Hypervelocity Stars Originating in the Andromeda Galaxy As a Probe of Primordial Black Holes

BIELEFELD UNIVERSITY

FACULTY OF PHYSICS

MASTERTHESIS

Hypervelocity stars originating in the Andromeda galaxy as a probe of primordial black holes

Author: Lukas Gulzow¨ [email protected] Bielefeld University

Supervisors: Prof. Malcolm Fairbairn [email protected] King’s College London

Prof. Dominik Schwarz [email protected] Bielefeld University

Sunday 31st January, 2021

Contents

1. Introduction5

2. Black holes and dark matter6 2.1. Black holes ...... 6 2.2. Gravitational waves and their recent detection ...... 8 2.3. Dark matter ...... 10

3. Primordial black holes 15 3.1. Formation mechanisms ...... 16 3.2. Constraints on primordial black hole dark matter ...... 17 3.3. Phenomena possibly explained by primordial black holes ...... 18

4. Hypervelocity stars 19 4.1. Hypervelocity stars from outside the Milky Way ...... 20 4.2. Selection of Gaia data ...... 22

5. Ejection of stars from Andromeda 26

6. Description of the simulation 27 6.1. Equations of motion ...... 28 6.2. Acceleration and mass terms ...... 29 6.3. Mass of the Local Group ...... 31 6.4. Dynamics of the two galaxies ...... 33 6.5. Initial conditions for the hypervelocity stars ...... 35 6.6. Coordinate transformations of the simulation results ...... 37

7. Results and discussion 42 7.1. Distance and velocity distributions ...... 43 7.2. Positions and velocity directions ...... 51

8. Conclusion 64

Acknowledgements 65

References 66

A. Coordinate transformations of the Gaia data 76

B. Distribution of component velocities 79

C. Simulation code 82

Declaration of authenticity 84

1. Introduction

A large part of the history of the Universe is also the history of dark matter. Ever since the first pieces of evidence for an invisible massive component of the energy density of the Universe were discovered [1, 2], there have been countless theories about its identity. Since then, many observations have been made to support the existence of dark matter [3, 4, 5, 6]. The attempts to learn more about dark matter have had varying levels of success, but to this day no consensus has been reached for any one theory. For a long time, weakly interacting massive particles (WIMPs) [7] were considered the prime candidate for dark matter. However, since none of many experiments have been able to detect them, they are falling out of favour [8]. Many other possibilities are being explored at the same time. They include axion [9, 10] and neutrino [11, 12] dark matter as well as candidates outside the realm of particle physics. This thesis investigates one such candidate, primordial black holes [13, 14, 15, 16]. A primordial black hole (PBH) is a theoretical kind of black hole that could have formed in the early Universe. PBHs were first considered by Hawking and Zeldovich in the 1970s [17, 18, 19]. If PBHs exist within galaxies, they are considered massive compact halo objects (MACHOs). MACHOs are objects in the halo of a galaxy that are not bound to any particular star. They typically emit only small amounts of radiation or none at all. Since PBHs form before Big Bang nucleosynthesis, they are considered nonbaryonic [20]. Due to this property, they are dark matter candidates in a MACHO dark matter scenario. Even though we have no evidence for their existence, PBHs are useful for a number of reasons in addition to their potential as a dark matter candidate. Low mass PBHs are the only objects that could emit a detectable amount of Hawking radiation by evaporation [21]. This is useful to place constraints on models of the early Universe and presents a possible explanation for certain phenomena, for example extragalactic [22, 23] and Galactic [24, 25] γ-ray backgrounds and γ-ray bursts [26, 27]. On the other hand, PBHs of higher mass that are not significantly affected by evaporation could serve as the seeds of supermassive black holes at the centres of galaxies [28, 29] or generate large-scale structure [30], amongst other things. However, it is important to note that all of these features have other possible explanations. In recent years, PBHs have become a more popular approach to explain dark matter [31, 32, 33] due to the detection of gravitational waves by LIGO/Virgo [34, 35]. The masses of the observed mergers of binary black holes are larger than expected and might be a sign of a significant population of PBHs with masses of order 10 M [31]. However, the most popular explanation considers the LIGO/Virgo detections to be the remnants of ordinary stars [36]. The topic of this thesis is inspired by a publication by Montanari, Barrado and Garc´ıa- Bellido on the topic of unbound hypervelocity stars (HVSs) and their connection to PBHs [37]. HVSs are stars that move with velocities of order 103 km/s. The HVSs we observe generally seem to originate in the inner parts of the Milky Way [38, 39]. Due to their high speed, they can be unbound to the Milky Way gravitational potential. Their origin is thought to be gravitational interactions with the supermassive black hole at the Galactic

5 centre (GC) [40, 41]. However, a small number of HVSs have trajectories pointing towards the Milky Way instead of away from it. This indicates a different point of origin [42]. As a possible explanation for this phenomenon, Montanari et al. [37] investigate the ejection of HVSs from nearby dwarf galaxies, globular clusters and the Andromeda galaxy caused by massive PBHs. For this purpose, they use data from the Gaia data release 2 (DR2) catalogue of the Gaia space telescope mission. The ejection mechanism as well as the speciﬁc PBH dark matter scenario were previously discussed by Clesse and Garc´ıa- Bellido [31, 32]. While Montanari et al. [37] mostly focus on HVSs from dwarf galaxies close to the Milky Way, this thesis lays its focus on HVSs ejected by massive PBHs within the An- dromeda galaxy. We randomly generate positions in the inner parts of Andromeda from which HVSs are ejected as well as velocity vectors just above escape velocity in random directions. We construct a dynamical simulation model of the gravitational system of Andromeda and the Milky Way in Python [43, 44, 45] and calculate the trajectory of each HVS in the potential of both galaxies. Trajectories sufﬁciently close to the Sun at present time t0 = 13.8 Gyr are analysed and the resulting data is transformed to Galactic coordinates. Similarly to Montanari et al. [37], we compare the trajectories to HVS data from the Gaia early data release 3 (eDR3) catalogue [46, 47, 48]. We transform th e Gaia data to Galactic coordinates as well. Finally, we evaluate the possibility of massive PBHs ejecting HVSs from Andromeda towards the Milky Way based on the results. Section 2 gives an overview about black holes, the recent LIGO/Virgo observations of binary black hole mergers, and dark matter. In Section 3 we introduce PBHs and discuss possible formation mechanisms, the constraints on PBH abundance in various mass ranges as well as problems in cosmology that PBHs could provide solutions for. In Sec- tion 4 we introduce HVSs, present a concise summary of the paper by Montanari et al. [37] on unbound HVSs and discuss our own selection of HVS data from the Gaia eDR3 catalogue. Section 5 discusses the ejection process by which HVSs are ejected from the Andromeda galaxy. In Section 6 we give a detailed description of the calculations involved in the simulation model. The results of the simulation are presented and discussed in comparison with Gaia data in Section 7. We make concluding statements in Section 8. Appendix A details the transformation of Gaia velocity data to Galactic coordinates. Appendix B contains additional results on the component velocities of HVSs from both simulation and Gaia data. Appendix C gives a description of the code used for the simulation.

2. Black holes and dark matter

2.1. Black holes Black holes were ﬁrst predicted as an exact solution of Einstein’s ﬁeld equations of general relativity in 1916 by Schwarzschild [49]. However, only in the 1960s it was proven by Penrose and Hawking that black holes appear in the Universe naturally [50]. A black

6 Fig. 1. Image of the supermassive black hole at the centre of the M87 galaxy and its shadow by the Event Horizon Telescope Collaboration [53]. hole is a region of spacetime in which the gravitational potential is so high that nothing can escape it. Its spherical boundary is called the event horizon. The mass of a black hole is predicted by general relativity to be concentrated in a point-like singularity in the center of the event horizon. The event horizon of a non-rotating black hole lies at distance

2GM R = (2.1) S c2 to the singularity. Here, G is Newton’s constant of gravity, M is the mass of the black hole and c is the speed of light. The radius RS is called the Schwarzschild radius [49]. Since black holes do not emit visible light or measurable amounts of any other radiation, they are not directly observable. Stellar black holes are produced by gravitational collapse. At the end of the life cycle of a massive star, its remnant can collapse into a black hole if it has sufﬁcient mass. If the mass of the remnant is not sufﬁciently large, it will collapse into a neutron star or white dwarf instead. Neutron stars can further collapse into black holes by accreting matter or merging with other stars, white dwarfs or neutron stars to reach the critical mass limit MTOV 2.3 M [51]. This limit is called the Tolman–Oppenheimer–Volkoff limit [52]. Black holes≈ created in this process have masses of the order of a few Solar masses up to a few tens of Solar masses, depending on the progenitor mass of the collapsing star. After their formation, they can increase their mass and grow in size by accreting matter or merging with other objects. While we cannot observe a black hole directly, we can detect the effects it has on its environment. We can detect the objects that orbit around it as well as the gravitational lensing effects it has on our observations of objects behind it. It is also possible to observe the “shadow” of a black hole in some cases [54]. Matter that slowly falls into a black hole

7 often forms an accretion disc around it. Friction between the particles in the disc leads to high temperatures as the density increases closer to the event horizon. This causes the disc to emit thermal radiation. The black hole casts a shadow on the radiating accretion disc that can be observed. The shadow is highly distorted due to the effect the gravitation of the black hole has on the radiation from the disc. At the centre of most galaxies we observe supermassive black holes (SMBHs) with masses of the order of millions or even billions of Solar masses [55]. SMBHs with massive accretion discs are called active galactic nuclei or AGNs. They are some of the brightest objects we have observed so far. There are multiple possibilities for the formation of SMBHs. The simplest possibility is massive stellar black holes that grew after their formation until today by accreting large amounts of matter. Alternatively, a theory suggests that before star formation began, large gas clouds were formed as gas was drawn in by already further collapsed dark matter halos. Dark matter collapses at earlier times than baryonic matter due to the lack of opposing forces like friction and photon pressure. These gas clouds ﬁrst collapse into a “quasi-star”. The “quasi-star” then directly collapses into a black hole without a supernova ejecting most of its mass [56]. In another scenario, the core-collapse of a dense star cluster is the origin [57]. Finally, PBHs, formed instants after the Big Bang, could serve as “seeds” for SMBHs [58]. We will discuss PBHs in more detail in Section 3. Stephen Hawking predicted in 1974 that black holes evaporate by emitting thermal radiation at a temperature inversely proportional to their mass [21]. This means low mass black holes radiate at higher energies than more massive ones. Black holes are also expected to slowly shrink over time due to this process if they do not accrete enough matter to counteract the loss of mass through evaporation. Theoretically, we can detect these black hole emissions. However, the emissions of a black hole of even stellar mass already have such low energy that it is virtually impossible to detect since it blends into the background noise. Black hole or Hawking radiation has not been detected so far. There has been a number of recent developments in black hole research. In 2019, the Event Horizon Telescope Collaboration produced an image of the SMBH at the center of the Messier-87 galaxy and its shadow using the radiation emitted by its accretion disk [54]. The image is shown in Fig. 1. The image matches predictions from general relativity about gravity in the vicinity of a black hole. Another major development in black hole and general relativity research was the direct detection of gravitational waves by LIGO/Virgo in 2015 [59].

2.2. Gravitational waves and their recent detection Einstein predicted gravitational waves (GWs) in 1916 as a consequence of his theory of general relativity [61]. Similar to how accelerated charges generate electromagnetic waves in electrodynamics, accelerated masses generate GWs. Energy is transported as gravitational radiation which manifests as oscillations of the curvature of spacetime. GWs travel at the speed of light. A large solitary mass does not emit GWs since it does not experience any acceleration. To create detectable amounts of emitted radiation, another close mass of similar size is required. For this reason, binary systems of compact objects

8 Fig. 2. Masses of all compact binaries and their mergers detected by LIGO/Virgo as of November 2020 [60]. Black holes of typical merger mass detected by GW emission are displayed in blue, neutron stars in orange. Stellar mass black holes and neutron stars discovered by electromagnetic radiation are shown in purple and yellow, respectively. Arrows go from the two primary black holes to the coalesced black hole. with large masses like black holes and neutron stars are the prime candidates for the detection of GW emission. The first indirect evidence for the existence of GWs was found in 1974 by Hulse and Taylor in the form of a pulsar binary [62, 63]. The decaying orbit of the pulsars is in good agreement with the energy loss due to gravitational radiation predicted by general relativity. In 1993 Hulse and Taylor received the Nobel prize for their discovery and its implications. In 2015, gravitational waves were first detected directly by the LIGO/Virgo collaborations. Their observatories use laser interferometers with arm lengths of 4 km to detect the slight spacetime disturbances caused by passing GWs. Since the detection∼ of GW150914, the first event in 2015, LIGO/Virgo has undergone multiple observing runs and detected several more GW events. Including the results of the first half of the latest run O3a (Observing run 3), published on 28 October 2020, LIGO/Virgo has now detected a total of 50 GW events [34, 35]. The masses of all detected binaries as well as the masses of the resulting coalesced black holes are displayed in Fig. 2. Most of the detected events are mergers of black holes that have masses of the order of 30 M . The mass of the coalesced black hole is typically a few Solar masses lower than∼ the sum of the two primary masses due to the energy loss from gravitational radiation

9 during the inspiraling. The ﬁrst detection and all further observations of GW activity are being used to make statements about the properties of the newly discovered black hole binaries. [35]. Most notable are the mass and the spin distribution of binary black holes as well as the global merger rate. It was not expected to discover a signiﬁcant population of black holes or black hole binaries of higher mass than is typically expected for black holes of stellar origin. The possibility stands that LIGO/Virgo detected a signature of dark matter in the form of a population of massive PBHs [33]. Before this possibility is discussed in detail in Section 3, we want to give an overview about dark matter.

2.3. Dark matter The standard consensus model of cosmology relies heavily on dark matter [64, 65]. It is called ΛCDM model. Λ stands for a cosmological constant representing “dark energy” with constant density, and CDM for “cold dark matter” [66]. According to ΛCDM, the analysis of the cosmic microwave background (CMB) radiation shows that most of the energy density of the Universe is accounted for by energy components we have not been able to detect directly so far. This is what we call “dark matter” and “dark energy” [4]. Stars, planets and interstellar gas, in other words visible, baryonic matter, only account for about 1/6 of the total matter we observe in the Universe. The remaining 5/6 are something that we cannot directly see or detect. Nonetheless, it behaves like matter and we can observe its gravitational effects. This unknown component does not interact with the electromagnetic ﬁeld in any way we can observe. We call this phenomenon “dark matter”. ΛCDM requires the additional gravity of dark matter to slow down the expansion of the Universe after the Big Bang. Without only baryonic matter, the content of the Universe would be spread much thinner and it would not have the same light element abundance [67]. However, baryonic and dark matter combined still only provide just over 30% of the total energy present in the Universe. We attribute the remaining 68% of the total energy present in the Universe to “dark energy”. This is the name of∼ the phenomenon that is thought to cause the accelerated expansion of the Universe. The density of dark energy is assumed to be uniform across all of space and does not dilute with its expansion. As such, its presence becomes more dominant as time goes on and the Universe expands further. The energy distribution of dark energy, dark matter and baryonic matter in the Universe at present time is displayed in Fig. 3. There is plenty of evidence for the existence of dark matter. All of it is completely based on the behaviour of baryonic matter which does not agree with models that are baryon-only. The ﬁrst pieces of evidence were found in the high velocities of galaxies within clusters [1] and in galaxy rotation curves [2]. In the case of high velocity galaxies, their velocities are too high to be gravitationally bound to the corresponding clusters if the clusters are only as heavy as their luminosities imply. Similarly, in galaxies with only baryonic matter the rotation speed of stars in the outer regions is expected to decrease anti-proportionally to the distance to the centre of the galaxy. However, in every galaxy

10 Fig. 3. Energy distribution in the Universe today [68]. Ordinary matter refers to baryonic matter. we have observed so far, the rotation speed is approximately constant at all distances above a certain distance to the centre. In both the clusters and the galaxies, their outer components have such high velocities that they should fall apart as a result if only baryonic mass is present. One explanation is the presence of large amounts of unseen, dark matter in galaxy clusters and galaxies themselves. A comparison between the expected and observed rotation curve of a galaxy is shown in Fig. 4. Gravitational lensing experiments also strongly suggest the existence of dark matter [3]. Gravitational lenses are a phenomenon predicted by both classical physics and general relativity. Large concentrations of matter bend the light rays passing them due to their gravitational potential. According to general relativity, they bend spacetime itself with their gravity which means they inﬂuence the path of light [70]. The resulting effect is similar to classical optical lensing although the mechanism behind it is wholly different. A gravitational lens can produce an image ring from a single luminous source if it is fortunately positioned. This phenomenon is called an Einstein–Wilson ring [71]. Fig. 5 schematically shows the geometry of a gravitational lens and the images it might produce. We can use gravitational lensing to assess the mass of the lensing object by looking at the separation of the resulting images in the sky. A galaxy can produce images that are a few arc seconds apart [72]. Galaxy clusters are capable of separating images by several arc minutes [73]. Comparing these measurements to the expected lensing effects of the visible mass shows a lensing effect that is about 90% higher than the mass evidence from radiation. This result, again, implies a matter component that does not interact electromagnetically and makes up much more of the total mass of galaxies and clusters than baryonic matter.

