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1.3 Self-Interacting Dark Matter Durham E-Theses The Cosmological Implications of Self-Interacting Dark Matter ROBERTSON, ANDREW How to cite: ROBERTSON, ANDREW (2017) The Cosmological Implications of Self-Interacting Dark Matter, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/12305/ Use policy This work is licensed under a Creative Commons Attribution 3.0 (CC BY) Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk The Cosmological Implications of Self-Interacting Dark Matter Andrew Alexander Robertson Abstract In this thesis I study how dark matter particles that interact through forces other than just gravity would affect the formation of structure in the Universe. This begins with a theoret- ical calculation of the location and rate at which these interactions take place throughout cosmic history. Giant galaxy clusters are expected to have the highest rates of dark matter interactions, at least for the simplest dark matter particle models. Predicting the formation of structure with non-standard dark matter requires the use of N-body simulations. I therefore introduce and test a set of modifications to the GAD- GET code that allow it to simulate a class of dark matter models known as self-interacting dark matter (SIDM). I focus particular attention on rarely discussed aspects of simulating SIDM; including how to handle particles scattering multiple times within a single time-step and how to implement scattering across processors. I also discuss how best to choose nu- merical parameters associated with the SIDM implementation and the range of numerical parameters that produce converged results. Because galaxy clusters should have particularly high rates of dark matter interactions, I use this code to perform simulations of a pair of merging galaxy clusters known as the ‘Bul- let Cluster’. At first these employ simple SIDM particle physics models for the dark matter. I demonstrate the importance of analysing simulations in an observationally motivated manner, finding that the way in which simulation outputs are compared with observations can have a significant impact on the derived constraints upon dark matter’s properties. I then look at what happens to these constraints for more complicated particle physics mod- els of SIDM. In isolated systems, the effects of a complicated scattering cross-sections can be modelled using an appropriately-matched simple cross-section, while in systems like the Bullet Cluster, complicated cross-sections lead to phenomenology not seen with simpler particle models. Overall I find that SIDM remains a viable class of dark matter models, consistent with current observations. The Cosmological Implications of Self-Interacting Dark Matter Andrew Alexander Robertson A thesis presented in accordance with the regulations for admittance to the degree of Doctor of Philosophy Institute for Computational Cosmology Department of Physics University of Durham April 2017 Contents List of Figures vi Declaration xi Acknowledgements xiii 1 Introduction1 1.1 Cosmology.......................................1 1.1.1 Hot Big Bang Model.............................2 1.1.2 The expansion history of the Universe...................3 1.1.3 Structure formation..............................7 1.2 Dark matter...................................... 10 1.2.1 Evidence for dark matter.......................... 10 1.2.2 Experimental searches for dark matter................... 12 1.2.3 The ΛCDM ‘concordance cosmology’................... 16 1.2.4 Challenges facing ΛCDM.......................... 21 1.3 Self-interacting dark matter............................. 25 1.3.1 Astrophysical motivation and constraints................. 25 1.3.2 Particle physics models........................... 30 1.4 Thesis outline..................................... 32 2 Self-interacting dark matter scattering rates through cosmic time 36 i 2.1 Introduction...................................... 36 2.2 Interaction rate over cosmic time.......................... 37 2.2.1 Mass function of collapsed structures................... 38 2.2.2 Interaction rates in collapsed structures.................. 41 2.2.3 DM’s cosmic scattering rate......................... 44 2.3 Sensitivity to astrophysical assumptions...................... 45 2.3.1 Concentration-mass-redshift relations................... 47 2.3.2 Mass function prescription......................... 48 2.3.3 Varying cosmological parameters..................... 50 2.3.4 Scatter in the concentration-mass relation................. 51 2.4 Velocity-dependent cross-sections.......................... 52 2.4.1 Particle model................................. 52 2.4.2 vdSIDM cosmic scattering rates....................... 