Dark Ma Er in the Earth and The

University of Southern Denmark

Doctoral Dissertaon Dark Maer in the Earth and the Sun – Simulang Underground Scaerings for the Direct Detecon of Low-Mass Dark Maer

Timon Emken

supervised by Chris Kouvaris

Centre for Cosmology and Parcle Physics Phenomenology arXiv:1906.07541v1 [hep-ph] 18 Jun 2019 Department of Physics, Chemistry, and Pharmacy

Submied to the University of Southern Denmark January 31, 2019

Inveniam viam aut faciam.

Contents

Abstract ...... iii Sammenfatning ...... iv List of Publicaons ...... v List of Figures ...... vii List of Tables ...... xi List of Acronyms ...... xiii Acknowledgments ...... xv

1 Introducon1

2 Dark Maer5 2.1 History and Evidence of Dark Maer in the Universe ...... 6 2.2 Parcle Dark Maer ...... 16

3 Direct Detecon of Dark Maer 27 3.1 The Historical Evoluon of Direct DM Searches ...... 28 3.2 The Galacc Halo ...... 33 3.3 Scaering Kinemacs ...... 36 3.4 Describing DM-Maer Interacons ...... 39 3.5 Recoil Spectra and Signal Rates ...... 49 3.6 Stascs and Exclusion Limits ...... 57

4 Terrestrial Eﬀects on Dark Maer Detecon 63 4.1 Monte Carlo Simulaons of Underground DM Trajectories . . . . . 65 4.2 Diurnal Modulaons ...... 80 4.3 Constraints on Strongly Interacng DM ...... 96

5 Solar Reﬂecon of Dark Maer 123 5.1 DM Scaerings in the Sun ...... 125 5.2 Direct Detecon of Reﬂected DM ...... 134

i 5.3 MC Simulaons of Solar Reﬂecon ...... 137

6 Conclusions and Outlook 149

A Constants and Units 155 A.1 Physical Constants ...... 155 A.2 Natural Units ...... 156

B Astronomical requirements 157 B.1 Constants ...... 157 B.2 Sidereal Time ...... 158 B.3 Coordinate Systems ...... 159 B.4 Velocity of the Sun, the Earth, and the Laboratory ...... 164 B.5 Modelling the Earth and Sun ...... 167

C Direct Detecon Experiments 173 C.1 DM-Nucleus Scaering Experiments ...... 173 C.2 DM-Electron Scaering Experiments ...... 175

D Numerical and Stascal Methods 179 D.1 Adapve Simpson Integraon ...... 179 D.2 Interpolaon with Steﬀen Splines ...... 180 D.3 Kernel Density Esmaon ...... 182 D.4 The Runge-Kua-Fehlberg Method ...... 184

E Rare Event Simulaons 187 E.1 Importance Sampling ...... 187 E.2 Geometric Importance Spling ...... 191

ii Abstract

Astrophysical and cosmological observaons provide compelling evidence that the majority of maer in the Universe is dark. Showing no interacons with electromagnec radiaon, this dark maer (DM) eludes direct observaons, and its nature and origin remains unknown to this day. Direct detecon experiments search for interacons between halo DM and nuclei inside a detector. So far, a variety of experiments were only able to set stringent limits on the DM parameter space. These constraints weaken for sub-GeV DM masses, as light parcles are not energec enough to trigger most detectors. New experimental efforts shi the focus towards lower masses, for example by looking for inelasc DM-electron scaerings. If scaerings between DM and ordinary maer are assumed to occur in a detector’s target material, collisions will naturally take place inside the bulk of planets and stars as well. For sufficiently large cross secons, these scaerings might occur in the Earth or Sun even prior to the detecon. In this thesis, we study the impact of these pre-detecon scaerings on direct searches of light DM with the use of Monte Carlo (MC) simulaons. By simulang the trajectories and scaerings of many individual DM parcles through the Earth or Sun, we determine the local distorons of the stascal properes of DM at any detector caused by elasc DM-nucleus collisions. Scaerings inside the Earth distort the underground DM density and velocity distribuon. Any detector moves periodically through these inhomogeneies due to the Earth’s rotaon, and the expected event rate will vary throughout a sidereal day. Using MC simulaons, we can determine the exact amplitude and phase of this diurnal modulaon for any experiment. For even higher scaering probabilies, collisions in the overburden above the typically underground detectors start to aenuate the incoming DM flux. The crical cross secon above which an experiment loses sensivity to DM itself is determined for a variety of DM-nucleus and DM-electron scaering experiments and different types of interacons. Furthermore, we develop the idea that sub-GeV DM parcles can enter the Sun, gain kinec energy by colliding on hot nuclei and get reflected with great speeds. By deriving an analyc expressions for the parcle flux from solar reflecon via a single scaering, we demonstrate the prospects of future experiments to probe reflected DM and extend their sensivity to lower masses than accessible by halo DM alone. We present first results for MC simulaons of solar reflecons. In- cluding reflecon aer mulple scaerings greatly amplifies the reflected DM flux and thereby the potenal of solar reflecon for direct searches for light DM.