11 Fig. 4. Schematic graph of the expected and observed rotation curves of a galaxy [69]. It shows the orbital velocity as a function of galactic radius. The expected curve of a √1 baryon-only galaxy with a r dependence is drawn in blue. The curve consistent with observations is drawn in green. The three dashed black curves show the behaviour of orbital velocity due to single components of a galaxy.

Fig. 5. Schematic sketch of a gravitational lens [74]. Left: General geometry of light paths close to a gravitational lens. Right: Split up images of a light source resulting from the geometry on the left. The dashed line represents an Einstein-Wilson ring. .

12 The behaviour of colliding galaxy clusters is another piece of evidence. A famous example for this is the so-called Bullet Cluster [5]. The galaxy cluster 1E0657-558 or the Bullet Cluster is the aftermath of the collision of two galaxy clusters. The smaller cluster penetrated and passed through the larger cluster and now they are moving away from each other. The components that make up the clusters, stars and gas and dark matter, behave differently in collisions and can be examined separately. The stars are spread out and were only slightly slowed down by gravitational interactions during the collision of the two clusters. The gas that constitutes most of the baryonic matter of the clusters was signiﬁcantly slowed down by the collision. If dark matter exists, it should have interacted only gravitationally during the collision and be at the same position as the stars afterwards. If it does not exist, the gravitational centre of each cluster would have shifted towards the new position of the gas. Fig. 6 shows both the visible light from the bullet cluster as well as gravitational lensing and X-ray observations. Looking at the gas in the X-ray spectrum in the bottom panel shows that it lags behind the stars after the collision, just as predicted. However, by looking at the gravitational lensing measurements, we can clearly see that the gravitational centres of both clusters are approximately at the same positions as the stars. This implies the presence and prevalence of dark matter which was not slowed down signiﬁcantly during the collision. The Bullet Cluster is also highly useful to put constraints on the self-interaction of dark matter [75]. From the information we have about the two clusters, it is possible to estimate the self-interaction cross-section of particle dark matter. Dark matter is also required for large-scale structure formation in the ΛCDM model. Dark matter is assumed to have stopped interacting non-gravitationally with Standard Model particles early on. Afterwards, small density perturbations in the Universe cause the dark matter to collapse into dense concentrations. Since dark matter only interacts gravitationally at this point, there are no forces to oppose the collapse. Baryonic matter, however, is prevented from collapsing at that time by its interaction with photons and the resulting radiation pressure. In addition, the density perturbations in the early Universe would be suppressed by expansion at the time of baryon-photon decoupling. It follows that the large-scale structure we see today would not have enough time to form if it ori- ginated only from the collapse of baryonic matter. It requires the already existing dark matter concentrations for the baryons to fall in once baryons decoupled from photons [6]. There are many theories about what exactly makes up the dark matter. When considering dark matter candidates, we differentiate between “hot”, “warm” and “cold” dark matter [2]. These labels correspond to the velocity or free streaming length of dark matter particles. Hot dark matter moves at relativistic speeds while the speed of cold dark matter is small compared to the speed of light. The free streaming length of warm dark matter is naturally somewhere in-between. Simulations show that hot and cold dark matter lead to different processes of structure formation. Hot dark matter leads to a “top-down” sequence in which large structures form by collapse at the beginning that later fragment into smaller, galaxy-sized structures. On the other hand, cold dark matter causes a “bottom- up” sequence in which smaller halos form at the beginning that later merge and form large scale structures. Comparing these simulations with galaxy surveys shows that hot dark matter cannot account for all dark matter in the Universe [11]. As previously mentioned,

13 Fig. 6. Measurements of the Bullet Cluster [76]. Top: Visible light image of the bullet cluster, overlaid in green with the gravitational centres of each cluster deduced by gravitational lensing. Bottom: X-ray image of bullet cluster gas clouds showing the location of the bulk of the baryonic mass of the cluster in blue, red and yellow while showing the overall gravitational centres at different locations. The white bar represents 200 kpc at the distance of the cluster. the current consensus is the ΛCDM model with cold dark matter. Since we can account for all baryonic matter that formed shortly after the Big Bang, we know that dark matter is non-baryonic. If baryogenesis had produced more baryonic matter than we can account for, the baryon fraction of the total energy density and the element distribution today would be different from observational values [4]. As such, many candidates are particles that do not fall under baryons. A former prime candidate for cold dark matter is a yet undiscovered elementary particle which only interacts via the weak force. These particles are called WIMPs (weakly interacting massive particles) [7]. However, due to the lack of any successful detections despite

14 increasingly sensitive experiments, WIMPs have started to fall out of favour [8]. Another possible candidate are axions, hypothetical particles that were introduced in quantum chromo dynamics (QCD) to solve the strong CP problem [9]. As they typically have low mass, they are mainly classified as hot dark matter, but their self-interaction as bosons allows a classification as cold dark matter as well [10]. Sterile, right-handed neutrinos are a dark matter candidate that is, to some degree, already represented in the Standard Model of particle physics [11]. They can be hot, warm or cold dark matter candidates, depending on their mass and the mechanism by which they are produced [12]. The dark matter candidate we want to look at in more detail is PBHs. PBHs are part of the massive compact halo object (MACHO) cold dark matter scenario [16]. MACHOs are objects in the halos of galaxies that are not associated with any planetary system. They are dark, meaning non-luminous. Possible candidates for MACHOs include black holes, neutron stars, brown dwarfs and, independent planets. Some alternatives to dark matter have been proposed, most notably modified Newto- nian dynamics (MOND) [77, 78]. The hypothesis of MOND allows for a modification of Newton’s laws that applies only for small values of acceleration. For example, MOND can account for the higher than expected galaxy rotation speeds. However, it is unable to explain some behaviours of galaxy clusters like the Bullet Cluster and does not account for cosmological phenomena like large-scale structure.

3. Primordial black holes

The study of gravitationally collapsed objects in the early Universe or PBHs began in the 1960s and 70s [17, 18, 19]. PBHs are a hypothetical kind of black hole that only could have formed in the time from instants after the Big Bang to a few minutes afterwards. PBHs are considered to be different from black holes that are able to form today from the core collapse of stars. They form before Big Bang nucleosynthesis (BBNS) and during the radiation-dominated era from the collapse of overdensities or other mechanisms. As such, they are not included by the constraint that baryons produced from BBNS provide at most 5% of the total energy in the universe [79]. For this reason, they are considered nonbaryonic [20]. They are a candidate for dark matter in a MACHO dark matter scenario due to this property [14, 15]. PBHs also have the potential to answer a number of other important questions in cosmology. However, there are many constraints on the abundance of PBHs in almost every possible mass range [14]. The black holes that can form today typically have masses at least of the order of a few Solar masses. Theoretically, the mass spectrum of PBHs can be much wider. In most models, the mass within the particle horizon at the time the PBH collapses, which is called horizon mass MH , determines the PBH mass MPBH

2 3 3 3 H c c t 15 t M MH ρR 10 g. (3.1) PBH ∼ ∼ H ∼ G H3 ∼ G ∼ 10−23 s

For example, a PBH formed at t = 10−10 s after the Big Bang has a mass of

15 24 −6 M 10 kg 10 M [13]. PBH ≈ ≈ 3.1. Formation mechanisms There are multiple mechanisms in the early Universe that can result in the formation of PBHs. The one that is most commonly referred to is the collapse of early Universe density perturbations that were magnified by cosmological inflation. Cosmological inflation is a hypothesized period in the early Universe in which the Universe underwent exponential expansion. It provides solutions for a number of problems in the ΛCDM model. Quantum fluctuations or primordial density perturbations are magnified to macroscopic scales which gives a possible explanation for the origin of large-scale structure in a ho- mogeneous, isotropic universe. Inflation also provides solutions for the horizon problem, which is the homogeneity of the CMB on causally disconnected scales, and the flatness problem. The flatness problem is the observational fact that the Universe appears to be flat which requires considerable fine-tuning [80]. When the primordial density perturbations re-enter the particle horizon at the end of the inflationary epoch during radiation domination, they can collapse to form PBHs [13]. The most important factor in this process is the density contrast δ δρ δ = . (3.2) ρ0

Here, δρ is the local density and ρ0 the average density of the universe. If a critical density contrast threshold δ = δc is reached, the formation of a PBH is possible. This threshold depends heavily on the shape and amplitude of the density perturbation [81, 82]. These two attributes of primordial density perturbations are determined by the form of the primordial power spectrum and hence the threshold for PBH formation and the abundance of PBHs are dependent on it [83]. The exact value of the threshold has been determined to be δc 0.41 at its lowest for broad shapes of the primordial power spectrum [84] and ≈ δc 0.67 at its highest for peaked shapes [85]. PBH≈ formation from collapsing density perturbations is possible during the matter- dominated era as well. Since the pressure in the cosmologic equation of state is set to zero during matter domination, PBHs can form from just a fraction of the matter within the horizon. However, the perturbation has to be sufficiently spherical in this case [13]. PBHs can also form from the collapse of cosmic string loops. Cosmic strings are one- dimensional topological defects that may form as a result of a phase transition in the early Universe. As multiple strings intersect, they may form loops. Such a cosmic string loop can collapse into a PBH if all of its dimensions are less than its Schwarzschild radius. The mass of PBHs created in this process is, again, roughly equal to the horizon mass [13]. Another formation mechanism are first order phase transition bubble collisions. Bubbles of the new phase form, expand and finally collide. The collision can result in the creation of a PBH with mass of order of the horizon mass. However, obtaining a notable abundance of PBHs through this process requires fine-tuning of the bubble formation mechanism [13].

16 3.2. Constraints on primordial black hole dark matter In the context of dark matter, there are many constraints on the mass spectrum of PBHs. The constraints determine how much of the dark matter can be made up of PBHs of a particular mass. We note that all masses given here are the PBH masses at the time of their formation. The evaporation of black holes due to Hawking radiation places a limit on the mass −18 of PBHs that could still exist today. PBHs with masses MPBH < 10 M , would have completely evaporated before today [86]. Slightly heavier PBHs, up to MPBH = −16 10 M , evaporate and emit γ-radiation that we can use to constrain their abundance [22]. −9 On the other hand, PBHs or any compact objects (COs) with mass MCO = (10 − 10) M are constrained by the microlensing of stars. Their lensing effect is not strong enough to produce separate images, but instead leads to a temporary, achromatic ampli- fication of the flux of the stars [87]. Quasar microlensing and strong lensing of fast radio bursts further constrain COs with 0.05 < MCO/M < 0.45 and MCO & (10 100) M respectively [88]. − The merger rate observations of gravitational waves (GW) from black hole mergers by LIGO/Virgo place constraints on PBH binaries in the mass range of 10 . MPBH/M . 300 [89]. A slightly lighter constraint from the same observations is placed on binaries down to MPBH 0.2 M [90, 91]. The difficulty with the GW constraints is the evolution of the binaries between∼ their formation and the merger. There are arguments being made for both stronger constraints [92] and lighter constraints [93] in the case of large initial PBH clustering. Compact objects in the same mass range MCO = (1 100) M are excluded from con- stituting all of the dark matter by dynamical effects in dwarf− galaxies (heating, broadening of the kinetic energy distribution) and the disruption of wide binaries[94, 95]. Lastly, accretion radiation from PBH accretion discs places constraints on PBHs of mass MPBH & 10 M at redshift z > 0 [96]. At present time, accretion discs of PBHs with MPBH > M would emit observable amounts of X-ray and radio emissions. [97]. Comparing the expected emissions to known sources in surveys of the GC leads to another constraint on PBHs of MPBH (30 100) M [98]. The multitude of different∼ constraints− on the fraction of DM that can be constituted by PBHs of various masses is shown in Fig. 7. Overall, the constraints only leave open 17 22 the mass range (10 . MPBH . 10 g) to make up all of the dark matter [14]. In addition to the direct constraints on the abundance of PBHs, there are multiple indirect constraints on PBHs formed collapsing density perturbations during radiation domination. They include constraints from the stochastic GWs background, the shape of the primordial power spectrum and from CMB spectral distortions. However, these constraints mostly overlap with the previously discussed ones [14].

17 MPBH [g] 1015 1018 1021 1024 1027 1030 1033 1036 100

1 10− DM Ω

/ GWs Microlensing Accretion Dynamical PBH 2 10− Evaporation = Ω PBH

f 3 10−

4 10− 18 15 12 9 6 3 0 3 10− 10− 10− 10− 10− 10− 10 10 MPBH [M ]

Fig. 7. Constraints on PBH dark matter from evaporation, microlensing, gravitational waves, accretion and dynamical effects [14]. The vertical axis shows the allowed fraction of dark matter fPBH that can be made up of PBHs. The horizontal axes shows PBH mass MPBH in Solar masses at the bottom and in grams at the top.

3.3. Phenomena possibly explained by primordial black holes The recent observations by LIGO/Virgo have opened up new possibilities for PBHs as dark matter [99]. While the observation results can be used to constrain the PBH dark matter fraction, as discussed above, these constraints depend on the predicted merger rate which varies strongly from model to model [100]. In addition, some of the other constraints on PBHs in speciﬁc mass ranges can be passed by as well. Efﬁcient merging of PBHs could have taken place in the early Universe [101], eventually leading to the formation of massive black holes with masses of the order of the observed mergers. By reaching this size not at their formation but through merging with each other, they pass the CMB distortion constraints as well as lensing constraints [31]. Following this model, massive primordial black holes (MPBHs) would pervade galactic halos which is consistent with a number of observations in nearby galaxies and notably near the centre of Andromeda [102, 103, 104]. MPBHs also possibly provide an explanation for the γ-ray excess seen by Fermi-LAT from unresolved γ-ray point sources towards the GC [105]. In the case of a broad PBH mass spectrum, there is the possibility of ex- plaining the origin of intermediate mass black holes (IMBHs) using MPBHs. They are

18 thought to be responsible for the ultra-luminous X-ray sources we observe near the GC [106, 107]. Moreover, PBHs of high mass are good candidates for seeds of SMBHs in galactic nuclei [28, 29]. In addition to the LIGO/Virgo detections, there has been a small number of claimed signatures of PBHs related to microlensing [108, 109, 110], as well as accretion and dynamical effects [111, 112]. Carr et al. [113] propose a scenario which may provide a uniﬁed explanation for all of these conundra. A number of the mentioned cosmological problems can still be resolved by PBHs if they only make up a small fraction of the dark matter. Most notably, PBHs of sufﬁcient mass provide a possible explanation for the origin of cosmological structures [30] and the seeds for the SMBHs even if they are not abundant enough to serve as the dominant part of dark matter. In addition, the possibility that PBHs explain the population of massive black holes detected by LIGO/Virgo does not require them do provide all of the dark matter either [15]. However, the most popular explanations for these problems are usually not related to PBHs. The phenomenon this thesis focuses on is unbound hypervelocity stars. Gravitational interactions with PBHs which eject stars from close galaxies have been raised as a possibility for their origin. We introduce hypervelocity stars and some of the previous work on unbound hypervelocity stars in Section 4. The proposed gravitational interaction between stars and PBHs that might be their cause is detailed in Section 5.

