Analytical Formalism to Study the 21 cm Signal from Cosmic Dawn and Epoch of Reionization A Thesis Submitted for the Degree of Doctor of Philosophy in the Faculty of Science by Janakee Raste Joint Astronomy Programme Department of Physics Indian Institute of Science Bengaluru { 560 012 (INDIA) June, 2019 c Janakee Raste June 2019 All rights reserved To my parents... Acknowledgements I was ten years old when I decided that I wanted to become an Astrophysicist. I am grateful to everyone who has since then supported my dream and has helped me fulfill it. I begin with thanking my parents for believing in me and supporting me in every way possible. Their love, support and wisdom have always been my biggest strength. I am grateful to my Ph.D. guide Prof. Shiv Sethi for insightful lectures and discussions on various areas of physics. I thank him for encouraging me to work on projects of my interest, bearing with my unusual working schedule and counselling me during challenging times. Over the past six years, I have been able to learn many things from him during many academic and non-academic discussions and arguments. He claims that one’s Ph.D. is not over until one comes to hate their guide; fortunately, it is only his opinion and not a formal requirement. I am really lucky to have him as my advisor. I thank Prof. Biman Nath and Prof. S. Sridhar for providing constructive criticism of my work as part of my academic advisory committee and for sending reference letters for me. I thank them as well as Prof. Tarun Deep Saini, Prof. Joseph Samuel and Prof. Pravabati Chingangbam for their excellent introduction to various fields of astrophysics and cosmology during course work. I have often turned back to their notes and lectures during my research work. I express my gratitude to Prof. Steven Furlanetto for providing valuable comments and criticism on my research work and for sending reference letters for post-doctoral applications. Saurabh, thanks for bearing with my weirdness for years and for helping me through every new phase of Ph.D. life. Akash, thanks for helping me through tough times. I admire you for your philosophy of life (or the lack of it) and I am fortunate to have been able to learn many things from you - from watching animes to solving quantum mechanics. I thank both of you for countless stimulating discussion on almost anything in and out of this world. Thank you for lending an ear, irrespective of whether I was rehearsing talks or having disheartened rant about life, and for always providing constructive criticism. Anirudh, I am grateful to you for being an amazing friend who has stood by me and helped me through every phase of life. I will miss the time spent with you over five years; without you, Ph.D. life would have been surely difficult. I thank my seniors Nazma, Meera, Abir, Karam, and Anjan for always providing advice and support. Ravi, thanks for discussions and occasional coffee during late working hours. Priyanka, Kshitija, Nancy, Sanhita, Ranita, Palak, Shruti, Samskruthi and Sindhu, I thank all of you multi-talented girls for friendship and inspiration over the years. From exotic dark chocolates to home-cooked food, from singing and dancing to movies, from trekking to shopping, I have been fortunate to have had your company. viii Acknowledgements Aditi, I admire you for teaching me to look at life from different perspectives and for clicking great pictures, writing great poems and cracking jokes (great and otherwise). Manami and Gunjan, thank you for always bringing a smile to people and making awesome coffee. Varun, Raj, Siddhartha and Chaitanya, I have had so much fun celebrating festivals and other special occasions with you, especially with the unique dancing and singing styles which all of you have. Mugundhan, thanks for having discussions on books and regular ice-cream trips. To everyone at RRI, I will miss our discussions at tea-time, side trips during conferences, visits to Sankey Tank, freshers trips, celebrating in-house cultural evening and festivals. Thank you all for helping me maintain my sanity and enthusiasm over the past six years. I thank Akanksha, Sandeep, and many many people whom I met during conferences and schools; you all made these events memorable. Prasun da and Nirupam da, thank you for being sources of inspiration, knowledge and occasional treats. Jordan, I enjoyed discussing my work with you; your comments were very helpful. Finally, I express my gratitude to all teachers, doctors, cooks, administrative and library staff, security guards, friends, family, and strangers, who have helped me through this awesome adventure of Ph.D. ix Synopsis The epoch of cosmic dawn and reionization (CD/EoR) is one of the most important time periods of the history of the universe when the first sources of radiation like stars and galaxies were formed. These sources emitted a wide spectrum of radiation and changed the properties of their surrounding medium. This epoch is not completely understood theoretically and is poorly constrained with observations. 