iii Sammenfatning

Astrofysiske observaoner giver overbevisende tegn pa,˚ at størstedelen af stof i uni- verset er mørkt. Dee mørke stof (DM) viser ingen observaonelle interakoner med elektromagnesk straling,˚ og dets natur og oprindelse er stadig ukendt. Di- rekte detekonseksperimenter søger eer interakoner mellem DM fra galakse- haloen og atomkerner i en detektor. Hidl har eksperimenter kun været i stand l at sæe strenge grænser i DM parameterrummet. Disse begrænsninger løsnes for sub-GeV DM-masser, da lee parkler ikke har nok energi l at udløse en maling˚ i de fleste detektorer. Ny eksperimentel indsats skier fokus mod lavere masser, for eksempel ved at lede eer uelasske sammenstød mellem DM og elektroner. Hvis sammenstød mellem DM og almindeligt stof antages at ske i detektoren, vil kollisioner ogsa˚ finde sted i hovedparten af planeter og stjerner. For lstrækkeligt store tværsnit kan disse sammenstød forekomme i Jorden eller Solen, endda før de males˚ i detektoren. I denne aandling undersøger vi virkningen af disse præ- detektor sammenstød i direkte søgninger af let DM ved brug af Monte Carlo (MC) simuleringer. Ved at simulere baner og sammenstød af mange individuelle DM- parkler gennem Jorden eller Solen bestemmer vi de lokale forvrængninger i de stasske egenskaber af DM forarsaget˚ af elasske DM-atomkernekollisioner i en given detektor. Sammenstød inde i Jorden ændrer den underjordiske DM-densitet og hasgheds- fordeling. Enhver detektor bevæger sig periodisk gennem disse inhomogeniteter mens planeten roterer, og den forventede hændelsesrate vil variere i løbet af en siderisk dag. Ved hjælp af MC-simuleringer kan vi bestemme den nøjagge amplitude og fase af denne døgnmodulering i givent eksperiment. For endnu højere sammenstødssandsynligheder begynder sammenstød i lagene ovenover det underjordiske eksperiment at dæmpe den indkommende DM-flux. Vi bestemmer det kri- ske tværsnit, hvor et eksperiment mister følsomheden overfor DM for en række atomkerne- og elektronspredningsforsøg samt forskellige typer interakoner. Desuden udvikler vi ideen om, at sub-GeV DM-parkler kan trænge ind i Solen, fa˚ kinesk energi ved at kollidere med varme atomkerner og blive reflekteret med høje hasgheder. Vi udleder analyske udtryk for parkelfluxen fra solreflekon via et enkelt sammenstød og demonstrerer udsigterne for fremdige eksperimenter l at lede eer reflekteret DM og udvide følsomheden over for lavere masser end hvad der er lgængelig ved halo-DM alene. Vi præsenterer de første MC-resultater. Vi inkluderer refleksion eer flere sammenstød, som forstærker den reflekterede DM-flux og derved potenalet ved solreflekon l direkte søgninger eer let DM. iv List of Publicaons

This thesis contains results, which were published in the following papers.

Paper V T. Emken, R. Essig, C. Kouvaris and M. Sholapurkar, Direct detecon of strongly interacng sub-GeV dark maer via electron recoils, (to appear),

Paper IV T. Emken and C. Kouvaris, How blind are underground and surface detectors to strongly interacng Dark Maer?, Phys. Rev. D97 (2018) 115047, [1802.04764]

Paper III T. Emken, C. Kouvaris and N. G. Nielsen, The Sun as a sub-GeV Dark Maer Accelerator, Phys. Rev. D97 (2018) 063007,[1709.06573]

Paper II T. Emken and C. Kouvaris, DaMaSCUS: The Impact of Underground Scaer- ings on Direct Detecon of Light Dark Maer, JCAP 1710 (2017) 031,[1706.02249]

Paper I T.Emken, C. Kouvaris and I. M. Shoemaker, Terrestrial Eﬀects on Dark Maer- Electron Scaering Experiments, Phys. Rev. D96 (2017) 015018,[1702.07750]

I will refer to these publicaons throughout this thesis as e.g. Paper II. Results reported in these papers were parally generated with scienﬁc codes I developed during the preparaon of this thesis. They were made publicly available alongside the corresponding publicaons.

DaMaSCUS-CRUST T.Emken and C. Kouvaris, DaMaSCUS-CRUST v1.1,[ascl:1803.001] Available at https: // github. com/ temken/ damascus-crust , 2018-2019

DaMaSCUS T. Emken and C. Kouvaris, DaMaSCUS v1.0, [ascl:1706.003] Available at https: // github. com/ temken/ damascus , 2017

v vi List of Figures

1.1 Simulated DM trajectories in the Earth and Sun with mulple scatterings...... 3

2.1 Zwicky and the Coma cluster ...... 7 2.2 A random selecon of ten galacc rotaon curves of the SPARC sample...... 9 2.3 Strong gravitaonal lensing of the galaxy cluster SDSS J1038+4849 (le) and an overlay of the opcal, X-ray, and weak gravitaonal lensing observaon of the bullet cluster (1E 0657-558) (right)...... 11 2.4 The power spectrum of the CMB, both the data and the best ﬁt of the cosmological ΛCDM model...... 13 2.5 N-body simulaon of structure formaon...... 15 2.6 Thermal freeze-out of WIMPs...... 22 2.7 The three main search strategies for non-gravitaonal DM interacons...... 24

