Hunting dark matter with faint radio halos Michael Sarkis Thesis Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in Physics, in the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa Supervisor: Dr Geoff Beck 26 October, 2023 Abstract The nature of Dark Matter (DM), the elusive substance that constitutes a significant amount of the total matter in the universe, remains an unsolved problem in modern physics despite a decades-long search effort. One approach to this problem has been to search for faint emission signatures that are produced indirectly from the DM present in large astrophysical structures, and thus infer properties about theoretical DM models from observational data. In recent years, the results from studies that use this type of indirect search have produced stringent constraints on the most popular DM particle candidate parameter spaces, ruling out swathes of viable DM models. These compelling results have been enabled by the arrival of sophisticated interferometric radio telescopes, which are excellent DM hunters due to their high sensitivity and resolution. In this thesis, we focus on the use of the latest data from the MeerKAT radio interferometry telescope, through the first public release of the MeerKAT Galaxy Cluster Legacy Survey, to search for DM emissions in a set of nearby galaxy clusters. Each step of this process, from the creation of theoretical DM emission models to the statistical analysis of the observational data, has been described in detail in this thesis. With this data, we find an almost universal improvement to results found with corresponding modelling scenarios in the literature. Since this work is among the first to use MeerKAT data in astrophysical DM searches, these results present a strong argument for continued work in this field. Another central focus of this thesis is the accurate modelling of the physical processes involved in the production of the DM-induced radio emissions, as the quality of current radio data requires theoretical models that are sufficiently accurate to describe the emission at such high resolutions. One aspect of the modelling that has lacked this accuracy has been the solution to the diffusion-loss equation, which is a crucial factor in determining indirect DM emissions. A new algorithm for solving this equation, which provides higher accuracy and computational efficiency than previous public methods, has thus been developed and presented in this thesis. These aspects of DM indirect detection study will become ever more important as we approach the inauguration of the Square Kilometre Array (SKA), which will provide data with unprecedented potential with which to continue the hunt for DM. i Declaration I declare that this thesis is my own, unaided work. It is being submitted for the Degree of Doctor of Philosophy at the University of the Witwatersrand, Johannesburg. It has not been submitted before for any degree or examination at any other University. Michael Sarkis 26 October 2023 ii Declaration - Publications The following are details of publications that include research presented in this thesis. The candidate undersigned declares that following data is correct and accurate. • Beck, G. and Sarkis, M. (2023a). “Galaxy Clusters in High Definition: A DarkMatter Search”. In: Physical Review D 107.2, p. 023006. doi: 10.1103/PhysRevD.107.023006. arXiv: 2210.00796 • Sarkis, M., Beck, G., and Lavis, N. (2023b). “A Radio-Frequency Search for WIMPs in RXC J0225.1-2928”. In: Letters in High Energy Physics 2023. doi: 10.31526/LHEP.2023.361. arXiv: 2303.00684 • Sarkis, M. and Beck, G. (2023a). “Simulating the Radio Emissions of Dark Matter for New High-Resolution Observations with MeerKAT”. in: The Proceedings of SAIP2022. The 66th Annual Conference of the South African Institute of Physics. UJ, ed. by Prof A. Prinsloo, pp. 408–413. doi: 10.48550/ARXIV.2301.03326 • Sarkis, M. and Beck, G. (2022). “Diffusing Assumptions in Astroparticle Physics”. In: The Proceedings of SAIP2021. The 65thAnnual Conference of the SouthAfrican Institute of Physics. UJ, ed. by Prof A. Prinsloo, pp. 316–321. url: http://saip.org.za/Proceedings/ Track%20D/51.pdf Michael Sarkis 26 October 2023 iii https://doi.org/10.1103/PhysRevD.107.023006 https://arxiv.org/abs/2210.00796 https://doi.org/10.31526/LHEP.2023.361 https://arxiv.org/abs/2303.00684 https://doi.org/10.48550/ARXIV.2301.03326 http://saip.org.za/Proceedings/Track%20D/51.pdf http://saip.org.za/Proceedings/Track%20D/51.pdf Acknowledgements I would firstly like to acknowledge and thank my supervisor, Dr Geoff Beck. The excellent advice that was freely given to me throughout my years as a postgrad, and the help and support during the the countless times I needed it, have been invaluable. He conceptualised this project, provided access to code and software that were necessary to find some of its results, and guided it towards its ultimate completion. He also provided financial support to attend several conferences and to cover the tuition fees of the final year of the the PhD, for which I am very grateful. I would also like to thank Natasha Lavis, for working with me to find results under pressure and especially for helpingme understand aspects of the observational data fromMeerKAT, and Dr Justine Tarrant for all the advice and motivation that helped me handle the ins and outs of PhD life. Then, I would like to acknowledgemy dear friends and family, who have helpedme throughout this PhD in one way or another. Particularly, to my family, Peter, Jennifer, Luke and Joshua, and to Cara, Tanita, Vladimir and Imaan – this work would not have been possible without your constant love and support. Thank you. This work makes use of MGCLS data products, which were provided by the South African Radio Astronomy Observatory and the MGCLS team and were derived from observations with the MeerKAT radio telescope. The MeerKAT telescope is operated by the South African Radio Astronomy Observatory, which is a facility of the National Research Foundation, an agency of the Department of Science and Innovation. This work is based on the research that was supported by the National Research Foundation of South Africa (Bursary No. 112332). iv Contents Glossary vii 1 Introduction 1 2 The Dark Matter Problem 5 2.1 Background review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Are WIMPs the solution? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Current status of the hunt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Indirect Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Physical Modelling 18 3.1 Dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.1 Particle models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.2 Halo models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Electron propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.1 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.2 Energy Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2.3 Magnetic fields and gas densities . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Observable emissions from DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.1 Synchrotron emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.2 Monochromatic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Solving the Diffusion-Loss Equation 36 4.1 Overview of solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 The Crank-Nicolson scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.3 Algorithm details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3.1 Symmetries and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3.2 Block matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3.3 Sparse matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3.4 𝛼-Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3.5 Initial and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3.6 Physical Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.3.7 Stability and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.4 Method comparisons and testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.4.1 Accuracy and benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.4.2 Comparisons to semi-analytical method . . . . . . . . . . . . . . . . . . . . . 52 4.4.3 RX-DMFIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5 Observational Data 63 5.1 Data from MGCLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2.1 Selection of RoI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2.2 Source-subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.3 Model Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.3.1 Position of DM halo centre . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.3.2 Model reprojection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 v 6 Dark Matter Hunting 76 6.1 Statistical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.1.1 Likelihood ratio tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.1.2 Model exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.2 Cross-section exclusion limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7 Conclusion 89 References 92 vi Glossary Common abbreviations Abbreviation Description ASKAP Australian SKA Pathfinder (radio interferometry telescope) CLT Central Limit Theorem (statistical theorem) DM Dark Matter DMID Dark Matter Indirect Detection dSph dwarf Spheroidal galaxy FITS Flexible Image Transport System (data file format) gNFW Generalised NFW (DM halo density profile) IC Inverse Compton scattering (particle interaction) LHC Large Hadron Collider (particle collider) MCXC Meta-Catalogue of X-ray-detected Clusters (observational catalogue) MGCLS MeerKAT Galaxy Cluster Legacy Survey (observational programme) MW Milky Way NFW Navarro-Frenk-White (DM halo density profile) OS Operator Splitting (numerical technique) RMS Root Mean Square (typically as an error estimate) RoI Region of Interest (subset of a data image) SKA Square Kilometre Array (radio interferometry telescope) SM Standard Model (of particle physics) WIMP Weakly Interacting Massive Particle (DM candidate) 2HDM+S Two-Higgs Doublet Model (SM extension, DM candidate) ΛCDM Λ + Cold DM (standard model of cosmology) Prominent software packages or tools Package URL PyBDSF https://github.com/lofar-astron/PyBDSF PPPC4DMID http://www.marcocirelli.net/PPPC4DMID.html RX-DMFIT https://github.com/alex-mcdaniel/RX-DMFIT astropy.reproject https://github.com/astropy/reproject SAOImageDS9 https://sites.google.com/cfa.harvard.edu/saoimageds9 WebPlotDigitizer https://automeris.io/WebPlotDigitizer/ LATEX-Cookbook https://github.com/alexpovel/latex-cookbook vii https://github.com/lofar-astron/PyBDSF http://www.marcocirelli.net/PPPC4DMID.html https://github.com/alex-mcdaniel/RX-DMFIT https://github.com/astropy/reproject https://sites.google.com/cfa.harvard.edu/saoimageds9 https://automeris.io/WebPlotDigitizer/ https://github.com/alexpovel/latex-cookbook CHAPTER 1 Introduction In the current standardmodel of cosmology, over a quarter of the energy density in the universe exists in the form of some type of Dark Matter (DM) (Planck Collaboration et al. 2020). Although there is strong evidence for its existence, including numerous observations of galaxy cluster dynamics, galactic rotation curves, anisotropies in the Cosmic Microwave Background (CMB) and gravitational lensing, as well as implications from the current layout of large-scale structure in the universe, its physical nature remains unknown (see for instance the excellent reviews in Bergstrom (2000), Bertone et al. (2005, sec. 2), and Bertone et al. (2018)). This evidence has come primarily from the gravitational effects of DM (Rubin 2004; Bertone et al. 2005; Clowe et al. 2006; Springel et al. 2006; Dietrich et al. 2012; Planck Collaboration et al. 2020), as it does not seem to directly interact with our usual astrophysical messengers. The inherently weak interactions of DM have thus made any kind of detection difficult, with more sophisticated and precise detection schemes being developed at an increasing pace, always to miss out on an unambiguous detection signal. Further suggestions of the existence of DM come from deficiencies within the Standard Model (SM) of particle