extracted from the metric, an event horizon, defined as the surface beyond which information cannot escape, exists at 푟 = 2퐺푀.

Essentially all macroscopic objects in the universe spin about some axis. The Schwarzschild solution, which describes a non-spinning black hole, cannot describe these spinning objects. A metric describing a spinning black hole was discovered by Roy Kerr and is given by

(︂ 2퐺푀푟)︂ Σ 푑푠2 = − 1 − 푑푡2 + 푑푟2 + Σ푑휃2 Σ ∆ (︂ 2퐺푀푎2푟 )︂ 4퐺푀푎푟 + 푟2 + 푎2 + sin2 휃 sin2 휃푑휑2 − sin2 휃푑푡푑휑 (1.4) Σ Σ

where Σ = 푟2 + 푎2 cos2 휃 and ∆ = 푟2 − 2퐺푀푟 + 푎2 [1]. The Kerr metric is now characterized by two parameters: the black hole mass 푀 and a spin parameter 푎 = 퐽/푀, the angular momentum per unit mass. Looking at the metric, it can be shown

13 that an event horizon also exists for spinning black holes, and is found to be the larger root of ∆ = 0. We can also note that the Kerr metric does not depend on 푡 or 휑. Using Noether’s theorem, we can deduce that there are two conserved quantities

in this system, the energy 퐸 and axial angular momentum 퐿푧. A third conserved quantity was discovered by Brandon Carter and is called the Carter constant, denoted 푄. These three conserved quantities determine the geometry of bound orbits in Kerr spacetime.

1.3 Orbits in General Relativity

Now consider a small body of mass 푚 orbiting around a larger body of mass 푀. Standard Newtonian gravity would dictate that the small body would orbit defined by Kepler’s laws, in closed elliptical orbits. These orbits are fully specified by the following parameters: the semi-latus rectum 푝, eccentricity 푒, and inclination 퐼. The radius of this elliptical orbit is then described by

푝푀 푟 = (1.5) 1 + 푒 cos 휓

where 휓 is an angle called the “true anomaly." We then see that the minimum and maximum radii are given by

푝 푝 푟 = 푟 = , 푟 = 푟 = (1.6) min 푝 1 + 푒 max 푎 1 − 푒

where the subscripts 푝 and 푎 refer to the radius at periapsis and apoapsis, respectively. The orbit oscillates in the angle 휃 between

휃min = 휋/2 − 퐼 , 휃max = 휋 − 휃min (1.7)

When 퐼 = 0, the orbit lies in the plane 휃 = 휋/2 for all times. In this case, the true anomaly 휓 is identical to the axial orbit angle 휑; for general inclination, the relationship between 휓 and 휑 is more complicated. The key point is that, up to initial

14 conditions, the parameters (푝, 푒, 퐼) fully characterize the orbits. For calculational purposes, we often use (푝, 푒, 푥), where 푥 = cos 퐼. When we move to general relativity, orbits no longer follow Kepler’s laws. Weak field orbits, which have 푟 ≫ 푀 at all times, are similar to Keplerian orbits, but exhibit general relativistic corrections which cause precessions such that the orbits do not close after a full period. Such orbits are still fully characterized by parameters (푝, 푒, 퐼) or (푝, 푒, 푥) which are defined in a way similar to their Newtonian orbit counterparts. When in the strong-field regime, although orbits will not generally be described by an ellipse, we still see that the orbits will be bounded by the minimum and maximum radii and angles described above. Noting that the radius and polar angle of the orbit oscillate between these bounds, we can define orbital frequencies for these parameters, denoted Ω푟 for the radial motion, Ω휃 for the polar motion, and Ω휑 for the axial motion. In Keplerian orbits, a closed elliptical orbit necessitates that these three frequencies are equal. However, in the strong-field regime we see that these frequencies diverge significantly fromeach other. This divergence in orbital frequencies gives rise to the significant deviation from the closed elliptical orbits of Newtonian gravity. The resulting motion of a body in the strong-field regime in Kerr spacetime is discussed in detail in a paper by Steve Drasco and Scott Hughes [2]. To summarize, we find four ordinary differential equations describing how the body moves through the Boyer-Lindquist coordinates (푡, 푟, 휃, 휑) per unit proper time 휏. Yasushi Mino introduced a new time parameter 휆, defined by 푑휆 = 푑휏/(푟2 + 푎2 cos2 휃) = 푑휏/Σ, which allows one to fully separate these coordinate motions, yielding

(︂ 푑푟 )︂2 = 푅(푟) , (1.8) 푑휆 (︂ 푑휃 )︂2 = Θ(휃) , (1.9) 푑휆 푑휑 = Φ(푟, 휃) , (1.10) 푑휆 푑푡 = 푇 (푟, 휃) (1.11) 푑휆 (1.12)

15 with

[︀ 2 2 ]︀2 [︀ 2 2 ]︀ 푅(푟) = 퐸(푟 + 푎 ) − 푎퐿푧 − ∆ 푟 + (퐿푧 − 푎퐸) + 푄

2 2 2 2 2 Θ(휃) = 푄 − cot 휃퐿푧 − 푎 cos 휃(1 − 퐸 ) (︂푟2 + 푎2 )︂ 푎2퐿 Φ(푟, 휃) = csc2 휃퐿 + 푎퐸 − 1 − 푧 푧 ∆ ∆ [︂(푟2 + 푎2)2 ]︂ (︂ 푟2 + 푎2 )︂ 푇 (푟, 휃) = 퐸 − 푎2 sin2 휃 + 푎퐿 1 − . ∆ 푧 ∆

where 퐸, 퐿푧, and 푄 are the constants of motion as mentioned in Section 1.2 and ∆ is the same as defined in Equation (1.4). These decoupled differential equations confirm that 푟 and 휃 are periodic functions of 휆 with periods given by

