University of the Witwatersrand Johannesburg School of Physics A dissertation submitted to the Department of Physics, University of the Witwatersrand in fulfilment of the requirements for the degree of Master of Science. The Effect of Dark Matter in the Epoch of Reionization Author: Mpho Kgoadi Date: 2021-09-13 Supervisors: Dr Geoff Beck Abstract The problem of dark matter has been of great importance in modern physics since its inception. Many theories have been proposed about the nature of dark matter but perhaps the most studied is the WIMP (Weakly Interacting Massive Particle). This particle has been favoured because it has the properties of dark matter that have been measured experimentally, so far. In this work we present an argument for studying the properties of dark matter in the Epoch of Reionization (EoR) using the redshifted 21 cm background. The 21 cm line of hydrogen provides great potential in studying the Universe at an early stage. This could provide rich information about the thermal and ionization history of the Universe as well as understanding the physics behind the formation of the first stars and galaxies. This will allow us to have a full picture of the global 21 cm background including the effects of WIMPs, if there are any. We also will demonstrate the potential power of HERA and the SKA to probe the high redshift Universe, being able to produce constraints that are quite optimistic against current benchmark models of indirect detection of dark matter. Declaration I, MPHO ARTHUR KGOADI , declare that this dissertation and the work presented in it are my own. I Confirm that: • The research reported in this dissertation, except where otherwise indicated, is my original research • This work has not been submitted to any other institution or for any other degree. • Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this research proposal is entirely my own work. • I have acknowledged all main sources used and have cited these in the reference section. Signed: Date: 2021-09-13 ii Dedication For my late grandmother iii Acknowledgements I would like to express my gratitude to my supervisor Dr Geoff Beck, thank you for your continuous support, your patience and understanding. I could not have asked for a better advisor. Special acknowledgement goes to Andrew Sam and the whole staff at the DRU, I would not be here if it weren’t for them. I would also like to thank my best friends in the whole wide world for their everlasting support and for being there for me through the highs and the lows. In addition, the financial assistance provided by my supervisor, the NRF Thuthuka grant no. 117969 is hereby acknowledged. Finally, I would like to thank my mother, Zintle, Shimi and the rest of my family for being there for me, and for their patience and love. iv Contents List of Figures iviv 1 Introduction 11 2 What is Dark Matter ? 55 2.1 Dark Matter Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.1.1 Dynamics of stars and galaxies . . . . . . . . . . . . . . . . . . . . . . . . 55 2.1.2 Gravitational lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.1.3 Cosmic Microwave Background . . . . . . . . . . . . . . . . . . . . . . . . 77 2.1.4 Big Bang Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.1.5 Structure formation and other sources . . . . . . . . . . . . . . . . . . . . 88 2.2 Dark Matter Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.2.1 Weakly interacting massive particle . . . . . . . . . . . . . . . . . . . . . . 1010 2.2.2 Dark matter halos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111 2.3 Dark matter detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313 2.3.1 Direct detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313 2.3.2 Collider experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515 2.3.3 Indirect detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1616 3 Reionization and the 21 cm Hydrogen Line 2222 3.1 Epoch of Reionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2323 3.2 21 cm line of hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2525 3.2.1 Determining the spin temperature . . . . . . . . . . . . . . . . . . . . . . 2828 3.2.1.1 Collisional coupling . . . . . . . . . . . . . . . . . . . . . . . . . 2929 i 3.2.1.2 Wouthuysen-Field effect . . . . . . . . . . . . . . . . . . . . . . . 3030 3.2.2 Thermal evolution of the IGM . . . . . . . . . . . . . . . . . . . . . . . . 3333 3.2.2.1 Compton Heating . . . . . . . . . . . . . . . . . . . . . . . . . . 3434 3.2.2.2 Lyman-α heating . . . . . . . . . . . . . . . . . . . . . . . . . . 3434 3.2.2.3 X-ray heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3535 3.2.3 Ionization History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3636 3.2.4 Global 21 cm signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3737 3.3 Exotic energy injection and dark matter . . . . . . . . . . . . . . . . . . . . . . . 4040 3.3.1 DarkHistory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4343 3.3.1.1 Thermal and ionization histories . . . . . . . . . . . . . . . . . . 4444 3.3.1.2 Calculating fc(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4646 3.3.1.3 Code structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4949 3.4 Observing the 21 cm signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5454 3.4.1 EDGES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5454 3.4.2 Can interferometers be used to measure the global 21 cm signal ? . . . . . 5555 3.4.3 SKA and HERA sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 5656 4 The Effect of Dark Matter in the Epoch of Reionization 5858 4.1 bb̄ annihilation channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5959 4.1.1 Global 21 cm signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5959 4.1.2 HERA vs SKA constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 6464 4.2 τ+τ− annihilation channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6767 4.2.1 Global 21 cm signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6767 4.2.2 HERA vs SKA constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 7171 4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7373 5 Conclusion 7575 5.1 Challenges and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7676 ii Bibliography 7777 iii List of Figures 2.1 left panel : Image of the Bullet Cluster obtained with gravitational lensing. The X-ray observations of the gas component is shown in pink and the lensing map is shown in blue. The image is taken from [11][22]. Right Panel: Mass density contours (green) of the Bullet Cluster and the distribution of baryonic matter. Taken from [33] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.2 A diagram showing annual modulation and directionality . . . . . . . . . . . . . 1515 2.3 Mediator mass reach of CMS searches for a selection of results targeting elec- troweak SUSY production. Adapted from [44] . . . . . . . . . . . . . . . . . . . . 1616 2.4 The upper limits of the dSph observations by VERITAS (black solid line) to the DM annihilation cross section into bb̄ (left) and τ+τ− (right) pairs. See legends for details of the comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2121 3.1 A schematic diagram of the history of the Universe from the big bang to now. After the big bang the Universe undergoes recombination when the photons get decoupled from the gas and the relic radiation is observed as the CMB at redshift = 1000. After the recombination epoch, the Universe is neutral in the period known as the Dark Ages. About 500 million years after the big bang, the first stars and galaxies form and heat up the Universe and starting the Epoch of Reionization. The Universe is fully ionized around 1 billion years after the big bang, at redshift ≈ 6. This image is credited to NAOJ [55]. . . . . . . . . . . . . . 2424 3.2 The Wouthuysen-Field effect energy levels. The forbidden transitions are in lines and the dotted lines are the allowed transition but do not contribute to spin flips. Image taken from Ref. [66] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3131 iv 3.3 The solid lines represent ionizing photons redshifting into Lyman-n resonance, which then cascade down via multiple possible decay channels. Some decay chan- nels lead to lyman-α emissions contributing to the Lyman-α background. This figure shows decay chains for Lyman-β and Lyman-γ. Dashed lines represents Lyman-n transitions and red-dashed line is the Lyman-α transition. The forbid- den transitions are depicted by the dotted lines. Figure taken from [77] . . . . . . 3333 3.4 The evolution of the global 21 cm signal. The signal transitions from the early phases of collisional coupling. There is no signal when the gas temperature couples to the background CMB temperature. Fluctuations in the later stages are dominated by various heating mechanisms such as Lyman-α, X-ray, and ionizing radiation background. Relevant redshifts accompany the subsequent radiation backgrounds. Image taken from [77] . . . . . . . . . . . . . . . . . . . . . . . . . . 3838 3.5 Constraints of WIMP annihilation cross-section to reionization for annihilation channels χχ → e+e− (left) and χχ → γγ. The hatched regions represents the parameter space ruled out by the optical depth constraints (orange) and the CMB power spectrum constraints measured by Planck (red). The contribution to the free electron fraction xe is depicted by the colour density plot with the black, dashed contours shown for a contribution to xe at z = 6. image taken from [88]. . 4242 3.6 The effects of decaying and annihilation WMD and LDM models on the 21 cm brightness temperature. The solid line shows δTb for a baseline model without any inclusion of DM effects. The dotted line as well as the long dashed and the short dashed lines represent δTb with 10 MeV annihilating DM, 25 keV decaying WDM and the 10 MeV decaying WDM, respectively. Image taken from [99] . . . 4343 3.7 Flowchart showing the calculation of ionization and thermal histories in Dark- History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5151 3.8 Matter temperature (left) and hydrogen ionization history (right) against redshift solved in the presence of dark matter annihilation into bb̄ pairs using DarkHistory [1010]. The blue line shows the histories with energy injection but no backreaction. The orange line shows dark matter annihilation with backreaction and the black, dashed line shows the baseline without any injection. . . . . . . . . . . . . . . . 5353 4.1 The global 21 cm brightness temperature for the baseline (black) case compared to the 100 GeV DM with 〈σv〉 = 3× 10−26 cm3 s−1 (orange) . . . . . . . . . . . 5959 v 4.2 Spin temperature and the matter temperature plot from ref.[1111] (Top) and the same plots for our Darkhistory case (bottom). The orange plot is the matter temperature, blue is the spin temperature and the black one is the CMB. . . . . 6060 4.3 Coupling coefficients from ref.[1111] compared to the Darkhistory ones. The solid orange and blue are the collisional coupling (yc) and the Lyman-α coupling (ya) for ref.[1111] respectively and the dashed green and red are the collisional coupling and the Lyman-α coupling for Darkhistory. . . . . . . . . . . . . . . . . . . . . . 6060 4.4 Brightness temperature for the 100 GeV DM case (blue) sensitivity against the EDGES noise (orange) with 6 hour integration time from ref.[1212]. . . . . . . . . . 6161 4.5 The global 21 cm brightness temperature for the baseline (black) case compared to the 100 GeV DM with 〈σv〉 = 4 × 10−26 cm3 s−1 (orange) with HERA noise (blue) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6262 4.6 The global 21 cm brightness temperature for the baseline (black) case compared to the 100 GeV DM with 〈σv〉 = 9×10−27 cm3 s−1 (orange) with SKA noise (blue) 6363 4.7 Sensitivity constraints on the DM annihilation cross section for DM annihilating into bb̄ final state for 1σ c.l with HERA and SKA after 1000 hours of integration time. The blue line represents the HERA constraints and the red line represents the SKA constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6464 4.8 Sensitivity constraints on the DM annihilation cross section for DM annihilating into bb̄ final state for 2σ c.l with HERA and SKA after 1000 hours of integration time. The blue line represents the HERA constraints and the red line represents the SKA constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6464 4.9 Sensitivity constraints on DM annihilation cross section for DM annihilating into bb̄ final state for 2σ c.l with HERA and SKA after 1000 hours of integration time compared to the constraint on the DM annihilation cross section for bb̄ final states from Ref.