4. Hypervelocity stars

We probe a dark matter scenario with MPBHs. For this purpose, we investigate unbound hypervelocity stars that are ejected from the Andromeda galaxy by interactions with MPBHs. Hypervelocity stars (HVS) were first predicted by Hills in 1988 [40]. The first HVS in the Milky Way was discovered in 2005 by Brown et al. [38]. Since then, the number of known HVSs in the Milky Way keeps increasing with every new probe of our galaxy [39]. HVSs are generally defined as stars that have velocities of order 1000 km/s. Since the Milky Way escape velocity is of the same order of magnitude [114], it is easy to see that they can be unbound to the Milky Way gravitational potential. Most of the known HVSs seem to originate near the GC since their trajectories point away from it. The cause for such high kinetic energies is thought to be gravitational interactions between binary stars and the SMBH in the GC [40, 41]. In this interaction, one of the stars is captured by the SMBH while the other is ejected at a high velocity. Another possible origin is the ejection of one half of a binary, caused by the supernova explosion of the other half [115]. The known HVSs are main sequence stars with masses of the order of a few Solar masses [116].

19 8000 Sun Stars 7000

6000

5000 [km/s] 2 z 4000 v + 2 x

V 3000

2000

1000

6000 4000 2000 0 2000

Vy [km/s]

Fig. 8. Toomre diagram of Galactocentric Cartesian velocities of Gaia DR2 HVSs [37]. Displayed is the selection of HVSs made by Marchetti et al. [42] with 68% conﬁd- ence error bars. The black semi-circle centred at the origin is given by the velocity of the Sun (black star).

4.1. Hypervelocity stars from outside the Milky Way Montanari et al. [37] investigate multiple HVSs from the Gaia DR2 catalogue [48, 117]. The Gaia DR2 catalogue is an all-sky survey covering about 1.7 billion objects. However, only about 7 million of them have an attached radial velocity which makes them eligible for a velocity analysis [42]. Montanari et al. [37] narrow this selection down further to 1649 HVSs with various data quality cuts. In addition, they ﬁlter the data according to Galactocentric velocity as well as the probability of being unbound from the Milky Way. Velocity attributes and positions of this selection of 1649 HVSs are shown in Fig. 8 and 9 in Galactocentric Cartesian coordinates. In this coordinate frame, the x-axis runs along the connection between the Sun and the GC with the positive direction pointing from the Sun towards the GC. The positive y-direction follows the rotation of the Galactic disc at the position of the Sun. The z-axis points towards the Galactic North Pole. Galactocentric coordinates are not to be mistaken for Galactic Cartesian coordinates which is a coordinate system with the Sun at its centre. The orientation of the axes of the Galactic Cartesian coordinate system are identical to those of Galactocentric Cartesian coordinates. An even smaller selection of HVSs results from examining the probabilities of the HVSs of being unbound to the Milky Way. The probabilities are provided by Marchetti et al. [42]. 20 HVSs remain with a probability > 80% of being unbound. Thirteen out of

20 Stars Globular clusters Sun

z [kpc] 0

30 20 10 0 10 20 x [kpc]

0 y [kpc] 10

30 20 10 0 10 20 x [kpc]

Fig. 9. Galactocentric positions of globular clusters and HVSs from Gaia DR2 selection [37]. Displayed is the selection of HVSs made by Marchetti et al. [42] with 68% conﬁdence error bars. Globular cluster error bars are smaller than the markers.

21 these 20 HVSs have trajectories that point towards the Milky Way disc. This indicates that their origin is not within the Milky Way, but outside of it. The trajectories of HVSs that originate inside the Milky Way typically point away from it. Montanari et al. [37] argue that they may originate in globular clusters and dwarf galaxies surrounding the Milky Way or even in the Andromeda galaxy. They propose the ejection of HVSs could be caused by interactions of stars with MPBHs. Montanari et al. [37] focus on dwarf galaxies and suggest that their low surface brightness and low star population could be explained by this mechanism in a PBH dark matter scenario The ΛCDM model predicts a large number of such ultra-faint dwarf galaxies orbiting galaxies. The fact that we have only observed a small number of them so far is called the missing satellite problem. ΛCDM also states that such large sub-structures have massive star formation due to their mass and should be luminous enough to be detected due to their visible light. This is called the too-big-to-fail problem. MPBHs as dark matter could potentially solve both problems. The lack of star formation in these dwarf galaxies could be due to a large part of the gas being absorbed by MPBHs instead of collapsing to form stars. In addition, the existing stars in the clusters could have been ejected from the shallow gravitational wells of the dwarf galaxies through slingshot events with MPBHs. Multiple of these events are required for the stars to reach escape velocity. PBHs of small size would have gone through the same process, leaving mainly MPBHs to populate the cluster [31, 32]. The ejection process is detailed in Section 5. Montanari et al. [37] consider Gaia HVSs with distances to the Sun of up to 13 kpc. They develop a simulation to trace back the HVS trajectories and a Bayesian framework to try and find correlations with the positions of dwarf galaxies and globular clusters. An impact parameter then determines the likelihood of a scattering event between the HVSs and a compact object within the globular clusters or galaxy in question. After finding multiple HVS candidates for scattering events with objects in the dwarf spheroidal galaxy Sagittarius dSph, they check the age of the HVSs to verify their consistency with the trajectory length. They find up to five suitable HVSs that might originate in Sagittarius dSph. Possible trajectories for two of them as well as Sagittarius dSph are shown in Fig. 10, la- belled by their source ID. No other matches are found in either other dwarf galaxies, globular clusters or Andromeda. While these events might correspond to PBH scattering events, there is only a statistically small number of results and significant uncertainties in the PBH mass distribution. Montanari et al. [37] are not able to set meaningful limits on the PBH properties necessary for the interaction to occur.

4.2. Selection of Gaia data In this thesis, we focus on HVSs from the Andromeda galaxy. In contrast to the dwarf galaxies in the immediate neighbourhood of the Milky Way, Andromeda and its satellites are much further away. For this reason, we only consider stars ejected from the inner region of Andromeda since we assume HVSs that reach the Milky Way are much more likely to originate in this region of high stellar density than in dwarf galaxies orbiting it. Instead of tracing back the trajectories of Gaia HVSs, we attempt to simulate realistic trajectories that match them. We randomly generate initial positions near the center of

22 61985942 62420658 Sagittarius dSph 0.0 2.5 5.0 7.5 10.0

z [kpc] 12.5 15.0 17.5 20.0 50 40 30 5 20 0 10 5 y [kpc] x [kpc] 10 0 15 10 20

Fig. 10. Possible trajectories of two HVSs are presented in blue and orange, Sagittarius dSph in black [37]. The trajectories are displayed in Galactocentric Cartesian coordinates. Line density is proportional to likelihood. The markers correspond to positions at t0.

Andromeda from which we send off HVSs. For the initial velocity vectors, we generate random directions. All velocities are of the order of the escape velocity of Andromeda. We calculate the trajectories of the ejected HVSs in order to determine whether such a scenario is possible and if it is compatible with data from the Gaia eDR3 catalogue. The simulation and calculation of trajectories is discussed in detail in Section 6. In Section 7, we compare the simulation results with Gaia HVS data. In order to pick out HVS data from the Gaia eDR3 catalogue, we select them according to their velocity in the Galactocentric rest frame. We use the notation vgal for velocities in the Galactocentric rest frame and vhel for velocities in the heliocentric rest frame. Since the Gaia data is in the heliocentric rest frame, we initially apply a wide selection of heliocentric velocity vhel > 80 km/s. We also set the same limit as Montanari et al. [37] at a distance of 13 kpc due to the inaccuracy of the Gaia data at long distances. After the initial selection, we transform the data to Galactic Cartesian coordinates and we boost all remaining HVSs to the Galactocentric rest frame using the velocity vector of the Sun ~vgal,Sun [117] in Galactocentric Cartesian coordinates

 11.1  ~vgal,Sun = 252.24 km/s. (4.1) 7.25

23 Gaia selection criteria Number of stars in selection Gaia eDR3 catalogue 1.8 109 ∼ × Gaia eDR3 entries with v 8.82 108 rad ∼ × d < 13kpc, vhel > 80 km/s 2,262,120

2σ certainty in vrad and ωRA, ωDec measurements 2,170,840

2σ certainty in p measurements 2,163,024

vgal > 600 km/s 259

d < 7 kpc 14

Tab. 1. Table of Gaia eDR3 HVS data selections. The selection criteria are applied cu- mulatively. d is the distance of the catalogue stars to the Sun. vhel and vgal, respectively, are the star velocities in the heliocentric and Galactocentric rest frame. vrad is the radial velocity. ωRA and ωDec are the proper motion velocities of the stars in the coordinate directions of the equatorial coordinate system. p is the parallax.

Afterwards, we apply the second selection of Galactocentric velocity vgal > 600 km/s. The remaining HVSs are transformed back to the heliocentric rest frame to be compared with the results from the simulation. The details of the coordinate transformations necessary to convert the Gaia velocity measurements to vectors in the Galactic Cartesian coordinate system are discussed in Appendix A. In addition to the selection based on the measurement values, we concurrently apply a selection according to the errors given by the Gaia eDR3 catalogue. We only allow values of the radial velocity vrad, proper motions ωRA, ωDec and parallax p with values within a 2σ-certainty interval. The different selection criteria for the Gaia data and the amount of stars in each selection step are shown in Table 1. Fig. 11 shows the velocities of the ﬁnal selection of 259 Gaia HVSs in the heliocentric rest frame and an additional, stricter selection of all 14 HVSs within 7 kpc of the Sun as Toomre diagrams. The green star represents the relative velocity between the heliocentric and Galactocentric rest frame. Due to the data selection criteria, the diagrams are only populated at Galactocentric velocities vgal > 600 km/s. The velocity of the Sun ~vgal,Sun and the circle of constant Galactocentric velocity it lies on are marked as a red star and circle. We see that the abundance of data points decreases with increasing Galactocentric velocity, signiﬁed by the black semi-circles of constant Galactocentric velocity. When considering the heliocentric rest frame at vhel,y = 0, we observe a large majority of HVSs to have a negative vy component. Most of them also have higher negative velocity in the

24 1000 Gaia HVSs (d < 13kpc) ]

s Sgr A* / 800

m Sun k [

) z

, 600 l e 2 h v

+ 400 x , l e 2 h v ( 200

0 1000 750 500 250 0 250 500 750 1000

vhel, y [km/s]

1000 Gaia HVSs (d < 7kpc) ]

s Sgr A* / 800

m Sun k [

) z

, 600 l e 2 h v

+ 400 x , l e 2 h v ( 200

0 1000 750 500 250 0 250 500 750 1000

vhel, y [km/s]

Fig. 11. Toomre diagrams of velocities of Gaia HVSs at t0. The velocity components (vhel,x, vhel,y, vhel,z) on the axes correspond to the velocity in the Galactic Cartesian coordinate system and in the heliocentric rest frame. The HVS velocities are shown as blue dots. The black semi-circles represent constant velocities vgal of 600, 700, 800 and 1000 km/s in the Galactocentric rest frame. The relative velocity between the heliocentric and Galactocentric rest frames is displayed as a green star and serves as the centre of the semi-circles. The velocity of the Sun in the Galactocentric rest frame is shown as a red star and the corresponding velocity magnitude as a red semi-circle with radius 250 km/s. Top: 259 HVSs within 13 kpc of the Sun. Bottom: 14 HVSs within∼7 kpc of the Sun.

25 y-direction than the relative velocity of the GC at vhel,y 250 km/s which means they are moving in the opposite y-direction compared to the the≈ − Sun in the Galactocentric rest frame. Due to the relative velocity between the two rest frames, these HVSs receive a sig- niﬁcant boost in the heliocentric rest frame. Excluding HVSs that are not within 7 kpc of the Sun leads to approximate symmetry between positive and negative vhel,y components in the heliocentric rest frame. However, HVSs with negative vhel,y component on average appear noticeably faster. We refer back to this observation in Section 7.

5. Ejection of stars from Andromeda

We brieﬂy touched upon the ejection of stars from the potential wells of ultra-faint dwarf galaxies by MPBHs in Section 4.1. In this chapter, we want to examine this process in more detail and consider it for the Andromeda galaxy. The mechanism discussed by Clesse and Garc´ıa-Bellido [31, 32] is a gravitational slingshot interaction between stars and MPBHs. An illustration of the mechanism is shown in Fig. 12. The resulting velocity magnitude v2 of the accelerated star is given by energy- momentum conservation to be

2U + (1 q)v1 v2 = − . (5.1) 1 + q

Here, U is the velocity of the MPBH and v1 is the velocity of the star before the interaction occurs. Both of these velocities are peculiar velocities. q = m/M is the mass quotient of the star mass m and the black hole mass M. Given a large enough velocity of the MPBH, the star can exceed the escape velocity of a dwarf spheroidal galaxy after a few slingshot events. Due to their shallow potential well, the escape velocity of dwarf spheroidals is of the order of tens of km/s. The above formula only covers head-on encounters between stars and black holes. If the two objects meet at an angle θ, the resulting velocity v¯2(θ) s 2 2 2 2U + (1 q) v1 cos θ v¯2(θ) = v sin θ + − (5.2) 1 1 + q will differ slightly, depending on θ. However, marginalising over angles results in the approximation

2U v¯2 (5.3) ≈ 1 + q which is nearly identical to Eq. (5.2) with the assumption v1 U. In the simulation, we mostly consider stars near the center of Andromeda due to the high stellar density in that region. The escape velocity of Andromeda is 900 km/s near the center [118]. Since multiple slingshot events are needed for HVSs∼ to reach escape velocity, we do not consider HVSs with much higher velocities at the beginning of their

26 Fig. 12. Schematic depiction of gravitational slingshot interaction between a MPBH of mass M and a star of mass m [31]. trajectory. There are alternative possibilities for the ejection of HVSs from Andromeda. Similarly to the Milky Way, we expect Andromeda to produces HVSs from interactions between stars or binaries and the SMBH at its centre that are fast enough to escape its gravitational potential. Other mechanisms that have been considered are the inspiraling of an IMBH or scattering off of stellar mass black hole clusters that orbit around the SMBH [119]. If these HVSs reach the Milky Way as well, it might be difﬁcult to differentiate between HVSs from “normal” interactions and HVSs ejected by MPBHs.

6. Description of the simulation

We use Python to model the gravitational system of the Milky Way and Andromeda galaxies. Speciﬁcally, we utilise the SciPy [43], NumPy [120] and AstroPy [44, 45] libraries. The simulation model covers the combined gravitational potential of both galaxies as well as their relative positions over time. The HVSs are sent off from Andromeda at random times within the interval of t = [10, 12] Gyr after the Big Bang into random directions with velocities just above escape velocity. Initial positions of the HVSs are generated randomly near the centre of Andromeda. Only a small number of ejected HVSs reach the Milky Way and even fewer come close enough to the Sun at present time t0 to compare them to Gaia measurement data. We calculate their trajectories from their send-off time until today and compare simulation HVS properties at present time t0 to HVS data from the Gaia eDR3 catalogue [48, 117]. We choose a Cartesian coordinate frame for the calculation of trajectories in the simulation. It is right-handed and uses the GC as its origin. For the sake of convenience, we choose the x-axis to point towards the position of Andromeda at t0. In addition, we choose the x-y-plane so it contains both the GC and the Sun. In order to compare the

27 simulation results to Gaia data, we transform the HVS position and velocity vectors from the simulation coordinate frame to the Galactic coordinate frame after the calculations are complete. For this purpose, we shift the coordinate origin to the position of the Sun and then perform two rotations in order to align the coordinate axes with Galactic Cartesian coordinates. A transformation to spherical coordinates gives us the vectors in Galactic coordinates. Since the simulation operates in the Milky Way Galactocentric rest frame, we boost the simulation velocity vectors to the heliocentric rest frame as a final step. All transformation steps are detailed in Section 6.6. We use the following notations for the coordinate frames in the different steps of the transformation in Section 6.6. Position and velocity vectors ~r and ~v are in the simulation coordinate frame. The notation ~r 0 and ~v 0 is used for vectors in the transition coordinate frame after the shift of the origin to the position of the Sun and the first rotation. Finally, vectors with notation ~r 00 and ~v 00 are in the Galactic Cartesian or Galactocentric Cartesian coordinate frame. The index G signifies the Galactocentric Cartesian coordinate frame in this case. For consistency, this notation scheme is also used for the vectors in the remaining parts of this section. Before we boost the simulation velocity vectors to the heliocentric rest frame, all velocity vectors are in the Galactocentric rest frame. The index notation for the rest frames we established in Section 4.2 is only used in this last step of the transformation.