∫︁ 푟푚푎푥 푑푟 Λ푟 = 2 1/2 (1.13) 푟푚푖푛 푅(푟) ∫︁ 휋/2 푑휃 Λ휃 = 4 1/2 (1.14) 휃푚푖푛 Θ(휃)

We can then find the frequencies in Mino time tobe

Υ푟,휃 = 2휋/Λ푟,휃 (1.15)

Converting to Boyer-Lindquist time 푡, we find that the orbital frequencies are given by

Ω푟 = Υ푟/Γ , Ω휃 = Υ휃/Γ , Ω휑 = Υ휑/Γ (1.16) where Γ, defined in [2], is an averaged value of the function 푇 (푟, 휃) defined above and describes the average rate of accumulation of coordinate time 푡 per unit Mino time 휆.

The quantity Υ휑 is similarly an appropriately averaged value of Φ(푟, 휃), and describes how much axial angle 휑 accumulates in an average orbit. Using the equations for the motion of the mass and the orbital frequencies, we can observe and analyze how a particle orbits around a black hole.

16 1.4 Gravitational Waves

Going back to the discussions in Section 1.2 where we discussed simplifying Einstein’s Field Equations, we find that, working in the linearized gravity regime where spacetime is a perturbation of flat spacetime, there are two radiative solutions to the field equations. Note that in this regime, the amplitudes of any waves resulting from the radiative solutions must have small amplitudes so we can ignore nonlinear effects. These waves, called gravitational waves, travel at the speed of light and carry energy and angular momentum similar to electromagnetic waves. Gravitational waves are produced by accelerating masses through spacetime. These waves propagate by stretching and compressing spacetime in two polarizations, one disturbing spacetime in the coordinate axes perpendicular to the direction of prop- agation (ℎ+) and another that is rotated 45 degrees from these axes (ℎ×). Because the gravitational waves carry energy, the accelerating mass that produces the gravitational waves is losing energy and angular momentum. This dissipation will cause the orbit to decay, evolving from one geodesic orbit to another in an inspiral as energy and angular momentum are carried from the binary. This inspiral is dictated by the strength of the interaction between the accelerating body and the spacetime; a body that is orbiting in the weak-field regime will not interact very strongly and thus the gravitational waves radiated will not carry off high energy such that the body inspirals very slowly. On the other hand, bodies that are orbiting in the strong-field regime will radiate away more energy and will inspiral faster. Gravitational waves have been directly detected by the ground-based gravitational wave observatories LIGO (Laser Interferometer Gravitational-Wave Observatory) and Virgo, confirming Einstein’s theory of general relativity and sparking a neweraof gravitational wave astronomy. Analyzing gravitational waves can lead to huge insights in galaxy formation, cosmology, and many other astronomical phenomena. Future space-based detectors, such as LISA (Laser Interferometer Space Antenna), plan to expand the region in which we can observe gravitational waves to learn even more about the universe.

17 18 Chapter 2

Extreme Mass Ratio Inspirals

Extreme mass ratio inspirals (EMRIs) are binary black hole systems formed when a stellar mass (10-100 solar masses) black hole is captured on an orbit of a massive (105 – 107 solar masses) black hole such as those typically found at the centers of galaxies. Low frequency gravitational waves (.03-100 millihertz) are produced when the small black hole interacts with the warped spacetime of the supermassive black hole in the strong-gravity regime. This is far outside the sensitive band of LIGO but will be observed by the space-based detector LISA, currently planned for launch in 2034. Measuring gravitational waves from EMRI events would be used to map the strong-field black hole spacetime near the large black hole, which can then beused to precisely measure black hole properties, test the strong-field nature of black hole gravity, and understand the types of objects at and near the centers of galaxies. Computing the most complete models of gravitational waves from EMRI systems is quite intensive because we must consider in detail how the small body contributes to the binary’s spacetime. However, a separation of timescales leads us to be able to use the “adiabatic approximation,” a perturbation theory that drastically simplifies the computations used to determine the small black hole’s trajectory and thus the gravitational waves produced for such systems. The adiabatic description ignores important effects that we know will have an important impact on real EMRI systems. In this project, I consider how adding in second order effects arising from radiation reaction and spin-curvature coupling will affect the small body’s trajectories and

19 observe the way these effects translate to the form of the gravitational waves.

2.1 Solution to the Teukolsky Equation

As mentioned in Section 1.4, we obtain gravitational-wave solutions when working in the linearized gravity regime and looking at the two radiative equations. When the source of the waves is a small body orbiting in the strong field of a Kerr black hole, we can compute these waves by linearizing the Einstein field equations with respect to the mass ratio of the binary and introducing a frequency and multipolar decomposition. The waves we compute using this method take the form

∞ 푙 ∞ ∞ ∑︁ ∑︁ ∑︁ ∑︁ 푖푚휑 −푖Φ푚푘푛 ℎ(푡) = ℎ+ − 푖ℎ× = 퐻푙푚푘푛 푆푙푚(휃) 푒 푒 (2.1) 푙=2 푚=−푙 푛=−∞ 푘=−∞

In this equation, 푆푙푚(휃) is a spheroidal harmonic, a function that serves as an angular basis function for fields in the non-spherical geometry of a rotating black hole. 퐻푙푚푘푛 is a complex amplitude computed by solving an ordinary differential with respect to the black hole’s radial coordinate. Φ푚푘푛 is a phase, given by

∫︁ 푡 ′ ′ Φ푚푘푛 = 휔푚푘푛(푡 )푑푡 , 휔푚푘푛 = 푚Ω휑 + 푘Ω휃 + 푛Ω푟 (2.2) 0 and is dependent on the orbital frequencies [3]. Each index 푙푚푘푛 contribute to a gravitational wave mode. We sum over the modes to get a complete gravitational wave solution.