physics, which is currently the most well-tested physical theory in the history of science. Several extensions to the SM that account for these deficiencies also predict the existence of particles which could constitute DM, and so when the much-anticipated solution to the DM problem does arrive, it is likely to bring new physics with it. Given the significant implications of this DM problem, there has been a global search effort dedicated to its solution for several decades, with the first studies on this topic dating back to the early 1930s (Bertone et al. 2018). Modern searches for DM are taking place in large interna- tional collaborations, like in the Large Hadron Collider (LHC) (Kahlhoefer 2017), in enormous underground detectors like XENONnT (XENON Collaboration et al. 2023), or with sophisticated astronomical observatories like the Fermi-LAT space telescope (Ackermann et al. 2017). These examples highlight the feats of modern technology and engineering that have been utilised in the search for DM, and give some indication of the true scale of this problem. As the nature of these detection schemes are also complementary to each other, with each probing different parameter spaces of potential DM candidates, we have cast a very wide net into the dark side of the universe. The most common candidate for DM considered in the literature is the generic Weakly Inter- acting Massive Particle (WIMP), which conforms to many of the most stringent requirements of a DM candidate set by observation, while also appearing in several models that extend the SM. These particles are neutral and weakly-interacting, and it is usually assumed that they formed a thermal relic population in the early universe (Steigman et al. 2012), which has resulted in the current abundance we observe today. They are often considered to be collisionless and cold, i.e. non-relativistic, which is a scenario implied by the evolution of large-scale structure in the universe (Springel et al. 2006; Kuhlen et al. 2012) and forms a core component of the ΛCDM model of cosmology. In many studies, WIMPs are treated independently of specific underlying particle physics models, possessing the bare qualities needed to satisfy DM candidature. While we do consider model-independent WIMPs in this thesis, we also consider a particular particle model that was recently proposed in von Buddenbrock et al. (2015) and von Buddenbrock et al. (2016), which is a variation of a two-Higgs-Double model, and is referred to as the 2HDM+S. This is an extension to the SM that was proposed to explain anomalous lepton production in the data gathered from Run 1 and Run 2 of the CMS and ATLAS experiments in the LHC (The CMS Collaboration et al. 2008; Aad et al. 2008), and the model posits the existence of a DM candidate particle, also outside the SM. The existence of WIMPs can be probed through several independent physical mechanisms, which are possible in all the example detection schemes referred to above. The first method relies on the production of WIMPs via particle collisions in large particle accelerators, and is referred to as collider detection. The second method attempts to detect the signatures of collisions between Page 1 of 104 1. Introduction M. Sarkis, PhD Thesis WIMPs and SM nuclei in large underground detectors, and is referred to as direct detection. The final method, relevant to this thesis, searches for emissions produced by WIMPs that reside in astrophysical structures through the use of telescope observations, and is referred to as indirect detection (ID/DMID). This detection scheme is enabled by the expectation that WIMPs have the ability to self-annihilate, which would lead to the production of a set of kinematically-accessible annihilation products within the SM, either directly or through some mediator that couples to the SM. There is the additional possibility that WIMPs have a finite lifetime and decay into SM particles, again producing a set of SM products. These are crucial mechanisms, as they could lead to a method for observing DM signatures indirectly, through the SM products of WIMP-WIMP interactions. As such, a primary goal of ID studies is to probe the annihilation cross-section parameter, which describes the amount of annihilation that occurs and is thus a direct factor in the emission produced by a region of WIMPs. The field of DMID has matured substantially over the past couple of decades, and now in- cludes comprehensive studies of multi-wavelength and multi-messenger emissions from DM that encompass the range of cosmic-ray antimatter to radio frequencies (for reviews, see Bertone (2010, part V), Cirelli (2013), and Gaskins (2016)). The most common form of DMID study in the literature focuses on the prompt gamma-ray emission from WIMP annihilations. Study of this messenger has been driven in popularity by several factors, including a set of observations with the Fermi-LAT gamma-ray telescope (Fermi/LAT Collaboration et al. 2009) and the fact that the calculation of the expected signal on Earth is relatively simple, since the gamma rays are not deflected by ambient environments like magnetic fields. Other messengers, like neutrinos or anti-protons, have also gained popularity with the operation of new, sensitive detectors like IceCube (Abbasi et al. 2012; Abbasi et al. 2023) or the Alpha Magnetic Spectrometer (AMS-02) on the International Space Station (Barao 2004; Calore et al. 2022). The particular focus of this thesis is however on radio frequency ID, which has a long history originating in the 1990s (Berezinsky et al. 1992) and in its modern form has produced some of the most advanced ID results, consisting of relatively strong constraints on the WIMP annihilation cross-section (Regis et al. 2021; Egorov 2022; Beck et al. 2023a). The stringent constraints from these studies are a consequence of the latest generation of radio interferometry observatories (Thompson et al. 2017), which are comprised of arrays of single radio dishes that correlate incoming signals, allowing for baselines on the order of tens to hundreds of kilometres. This sophisticated method of determining the incoming flux of radio emission can result in extremely high sensitivity and spatial resolution, which are both crucial aspects of DM hunting. The Square Kilometre Array (SKA), a multinational radio observatory with a planned ob- serving area of a square kilometre, will be the largest radio interferometry device built to date, and will be the flagship radio telescope of its generation (Dewdney et al. 2009). While we await its inauguration, precursor instruments like the South African MeerKAT telescope (Jonas et al. 2018) and the Australian SKA Pathfinder (ASKAP) (Johnston et al. 2008) have already had their first-light observations. There is now consensus in the literature that the SKA and its precursors have excellent potential to further advance the DMID search effort (Colafrancesco et al. 2015a; Colafrancesco et al. 2015b; Beck et al. 2017; Storm et al. 2017; Beck 2019b; Chen et al. 2021). In particular, the high resolution of these instruments is important for distinguishing the emission from different regions of DM density and for disentangling any compact source emissions from diffuse fluxes, and the high sensitivity is necessary to observe the relatively faint DM signal; factors which both lend to a better overall search potential compared to previous or single dish instruments. Indications of this potential have already begun to be realised with initial studies that use data from precursor observation programmes, through the ASKAP Evolutionary Map of the Universe (EMU) survey (Norris et al. 2021) and the MeerKAT Galaxy Cluster Legacy Survey (MGCLS) (Knowles et al. 2022), respectively (Regis et al. 2021; Sarkis et al. 2023b). In this thesis we focus on the MGCLS in particular, which had its first public data release (DR1) on the 11th November 2021. This survey consists of 115 galaxy clusters, positioned between 0° and −85° declination and at a median redshift of 0.14, with a total observing time of ∼ 1000 hours. The use of galaxy clusters as test-beds for DM study has a rich history, with some of the very first evidence for DM coming from observations of the Coma galaxy cluster (Zwicky 1933; Andernach et al. 2017), and more recent studies of the Coma cluster being seminal works in the field of radio-based DMID (Colafrancesco et al. 2001; Colafrancesco et al. 2006). Since then, Page 2 of 104 1. Introduction M. Sarkis, PhD Thesis there have also been a number of searches for specific models of DM, or studies of the general morphology of the radio emissions from galaxy clusters (Colafrancesco et al. 2006; Colafrancesco et al. 2011; Marchegiani et al. 2018; Marchegiani et al. 2019), and studies searching for constraints on the annihilation cross-section (Storm et al. 2013; Storm et al. 2017; Kiew et al. 2017; Chan et al. 2020; Chan et al. 2022; Beck et al. 2023a). Despite the above-mentioned sources, the use of galaxy clusters in radio DMID is still generally a less popular choice in the literature compared to small satellite galaxies of the MW (particularly those referred to as dwarf spheroidal galaxies (dSphs), see the review in Beck (2019b) for instance). This is primarily because of the presence of large background emissions from non-DM sources, often in the form of large diffuse radio halos, that make isolating the DM signal from other sources difficult. Since dSphs are nearby objects with masses dominated by DM, and contain low amounts of stars and gas (implying relatively low background emission from baryonic sources and thus easier isolation of pure DM emission), they are attractive targets for DM hunting. However, dSphs possess several sources of uncertainty, such as their internal magnetic fields and WIMP density distributions, which as we will see later in this thesis, play an important role in the determination of the radio emissions from WIMP annihilation products. These physical environments are significantly more constrained in galaxy clusters, while still being DM-dominated. Further, the selection of galaxy clusters that lack bright radio halos as search targets can mitigate (to some degree) the afore-mentioned drawback of high background baryonic emissions. These points then generally allow for search results that are based on less uncertainty than those found using dSphs, and suggest that they are deserving of further detailed investigation. A crucial factor in the reliability and accuracy of DMID studies is themodelling of the predicted emissions. In radio studies, this modelling process is a complicated one. The scenario we consider in this thesis is the creation of electrons and positrons (referred to as just electrons) as final- state products of WIMP annihilations, which due to the large masses of allowed WIMP models, are highly-relativistic. Being charged, these electrons are also subject to many interactions with the ambient astrophysical environment, and the interaction with magnetic fields is specifically what leads to the emission of radio waves, through the process of synchrotron radiation. This synchrotron radiation is the ultimate target of DMID searches, and its precise determination relies on accuratemodels of the spatial number density and energy distribution of the produced electrons. The evolution of these properties is usually described by a general cosmic-ray propagation equation (for example in Strong et al. (1998, app. eq. 1) and Bertone (2010, eq. 27.2)), though in the astrophysical environments considered here, it is sufficient to consider a truncated form of this equation which encapsulates the dominant effects of spatial diffusion and energy losses – a form generally referred to as the diffusion-loss equation. A recurring theme of this thesis is a focus on the accuracy of the above modelling scenario. The diffusion-loss equation is not trivial to solve, which has led to the development of several mathematical techniques that have various degrees of accuracy and ease of use. The most common technique employed in the literature uses a semi-analytical approach, popularised byColafrancesco et al. (2006) and based on the method shown in Baltz