[1313]. The blue line represents the HERA constraints and the red line represents the SKA constraints. The black dotted lines represents the constraints of DM annihialtion cross section corresponding to the current optical depth measurements of ∆τ = 0.012. . . . . . . . . . . . . . . . . . . . . . . . . . 6666 vi 4.10 Sensitivity constraints on DM annihilation cross section for DM annihilating into bb̄ final state with HERA and SKA for 1000 hours of integration time compared to the upper limits on the DM annihilation cross section from Ref.[1414] (solid black line) and Ref.[1515] (solid red line) for bb̄ channel at 95% confidence interval. . . . . 6666 4.11 The computed matter temperature Tm and hydrogen ionization fraction xHII (right) in the presence of 100 GeV WIMP DM particle annihilation into τ+τ− with cross section of 3 ×10−26 cm3 s−1. . . . . . . . . . . . . . . . . . . . . . . . 6767 4.12 The global 21 cm brightness temperature for the baseline (black) case compared to the 100 GeV DM with 〈σv〉 = 3× 10−26 cm3 s−1 (orange). . . . . . . . . . . . 6868 4.13 The global 21 cm brightness temperature for the baseline (black) case compared to the 100 GeV DM with 〈σv〉 = 1 × 10−25 cm3 s−1 (orange) with HERA noise (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6969 4.14 The global 21 cm brightness temperature for the baseline (black) case compared to the 100 GeV DM with 〈σv〉 = 1 × 10−26 cm3 s−1 (orange) with SKA noise (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7070 4.15 Sensitivity constraints on the DM annihilation cross section for DM annihilating into τ+τ− final state for 1σ c.l with HERA and SKA after 1000 hours of inte- gration time. The blue line represents the HERA constraints and the red line represents the SKA constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7171 4.16 Sensitivity constraints on the DM annihilation cross section for DM annihilating into τ+τ− final state for 2σ c.l with HERA and SKA after 1000 hours of inte- gration time. The blue line represents the HERA constraints and the red line represents the SKA constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7171 4.17 Sensitivity constraints on DM annihilation cross section for DM annihilating into bb̄ final state for 2σ c.l with HERA and SKA after 1000 hours of integration time compared to the constraint on the DM annihilation cross section for τ+τ− final states from Ref.[1313]. The blue line represents the HERA constraints and the red line represents the SKA constraints. The black dotted lines represents the constraints on the annihilation cross section corresponding to the current optical depth measurements of ∆τ = 0.012. . . . . . . . . . . . . . . . . . . . . . . . . . 7272 vii 4.18 Sensitivity constraints on DM annihilation cross section for DM annihilating into bb̄ final state with HERA and SKA for 1000 hours of integration time compared to the upper limits on the DM annihilation cross section from Ref.[1414] (solid black line) and Ref.[1515] (solid red line) for bb̄ channel at 95% confidence interval. . . . . 7272 viii Chapter 1 Introduction The problem of dark matter (DM) has become one of the most exciting endeavours in modern physics. The discovery of dark matter forced phyisicists to examine the current understanding of cosmology. Dark matter forms a fundamental part of the current standard model of the uni- verse known as the Λ-Cold Dark Matter (ΛCDM). This is a model that describes the expanding universe with three major fluid components: a cosmological constant Λ which behaves like a fluid with negative pressure and therefore constitutes a form of so-called ”dark energy”, the non-relativistic (cold) dark matter and finally baryonic matter. Fritz Zwicky originally hypoth- esised the idea of dark matter in an attempt to explain the orbital velocity of galaxy clusters [1616], since then there has been a hunt for this elusive matter. Although there has been lack of evidence of the nature of dark matter, there are numerous observational experiments indicat- ing its presence. Some of these observations include the fluctuations in the Cosmic Microwave Background [1717], structure formation [1818], gravitational lensing measurements [1919], Big Bang nucleosynthesis [2020] and galaxy rotation curves [2121]. All these experiments have confirmed that dark matter might be of particle nature, however most of these experiments are indirect and mostly rely on gravitational effects of dark matter. This does not allow us to learn about the exact nature of dark matter and its behaviour, in fact, some scholars have challenged the idea of dark matter and instead hypothesize the idea of Modified Newtonian Dynamics (MOND) to explain the missing mass problem [2222]. 1 Since the evidence from observations pointed towards dark matter being of particle nature, it is important to examine which standard model or beyond standard model particles fit the description. Some of these particles include weakly interacting massive particles (WIMPs), axions, sterile neutrino and supersymetric particles (SUSY). WIMPs are thought to be one of the most promising candidates for DM for a number of reasons. The first one being that they arise naturally in a number of theoretically well-motivated models [2323, 2424, 2525, 2626]. Secondly, for reasonable ranges of WIMP mass and annihilation cross section the relic abundance of DM can be obtained through mechanisms of thermal freeze-out as well as other production mechanisms [2727, 2828]. Lastly, WIMPs present a promising target for direct detection experiments of DM because a large fraction of their typical detection rates are within reach for current and future detectors, making them testable. This is why WIMPs are the main focus of this work. WIMPs interact with standard model fields only via the weak nuclear force, they are non- baryonic and electrically neutral. They are are stable on cosmological timescales, this means that they carry some sort of conserved quantum number [2323]. An example includes the Kaluza- Klein particle [2929], for which the lightest member of a matter sector is charged under some discrete symmetry. These WIMPs are members of the SU(2)L multiplets, if there are no other symmetries which might force their masses down, they should naturally acquire masses mχ within a few orders of magnitude of the SU(2)L × U(1)Y electroweak symmetry-breaking scale (i.e GeV or TeV). Furthermore, this makes WIMPs sufficiently heavy to constitute cold dark matter even if they have been produced thermally in the early Universe. Some popular example of WIMPs include the neutralino, which is the lightest particle from the minimal supersymmet- ric extension of the standard model (MSSM) [3030]. WIMPs can annihilate/decay into Standard Model (SM) particles, particularly in regions of high DM density. Some of these secondary particles include gamma rays, neutrinos, electron- positron pairs and proton-antiproton pairs. Each annihilation process produces a distinctive signature such as gamma-rays from the center of galaxies such as the Milky Way, or neutrinos from the Sun. Notably, WIMPs will produce gamma-rays almost every time through various channels, i.e through pion production for hadronic channels or through inverse Compton and bremsstrahlung for leptonic channels. This could be measured through indirect detection and could potentially provide strong constraints on the dark matter mass and annihilation cross 2 section or life-time for decays. Recent indirect astrophysical detections have provided strong limits on possible WIMP models through studies of the CMB [1717], gamma-ray studies of galax- ies with the Fermi-LAT [1515, 3131] as well as some studies of positron fractions from PAMELA and AMS-02 [3232, 3333]. The galactic center excess observations from the Fermi-LAT data remain the most intriguing hints of DM detection from the past few years [1414]. There have been several claims from the galactic excess center observations that favour a WIMP of mass ≥ 30 GeV and cross section of 〈σv〉 ∼ 3 × 10−26. However, there have been competing arguments that explain the galactic center excess without the presence of dark matter [3434, 3535, 3636]. This work will focus on detecting dark matter through radio observation of the 21 cm spectrum from the Epoch of Reionization using future telescopes, namely, the Hydrogen Epoch of Reionization Array (HERA) and the Square Kilometer Array (SKA). The Epoch of Reionization (EoR) is a period in the evolution of the Universe which occurred between z ∼ 6-13, where intergalactic hydrogen changes from being completely neutral to being completely ionized. The EoR represents the moment in cosmic history when structure formation has progressed far enough to impact the global state of the Universe. The measurements of the conditions during EoR promises a wealth of information about structure evolution, as such it could form a bridge between fundamental physics and cosmology and the complex astrophysics associated with galaxy formation [3737, 3838]. In principle, the EoR can be probed in many different ways. In this work, the focus is on the observations based on the redshifted 21 cm line of neutral hydrogen. The 21 cm line is the hyperfine spin-flip transition of neutral hydrogen caused by the interaction between the electron and proton magnetic moments [77]. This technique provides a promising way to study the high redshift Universe, However, observing the EoR is still a considerable challenge since there is still an issue extracting and identifying cosmological signals from data and interpreting it correctly. The reason is that the detectable signal in the relevant frequency from the EoR is tainted by a number of components - ionospheric distortions, instrument noise and response as well as extragalactic and galactic foreground [3939]. 3 We begin with Chapter 2, providing a brief introduction to the problem of DM, including a detailed review of the observational evidence that proves the particle nature of DM, specifically a non-baryonic nature of DM. A description of particle solution of DM is given and an overview of direct and indirect methods of searching for WIMP DM. Chapter 3 details the EoR and the full description of 21 cm physics. This leads to a full global 21 cm cosmology picture which details the complicated astrophysics involved in the analysis. Chapter 3 also contains a brief overview of the influence exotic energy injection and dark matter as well as evidence from previous work. It ends with a section on DarkHistory, a code package for calculating the evolution of thermal and ionization histories with DM injection included. Chapter 4 is the results section, where detailed findings of our work has been provided. The section begins with an analysis of temperature and ionization histories for two annihilation channels, bb̄ and ττ , this provides a framework for the subsequent calculations. The brightness temperature is then plotted for the baseline and dark matter cases to see the effect that dark matter has on the global 21 cm signal. The constraints for DM masses and annihilation cross sections are then derived for HERA and SKA cases. Finally the constraints are compared against previous work. Chapter 5 provides the concluding summary. 