2.2 Perturbation Theory

The adiabatic approximation uses the fact that EMRI systems can be described as a perturbative expansion with a two-timescale parameter, governed by the time it takes for the small body to inspiral and the much quicker time for the body to complete one orbit, which are related to the mass ratio of the binary system. To guide our analysis, . it is useful to introduce angle variables 푞훼 = (푞푡, 푞푟, 푞휃, 푞휑) that parameterize the small

20 . 2 body’s coordinate-space motion, and actions 퐽푖 = (퐸/푚, 퐿푧/푚, 푄/푚 ) which are simply related to the geodesic constants of motion. These quantities are governed by evolution equations whose form may be written schematically by

푑푞 훼 = 휔 (J) + 휀푔 (푞 , 푞 , J) + 푂(휀2) (2.3) 푑휆 훼 훼 휃 푟 푑퐽 푖 = 휀퐺 (푞 , 푞 , J) + 푂(휀2) (2.4) 푑휆 푖 휃 푟 where 휀 = 푚/푀 is a parameter which allows us to keep track of the perturbative order of each term [4]. To zeroth order in 휀, we find that Equations (2.3) and (2.4) describe geodesic motion, with the generalized frequency 휔훼(J) describing how each angle accumulates as a function of Mino time 휆. The quantities 푔훼 and 퐺푖 are forcing terms which describe corrections to geodesic motion.

Further analysis [4] shows that 퐺푖 describes the average dissipative decay of orbits due to gravitational-wave emission, and 푔훼 on average describes a small shift to the orbital frequencies arising from the small body’s impact on the binary. Both quantities also introduce short-timescale oscillations, which tend to average away. In the two-timescale expansion, the time scale describing these two phenomena are very different, and thus they can be separately analyzed. In particular, we can treatthe system’s trajectory as a flow between geodesics, which can be modelled by treating the “constants” of the motion as slowly varying parameters.

2.3 Self-Force Corrections

The leading order corrections to the geodesic motion of an orbiting body, or “self- force” corrections, describe how the orbiting body deviates from its geodesic motion. The orbit-averaged self force yields the radiation reaction which drives the small body’s inspiral into the black hole. In addition, if the small body itself spins about an axis, this spin couples to spacetime curvature, producing further forces which push the motion away from the geodesic limit. In this project, we analyze both radiation reaction and spin-curvature corrections to the motion of small body orbiting a Kerr

21 black hole.

2.3.1 Radiation Reaction

Due to the backreaction of gravitational waves on the EMRI system, the “constants” of the motion are not actually constant, but instead slowly evolve. Using the coefficients

퐻푙푚푘푛 in Equation (2.1) which describe the gravitational waves the system emits, as well as a similar set of coefficients which describe how the black hole is tidally stretched and squeezed by the orbiting body, we can compute formulas describing the evolution of 퐸, 퐿푧, and 푄 due to radiation emission. For Kerr black holes, there is a relationship between (퐸, 퐿푧, 푄) and (푝, 푒, 푥) such that we can then change the rates of change of the constants of motion to rates of change of these three orbital parameters.

2.3.2 Spin-Curvature Coupling

The second self-force correction we are concerned with is the spin-curvature correction. This case arises if the small black hole also has spin. When the small black hole spins, it couples to the spacetime curvature in a way that introduces a force that modifies the geodesic orbit used in the adiabatic approximation. The force on the smaller body due to the spin-curvature effect is written as

퐷푝휇 1 퐹 휇 = = − 푅휇 푢휈푆휆휎 (2.5) 푑휏 2 휈휆휎 where 퐷푝휇/푑휏 denotes a covariant derivative of the 4-momentum along the small

휇 휈 body’s trajectory, 푅휈휆휎 is the Riemann curvature tensor, 푢 is the small body’s 4- velocity, and 푆휆휎 is the spin tensor [6]. Knowing that the constants of motion are

22 related to the small body’s 4-momentum by

퐸 = −푝푡 (2.6)

퐿푧 = 푝휑 (2.7)

휇 휈 퐾 = 퐾휇휈푝 푝 (2.8)

2 푄 = 퐾 − (퐿푧 − 푎퐸) (2.9) where

2 2 퐾휇휈 = Σ(푚휇푚¯ 휈 +푚 ¯ 휇푚휈) − 푎 cos 휃푔휇휈 (2.10)

푚휇 ˙= [푚푡, 푚푟, 푚휃, 푚휑] (2.11) 1 ˙= √ [−푖푎 cos 휃, 0, Σ, 푖(푟2 + 푎2) sin 휃] (2.12) 2(푟 + 푖푎 cos 휃) we can take a derivative with respect to Mino time to find the rates of change of the constants of motion due to spin-curvature coupling [6]. However, because observables are measured using Boyer-Linquist time, it is useful to use the relationship between Boyer-Lindquist time 푡 and Mino time 휆,

푟2 + 푎2 푑푡/푑휆 = [︀퐸(푟2 + 푎2) − 푎퐿 ]︀ − 푎2퐸 sin2 휃 + 푎퐿 (2.13) ∆ 푧 푧

to find the rates of change 푑퐸/푑푡, 푑퐿푧/푑푡, and 푑푄/푑푡 [7]. For example, we find that the rate at which an orbit’s energy changes due to the spin-curvature force is simply related to the timelike component of the force in Equation (2.5). Converting from proper time to coordinate time, we find that 푑퐸/푑푡 is directly related to −퐹푡.