et al. (1998) (see also Vollmann (2021) for a variation on this method), and while directly producing a solution to the equation, it requires the approximation that all involved physical quantities are spatially independent. This requirement is typically satisfied by substituting the spatially-dependant quantities, like magnetic fields and thermal gas densities, with scalar averages. This loss of spatial information is the primary motivation for the use of numerical techniques, which are based on finite-differencing schemes and used extensively in the field of galactic cosmic-ray physics (Strong et al. 1998; Porter et al. 2022). Solutions found with the numerical method are more accurate and computationally efficient than their semi-analytical counterparts, however these benefits come at the cost of a significantly more complicated implementation. This has caused a relatively low adoption of this technique in the literature, with the only uses of it by the authors Regis et al. (2015b), Regis et al. (2017), and Regis et al. (2021) with a proprietary implementation, and Egorov (2022) with a modified version1 of the popular GALPROP package (Strong et al. 1998). The practical and theoretical aspects of each solution method discussed above are investigated in detail in this thesis. As the accuracy of the electron distribution calculated with the diffusion- 1Online repository Github: https://github.com/a-e-egorov/GALPROP_DM Page 3 of 104 https://github.com/a-e-egorov/GALPROP_DM 1. Introduction M. Sarkis, PhD Thesis loss equation directly impacts the expected DM-induced synchrotron radiation, it is a salient factor for DMID studies performed with high-resolution observational data. The anticipation of radio interferometry data has motivated the development of a numerical solution method that is accurate and computationally efficient, while also being accessible and easy to integrate into a general radio-based DMID framework. While the two instances of the use of numerical methods given above do exist, the current implementations are either completely inaccessible or based on code that was designed specifically for galactic cosmic-ray propagation, which is not fully aligned with general DMID studies (particularly with galaxy cluster search targets). The numerical solution method developed and presented in this work is expected to be released, together with all necessary computational tools for modelling DM emission, in a public release sometime in the near future. The overall structure of the rest of the thesis is as follows. In Chapter 2 we provide a brief outline of the DM problem. The main focus of this Chapter is a review of the current status of the radio-based DMID field, though it also contains a summary of ID schemes using other astrophysical messengers. We also give a cursory description of some historical aspects to the DM problem, briefly summarising the most prominent pieces of evidence and prevailing DM candidate models. Chapter 3 then provides a detailed account of the physical modelling scenarios that are used to predict the radio emissions from WIMP DM. This Chapter contains the bulk of the theoretical aspects involved in the work, which includes the modelling of WIMP annihilations and DM halos, the relevant physical effects involved in the propagation of electrons (particularly the various aspects of diffusion and energy loss), and the calculation of synchrotron emissions from a population of high energy electrons. Chapter 4 is dedicated to the analysis of the diffusion-loss equation. It contains a review of solution methods that are commonly used in the literature, which is followed by a rigorous presentation of a computational algorithm, developed in this project, to solve this equation. It also includes a set of comparisons between results found with this algorithm and semi-analytical methods, with a comprehensive discussion around the use and merits of each method. Given the rigorous nature of this Chapter, a brief summary of all pertinent points has also been provided. In Chapter 5 we provide a showcase of the observational data obtained by the MGCLS pro- gramme, and discuss in detail the relevant data processing techniques that were used to prepare the radio images for analysis. We also discuss the preparation of DM emission models, found by the methods in the previous Chapters, for comparisons to the data. Chapter 6 presents the main results of this work, in terms of the constraints found for the WIMP annihilation cross-section based on the data from the MGCLS prepared in the previous chapter. It also contains a detailed explanation of the statistical analysis that was employed to find the constraints. A comprehensive discussion on these results is provided at the end of the Chapter, which also contains a comparison to comparable work in the literature. The thesis is then finally concluded with a brief summary of the main outcomes of this work in Chapter 7. Page 4 of 104 CHAPTER 2 The Dark Matter Problem The problem of DM has a long and rich history of research, which has allowed it to grow into a global collaborative and interdisciplinary field. Finding the solution to the DM problem is now one of the most pressing priorities in modern physics, and it is likely to have far-reaching implications in fields from particle physics to cosmology. The aim of this Chapter is to review the overall status of the field, with a particular focus on the DM paradigm that is used in this thesis. As such, the main focus will be on reviewing WIMPs and the various indirect detection schemes that are used in the attempt to detect them. As the field of DM research is vast, for other DM models and phenomenology not used here we have provided sources to excellent works in the literature for further reading. The precise physical modelling of the various aspects of DM detection will be described in Chapters 3, 4 and 5. 2.1 Background review Before the modern understanding of the DM problem, there were a number of astronomers and mathematicians in the 19th and 20th centuries that observed behaviour which appeared anomalous to their understanding of the celestial bodies (Bertone et al. 2018, cha. II). Indeed, if there was an indication of a massive object that was undetectable in space, the object was often referred to as ‘dark matter’. However, these astronomical phenomena were often attributed to known objects that were just too dark to be observed, like dark stars or planets. In the early 1930s, the Swiss astronomer Fritz Zwicky produced a now-famous piece of work (Zwicky 1933; Andernach et al. 2017), which showed a large scatter in the line-of-sight velocities of galaxies within the Coma galaxy cluster, as measured by their redshift. When the virial theorem was applied to this cluster, it was found that the mass inferred from the mass-to-luminosity ratio was only 2% of the total calculated mass, signifying a significant amount of unaccounted-for dark matter. This is typically considered as the beginning of the modern DM problem in the literature, though it should be noted that there was a significant contribution to the field at this time from a host of other scientists, including Sinclair Smith, Jan Oort and Edwin Hubble, for example (Bertone et al. 2018). Another prominent piece of historical literature is based on the measurement of the stellar rotation curves of galaxies. A particularly famous piece of work was authored by Vera Rubin and Kent Ford Jr in 1970 (Rubin et al. 1970) for the rotation curve of the Andromeda Galaxy, with further notable works that studied more galaxies coming in the following years (Rubin et al. 1978; Rubin 1983). The conclusion from these studies was that the orbital velocity of stars far from the central (and luminous) bulge of the galaxy were independent of radius, resulting in a ‘flat’ curve at higher radius. Since Newtonian gravity predicts a decrease in the stellar orbital velocity at large radii, there was again implication for some unseen contribution to the total mass of the galaxies. By this time, the consensus in the astronomical community was that there had to be some explicit component of missing matter, which was now based on a large body of work from analyses of astronomical observations (Bertone et al. 2018). At roughly the same time as the above works, the theoretical effects of gravitational lensing were being discovered. The ability for matter to act as a lens for background photons, within the framework of the theory of General Relativity, provided another probe for finding potentially non-luminous matter. Since then, there have been a number of modern studies that, through observations of gravitational lensing, suggest a significant amount of DM in galaxies and galaxy clusters (Bergmann et al. 1990; Becker et al. 2004; collaboration et al. 2000; Tisserand et al. 2007). A result of observations of weak-lensing in the merging of two galaxy clusters 1E0657-558 published in 2006 (Clowe et al. 2006), known colloquially as the ‘Bullet Cluster’, is often cited as more strong evidence for the existence of DM in large amounts. In this cluster merger, the X-ray emitting Page 5 of 104 2. The Dark Matter Problem M. Sarkis, PhD Thesis intracluster gas – which traces the dominant baryonic component of the cluster – has become separated from the centre of mass of each cluster determined through gravitational lensing, due to ram pressure of the intracluster gas (see Figure 2.1). This spatial separation was measured at a significance level of 8𝜎, and suggests that the centre of mass is following a large, unseen contributor to the gravitational potential. These observations are explained well by the presence of collisionless DM in the cluster, and the DM interpretation of this phenomenon is widely accepted in the literature. Figure 2.1: The famous composite image of the ‘Bullet Cluster’ (1E0657-558) galaxy cluster merger, taken from the Chandra photo album1. This composite shows the diffuse central X- ray emission (pink) overlaid on the optical image which shows the distinct member galaxies (white/orange) of the merging clusters, and an estimate of the mass densities of the clusters found from lensing measurements (blue). As the X-ray emission traces the baryonic content of the region, the separation between the pink and blue regions in this image indicates the presence of a large, unseen gravitational component, which is typically attributed to DM. 1https://chandra.harvard.edu/photo/2006/1e0657/ Some of the strongest evidence for DM has come from observations on a cosmological scale, which possess the additional feature of being able to constrain the total amount of DM the universe, compared to observations of localised targets. These observations have been enabled by the extraordinary precision of modern instruments, in particular the Planck space telescope (Planck Collaboration et al. 2020), which superseded the Cosmic Background Explorer (COBE) (Bennett et al. 1993), WilkinsonMicrowave Anisotropy Probe (WMAP) (Spergel et al. 2003). Measurements of the temperature anisotropies present in the Cosmic Microwave Background (CMB) performed with the above instruments have shown that the amplitude of primordial fluctuations, which relate to the early state of the universe, are at the scale of a factor of 10−5. By analysing the power spectrum obtained from the CMB anisotropies, a set of constraints on the cosmological parameters that govern the evolution of the universe can be obtained. From the Planck 2018 results (Planck Collaboration et al. 2020) for example, the total matter density parameter was determined to be Ω𝑚ℎ2 = 0.1432 ± 0.0013 (where ℎ is the usual Hubble parameter and the factor Ω represents the fraction of total matter in the universe). However, the baryonic density parameter was determined to be Ω𝑏ℎ2 = 0.02236 ± 0.00015. This discrepancy is a clear indication for the existence of a matter component that is non-baryonic. Further, the ‘missing’ component needs to account for ∼ 84% of all matter in the universe. As the results from these experiments have found no significant evidence for cosmological parameters that vary from the ΛCDM model of cosmology, the existence of cold DM in