3.1 The local speed distribuon of DM and its annual modulaon. A Mollweide projecon of the direconal distribuon with the characterisc dipole of the DM wind is shown as well...... 34 3.2 Elasc scaering kinemacs between a DM and a target parcle. . . 36 3.3 The nuclear Helm form factor (solid line) and its exponenal approximaon (dashed line) for oxygen and xenon...... 42 3.4 The atomic form factor, given in eq. (3.4.30), which describes the screening of the nuclear electric charge for silicon, argon, germanium, and xenon nuclei...... 47 3.5 Examples of η(vmin(ER)) with varying DM masses for a 131Xe target. 50 3.6 Generic exclusion limits from direct detecon and their scaling. . . 59 3.7 Yellin’s Maximum Gap method for ﬁnding upper limits...... 60

vii 4.1 Examples of inverse transform sampling (le) and rejecon sampling (right) for a normal distribuon with mean µ = 5 and standard deviaon σ = 1...... 67 4.2 Recursive algorithm to sample the free path length and therefore the locaon of the next scaering with different regions and layers of sharp boundaries...... 73 4.3 Distribuon of the scaering angle cos θ for SI interacons for different DM masses...... 75 4.4 Distribuons of the scaering angle for contact, electric dipole, and long range interacons...... 76 4.5 Flow chart for the trajectory simulaon algorithm...... 78 4.6 Sketch of the inial posion...... 83 4.7 Illustraon of the isodetecon angle and rings and their projecon onto the Earth surface at 0:00 and 12:00 with ∆Θ = 5°...... 84 4.8 Time evoluon of the isodetecon angle for different laboratories around the globe over the duraon of three days...... 85 4.9 Comparison of the analyc EarthShadow and the MC results of DaM- aSCUS...... 91 4.10 DM speed distribuons across the globe for our four benchmark −3 points. Note that they are normalized to 0.3 GeV cm . The black dashed line shows the speed distribuon of free DM...... 93 4.11 Results of the MC simulaons: The first three plots depict the local DM density, average speed and event rate respecvely as a funcon of Θ. The last panel shows the corresponding diurnal modulaon in the local sidereal me for experiments in the northern and southern hemisphere...... 94 4.12 The percenle signal modulaon as a funcon of latude...... 95 4.13 The underground DM speed distribuon of eq. (4.3.14) for mχ = 1 GeV SI -30 2 and σp = 10 cm at different depths...... 103 4.14 Sketch of the MC simulaons of parallel shielding layers (not to scale).105 4.15 Comparison of the different methods and the predicted number of events for XENON1T (solid lines) and DAMIC (2011) (dashed lines) for a DM mass of 10 GeV...... 110 4.16 The constraints (90% CL) by various direct detecon experiments, together with limits from XQC and the CMB...... 112 viii 4.17 Exposure scaling of the constraints (95% CL) for a generic semiconductor detector...... 115 4.18 Underground depth scaling of the constraints (95% CL) for a generic semiconductor detector...... 116 4.19 Direct detecon constraints (95% CL) by various DM-electron scattering experiments...... 117 4.20 Projected constraints (95% CL) for the planned experiments DAMIC- M and SENSEI...... 118 4.21 Orbital modulaon of the DM speed distribuon and detecon rates for a balloon and a satellite experiment for mχ = 100 MeV, σe = 2 2 10-23cm , and a light mediator (FDM ∼ 1/q )...... 119 4.22 Constraints on DM with an ultralight dark photon mediator...... 121

5.1 Gravitaonal acceleraon of DM close to the Sun ...... 125 5.2 Sketch of a DM trajectory crossing a solar shell of radius r twice. . . 130 5.3 Average probability hPleave(v, r)iJ02 to escape the Sun without re- scaering for mχ = 100 MeV and increasing cross secons...... 133 5.4 Differenal DM parcle flux for the halo and solar reflecon for DM masses of 10 GeV (solid), 1 GeV (dashed), and 100 MeV (doed).134 5.5 The red (yellow) contours are projected constraints for a CRESST-III type (idealized sapphire) detector with exposures of 1, 10, and 100 ton days (10, 100, and 1000 kg days) in solid, dashed, and doed lines respecvely...... 135 5.6 Sketch of the inial condions for DM parcles entering the Sun (not to scale)...... 142 5.7 Comparing the equal-angle isodetecon rings for the Earth simulaons and the equal-area isoreflecon rings for the Sun simulaons. 144 5.8 Histogram of the radius and final DM speed of the first scaering in dS the Sun. The black contours trace the analyc scaering rate dv dr . 145 5.9 Time evoluon of the total DM energy (solid lines) and the radial coordinate (dashed lines) for two example trajectories...... 146 5.10 Contribuon of mulple scaerings to the solar reflecon spectrum. 147

B.1 Flowchart for transformaons between the coordinate systems. . . 163 B.2 The Earth’s orbital velocity in the galacc and heliocentric eclipc frame and the laboratory’s rotaonal velocity in the equatorial frame.165 B.3 The Preliminary Reference Earth Model...... 167

ix B.4 Density proﬁle of the US Standard Atmosphere and diﬀerent layer approximaons for the MC simulaons...... 168 B.5 The solar model AGSS09...... 171

D.1 Example for Steﬀen interpolaon. Ten equidistant points (x, f(x)) with f(x) = cos(x) sin(x) are chosen and interpolated. Note that the extrema of the interpolaon funcon coincide with the data points...... 180 D.2 KDE for a bounded domain with and without the pseudo data method of Cowling and Hall (C&H)...... 183