We really wish to find the change in the orbital parameters 푝, 푒, and 푥. To do this, we must first realize that the rates of change 푑푟/푑푡 and 푑휃/푑푡 at the turning points 푟푎, 푟푝, and 휃min must equal zero. Using the definitions given in Equations (1.8)

23 and (1.9), we can take a time derivative to find that

푑 푅(푟 ) = 0 (2.14) 푑푡 푝 푑 푅(푟 ) = 0 (2.15) 푑푡 푎 푑 Θ(휃 ) = 0 (2.16) 푑푡 푚푖푛

Given these equations and rearranging, we can find a Jacobian that relates the rates of change of the constants of motion to the rates of change of the orbital parameters:

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 퐽퐸푟푎 퐽퐿푧푟푎 퐽푄푟푎 푑퐸/푑푡 푑푟푎/푑푡 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ (2.17) ⎜ 퐽퐸푟푝 퐽퐿푧푟푝 퐽푄푟푝 ⎟ ⎜ 푑퐿푧/푑푡 ⎟ ⎜ 푑푟푝/푑푡 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 퐽퐸푥 퐽퐿푧푥 퐽푄푥 푑푄/푑푡 푑푥/푑푡

We can easily extract the equation for the rate of change of the inclination parameter 푥, given by √ 2 1 − 푥2√︀푄 − 푎2(1 − 퐸)2(1 − 푥2)퐿˙ − 푥푄˙ − 2푥푎2퐸(1 − 푥2)퐸˙ 푑푥/푑푡 = (1 − 푥2) 푧 2[푄 − 푎2(1 − 퐸2)(1 − 푥2)2] 2 ˙ 3 ˙ 2 2 ˙ 2푥(1 − 푥 )퐿푧퐿푧 − 푥 [푄 + 2(1 − 푥 )푎 퐸퐸] = 2 4 2 2 (2.18) 2[퐿푧 + 푥 푎 (1 − 퐸 )]

These two expressions are equivalent, with the first expression being well-behaved when |푥| is small and the second expression being well-behaved when |푥| is near 1.

The other two rates of change are given by

푑푟 * = 퐽 (푟 )퐸˙ + 퐽 (푟 )퐿˙ + 퐽 (푟 )푄˙ (2.19) 푑푡 퐸 * 퐿 * 푧 푄 * 24 with * being either 푎 or 푝, where

2 2 2 2 2 3 2 풟(푟) ≡ 2푀[푄 + (퐿푧 − 푎퐸) ] − 2푟[퐿푧 + 푄 − 푎 (1 − 퐸 )] + 6푀푟 − 4푟 (1 − 퐸 ) 4푎푀(퐿 − 푎퐸)푟 − 2퐸푟2(푎2 + 푟2) 퐽 (푟) ≡ 푧 퐸 풟(푟) 4푀(푎퐸 − 퐿 )푟 + 2퐿 푟2 퐽 (푟) ≡ 푧 푧 퐿 풟(푟) 푟2 − 2푀푟 + 푎2 퐽 (푟) ≡ . (2.20) 푄 풟(푟)

Using the definitions of 푟푎 and 푟푝 given in Equation (1.6), we can convert the rates of change of the apoapsis and periapsis to rates of change of 푝 and 푒, finding that

푑푝 (1 − 푒)2 (1 + 푒)2 = 푟˙ + 푟˙ (2.21) 푑푡 2 푎 2 푝 푑푒 (1 − 푒2) = [(1 − 푒)푟 ˙ − (1 + 푒)푟 ˙ ] . (2.22) 푑푡 2푝 푎 푝

Our ultimate goal is to calculate the small body’s trajectory due to both radiation reaction and spin-curvature effects. To get the full contribution of both to the rates of change of the orbital parameters, we simply add the rates of change such that

푑푝/푑푡 = (푑푝/푑푡)푅푅 + (푑푝/푑푡)푆퐶 (2.23)

푑푒/푑푡 = (푑푒/푑푡)푅푅 + (푑푒/푑푡)푆퐶 (2.24)

푑푥/푑푡 = (푑푥/푑푡)푅푅 + (푑푥/푑푡)푆퐶 (2.25)

where 푅푅 is the contribution due to radiation reaction effects and 푆퐶 is the contribution due to spin-curvature coupling.

25 26 Chapter 3

Implementation

Now that the mathematics have been introduced, I use the physics that is already understood about EMRI mergers to create Python and Mathematica pipelines for constructing inspiral worldlines and the resulting gravitational waveforms. The inspiral construction takes into consideration both radiation reaction and spin-curvature coupling effects, and the worldline that is created is then fed into the gravitational wave generator. Although the generation is fairly slow, the pipelines create a con- densed way to obtain the gravitational waves produced for any initial conditions in the parameter space.

3.1 Creating Inspiral Worldlines

To create inspiral worldlines, we utilize the adiabatic approximation to treat the trajectory of the stellar-mass black hole as a flow between geodesics. Section 2.3.1 describes how the rates of change of the constants of motion are computed due to radiation reaction effects. The program GREMLIN, created by Scott Hughes, computes these rates of change at a given 푝, 푒, and 푥 value [3]. However, generating the rates of change is computationally expensive, so instead of computing the rates of change at every step along an inspiral, we instead create a grid of the rates of change. A grid is created for a given massive black hole spin parameter 푎 in a range of 푎 ∈ [0, 1), stepped by 0.1. Each grid is structured, sampling from the last stable orbit 푝퐿푆푂 to

27 10 + 푝퐿푆푂, with higher sampling closer to 푝퐿푆푂 to resolve the more rapid variation in quantities in the strong field. The values of the rates of change of the constants of motion and of the orbital parameters for each grid point is imported as an HDF5 file into the inspiral generation code so these values can be stored and usedasmany times as we need.