our universe is therefore well-established. Another aspect of DM research, the theoretical simulation of the effects of DM through large 𝑁-body simulations, has helped to define our understanding of DM on a huge variety of physical scales. These simulations are ever-improving alongside the development of computer technology, Page 6 of 104 https://chandra.harvard.edu/photo/2006/1e0657/ 2. The Dark Matter Problem M. Sarkis, PhD Thesis and modern simulations have been performed with trillions of particles in simulation volumes that span cubic Gpc scales. In particular, simulations of the large-scale evolution of structure in the universe have shown that the inclusion of cold DM models more accurately represents the observations from large galaxy surveys, such as the Sloan Digital Sky Survey (SDSS) (York et al. 2000) (or more recently those performed with the Baryon Oscillation Spectroscopic Survey (BOSS) (Tröster et al. 2020)), compared to simulations with the absence of DM (Springel et al. 2006; Garrett et al. 2011; Kuhlen et al. 2012). These results provide extremely useful information when modelling DM in various astrophysical scenarios, and help drive our understanding of DM in a complementary way to the other astrophysical pieces of evidence we have discussed. When combining all pieces of modern observational evidence, the existence of some kind of non-baryonic DM in large quantities in the universe is strongly supported. There have thus been a number of proposed particle models that have the necessary physical traits to account for these effects, each with various theoretical motivations. For a comprehensive review of the proposed models, we refer the reader to the studies by Bertone et al. (2005, cha. 3), Bertone (2010, part II), Feng (2010), Garrett et al. (2011), and Bertone et al. (2018, cha. V). As the focus of the work in this thesis is on a particular candidate particle known generally as a Weakly Interacting Particle (WIMP), we continue by focusing our discussion on this candidate. 2.2 Are WIMPs the solution? By far the most popular candidate particle model for solving the DM problem in the literature is the WIMP. This term could technically refer to any number of specific particle models, as it only defines a set of phenomenological aspects necessary to account for the observational evidence of DM. Of these, it must (i) be massive enough to explain observations of gravitational effects, (ii) interact weakly with regular matter (and without electromagnetic interactions), to explain the lack of detected photons from DM, and (iii) have been created in a sufficient number so that after thermal freeze-out their abundance can account for the current observed abundance of DM. These points are usually supplemented by a further requirement on the average velocity of the particles, with non-relativistic particles being referred to as ‘cold’, relativistic particles as ‘hot’, and those with intermediate velocities as ‘warm’. The current layout of large-scale structure in the universe (Springel et al. 2006) suggests that massive structures formed through a bottom- up process, which entails small objects collapsing first and then gradually merging with other collapsed objects to create complex structures. This scenario is only possible with cold WIMPs, as highly-relativistic particles would be able to stream further than the size of small-scale structures, prohibiting their growth. There are a variety of specific particle models that conform to the above WIMP requirements. Themost commonmodels considered in the literature are usuallymembers of extensions to the SM, particularly in the supersymmetric extensions like the Minimal Supersymmetric Standard Model (MSSM) (such as neutralinos, gravitinos, sneutrinos, axinos) (Bertone et al. 2018). However, more exotic models like Kaluza-Klein particles that are predicted in models that contain extra spatial dimensions, are also viable. (Feng 2010). In many studies in the literature, and in this thesis, WIMPs are considered as model-independent generic particles that have the required properties. For example, the DM cookbook (Cirelli et al. 2011) which is popular in the literature has computed a number of particle characteristics, like annihilation and decay pathways, for completely model-independent WIMPs. The reason that generic WIMPs are considered favourites in the pool of candidate particles is due to theoretical motivations, and in particular a deficiency in the Standard Model (SM) of particle physics known as the Gauge Hierarchy Problem (Feng 2010). This problem is sometimes framed in terms of the discrepancy between the mass of the Higgs Boson, at 125 GeV, and the expected mass of dimensionful parameters in the case of electroweak symmetry breaking, which is the Planck mass of ∼ 1 × 1019 GeV. A solution to this problem is the existence of particles at the weak scale, i.e. 10 GeV – 10 TeV, which also happen to fit the characteristics of the described WIMPs from astrophysical evidence. In the simple case of WIMPs being created thermally after the Big Bang, the number density (𝑛) can be described by the Boltzmann equation (Feng 2010, Page 7 of 104 2. The Dark Matter Problem M. Sarkis, PhD Thesis eqn. 5) 𝑑𝑛 𝑑𝑡 = −3𝐻𝑛 − ⟨𝜎𝑣⟩(𝑛2 − 𝑛2 eq) , (2.1) where 𝐻 is theHubble constant, 𝑛eq is the thermal equilibrium number density ofWIMPs, and ⟨𝜎𝑣⟩ represents the thermally-averaged cross-section for WIMP annihilations. In the early primordial environment, the production of WIMPs through interactions of SM particles would be equalled by their own annihilations, so that WIMPs would be in thermal equilibrium with all other particle species. However, the expansion of the universe would cause the average temperature in this environment to drop below the WIMPs mass, which would suppress the creation of new WIMPs through SM interactions. At this stage, the number density of WIMPs in the expanded universe is low enough that their annihilations effectively cease, creating the constant density that has persisted until the current epoch. The annihilation cross-section is an important parameter in this process, and as it determines the rate of WIMP annihilations, it is the primary parameter of interest in indirect detection studies (discussed in the following Section). 2.3 Current status of the hunt The attempt to detect a WIMP has been a huge scientific endeavour, and the current experimental landscape can be divided into three distinct classes. The first of these, and the most direct, attempts to search for indications of the scattering of a WIMP with a target nucleus. The primary parameter of interest in these studies is therefore the WIMP-nucleon interaction cross-section. The detectors that are sensitive enough to detect the low energy signatures of these collisions are usually embedded in large quantities of material that maximise any potential signals, such as liquid Xenon. They are also commonly built in underground facilities to reduce the background signals from non-DM scattering events. Some examples of state-of-the-art direct detection experiments are the PandaX-4T(Meng et al. 2021), LUX-ZEPLIN (LZ) (Aalbers et al. 2022), XENON1T(Aprile et al. 2018) or the recently expanded version XENONnT (XENON Collaboration et al. 2023), and a set of upper limits on the WIMP-nucleon cross-section determined in XENON Collaboration et al. (2023) have been displayed in Figure 2.2a. The results from the latest set of experiments are all relatively close, with the strongest limits coming from WIMP masses ≳ 10GeV, with a steep increase in the limits below this value. The second of these search techniques is known as collider detection, and relies on the creation of WIMPs in particle collider experiments through the scattering of highly energetic particles. Even though the detectors within these experiments would be unable to detect the created WIMPs, the creation process would result in missing energy in the detectors, which could be analysed to implicate a WIMP signature. There have been a number of WIMP searches at the LHC (see Kahlhoefer (2017) for a review), which also attempt to determine the WIMP-nucleon cross-section. In Figure 2.2b we show a set of recent upper limits on the WIMP-nucleon cross-section determined by ATLAS Collaboration (2022), using proton-proton collisions with the LHC and detected by the ATLAS experiment. The value of complementary detection schemes can be seen directly here, with the collider experiment providing strong limits, almost competitive with direct detection results, in the WIMP mass range that is only weakly probed by direct detection (i.e. ≲ 10GeV). 2.3.1 Indirect Detection We now shift our focus to the detection scheme that is relevant for our work – ID. For reviews that contain excellent accounts of the various aspects this field, see Bertone (2010, part V), Cirelli (2013), and Gaskins (2016). The common thread between all DMID schemes is the search for DM that resides in astrophysical structures, and produces some kind of observable emission. In the case of WIMP DM, this emission is generated either as a result of WIMP-WIMP annihilations or decay of individual WIMPs that produce a set of SM particles. The detection, or so far non-detection, of these products then allows us to infer properties about the underlying DM models, which can act as useful complementary datasets to the alternative detection schemes and particle parameters already discussed. As the emission from these processes is stronger in regions where there is a Page 8 of 104 2. The Dark Matter Problem M. Sarkis, PhD Thesis (a) (b) Figure 2.2: Examples of state-of-the-art results from direct and collider detection experiments, showing sets of upper limits (at 90% confidence level) on viable spin-independent WIMP-nucleon cross-sections. (a): Upper limits from studies of various direct detection experiments (see text for details). Figure taken from latest (2023) XENONnT results (XENON Collaboration et al. 2023, fig. A.4). (b): Upper limits found with proton-proton collisions at the LHC, detected by the ATLAS experiment. Also shown are some recent limits from direct detection searches, for comparison. Figure taken from latest (2022) ATLAS results (ATLAS Collaboration 2022, fig. 7). larger abundance of WIMPs, astrophysical structures that are dominated by DM or contain dense regions of DM are generally favourable as search targets. Another commonality in all ID schemes is that the defining parameter of the searches, the annihilation cross-section, is currently constrained by the thermal relic WIMP population that is required to reproduce the current observed cosmological abundance of DM (Steigman et al. 2012). The precise calculation of this cross-section in Steigman et al. (2012) results in a value of ⟨𝜎𝑣⟩ ≈ 2.2 × 10−26 cm3 s−1 in the regions of WIMP mass that are generally considered in the literature, which is slightly – but importantly – different from the previously popular trend of using a canonical value of ⟨𝜎𝑣⟩ ∼ 3.0 × 10−26 cm3 s−1. This small value of the cross-section further motivates the use of search targets that contain high densities of DM, as low densities simply interact too infrequently to produce the necessary SM products for an observable signal on Earth. Since the expected emissions from WIMP annihilations or decay are directly proportional to this cross-section, it is often treated as a free parameter in ID studies. This treatment results in a set of curves within the ⟨𝜎𝑣⟩ − 𝑀𝜒 parameter space (where 𝑀𝜒 is the mass of the WIMP), which define the upper limit on ⟨𝜎𝑣⟩ which is required to produce emission compatible with observational data. These curves can then be compared to the precise value of the thermal relic ⟨𝜎𝑣⟩ value calculated in Steigman et al. (2012), and models that overpredict this value (based on some statistical confidence level) would be considered unable to produce the total current cosmological abundance of DM (this process will be described in further detail in Chapter 6). As alluded to in Chapter 1, the field of DMID study has matured into a strong global search effort, spanning multiple wavelengths and astrophysical messengers. The most popular of these schemes in recent