4 Chapter 2 What is Dark Matter ? This chapter contains an introduction to the problem of dark matter. The particle solutions to the problem are outlined and furthermore, a detailed review of the detection strategies of dark matter is provided, especially focusing on the indirect detection method. Then section 2.1section 2.1 provides a brief outline of the observational evidence of dark matter, indicating a particle solution to the problem. This is followed by section 2.2section 2.2, by a detailed discussion of DM particles, with particular emphasis on Weakly Interacting Massive Paricles (WIMPs). This chapter closes with section 2.3section 2.3 which focuses on detection methods of dark matter, including direct, collider and indirect detection. 2.1 Dark Matter Observations 2.1.1 Dynamics of stars and galaxies The problem of dark matter began in the early 20th century with astronomer Fritz Zwicky [1616]. He was studying the velocity dispersion of galaxies within clusters when he noticed that the outer members of the Coma Cluster were moving faster than anticipated, with some exceeding 2000 km/s. Although the velocity dispersion of the Coma clusters were observed by scientists before him, Zwicky was the first one to use the well studied virial theorem to further investigate this problem. He noticed that the only way to account for the observed orbital velocities of the outer members of the cluster with the virial theorem was to propose that Coma cluster 5 contained a non-luminous, very large mass component. He would later coin this missing mass the term ”dark matter”. The idea of the missing mass in galaxy clusters was met with skepticism up until 1970 when Rubin and Ford [4040] studied a similar effect for galaxies. Similar to galaxies in the outskirts of a cluster, stars in the outer regions of galaxies orbit faster than expected. They found that the rotation curves of galaxies are flat rather than dropping off at long distances from the centre of the galaxy. This flat behaviour was another hint that there was missing matter in galaxies that was holding the stars in orbit and the mass distribution spans much larger distances. The work by Rubin et al was enough to convince the scientific community that a large amount of unseen mass is present in the universe and throughout the years there have been numerous observations at all cosmological scales that support the existence of DM. 2.1.2 Gravitational lensing In Einstein’s theory of General Relativity (GR) [4141], gravity is not a force but the curvature of space-time as a result of the density distribution of various matter-energy components. GR pre- dicts the deflection of light around massive bodies because light follows a geodesic in space-time. This produces a distorted (weak lensing) [33] or multiple (strong lensing) [4242] images of the light source for the observer, This is refered to as gravitational lensing. This phenomenon provides a way to measure the mass and the matter distribution of the foreground object from the angles of deflection since the amount of lensing is sensitive to the strength of the gravitational field generated by the massive object [4343]. This method can be used to calculate DM mass in the foreground object by comparing it to the visible mass since the angle of lensing is affected by the total mass distribution, not only the visible mass. Observations of the well-known Bullet cluster [11, 1919], a collision of two galaxy clusters, provides the best evidence of DM using gravitational lensing, shown in the left panel of Figure 2.1Figure 2.1. Not only does the lensing map (in blue) exhibit large amounts of DM not evident in the X-ray gas map (in pink), but the distribution of the two components provides crucial insight into the properties of dark matter. The interstellar gas of the two clusters, seen in X-ray, lags behind 6 Figure 2.1: left panel : Image of the Bullet Cluster obtained with gravitational lensing. The X-ray observations of the gas component is shown in pink and the lensing map is shown in blue. The image is taken from [11][22]. Right Panel: Mass density contours (green) of the Bullet Cluster and the distribution of baryonic matter. Taken from [33] because it interacts electromagnetically. Gravitational lensing observations show that the center of mass of both clusters is in the region near the visible mass, this is seen from the contour map in the right panel Figure 2.1Figure 2.1 , indicating that there is a large amount of additional mass in both clusters that is appears undisturbed after the collision, a weakly interacting massive component which is a property of DM. 2.1.3 Cosmic Microwave Background The Cosmic Microwave Background (CMB) plays a further role in determining the DM abun- dance in the universe. The CMB is radiation emitted in the early universe during a period called recombination, when the temperature had become low enough for neutral hydrogen to form so that photons could escape. Observations from the Wilkinson Microwave Anisotropy Probe (WMAP) [4444] and the Planck telescope [1717] show the universe to be anisotropic in the early universe to a level of about 1 in 105. There are fluctuations on small scales coming from from the pressure waves in the early universe plasma that were ”frozen in” in the CMB. These fluctuations are thought to be originating from the fluctuations of a quantum field responsible for inflation and are relevant since they are the initial gravity wells around which the matter in the universe formed structures we see today. The sizes of the fluctuations give a precise mea- surement of the amount of DM in the universe. The CMB power spectrum reveal that the first peak indicates that the geometry of the Universe is flat, the remaining peaks implies that the matter component is dominated by DM with ordinary baryonic matter contributing to about 7 5% while DM is about 26%. The remaining dominant matter-energy contribution comes from dark energy with about 69%. to this day, the CMB provides the most precise measurement of the amount of DM in the Universe. 2.1.4 Big Bang Nucleosynthesis The early universe contained an abundance of light elements produced via the Big Bang Nu- cleosynthesis (BBN), this is another important evidence for DM. After a few minutes after the Big Bang, when the universe sufficiently cooled to about a temperature of ten billion degrees, neutrons and protons were able to combine to form stable deuterium in the following process: p + n → D + γ. Subsequently, helium and lithium began to form as well [4545]. Elements heavier than helium had to wait much longer, for star formation to take place in order to appear. BBN theory predicts that roughly 25% of the baryonic mass in the universe is made of Helium-4, 0.01% deuterium and even smaller quantity of Li-7; about 10−10 [4646]. These predictions are in agreement with data as long as baryonic matter can only account for about 5% of the total energy density of the universe, or Ωb ∼ 0.05 [4747]. 2.1.5 Structure formation and other sources More evidence for dark matter is derived from a number of sources. These include the cos- mic evolution and the growth of structure; as discussed in subsection 2.1.3subsection 2.1.3 the early universe produced overdense regions, they were dominated by dark matter. The overdensities acted as gravitational seeds that would ultimately be responsible for the formation of stars and galaxies. Without DM in the early universe, structures would start to grow at a much later stage [4848] but this is in disagreement with observation [4949, 5050]. Large-scale ’N-body’ simulations support this claim and demonstrates that large scale structures observed today could have only formed in the presence of a large amount of cold and non-dissipative dark matter. Measurements from the type 1A supernaovae also favours the argument for dark matter. This is the process of standard candles with known luminosity. The geometry of the universe, which is dependent on the energy content of the universe can be determined by fitting observed luminosities as a function of redshift. This analysis shows that the matter content (dark and baryonic matter) of the universe is Ωm . 0.3 [5151]. The luminosity distance-redshift relation in combination with 8 the BBN arguments gives a strong indication that the matter content in the universe is mostly non-baryonic. With observational evidence from primordial BBN and the CMB anisotropy data to the evolu- tion of structures and the dynamics of galaxies, it has become clear that the universe contains a large amount of non-baryonic matter component. All these sources agree that this matter component is weakly interacting and makes up a large amount of all the matter content in the universe. It is most likely an exotic particle not found in the standard model [5252]. 2.2 Dark Matter Particles Various models have been proposed in search of a solution to the missing mass problem. The issue is that the information from the observations in section 2.1section 2.1 gave no clue to the exact nature of DM but it does suggest that it should be a new and exotic particle. This idea grew successively since the 1970s when the main alternatives were slowly ruled out as the experi- ments became more sensitive and powerful. These alternatives featured astrophysical objects known as the Massive Astrophysical Compact Halo Objects (MACHOs). A MACHO is any astrophysical body that emits little to no light and occupied the outer halos of galaxies. These objects were thought to be white and brown dwarfs, black holes and planets. MACHOs can be detected via gravitational lensing as they pass in front of stars or galaxies. There have been several measurements of MACHOs, particularly Ref.[5353] claimed to have found microlensing to predict the existence of MACHOS, enough to make up 20% of the DM in the galaxy. However, searches after that have ruled out the possibility of MACHOs making up a considerable amount of DM. since then, MACHOs were quickly ruled out since they were baryonic matter, violating the non-baryonic requirement. The requirements for a good dark matter candidate include a particle that is massive, it should carry no electric and colour charge, it should interact only weakly with ordinary matter and be long-lived (stable) or at least have a lifetime that exceeds the age of the universe. a few standard model particles meet this requirement, so the distinguishing factor is the velocity of 9 the particles. Good DM candidates should be non-relativistic or ”cold” according to the density fluctuations in the CMB that are responsible for structure formation. Particles with velocities that are ultra-relativistic or ”hot” during the time of decoupling due to the large mean free path were unable to form structures. There are still popular ”hot” dark matter candidates [5454, 5555], which emerges from the Pecci-Quinn solution to the strong CP problem that are still considered viable candidates. However, despite all of this, one candidate still seem to fit the available evidence most satisfactorily; the Weakly Interacting Massive Particle (WIMP), which is the main focus of this work. 2.2.1 Weakly interacting massive particle WIMPs are strongly favoured as candidates for dark matter because they meet most of the requirements from section 2.1section 2.1 and for the the fact that they arise naturally in supersymmetry (SUSY) and Kaluza-Klein (KK) theories. They can interact with other nucleons and with themselves. The scattering cross section is used to measure the probability of WIMPs interacting with other nucleons and the annihilation cross section is used to calculate the rate at which they self-annihilate. WIMPs are produced in the early hot universe, when the DM particles were in thermal equilibrium with standard model particles in a hot plasma, with DM particles and anti-particles annihilations balancing each other out. The rate at which the DM particles self-annihilate decreases as the universe expands because the number density of the particles also decreases. This is given by the Boltzmann equation with number density n for a dark matter particle [2323]: dn dt + 3Hn = −〈σv〉(n2 − neq2) (2.1) where neq and n are the equilibrium and actual number densities of the species, H ≡ ȧ/a is the hubble expansion with a being the scale factor and 〈σv〉 is the velocity-averaged annihilation cross-section. The DM particles stop annihilating once H ∼ 〈σv〉n. Equation 2.1Equation 2.1 can be solved 10 numerically to obtain relic density of different DM particle species. [2323, 5656] gives the relic density for WIMPs as Ωχh 2 = mχnχ ρc ≈ 3× 10−27cm3s−1 〈σv〉 (2.2) where h is the reduced hubble constant and ρc is the critical density. Equation 2.2Equation 2.2 shows that the relic density has an inverse relation to its annihilation cross-section. This means that specific DM particles freeze out later if the annihilation cross-section is longer and this exhibits a lower relic density. The current accepted limit for the annihilation cross-section is 〈σv〉 = 3×10−26cm3s−1. If there is an assumption that a new DM particle that interacts weakly exists, then its anni- hilation cross-section can be estimated by 〈σv〉 ≈ α2/E2 weak where α = 10−2 and Eweak is the electroweak scale given by Eweak ≈ 100 GeV and this results to 〈σv〉 ≈ 10−8GeV = 10−25cm3s−1. By assuming that DM is a stable weakly-interacting particle, the relic density is found to be within one order of magnitude from the calculated value. This is known as the ”WIMP miracle” and gives a strong possibility that WIMPs might be the solution to the DM problem. 