E.1 Importance Sampling modiﬁcaons of the simulaon’s PDFs. . . . . 190 E.2 Illustraon of two DM trajectories using GIS with three importance domains of equal size. The parcle can split into either 2 or 4 copies. The weight development along the paths is shown as well...... 193

x List of Tables

3.1 Some soluons of eq. (3.6.2) for diﬀerent conﬁdence levels and event numbers...... 58

4.1 The four benchmark points to study the impact of DM-nucleus scatterings and diurnal modulaons at direct detecon experiments. . . 91

A.1 Physical constants and masses in SI units ...... 155 A.2 Conversion between SI and natural units ...... 156

B.1 Astrophysical quanes ...... 157 B.2 Coordinate systems ...... 160 −3 B.3 The density proﬁle coeﬃcients in g cm and radii of the mechanical layers in the PREM...... 168 B.4 Relave element abundances in wt% of the Earth core, mantle,crust, and atmosphere...... 169

C.1 Summary of direct detecon experiments based on nuclear recoils included in our analyses...... 175 nl (2) C.2 Binding energy EB and number of secondary quanta n of xenon’s atomic shells. For the limits, the lowest number of secondary quanta is used...... 176 C.3 Parameters for the analysis of XENON10 and XENON100...... 176 C.4 Binned data for XENON10 and XENON100...... 177 C.5 Eﬃciencies and observed signals in the electron bins for SENSEI (2018) and SuperCDMS (2018)...... 177 C.6 Nuclear composion of concrete in terms of the isotopes’ mass fracons fi...... 178

xi xii List of Acronyms

BBN Big Bang Nucleosynthesis ...... 14 CL Confidence Level ...... 57 CMB Cosmic Microwave Background...... 12 CMS Center of Mass System ...... 37 CDF Cumulave Distribuon Funcon ...... 58 COBE Cosmic Background Explorer ...... 13 DaMaSCUS Dark Maer Simulaon Code for Underground Scaerings ...... 3 DM DarkMaer...... 1 EFT Effecve Field Theory...... 39 GIS Geometric Importance Spling ...... 107 GAST Greenwich Apparent Sidereal Time ...... 159 GMST Greenwich Mean Sidereal Time ...... 158 IS Importance Sampling...... 107 KDE Kernel Density Esmaon...... 108 LAST Local Apparent Sidereal Time...... 95 LHC Large Hadron Collider ...... 25 LIGO Laser Interferometer Gravitaonal-Wave Observatory ...... 17 LNGS Laboratori Nazionali del Gran Sasso ...... 30 LSM Laboratoire Souterrain de Modane MACHO Massive Compact Halo Object ...... 16 MC Monte Carlo ...... 3 MET Missing Transverse Energy ...... 25 MOND Modified Newtonian Dynamics ...... 10 MSSM Minimal Supersymmetric Standard Model ...... 20 NREFT Non-Relavisc Effecve Field Theory ...... 40 PBH Primordial Black Hole...... 17 PDF Probability Density Funcon ...... 65 PE Photoelectron ...... 54 xiii PMF Probability Mass Funcon...... 57 PMT Photomulplier Tube ...... 30 PREM Preliminary Reference Earth Model ...... 86 QCD Quantum Chromodynamics ...... 21 RHF Roothaan-Hartree-Fock ...... 54 SD Spin-Dependent ...... 43 SHM Standard Halo Model ...... 33 SI Spin-Independent ...... 40 SIMP Strongly Interacng Massive Parcle ...... 31 SM Standard Model of Parcle Physics ...... 1 SSM Standard Solar Model ...... 124 SUPL Stawell Underground Physics Laboratory ...... 93 SURF Sanford Underground Research Facility...... 30 SUSY Supersymmetry ...... 19 TT Terrestrial Time ...... 158 UT Universal Time ...... 95 WIMP Weakly Interacng Massive Parcle ...... 20 WMAP Wilkinson Microwave Anisotropy Probe ...... 13

xiv Acknowledgments

No dark maer detector exists in isolaon, and neither do PhD students. This dissertaon would not have been possible without a great number of people whose direct and indirect support I am deeply grateful for. I am grateful to Chris Kouvaris for the opportunity to work in this excing field of research and for a fruiul collaboraon. His supervision allowed me to grow as an independent researcher. I would also like to thank the Faculty of Science at SDU and in parcular the PhD School Secretariat for assistance especially during the final sprint. More thanks go to my collaborators, namely Rouven Essig, Niklas G. Nielsen, Ian M. Shoemaker, and Mukul Sholapurkar for their contribuons to the research in this thesis and countless valuable discussions. I learned a lot from them, and it is a pleasure to work with them. During my PhD studies, I had the privilege to visit and work at the C.N. Yang Instute for Theorecal Physics in Stony Brook, NY. For the hospitality and kindness that I encountered during my stay there, I am deeply grateful. The computaons of this thesis involved months of coding1 and simulaons running on the ABACUS 2.0 supercomputer. I would like to express my gratude to the team of the DeiC Naonal HPC Center at SDU for their always swi and helpful assistance. At CP3, I was privileged not only to carry out my research, but also meet amazing colleagues, many of which have become dear friends. I especially appreciated the atmosphere and acvies among the PhD students and postdocs. In parcular, I would like to thank my collaborator, office-buddy, and friend Niklas for the me and work we shared in the party office. A special thanks goes to my enre family, who I oen missed during my me in Odense. I thank especially my parents for their uncondional love and support. When I came to Denmark three years ago, lile did I know who I would meet. I want to thank you, Majken, for your words of love, encouragement, and wisdom. Your humour and support never fail to brighten even the most stressful day. There are without a doubt many names missing on this page, names of people who deserve to be menoned here. You know who you are. You made my me here at CP3 worthwhile. Thank you.