years searches for the prompt gamma-ray photons originating from WIMP annihilations. This popularity was driven by a wealth of gamma-ray data from observatories like the Fermi-LAT space telescope (Fermi/LAT Collaboration et al. 2009), and Imaging Atmospheric Cherenkov Telescopes (IACTs) such as H.E.S.S., MAGIC and VERITAS (H. E. S. S. collaboration et al. 2006; Sitarek et al. 2013; Park 2016), and the water Cherenkov array HAWC (Abeysekara et al. 2017). However, the use of gamma rays also has a major benefit, in that the expected emission from WIMP annihilations can be determined relatively simply compared to other wavelengths. The simplicity of gamma-ray emission is a result of the neutral high-energy photons being able to propagate directly from the source of the emission to the detectors, without complicated interme- diary physical interactions with environments like magnetic fields. This allows the calculation of the expected emission to be neatly split into two factors that contain independent information Page 9 of 104 2. The Dark Matter Problem M. Sarkis, PhD Thesis relevant to the WIMP particle models and the astrophysical environment. The separation of these two factors has allowed for simple estimates of the expected emission from various astrophysical targets in what are commonly referred to as 𝐽-factors, without the need for specifying information about the particle model, which can make gamma-ray ID studies easily accessible for analysis and comparison. In gamma-ray ID studies, there are two principle astrophysical search targets. The first is the Galactic Centre (GC), which has had arguably the most attention of any DMID study in the literature. In particular, studies showed that observations of the region with the first two years of the Fermi space telescope were underspecified by models of astrophysical gamma-ray backgrounds, but could be compatible with an extra component of annihilating DM (Goodenough et al. 2009; Hooper et al. 2011). This led to a significant effort to characterise and analyse these observations in a DM context, represented by the following list of noteworthy papers (Calore et al. 2015b; Gordon et al. 2013; Calore et al. 2015a; Daylan et al. 2016; Ackermann et al. 2017; Karwin et al. 2017; Cholis et al. 2022). As shown in the work by Di Mauro (2021) and Cholis et al. (2022) for example, interest in this region is still strong. The main setback and cause for debate in this discovery has been the astrophysical background associated with the GC, and the accurate modelling that is needed to match the characteristics of the excess signal. The study by Di Mauro (2021) has placed a set of new constraints on various physical parameters and DM modelling scenarios for the DM interpretation of the excess using 11 years of Fermi-LAT data, although conclude that even more accurate modelling of the GC region is needed to improve the results. In Cholis et al. (2022), some new modelling techniques for this GC background emission were employed, using cosmic-ray data from the AMS-02 (Barao 2004) on board the International Space Station and the Voyager 1 probe (Stone et al. 1977). The results of this detailed analysis confirm previous indications that the known abundance of millisecond pulsars around the GC are unable to independently explain the signal detected by the Fermi telescope (Zhong et al. 2020), and provide a set of best-fit DM parameters that constitute a DM interpretation of this signal. This result is shown in Figure 2.3a, which contains the best-fit contours for 1,2 and 3𝜎 limits of the DM interpretation, corresponding to a model of (𝑀𝜒 ≈ 40GeV, ⟨𝜎𝑣⟩ ≈ 1.4 × 10−26 cm3 s−2) and annihilation into gamma-ray photons through the bottom quark channel. The other prominent astrophysical search target in gamma-ray ID studies is of the satellite dSph galaxies of the MW (see Hooper et al. (2015) or Fermi-LAT Collaboration (2015) for an overview). The motivation for these sources are that they are extremely DM-dominated and contain little quantities of baryonic matter, which makes disentangling DM-induced gamma- ray signals from background sources much easier. There have been a multitude of studies that investigated gamma-ray emissions from these dSphs with instruments like Fermi-LAT (see Hoof et al. (2020) for a comprehensive list of references). In this thesis we draw particular attention to two recent studies: Hoof et al. (2020) and Armand et al. (2021), which are both noteworthy for their completeness compared to previous studies – they combine datasets for a number of different dSphs (27 and 20, respectively), and in the case of Armand et al. (2021), 5 different gamma-ray observatories (with the caveat that the published results are preliminary). The combination of datasets used in these works provide a robust analysis, which also leads to strong limits on the annihilation cross-section. However, it should be noted that recent work by Ando et al. (2020) has shown the use of updated priors on the DM halo density in dSphs, motivated by physical models of structure formation, leads to annihilation cross-section constraints that are up to a factor of 7 times weaker than when using the priors previously considered in the literature. An example set of results for DM annihilations through the bottom quark channel from Hoof et al. (2020) (the more detailed of the two publications) is displayed in Figure 2.3b, which also contain some comparisons to previous limits in the literature. In general, the trend that emerges when considering the full set of gamma-ray ID result is that lower massWIMPs, with the exact values dependant on the annihilation channel, are incompatible with the value of the thermal relic cross-section. The preferred parameter space from analysis of the GC excess is in tension with ⟨𝜎𝑣⟩ upper limits from dSphs, though some uncertainty regarding the modelling of the DM density in these objects could resolve this tension. Due to the points mentioned before regarding the benefits of gamma-ray searches, and the sheer amount of analysis available in the literature, the constraints on the WIMP parameter space from these studies are strong, especially compared to other messengers like neutrinos or antiprotons. This can be seen in Page 10 of 104 2. The Dark Matter Problem M. Sarkis, PhD Thesis (a) (b) Figure 2.3: Examples of current state-of-the-art results from gamma-ray ID studies. (a): The preferred parameter space for WIMP annihilations into bottom quarks, given a DM interpretation of the excess GC gamma-ray signal. The solid, dashed and dotted lines correspond to 1,2 and 3𝜎 contours, and the three colours correspond to different regions of interest considered in the data. The considered regions were a full 40° × 40° area around the GC, and then that area confined to only the northern and southern hemispheres centred on the GC. Although the excess signal was found to be present in all regions, the energy spectrum of the excess signal differs between the north and south, which leads to the tension in the best fit values seen in the Figure. Figure taken from Cholis et al. (2022, fig. 18). (b): A set of upper limits, calculated using a profile likelihood technique, for WIMP annihilations into bottom quarks. The star represents the best-fit parameters, and several curves representing results from the literature have been included (shown on the figure). Figure taken from Hoof et al. (2020, fig. 9). Figure 2.4, in which a few noteworthy results have been collated and compared to othermessengers (see below). In this plot the results from the combined analysis of dSphs (Hoof et al. 2020; Armand et al. 2021) seem to produce the strongest constraints on the cross-section when compared to the thermal relic value, ruling-out WIMP masses below ≲ 100 − 200GeV. Note that for the sake of clarity, most of the uncertainty bounds in these results have not been displayed (with the more constraining cases selected here), and since there are many results for varying parameters (like analysis techniques, confidence levels and annihilation channels), we have chosen only compatible results as far as possible. Figure 2.4 is thus for comparative purposes and the observation of general trends only. The use of other astrophysical messengers in ID studies, like neutrinos and cosmic rays, has seen significant advancement in the past several years, primarily due to the improving capabil- ities of detectors like Super-Kamiokande (SK) (Walter 2008), IceCube (Abbasi et al. 2012) and ANTARES (Margiotta 2009), or the AMS-02 instrument (Barao 2004). In the case of neutrino-based ID, the neutrinos produced during the WIMP annihilation process (similarly to gamma-rays) propagate directly towards the Earth without significant interactions due to the intermediate environment. A popular source target in the literature for these searches is the DM halo within the MW itself, with a particular focus on the GC due to the increased density of WIMPs (Mijakowski 2020). However, there is also interest in the detection of neutrinos via the process of WIMP capture by astrophysical bodies such as the Sun or the Earth. In this process, WIMPs become bound in the gravitational potential wells of large bodies when they lose energy after scattering with resident nuclei within these objects, accumulating until their density is high enough to annihilate in sufficient numbers. Unlike other SM annihilation products, neutrinos would be able to escape these environments due to their own weakly-interacting nature, which would allow them to be detected on Earth (Mijakowski 2020). There have been several neutrino-based searches forWIMPs, most prominently at the SK, IceCube and ANTARES detectors (some large and noteworthy studies include Aartsen et al. (2017), Albert et al. (2020), and Abe et al. (2020)). Another interesting Page 11 of 104 2. The Dark Matter Problem M. Sarkis, PhD Thesis 101 102 103 104 (Mχ /GeV) 10−30 10−28 10−26 10−24 10−22 (〈σ v 〉/ cm 3 s− 1 ) Thermal relic Hoof+ 2020 Armand+ 2021 Ackermann+ 2021 Abdallah+ 2021 Cholis+ 2021 Abbasi+ 2023 Calore+ 2022 Figure 2.4: A non-exhaustive list of several noteworthy ID studies, and the constraints on the annihilation cross-section they have produced. The results shown here are limited to non-radio- based studies (particularly gamma-ray shown by the solid lines, neutrino by the connected scatter plot and antiproton by the dot-dashed line) that investigate the bottom quark annihilation channel, with the exception of those from Abbasi et al. (2023) which show a neutrino channel. The value of the thermal relic cross-section, calculated in Steigman et al. (2012), is shown as a dotted black line. For a corresponding summary of constraints from radio studies, see Figure 2.6. For information about each of the publications shown here, see Table 2.1 or the text. detection scenario is possible with neutrinos, in that a monochromatic flux is expected from certain annihilation processes from cold WIMPs (see Abbasi et al. (2023) for details). As this monochromatic flux is not expected from any other known astrophysical processes, its detection would be a so-called ‘smoking gun’ detection, unambiguously implying a DM interpretation. With all detection scenarios considered however, the limits from neutrino-based ID are significantly weaker than other messengers. This can be seen in Figure 2.4, where we have plotted the current strongest limits from Abbasi et al. (2023, fig. 9) (note that we have used the results for the 𝜈𝑒𝜈𝑒 channel, as the results from the 𝑏𝑏 channel, shown in all other results, are roughly two orders of magnitude weaker). A summary of the limits obtained from a variety of neutrino experiments can be found in Figure 2.5a. The last of the major astrophysical messengers discussed in this thesis outside radio ID studies is of cosmic rays, with the bulk of studies in the field typically investigating electrons/positrons and protons/antiprotons. Cosmic-ray ID is usually performedwith space-bound particle detectors, like the afore-mentioned AMS-02 instrument and the specialised Dark Matter Particle Explorer (DAMPE) instrument (Chang 2014). The use of cosmic rays as astrophysical messengers is made more