2.2.2 Dark matter halos WIMPs are massive, this can lead to formation of dense viralized clumps of DM bound together by gravity, known as ”halos”. These provide gravitational potential wells to which structures such as galaxies form. Detailed large-scale N-body simulations of structure formation can provide valuable information about the density profiles of DM which can be combined with benchmark models of particle physics to give a meaningful way to indirectly detect WIMPs. This is still a work in progress since even after years of Monte Carlo simulations and a number of different experiments, the scientific community has yet to reach a consensus of which is the most accurate DM density profile. Stellar velocity measurements and more advanced simulations of cold DM and baryonic physics continue to change the current understanding of halo profiles. Studying the properties of DM halos is important since the highly dense halo regions in galaxies or galaxy clusters have an increased chance of WIMP pair annihilation which increase a chance 11 of its detectablity. These DM halos are the missing mass that was mentioned in section 2.1section 2.1. The shapes for density profiles can be inferred from observations of the path stars take in the gravitational potential well of the halo and from N-body simulations. The most favoured profile is a cusped (ρ ≈ r−1) profile but some mini-galactic sized objects are described better with a cored (central density is constant) profile. The most commonly used profile is the Navarro- Frenk-White (NFW) [5757] profile which provides a good fit for a wide range of halo masses and it is given by the equation: ρNFW (r) = ρ0 (r/rs)(1 + r/rs)2 (2.3) where r is the distance from the center of the halo, rs is the scale radius and ρ0 is the scale density which can be written as a ratio of the scale virial radius Rvir to rs and is defined as the concentration parameter c or cvir = Rvir rs . The virial mass can be shown to be: M = 4πρ0r 3 s [ ln (1 + c)− c 1 + c ] (2.4) [5858, 5959, 6060] have employed this method on various situations including modelling the dwarf Spheroidal (dSph) satellite of the Milky Way which shows that the fit is less concentrated than expected, means that there needs to be some profiles that can explain small sized dwarf galaxies. Another profile that is a good fit as the NFW, and might be even better at galactic-size is the Einasto profile [6161]. It is given by the following equation: ρeinasto(r) = ρ0e −2n[(r/rs)1/n−1] (2.5) 12 where n = 1/α is a parameter that describes the degree of curvature of the distribution. The Einasto profile has a logarithmic slope that vary with radius and can be written as: dlnρ dlnr = −2 ( r r−2 )1/n (2.6) where r−2 is the radius from which the slope is -2. The Einasto and the NFW profiles are both ”cusped” profiles because of the steep slope around r=0. a more commonly used ”cored” profile is called the Burkert profile [6262] ρBurkert = ρ0 (1 + r/rs)(1 + r/rs)2 (2.7) As mentioned above, the Burkert profile best fits rotation curves of smaller dwarf sized galaxies. Detection of stellar population of dSph provide mass estimates at different radii that are in agreement with the cored profiles and are incompatible with cusped profiles [6363, 6464]. 2.3 Dark matter detection This section will cover how WIMPs can be detected using three different methods. The first one being direct detection, this involves searching for the recoil of WIMPs scattering on nuclei using very sensitive detectors. This is followed by collider searches which is an attempt to create WIMPs directly by smashing standard model particles together in accelerators like the Large Hadron Collider (LHC). Lastly, indirect detection where one searches for annihilation products or decays of WIMPs in astrophysical environments of high dark matter density like halos. Indirect detection is the main focus of this work thus it will be covered in detail in this chapter. 2.3.1 Direct detection Goodman and Witten [6565] came up with an idea of detecting WIMPs directly by looking for nuclear recoils in target materials of large volumes on Earth. WIMPs interact with standard 13 model particles weakly, in most cases they will not interact with electrons but elastically scatter off the nucleus. When this happens, they will transfer some of their energy to the particles they are interacting with, this momentum transfer will lead to a recoil which could be detected by the experiment. The detection processes include ionisation of target atoms, scintillation, and phonon production generated in crystals. A WIMP with mass 10 GeV to 1 TeV that has undergone elastic scattering would produce nuclear recoil energy of 1 to 100 keV [6666]. This energy is really small, the measurements need to be very sensitive and precise, the advantage of direct detection is its ability to differentiate between galactic signal and Earth based background signals. The expected rate of WIMPs scattering off a nucleus is given by the differential recoil spectrum dR dE (E, t) = ρ0 mχmn ∫ vf(v, t) dσ dE (E, v)d3v (2.8) where E is the nuclear recoil energy, ρ0 is the local dark matter density, mχ is the mass of the WIMP, the target nucleon mass is given by mn. dσ dE is the differential cross-section and the function f(v, t) is the normalized WIMP velocity distribution. The velocity is defined in the reference frame of the detector. The minimum velocity needed for a WIMP to induce a recoil is vmin = √ Emn 2 (mn +mχ)2 (mnmχ) = √ Emn 2 1 µ2 (2.9) where µ is the reduced mass of the system. WIMPs that have a velocity exceeding that of the escape velocity will not be bound by the Milky Way’s gravitational potential and will not be detected. Directionality and annual modulation play a key role in direct detection. The Sun revolves around the center of the galaxy, this motion around the DM halo creates a DM wind. The motion of the Earth around the Sun adds up as a factor to create the annual modulation of the WIMP wind velocity which leads to the recoil rate as mentioned in Equation 2.8Equation 2.8 as seen in the figure below. 14 Figure 2.2: A diagram showing annual modulation and directionality This shows that the velocity distribution of the WIMP should peak in June and be at its minimum in December when Earth is moving in the opposite direction of the galactic disk. There are numerous direct detection experiments that are currently ongoing in search of WIMPs. These include the Dark Matter Large Sodium Iodide Bulk for RAre processes (DAMA/LIBRA) [6767], the Coherent Germanium Neutrino Technology (CoGeNT) [6868] and the XENON1T which has the strongest limits placed on the spin-independent cross-section at about 4.1 × 10−47cm2 for a WIMP mass of 50 GeV at 90% confidence level [6969]. 2.3.2 Collider experiments Collider searches is the use of particle accelerators to smash protons with a center of mass energy of 13 TeV together in hope of finding signals of dark matter particles in the collision. One possible way of detecting WIMP signal would be to find the missing transverse energy (missing ET ), which refers to the undetected energy expected in collisions due to the conservation of energy and momentum. WIMPs can go through the detector without being detected because they interact weakly with standard model particles, this is the missing energy. This method of searching for missing energy in collisions has been interpreted in roughly three classes of models at the Large Hadron Collider (LHC) - complete models, such as the MSSM; effective field theories, where new high mass particles are not included but their effect is encoded in low-energy constants; and simplified models, which are a compromise of the former two, not complete particle physics models but also not effective models as they include new particles [4545, 4646]. 15 ATLAS and CMS have been participating in experiments looking for SUSY particles. Searches have produced strong constraints on SUSY particles with neutralino masses of about 2 TeV [7070]. Direct production of SUSY particles that are weakly coupled have a smaller production rate and the constraints on them are weak. Third generation squarks are constrained to be a few hundred GeV for neutralinos with a similar mass as seen in Figure 2.3Figure 2.3. Figure 2.3: Mediator mass reach of CMS searches for a selection of results targeting electroweak SUSY production. Adapted from [44] Colliders will help most in identifying and characterising the theory to which the dark matter particles belong whilst direct and indirect searches will provide information about how stable the particle are and their cosmological abundance. 2.3.3 Indirect detection Indirect detection refers to observation of dark matter annihilation/decay products in a given astrophysical environment such as the Sun [7171] or galactic dark matter halos [1515] mentioned in subsection 2.2.1subsection 2.2.1 . These annihilation products come in the form of photons, neutrinos, electron positron pairs, proton antiproton pairs and even antideuterium. Various annihilation chan- nels could produce specific signatures, an example includes detection of high energy neutrinos coming from the Sun or gamma-rays coming from the centers of galaxies. It is important to note that nearly all annihilation channels will produce gamma-rays at some point, for hadronic channels it could be through pion production and for leptonic channels it could be via inverse Compton or bremsstrahlung [7272]. This method is very complicated because backgrounds from ordinary astrophysical processes make it difficult to estimate where the sources originate from. Dwarf spheroidal galaxies become a good option since they are mostly dominated by DM which 16 leads to very low background but they also have low predicted fluxes which could be concerning [7373]. Notable observations of cosmic rays and gamma-rays fluxes have provided possible signals of an- nihilation or decay products of dark matter particles. Some of the observations include WMAP and Fermi satellites revealing the excess of gamma-rays and microwaves in the galaxy [7474]. The 511-keV line emission detected by the SPI spectrometer using the INTEGRAL satellite on the bulge of the galaxy [7575]. PAMELA and AMS-02 observed a rise in the positron fraction above 10 GeV spectral line [3333, 3232]. Perhaps the toughest constraints for WIMPs come from the observations of dwarf spheroidal galaxies for masses just below 100 GeV provided by the Fermi-LAT and MAGIC experiments [7676]. It is worth noting that given the limited statistics of the fermi line or the astrophysical backgrounds or large systems of the 511-keV emmision and positron fraction, a dark matter interpretation from these signals is not clear. A boost factor is required to explain the excess flux predicted for the most generic and conserva- tive models such as the steepening of the halo profile. In fact, a number of important theoretical studies have suggested a need for a boost to associate the above mentioned signals with dark matter e.g since the fluxes from DM annihilations depend on the square of the density, some processes could change the inner profile of the galactic center, so strong boosts near the center of galaxies are possible. Another boost could result if the annihilation cross section today is larger than it was at decoupling, especially if there is a strong dependence on relative velocity between DM particles. A p-wave velocity dependent cross section of σv ≈ bv2 could explain the 511 keV emission since it is well known that DM particles that were initially moving at relativistic speeds at decoupling are now moving at slower speeds and that could result in the relic density for a light MeV DM particle that annihilates into non-relativistic electron-positron pairs with σv ≈ 10−5 [7777, 7878]. Similarly, a significant boost is required to explain the positron excess. AMS-02 data suggests that the thermal cross section falls short by two orders of mag- nitude but this can be reconciled if the annihilation follows σv ≈ 1/v. This is the Sommerfeld enhancement and it comes from the exchange of new light particles and it explains the rising positron fraction at high energies [7979, 8080, 8181]. 