1I should also acknowledge the thousands and thousands of faceless heroes of Stack Overﬂow who unknowingly and without reward solved a vast number of my everyday problems and ‘taught’ me the art and ordeal that is programming.

xv xvi Chapter 1

Introducon

The nature of dark maer is one of the most excing open quesons of natural science in general and astro- and parcle physics in parcular. The field of high-energy physics finds itself in a peculiar situaon. With the discovery of the Higgs parcle at the Large Hadron Collider at CERN in 2012 [8,9], the Standard Model of Parcle Physics (SM) was confirmed to describe the behavior of fundamental parcles on all tested energy scales with remarkable precision. The physics of visible maer, fundamentally composed of leptons, quarks, and their interacons, seems very well understood. The success of the SM clashes at the same me with a series of astrophysical observaons, all of which substanate the noon that the visible maer, the maer we observe in forms of stars, galaxies, gas, or planets, the maer we can describe so accurately, can only account for about 15% of the total maer of our Universe. In order to make sense of various independent astrophysical measurements from galacc to cosmological scales, it seems vital to make the astounding assumpon that 85% of maer is dark. Showing no interacons with electromagnec radiaon, this Dark Maer (DM) eludes all direct observaons, yet affects and dominates gravitaonal dynamics on astronomical scales. DM is the umbrella term to capture the unidenfied explanaon of these observaons which can not be aributed to any parcle of the SM, as its ulmate origin is enrely unknown. Ordinary maer is fundamentally composed of parcles, and it is not farfetched to assume that this applies to the dark sector of maer as well. If so, the Earth would consequently get traversed by a connuous stream of a vast number of DM parcles at any moment. Should these parcles interact with ordinary matter through some interacon besides gravity and occasionally scaer with atoms, it could be possible to observe these collisions inside a detector. Experiments on Earth should be able to discover DM, provided that some “portal” between the light

1 CHAPTER 1. INTRODUCTION

and the dark sector exists. Many of such direct detecon experiments have been conducted in the last three decades, thus far unable to discover DM on Earth. If we expect these scaerings to occur inside a detector at a non-vanishing rate, they should also happen without being detected inside the Earth’s or Sun’s bulk mass. For sufficiently strong interacons, this might even happen prior to detecon. Underground scaerings before passing through a detector affect the expected outcome of the experiment. Elasc collisions on nuclei change the trajectory and speed of DM parcles on their way through the medium, with potenally strong implicaons for direct detecon experiments. Nuclear scaerings modify the underground spaal and energec distribuon of DM parcles inside the Earth through deflecon and deceleraon. For a significant scaering probability, the expected signal rate for any detector would depend on its exact locaon, because the average underground distance for DM to reach the detector and therefore also its scaering probability vary periodically. Since the experiment does not stand sll but rotates around the Earth axis, the signal rate will show a diurnal modulaon. The phase and amplitude of this modulaon, which we will predict over the course of this thesis, depends on the DM model and the experiment’s locaon on Earth. Diurnal modulaons would not only be a clean signature disnguishing a signal from background, they could also tell us something about the interacon itself. Direct detecon experiments are usually set up deep underground in order to shield off background sources. However, if the DM-maer interacons are so strong that incoming DM parcles from the halo collide on nuclei of the experiment’s overburden already, the shielding layers (typically ∼ 1 km of rock) could weaken the DM signal itself, up to the point where terrestrial experiments lose sensivity to strongly interacng DM parcles enrely. Their scaerings on nuclei in the Earth crust or atmosphere would then aenuate the observable flux below detectability. This is a natural limitaon of any direct DM search on Earth and needs to be quanfied. It turns out that pre-detecon scaerings can also extend an experiment’s sensivity. Through collisions with highly energec nuclei of the hot solar core, low-mass DM parcles could gain energy. These parcles fall into the Sun’s gravitaonal well, get further accelerated by elasc collisions and leave the star much faster than the inial speed. The solar reflecon flux of DM can extend an experiment’s sensivity to lower masses, since faster DM parcles can deposit more energy in a detector. A powerful tool to invesgate the effect of many underground scaerings are

2 Figure 1.1: Simulated DM trajectories in the Earth and Sun with mulple scaerings.

Monte Carlo (MC) simulaons. By simulang individual trajectories of parcles passing through the Earth or Sun while colliding with terrestrial or solar nuclei, we can quanfy the phenomenological impact by numerical and stascal methods. Examples of simulated trajectories in the Earth and the Sun are shown in ﬁg- ure 1.1. Most of the results of this thesis have been obtained by seng up ded- icatedMC codes and running simulaons on a supercomputer. The code used to generate the published results have been released together with the corresponding papers. In parcular, the Dark Maer Simulaon Code for Underground Scat- terings (DaMaSCUS)[7] and DaMaSCUS-CRUST[6] are publicly available.