challenging than gamma rays or neutrinos by the fact that the charged particles are highly susceptible to interactions with ambient media, which makes the accurate modelling of their propagation difficult (see Chapter 4 for a discussion of relevant effects). Additionally, the modelling of background or primary sources of cosmic rays can have significant uncertainty, which makes the disentanglement of a pure dark matter signal from the background challenging, especially when using instruments with poor resolution or high systematic uncertainty (Heisig 2021). Nonetheless, there has been keen interest in this field in the literature, with the recent history of cosmic-ray ID searches being dominated by claims of excess detections whichwere accompanied Page 12 of 104 2. The Dark Matter Problem M. Sarkis, PhD Thesis (a) (b) Figure 2.5: Current state-of-the-art results from multi-messenger ID studies, showing results from neutrinos, antiproton, gamma-ray and radio observations. (a): Upper limits on ⟨𝜎𝑣⟩ at a 90% confidence level for a number of neutrino-based ID studies, using data from the SK, IceCube and ANTARES detectors. Figure taken from Abe et al. (2020, fig. 8). (b): Upper limits on ⟨𝜎𝑣⟩ at a 95% confidence level using antiproton data from the AMS-02 instrument. Also shown are results from gamma-ray searches of the GC and a radio search of the LMC. Figure taken from Calore et al. (2022, fig. 3). by DM interpretations. Two notable examples are of the excess antiproton ( ̅𝑝) signal discovered by the AMS-02 instrument (Aguilar et al. 2016) and the ∼ 1.4TeV excess flux of electrons/positrons discovered with the DAMPE instrument Ambrosi et al. (2017), with examples of the posited DM interpretations coming from Cholis et al. (2019) and Yuan et al. (2017), Yang et al. (2017), and Fan et al. (2018), respectively (also see Beck (2019a) for an analysis of these claims, specifically in the context of multi-wavelength DMID, or Cirelli (2013) for phenomenological aspects). In the case of the ̅𝑝 excess, recent work in Calore et al. (2022) (based originally on earlier work in Giesen et al. (2015)) has shown that with updated modelling techniques and data analysis, the DM interpretation of this excess is unnecessary. This work has also produced a set of limits that are competitive with other ID messengers, seen in Figures 2.4 and 2.5b. Radio searches We now focus on the DMID scheme relevant to this work, which is based on the radio emission from WIMP annihilations. Unlike the messengers described above, the observable radio waves are not a result of prompt emission from WIMP annihilations, instead being radiated from phys- ical interactions of the charged annihilation products. In ID studies the products are typically considered to be electrons and positrons, as these are produced more abundantly than hadronic final states in WIMP annihilations (Colafrancesco et al. 2006; Cirelli et al. 2011). In this case, the emission from the high-energy electrons is dominated by synchrotron radiation (Bertone 2010, cha. 27) which is produced when the electrons interact with a magnetic field. We have discussed all relevant physical interactions and characteristics of this process, from the creation of electrons by WIMP annihilations to the emission of synchrotron radiation, in detail in Chapters 3 and 4. Some notable early works in the field of radio ID include Baltz et al. (1998) and Baltz et al. (2004), which were the first studies to provide a detailed account of the diffusion of the electrons produced byWIMP annihilations in theMWand its satellite galaxies, and their resulting emissions. Later, the studies by Colafrancesco et al. (2001) and Colafrancesco et al. (2006) expanded upon this work by creating a detailed picture of the potential multi-frequency nature of emissions from DM annihilations in the context of galaxy clusters. In particular, Colafrancesco et al. (2006) became a standard reference in radio-based ID studies, given the detailed mathematical description of Page 13 of 104 2. The Dark Matter Problem M. Sarkis, PhD Thesis the radio emissions produced by annihilating WIMPs. An important factor in this work was the prescription for a semi-analytical solution to the electron propagation equation, commonly known as the diffusion-loss equation, which was necessary to model the spatial and energy distribution of the electrons and ultimately determine the synchrotron flux emitted from them. The presented solution method became the basis for a vast array of radio ID studies, and will be discussed in more detail later in the thesis. Alongside this solution method was an analysis of the primary physical effects involved in the equation, namely diffusion and energy loss, which showed the general behaviour and characteristics of each in various astrophysical scenarios. Since then, the interest in radio-based DMID has grown substantially, with numerous studies being performed on different targets and with data from different observatories. Some recent review articles like Beck (2019b) and Chan (2021) contain comprehensive analyses of the literature, and in this section we focus on a few of the most recent and noteworthy results from various targets and instruments. Because of the significant gain in spatial resolution and sensitivity when using an array of radio dishes through interferometry (Thompson et al. 2017; Beck 2019b), the strongest results on the WIMP annihilation parameter space have come from array telescopes. The prominent observatories used in radio ID studies are the LOw Frequency ARray (LOFAR) (van Haarlem et al. 2013), the Australia Compact Telescope Array (ATCA) (Wilson et al. 2011), the Karl G. Jansky Very Large Array (JVLA) (Napier et al. 1983; Lane et al. 2014), and most recently the SKA precursor instruments MeerKAT (Jonas et al. 2018) and the Australia SKA Pathfinder (ASKAP) (Johnston et al. 2008); although observations from single dish instruments like the Green Bank Telescope (GBT) (Prestage et al. 2009), Giant Meterwave Radio Telescope (GMRT) (Swarup et al. 1991) or the Effelsberg telescope (Hachenberg et al. 1973) are also sometimes used. In terms of observational targets, the most commonly studied objects have been the satellite dSphs of the MW, which benefit from the same characteristics discussed above in the context of gamma rays. The first noteworthy set of results to be obtained from dSphs were presented in Colafrancesco et al. (2007) for the Draco dSph (using VLA data and projections for LOFAR, but also considered through a multi-wavelength approach). The constraints found in this work were relatively weak, with even the most optimistic projections failing to probe the thermal relic cross- section. Further analysis of dSphs came from Spekkens et al. (2013) and Natarajan et al. (2013) using the GBT and VLA instruments. Although these results suffer from significant uncertainties in the modelling of the magnetic field, diffusive environment and DM halo within the considered targets (a common feature in searches of dSph targets), the thermal relic value was probed in some favourable modelling scenarios, for example in the case of annihilation through the 𝑒−𝑒+ channel and strong magnetic fields. Further searches in six dSphs: Carina, Fornax, Sculptor, Hercules Segue 2 and Bootes II, were then carried out in the series of papers Regis et al. (2014), Regis et al. (2015a), and Regis et al. (2015b), with the limits on ⟨𝜎𝑣⟩ presented in Regis et al. (2014, figs. 4-5). The shown limits again vary drastically between modelling scenarios, with the optimistic scenario strongly ruling out the entire WIMP parameter space, while the pessimistic scenario not getting close to probing the thermal relic value of ⟨𝜎𝑣⟩. The average modelling scenario is able to reach the thermal relic value for a small range of WIMP masses (∼ 20 − 40GeV) when annihilating through the electron channel, but not at all for the 𝑏𝑏 channel. These limits, in the case of an average modelling scenario, were improved by roughly an order of magnitude when the Reticulum II dSph was used as a search target, again with the use of ATCA data in (Regis et al. 2017). The results found in Regis et al. (2014) and Regis et al. (2017) are of particular relevance to this thesis, since they were the first to employ a numerical approach in the solution to the diffusion-loss equation, as opposed to the semi-analytical method popularised by Colafrancesco et al. (2006). This approach offers an improvement to the accuracy of the solution in terms of the spatial dependance of the electron population, and all aspects surrounding this choice of solution method will be discussed in detail in Chapter 4. The recent study on the Canes Venatici I dSph in Vollmann et al. (2020) is noteworthy, as, like in Regis et al. (2014) and Regis et al. (2017), it forewent the usual solution method of the diffusion- loss equation. Instead, it employed a solution technique based on approximations of the diffusion environment in various ‘regimes’ that ultimately result in a universal profile for the expected synchrotron emissions in dSphs. The mathematical description of this method was presented in detail in a follow-up paper (Vollmann 2021), and is summarised in this thesis in Chapter 4 (it should also be noted that in Vollmann (2021) a correction was made to the commonly-used Page 14 of 104 2. The Dark Matter Problem M. Sarkis, PhD Thesis semi-analytical method in Colafrancesco et al. (2006)). Again, uncertainties in the modelling of diffusion parameters lead the results in Vollmann et al. (2020) to vary significantly, with the average case displayed in Figure 2.6. The only other published set of results in the literature that use the ‘universal profile’ solution method are from Gajović et al. (2023), which used data from the LOFAR Two-metre Sky Survey (LoTSS) (Shimwell et al. 2022) to obtain upper limits on six dSphs, namely: Canes Venatici I, Ursa Major I, Ursa Major II, Ursa Minor, Willman I, and Canes Venatici II. Although the average modelling scenario presented therein results in limits that probe the ⟨𝜎𝑣⟩ at low masses (with similar characteristics to the results in Vollmann et al. (2020)), the modelling of the DM halos in this work was done with a cuspy DM density profile. Cuspy density profiles are currently disfavoured for the modelling of dSphs, based on the analysis of simulations which instead suggest these halos have flatter central densities (see Chapter 3 or de Blok (2010)). 