17 In order to determine the photon flux coming from DM annihilation or decay, one needs to take into account both contribution from particle physics for the relevant cross-section or decay rates and for astrophysics for dark matter density distribution. This flux resulting from annihilation of a WIMP with mass mχ to gamma-rays or neutrinos is defined as dφ dE = PJ (2.10) where P is the particle physics term and J is the astrophysical term. Let the volume in which WIMPs annihilate be dV = l2dldΩ where l is the line-of-sight distance and dΩ is solid angle. We can define the thermally averaged cross-section for the annihilation channel i as 〈σiv〉 such that the annihilation rate per particle is written as channels∑ i ρ(r) Mχ 〈σiv〉 (2.11) The sum with the index i is taken over all annihilation channels. We can get the total anni- hilation in a volume dV by multiplying the annihilation rate per particle with the number of particle pairs in dV since there are two particles that participate in every reaction thus ( channels∑ i ρ(r) Mχ 〈σiv〉 )( ρ(r)dV 2Mχ ) (2.12) . let dNγ,i/dE be the differential annihilation spectrum, the differential flux coming from a volume element dV = l2dldΩ at a distance l is then 1 4πl2 channels∑ i dNγ,i dE ρ(r)2〈σiv〉 2M2 χ l2dldΩ (2.13) 18 Taking an integral over the whole volume in the cone we get the differential photon flux from annihilating dark matter that is expected to be measurable: dφ dE = ∫ ∆Ω dΩ ∫ l.o.s dl channels∑ i dNγ,i dE ρ(r)2〈σiv〉 8πM2 χ (2.14) = 1 8π channels∑ i dNγ,i dE 〈σiv〉 M2 χ × ∫ ∆Ω dΩ ∫ l.o.s dlρ(r)2 (2.15) from this we can see that the particle physics term is P = channels∑ i dNγ,i dE 〈σiv〉 M2 χ (2.16) and the astrophysical term is J = ∫ ∆Ω dΩ ∫ l.o.s dlρ(r)2 (2.17) In a similar fashion, the flux from decaying dark matter can be derived as dφ dE = 1 4π channels∑ i Γi Mχ dNγ,i dE × ∫ ∆Ω dΩ ∫ l.o.s dlρ(r) (2.18) where Γi is the decay width of channel i. For dark matter decay, the signal is proportional to the sight integral of the density, unlike annihilation which is proportional to the square of the density, this gives slightly less sensitivity to the detailed halo model. To get the full picture, it is necessary to understand the full phase-space density which includes not only the density profile but the velocity distribution in order to derive a self-consistent halo model that includes dynamical measurements such as the stellar velocities and rotation curves. the phase-space density of DM is defined as: dN = f(x,v, t)d3xd3v (2.19) 19 where dN is the number of particles within a phase-space element d3xd3v in coordinates (x,v). Assuming that the phase-space is separable such that it can be written as f(x,v) = n(x)fv(v), and DM is self-annihilating, then the rate of annihilation of DM in a halo is Γ = 〈σAvrel〉 ∫ d3xn2x (2.20) the velocity-averaged annihilation cross section 〈σAvrel〉 is taken over relative velocities of par- ticles interating with each other. We assume a velocity independent s-wave annihilation cross section, however, in some cases where the phase-space density is not separable and have a p- wave cross sections like the simulated halos defined in subsection 2.2.1subsection 2.2.1 then it is not possible to simplify the above expression and one must calculate the collision term in the Boltzmann equation [8282]. The annihilation to charged particles situation is slightly different because the diffusion and energy loss also affect the observed signal [8383, 8484]. One needs to convolve the source function for annihilation with the Green’s function for cosmic-ray propagation including diffusion and energy loss mechanisms such as inverse-Compton and synchrotron radiation for electrons. Both of these mechanisms have a dependency on energy and it can combined to give dE dt = −bE2 (2.21) If we include the diffusion constant κ, the transport equation can be solved to give the Green’s function which will gives the intensity of positrons and electrons at position x, time t and energy E due to an impulse at position t and energy E0 G(x, t, E) = (4πκt)−3/2(1− bEt)2exp ( − x2 4κt − t/τ ) δ ( E0 − E −1− bEt ) (2.22) where τ is the lifetime of the WIMP and δ is the Dirac-delta function. This equation shows that for a emitting source at a distance x, that admits a power-law spectrum, there will be a 20 sharp steep in the spectrum beyond E ≥ 4κ br2 . If we combine this with the source function we can get the annihilation rate Q(E, x) = 4π dNe+ dEd3xdt = channels∑ i 〈σiv〉 M2 χ dNe+,i dE × ρ(r)2 (2.23) and we can get the observed, nearly isotropic, electron spectrum by integrating over time and over the distribution of DM: dNe+ dEe+ = ∫ dt ∫ d3x ∫ dE′G(x, t, E,E′)Q(E′, x) (2.24) This can be solved numerically with codes such as the GALPROP [2525]. Some current limits from indirect detection are included in Figure 2.4Figure 2.4 taken from [8585]. This shows the limits obtained by the VERITAS of DM annihilation cross-section into bb̄ and τ+τ− pairs compared with other gamma-ray instruments of the dSph observations. The 95% confi- dence level upper limits to the channel bb̄ is shown on the left of the figure and the τ+τ− limit on the right. For bb̄ the constraints reach < σv > ≈ 10−23cm3s−1 at mχ ≈ 1 TeV and the τ+τ− limits are < σv >≈ 3× 10−24cm3s−1 at mχ ≈ 300 GeV Figure 2.4: The upper limits of the dSph observations by VERITAS (black solid line) to the DM annihilation cross section into bb̄ (left) and τ+τ− (right) pairs. See legends for details of the comparison 21 Chapter 3 Reionization and the 21 cm Hydrogen Line This chapter will review the Epoch of Reionization and how it can be detected with the method of the redshifted 21-cm line of hydrogen. This chapter will briefly explain the physics behind the 21-cm hydrogen hyperfine line and its detectability. An overview of how dark matter can alter the reionization history will be provided in section 3.3section 3.3 as well as how it affects the 21-cm spectrum. Then subsection 3.3.1subsection 3.3.1 will discuss a code package called DarkHistory by Ref.[1010] which will be used as a method of acquiring results in the following chapter. We then proceed to give a brief outline of how we calculate the sensitivity of the Square Kilometer array (SKA) and Hydrogen Epoch Of Reonization Array (HERA). 22 3.1 Epoch of Reionization The Epoch of Reionization (EoR) is one of the most exciting problems in modern physical cosmology. The possibility of observing the events that lead to EoR promises rich information about the complicated astrophysics that is responsible for galaxy formation and it could shape our current understanding of fundamental physics. Scientists have been able to use the CMB as a means to explain how complex and non-linear structures formed from the observed initial perturbations that are inhomogeneous in 1 part in 100 000. This paradigm has been very successful explaining the early Universe, however, the events after that: the Cosmic Dark Ages and the Cosmic Dawn have been untested by observation. These periods will provide a better understanding of the phase transition of a neutral Universe to a completely ionized one. About 380 000 years after the big bang, the Universe sufficiently cooled for protons and electrons to combine to form neutral hydrogen, this is the epoch of recombination. During this time, photons decoupled from the gas and this relic radiation can be observed today in the microwave spectrum and it is what we refer to as the CMB. This is followed by the cosmic dark ages where the universe was filled with neutral hydrogen and had no luminous sources. It is during this time that larger non-linear structures begin to form and collapse to form sheets, filaments and halos. These eventually form stars and galaxies which then emit photons ending the dark ages, bringing about the cosmic dawn at a redshift of about z ≈ 30. The first stars heated the surrounding intergalactic medium (IGM) forming near-infrared and non-ionizing ultra-violet (UV) and X-ray backgrounds which raised the temperature of the gas in the Universe and started the reionization process where the ionization radiation escapes the galaxies and ionizes their surrounding IGM. This is called the Epoch of Reionization [77, 3737, 3939]. A brief history of the Universe is shown in the Figure 3.1Figure 3.1. The main contributors of the reionization process are considered to be the young, star-forming galaxies but some theories suggests that dark matter annihilation/decay may have played a role in this [8686, 8787, 8888], we try to investigate this claim in this research. Observing the EoR has been a considerable challenge, to this day, the majority of EoR related observations provide weak and model dependent constraints on reionization. Some observations cannot agree as to when the reionization process was completed. This includes the absorption spectra of high redshift quasars suggesting that the universe was neutral by redshift z ≈ 6 [8989, 9090] 23 Figure 3.1: A schematic diagram of the history of the Universe from the big bang to now. After the big bang the Universe undergoes recombination when the photons get decoupled from the gas and the relic radiation is observed as the CMB at redshift = 1000. After the recombination epoch, the Universe is neutral in the period known as the Dark Ages. About 500 million years after the big bang, the first stars and galaxies form and heat up the Universe and starting the Epoch of Reionization. The Universe is fully ionized around 1 billion years after the big bang, at redshift ≈ 6. This image is credited to NAOJ [55]. whereas the CMB radiation obtained by the Planck satellite suggesting that reionization ended at z ≈ 8.8 ± 1.1 [1717], It is important to note that this is model dependent. This means that most of the information that is available comes from theory and simulations, however, there are now up and running telescopes that try to rectify this issue. Out of the many probes and techniques the most promising one and the focus of this dissertation is the measurement of the redshifted 21 cm line of hydrogen. There are a number of efforts dedicated to detecting the EoR by mapping the 21 cm line with radio instruments such as the Precision Array to Probe the Epoch of Reionization (PAPER) [9191], Low Frequency Array (LOFAR) [9292], Murchison Widefield Array (MWA) [9393] as well as the Hydrogen Epoch of Reionization Array (HERA) [9494] and the Square Kilometer Array (SKA) [3737]. The last two instruments will be used in this work to constrain some results. Since the universe is mostly filled with hydrogen, this technique will prove to be useful and the aim is to measure the 21 cm intensity from the intergalactic neutral hydrogen with the CMB being a background source at redshifts of about z 6 12 but the SKA will take this a step further by going beyond reionization and making detailed intensity maps to a redshift of z = 30 while HERA will also be able to probe EoR to z ∼ 18. The results of 24 this could provide great insight about the evolution of the EoR, the sources that ionized the gas as well as the sizes of the ionized regions. 