When we generate an inspiral, we begin at an initial 푝, 푒, and 푥 point and use the rates of change at that point to step to the next point along the inspiral, where we then use the rates of change at that next point to step again, and so on. To find the exact values of the rates of change of the orbital parameters at a given point in the gridded space, we interpolate the grid using a cubic spline interpolation we created following [5]. The smooth behavior of flux values in the grid allows us to accurately interpolate to any point in the grid. Currently, the cubic spline interpolation function we created can only interpolate on 2D grids, so our current analysis is limited to either equatorial eccentric or inclined circular orbits.

Section 2.3.2 discusses how we can compute the rates of changes of the orbital parameters due to spin-curvature coupling. A Mathematica notebook created by graduate student Lisa Drummond computes the force due to spin-curvature coupling 퐹 휇 given in Equation (2.10) to obtain analytical functions for the rates of change of the constants of motion. Using Equations (2.18)-(2.27) we can then find analytical functions for the rates of change of orbital parameters. These functions are completely generic and depend on the spin of the massive black hole 푎 along with the orbital parameters 푝, 푒, and 푥. These analytical functions are too large for generic Kerr orbits to import into another Mathematica notebook to use for inspiral generation. Taking the Schwarzschild limit (푎 = 0) simplifies the analytical functions enough to make it possible for them to be imported into an inspiral generator. For this first analysis and exploration, we have confined our study to the influence of spin-curvature coupling on Schwarzschild inspirals. As we will discuss in Chapter 5, we plan in future work to import this functionality into a computational efficient C code which will make it possible to explore how spin-curvature coupling affects inspirals in the more general Kerr case.

28 It is important to note in what parameter space these functions can be used in. As mentioned previously, the radiation reaction data we have available cover a range of Kerr spin parameter 푎, including the Schwarzschild limit. At the moment, these data sets are only useful for orbits that are either eccentric but confined to the equatorial plane (cos 퐼 = ±1), or are inclined but circular (푒 = 0). Because the Schwarzschild limit is spherically symmetric, analysis in the equatorial plane is sufficient to describe the generic problem. Since practical considerations inour analysis of the spin-curvature force drove us to focus our analysis on Schwarzschild in this thesis, we only consider equatorial and eccentric orbits in this initial analysis. Now that we have methods to compute both radiation reaction and spin-curvature values for the rates of change of the orbital parameters, we can begin generating inspiral worldlines. As touched on previously, the way we create inspirals is to begin at an initial point and step through the worldline. We use Runge-Kutta intergration [5] to step along the inspiral. The inspiral ends when the worldline has crossed the last stable orbit. At this point, we have obtained the full trajectory that the small body travels from the initial point until it crosses the last stable orbit and plunges into the massive black hole.

3.2 Generating Gravitational Waves

After modeling the inspiral, we use the solution to the frequency-domain Teukol- sky equation given in Equation (2.1) to model gravitational waves produced along an inspiral. Along with computing the rates of change of the orbital parameters,

GREMLIN also computes the complex amplitudes of gravitational waves 퐻푙푚푘푛 and the orbital frequencies Ω푟,휃,휑 [3]. As mentioned previously, GREMLIN is computationally expensive; therefore, we also create grids for the complex amplitudes and orbital frequencies, which depend on the location on the grid in 푝, 푒, and 푥, along with the given mode 푙, 푚, 푘, 푛. We use the same cubic spline interpolation functions as when we were computing the worldline to find the values of the complex amplitudes and orbital frequencies at the 푝, 푒, and 푥 values along the worldline.

29 The gravitational wave generator inputs an inspiral worldline and computes the gravitational wave ‘voice’ for each mode produced for the whole inspiral, summing up each voice to obtain the actual gravitational wave produced for the whole inspiral. Although Equation (2.1) says that 푛 and 푘 should be summed from negative infinity to infinity, this is computationally unrealistic. Because lower order modes contribute more to the waveform, we see that it is satisfactory to only sum 푛 and 푘 to a finite amount. Note for equatorial orbits, 푘 is equal to zero and does not need to be summed over. In this instance, for lower eccentricity, it is sufficient to sum 푛 from around -10 to 10 and for higher eccentricity, it is sufficient to sum 푛 from around -30 to 30. In all instances, we choose a high enough range so that we can resolve the waveform. We can then analyze the gravitational waves produced to see how inspirals created with radiation reaction and spin-curvature effects modify the waveforms we detect.

3.3 Performance

While we have taken efforts to optimize the inspiral and gravitational wave generators, the computations are still fairly time-consuming. For radiation reaction only inspirals in Schwarzschild spacetime, the worldlines take around 1 second at low eccentricity and around 80 seconds at high eccentricity for the Python generator. For the Mathematica generator, the same low eccentricity inspiral takes around 270 seconds to produce. When spin-curvature effects are added, the Mathematica generator becomes much slower. At low eccentricities for a system mass ratio of 10−3, the inspiral is created in approximately 13 hours, and high eccentricities are created in around 45 hours. We find that the computation time scales with mass ratio. A smaller mass ratio of 5 × 10−3 takes around 2 to 2.5 hours and a higher mass ratio of 10−4 takes around 40 hours for the low eccentricity case. Changing the step size of the Runge- Kutta integrator helps marginally. Changing the step size from 10 to 15 reduces the low eccentricity inspiral from 13 hours to around 8 hours. We must be careful not to increase the step size so much as to lose the resolution on the spin-curvature oscillations.