101 102 103 104 (Mχ /GeV) 10−30 10−28 10−26 10−24 10−22 (〈σ v 〉/ cm 3 s− 1 ) Thermal relic Vollmann+ 2020 Regis+ 2017 Egorov 2022 Kiew+ 2017 Regis+ 2021 Figure 2.6: A non-exhaustive list of several noteworthy ID studies, and the constraints on the annihilation cross-section they have produced. The results shown here are limited to radio-based studies that investigate the bottom quark annihilation channel. The value of the thermal relic cross-section, calculated in Steigman et al. (2012), is shown as a dotted black line. Similar to Figure 2.4. For information about each of the publications shown here, see Table 2.1 or the text. Although less common, notable results have come from search targets other than dSphs. The first of these is Egorov (2022), which presented updated limits on ⟨𝜎𝑣⟩ from the M31 galaxy from an earlier work (Egorov et al. 2013). In this work, a variety of radio datasets were used, including survey data from the LoTSS and the VLA Low-frequency Sky Survey Redux (Lane et al. 2014), as well as data from the Effelsberg telescope and the VLA. This work was also the first to solve the diffusion-loss equation in two spatial dimensions, which required a detail modelling of the magnetic field in the spiral galaxy, and was carried out with a numerical implementation of the solution to the diffusion-loss equation. This implementation involves the adaptation of the GALPROP galactic cosmic-ray propagation solver code package, which is discussed in detail in Chapter 4. We also note that an error in the calculation of the energy loss rate due to bremsstrahlung radiation, common in many previous studies, was corrected in this paper (discussed further in Section 3.2.2). The resulting limits from this analysis produce strong constraints on the ⟨𝜎𝑣⟩, which are plotted in Figures 2.7a and 2.7 and rule out the thermal relic cross-section for WIMP masses ≲ 70GeV for annihilation through the 𝑏𝑏 channel and the median modelling scenario. Another notable result has been presented in Regis et al. (2021), which was the first study Page 15 of 104 2. The Dark Matter Problem M. Sarkis, PhD Thesis (a) (b) Figure 2.7: Current state-of-the-art results from radio-frequency ID studies, showing 95% con- fidence levels on the upper limits of ⟨𝜎𝑣⟩ when annihilation occurs through the bottom quark channel. (a): Upper limits found from various radio observations of the M31 galaxy. The uncer- tainty bands represent variations in the considered modelling scenario. Also shown are select results from other studies. Figure taken from Egorov (2022, fig. 10). (b): Upper limits found from observations of the LMC performed by the EMU survey, with the ASKAP instrument. Uncertainty bands represent results found using different DM halo density profiles. Also shown are select results from other studies. Figure taken from Regis et al. (2021, fig. 4). to use data from an SKA precursor instrument, ASKAP, in an ID search of the Large Magellanic Cloud (LMC). This data was obtained through the Evolutionary Map of the Universe (EMU) survey programme (Norris et al. 2021), and has resulted in the strongest constraints on the thermal relic cross-section yet obtained, for any DMID study. The upper limits on ⟨𝜎𝑣⟩ from this work, which even in the least constraining case rule out the thermal relic cross-section for WIMP masses ≲ 480GeV, are shown in Figures 2.7b and 2.7. The large improvement in the constraining power of these results were attributed to the more sensitive radio observations from ASKAP compared to ATCA, as well as an improved statistical analysis. The statistical analysis adopted in Regis et al. (2021), which entails a detailed comparison of two-dimensional image data to the model, is similar to the one used in this thesis, which will be discussed in detail in Chapter 6. The final observational target used in radio-based ID is of galaxy clusters, which have been comparatively less common in recent literature (outside our own work) (Storm et al. 2013; Storm et al. 2017; Kiew et al. 2017; Chan et al. 2020). Of these studies, those in Storm et al. (2013) and Kiew et al. (2017) have considered a set of galaxy cluster targets (11 and 6, respectively) and used a variety of archival radio datasets and upper limits on the fluxes observed in radio halos to find upper limits on the annihilation cross-section. The resulting upper limits on ⟨𝜎𝑣⟩ obtained in these studies are noticeably weaker than other observation targets, with the slightly stronger limits from Kiew et al. (2017, fig. 3) shown in Figure 2.6. In Chan et al. (2020), a search using radio data from the GMRT of the Abell 4038 cluster (Kale et al. 2012) is made, wherein they find a best-fit value of 𝑀𝜒 = 40GeV for annihilation through bottom quarks. A feature of the results in Storm et al. (2013) and Chan et al. (2020) is that the effect of substructure in the DM halo (Zavala et al. 2019) significantly improves the limits; however with the simplified modelling scenarios used for the DM halo density (especially in Chan et al. (2020)) and the lack of observational evidence for the substructure models considered, this could be considered as an extra source of uncertainty in the results. Finally, the study performed in Storm et al. (2017) produces a set of predicted upper limits on ⟨𝜎𝑣⟩ based on the projected instrument capabilities of several telescope arrays, namely: ASKAP (through the EMU programme), LOFAR and the APERTIF array (Oosterloo et al. 2009), for a set of generic galaxy cluster models. The upper limit results probe the thermal relic annihilation cross-section at some WIMP masses for most of the considered annihilation channels, but are not as constraining as the results from M31 or the LMC (Egorov 2022; Regis et al. 2021). We have discussed the results from the above sources, in the context of the constraints found on Page 16 of 104 2. The Dark Matter Problem M. Sarkis, PhD Thesis the cross-section in this work, in Chapter 6. The overall picture of radio-based ID studies is therefore a promising one, given the stringent constraints on the annihilation cross-section obtainedwith several recent studies. However, there is also a prevailing trend of large uncertainty in these results, due either to underspecified modelling scenarios of the diffusive environment or a lack of sophistication in the modelling of DM halos. With the availability of high-quality interferometric radio data from SKA precursors like MeerKAT and ASKAP, and eventually from the SKA itself, accurate solutions to the diffusion-loss equation that are supplemented by accuratemodels of the diffusive environmentwill be especially important factors to consider for reliable results. Reference Observational target Instruments Messenger Kiew et al. (2017) Galaxy Clusters Misc. Radio Regis et al. (2017) Reticulum II dSph ATCA Radio Vollmann et al. (2020) Canes Venatici I dSph LOFAR (LoTSS DR2) Radio Egorov (2022) M31 LOFAR, VLA, Effelsberg Radio Regis et al. (2021) LMC ASKAP (EMU) Radio Abdallah et al. (2016) GC H.E.S.S. Gamma-ray Ackermann et al. (2017) GC Fermi-LAT Gamma-ray Hoof et al. (2020) 27 dSphs Fermi-LAT Gamma-ray Armand et al. (2021) 20 dSphs Misc. (see text) Gamma-ray Cholis et al. (2022) GC (best fit) Fermi-LAT Gamma-ray Abbasi et al. (2023) MW IceCube (DeepCore) Neutrinos Calore et al. (2022) MW AMS-02 Antiprotons Table 2.1: Description of the highlighted DMID studies used in Figures 2.4 and 2.6. Page 17 of 104 CHAPTER 3 Physical Modelling A core component of indirect detection studies is the correct prediction of the emissions that are produced by DM interactions. This typically involves choosing a particular DM candidate particle model, which determines the interaction rate and output spectra of individual particles, and a halo model, which describes the overall structure and distribution of virialised clumps of DM. For some indirect detection studies (those that investigate only the prompt emission of gamma rays, for example), the above factors can be sufficient to determine the expected emission, if the products of the DM interactions do not interact with ambient astrophysical environments. The modelling of radio emission, however, involves various physical processes that affect the DM products and can entail additional calculations of magnetic fields, thermal gas populations, particle diffusion and energy losses. In the following Chapter, we present the complete radio emission modelling process used in this project, from the DM particle candidates to the expected observable emission here on Earth. However, the solution of the diffusion-loss equation, which is also a prominent part of the modelling process, requires a more comprehensive description and is the sole topic of Chapter 4. 3.1 Dark matter 3.1.1 Particle models In the vast sea of viable, theoretically-motivated DM particle models, many of the most pop- ular candidates have been studied extensively by large interdisciplinary teams of particle and astrophysicists. The phenomenology of these models has thus been captured in software that is available to the community, making the modelling of otherwise complicated particle interactions and processes relatively easy to perform. Notable software packages in this respect (among others) are the micrOMEGAs (Belanger et al. 2002) and DarkSUSY (Gondolo et al. 2004) packages (the latter of which is also used by the RX-DMFIT package, which is relevant to this work and will be discussed in Detail in the next Chapter). An alternative to these computational solutions, which has become popular for DM indirect detection studies in recent years, is the Poor Particle Physicist Cookbook for Dark Matter Indirect Detection (PPPC4DMID) (Cirelli et al. 2011). This cookbook provides pre-computed numerical tables of particle spectra (ingredients), functions for computing the propagation of charged annihilation products through the galaxy (recipes), and even results of flux calculations that are ready for consumption. All of the relevant tables and results are hosted in an online repository1, which is where the relevant material for this project has been obtained. The aspect of the PPPC4DMIDwe aremost interested in here is the energy spectrum of particles produced during DM self-annihilations (which are denoted hereafter as DM-DM or 𝜒𝜒). The form of this spectrum is usually taken to be d𝑁𝑖 /d𝐸, where 𝑁𝑖 is the multiplicity, or number of produced particles, and 𝐸 is their corresponding energy. The subscript 𝑖 represents the stable SM particle that is the final state of the annihilation, i.e. 𝑖 ∈ {𝑒+, 𝑒−, 𝛾, 𝑝, 𝑝, 𝜈, 𝜈, 𝑑}. The final state can be reached through various intermediate channels, denoted by 𝑓, through the general reaction 1Website homepage: http://www.marcocirelli.net/PPPC4DMID.html, Release 6.0 Page 18 of 104 http://www.marcocirelli.net/PPPC4DMID.html 3. Physical Modelling M. Sarkis, PhD Thesis pathways of 𝜒𝜒 → 𝑓 → 𝑖.2 A large set of channels, given by 𝑓 = { 𝑒+ 𝐿 𝑒− 𝐿 , 𝑒+ 𝑅𝑒− 𝑅 , 𝜇+ 𝐿 𝜇− 𝐿 , 𝜇+ 𝑅𝜇− 𝑅, 𝜏+ 𝐿 𝜏− 𝐿 , 𝜏+ 𝑅 𝜏− 𝑅 , 𝑞𝑞, 𝑐𝑐, 𝑏𝑏, 𝑡𝑡, 𝛾𝛾, 𝑔𝑔, 𝑊+ 𝐿 𝑊− 𝐿 , 𝑊+ 𝑇 𝑊− 𝑇 , 𝑍𝐿𝑍𝐿, 𝑍𝑇𝑍𝑇, ℎℎ, 𝜈𝑒𝜈𝑒, 𝜈𝜇𝜈𝜇, 𝜈𝜏𝜈𝜏, 𝑉𝑉 → 4𝑒, 𝑉𝑉 → 4𝜇, 𝑉𝑉 → 4𝜏 } (3.1) has been used in the PPPC4DMID, where 𝑔 = 𝑢, 𝑠, 𝑑 denotes light quarks, the subscripts 𝐿, 𝑅 and 𝐿, 𝑇 refer to the polarisation of each lepton (Left, Right) or vector boson (Longitudinal, Transverse), and 𝑉 represents a new light boson in some proposed DM models (Pospelov et al. 2009; Arkani- Hamed et al. 2009). Although the channels shown in Equation (3.1) also undergo a range of interactions before reaching the final state, they are simply used and referred to in the literature by the labels of their primary products, as in Equation (3.1). It has also become common practice for indirect detection studies to only consider a subset of the channels shown above, usually selecting one channel from a group with similar particle characteristics, whose spectra are then taken as representative of the wider set. Further, since these channels represent the particle pairs that are produced directly from the DM-DM collision, only channels with a total mass less than 2 times the individual DM particle mass are viable within a certain DM model. The energy spectra in PPPC4DMID were calculated using two Monte Carlo event-generator programs, PYTHIA (Sjöstrand et al. 2008) and HERWIG (Corcella et al. 2001), which simulate the parton showers and hadronisation processes that are necessary for the production of SM particles in a DM-DM collision event. While both programs perform the same function, they do so using independent techniques, which allows Cirelli et al. (2011) to compare the results of each and obtain uncertainty estimates related to theMonte Carlo method itself. For further details regarding the difference between these programs, we refer the reader to Cirelli et al. (2011, sec. 3.2), and note that the final spectra in the cookbook are found using the PYTHIA code. We also explicitly make note of the presence of electroweak radiative corrections in the cookbook results, applied by the authors following the prescription in Ciafaloni et al. (2011). These corrections can have a large impact on the final energy spectrum, changing both the multiplicity and the set of SM particles available in the final state. This effect is especially significant when the energy of the particles is much larger than the typical electroweak scale (relevant for DM particle models with large masses), and is due to the radiation of electroweak gauge bosons from the highly energetic products of the DM-DM annihilations (for further details about these electroweak interactions, we refer the reader to Ciafaloni et al. (2011)). An example set of results from the PPPC4DMID, showing the final energy spectrum, is shown in Figure 3.1. We download these quantities directly from the online repository, and import them into our modelling pipeline after some minor unit conversions. A large emphasis has been placed on the model independent nature