Concerning the thesis’ structure, the ﬁrst two chapters introduce dark maer and the aempts to directly detect it. In chapter2, we review the evidence for DM and follow its history over the course of the 20th century. The evoluon from DM as a purely astronomical queson to an acve ﬁeld of parcle physics in the century’s second half is emphasized. The direct detecon of DM is the main topic of chapter3, where we will summarize previous detecon aempts, priorizing direct searches for low-mass DM. This chapter also reviews the basics and essenal computaons of recoil spectra and signal rates for both convenonal direct detecon via nuclear recoils and electron-scaering experiments. The next two chapters contain the main results of this thesis. The foundaons and results of the simulaons of DM inside the Earth are compiled in chapter4. Therein, we formulate the general algorithms for theMC simulaons of under-

3 CHAPTER 1. INTRODUCTION

ground trajectories. This is followed by the applicaon of the algorithm to the enre Earth to quanfy diurnal modulaon of detecon rates. The second applicaon of the terrestrial simulaons concerns trajectories through the overburden of a given experiment, e.g. the Earth crust or atmosphere. This allows to determine the exact constraints on strongly interacng DM. In chapter5, we focus the aenon on DM parcles scaering and geng accelerated inside the Sun. The theorecal framework to describe DM scaerings in a star is formulated and applied to study the detecon prospects of solar reﬂecon of DM via a single scaering with analyc methods. Furthermore, theMC algorithms are extended for DM trajectories inside the Sun by including its gravitaonal force and thermal targets. These simulaons can shed light on the contribuon of mulple scaerings to solar reﬂecon. Finally, we conclude in chapter6. In addion, a number of appendices are included in this thesis containing details on the astronomical prerequisites of the simulaons, on the experiments, various numerical methods, and more. These appendices are supposed to give a broad and extensive overview of the more technical, yet essenal fundamentals and techniques applied throughout the thesis.

4 Chapter 2

Dark Maer

“The weight of evidence for an extraordinary claim must be propor- oned to its strangeness.” –Pierre-Simon Laplace (1749-1827) [10].

The claim that our Universe is dominated by a form of maer which we cannot see nor directly measure is indeed extraordinary and requires jusficaon. Although the different pieces of evidence in favour of dark maer have been presented and reviewed in a plethora of publicaons, books, and presentaons, we believe it is vital to keep in mind the compelling reasons why a lot of sciensts spend a great amount of me and resources on the search for dark maer. This is why we will once more review the evidence for dark maer in the Universe and also shed some light on the rich history of dark maer research. While it started as a purely astronomical discipline, over me it evolved into a large interdisciplinary field of research bringing together astrophysicists, cosmologists, high energy physicists, and many more. The details of the evidence and its historic development are by no means complete. For further reading on this subject, we recommend a series of informave reviews [11–17].

5 CHAPTER 2. DARK MATTER 2.1 History and Evidence of Dark Maer in the Universe

When the term ‘dark maer’ first started to appear in the astronomical literature during the early 20th century, it had a very different meaning than it does today. ‘Dark maer’ was used descripvely, simply to refer to ordinary maer which neither shines nor reflects light – stars, which are too distant or too cool and faint to be observed, dim gas clouds, and other solid objects. At this point, no one had any reason to entertain the idea of dark maer as some new, exoc form of maer, since lile was known about the non-stellar mass in the Milky Way. In order to esmate the total mass of the galaxy, which could be compared to the observed amount, Henri Poincare´ applied Lord Kelvin’s idea to treat the galaxy as a thermodynamic gas of gravitang stars [18]. Furthermore, two Dutch astronomers, Jacobus Kapteyn in 1922 [19] and his student Jan Oort in 1932 [20], analysed stellar velocity in our galacc neighbourhood to esmate the local dark maer density. While these studies, among many others, showed no evidence for a large discrepancy between bright and dark maer1, newer observaons on intergalacc and galacc scales started to indicate otherwise.

2.1.1 The ‘missing mass’ problem of galaxies

Galaxy clusters Following Lord Kelvin’s and Poincare’s´ approach, the astronomer Fritz Zwicky applied the virial theorem to astronomical observaons in 1933 [21]. The virial theorem relates the average total kinec energy hT i and the average potenal energy hV i of a stable system, 1 hT i = − hV i . 2 (2.1.1)

Zwicky used the virial theorem to the Coma galaxy cluster in order to esmate its mass. For this purpose, he measured the Doppler shis of spectral lines to measure the galaxy’s velocies in the line of sight. Furthermore, he esmated the total mass of the Coma cluster to be the sum of all stars mes the solar mass,

9 42 M ≈ 800 × 10 ×M ≈ 1.6 × 10 kg . |{z} |{z} (2.1.2) number of galaxies stars per galaxy

1Oort did indeed ﬁnd a discrepancy between the total and stellar density, but aributed this to neglecng faint stars close to the galacc plane.

6 2.1. HISTORY AND EVIDENCE OF DARK MATTER IN THE UNIVERSE

(a) The Coma cluster [23]. (b) Fritz Zwicky (1971) [24].