3.2 21 cm line of hydrogen The 21 cm line is a hyperfine transition line of neutral hydrogen between two states caused by the interaction of the magnetic moments of the electron and the proton. The lower singlet state arises when the spins are antiparallel and the excited triplet state when the spins are parallel. The difference in energy between the two states is ∆E10 = 5.9× 10−6 eV, this is a frequency of 1420 MHz, corresponding to a wavelength of λ = 21.1061 cm [9595]. The 21 cm line is forbidden by selection rules of quantum mechanics, meaning that the probability for a spontaneous 1 → 0 transition is given by the value A10 = 2.85× 10−15sec−1, known as the Einstein’s coefficient. This very low spontaneous emission rate corresponds to a half-life of τ1/2 = A−1 10 = 3.5× 1014s ≈ 11Myrs (3.1) Despite this, the 21 cm transition line still remains one of the most important astrophysical probes because of three main reasons. The first being that neutral hydrogen is ubiquitous in the early universe, some fraction of it will be in the triplet state so the spontaneous emissions are not rare. Secondly, The optical depth of HI gas is very small at relevant 21 cm frequencies, so transmission rates are larger. Lastly, the population of the triplet states are established by the efficiency of collisions and Lyman-α pumping. The brightness temperature Tb is the fundamental observable for telescopes in detecting the redshifted 21 cm line. The brightness temperature can be associated with the temperature of a blackbody emitting at an observed Intensity at appropriate frequencies derived from the Rayleigh-Jeans limit such that: Iν = Bν(Tb) = 2ν2kbTb c2 (3.2) 25 where kb is the Boltzmann’s constant and c is the speed of light. The radiative transfer equation normally written as Iν = dI dν can be used to derive the brightness temperature. For thermally emitting material at a temperature T, the transfer equation can be written in terms of the optical depth for absorption as, dIν dτν = −Iν +Bν(T ) (3.3) τν is the optical depth through a cloud at a specific frequency and Bν is the Planck func- tion. Using the relations of Intensity and brightness temperature in Equation 3.2Equation 3.2, brightness temperature can be expressed as dTb dτν = −Tb + TCMB (3.4) TCMB represents the background CMB temperature. The differential equation above can be solved to find the temperature of the radiation at frequency ν [3838] Tb(ν) = Ts(1− e−τν ) + TCMB(ν)e−τν (3.5) This equation provides information about specific conditions that influence the brightness tem- perature signal. The transmission probability is given by the background radiation factor exp(−τν) and the emission probability is given by the factor 1-exp(−τν). The brightness tem- perature depends on two factors, namely the spin temperature and the optical depth. The spin temperature Ts is the ratio between the occupation of the two hyperfine levels expressed as, n1 n0 = g1 g0 e−T?/Ts (3.6) where n0 and n1 are the number densities of electrons in the singlet and triplet states respec- tively. The statistical weights of the energy levels are given by g1 = 3 and g0 = 1. T? = 0.0681K is the temperature associated with the 21 cm wavelength. 26 The optical depth can be derived using the fact that the 21 cm line is an ideal transition that can be described by the Einstein’s coefficients and their relations. The Einstein Coeffiecient B01 corresponding to absorption and the emission coefficient B10 can be produced by the 21 cm radiation incident on an atom. Their probabilities are given by Ref [9696], IνB01 = g1 g0 B10Iν (3.7) and IνB10 = A10 λ2Iν 2hν10 (3.8) The absorption cross section of the 21 cm line is σν = σ01φ(ν) = 3c2A10 8πν2 φ(ν) (3.9) φ(ν) is known as the line profile and is normalised such that ∫ dνφ(ν) = 1. The optical depth of a cloud of hydrogen is then τν = ∫ ds(1− e−E10/kBTs)σ0φ(ν)n0 (3.10) Using the fact that the optical depth is small at required frequencies, the observed 21 cm brightness temperature is δTb = Ts − TCMB 1 + z (1− e−τν ) (3.11) ≈ Ts − TCMB 1 + z τ (3.12) 27 ≈ 27xHI(1 + δb) ( Ωbh 2 0.023 )( 0.15 Ωmh2 1 + z 10 )1/2(Ts − TCMB Ts )[ ∂rvr (1 + z)H(z) ] mK, (3.13) xHI is the neutral fraction of hydrogen, δb is the fractional overdensity in baryons and ∂rvr is the velocity gradient along the line of sight [3939]. This result shows that in order for a brightness temperature signal to be observed, the spin temperature needs to deviate from the CMB. The physics that governs the spin temperature is thus important in understanding the signal. The ionization history of hydrogen also plays a vital role, which makes this the ideal probe for reionization. 3.2.1 Determining the spin temperature The following mechanisms are responsible for determining the spin temperature: a) Absorption of background radiation and stimulated emission; b) collisions with other particle species such as HI atoms, protons and free electrons; and c) lyman-α scattering that involve intermediate excited states. The equation that describes the spin temperature is given by [77] T−1 s = T−1 γ + xcT −1 k + xαT −1 c 1 + xc + xα (3.14) xc is the collisional coupling coefficient and xα is the Lyman-α coefficient. Tγ is the background temperature, in this case the CMB and Tk is the gas temperature. Tc is the colour temperature of the Lyman-α radiation. The coupling coefficients determine whether collisions or Lyman-α photons affect the signal and provides information about how strongly coupled is the spin tem- perature to the gas temperature. The full evolution of the spin temperature can be established once the five parameters in Equation 3.14Equation 3.14 can be calculated. The underlying physics leading to the contributing factors is explored below. 28 3.2.1.1 Collisional coupling Collisions between particles induce a spin-flips in a hydrogen atom and can affect the spin temperature in the early Universe when the particle densities were high. There are three collision channels available: collisions between H-H atoms [9797], H-e [9898] and H-p [9999]. The collisional coupling for a species i can be written as xic ≡ C10 A10 T? Tγ = niκ i 10 A10 T? Tγ (3.15) where κi10 is the specific rate coefficient for spin deexcitation by collisions with species i and C10 is the collisional excitation rate. The total coupling coefficient can be written to include the three channels, xc = xHHc + xeHc + xpHc (3.16) xc = T? A10 ( nHκ HH 10 + neκ eH 10 + npκ pH 10 ) (3.17) where κHH10 represents the scattering rate for hydrogen atoms, κeH10 is for electrons and hydrogen and lastly κHH10 is between protons and hydrogen atoms. The collision rates (with units of cm3s−1) are obtained through quantum mechanical calculations for relevant particle interaction and the values are tabulated in Ref. [9797, 9898, 9999, 100100]. The general form of the scattering rate for the three collision channels is κi10 = √ 8kBTk πMi σ̄i (3.18) where the average spin transition cross section is σ̄i = 1 (kBTk)2 ∫ dEσ̄i(E)Ee E kBTk 29 This can be interpreted as the average collision velocity for each channel. Collisional coupling is the most prominent coupling during the cosmic dark ages and the details of the process become important. The calculations above, for example, assumes that the collisional cross-sections do not depend on velocity since the velocity dependency leads to a non-thermal distribution for the hyperfine occupation [77]. This can cause a suppression of the 21 cm signal by at least 5 %, which is important in the context of making use of the 21 cm signal for precision cosmology [9898]. 3.2.1.2 Wouthuysen-Field effect The Wouthuysen-Field effect becomes more important during the formation of the first stars in the Cosmic Dawn. It describes the coupling of the spin temperature to the Lyman-α photons i.e UV photons. This process is illustrated in Figure 3.2Figure 3.2 and it shows the hyperfine structure of the hydrogen 1S and 2P levels. If hydrogen is initially in the singlet state then the absorption of Lyman-α photons will excite the atom to the 2P state and follows the dipole selection rules which only allows for transitions when ∆F = 0 ± 1 and no F = 0 → 0 which makes the other levels inaccessible. The only accessible states are the 21P1/2 or the 21P3/2 and from there the atom can deexcite down to the 11S1/2 state. The rest of the transitions are illustrated in Figure 3.2Figure 3.2. The coupling can be written as xα = 4Pα 21A10 T? Tγ (3.19) Pα is known as the scattering rate of Lyman-α photons. The relation P01 = 4Pα/27 was used [101101]. The total rate per atom at which Lyman-α photons scatter is given by Pα = 4πχα ∫ dνJν(ν)φα(ν) (3.20) 30 where σν = χαφα(ν) is the local absorption cross section, the oscillation strength of the Lyman- α transition is given by χα ≡ πe2fα/mec, φα is the Lyman-α absorption profile and Jν(ν) is the angle-averaged specific intensity of the background radiation field by number. Figure 3.2: The Wouthuysen-Field effect energy levels. The forbidden transitions are in lines and the dotted lines are the allowed transition but do not contribute to spin flips. Image taken from Ref. [66] Combining Equation 3.19Equation 3.19 and Equation 3.20Equation 3.20, an expression for the coupling can be described as xα = 16π2T?e 2fα 27A10Tγmec SαJα = Sα Jα JCα (3.21) 31 The suppression factor which describes the photon distribution close to the Lyman-α resonance is, Sα = ∫ dxφα(x)J(x) (3.22) The specific flux Jα can be evaluated at the Lyman-α frequency. Equation 3.21Equation 3.21 can be used to calculate the critical flux needed to produce xα = Sα where JCα = 1.165 × 1010[(1 + z)/20]cm−2s−1Hz−1sr−1. The critical flux in terms of the number of Lyman-α photons per hydrogen atom can be expressed as JCα /nH = 0.0767[(1 + z)/20]−2. The physics described above describing the Wouthuysen-Field effect couples the spin tempera- ture to the colour temperature of the UV radiation field. Tc is a measure of the shape of the radiation field as a function of frequency and is defined by h kBTc = −dlognν dν (3.23) where nν = c2Jν/2ν 2 is the photon occupation number. The shape of the Lyman-α profile will be similar to that of a black body at the gas temperature Tk for an optically thick medium since the Lyman-α photons scatter at a higher rate. Therefore Tc ≈ Tk for frequencies of interest. This means that in astrophysics, the primary interest lies with photons that redshift into the Lyman-α resonance from frequencies below the Lyman-β resonance. Figure 3.3Figure 3.3 shows the Lyman-α photons that can be produced by atomic cascades from photons redshifting into higher Lyman-n series resonances. The probability of converting a Lyman-n into a Lyman- α photon is given by the atomic rate coefficients in the table found in [102102]. About 30% of conversion is typical for large n which changes the contribution of photons that are injected to the Lyman-α line to the Wouthuysen-Field coupling. 32 Figure 3.3: The solid lines represent ionizing photons redshifting into Lyman-n resonance, which then cascade down via multiple possible decay channels. Some decay channels lead to lyman-α emissions contributing to the Lyman-α background. This figure shows decay chains for Lyman-β and Lyman-γ. Dashed lines represents Lyman-n transitions and red-dashed line is the Lyman-α transition. The forbidden transitions are depicted by the dotted lines. Figure taken from [77] 3.2.2 Thermal evolution of the IGM The next important ingredient of the spin temperature is the thermal evolution of the IGM parametrized by the gas temperature Tk and free electron fraction xe. The evolution of the gas temperature Tk can be modelled as dTk dt = 2Tk 3n dn dt + 2 3kB ∑ i εi n (3.24) Where n is the number density of neutral gas particles, kB is the Boltzmann’s constant. The first term is a cooling term due to the Hubble expansion, the second term describes the heating of the IGM determined by the factor εi which is the heating rate per unit volume taken over processes i. To solve the above equation, specific heating mechanisms that play a role at different times in the history of the Universe need to be dissected. 