of these results, through both the event generators and the electroweak corrections in Ciafaloni et al. (2011), which allows us to very conveniently use the full set of results from the cookbook. The models considered in this work, still of a generic WIMP nature, will then be paramaterised by the annihilation cross-section (⟨𝜎𝑣⟩) and the WIMP mass (𝑀𝜒). The energy spectrum, in its most general form, will be a combination of the spectra from all relevant intermediate channels 𝑓, i.e. d𝑁𝑖 d𝐸 = ∑ 𝑓 𝐵𝑓 d𝑁𝑓 →𝑖 d𝐸 , (3.2) where 𝐵𝑓 represents a branching ratio that determines the overall contribution from each channel. In indirect detection studies, it is standard to consider results for individual channels, which implies setting 𝐵𝑓 = 1 and selecting a single channel from the set in Equation (3.1). Following the points made above, we consider 3 channels in this work: bottom quarks (𝑏𝑏), muons (𝜇−𝜇+) and tau leptons (𝜏−𝜏+). In terms of the final state 𝑖, we consider both electrons (𝑒−) and positrons (𝑒+), which have effectively identical behaviour for the processes relevant in our modelling. Henceforth, 2The notation of these quantities in the literature has been defined in a number of different ways, so we choose to generally follow the notation of Cirelli et al. (2011) in this Section. Page 19 of 104 3. Physical Modelling M. Sarkis, PhD Thesis we may refer to the final state as just ‘electrons’, but account for both products in the calculations. The relevance of this choice of final state, in the context of radio emissions, will be made clear in Section 3.2. Then, with a given DM model (𝑀𝜒, ⟨𝜎𝑣⟩) and intermediate channel, we define a source function, 𝑄(x, 𝐸), which describes the continual production of final state electrons in a region of annihilating WIMPs, as follows: 𝑄(x, 𝐸) = 𝒩𝜒(x)⟨𝜎𝑣⟩d𝑁𝑒± d𝐸 . (3.3) In this equation the factor 𝒩𝜒(x) represents the number density of WIMPs which can annihilate in the region of interest. So far we have discussed the case of annihilating WIMPs, which is the relevant phenomenon for our work. However, there also exists the possibility that individual WIMPs decay with a finite lifetime, with the decay chain 𝜒 → 𝑓 → 𝑒±. In this case, the annihilation cross-section in Equation (3.3) should be replaced by the decay rate Γ for the respective channel (as we will see in Chapter 6, these quantities are usually treated as the free parameters in the model to be constrained through observation). The relevant WIMP number density is then simply found by 𝒩𝜒(x) = 1 2 ⎛⎜ ⎝ 𝜌𝜒(x) 𝑀𝜒 ⎞⎟ ⎠ 2 (annihilation) , (3.4) 𝒩𝜒(x) = ⎛⎜ ⎝ 𝜌𝜒(x) 𝑀𝜒 ⎞⎟ ⎠ (decay) , (3.5) where 𝜌𝜒 is the DM density, discussed further in Section 3.1.2. The factor of 1/2 in the annihilation term assumes that theWIMPs are Majorana fermions, allowing all particles to annihilate with each other. If they are instead treated as Dirac fermions, which would only permit particle-antiparticle annihilations, this factor would change to 1/4. Figure 3.1: Example energy spectrum of positrons produced during a single DM-DM annihilation, via different intermediate channels. In Cirelli et al. (2011), quantities are expressed in terms of the dimensionless energy fraction 𝑥 = 𝐾/𝑀𝜒, where K is the kinetic energy of a single DM particle of mass 𝑀𝜒, so the expression d𝑁𝑓 /d log 𝑥 represents the energy spectrum of the produced final state particles. This plot has been normalised to the average multiplicity in the sample. Figure from Cirelli et al. (2011, fig. 3). Page 20 of 104 3. Physical Modelling M. Sarkis, PhD Thesis 2HDM+S and the Madala hypothesis In the above section we have described the particle modelling scenarios that are usually considered in indirect detection literature. Here, we briefly outline the phenomenology of a relatively new model, which also provides a DM candidate particle. The model is based on a generic extension to the SM, known as the two-Higgs Doublet Model (2HDM) (proposed in Lee (1973), see also Branco et al. (2012)), which simply introduces the possibility of a second doublet of the Higgs field. This type of model is well-motivated by some theoretical deficiencies of the SM, and an extra Higgs doublet is necessary in some models of supersymmetry or models that contain axions. A specific type of 2HDM model, which is known as a Type-II 2HDM (the most similar to supersymmetric models, and popular in the literature) that also permits the existence of a new Higgs-like particle, 𝑆, and a DM candidate particle, 𝜒, was originally proposed in von Buddenbrock et al. (2015), and then later expanded upon in von Buddenbrock et al. (2016). The addition of 𝑆 to the generic 2HDM has motivated the naming convention of ‘2HDM+S’ for this model, which we adopt for this thesis. The model has also been referred to as the Madala hypothesis in the literature, which is a word in the isiZulu language that translates to ‘old person’. This name is a reference to the extra Higgs scalar (denoted 𝐻, the Madala boson) that is introduced by the second Higgs doublet, since 𝐻 is conventionally heavier (due to being ‘older’) than the SM Higgs (von Buddenbrock 2017). An observable signature of the 2HDM+S model is the production of a number of leptons in particle colliders that would be considered anomalous, or in excess, of SM theories. In this regard, the authors of von Buddenbrock et al. (2016) found that the relevant parameters of the 2HDM+S model were compatible with data gathered from Run 1 of the CMS and ATLAS experiments in the Large Hadron Collider (LHC) (The CMS Collaboration et al. 2008; Aad et al. 2008). Then, after analysis of further Run 2 data, the model parameters were constrained, but still viable (von Buddenbrock et al. 2019) (see also von Buddenbrock et al. (2017), von Buddenbrock (2017), and von Buddenbrock et al. (2018)). Of these constraints, we only focus on a few that are most relevant to this work. In particular, we follow the general analyses of Beck et al. (2023b) and Beck et al. (2017), which were the first to investigate the possibility of using this new model in indirect DM searches. In these works, a fixed mass of the Madala boson of 𝑚𝐻 = 270GeV and the scalar S of 𝑚𝑆 = 150GeV are used. Since the 𝜒 particle is still unconstrained by LHC data, results were found for a range of possible masses. However, as the particle 𝜒 is assumed to couple directly to 𝑆, there are still some theoretical limits on its mass which are determined by the associated interaction chains. In the cases of the 𝜒 particle having spin of 0, 1/2, or 1, the Lagrangians that define these interactions are given by (Beck et al. 2023b, eqs. 7-9): ℒ0 = 1 2𝑀𝜒𝑔𝑆 𝜒𝜒𝜒𝑆 , ℒ1/2 = 𝜒(𝑔𝑆 𝜒 + 𝑖𝑔𝑃 𝜒 𝛾5)𝜒𝑆 , ℒ1 = 𝑔𝑆 𝜒𝜒𝜇𝜒𝜇𝑆 . (3.6) In these expressions the 𝑔𝑆 𝜒 factor represents the strength of the scalar coupling between 𝑆 and 𝜒, 𝑔𝑃 𝜒 is the strength of the pseudo-scalar coupling, and 𝛾5 = 𝑖𝛾0𝛾1𝛾2𝛾3, where 𝑖 is the complex unit and the 𝛾’s are the standard gamma (or Dirac) matrices. We note that in the hypothesis that the DM candidate is a vector boson, it would need to interact with the Higgs/Scalar field to acquire its mass, in a similar manner to other SM vector bosons. Without these interaction terms in the Lagrangian, a vector DM candidate could break the symmetry that gives mass to vector bosons. Any potential interactions with the Higgs field in different particle hypotheses would be specified by the underlying particle physics framework, and are outside the scope of this thesis (for further details on vector DM, see for example Arcadi et al. (2020)). For this work, we consider the following processes (similar to the annihilation channels described in the Section above) defined in Beck et al. (2023b): 𝜒𝜒 → 𝑆 → 𝑒+,− (3.7) and 𝜒𝜒 → 𝑆 → 𝐻 𝑆/ℎ → 𝑒+,− . (3.8) Page 21 of 104 3. Physical Modelling M. Sarkis, PhD Thesis Clearly, for the above reactions to be possible, the mass of 𝜒 needs to obey 𝑀𝜒 > 1/2 𝑚𝑆 = 75GeV in Equation (3.7) and 𝑀𝜒 > 1/2 (𝑚𝐻 +𝑚ℎ) ∼ 200GeV for Equation (3.8). The energy spectra from these channels, in a similar manner to the PPPC4DMID, were then found using Monte Carlo event generators and presented in Beck et al. (2023b, figs. 1, 2). Specifically, the MG5_@MC (Alwall et al. 2011) tool was interfaced with the PYTHIA package to produce results for a range of 𝜒 masses and final states. An example spectrum for positrons, most relevant to this work, is shown in Figure 3.2. These results use the reaction in Equation (3.7), with a spin-0 𝜒 particle. In the full set of results, the difference in the spectra for different spin parameters of 𝜒 are not significant, but the process in Equation (3.7) leads to spectra that are several orders of magnitude larger than those found from Equation (3.8). Thus, the use of the 2HDM+S model in this work is restricted to the case of a spin-0 𝜒 particle annihilating through the process in Equation (3.7). Figure 3.2: Example energy spectra of positrons produced during a single DM-DM annihilation, with varying 𝜒 mass. These spectra are found through the annihilation process described in Equation (3.7), for a spin-0 𝜒 particle. Figure from Beck et al. (2023b, fig. 1). 3.1.2 Halo models With the characteristics and behaviour of the individual DM particles set, we then need to model the distribution of these particles in the regions we are interested in. There is substantial evidence to suggest that DM particles have a tendency to clump together and form gravitationally-bound halos, coming from both analytical arguments (see for instance Silk et al. (1993)) and from a host of large 𝑁-body simulations (see Kuhlen et al. (2012) or Vogelsberger et al. (2020) for excellent reviews). Since the properties of these halos – including their formation histories, inner structures and mass distributions – have been the topic of several dedicated studies in recent years, the current paradigm of DM halo modelling has become quite robust. In this Section we will provide a brief outline of some pertinent aspects of DM halo modelling, and present the specific equations that are used in this work. For more comprehensive summaries or reviews on this topic, we refer the reader to the following works: Navarro et al. (2010), Bertone (2010, ch. 2), Kuhlen et al. (2012), Zavala et al. (2019), and Vogelsberger et al. (2020). In this work, only the formation of Cold, non-relativistic DM (CDM) halos is considered, as the particle models used here all behave like CDM. We note that although other DM scenarios, like WarmDM(WDM)or Self-InteractingDM(SIDM), can also lead to the formation of halos, there can be significant departures from CDM halo characteristics (especially in the case of SIDM) (Kuhlen et al. 2012; Zavala et al. 2019). The majority of these halo characteristics have been sourced from analyses of modern 𝑁-body simulations, which are run on high-performance supercomputers Page 22 of 104 3. Physical Modelling M. Sarkis, PhD Thesis utilising thousands of CPU cores and taking millions of CPU hours. These simulations usually track an ensemble of 𝑁 particles (consisting of dark or baryonic matter, or both) from the very beginnings of structure formation, seeded by tiny density perturbations that grow with inflation and produce large self-gravitating halos of DM, up to the current epoch. Treating the evolution of these particles in a numerical fashion also allows the non-linear regimes of standard structure formation theories to bemore accurately probed, avoidingmany of the assumptions and limitations of analytical methods. Many such simulations have been performed over the past three decades, and a complete review of them could be the subject of an entire thesis. Since we are only focused on a few very particular outcomes from these simulations, we provide just a brief list of some of the more notable projects, which can serve as a starting point for further reading in this subject. The most long-standing of these simulation projects is the Millennium project, which, in its various forms, has been running since 2005. The inaugural simulation, now known as Millennium I (Springel et al. 2005), tracked the evolution of 10 billion DM particles in a (685Mpc)3 volume. Since then, the Millennium II, XXL and most recent MillenniumTNG simulations (Boylan-Kolchin et al. 2009;