21 cm radiation emitted due to the hyperfine splitting of neutral hydrogen (HI) is one of the most important probes to study the state and dynamics of the neutral medium during this epoch. In this thesis, we present an analytical formalism to study the fluctuating component of the HI signal from the epoch of cosmic dawn and reionization. We use excursion set formalism to calculate the size distribution of randomly distributed self-ionized regi- ons. These ionization bubbles are surrounded by isotropically heated neutral regions. The spin temperature (TS) of HI in these regions might be fully or partially coupled to the kinetic temperature of the medium. We model the ionization, X-ray heating, and Lyman-a coupling using five parameters: efficiency of ionization (z), number of X-ray photons emitted per stellar baryon (Nheat), the spectral index of X-ray photons (a), minimum frequency of X-ray photons (nmin), and the ratio of source luminosity of Lyman-a to ionizing photon ( fL). We develop a formalism to compute the two-point correlation function for this topology, taking into ac- count the autocorrelation of the ionization field and spin temperature field, and cross-correlation between ionization and spin temperature. Using geometric and probabilistic arguments, we compute the global HI signal, its autocorrelation and power spectrum in the redshift range 10 z 30 for the LCDM model. We ≤ ≤ check the validity of this formalism for various limits and simplified cases and develop a few toy models to explore the results more intuitively. Our results agree reasonably well with existing results from N-body simulations in spite of following an en- tirely different approach and requiring orders of magnitude less computational power and time. We further apply our formalism to study the fluctuating component corresponding to the recent EDGES observation that shows an unexpectedly deep absorption trough in global HI signal in the redshift range 15 < z < 19. We show that, generically, this observation predicts larger signal in this redshift range, but smaller signal at higher redshifts. We also explore the possibility of negative real-space auto-correlation of spin temperature in the early universe. x Synopsis Chapter 1: Introduction In the first chapter, we explore the current observational bounds on the early history of the universe and their theoretical implications. We briefly present current theoretical understanding of how the large scale structures collapsed and first sources of radiation were formed within them. We list the expected properties of these first sources and their effect on their surrounding medium. We summarise the excursion set for- malism used in our work to find the size distribution of isolated ionization bubbles at any redshift. We also discuss the hyperfine line splitting of neutral hydrogen and the mechanisms that affect the population of these two states, and hence the spin temperature of HI; these mechanisms are background radiation, collisi- ons, and Lyman-a photons. We conclude with listing ongoing effort to theoretically understand the epoch of CD/EoR through simulations and analytical methods. Chapter 2: Temperature Evolution of the IGM In this chapter, we compute X-ray heating of the intergalactic medium (IGM) surrounding the ionization bubbles due to photoionization of neutral atoms. We calculate the temperature evolution at a point as a function of its distance from the centre of the X-ray source. We also compute the background temperature evolution due to the combined effect of all the faraway sources. Since the X-rays have a large mean free path, the temperature profiles are large; therefore, we explore various possible methods to truncate these profiles while allowing them to overlap with one another and ensuring that they merge smoothly with the background values far away from any source. We also study a fiducial model where the baryonic gas might be coupled to cold dark matter through velocity-dependent interaction. Chapter 3: Spin Temperature Coupling In this chapter, we present the physics of HI spin temperature TS coupling with the matter kinetic tempe- rature TK through Lyman-a photons and collisions. We calculate the number density of Lyman-a photons as a function of distance from a source, taking into account the effect of higher-order Lyman-n transitions. We also compute the background value of Lyman-a number density far away from any sources. We also calculate the Lyman-a mean free path in the neutral medium and the average number of scattering expe- rience by a typical Lyman-a photon before it is redshifted out of the resonance width. Combined with the kinetic temperature profiles calculated in the previous section, we can use the value of coupling coefficients to calculate the final spin temperature profiles around ionization bubbles.