6.1. Equations of motion To calculate a HVS trajectory, we need to solve the equations of motion. In this case, the equations of motion are ordinary differential equations (ODEs) for the position ~r and velocity ~v of the star

d~r(t) d~v(t) = ~r˙(t) = ~v(t) and = ~v˙(t, ~r(t)) = ~a(t, ~r(t)). (6.1) dt dt In three dimensions, we have a total of six equations, one for each of the three components of the position and velocity vectors. The DEs are solved in the simulation code by the ODEint function from the SciPy library. It integrates provided velocity and acceleration components to ﬁnd position and velocity components, respectively. ODEint utilises the Runge-Kutta method [121, 122]. The DEs for the position components depend on the initial velocity vector ~v0 = (vx, vy, vz) in addition to the initial position vector ~r0 = (rx, ry, rz). The corresponding DEs for the velocity components depend on the acceleration components which in turn depend on the initial position of the star. We need an equation for each acceleration component as well as a send-off time and initial conditions for the components of the position and velocity vectors to solve Eq. (6.1). Given all this, ODEint integrates the DEs and returns the HVS trajectory as a list of positions and velocities for the time between the send-off time and t0.

28 6.2. Acceleration and mass terms We acquire the equations for the acceleration components by utilising Newton’s second law of motion and Newton’s law of gravitation. Below, F~ denotes the gravitational force and ~a the acceleration. M and m represent the mass of the galaxy and the mass of the HVS, respectively. ~r is the position vector of the HVS with respect to the centre of the galaxy. G is Newton’s gravitational constant. We set as equal Newton’s second law of motion

F~ = m ~a (6.2) · and Newton’s law of gravity

Mm ~r F~ = G (6.3) − ~r 2 ~r | | | | which results in an equation for the acceleration a due to gravity

GM ~r ~a = . (6.4) − ~r 2 ~r | | | |

We write for each of the three components ai

GM ri ai = . (6.5) − ~r 2 ~r | | | | We note that we eliminate the HVS mass m from the equation and, as such, it plays no role in the simulation. While Eq. (6.5) is rather simple, the HVS trajectories start within Andromeda and possibly pass through the Milky Way. The gravitational potential accelerating the HVSs varies accordingly. Therefore, we need an expression for the mass M of each galaxy up to the distance of the HVS to the corresponding galactic centre. Further, the mass of each galaxy as well as the distance of the HVS with respect to the centres of the two galaxies need to be in two separate terms of the equation. For the mass term of a single galaxy we use two different density proﬁles. The Navarro- Frenk-White (NFW) proﬁle [123] is used for the dark matter halo of a galaxy, making up the bulk of its mass. It is a spherically symmetric halo model based on N-body simulations in a cold dark matter scenario which follows the density relation ρ 0 (6.6) ρNFW(r) = 2 . r 1 + r Rs Rs

Here, ρ0 is the characteristic density and Rs denotes the scale radius which both differ for every halo. The scale radius is determined from the virial radius Rvir of a galaxy which can then be used to calculate ρ0 (see Section 6.3).

29 From the NFW model density ρNFW, we derive the mass MDM of the dark matter halo

Z Rmax Z π Z 2π 2 MDM(Rmax) = r sin(θ)ρNFW(r)dr dθ dφ 0 0 0 Z Rmax 2 = 4πr ρNFW(r)dr (6.7) 0 3 Rs + Rmax Rmax = 4πρ0Rs ln Rs − Rs + Rmax by integrating over its radius up to a maximum radius Rmax. We add a second component of mass MP resulting from the Plummer density proﬁle [124]. The Plummer proﬁle describes the distribution of baryonic matter in a galaxy. It is a spherically symmetric model as well which means it is at its most accurate near the centre of a galaxy. In the Plummer model density relation

− 5 3M r2 2 ρ (r) = Bar 1 + , (6.8) P 4π a3 a2 · P P

MBar is the total baryonic mass of the galaxy and aP denotes the Plummer radius which determines the core radius of the galaxy. We acquire the corresponding Plummer mass term MP Z Rmax 2 MP (Rmax) = 4πr ρP (r)dr with r < Rmax 0 3 (6.9) Rmax = MBar 2 2 3/2 (Rmax + aP ) up to a maximum radius Rmax in the same way as previously for the NFW proﬁle in Eq. (6.7). For the total mass of a galaxy up to a radius R, we simply add up both terms to ﬁnd

Mtot(R) = MDM(R) + MP (R). (6.10)

For the sake of simplicity, we approximate the Milky Way and Andromeda to have the same total mass since their masses are on the same order of magnitude [125, 126, 127]. This means MBar, ρ0, Rs and aP have the same value in the mass terms of both galaxies. Now we can construct a total mass term for each galaxy with Eqs. (6.7) and (6.9) and insert them into the acceleration equation (6.5) to find the acceleration due to the mass of each galaxy. Adding up both acceleration terms gives the total acceleration ~atot influencing the HVSs caused by the gravitational influence of both galaxies. It is split up into the three

30 spacial components atot,i = atot,x, atot,y, atot,z 3 ~r Rs + ~r ri MBar ri atot,i =4πG ρ0Rs | | ln | | 3 3/2 Rs + ~r − Rs ~r − 4π 2 2 ( ~r + aP ) | | | | | | (6.11) 3 ~rA Rs + ~rA rA,i MBar rA,i +ρ0Rs | | ln | | . 3 2 2 3/2 Rs + ~rA − Rs ~rA − 4π ( ~rA + a ) | | | | | | P ri is the i-component of the position vector of the HVS with respect to the centre of Milky Way ~r , and rA,i is the i-component of the position vector of the HVS with respect to the centre of Andromeda ~rA. We have replaced Rmax in each mass term with the magnitude of the relevant position vector, respectively ~r or ~rA . The three components of the acceleration are completely analogous to each| other.| | |

6.3. Mass of the Local Group There are still some constant parameters with unknown values in Eq. (6.11), namely MBar, ρ0, Rs and aP . We ﬁnd MBar by calculating the combined mass of the Milky Way and Andromeda. It is approximately equal to the mass of the Local Group of galaxies MLG

M M + M . (6.12) LG ≈ MW And which we estimate using the Local Group timing argument [128]. It sets up a simple model of two masses on a radial orbit to describe the system of the Milky Way and An- dromeda. At the time of the Big Bang tBB = 0, the masses have a separation distance of SBB = 0 and later separate due to the expansion of the Universe. The fact that the Milky Way and Andromeda are currently approaching each other implies that the maximum separation Smax between the two galaxies is larger than it is at present time t0 and that they are moving on a cycloid trajectory. We assume Andromeda is on its ﬁrst approach to the Milky Way since tBB. We obtain the position vector of Andromeda with respect to the Milky Way centre 00 00 ~r G,And and the velocity vector of Andromeda ~v G,And at t0 from van der Marel et al. [129] as  378.9 66.1 00 − 00 ~r G,And =  612.7  kpc and ~v G,And 76.3 km/s. (6.13) 283.1 ≈ 45.1 − The vectors are both written in Galactocentric Cartesian coordinates and in the Galacto- centric rest frame. We calculate the present day value of the separation distance S0 00 between the two galaxies from the position vector ~r G,And by taking its absolute value

00 ~r = S0 774.0 kpc. (6.14) | G,And| ≈

The approximate radial approach velocity of Andromeda vLG,0 today is found by trans- 00 forming ~v G,And to the the velocity vector of Andromeda in the simulation coordinate frame

31 ~vAnd. For this purpose, we apply the inverse of the two rotations of the transformation de- 00 tailed in Section 6.6 to ~v G,And

 109.2 00 − ~v G,And ~vAnd  14.5  (6.15) → ≈ −8.9 and subsequently take the velocity magnitude of ~vAnd

~v = v 110.6 km/s. (6.16) | And| LG,0 ≈

We note that ~vAnd is still in the Galactocentric rest frame. The radial approach model is wholly described by the parametric form of Kepler’s laws for an orbit with zero angular momentum. We can write the separation distance S and cosmic time t which depend on the angular parameter χ as S S(χ) = max (1 cosχ) (6.17) 2 − s S3 t(χ) = max (χ sinχ) . (6.18) 8 GMLG −

χ is called cycloid parameter. We derive and combine these two equations to ﬁnd the χ-dependent relative velocity vLG between the two masses

dS dS dt r2 GM sinχ v (χ) = = / = LG . (6.19) LG dt dχ dχ S 1 cosχ max −

Substituting Eqs. (6.17) and (6.18) for Smax and GMLG in Eq. (6.19) gives

v (χ)t(χ) sinχ(χ sinχ) LG = − . (6.20) S(χ) (1 cosχ)2 −

We insert the present day values of the separation distance S0, vLG,0 and the age of the Universe t0 13.8 Gyr into Eq. (6.20) to ﬁnd the present value of the cycloid parameter ≈ χ0 4.201. Now we can calculate the total mass of the Local Group M by obtaining ≈ LG Smax from Eq. (6.17) and substituting into Eq. (6.18). We rearrange to ﬁnd

3 Smax 2 12 MLG = 2 (χ0 sinχ0) 4.211 10 M . (6.21) 8 G t0 − ≈ × The result is on the same order of magnitude as the value from Li and White [130] and agrees with the value from Gonzalez et al. [131]. Using this result, we ﬁnd 1 1 M M (6.22) Bar ≈ 6 × 2 LG

32 for the baryonic mass MBar of each galaxy since dark matter makes up about 5/6 of matter and we assume both galaxies to have the same mass. Analogously, we ﬁnd the mass of dark matter M 5/12 M in each galaxy. We use this value to calculate ρ0. DM ≈ LG In addition to being dependent on the speciﬁc dark matter halo, the scale radius Rs and the characteristic density ρ0 are also mutually dependent on each other. The scale radius Rs R R = vir . (6.23) s c can be determined from the virial radius Rvir. Here, c is the so-called concentration parameter which can take values from 4 to 40 for halos of various sizes. A typical value for galaxies like the Milky Way and Andromeda is c = 10 which we use for both galaxies. We choose R 200 kpc since the virial radii of both galaxies are of this order vir ≈ [132, 133]. The scale radius of each dark matter halo follows as Rs 20 kpc. With this ≈ value, we can calculate the characteristic density ρ0 by integrating over the halo up to the virial radius analogously to Eq. (6.7)

Z Rvir 2 MDM = 4πr ρNFW(r)dr 0 (6.24) 3 c = 4πρ0R ln (1 + c) s − 1 + c and subsequently solve the equation for ρ0 −1 MDM c 7 −3 ρ0 = 3 ln(1 + c) 1.172 10 M kpc . (6.25) 4πRs − 1 + c ≈ ×

Lastly, we choose a best guess value for the Plummer radius aP = 5 kpc without deeper consideration because it gives us a distribution of baryonic matter that is mostly concentrated within 5 kpc of the centre of each galaxy. In the outer regions of a galaxy the Plummer density ρP falls off sharply with

−5 ρP r . (6.26) ∼

The constant parameters Mbar, ρ0, Rs and aP that are necessary to calculate the total acceleration (see Eq. 6.11) and their values are shown in Table 2.

6.4. Dynamics of the two galaxies

In the equation for the components of the total acceleration ~atot (see Eq. 6.11), we need not only the position of the HVS in the simulation coordinate frame, but also its position with respect to Andromeda at every point in time in the same coordinate system. The position of Andromeda itself is not constant in simulation coordinates since the two galaxies have been steadily approaching each other for the last few billion years. To account for

33 Parameter value 11 Mbar 3.509 10 M ×7 −3 ρ0 1.172 10 M kpc × Rs 20 kpc aP 5 kpc

Tab. 2. Constant parameters in total acceleration equation (6.11) this movement, we introduce functions that return the position and velocity vectors of Andromeda depending on cosmic time t. To construct these functions, we solve the two- body problem of the Milky Way and Andromeda in much the same way as for the HVS trajectories. In the place of a HVS, we instead calculate the trajectory of Andromeda in the rest frame of the Milky Way. Since the two galaxies do not collide during the time frame that is relevant to us, we can apply a simpler equation by using their respective total masses in the acceleration equation (6.5). The gravitational force between the two galaxies is given by

M M ~r F~ = G MW And And . (6.27) − ~r 2 ~r | And| | And|

Here, ~rAnd is the position vector of Andromeda in simulation coordinates. We then introduce the reduced mass µ with the total mass of the Local Group MLG that we calculated in Section 6.3

MMWMAnd µ = , with MLG = MMW + MAnd (6.28) MLG to simplify this two-body problem to a one-body problem of the movement of Andromeda relative to the Milky Way which is at rest. We again use Newton’s second law of motion (see Eq. 6.2) to ﬁnd the acceleration ~aAnd

F~ M ~r ~a = = G LG And And µ − ~r 2 ~r | And| | And| (6.29) MLG a ,i = G r ,i ⇒ And − ~r 3 And | And| that acts on Andromeda depending on its position relative to the Milky Way. Given initial conditions for Andromeda and Eq. (6.29) for the acceleration~aAnd, ODEint solves the system of ordinary differential equations given by Eq. (6.1) to produce a list of values for the position and velocity of Andromeda over the relevant time frame. Since we can only measure the position of Andromeda today, it is useful that we can apply this method forwards and backwards in time. We use the initial conditions for Andromeda at present time from van der Marel et al. [129], which we used in Section 6.3, to calculate the trajectory backwards in time. We then interpolate the list returned by ODEint to ﬁnd

34 the simulated position vector ~rAnd, Sim(t) and velocity vector ~vAnd, Sim(t) of Andromeda as functions of time for the last few billion years. Using this result, we calculate the position vector ~rA

~rA = ~r ~r (6.30) − And, Sim of the HVS with respect to Andromeda by subtracting ~rAnd, Sim from the HVS position vector ~r. All vectors used are in the simulation coordinate frame. Now, we only need initial conditions for the HVS position and velocity vectors to use Eq. (6.11) to calculate HVS trajectories with ODEint.

6.5. Initial conditions for the hypervelocity stars We need six separate initial conditions for the six ordinary differential equations in Eq. (6.1), three components each for the position and velocity vector. For each HVS, they are generated randomly with weighting conditions determining the range in which they can be found. In addition, we randomly generate an initial send-off time tSO in the interval between tSO, min = 10.0 Gyr and tSO, max = 12.0 Gyr (6.31) for each HVS. We choose this interval to make sure the HVSs are able to reach the Milky Way within their lifetime. The intial position and velocity vectors need to be in the simulation coordinate system. The vector components are generated in the corresponding spherical coordinate system. We generate positions near the center of Andromeda where the orientation of the galactic disc has little impact. As such, we neglect the orientation of Andromeda and assume its central region is spherically symmetric for our purposes. The radial coordinate r is determined by rearranging the Plummer model density relation in Eq. (6.8). It results in the radius-density relation

s 2 3 − 5 4πa ρP r(ρP ) = a 1. (6.32) · 3Mbar − We randomly generate a density value between the Plummer model maximum density ρP (r = 0) and ρP = 0 and substitute it in Eq. (6.32) to ﬁnd the corresponding radius which we use as the radial coordinate. This gives us more initial radial coordinates towards the center of the Andromeda where more stars are located. The azimuthal angle ϕ and the inclination angle θ are randomly generated in a way to give positions isotropically distributed around the centre of Andromeda. To generate initial velocity vectors, we choose an upper and lower limit for the velocity magnitude. For the lower limit, we choose the escape velocity of Andromeda vEsc near its centre [118]

v = v 900 km/s. (6.33) min Esc ≈

35 8.0%

6.0%

4.0% % of HVSs

2.0%

0.0% 900 925 950 975 1000 1025 1050 1075 1100 Initial velocity magnitude [km/s]

Fig. 13. Distribution of initial HVS velocities after slingshot events with MPBHs. Ran- domly generated within the interval v = [900, 1100] km/s with exponential weight function f(x).