53 2.5 Conclusions...................................... 58 3 Simulating self-interacting dark matter 61 3.1 An Introduction to N-body techniques....................... 61 3.1.1 The N-body method............................. 61 3.1.2 Two-body interactions............................ 62 3.1.3 The Collisionless Boltzmann Equation................... 65 3.1.4 Gravitational softening............................ 67 3.1.5 The GADGET N-body code.......................... 67 3.2 Collisional dynamics................................. 72 3.2.1 Calculating the interaction rate....................... 73 3.2.2 Calculating the post-scatter velocities................... 74 3.3 Methods of simulating SIDM............................ 74 3.3.1 Fluid-like description of SIDM....................... 75 3.3.2 Individually resolved DM interactions................... 76 3.4 Implementation.................................... 78 3.4.1 Scattering Rate................................ 78 ii 3.4.2 Scattering kinematics............................. 80 3.4.3 Multiple scatters within a time-step.................... 81 3.4.4 Scattering within leapfrog integration................... 84 3.4.5 Scattering in cosmological simulations................... 85 4 Testing the self-interacting dark matter implementation 89 4.1 Scattering from a uniform background....................... 89 4.1.1 Scattering rates................................ 90 4.1.2 Post-scatter kinematics............................ 94 4.2 Scattering in an isolated halo............................ 94 4.2.1 Generating Hernquist profile initial conditions.............. 96 4.2.2 Scattering rates in an isolated halo..................... 98 4.2.3 How to choose the neighbour-search radius............... 101 4.2.4 The evolution of density profiles with SIDM............... 106 4.2.5 Core collapse................................. 113 4.3 Scattering in a cosmological simulation...................... 115 5 What does the Bullet Cluster tell us about self-interacting dark matter? 120 5.1 Introduction...................................... 120 5.2 Simulation initial conditions............................. 122 5.2.1 Density profiles................................ 123 5.2.2 Relative velocity of the DM haloes..................... 124 5.2.3 Summary of initial conditions........................ 125 5.2.4 Comparison to other SIDM studies..................... 125 5.3 Measuring Positions................................. 128 5.3.1 Shrinking Circles............................... 128 5.3.2 Parametric fits to 2D density maps..................... 130 5.3.3 Parametric fits to shear maps........................ 132 5.4 Results......................................... 136 5.4.1 Offsets with different cross-sections.................... 136 5.4.2 Sensitivity to varying initial conditions.................. 138 iii 5.4.3 Offsets with different position measures.................. 145 5.4.4 Offsets including gas............................. 150 5.5 Conclusions...................................... 154 6 Simulations of self-interacting dark matter with anisotropic scattering 160 6.1 Introduction...................................... 160 6.2 Angular Dependent Scattering........................... 162 6.2.1 Particle physics of angular dependent scattering............. 162 6.2.2 Integrated cross-sections........................... 163 6.2.3 A velocity-independent, anisotropic cross-section............ 164 6.2.4 Yukawa-potential SIDM........................... 166 6.3 Implementing DM scattering............................ 167 6.3.1 Implementation of velocity-dependent scattering............ 168 6.3.2 Implementation of angular-dependent scattering............ 168 6.3.3 Implementation of velocity-dependant angular-dependence...... 169 6.3.4 Testing generalised scattering........................ 169 6.4 Core growth in isolated haloes........................... 170 6.4.1 Determining core sizes............................ 172 6.4.2 KVI scattering................................. 172 6.4.3 Yukawa-potential scattering......................... 174 6.5 DM-galaxy offsets in the Bullet Cluster....................... 176 6.5.1 KVI scattering................................. 178 6.5.2 Yukawa-potential scattering......................... 181 6.6 Conclusions...................................... 187 7 Conclusions 191 A The expected number of scattering events for a cube moving through a uniform background 197 B A guide to SIDM with a Yukawa potential 199 B.1 Rutherford scattering................................. 199 iv B.2 Scattering through a Yukawa potential....................... 202 C The contribution of mass at different radii to the projected density and shear sig- nals 206 Bibliography 208 v List of Figures 1.1
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