Figure 2.1: Zwicky and the Coma cluster

Based on this esmate, the average velocity and velocity dispersion were esmated to be

3 M 1/2 phv2i = G ≈ 82 km s−1 , 5 N R (2.1.3) r hv2i σ = ≈ 47 km s−1 , v 3 (2.1.4) where he esmated the cluster’s radius R to be of order 106 ly. This esmate was however in direct conﬂict with Zwicky’s observaons. He measured the apparent −1 velocies of eight galaxies and found a large velocity dispersion of 1100 km s . Zwicky concludes that, if we want to obtain a velocity dispersion of the same order from the virial theorem, we have to assume maer densies of at least 400 mes larger than the stellar density. The Coma cluster could otherwise not be considered a bound system and would disperse over me. Only a small fracon of the mass was observable, most of it seemed missing. Four years later, Zwicky speculated that this ‘dark maer’ should be made oﬀ cold stars, gases, and other solid bodies, which might also absorb background light and thereby reduce the observed luminosies further [22]. The mass-to-light rao refers to rao between the total mass and its luminosity. Many observaons of large mass-to-light raos of galaxy clusters were published in the following years2. But many astronomers did not accept the hypothesis of large amounts of dark maer in galaxy clusters, quesoning whether galaxy clus-

2The ﬁrst measurements of large galacc mass-to-light raos actually preceded even Zwicky’s observaons by three years and were made by the Swedish astronomer Knut Lundmark in 1930 [25].

7 CHAPTER 2. DARK MATTER

ters are truly bound systems. This interpretaon was disfavoured by the age of the universe, as unbound clusters should have disintegrated by now. Clusters were indeed gravitaonally bound systems. Others tried to ﬁnd the missing mass not in the galaxies, but in the intergalacc space, as proposed already in 1936 by the astronomer Sinclair Smith3 [26]. They looked e.g. for hydrogen gas or ions outside galaxies. None of these observaons showed a suﬃcient amount of ordinary matter to explain the large mass-to-light raos, and the missing mass problem of galaxy clusters remained.

Galacc rotaon curves Historically, the most important evidence in favour of large amounts of dark maer in the Universe came from galacc rotaon curves, i.e. the speed v of visible maer orbing the galacc center as a funcon of the galactocentric distance r [27, 28]. Assuming circular orbits, Newtonian dynamics predicts the rotaon curve, r G M(r) v(r) = N , r (2.1.5)

where GN is Newton’s constant and M(r) denotes the mass found within a sphere of radius r, which follows from the mass density distribuon ρ(r) of the galaxy, ZZZ 3 0 0 0 M(r) = d r ρ(r )Θ(r − r ) . (2.1.6)

For stars well outside the bulk mass, M(r) is approximately constant, and the predicted rotaon curve should follow 1 v(r) ∼ √ . r (2.1.7)

This Keplerian speed drop is most notably observed in the planetary orbits of the solar system. Since it is difficult to infer the rotaon curve of our own galaxy, the Milky Way, the first observaons of galacc rotaon curves were obtained for the Andromeda galaxy (M31). The first astronomer to make spectrographic observaons of its rotaon curve and extract informaon about the galaxy’s mass distribuon, was the American Horace Babcock in 1939 [29]. He failed to observe the expected Kep-

3Just as Zwicky for the Coma cluster, Smith found a high mass-to-light rao for the Virgo cluster early on. Instead of the virial theorem, he used the circular orbits of the outermost galaxies to esmate the cluster’s total mass.

8 2.1. HISTORY AND EVIDENCE OF DARK MATTER IN THE UNIVERSE

350

300

] NGC 5055 s / NGC 5985 km

[ 250 NGC 6195 obs 200 NGC 6946 NGC 7331 150 UGC 00128 UGC 02916 100 UGC 02953 rotational velocity v 50 UGC 09133 UGC 06614 0 0 10 20 30 40 50 60 70 radius r [kpc]

Figure 2.2: A random selecon of ten galacc rotaon curves of the SPARC sample. lerian behavior, instead Babcock found that the orbital velocies approach a constant value for the outer spiral arms and concluded that there must be much more mass at large radii than observed. Ulmately, he tried to explain the large mass- to-light rao with light absorpon with addional material in the outer regions of Andromeda or some new modiﬁcaon of the galacc dynamics. Even though Bab- cock’s measurements turned out inconsistent with newer measurements, his descripon of the qualitave behavior of the rotaon curve was correct. One year later, Oort reported a similar discrepancy between the light and mass distribuon in the galaxy NGC 3115, where he found a mass-to-light rao of 250 at large galactocentric distances [30]. Just as Babcock, Oort speculated on light absorpon and diﬀusion by interstellar gas and dust, as well as the existence of faint dwarf stars, as the source of this puzzle, but also menoned the idea of the galaxy being embedded in a larger dense mass. In the last sentences of his paper, Oort states the need for rotaon speed observaons at larger radii. These was made possible by the advent of radio astronomy. The 21cm spectral line of neutral hydrogen, predicted by Oort’s student Hendrik van de Hulst in 1944 [31] and discovered in 1951 by Ewen and Purcell [32], allowed the measurements of the rotaon curve to much higher radii. Van de Hulst himself, among many others, was involved in measuring both the Milky Way’s [33] and Andromeda’s [34] rotaon curve by means of 21cm line observaons. When Vera Rubin and Kent Ford revisited Andromeda in 1970 and measured the opcal rotaon curve with high accuracy [35], they found a constant rotaonal speed at large radii far exceeding the galacc disk in agreement with radio observaons. They concluded that the mass of the galaxy increases approximately linear with radius in the outer regions. We can see from eq. (2.1.5) that M(r) ∼ r would indeed explain