Due to the low probability of additional slingshot events after a HVS reaches escape velocity, we choose an upper limit of vmax = 1100 km/s which is not signiﬁcantly larger than the lower limit. In addition, we use an exponential weighting function f(x)

f(x) = e−12x (6.34) to shift the distribution towards the escape velocity. This is done to account for the higher likelihood of HVSs only barely exceeding escape velocity. The exponent 12 is chosen − arbitrarily. We generate a random number N between the numerical values Nmin and Nmax of vmin and vmax. We also generate a second random number Ncrit between zero and f(Nmin). If f(N) > Ncrit, then N is a viable value for the initial velocity magnitude. The distribution of initial velocity magnitudes is displayed in Fig. 13. The velocity directions are generated so that all directions have the same probability, analogously to the isotropic distribution of the angular components of the position vectors. After we obtain three components each for the position and velocity vector, we transform them to Cartesian vectors. Since we generate the initial conditions with respect to Andromeda, we add its position and velocity at the time of ejection to the vectors to ﬁnd the initial conditions in the simulation coordinate system. This is done by utilising the functions established in Section 6.4.

36 6.6. Coordinate transformations of the simulation results Given the differential equations (6.1), the acceleration equation (6.11), the Milky Way- Andromeda dynamics (see Section 6.4), and the HVS initial conditions (see Section 6.5) the simulation returns the trajectory of a HVS. At this point in the calculation, the data of the simulated HVS trajectory is still in the Cartesian simulation coordinate system. Due to our choice of axes, it is not aligned with the Galactocentric Cartesian coordinate frame. In addition, the Gaia data is mapped in equatorial coordinates. It is a coordinate system commonly used in astrophysics that is based on the celestial sphere of the Earth. However, we choose to transform both data sets to Galactic coordinates in order to keep the GC in a prominent position. Galactic coordinates are the corresponding spherical coordinate system to Galactic Cartesian coordinates, introduced in Section 4.1. We now need to perform multiple coordinate transformations in order to transform the simulation data to Galactic coordinates. Since the Gaia data is mapped at present time t0 we only consider the simulation results at this time and perform the transformation with values at t0 for all involved objects. The corresponding transformation of the Gaia data to Galactic coordinates is discussed in Appendix A. In the simulation coordinate system, the origin lies at the centre of the Milky Way. We chose the axes so that Andromeda lies on the x-axis and the Sun on the x-y-plane at t0. The first step of the transformation to Galactic coordinates is to move the coordinate origin along the position vector of the Sun ~rSun in the simulation coordinate frame. For this purpose, we construct a triangle between the position of the SMBH Sagittarius A* (Sgr A*) at the coordinate origin of the Galactocentric Cartesian coordinate system, and the positions of the Sun and of Andromeda at t0in the same coordinate frame. The triangle is shown schematically in Fig. 14. To construct ~rSun, we now need to find the angle β in Fig. 14. β is the angle between the primary direction of the simulation coordinate frame and the connection between the Sun and the simulation coordinate origin Sgr A*. We find β by calculating the angle between 00 00 the position vectors of Andromeda ~r G,And and the Sun ~r G,Sun in Galactocentric Cartesian coordinates. They respectively represent the edges a and b of the triangle. We again use 00 00 ~r G,And from van der Marel et al. [129] and obtain ~r G,Sun from Abuter et al. [134] as

 378.9  8.178 00 − 00 − ~r G,And =  612.7  kpc and ~r G,Sun =  0  kpc. (6.35) 283.1 0 − Now, we simply obtain the angle β between the two vectors by calculating the scalar product

~r 00 ~r 00 β = arccos G,And · G,Sun 60.691◦. (6.36) ~r 00 ~r 00 ≈ | G,And|| G,Sun| Since β is the angle between the x-axis of the simulation coordinate system and the connection between the Sun and coordinate origin Sgr A*, we can calculate the position

37 Sgr A* z e Sgr A* y x b

r / r´´ b Sun G,Sun

r / r´´ And G,And

a Sun

c x´´

y´´ Sgr A* z´´

Andr

Fig. 14. Schematic illustration of the triangle between Sgr A*, the Sun and Andromeda (Andr) at t0. The coordinate system (x, y, z) in the top middle is the simulation coordinate frame. The coordinate system (x00, y00, z00) on the lower right is the Galactocentric Cartesian coordinate frame with slightly tilted axes for visual clarity. We calculate the angle β using the scalar product between the vectors 00 00 ~r G,Sun and ~r G,And. They are both in Galactocentric coordinates and represent the edges a and b of the triangle. The vectors ~rSun and ~rAnd are the identical vectors in the simulation coordinate frame. Given β, we calculate the vector ~rSun. Angles and distances are not to scale.

vector of the Sun ~rSun in simulation coordinates with

 00    ~r G,Sun cos β 4.003 | 00 | ~rSun =  ~r G,Sun sin β  7.131 kpc. (6.37) | z| ≈ 0

The z-coordinate is approximately zero because we chose the Sun to lie in the x-y-plane. To complete the ﬁrst part of the transformation to Galactic coordinates, we move the origin of the simulation coordinate frame to the position of the Sun by moving it along ~rSun.

38 The next steps are to rotate the coordinate frame so that the primary direction points towards the GC and the x-y-plane coincides with the Galactic plane which is the x-y-plane of the Galactic Cartesian coordinate system (x00, y00, z00). For this process, we neglect the elevation of the Sun above the Galactic plane since it is of order 10−2 kpc [135]. We now perform two rotations, one each around the z-axis and the x-axis. The angle for the rotation around the z-axis to align the x-axis with the primary direction of the Galactic Cartesian coordinate frame is simply ε = π β. We insert it into the generic rotation − matrix around the z-axis Rz in three dimensions and apply it to the HVS position vector ~r for the ﬁrst rotation cos ε sin ε 0 0 − 0 Rz ~r = ~r sin ε cos ε 0 ~r = ~r . (6.38) · ⇔ 0 0 1 ·

The shift of the coordinate origin in the first step as well as the first rotation are schematically shown in Fig. 15. Since the Galactic Cartesian coordinate frame is only one rotation away, we can find the angle φ around the x-axis between the x-y-plane of the transition coordinate sys- 0 0 0 00 tem (x , y , z ) and the Galactic plane by looking at the position of Andromeda ~r G,And in 00 Galactocentric Cartesian coordinates. We subtract the position vector of the Sun ~r G,Sun 00 00 from ~r G,And to find the position vector of Andromeda ~r And in Galactic Cartesian coordinates  378.9  8.178 00 00 00 − − ~r And = ~r G,And ~r G,Sun =  612.7   0  kpc − 283.1 − 0 − (6.39)  370.7 −  612.7  kpc. ≈ 283.1 − 00 Now, we use ~r And as the solution of the system of linear equations given by the generic rotation matrix around the x-axis Rx in three dimensions

0 00 Rx ~r = ~r · And And

1 0 0  (6.40) 0 00 0 cos φ sin φ ~r And = ~r And ⇔ 0 sin φ −cos φ ·

0 where ~r And is the position vector of Andromeda after the shift of origin to the Sun and 0 the first rotation around the z-axis are applied. To find ~r And, we apply these steps to the position vector of Andromeda ~rAnd in the simulation coordinate frame. Since Andromeda lies on the x-axis of the simulation coordinate frame at t0, we find ~rAnd by setting the 00 magnitude of ~r And as its x-coordinate

39 x´

z/z z/z´ Sgr A*

r Sun y e Sun Sun x y´

y y Andromeda x x

Fig. 15. Illustration of the ﬁrst two steps of the transformation. The coordinate system (¯x, y,¯ z¯) is another transition system after the shift of origin and before the rotation around the z/z¯-axis. This notation is only used in this ﬁgure. The coordinate systems (x, y, z) and (x0, y0, z0) follow the established notation scheme. The z/z/z¯ 0-axes all have identical orientation. Left: Shift of the coordinate origin from Sgr A* to the Sun with ~rSun. Right: Rotation by ε around the z-axis to align the x¯-axis with the x0-axis. Angles and distances are not to scale.

 ~r 00  774.023 | And| ~rAnd =  0   0  kpc. (6.41) 0 ≈ 0

Now we subtract the position vector of the Sun ~rSun in the simulation coordinate frame 0 and apply the rotation in Eq. (6.38) to ﬁnd ~r And

0 ~r = Rz (~r ~r ) And · And − Sun  370.718 0 − (6.42) ~r And  674.944  kpc. ⇒ ≈ 0

Using this result, we solve the system of linear equations given by Eq. (6.40) to ﬁnd φ 24.799◦ for the rotation angle around the x-axis. Now we insert φ into Eq. (6.40) ≈ − 0 and apply the rotation to ~r And to ﬁnd the position of Andromeda in Galactic Cartesian 00 coordinates ~r And,Sim 00 0 ~r = Rx ~r And,Sim · And  370.718 00 − (6.43) ~r And,Sim =  612.702  kpc, ⇒ 283.101 −

40 which is transformed from the position of Andromeda ~rAnd in the simulation coordinate 00 system. This result is in good agreement with the position of Andromeda ~r And in Galactic Cartesian coordinates  370.7 00 − ~r And =  612.7  kpc, (6.44) 283.1 − which we found by combining the observational values from van der Marel et al. [129] and Abuter et al. [134] used in Eq. (6.39). All that remains is the transformation to the Galactic coordinates x00 00 ~r = y00 00 z (6.45) ! y00 z00 g = (l, b) = arctan 00 , arcsin(p , → x x00 2 + y00 2 + z00 2 which are simply the angular components of the standard spherical coordinates corresponding to Galactic Cartesian coordinates. 00 Applying the spherical transformation to ~r And,Sim gives us the position of Andromeda gAnd,Sim in Galactic coordinates

 370.718 00 − ◦ ◦ ~r And,Sim =  612.702  kpc gAnd,Sim = (121.176 , 21.570 ). (6.46) 283.101 → − −

This also shows good agreement with the position of Andromeda gAnd in Galactic coordinates

g = (121.174◦, 21.573◦) (6.47) And − from the Simbad data base [136]. Before the transformation to Galactic coordinates, we perform a last step for the velocity vectors in addition to the transformation to Galactic Cartesian coordinates. This is done because the simulation results are calculated in the Galactocentric rest frame and the Gaia data is mapped in the heliocentric rest frame. We use the notation for the heliocentric and Galactocentric rest frames we established in Section 4.2. For the transformation of the simulation coordinate system velocity vectors ~vgal,HVS to the corresponding vectors 00 ~v gal,HVS in Galactic Cartesian coordinates, we only apply the two rotations since velocity 00 vectors do not depend on the origin. Subsequently, we boost ~v gal,HVS to the heliocentric 00 rest frame velocity vector ~v hel,HVS

~v 00 = ~v 00 ~v 00 , (6.48) hel,HVS gal,HVS − gal,Sun

41 00 with the velocity vector of the Sun ~v gal,Sun in the Galactocentric rest frame

 11.1  00 ~v gal,Sun = 252.24 km/s. (6.49) 7.25

00 We obtain ~v gal,Sun from Eq. (4.1) and write it with the notation for Galactic Cartesian coordinates. We brieﬂy summarise the transformations applied to the simulation position and velocity vectors. A simulation HVS position vector ~rHVS is transformed to the position vector 00 ~r HVS in Galactic Cartesian coordinates with

00 ~r = RxRz (~r ~r ). (6.50) HVS · HVS − Sun

The simulation velocity vector ~vgal,HVS in the Galactocentric rest frame is transformed 00 and boosted to the velocity vector ~v hel,HVS in Galactic Cartesian coordinates and in the heliocentric rest frame with

00 00 ~v = (RxRz ~v ) ~v . (6.51) hel,HVS · gal,HVS − gal,Sun In both cases, we respectively use the angles β and ε for the two rotations. 00 00 Afterwards, we transform ~r HVS and ~v hel,HVS to Galactic coordinates with the transformation in Eq. (6.45). By applying these transformation steps to the position and velocity vectors resulting from the simulation, we ﬁnd the HVS positions and velocity directions in Galactic coordinates and in the heliocentric rest frame. In this form, we can compare the simulation results to the Gaia HVS data.

7. Results and discussion

The simulation produces data for every HVS at the end of its calculated trajectory at present time t0 = 13.8 Gyr. This data set includes the distance between the HVS and the Sun, its velocity magnitude and velocity vector in Galactic Cartesian and Galactic coordinates, the initial send-off time as well as the position of the HVS in Galactic and equatorial coordinates. Due to the isotropic distribution of initial velocity directions, an overwhelming majority of HVSs do not come anywhere near our galaxy. We need to calculate a large number of trajectories to find a significant amount HVSs that are close enough to us at present time to compare them to Gaia measurement data. In total, we calculate 1.8 108 trajectories with randomly generated send-off times between 10 and 12 Gyr after∼ the× Big Bang. We apply multiple filters to only take HVSs into consideration that are sufficiently close to the Sun at present time. By filtering out HVSs that have a distance of 500 kpc or more to the centre of the Milky Way, we eliminate all HVSs that are either recaptured by Andromeda or move away from both galaxies entirely. In order to find the HVSs that reach the inner parts of the Milky

42 Way, we apply another filter d < dFilter = 50 kpc (7.1) with an allowed maximum distance dFilter to the GC. This selection results in 6862 HVSs. As the last step of the selection process, we apply another filter to only consider HVSs within 13 kpc of the Sun in order to find results that are close enough to be compared to data from the Gaia eDR3 catalogue. The catalogue has limited accuracy at long distances. In the end, only 521 HVSs are left after the entire selection process. At this point, it is important to note that this amount of simulation HVS is not expected to be representative of the actual amount of HVSs from Andromeda in the Milky Way. It is rather a sufficiently large amount of data to analyse. As previously mentioned, we only consider HVSs that come closer to the GC than dFilter at present time t0. Other methods were considered to find the point in time in recent cosmic history at which a HVS has its minimum distance to the Sun. A method that returns the minimum distance in the time interval [13.75, 13.80] Gyr yields results that have are most likely to be at either end of the interval. Results at t = 13.75 Gyr are too far in the past to be compared with measurement data at t0. The same method with a smaller time interval of [13.79, 13.80] Gyr yields results with the same behaviour. A comparison with the method of simply returning the distance between the HVSs and the Sun at t0 shows that it picks up nearly all of the results from the time interval method. The only difference are HVSs that have a minimum distance dmin > 10 kpc to the Sun at t = 13.79 Gyr. These HVSs are the most distant and furthest in the past and have the least accuracy as a result. As such, we decided to use the method of returning the data at present time t0. Due to the considerable number of assumptions and approximations we made during the construction of the model and the processing of the simulation results, we neglect error analysis. None of the simulation results are to be taken as exact.