33 3.2.2.1 Compton Heating Before star formation, the dominant mechanism is the heating of the gas from Compton scat- tering between the CMB photons and the residual free electron fraction. The energy of the electrons increases during the scattering process which can transfer the excess energy from the CMB to other particles in the gas. The heating rate per particle due to Compton scattering is calculated in [103103, 104104] 2 3 εcompton kBn = xe 1 + fHe + xe ( Tγ − Tk Tγ ) µγ µ̄γ (1 + z)4 (3.25) where µγ is the energy density of the CMB, fHe is the helium fraction (by number) and we define the Compton cooling time as tγ ≡ 3mec 8σTµγ (3.26) where σT = 6.65×10−25cm2 is the Thomson cross section. At high redshifts when astrophysical sources are not yet prevalent, Compton heating couples the gas temperature to the CMB such that the IGM cools at the same rate as the CMB i.e Tk ∝ (1+z). At a later stage, the IGM decouples from the gas and cools adiabatically with the expansion thereafter. 3.2.2.2 Lyman-α heating Lyman-α plays a secondary heating mechanism after X-ray heating which is thought to be the largest contributing mechanism. Lyman-α requires large fluxes of Lyman-α photons so it is most important at later stages of the evolution and may be insufficient to the gas to the CMB temperature alone. X-ray heating leads to the suppression of the Lyman-α background. The energy loss in the radiation background therefore goes into heating the IGM. The heating rate can be found in [105105] 2 3 εα kBnHH(z)Tk ≈ 0.8 T 3/4 k xα Sα ( 10 1 + z ) (3.27) 34 3.2.2.3 X-ray heating X-ray heating of the gas is the most important mechanism contributing to the gas temperature. As star formation takes place, high intensity X-ray sources such as population III stars, black holes and quasars fill the Universe [3838, 106106, 3939]. X-ray photons emitted by these sources are able to heat the IGM far from the source since they have a long mean free path, this ensures homogeneous heating of the gas. The comoving mean free path of an X-ray photon is given by [9898] λx = 4.9x̄ 1/3 HI ( 1 + x 15 )−2( E 300eV )3 Mpc (3.28) The Universe will be optically thick for photons below the energy E = 2 [ (1 + z) 15 ]1/2 x̄1/3keV (3.29) The E−3 dependence means that soft X-rays which fluctuate at small scales dominates the heating while there is an existing uniform harder X-rays also contributing to the heating. The relationship between star formation rate, SFR, and X-ray luminosity, Lx, can be extrapolated at high redshift as [9898] Lx = 3.4× 1040fx ( SFR 1M�yr−1 ) (3.30) where fx is some renormalization factor. Understanding the high redshift X-ray sources is crucial in determining fx. The high energy X-ray photons ionize HI and HeI, this creates the energetic free electrons which heat the IGM through three main channels. the first being collisional ionization which produce secondary electrons, the second is collisional excitation of hydrogen and helium, this produces photons capable of ionizing HI and Lyman-α background. Lastly, Coulomb collisions with free electrons. The fraction of the X-ray energies going into 35 heating, ionization and excitation are determined by their relative cross sections and are related by fx,h ∼ 1 + 2x̄i 3 (3.31) fx,ion ∼ fx,coll ∼ 1− x̄i 3 (3.32) The X-ray energy input to the gas can be determined by assuming SFR is proportional to the rate of the gas collapse into virialized halos [9898], 2 3 εx kBnH(z) = 103Kfx ( f? 0.1 fx,h 0.2 dfcoll/dz 0.01 1 + z 10 ) (3.33) where f? is the star formation efficiency. This equation only takes star formation rate into account, the other sources such as quasars need to be studied in greater detail in order to increase the accuracy of X-ray heating. There is also heating closer to the halo from gravitational collapse and a contribution from the UV photoionization, however this is subdominant in the IGM. 3.2.3 Ionization History The ionization history is the next important step towards a full understanding of the spin temperature and the brightness temperature. The mechanisms mentioned above as well as the collisional coupling mostly depend on the ionization state of the the Universe. The production rate of ionizing photons is assumed to be coupled to the star formation rate, with an average ionizing efficiency defined as [107107] x̄i = ζfcoll 1 + n̄rec (3.34) 36 where n̄rec is the average number of recombinations per ionized hydrogen atom and ζ is the ionization efficiency given by ζ = AHef?fescNion (3.35) where Nion is the number of ionizing photons produced per stellar baryons, AHe is a correction factor for the presence of Helium and fesc is the fraction of ionizing photons that escape the halos. At the late stages of ionization, recombination rates become important, ionized bubbles create dense clumps that act as sinks of ionizing photons which slows down the expansion of the bubble. this term that opposes ionization is defined to be ( dx̄i dt rec ) = −αC(z, x̄i)x̄i(z)ne(z) (3.36) with α denoting the recombination coefficient and C ≡ 〈 n2 e 〉 / 〈ne〉2 being the clumping factor and ne is the mean electron density in ionized regions. The overall neutral fraction is given by dx̄i dt = ζ(z) dfcoll dt − αC(z, x̄i)x̄i(z)ne(z) (3.37) The full analysis of the above equation depends on parameters which are poorly constrained by observation and are detailed in Ref. [77, 9898] 3.2.4 Global 21 cm signal We have described the physics that determines the evolution of the spin temperature, gas temperature, ionization history and the overall brightness temperature in the previous section, now we are ready to provide a general prediction of the global evolution of the 21 cm signal. The 21 cm signal can be summarized in Figure 3.4Figure 3.4 below. The important regimes are described below. 37 Figure 3.4: The evolution of the global 21 cm signal. The signal transitions from the early phases of collisional coupling. There is no signal when the gas temperature couples to the background CMB temperature. Fluctuations in the later stages are dominated by various heating mechanisms such as Lyman-α, X-ray, and ionizing radiation background. Relevant redshifts accompany the subsequent radiation backgrounds. Image taken from [77] • 1100 ≥ z ≥ 200: after recombination, high gas densities collisionally couple the spin and gas temperature to the CMB, Ts = Tk = TCMB ∝ (1+z) and no 21 cm signal is expected, T̄b = 0. • 200 ≥ z ≥ 40: In this regime, the gas decouples from the CMB since the Compton scattering rate falls below the expansion rate. This leads to adiabatic cooling of the gas, Tk ∝ (1 + z)2 leading to T̄b < 0 and an early absorption signal. • 40 ≥ z ≥ z?: As the Universe continues to expand, the gas density decreases, making collisional coupling ineffective. The spin temperature couples to the CMB again, Ts = TCMB and the brightness temperature approaches 0 and there is no 21 cm signal that can be detected. • z? ≥ z ≥ zα: The process of star formation starts at z?, These first sources emit both Lyman-α and X-rays, heating the gas significantly. The Wouthuysen-Field effect couples the spin temperature back to the gas such that Ts ∼ Tk < TCMB and absorption signals can be detected. 38 • zα ≥ z ≥ zh: As more stars form, Lyman-α coupling saturates, when xα >> 1. Fluctu- ations in the Lyman-α flux no longer affects the 21 cm signal. At this point, brightness temperature fluctuations are sourced by gas temperature fluctuations. Absorption is seen while the gas temperature remains below the CMB temperature, but hotter regions may begin to be seen in emission as the gas temperature approaches the CMB temperature. • zh ≥ z ≥ zT : 21 cm signal can be seen in emission after the gas is heated above the CMB temperature, Tk > TCMB. By this time, the ionization fraction has likely risen to the 1% level. The signal is sourced by a mixture of fluctuations in the ionization fraction, density and gas temperature. • zT ≥ z ≥ zr: As more sources turn on, heating continues and drives the gas temperature well above the CMB temperature, Tk >> TCMB. At a redshift zT the temperature fluctuations become unimportant, Ts ∼ Tk >> TCMB and the dependence on the spin temperature in Equation 3.13Equation 3.13 can be ignored. Fluctuations in the ionized fraction become dominant at this point. • zr ≥ z ≥ z: After reionization, the remaining 21 cm signal would have to originate from collapsed islands of neutral hydrogen It is important to note that most of these epochs are not well defined and there may be overlaps between them. The exact value of the redshifts stated above are model dependent. This dependency lies in the physics of the first stars, which is still insufficiently understood. The measurement of the global 21 cm signal could therefore constrain a large set of parameters. 39 3.3 Exotic energy injection and dark matter The discussion of the 21 cm global signal provides a clear picture of the reionization process. Early stars and quasars play a central role in reionization but their energy contribution is still a matter of debate in the research space. A number of studies have considered beyond standard model physics approach to make predictions for exotic heating of the IGM. Potential candidate sources of energy have been explored and a few present a compelling solution. Among these candidates, the most promising is dark matter. It is thought that DM annihi- lation in the early Universe can act as a source of X-rays which lead to ionization, heating and other processes [88]. DM annihilation rate scales as the square of the density, n2 DM , this indicates that it rises with the onset of structure formation and that the collapse of DM into halos can have a significant energy injection in the process of reionization. Annihilation and decay products have the potential to alter the levels of ionization, can heat up neutral gas and increase the production of Lyman-α photons. Figure 3.5Figure 3.5 shows the potential WIMP annihila- tion cross-section constraints coming from CMB limits on the ionized fraction xe at z ∼ 6 for the scenarios of χχ→ e+e− and χχ→ γγ [88]. It is important to note that these constraints are stronger than those from the Planck experiment [1717] and they go beyond the two annihilation presented above. We can understand the effects of DM decays/annihilations on the thermal and ionization evolution of the IGM, which in turn can make us understand how DM affects the 21 cm background signal. The evolution of the ionized fraction xe is given by the equation from Ref.[108108] −dxe dz = 1 H(z)(1 + z) [Rs(z)− Is(z)− Ix(z)] (3.38) where Ix = Ėx/E0 represents the ionization rate due to dark matter. Is and Rs are the standard ionization and recombination rates per baryons. Ref.