30 Generating the gravitational waveforms is also fairly time-consuming. We first try generating waveforms for a step size of 10−5 between consecutive inspiral data points, which produces the waveforms in approximately 30 minutes. However, these waveforms are not resolved enough to see the small scale structure. A step size of 10−6 between consecutive data points begins to resolve the structure of the waveforms, but the generating time is increased to approximately 5 hours. There is a lot of potential to optimize these pipelines to obtain faster inspiral and waveform generators, which will be touched upon in Chapter 5.

31 32 Chapter 4

Results

In this chapter, I present the resulting EMRI inspiral worldlines and gravitational waveforms for a set of initial conditions that were produced with the generators described in Chapter 3. For the reasons discussed there, we have confined our present analysis to the study of the trajectories of spinning bodies on equatorial orbits around Schwarzschild black holes. Our analysis shows that for much of the inspiral, the spin-curvature coupling imprints itself as a fairly low amplitude oscillation on the inspiral; both the semi-latus rectum 푝 and eccentricity 푒 oscillate as the spinning body precesses while moving along its orbit. We find that these oscillations grow quite large as the small body inspirals into the strong field of the black hole, reaching a peak before becoming small again just before crossing the last stable orbit. As we discuss in this chapter, this appears to be a resonance phenomenon. These features imprint themselves on the waveforms, demonstrating that the spin-curvature coupling produces important observable effects.

4.1 Accuracy Test

First, we wish to check that our code creates accurate inspiral worldlines compared to other worldline generators that are available. The generators that are available only consider radiation reaction effects. Therefore, we take our two inspiral generators, one in Python and one in Mathematica, and compare the outputs for radiation reaction

33 Figure 4-1: Comparison of an EMRI inspiral with a mass ratio of 10−3 due to radiation reaction effects starting at an initial point (푝, 푒, 푥) = (8, 0.2, 1). Our Python and Mathematica generators were compared to Scott Hughes’ C code. Hughes’ code uses slightly different stopping criteria and stepping size, which appears to be whythe C code trajectory deviates slightly from the other two. Other than this, the three generators agree to excellent precision. effects with a sample condition, 푝 = 8, 푒 = 0.2, 푥 = 1, 푎 = 0, with a mass ratio of 휇/푀 = 10−3 with the output from a C code that Scott Hughes had developed. The comparison graph is shown in Figure 4-1. As shown in Figure 4-1, we see that our inspiral generators agree to excellent precision to other worldline generators when using radiation reaction effects, other than a very minor deviation due to different stopping criteria and stepping size.

4.2 Radiation Reaction and Spin-Curvature Effects on Inspiral Worldlines

Now that we have confirmed our programs agree with other worldline generators, we next wish to understand how spin-curvature coupling modifies the radiation reaction

34 Figure 4-2: Comparison of radiation reaction (RR), spin-curvature (SC), and com- bined radiation reaction and spin-curvature (RR+SC) effects on 푝 and 푒 as a function of time 푡 for the case of initial condition (푝, 푒, 푥) = (8, 0.2, 1). Note that adding in spin-curvature effects leads to an oscillation about the radiation reaction inspiral. inspiral worldlines we produce. As an example, we take a small body with an initial point (푝, 푒, 푥) = (8, 0.2, 1) and look at how spin-curvature (SC), radiation reaction (RR), and both effects together (RR+SC) influence the changes in 푝 and 푒 as a function of Boyer-Lindquist time 푡. The results for this case are shown in Figure 4-2. The immediate effect we notice is the spin-curvature coupling creates an oscillation around the radiation reaction only inspiral, as expected.

4.2.1 Effect of Initial Conditions

Next, we wish to consider how different initial parameters influence the spin-curvature effects. We choose initial conditions (푝, 푒, 푥) = (8, 0.2, 1), (9, 0.5, 1), (10, 0.8, 1) with an EMRI mass ratio of 10−3. Note that all the given initial conditions are in equatorial orbits since they are in a Schwarzschild background spacetime. Figure 4-3 shows the resulting worldlines in 푝 vs 푒, comparing a trajectory with just radiation reaction (RR) effects with a trajectory with both radiation reaction and spin-curvature effects

35 (RR+SC). A few observations are immediately clear. First, we see that the case with initial condition 푒 = 0.8 grows to a larger eccentricity than 0.8 for a semi-latus rectum 푝 around 8.7. The inspiral in this case cannot be trusted for 푝 . 8.7 since the oscillations in 푒 go beyond the domain of data that we have and thus gives us something unphysical when 푒 gets large. We still include this case to demonstrate the other trends we observe. The first trend we observe is that when initial eccentricity increases, the oscillations created when adding spin-curvature effects deviates from a sine-wave like pattern to more pronounced, peaked waveforms with long smooth stretches in between. This effect is expected; the spin curvature force involves coupling of motion to curvature, with the strength of the curvature falling off as 1/푟3. Using Equation (1.5) with

휓 = 휒푟 being the radial Darwin phase, which increases smoothly as the orbit’s radius oscillates between periapsis and apoapsis, we expect contributions to the force to vary

3 3 as (1 + 푒 cos 휒푟) /푝 . As the orbit varies between periapsis and apoapsis, this term varies by a large amount; for eccentricities of 0.2, 0.5, and 0.8, this contribution varies by a factor of 3.375, 27, and 729, respectively. Along the same lines, when spin-curvature coupling is weak, we expect the orbit to spend more time near apoapsis, so we do not expect the oscillations to be sine-like unless the initial eccentricity is small. Instead, as we see in the 푒 = 0.5 and 푒 = 0.8 cases, we expect long stretches of inactivity in between large “punches” of spin-curvature coupling as the small body passes periapsis. Another observation found is that there is a resonance-like feature during stages of the orbit such that there is a deviation from the radiation reaction inspiral. This deviation is larger for increasing eccentricity.