7.1. Distance and velocity distributions The small number of remaining HVSs after the selection process is due to the randomly generated initial velocity directions of the trajectories. A single trajectory is much more likely to point towards any other direction than towards the Milky Way. Due to the gravitational potential of the Milky Way, some HVSs with initial trajectories not directly pointing at the Milky Way are attracted strongly enough to pass through it. The shape of the combined gravitational potential of the Milky Way and Andromeda in the simulation model is shown in Fig. 16. The potential well of the Milky Way is at the coordinate origin and the identical potential well of Andromeda is at its position at t = 10 Gyr after the Big Bang. The potential has a saddle point between the two galaxies. The top panel of Fig. 17 shows the distribution of distances between the simulation HVSs and the Sun at t0 on a logarithmic scale. Displayed is a randomly selected sample of 2.7 106 trajectories out of all calculated trajectories. We see the fraction of HVSs that∼ stay bound· to Andromeda is approximately equal to the fraction of HVSs that ﬂy off in different directions than towards the Milky Way. They are represented respectively by

43 0.025

750 0.045

500 0.065

250 0.085

0 0.105

Distance [kpc] 250 0.125

500 Gravitational potential [a.u.] 0.145

750 0.165

250 0 250 500 750 1000 1250 Distance [kpc]

Fig. 16. Contour plot of the shape of the gravitational potential of the Milky Way- Andromeda system at t = 10.0 Gyr after the Big Bang. The Milky Way is positioned at the origin. The potential is shown in arbitrary units. peaks at 770 kpc and 1000 kpc. The former is the distance between Andromeda and ∼ ∼ the Milky Way at t0 and represents the HVSs that stay bound to Andromeda. The second peak shows the HVSs that have an initial velocity direction away from both galaxies. It is less defined than the first peak which is owed to the varying distance between Andromeda and the Milky Way at different send-off times. Both peaks combined make up close to 90% of all calculated trajectories. Only about 1% of all HVSs reach a radius of 200 kpc around the Milky Way centre. The bottom panel of Fig. 17 presents the same information on the 521 HVSs within 13 kpc of the Sun found by the selection process. An r3 function is fitted on the distribution. It shows approximately an r3-dependency of the abundance of HVSs as the distance r increases. This indicates an approximately constant number density of HVSs within the regarded radius. Before we can take a look at the distribution of velocity magnitudes of the simulation HVSs, we have to consider the difference between the heliocentric and Galactocentric rest frames. Fig. 18 displays the heliocentric velocities of the simulation HVSs within two separate radii around the Sun as Toomre diagrams, analogously to the display of Gaia HVS velocities in Fig. 11. In contrast to the Gaia HVSs, the entire selection of simulation HVSs in the top panel is concentrated in one section of the diagram. The concentration indicates a narrow distribution of velocity directions. We discuss the simulation and Gaia

44 HVS velocity directions in detail in Section 7.2. Galactocentric velocities of the HVSs rarely exceed 1000 km/s as shown by the black circles of constant velocity in the Galactocentric rest frame. This is expected since the HVS are typically ejected from Andromeda at velocities below this value. Similarly, the higher density of data points at low velocities in comparison to the data points at high velocities is accounted for by the initial distribution of velocity magnitudes (see Fig. 13). Further narrowing down the selection to a radius of 7 kpc around the Sun that only contains 69 simulation HVSs, shows the HVSs concentrated in an even smaller area of the diagram. This is due to the smaller size of the considered region around the Sun. The gravitational potential in this region has less inhomogeneity than over the complete region deﬁned by the radius of 13 kpc around the Sun. This means that the HVSs, which all come in from approximately the direction of Andromeda, are accelerated with a similar force and change their direction in the same way. The HVS velocities, the relative velocity of the GC, and the velocity of the Sun are respectively marked as blue dots, and green and red stars. A comparison between them shows that all simulation HVSs move in the opposite y-direction of the Sun in the Galacto- centric rest frame. This is expected as the Andromeda galaxy has a positive y-coordinate in Galactic Cartesian coordinates (see Eq. (6.39)) and, as such, HVSs moving towards the Milky Way have a negative vy component. As seen in Eq. (4.1), the Sun has a positive velocity in the y-direction in the Galactocentric rest frame. This causes all simulation HVSs to appear faster when viewed in the heliocentric rest frame. The distributions of component velocities discussed in Appendix B show the preferred directions of the simulation HVSs as well. With this in mind, we now look at the velocity magnitude distributions.

45 10.0%

1.0% Fraction of HVSs

0.1%

0 200 400 600 800 1000 Distance r to the Sun [kpc]

r3 function fit 70

40 # of HVSs 30

0 2 4 6 8 10 12 Distance r to the Sun [kpc]

Fig. 17. Distribution of distance r to the Sun at t0 of the simulation HVSs. Top: Random sample of 2.7 106 simulation HVSs. Bottom: 521 HVSs within 13 kpc of the Sun. An r3∼function· is ﬁtted to the distribution.

46 1000 Simulation HVSs (d < 13kpc) ]

s Sgr A* / 800

m Sun k [

) z

, 600 l e 2 h v

+ 400 x , l e 2 h v ( 200

0 1000 750 500 250 0 250 500 750 1000

vhel, y [km/s]

1000 Simulation HVSs (d < 7kpc) ]

s Sgr A* / 800

m Sun k [

) z

, 600 l e 2 h v

+ 400 x , l e 2 h v ( 200

0 1000 750 500 250 0 250 500 750 1000

vhel, y [km/s]

Fig. 18. Toomre diagrams of velocities of simulation HVSs at t0. The velocity components (vhel,x, vhel,y, vhel,z) on the axes correspond to the velocity in the Galactic Cartesian coordinate system and the heliocentric rest frame. The HVS velocities are shown as blue dots. The black semi-circles represent constant velocities vgal of 600, 700, 800 and 1000 km/s in the Galactocentric rest frame. The relative velocity between the heliocentric and Galactocentric rest frames is displayed as a green star and serves as the centre of the semi-circles. The velocity of the Sun in the Galactocentric rest frame is shown as a red star and the corresponding velocity magnitude as a red semi-circle with radius 250 km/s. Top: 521 HVSs within 13 kpc of the Sun. Bottom: 69 HVSs within∼7 kpc of the Sun.

Fig. 19 shows the distribution of velocity magnitudes of the simulation HVSs in the heliocentric rest frame. The top panel shows the velocities of all HVSs that reach a radius of 50 kpc around the GC, organised by send-off time. In total, 6862 HVSs made it to

47 this part of the selection process. Overall, the distribution shows a peak at 800 km/s. However, the first two send-off time windows peak at slightly lower velocities∼ while the last two windows peak at a slightly higher velocity. This is most likely due to due to the shorter distance travelled by HVSs sent off at later times. They lose less velocity while receding from Andromeda until they start accelerating due to the gravitational potential of the Milky Way. We also note that the majority of HVSs have total velocities slower than their initial velocity, especially if we consider that they appear faster than they are in the Galactocentric rest frame. The velocity distribution of the final selection of 521 simulation HVSs within 13 kpc of the Sun is displayed in the bottom panel, organised in the same way as the top panel. It is less smooth due to the smaller sample size. As in the top panel, later send-off times generally correspond to higher velocities. The distribution peaks at a velocity of 950 km/s, higher than in the top panel. We account this to the fact that the average distance∼ to the GC for this selection of HVSs is smaller in comparison to the selection within 50 kpc of it. This means they have already experienced more gravitational acceleration from the mass within the Milky Way. For both selections, we observe the earliest send-off time window to contain the most HVSs. Respectively, the first window contains 30.7% and 33.8% of all results for the selections in the top panel and the bottom panel. The number of objects per window decreases in a linear fashion until the latest window only makes up 20.3% (top panel) and 18.2% (bottom panel) of each selection. Since the initial send-off times are evenly distributed across all four windows, this indicates that HVSs are more likely to be in the inner regions of the Milky Way and close to the Sun at t0 if sent off from Andromeda at earlier times. We account this to the high velocity of HVSs that are sent off at later times and to the shorter distance they have to travel to reach the Milky Way due to the mutual approach of the galaxies. This means at present time more of those HVSs have already passed through the region we are considering and are too far away to be picked up by the selection process. Analogously to the previous figure, Fig. 20 shows the velocity magnitude distribution of the entire selection of 259 Gaia HVSs on a logarithmic scale. Velocities vgal in the top panel are in the Galactocentric rest frame while the corresponding heliocentric velocities vhel are displayed in the bottom panel. The sharp cut-off at vgal = 600 kpc in the top panel is due to the velocity criterium of the Gaia data selection process, previously detailed in Section 4.1. The gap in the curve at 900 km/s is due to Gaia data at high velocity often having large errors. These values are∼ then eliminated in the selection process. Otherwise, the sharp decline in the abundance of HVSs with increasing velocity is expected. The distribution of velocities in the heliocentric rest frame in the bottom panel shows the peak at vhel 900 km/s instead. This significant shift can be explained if we look at the Galactocentric≈ velocities. A large fraction of the HVSs in the selection moves in the opposite direction that the Sun is moving in the Galactocentric frame. We observed this in Section 4.2 using Fig. 11. In the heliocentric frame, all these HVSs receive a boost of up to 250 km/s which moves the peak to the right as we observe in the bottom panel. This phenomenon∼ is also apparent in Fig. 26 and 27 which we discuss in Section 7.2.

48 Send-off-time t [Gyrs]: 16.0% 10.0 t < 10.5 10.5 t < 11.0 14.0% 11.0 t < 11.5 11.5 t < 12.0

12.0%

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6.0%

4.0% Fraction of HVSs

2.0%

0.0% 800 900 1000 1100 1200

Velocity vhel [km/s]

Fig. 19. Distribution of simulation HVS velocities vhel in the heliocentric rest frame at t0, organised by send-off time t. Top: 6862 HVSs within 50 kpc of the GC. Bottom: 521 HVSs within 13 kpc of the Sun.

49 10.0% Fraction of HVSs

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600 650 700 750 800 850 900 950 1000

Galactocentric velocity vgal [km/s]

10.0% Fraction of HVSs

1.0%

400 600 800 1000 1200

Heliocentric velocity vhel [km/s]

Fig. 20. Distribution of HVS velocities from Gaia data selection within 13 kpc of the Sun at t0. In total, 259 HVSs are shown. Top: Velocities vgal in the Galactocentric rest frame. Bottom: Velocities vhel in the heliocentric rest frame.

50 7.2. Positions and velocity directions Building on what we established in the previous section, this section discusses the positions and velocity directions at t0 of HVSs within 13 kpc of the Sun from both the simulation and Gaia data. Their positions on the sky as well as the directions of their velocity vectors are displayed on sky maps. The simulation HVSs are presented in Fig. 21 and 22, Gaia HVSs in Fig. 25 and 26. A point on a position sky map simply shows the point on the sky at which we observe the HVS. A point on a velocity direction sky map represents the point on the sky the corresponding velocity vector pointed at if it ran through the coordinate origin. The data on all sky maps is in Galactic coordinates and in the heliocentric rest frame. The positive angular direction for Galactic longitude points to the right in all plots. In addition, 3D plots show the positions and velocities of the simulation and Gaia HVSs within 13 kpc of the Sun. The simulation HVSs are displayed in Fig. 23 and Fig. 27 shows the corresponding plot for the Gaia HVSs. Fig. 28 shows a stricter selection of Gaia HVSs within 7 kpc of the Sun. A clearer view of the HVS positions without the velocity vectors is presented in x-y- and x-z-plane projection plots in Fig. 24 (simulation) and Fig. 29 (Gaia). All velocities are presented in the heliocentric rest frame. Fig. 21 displays the positions of the simulation HVSs on the sky. The distribution looks nearly isotropic around the Sun with the exception of tendencies towards the GC at (0◦, 0◦) as well as the Galactic disc at 0◦ Galactic latitude. The send-off time, shown with a colour map in the top panel, does not show any discernible correlation with the positions. The bottom panel instead uses a colour map of the velocity magnitude of the HVS velocity vectors corresponding to the positions. We see clearly that the fastest HVSs are only found towards the GC. Further, the longer the angular distance between the GC and any HVS position, the slower the HVS tends to be. This effect is due to the physical distance between the HVSs and the GC. For HVSs within 13 kpc of the Sun the angular distance to the GC on the sky is closely related to the physical distance between the two objects. Since all considered HVSs move towards the GC, HVSs close to it have higher kinetic energy and have already undergone more acceleration than HVSs further away from it. This explains the difference in velocity. Fig. 22 shows the velocity directions of the same HVSs as displayed in Fig. 21. As previously, the top panel is colour mapped according to send-off time while the bottom panel shows the distribution of velocity magnitudes. We see that the velocity directions of all HVSs that come close to the Sun are concentrated in one area of the sky. Without exception, they all point towards an approximately circular patch around ( 58◦, 16◦). It has an angular diameter of about 60◦. This is almost directly opposite∼ the− position of Andromeda at (121.174◦, 21.573◦) which is displayed as a black star in all simulation sky maps. Since Andromeda− approaches the Milky Way with predominantly radial velocity, this position has not changed signiﬁcantly since the HVSs were ejected. This result is to be expected from trajectories originating in the Andromeda galaxy.

51 12.00 Positions of simulation HVSs at t0 75° (d < 13 kpc, Gal. coord.) 60° 11.75 Andromeda 45° 11.50

30° ] r y G

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1100 Positions of simulation HVSs at t0 75° (d < 13 kpc, Gal. coord.) 60° Andromeda 45° 1050

30° ] s / m

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y t i

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850

Fig. 21. Positions on the sky at t0 of the 521 simulation HVSs within 13 kpc of the Sun in Galactic coordinates. The black star shows the position of Andromeda at t0. Top: Colour mapped by send-off time t. Bottom: Colour mapped by heliocentric velocity vhel. HVSs with velocities above or below the boundaries of the colour scale take on the colour of the upper or lower bound, respectively.

52 12.00 Velocity directions of simulation HVSs at t0 75° (d < 13 kpc, Gal. coord.) 60° 11.75 Andromeda 45° 11.50

30° ] r y G

11.25 [ 15° t

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Velocity directions of simulation HVSs at t0 1100 75° (d < 13 kpc, Gal. coord.) 60° Andromeda 45° 1050

30° ] s / m

15° 1000 k [

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850

Fig. 22. Velocity directions at t0 of the 521 simulation HVSs within 13 kpc of the Sun in Galactic coordinates and the heliocentric rest frame. The black star shows the position of Andromeda at t0. Top: Colour mapped by send-off time t. Bottom: Colour mapped by heliocentric velocity vhel. HVSs with velocities above or below the boundaries of the colour scale take on the colour of the upper or lower bound, respectively.

53 The top panel of Fig. 22 shows the HVS velocity directions to be slightly more concentrated if the HVSs are sent off at later times which means their trajectories start off closer to the Milky Way. These HVSs experience stronger gravitational attraction from the Milky Way at the start of their trajectory than HVSs that are sent off from longer distances at earlier times. We attribute the difference in concentration for the different send-off times to this fact. By comparison with the bottom panel, we observe that many of the HVSs with late send-off times correspond to the fastest HVSs. This agrees with the observation about the relationship between send-off time and velocity magnitude at t0 made in Section 7.1. The velocity direction distribution in the bottom panel also shows that the fastest HVSs generally deviate further from the direction of the GC than the HVSs at the lower end of the velocity spectrum. We can explain this phenomenon by taking a look at Fig. 23 and 24. Fig. 23 shows two points of view of the 3D plot displaying some of the 521 simulation HVSs within 13 kpc of the Sun. We choose to display a random selection of 120 HVSs in order for the plot to stay visually clear. The plot displays HVS positions and velocity directions as well as velocity magnitude as a colour map. The vector length is normalised and the deﬁning colour is the colour of the vector body. The positions of the HVSs are at the beginning point of the displayed velocity vectors. The data is displayed in Galactic Cartesian coordinates which means the Sun is at the origin. Sgr A* at the GC lies at 8.178 kpc on the x-axis. They are represented by a black and a green star, respectively. Fig. 24 displays the positions of the whole selection of 521 simulation HVSs projected on the x-y- and x-z-planes in Galactic Cartesian coordinates. The same colour map for the velocity magnitude of the HVS trajectories as in Fig. 23 is used. The positions of the Sun and Sgr A* are represented by black and green stars. The ﬁgure clearly shows the near-isotropy of the distribution of HVS positions in the considered region around the Sun except for the region around the GC. We have previously discerned and see in both Fig. 23 and 24 that the HVSs close to the GC have the highest velocities. As seen in Fig. 22, they tend more strongly to point towards the negative azimuth angular direction than the HVSs with lower velocity. This is the negative y-direction in Fig. 23 and in the top panel of Fig. 24. In the bottom panel of Fig. 23, we can clearly see this population of HVSs in the vicinity of the GC on the opposite side of the Sun. The deviation of their velocity direction when compared to the remaining HVSs can be attributed to their approach direction of the GC. When viewed from the direction of Andromeda which is on the top left, they approach the GC from the direction opposite to most of the other HVSs in the selection and from where the Sun is located. Due to this circumstance, the acceleration in the negative y-direction they experience when they close in on the GC is stronger.