[99] shows that we can have Ix = χi(z) Ėx E0 = fabs(z)χi(z)Γxfx mpc 2 E0 (3.39) 40 for DM decays, and Ix = χi(z) Ėx E0 = fabs(z)χi(z)fx mpc 2 E0 nDM,0Nb(z)〈σv〉 (3.40) for DM annihilation. Where fabs(z) is the energy absorbed fraction, which is the fraction of the DM particle rest mass that is absorbed by the gas at any redshift. E0 = 13.6 eV is the hydrogen ionization energy, mp is the mass of the proton, χi is the fraction of the energy absorbed by the IGM from dark matter decay/annihilation that goes into ionization. Γx = 1/τDM where τDM is the lifetime of DM particles and finally nDM,0 is the current number of DM particles per baryon. The equation that govern the evolution of the IGM temperature can be written as (1 + z) dTk dz = 2Tk + Iγxe H(z)(1 + fHe + xe) (Tk − TCMB)− 2χhĖx 3kbH(z)(1 + fHe + xe) (3.41) where χh is the fraction of the absorbed dark matter energy deposited into the IGM as heating [109109]. Iγ = (8σTaRT 4 CMB)/(3mcc) and fHe is the helium fraction by mass. One last equation needed for completion is the Lyman-α background intensity Jα written as Jα(z) = N2 Hhc 4πH(z) [ xexpα eff 22p + xexHIγeH + χα ˙Ex(z) NHhνα ] (3.42) where the first and the second term are contributions from recombination and collisional exci- tations, while the third term is the dark matter contribution. NH is the number density of HI atoms, αeff 22p represents the effective recombination coefficient to the 22P level. Lastly, γeH is the collisional excitation rate of hydrogen atoms. This formalism allows us to obtain the 21 cm radiation intensity in the presence of DM. In this work, exotic energy sources such as WIMPs will be injected in our model universe. We are going to be looking at two different annihilation channels, specifically bb̄ and ττ . This will allow us to observe how the global 21 cm signal gets altered in the presence of WIMPs of varying masses and annihilation cross sections. An example of DM effects on heating and reionization of neutral gas can be traced via the 21 cm brightness temperature spectrum as shown in Figure 3.6Figure 3.6 41 [99]. This shows the fiducial δTb without any DM annihilation or decay in solid lines compared to the 25-keV decaying warm dark matter (WDM) (long dash) and 10 MeV decaying light dark matter (LDM) (short dashed) and 10 MeV annihilating LDM (dotted lines). The figure demonstrates that light DM is very effective at influencing the brightness temperature while the WDM is less effective. This means that measurements of the 21-cm background will be able to effectively constrain many models of decaying and annihilating dark matter. Figure 3.5: Constraints of WIMP annihilation cross-section to reionization for annihilation channels χχ→ e+e− (left) and χχ→ γγ. The hatched regions represents the parameter space ruled out by the optical depth constraints (orange) and the CMB power spectrum constraints measured by Planck (red). The contribution to the free electron fraction xe is depicted by the colour density plot with the black, dashed contours shown for a contribution to xe at z = 6. image taken from [88]. 42 Figure 3.6: The effects of decaying and annihilation WMD and LDM models on the 21 cm brightness temperature. The solid line shows δTb for a baseline model without any inclusion of DM effects. The dotted line as well as the long dashed and the short dashed lines represent δTb with 10 MeV annihilating DM, 25 keV decaying WDM and the 10 MeV decaying WDM, respectively. Image taken from [99] 3.3.1 DarkHistory This section will cover a package called DarkHistory by [1010] as a method to calculate the exotic energy injection as well as ionization histories to effectively constrain annihilating WIMPs and see how much they affect reionization. DarkHistory is a public code package that allows fast and accurate computation of the effects of exotic energy injection on astrophysical and cosmological observables. This work will focus on the interactions that allow DM to annihilate to electromagnetically interacting SM particles. DarkHistory allows users to adjust inputs to the calculations such as changing reionization models or the DM annihilation or decay spectrums. The most striking features of the package include: • The first fully self-consistent treatment of exotic energy injection. The evolution of the IGM temperature TIGM and free electron fraction xe can be modified by exotic energy injections and previously, this modification was assumed to be perturbative and that the backreaction effects were due to these modifications were negligible. Backreaction is the modification to the ionized /temperature history which in turn modify the energy-loss processes for injected particles. This assumption breaks down before the cosmic dawn [88]. 43 DarkHistory can simultaneously solve for the evolution of the temperature and ionization as well as cooling of the injected particles. • A self-consistent treatment of astrophysical sources of heating and reionization, this allows for the flexibility of switching between ordinary sources of energy and exotic sources of energy. • Allows for a significant increase in calculations since there is a pre-computation of relevant transfer functions of particle energy, redshift and ionization levels. • treatment of helium ionization and recombination of helium • A novel approach for treatment of inverse Compton scattering (ICS) for high speed elec- trons. These improvements allow DarkHistory to rapidly scan over a number of different prescriptions for reionization in the form of photoheating or photoionization rates, or a hard-coded back- ground evolution for the free electron fraction xe. In the subsections below, we will outline the theoretical calculations needed to get the results as well as demonstrate as an example how to use the package. 3.3.1.1 Thermal and ionization histories DarkHistory can calculate the evolution of the temperature and ionization parameters with exotic energy injections included such as DM using an updated three-level atom (TLA) model for HI and He from RECFAST [104104, 110110]. Without exotic energy injections, the differential equations that describes the matter temperature and ionization fraction can be modelled as Ṫ (0) m = −2HTm + ΓC(TCMB − Tm) (3.43) ẋ (0) HII = −C [ nHxexHIIαH − 4(1− xHII)βHeE21/TCMB ] (3.44) 44 where H is the Hubble parameter, nH is the total number density of both neutral and ionized hydrogen. nHII ≡ nHII/nH where nH is the number density of free protons. xe is the free electron fraction and E21 is the Lyman-α transition energy. αH and βH are case-B recombination and photoionization coefficients for hydrogen respectively. The Peebles-C factor C is defined as the probability of a hydrogen atom decaying from a n=2 state to the ground state before photoionization occurs. The Compton scattering rate ΓC is given by [111111] ΓC = xe 1 + FHe + xe 8σTarT 4 CMB 3me (3.45) where ar is the radiation constant, σT is the Thompson cross-section, me is the electron mass and FHe is the relative abundance of helium nuclei by number. Exotic sources may inject energy on top of the baseline temperature and ionization histories, Ṫ (0) m and ẋ (0) HII which can alter the thermal and ionization evolution in the Universe. The rate at which energy is injected from DM annihilating with a velocity averaged cross-section 〈σv〉 or decaying with some lifetime τ much longer than the age of the Universe is given by ( dE dV dt )inj =  ρ2 χ,0(1 + z)6 〈σv〉 /mχ, annihilation ρχ,0(1 + z)3/τ, decay (3.46) where ρχ,0 is the mass density of DM today. It is important to note that this injected energy does not manifest itself instantaneously as ionization, excitation or heating of the gas. Instead, the primary particles injected into the Universe cools over timescales much larger than the the Hubble time, producing secondary photons that may redshift significantly before depositing their energy into the gas. The energy deposited into a channel c (hydrogen ionization, excitation or heating) can be parametrized once the cooling of injected primary particles can be determined, and it is given by ( dE dV dt )dep c = fc(x,x) ( dE dV dt )inj (3.47) The energy deposition also depends on the ionization fraction and the redshift of all the appro- priate species in the gas, which can be denoted as x ≡ (xHII , xHeII , xHeIII). For simplification, 45 helium is ignored, which makes the ionization dependence of these fc(z,x) only depend on hy- drogen, xHII = xe. With additional source terms included, the results of the injected energy on the ionization and thermal histories can be written as Ṫ injm = 2fheat(z,x) 3(1 + fHe + xe)nH ( dE dV dt )inj (3.48) ẋinjHII = [ fHion(z,x) RnH + (1− C)fexc(z,x) 0.75RnH ]( dE dV dt )inj (3.49) where R = 13.6 eV is the ionization potential of hydrogen. fc(z,x) has been previously calcu- lated with the assumption that the standard ionization history was computed by recombination codes, xstd(z), making redshift the only independent function of fc. Therefore, these calculations are applicable as long as the purtabations to the assumed ionization history are significantly small. This assumption holds true on the onset of recombination where the ionization history is well constrained by CMB measurements and large perturbations to the free electron fraction are not highly favoured. There is an exception for redshift below 100 where the ionized levels exceed the standard value of xe ∼ 2× 10−4 by quite a large margin [88]. During the reionization process, star formation causes the ionization and heating processes to happen at an accelerated rate, this makes the thermal and ionization history to separate from the baseline. This in turn decreases a portion of the injected energy that falls into ionizing these species while an increase in xe increases the number of charged particles available for low-energy electrons to heat the IGM, increasing the energy fraction that goes into heating. Calculating the full x-dependence of fc(z,x) allows for consistent calculation of temperature and ionization histories with both exotic energy injection processes and reionization. 3.3.1.2 Calculating fc(z) Energy from low-energy photons and low-energy electrons can be transferred into the excita- tion and ionization of atoms, it can also go into the heating of the IGM, as well as the addition of free-streaming photons to the CMB continuum. DarkHistory has the capability to moni- tor the amount of energy that low-energy photons and electrons deposit into a channel c ∈ (Hion, Heion, exc, heat, cont), which represent hydrogen ionization, helium ionization, hydrogen 46 excitation, heating of the IGM and sub-10.2 eV continuum photons respectively. The function fc for each channel is found by normalizing the total energy deposited into channel c within a redshift step by the total injected energy according to Equation 3.47Equation 3.47. We summarize the important results from [1010] for each channel below. The total amount of energy deposited per unit time and volume is ( dE dV dt )dep = ( dE dV dt )dep low + ( dE dV dt )dep high (3.50) where the energy deposited for a low-energy photon and electron spectra Nγ low[Ei] is given by ( dE dV dt )dep low = 1 G(z) ∑ α ∑ i E′iN α low[Ei] (3.51) and for the high energy photons and electrons: ( dE dV dt )dep high = 1 G(z) ∑ c Ehighc (3.52) where G(z) is the conversion factor between the spectra containing the number number of particles produced per unit time and volume and is written as G(z) ≡ ∆log(1 + z) nB(z)H(z) (3.53) This provides glimpses into the splitting of the deposited energy into separate channels. 47 For photons, fc is computed for low-energy, starting with energy deposition into continuum photons. The photons carry an energy that is ≤ 10.2 eV and they cannot efficiently transfer their energy into free electrons or atoms so they just stream freely. Their energy deposition is ( dEγ dV dt )dep cont = 1 G(z) 3R/4∑ Ei=0 EiN γ low[Ei] (3.54) The amount of energy deposited into hydrogen excitation, assuming that all photons with energies between 3R/4 = 10.2 eV and R = 13.6 eV deposit their energy instantaneously into hydron Lyman-α according to [3131]: ( dEγ dV dt )dep exc = 1 G(z) R∑ Ei=3R/4 EiN γ low[Ei] (3.55) for ionization, photons above the ionization potential R in the above equation have the ability to photoionize one of the species HI, HeI and HeII. The depostion energy for hydrogen ionization is given by ( dEγ dV dt )dep Hion = R G(z) ∑ Ei>R qγH [Ei]N γ low[Ei] (3.56) and for helium ionization: ( dEγ dV dt )dep Heion = RHe G(z) ∑ Ei>RHe qγHe[Ei]N γ low[Ei] (3.57) For electrons the results obtained from the MEDEA code [112112] are used to track high-energy electrons as they are injected into the IGM and a similar treatment as in Ref [113113] was used. The energy deposition from electrons is given by ( dEe dV dt )dep c = 1 G(z) ∑ i pc(Ei, xe)EiN e low[Ei] (3.58) 48 It is important to note that Ne ion has already been added to Ne low. The energy of 14eV is used where collisional excitations of hydrogen are possible but not ionization. Below 10.2 eV, electrons can only deposit their energy through Coulomb heating. Finally, for high-energy deposition, a component of the total energy deposited is given by ( dEhigh dV dt )dep c = 1 G(z) Ehighc (3.59) where the channel c ∈ [ion, exc, heat]. The high-energy excitation and ionization component is added to the Lyman-α excitation and hydrogen ionization is added f