4.2.2 Effect of Inspiral Mass Ratio

To further explore this resonance-like feature in inspirals considering spin-curvature effects, we next wish to explore how different mass ratios will alter this resonance feature. As an example, we take two initial conditions, (8, 0.2, 1) and (9, 0.5, 1) and

36 Figure 4-3: Comparison of radiation reaction and spin-curvature effects for various initial conditions. The first row is for an initial condition of (8, 0.2, 1), the second row for (9, 0.5, 1), and the final row for (10, 0.8, 1). The left column displays the entire inspiral while the right column zooms in on only the beginning of the inspiral. We see that an increase in eccentricity leads to a larger deviation from the radiation reaction only inspiral and changes the spin-curvature oscillations from sine-like to more spread out, peaked waves. Note that the case with 푒 = 0.8 grows to an eccentricity larger than 0.8, which is beyond the domain of our data set and thus cannot be trusted for 푝 . 8.7. This inspiral is still shown to emphasize the trends we witness as eccentricity increases.

37 Figure 4-4: Comparison of spin-curvature coupling for increasing mass ratios of 휇/푀 = 5 × 10−3, 10−3, and 10−4 for the (푝, 푒, 푥) = (8, 0.2, 1) initial condition. look at mass ratios of 5 × 10−3 and 10−3, along with 10−4 for the 푒 = 0.2 case. A mass ratio of 5 × 10−3 is beginning to abuse perturbation theory, but is still useful for exploring how these resonant-like phenomena behave. Figures 4-4 and 4-5 display how the spin-curvature coupling changes due to mass ratio for these two eccentricities.

We can immediately see that the mass ratio does impact the spin-curvature coupling oscillations as well as the resonance effect. Firstly, the oscillations about the radiation reaction inspiral scale with the mass ratio. As the mass ratio increases, the amplitudes of the oscillations decrease, as expected. The ratio of the amplitude of the feature to the oscillations and the width of the feature increases as the mass ratio increases. It appears that the oscillations get more spaced out as the inspiral approaches the feature, leading to a big jump, with more smaller oscillations after the feature. We expect the duration of the resonance feature to scale with the square root of the mass ratio, with a longer duration for smaller mass ratio, which is consistent with our results.

38 Figure 4-5: Comparison of spin-curvature coupling for increasing mass ratios of 휇/푀 = 5 × 10−3 and 10−3 for the (푝, 푒, 푥) = (9, 0.5, 1) initial condition.

We also see that the phase effects of the features all look different for thethree ratios. We expect how the system exits the resonance to vary depending on some accidental phase at which the system enters resonance. To explore the dependence of this accidental phase, we will next consider different spin axis phases. These effects, along with the width of the features, are all consistent with a resonance phenomenon. A possible model to explain this resonance-like feature may be that different Fourier modes of the forcing terms are combining in phase when certain orbital frequencies become commensurate during the inspiral.

4.2.3 Effect of Spin Axis Phase

For all of the cases we have considered this far, we have been using a case where the spin axis of the smaller body is oriented as (푆푟, 푆휃, 푆휑) = (1, 1, 1). To further explore the resonance-like effect we discovered, we now wish to consider cases with different spin axis orientation. We will now let the spin axis (푆푟, 푆휃, 푆휑) = √ √ ( 2 cos 휑, 1, 2 sin 휑), with the phase 휑 corresponds to a rotation of the axis nor-

39 Figure 4-6: Comparison of different small body spin axis orientations for the initial condition of (푝, 푒, 푥) = (8, 0.2, 1) with a mass ratio of 휇/푀 = 10−3. We consider axis phases of 휑 = 휋/2, 휋/4, and 휋/8.

mal to the orbital plane. The original (1, 1, 1) case corresponds to an axis phase of 휋/4. To explore the effect of spin axis phase, we will choose initial conditions of (푝, 푒, 푥) = (8, 0.2, 1) with a mass ratio of 휇/푀 = 10−3, and change the phase from 휋/4 to 휋/8 and 휋/2. Figure 4-6 shows the effects of varying the spin axis phase on the spin-curvature coupling inspiral.

From Figure 4-6, we can deduce that as the phase increases, the resonance feature shifts and the amplitude of the effects is enlarged. We suspect that if we were to continue to increase the phase, the feature will rise and then fall corresponding to the angle that goes into the spin components. In other words, the spin axis phase modulates terms in the spin-curvature force that combines in resonance such that the integrated impact of the resonance will increase and then decrease as the phase increases.

40 Figure 4-7: The evolution of eccentricity 푒 and Mino-time frequency 2Υ푟 − Υ푠 for the case (8, 0.2, 1) with a mass ratio of 10−3 and spin axis phase 휋/2. We see that the intersection of 2Υ푟 − Υ푠 = 0 corresponds to the time right before the resonance feature begins.

4.3 Orbital Frequencies During Resonance

To test our theory that the feature we observe is indeed a resonance, we now explore how the orbital frequencies compare to each other. We know that the spin-curvature force is axisymmetric (meaning it does not have any 휑 dependence), so we know that a resonance feature must be coincident with the moment when a combination of the radial and spin frequencies passes through zero. Figure 4-7 demonstrates how the

combination 2Υ푟 − Υ푠 and the eccentricity evolve as a function of coordinate time for the initial system (푝, 푒, 푥) = (8, 0.2, 1) with a mass ratio of 10−3 and spin axis phase of 휋/2, where Υ is the Mino-time frequency.