54 Simulation HVSs Sun Sgr A*

0 -axis [kpc] -axis z 5

10 10 5 5 0 y-axis [kpc]0 5 5 x-axis [kpc] 10 10

z-axis [kpc]

y-axis [kpc]

10 5 0 5 10 x-axis [kpc]

850 900 950 1000 1050 1100 Velocityv hel [km/s]

Fig. 23. 3D plot of 120 out of the 521 simulation HVSs with positions and velocity vectors at t0 in Galactic coordinates and in the heliocentric rest frame. Two points of view are displayed. The vector heads vanish in the bottom panel due to perspective. The HVSs are colour mapped by heliocentric velocity vhel. The colour of the vector body is deﬁning for the velocity. HVSs with velocities above or below the boundaries of the colour scale take on the colour of the upper or lower bound, respectively. Sgr A* in the GC is displayed at 8.178 kpc on the x-axis as a green star and the Sun as a black star at the origin.

55 1100 Simulation HVSs 10 Sun Sgr A* 1050

5 ] s /

1000 m k [

l e

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5 ] s /

1000 m k [

l e 0 h v

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5 V

900 10

850 10 5 0 5 10 x [kpc]

Fig. 24. Projection plots of the positions of the 521 simulation HVSs in the x-y- and x- z-planes in Galactic Cartesian coordinates. The black and green star markers respectively show the positions of the Sun and Sgr A* at the GC. The position markers of the HVSs are colour mapped according to heliocentric velocity vhel. HVSs with velocities above or below the boundaries of the colour scale take on the colour of the upper or lower bound, respectively.

56 We look at the corresponding sky maps to Fig. 21 and 22 produced from the Gaia data selection in order to compare them to the simulation results. They are displayed in Fig. 25 and Fig. 26. The former shows the Gaia HVS positions on the sky while the latter shows the corresponding velocity directions. We consider HVSs within two different regions deﬁned by radii around the Sun. First, we look at the HVSs within the same radius of 13 kpc around the Sun as in the previous analysis. Subsequently, we only take HVSs within 7 kpc of the Sun into account. The top panel in both Fig. 25 and 26 shows the larger radius while the smaller radius is displayed in each bottom panel. All sky maps are colour mapped by velocity magnitude. However, different scales are chosen for the two radii. As to be expected, the positions of HVSs within 13 kpc are heavily concentrated around the GC due to the high stellar density in this region. Approximately equal amounts are located below and above the Galactic plane, but there is a higher number of HVSs near the pole on the Northern Galactic hemisphere. This is also visible in the x-z-plane projection graph in the bottom panel of Fig. 29. The HVS positions within 7 kpc only seem to show a higher concentration in the area of the Galactic disc. However, the sample size is too small to make a meaningful statement about them. The velocity directions of HVSs within 13 kpc of the Sun show a heavily preferred area of the sky. This relatively wide area is centred around approximately ( 96◦, 2◦) and covers a big part of this half of the sky. As we have previously established,− this indicates that these HVSs are all moving approximately in the opposite direction of the Sun in the Galactocentric rest frame. This Solar velocity direction is approximately at (90◦, 0◦) on the sky maps. We can clearly see the velocity boost the HVSs receive due to observing them in the heliocentric rest frame. In turn, we see that HVSs with velocity direction similar to the Sun appear slower. Taking a further look at the population of fast HVSs in the 3D plots in Fig. 27 and the projection plots in Fig. 29 shows the majority of them on the opposite side of the GC as a cloud of objects with velocity vectors pointing to the right. They are presumably part of the Galactic bulge. We assume this heavily preferred direction is due to selection bias in the method of selection from the Gaia catalogue. Presumably, a similar population of HVSs exists on the same side of the GC as the Sun with velocity directions similar to the Sun. Furthermore, the slower HVSs have a different, more weakly preferred direction. Their velocity vectors have a slight tendency to point towards the southern Galactic hemisphere, as seen in the top panel of Fig. 26. Averaging over all velocity vectors in the top panel of Fig. 26 results in a velocity direction of ( 95.92◦, 1.59◦) which lies almost exactly in the center of the region preferred by the fast HVSs.− It is also very close to the velocity direction of the GC relative to the Sun. The averaged velocity directions are displayed as black dots on both velocity direction sky maps.

57 Positions of HVSs from GAIA eDR3 measurements 1050 75° (d < 13 kpc, vgal > 600 km/s, Gal. coord.) 60° 1000 45° 950

30° ] s 900 / m

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Positions of HVSs from GAIA eDR3 measurements 1050 75° (d < 13 kpc, vgal > 600 km/s, Gal. coord.) 60° 1000 45° 950

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Fig. 25. Positions on the sky of Gaia HVSs at t0 in Galactic coordinates. Displayed are all HVSs with Galactocentric velocities vgal > 600 km/s. The HVSs are colour mapped by heliocentric velocity vhel. HVSs with velocities above or below the boundaries of the colour scale take on the colour of the upper or lower bound, respectively. Top: 259 HVSs within 13 kpc of the Sun. Bottom: 14 HVSs within 7 kpc of the Sun.

58 Velocity directions of HVSs from GAIA eDR3 measurements 1050 75° (d < 13 kpc, vgal > 600 km/s, Gal. coord.) 60° 1000 45° Averaged velocity direction 950

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Velocity directions of HVSs from GAIA eDR3 measurements 1050 75° (d < 13 kpc, vgal > 600 km/s, Gal. coord.) 60° 1000 45° Averaged velocity direction 950

30° ] s 900 / m

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Fig. 26. Velocity directions of Gaia HVSs at t0 in Galactic coordinates and in the heliocentric rest frame. Displayed are all HVSs with Galactocentric velocities vgal > 600 km/s. The HVSs are colour mapped by heliocentric velocity vhel. HVSs with velocities above or below the boundaries of the colour scale take on the colour of the upper or lower bound, respectively. The black dot shows the averaged velocity directions of all HVSs combined. Top: 259 HVSs within 13 kpc of the Sun. Bottom: 14 HVSs within 7 kpc of the Sun.

59 If we move closer to the Sun in the bottom panel of Fig. 26, the HVSs within 7 kpc often have velocity vectors that run close to parallel to the Galactic plane. We see this in the corresponding 3D plot in Fig. 28 as well. The averaged velocity for this selection points towards ( 109.06◦, 16.06◦). Similarly to the positions however, the sample size is too small to make− meaningful statements. Appendix B brieﬂy discusses the distributions of component velocities of the Gaia HVSs within both considered radii. There, the preferred velocity directions are apparent as well.

The positions of simulation and Gaia HVSs show a small degree of compatibility with each other due to the concentration around the GC. In addition, the preferred area of the Gaia HVS velocity directions in Fig. 26 overlaps with the concentrated area of velocity directions displayed by the simulation HVSs in Fig. 22. While this indicates some correlation between the simulation and Gaia velocity directions, the correlation appears to be coincidental since nearly all Gaia HVSs with matching trajectories are located in only one part of the considered region. As we established, the Gaia HVSs that overlap with the simulation HVSs are part of the population we observe in the vicinity the GC in Fig. 27 and 29. The velocity directions of the fraction of those HVSs above and to the right of the GC in the bottom panel of Fig. 27 and the top panel of Fig. 29 agree with our predictions for HVSs originating in Andromeda. However, a comparison with Fig. 23 shows that we should observe approximately equal amounts of such trajectories in every part of the region deﬁned by the 13 kpc radius around the Sun. In the remaining parts of the regarded region, we only observe a small number of velocity vectors with matching directions. We note three examples in Fig. 27. The HVSs and velocity vectors in question are marked with orange boxes in both panels. The velocity vectors show approximately the same direction as the trajectories in Fig. 23. Two further examples are visible in Fig. 28, also marked with orange boxes. The top panel of Fig. 26 shows a small number of additional matching trajectories. However, not all of them are displayed in the 3D plot since Fig. 27 only shows a random selection of 150 HVSs from the selected catalogue of Gaia HVSs for visual clarity. In total, the small number of matching trajectories in the regarded region as a whole indicates that the HVSs above and to the right of the GC in the bottom panel of Fig. 27 belong to a population that only coincidentally shares similar velocity directions with HVSs originating in the Andromeda galaxy. As a whole, this population moves in the opposite direction of the Sun and appears to form an arc around the GC. We can see this arc in the bottom panel of Fig. 27 above the green star representing the GC and in both panels of Fig. 29. Following these observations, we can say with some certainty that HVSs from the Andromeda galaxy do not constitute a big fraction of the HVSs within the Milky Way. However, we cannot rule out the possibility of their existence entirely.

60 Gaia HVSs with vgal > 600km/s Sun Sgr A*

0 10

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10 5 x-axis [kpc] 5 0 y-axis [kpc] 10 5 10 z-axis [kpc]

-axis y [kpc]

10 5 0 5 10 x-axis [kpc]

600 700 800 900 1000 Velocityv hel [km/s] Fig. 27. 3D plot of a random selection of 150 positions and velocity directions from the 259 Gaia HVSs within 13 kpc of the Sun in Galactic Cartesian coordinates and in the heliocentric rest frame. Two different points of view are shown. The vector heads vanish in the bottom panel due to perspective. The HVSs are colour mapped by heliocentric velocity vhel. The colour of the vector body is deﬁning for the velocity. HVSs with velocities above or below the boundaries of the colour scale take on the colour of the upper or lower bound, respectively. Sgr A* in the GC is displayed at 8.178 kpc on the x-axis as a green star and the Sun as a black star at the origin.

61 Gaia HVSs with vgal > 600km/s Sun Sgr A*

2 6 0 4

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x-axis [kpc] 6 4 4 2 0 2 6 y-axis [kpc] 4 6

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500 600 700 800 Velocityv hel [km/s] Fig. 28. 3D plot of the positions and velocity directions of the 14 Gaia HVSs within 7 kpc of the Sun in Galactic Cartesian coordinates and in the heliocentric rest frame. Two different points of view are shown. The vector heads vanish in the bottom panel due to perspective. The HVSs are colour mapped by heliocentric velocity vhel. The colour of the vector body is deﬁning for the velocity. HVSs with velocities above or below the boundaries of the colour scale take on the colour of the upper or lower bound, respectively. Sgr A* in the GC is displayed at 8.178 kpc on the x-axis as a green star and the Sun as a black star at the origin.

62 10.0 1050 Gaia HVSs Sun 7.5 1000 Sgr A* 950 5.0

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Fig. 29. Projection plots of the positions of the 259 Gaia HVSs in the x-y- and x-z-planes in Galactic coordinates. The black and green star markers respectively show the positions of the Sun and Sgr A* at the GC. The position markers of the HVSs are colour mapped according to heliocentric velocity vhel. HVSs with velocities above or below the boundaries of the colour scale take on the colour of the upper or lower bound, respectively.

63 8. Conclusion

We have developed a simulation model of the gravitational system of the Milky Way and Andromeda galaxies. Using this model, we have calculated the trajectories of 1.8 108 HVSs that were randomly generated near the centre of Andromeda and ejected in random· directions. Out of these HVSs, we have analysed all 521 HVSs that arrive within 13 kpc of the Sun at present time t0. They are distributed almost isotropically throughout the considered region around the Sun, but are, expectedly, slightly more concentrated around the GC due to its gravitational attraction. As we expected, the directions of the HVS velocity vectors show a high degree of concentration around the point in the sky that is opposite to Andromeda. We have similarly analysed a selection of 259 HVSs from the Gaia eDR3 catalogue within 13 kpc of the Sun. We have selected them based on their velocity and the accuracy of their measurement data. The distribution of Gaia HVS positions is very dense around the GC, but significantly higher on the opposite side of the GC from where the Sun is located. The rest of the considered region around the Sun is only sparsely populated with HVSs. We suspect the asymmetric distribution around the GC is due to selection bias in the selection process based on heliocentric velocities of the Gaia stars. We observe preferred directions for the velocity vectors of both the slowest Gaia HVSs in the selection as well as the fastest ones. The fast Gaia HVSs strongly tend to move in the opposite direction of the movement of the Sun in the Galactocentric rest frame and are mostly located in an arc around the GC. The slow HVSs have velocity directions that have a slight tendency to point towards the Southern Galactic hemisphere. If we only consider HVSs within 7 kpc of the Sun, we only observe a preference of velocity vectors to run close to parallel to the Galactic disc. However, the small sample size of this stricter selection prevents us from making meaningful statements. Both the position distributions and the velocity directions of the simulation and Gaia HVSs show limited correlation with each other. Notably, a significant fraction of the Gaia HVSs shares the direction of HVSs coming from Andromeda. However, they are located close to the GC in only a small part of the considered region. We observe a much smaller number of matching Gaia HVS trajectories in the other parts of the considered region around the Sun. Since the simulation results predict HVSs with matching trajectories in the entire region, the population close to the GC most likely has matching directions coincidentally. Due to the small number of the remaining matching trajectories, we cannot make meaningful statements about them. We conclude that we cannot dismiss the possibility of HVSs from Andromeda reaching the Milky Way. However, we can state, according to the results of our calculations, they do not account for a significant fraction of HVSs in the Milky Way. Based on these findings, we are unable to make meaningful statements about the possibility of HVSs being ejected from Andromeda by interactions with MPBHs. We note that even in the case we had observed a matching distribution of HVSs in the Milky Way, we would likely be unable to distinguish them from HVSs ejected by interactions with the SMBH in the centre of Andromeda. This is because we expect similar production of HVSs in both galaxies. In retrospect, it becomes clear that some aspects of the simulation and the analysis of

64 results have not been realised optimally. A more optimised selection process of the Gaia data could eliminate the selection bias we most likely observe in our selected data set. In addition, the exponential weighting function f(x) for the initial velocity magnitude is arbitrarily chosen. In hindsight, a physically motivated weighting function, for example a Maxwell-Boltzmann distribution, is a better choice. A more comprehensive simulation is called for to determine the amount of HVSs from Andromeda more accurately. It should include the production of HVSs in the Milky Way. We expect the component velocities of these HVSs to show behaviour similar to the component velocity distributions of the Gaia HVSs shown in Appendix B. The vz component is expected to show symmetry around zero while the vx and vy components might have more complex behaviour due to the Milky Way spiral arms and rotation. The HVSs produced in the Andromeda satellites should be included as well, assuming they are not negligible. An accurate simulation of both the Milky Way HVS production and the incoming HVSs from Andromeda could reproduce the distributions from measurement data. The amount of HVSs from Andromeda could be estimated by analysing the combined populations of Milky Way HVSs and Andromeda HVSs. This also requires a more accurate reference data set than the selection of Gaia data used in this thesis. Under these conditions, it might be possible to make more conclusive statements about PBHs ejecting HVSs from Andromeda.

Acknowledgements

I am grateful to Malcolm Fairbairn for supervising this project during my time at King’s College London and for the continued supervision and support after my return to Biele- feld University. I am also grateful to Dominik Schwarz for supervising this project and for organisational help regarding my stay in the UK. I am grateful to my parents for sup- porting me during my undergraduate and graduate studies as well as for enabling my stay abroad. I thank Ferdinand Junemann¨ and Marius Neumann for useful discussions and comments. I thank Marcel Rodekamp for his help with high performance computing. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/ dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This research made use of Astropy (http://www.astropy.org), a community- developed core Python package for Astronomy [44, 45], numpy [120] , scipy [43] and matplotlib [137]. The author gratefully acknowledges the funding of this project “hpc-prf-hypvel” by computing time provided by the Paderborn Center for Parallel Computing (PC2). I am grateful to the ERASMUS+ programme, Bielefeld University and King’s College London for enabling me to start this project during my stay in the UK.