We see that 2Υ푟 − Υ푠 passes through zero right before the resonance feature, showing that the resonance-like feature in the inspiral is roughly coincident with the moment when this combination of frequencies is zero. A postulate for what is happening is that we are seeing a resonance between the geometry of the precessing

41 body and the radial location of the orbit. The spin-curvature force is largest at periapsis; if there were no relation between the two orbital frequencies, then when the orbit passes through periapsis, the force would kick the orbit with a random phase. However, when the two frequencies are in resonance, the precessing body would have the same orientation at each periapsis passage such that the spin-curvature force kicks the orbit in the same way. This effect builds up with each pass through periapsis until the orbit evolves further and the frequencies are no longer in phase. Although this is a valid hypothesis for what may be happening, a followup with more analysis is needed to investigate why this feature occurs.

4.4 Effects Imprinted on Waveforms

Now that we have fully investigated the effect spin-curvature coupling has on EMRI inspiral worldlines, we now wish to see how these effects appear in the gravitational waveforms we produce. To explore how spin-curvature coupling affects the waveforms, we consider the initial conditions of (푝, 푒, 푥) = (8, 0.2, 1) for spin phases of 휋/2 and 휋/4 to understand the impact that the spin axis phase has on the results. Figures 4-8 - 4-10 show the resulting waveforms for the different spin axis phases for beginning, during resonance, and late times in the inspiral. We see from Figure 4-8 that, in the beginning of the inspiral, the waveforms that contain spin-curvature coupling are very similar and deviate slightly from the purely radiation reaction waveform. This is consistent with Figure 4-6, as the beginning of the inspirals for each of the spin axis phases are very similar in the early stages of the inspiral. Near the resonance feature of the inspiral shown in Figure 4-9, we see that the spin-curvature gravitational waves vary significantly from the radiation reaction wave. Each of the phases are similar but distinctly different from the others. Then, at the very end of the inspiral, shown in Figure 4-10, we see that each waveform looks very different. Each inspiral ends at a variety of different 푝 and 푒 values, and thus their waveforms end at different times. We see that the radiation reaction only waveform ends first, followed by the 휋/4 and 휋/2 waveforms. From this example,

42 Figure 4-8: Gravitational waves produced at the beginning of the inspiral for initial condition (8, 0.2, 1) comparing radiation reaction (top) and spin-curvature effects at spin axis phases 휋/2 (middle) and 휋/4 (bottom). We see that both spin-curvature waveforms are similar at beginning times, but are different from the radiation reaction only waveform.

43 Figure 4-9: Gravitational waves produced during the resonance feature of the inspiral for initial condition (8, 0.2, 1) comparing radiation reaction (top) and spin-curvature effects at spin axis phases 휋/2 (middle) and 휋/4 (bottom). We see that all three waveforms have different behavior during resonance.

44 Figure 4-10: Gravitational waves produced at the end of the inspiral for initial condition (8, 0.2, 1) comparing radiation reaction (top) and spin-curvature effects at spin axis phases 휋/2 (middle) and 휋/4 (bottom). We see that each waveform ends at a different coordinate time and have different forms at the end of the inspiral.

45 we see that spin-curvature coupling corrections do lead to a significant impact in the gravitational waves we detect.

46 Chapter 5

Conclusions and Future Work

In this report, I have demonstrated how we can include higher order corrections of perturbation theory to create EMRI trajectories and waveforms. When considering both radiation reaction and spin-curvature effects, we find that spin-curvature coupling creates oscillations about the radiation reaction inspiral. These oscillations are more sine-like at low eccentricities and more peaked and spaced apart at higher eccentricities. Furthermore, we find that there is a resonance-like behavior during the middle to late stages of the inspiral that is created due to the orbital frequencies of the system combining in a way to create a resonance-like feature in the inspiral. This feature is dependent on the mass ratio of the EMRI system and the spin axis phase of the smaller body. We see that these effects imprint themselves in the gravitational waves produced. When spin-curvature coupling effects are added, the waveforms deviate significantly from the waveform created solely from a radiation reaction effect. We see that the resonance effect that we observe in the inspiral worldline likewise is imprinted on the waveforms we detect. As mentioned in Section 3.3, the code takes a while to run when the spin-curvature effects are incorporated. A natural next step in this work is to optimize ourprograms so that we are able to generate inspiral worldlines and waveforms in a timely fashion. Along with optimizing the current code, we will also translate some of the tools we created to C/C++ to produce a faster generator. In future stages of this project, we also wish to expand the parameter space that our code works for. As mentioned

47 previously, the code combining spin-curvature and radiation reaction effects is only available for Schwarzschild background spacetime. When we incorporate the possi- bility of a Kerr background, we can also model the evolution of inclination as the small body inspirals. We have already incorporated Kerr spacetime for the radiation reaction effects. However, our cubic spline interpolation function currently only works in two-dimensional grid spaces. A future version of this interpolation function will be able to take in three-dimensional spaces, allowing us to have completely generic orbits, as opposed to the equatorial eccentric or inclined circular orbits we are currently limited to. Once we incorporate these additions, we will be able to model any generic EMRI system in both Schwarzschild and Kerr backgrounds. The resonance-like effect described in Chapter 4 was not anticipated, and deserves much more attention. We intend to investigate this feature in more detail by decom- posing the terms which drive the oscillations in 푝 and 푒 in the Fourier domain and identifying how different harmonics evolve as the system evolves through this feature. We also intend to more thoroughly examine how the feature depends on orbit geometry and on the orientation of the inspiraling, spinning body, as well as to see whether this behavior persists for inspirals into Kerr black holes. If further work shows that this phenomenon occurs for a wide range of eccentric EMRI inspirals, this could have very important ramifications for the analysis of these signals using future LISA measurements.

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