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75 A. Coordinate transformations of the Gaia data

The data from the Gaia eDR3 catalogue is given in the equatorial coordinate system. It utilises the celestial sphere of the Earth. We use the AstroPy function Skycoord to transform the HVS positional data to Galactic coordinates. In order to obtain the distance to the Sun dS and velocity vectors ~vGaia of the measured HVSs, we use the given parallax P and radial velocity vrad as well as the proper motion velocities ωRA, ωDec in the direction of the equatorial coordinate components, right ascension RA and declination Dec. The parallax P in mili-arcseconds (mas) is simply converted to the distance to the Sun dS in kiloparsec (kpc) by taking the inverse. To ﬁnd the velocity vector of a given HVS in Galactic coordinates, we have to perform multiple coordinate transformations since we only have the previously mentioned measurement values. First, we determine the velocity vector of a HVS in equatorial Cartesian coordinates from the Gaia data using two coordinate rotations. This process is schematically displayed in Fig. 30. After these rotations and a transformation to spherical coordinates, we can again use the Skycoord function to convert the calculated velocity direction in equatorial coordinates to Galactic coordinates. The notation scheme in this section is unrelated to the scheme used in Section 6. 00 Initially, we construct the velocity vector ~v Gaia of the HVS in a Cartesian coordinate 00 00 00 frame (x , y , z ) that uses the position vector ~rGaia of the HVS in equatorial Cartesian coordinates as its x00-axis and the equatorial coordinate directions RA and Dec at the coordinates of the HVS as its y00- and z00-axes, respectively (see Fig. 30, top right). The 00 00 velocity component vGaia,x along the x -axis is simply the radial velocity of the HVS. We calculate the y00- and z00-components using the parallax and proper motion of the HVS with ω v00 = RA cos(Dec) C Gaia,y P · · ω (A.1) v00 = Dec C. Gaia,z P · 00 We need to multiply by cos(Dec) in vGaia,y since proper motion in the horizontal direction near the North pole of a spherical coordinate frame covers more real distance than near the equator of the coordinate frame in the same time frame. This is already included in the Gaia catalogue value of ωRA. C 4.744 is the constant needed for the conversion to km/s. ≈ To ﬁnd the velocity vector ~vGaia in the equatorial Cartesian frame (x, y, z) aligned with the standard right-handed equatorial coordinate frame, we rotate the coordinate frame 00 00 00 00 (x , y , z ) to match (x, y, z). This action is equivalent to rotating ~v Gaia with transposed T rotation matrices. We use the transpose Rz of the rotation matrix already introduced in 00 T Eq. (6.38) to rotate around the z -axis and the corresponding matrix Ry for the rotation around the y0-axis of the resulting intermediate coordinate frame (x0, y0, z0) (see Fig. 30,

76 z´´ y´´ x´´ Star

r Gaia q z´ y´ Sun x´ z EQ

x y f

celestial equator

Fig. 30. Schematic illustration of the transformation of the Gaia catalogue velocity data in hour angle equatorial coordinates. We transform the coordinate system 00 00 00 (x , y , z ) determined by the position vector ~rGaia of the HVS by rotating by θ = Dec around the y00-axis and by φ = RA around the z0-axis of the trans- itional coordinate system (x0, y0, z0). The z−0-axis is slightly tilted for visual clarity. Now it matches the coordinate system (x, y, z) at the position (0◦, 0◦), the vernal equinox (EQ). This coordinate system is aligned with the standard equatorial coordinate frame and gives us the velocity vector ~vGaia. middle right)

cos α 0 sin α  cos α sin α 0 T − T Ry =  0 1 0  and Rz =  sin α cos α 0 . (A.2) sin α 0 cos α − 0 0 1

We use the position coordinates ~rGaia = (RA, Dec) of the corresponding HVS as rotation angles. We rotate ﬁrst around the y-axis by the angle θ = Dec and subsequently around the z-axis by the angle φ = RA. We need to to use the negative of RA because the data is given in the hour angle− system which is the left-handed equivalent to the standard equatorial frame. The rotations are schematically displayed in Fig. 30. Now, we can simply perform the transformation from Cartesian to spherical coordinates (see Eq. (6.45)) to ﬁnd the point on the sphere in equatorial coordinates the velocity vector pointed at if it was placed in the coordinate origin. Applying Skycoord to it results in the velocity direction in the Galactic coordinate frame.

77 We use a second calculation to be sure the result is correct. We find the HVS velocity vector ~vGaia in equatorial Cartesian coordinates by directly calculating the first time derivation of the HVS position vector ~rGaia in the same frame. Using dS = 1/P , φ = RA and θ = π/2 Dec , we obtain ~r − Gaia     x dS sin θ cos φ · · ~rGaia = y = dS sin θ sin φ (A.3) · · z dS cos θ · We use θ = π/2 Dec to fit the standard definition of spherical coordinates with inclination from the− zenith, instead of declination from the equator which is used in the equatorial system. Since Dec is negative here, we give RA a positive sign due to the left-handed measurement data. d We find the first derivation by time of the position vector dt ~rGaia

 d d d  dt dS sin θ cos φ + dS dt θ cos θ cos φ dS dt φ sin θ sin φ d ~rGaia d d − d =  dt dS sin θ sin φ + dS dt θ cos θ sin φ + dS dt φ sin θ cos φ (A.4) dt d d dS cos θ dS θ sin θ dt − dt and we substitute d d = v dt S rad d θ = ω (A.5) dt − Dec d φ = ω dt RA to ﬁnd the velocity vector ~vGaia in the (x, y, z) coordinate frame   v sin θ cos φ dS ω cos θ cos φ dS ω sin θ sin φ rad − Dec − RA ~vGaia = vrad sin θ sin φ dS ωDec cos θ sin φ + dS ωRA sin θ cos φ (A.6) − vrad cos θ + dS ωDec sin θ. in terms of the distance dS and the proper motion velocities ωRA, ωDec. Both calculations of ~vGaia give the same result. We are now able plot the Gaia HVS positions and velocity directions in Galactic coordinates. In addition, we can transform the velocity vectors to Galactic Cartesian coordinates to perform the boost to the Galactocentric rest frame required for the selection process of the Gaia data described in Section 4.2.

78 B. Distribution of component velocities

This section shows the distributions of component velocities of both simulation and Gaia HVSs. All velocities are in the Galactic Cartesian coordinate system and in the heliocentric rest frame.

Simulation results Fig. 31 shows the distributions of the component velocities of the simulation HVSs. Two regions deﬁned by radii of 13 kpc and 7 kpc around the Sun are displayed in the left and right column, respectively. The former contains 521 HVSs while the latter contains only 69 HVSs. We observe the component velocities of the simulation HVSs to be completely asymmetric around zero. For both long and short distances, all three components are in agreement with trajectories coming from Andromeda which lies at negative x-, positive y- and negative z-coordinates below the Galactic plane at t0. The only exception is a small fraction of HVSs in the bottom left panel of Fig. 31 which have negative velocities in the z-direction. This is due to them already having crossed the Galactic plane and being accelerated down towards the GC afterwards. The distributions for simulation HVSs within 13 kpc show peaks at

vx 550 km/s ≈ vy 750 km/s (B.1) ≈ − vz 220 km/s. ≈ The HVSs within 7 kpc show peaks at similar values. However, due to the much smaller sample size, the distributions are signiﬁcantly less smooth.

Gaia data In contrast to the complete asymmetry of the simulation HVS component velocities, we look at the component velocities of the Gaia HVSs in Fig. 32. The HVSs show a slightly preferred negative direction in the vx component for both regarded radii. The vy components show an overwhelming majority of negative values for HVSs within the larger radius. The HVSs within 7 kpc of the Sun only show a slight tendency towards negative values. As discussed in Section 7.2, this degree of asymmetry is most likely due to to selection bias in the Gaia data set. Velocities vz along the z-axis are almost symmetric within 13 kpc, but show a slight tendency towards negative values for the HVSs in the smaller region. As we established in the discussion of Fig. 26, the near-symmetry of the vz component in the larger region is due to the population of faster HVSs around the GC. The HVSs on the slower end of the spectrum slightly prefer the negative z-direction which we see in Fig. 26. Due to the small sample size, the distributions of component velocities for the HVSs in the small radius are not sufﬁcient to make meaningful statements.

79 16.0% 14.0% 14.0% 12.0% 12.0% 10.0% 10.0% 8.0% 8.0% 6.0% 6.0%

4.0% Fraction of HVSs

Fraction of HVSs 4.0%

2.0% 2.0%

0.0% 0.0% 100 200 300 400 500 600 700 800 500 550 600 650 700 750 vx (radial) [km/s] vx (radial) [km/s]

17.5% 16.0%

15.0% 14.0%

12.0% 12.5% 10.0% 10.0% 8.0% 7.5% 6.0% 5.0% Fraction of HVSs

Fraction of HVSs 4.0%

2.5% 2.0%

0.0% 0.0% 1000 800 600 400 1000 950 900 850 800 750 700 650 600 vy (disc) [km/s] vy (disc) [km/s]

12.0% 12.0%

10.0% 10.0%

8.0% 8.0%

6.0% 6.0%

4.0% 4.0% Fraction of HVSs rcinof HVSs Fraction 2.0% 2.0%

0.0% 0.0% 200 0 200 400 600 100 200 300 400 vz (polar) [km/s] vz (polar) [km/s]

Fig. 31. Distribution of simulation HVS component velocities in Galactic Cartesian coordinates and in the heliocentric rest frame. The left column and right column respectively show the 521 HVSs within 13 kpc and the 69 HVSs within 7 kpc of the Sun. Top row: vx component. Middle row: vy component. Bottom row: vz component.

80 30.0% 10.0% 25.0% 8.0% 20.0%

6.0% 15.0%

4.0% 10.0% Fraction of HVSs Fraction of HVSs 2.0% 5.0%

0.0% 0.0% 600 400 200 0 200 400 600 600 400 200 0 200 400

vx (radial) [km/s] vx (radial) [km/s] 30.0%

20.0% 25.0%

20.0% 15.0%

15.0% 10.0% 10.0% Fraction of HVSs 5.0% Fraction of HVSs 5.0%

0.0% 0.0% 1250 1000 750 500 250 0 250 500 800 600 400 200 0 200 400

vy (rotation) [km/s] vy (rotation) [km/s] 22.5%

20.0% 8.0% 17.5%

15.0% 6.0% 12.5%

10.0% 4.0% 7.5% Fraction of HVSs Fraction of HVSs 2.0% 5.0%

2.5%

0.0% 0.0% 600 400 200 0 200 400 600 200 0 200 400 600 vz (polar) [km/s] vz (polar) [km/s]

Fig. 32. Distribution of Gaia HVS component velocities in Galactic Cartesian coordinates and in the heliocentric rest frame. The left column and right column respectively show the 259 HVSs within 13 kpc and the 14 HVSs within 7 kpc of the Sun. Top row: vx component. Middle row: vy component. Bottom row: vz component.

81 C. Simulation code

There are three files necessary for the calculation of HVS trajectories from Andromeda. The files are named M31-trajectory.py, RNG-initial-conditions.py and HVS-trajectory-simulation.py. The former two files produce data files that the latter file uses for the main calculation. This section briefly describes these three files and what they do. The files can be found at https://github.com/lguelzow/HV S-trajectories.

Calculation of the Andromeda trajectory The calculation of the trajectory of Andromeda in the last few billion years is done separately in the file M31-trajectory.py. We use the observational values for the present position and velocity of Andromeda from van der Marel et al. [129] as initial conditions. Since we use measurements of Andromeda at present time, the time interval for the calculation runs backwards from t0 = 13.8 Gyr to t = 10.0 Gyr to enable us to calculate the position and velocity of Andromeda in the past. Next, the differential equations are defined as a function. As discussed in Section 6.4, we use the system of ordinary differential equations given by Eq. (6.1). For the acceleration we use what we established in Eq. (6.29). The magnitude of the position vector of Andromeda is calculated every time the function is called since it is needed for the calculation of the acceleration. After the function is defined, ODEint is called and given the initial conditions, time interval and the function containing the DEs. ODEint calls the DE function in every step of the time interval. It uses integrates the velocity and acceleration components calculated from the position and velocity vector from the previous step to find the position and velocity vector at the current time step. For the first step, it integrates the values calculated from the initial vectors. At the end, ODEint returns an array containing the position and velocity vectors of Andromeda at every step of the time interval. The results are written into the data file M31-trajectory.txt. The main file of the simulation uses this data file to interpolate functions of the position and velocity of Andromeda depending on time.

Generation of random initial conditions In addition to the trajectory of Andromeda, we also generate the initial conditions separately in the file RNG-initial-conditions.py. At first, we establish the amount of initial conditions we generate. This determines how long the resulting data file is and how many trajectories are calculated in the simulation. At the same time, we set the constants and parameters we need for the generation of the HVSs initial conditions. Namely, these are the parameters used in the Plummer profile (see Eq. (6.8)) and the minimum and maximum values of the initial velocity. Given these parameters, we generate random position and velocity vectors with the steps detailed in Section 6.5. Since we generate them in spherical coordinates, we subsequently transform them to Cartesian coordinates. In this form, the initial conditions are written into the data file HVS-initial-conditions.txt.

82 The main simulation file calls this file to read the HVS initial conditions and adds the position and velocity of Andromeda to find the initial conditions in the simulation coordinate systems.

Simulation of HVS trajectories The main file of the simulation is HVS-trajectory-simulation.py. At the beginning, we establish the amount of time steps in the calculation, the constants discussed in Section 6.3 as well as the vectors, rotation matrices and angles discussed in Section 6.6. Further, we create the data files the results are written into at the end of the file. In the next section we call on the previously generated data file M31-trajectory.txt for the position and velocity of Andromeda over the last few billions years. We use this data to define time dependent functions for each component of the position and velocity vectors of Andromeda. Following this, the data file HVS-initial-conditions.txt containing the initial conditions for the HVSs is called and read to determine the amount of HVS trajectories that are calculated. The for-loop that entails solving the differential equations (see Eq. (6.1)) is repeated for each trajectory. At first, we define the time interval for the ODEint function according to the initial send-off time of the HVS. Subsequently, we calculate the HVS initial position and velocity vectors in the simulation coordinate system by adding the position and velocity of Andromeda at send-off time to the initial conditions read from the initial conditions file. Now, we define the differential equations as a function. This is done similarly as previously described for the trajectory of Andromeda. We use Eq. (6.11) for the total acceleration of a HVS in the gravitational potential of both galaxies. The HVS position vectors with respect to the centre of each galaxy are calculated every time the functions is called. They are necessary to calculate the respective acceleration of each galaxy. As before, ODEint calls the DE function in every time step and finds the position and velocity vectors by integration. After running through the time interval, ODEint returns the HVS trajectory as a table containing the time, position vector components and velocity vector components in each time step. The distance of the HVSs to the Milky Way centre as well as the Sun at t0 is used to select trajectories that pass through the Milky Way and close to the Sun. The selection process is discussed at the beginning of Section 7. After the first steps of the selection process in the simulation coordinate frame, we transform the position and velocity vectors of the remaining trajectories at t0 to Galactic coordinates in the heliocentric rest frame with the transformation established in Section 6.6. The remaining steps of the selection process are performed to find the HVSs that come within 13 kpc of the Sun. Finally, we write the the position and velocity vectors of the final HVS selection into the data file smallest-min-distances.txt. Further, we record secondary results into two additional data files. The velocity data of the HVS selection within 50 kpc of the GC is written into MW-centre-vel.txt to analyse their velocity distribution in comparison to the final HVS selection. Further, we write the distance to the Sun at t0 of every calculated trajectory into distances-to-sun.txt in order to be able to find the initial conditions of a specific HVS trajectory.