9 CHAPTER 2. DARK MATTER

the flat rotaon curve. Many more galaxies were analysed more systemacally in the late ’70s and ’80s, on the basis of opcal light and in parcular using the 21cm spectral line. The puzzling conclusion was that not a single observed rotaon curve showed the Keplerian speed drop, but instead had a flat rotaon curve [36] Confronted with these new observaons, the general interpretaon started to shi [37]. Astronomers started to appreciate that the ‘missing mass’ problem was indeed a real issue, that galaxies are bigger, and the outer galacc regions much more massive than they appear [38–41]. Others started to see a connecon to Zwicky’s ‘missing mass’ problem on cluster scales [42]. Up to today, thousands of galacc rotaon curves have been measured, a small selecon taken from the SPARC sample is shown in figure 2.2[43]. Their flatness is one of the most convinc- ing arguments that galaxies are embedded in a large DM halo4.

Gravitaonal lensing One of the predicons of Einstein’s general theory of relavity was the effect that light gets deflected by large masses, called gravitaonal lensing, which was first observed during a solar eclipse in 1919 [47]. Zwicky proposed already in 1937 that galaxies and galaxy clusters would act as huge gravitaonal lenses with observable consequences [48]. But it took 42 years before strong gravitaonal lensing was first observed [49]. A beauful example of strong gravitaonal lensing due to a galaxy cluster is shown in figure 2.3a. The mass of a heavy object is the crucial parameter determining the lensing effect and can be inferred this way. This was achieved for a galaxy cluster by e.g. Fischer and Tyson in 1997, who observed a mass-to-light rao of around 200 [50]. During the ’90s, it became more and more clear that the total masses of galaxy clusters obtained from gravitaonal lensing was consistent with independent measurements based on e.g. velocity dispersions [51, 52]. This consistency solidified the need for large amounts of undetected maer in clusters. Based on the idea by Kaiser and Squires in 1993 [53], weak lensing observaons allowed to directly map the spaal distribuon of DM in clusters in the following years, without any assumpons about its nature [54, 55]5.

The bullet cluster Another famous, more recent piece of evidence for DM on cluster scales is the observaon of the ‘bullet cluster’ [59–61], shown in ﬁgure 2.3b.

4We brieﬂy menon a prominent alternave to DM, which can reproduce the galacc rotaon curves by modifying Newton’s laws of moon, Modiﬁed Newtonian Dynamics (MOND)[44], and its relavisc realizaon [45]. A review can be found in [46]. 5For more details on lensing evidence for dark maer, we recommend the review by Massey et al. [56].

10 2.1. HISTORY AND EVIDENCE OF DARK MATTER IN THE UNIVERSE

(a) The ‘smiling cluster’ [57]. (b) The bullet cluster [58].

Figure 2.3: Strong gravitaonal lensing of the galaxy cluster SDSS J1038+4849 (le) and an overlay of the opcal, X-ray, and weak gravitaonal lensing observaon of the bullet cluster (1E 0657-558) (right).

The bullet cluster consists of two sub-clusters, which are driing apart aer having passed through each other. During this collision, the X-ray eming gas, visible in red in the figure, was separated by the galaxies due to their electromagnec interacons. The galaxies act like collision-less parcles and simply passed by unaffected. Without the presence of DM in this cluster, the predicted mass distribuon of this system should follow the X-ray observaons, as the gas makes up the majority of baryonic mass 6. However, the simultaneous measurement of weak gravitaonal lensing allowed to directly map the gravitaonal potenal of the bullet cluster (visible in blue in the figure), which traces not the gas, but the galaxies. It strongly suggests that the majority of mass is in the form of an undetected and collision- less maer. The spaal separaon of gravitaonal and visible mass, which was now observed in mulple instances [62], not only provides addional evidence for the existence of DM, but also challenges alternave proposals of modified gravity such as MOND.

Despite the strong evidence on the scales of galaxies and galaxy clusters, the most compelling evidence emerged on even larger scales.

6In astrophysics, ‘baryonic maer’ or ‘baryonic mass’ is oen used rather loosely to refer to ordinary maer including the non-baryonic electrons, as the protons and neutrons contribute most to the mass.

11 CHAPTER 2. DARK MATTER

2.1.2 Cosmological evidence

Cosmic Microwave Background During the early universe, all maer and radiaon made up an almost homogeneous plasma. Baryons and electrons were undergoing Thomson scaering and were in thermal equilibrium. As such, the universe was opaque to photons. This changed abruptly during recombinaon, when the universe cooled down and electrons and baryons formed neutral atoms. The universe became transparent rapidly, and photons could propagate freely aer their last scaering on a proton or electron. These photons form a radiaon background present throughout the cosmos, which is present to this day. This cosmic background radiaon was first predicted for the hot big bang model in 1948 by Alpher, Hermann and Gamow [63–65]. In 1965, Penzias and Wilson accidentally detected an isotropic source of microwave radiaon with a temperature of around 3.5 K [66]7. In the same year, Robert Dicke and his collaborators, who were scooped by the discovery, idenfied this radiaon as the cosmic microwave background radiaon, dang back to the me of recombinaon, and predicted 17 years prior [68]. Since the photons were in thermal equilibrium prior to their last scaering, the background radiaon should follow a Planckian black body spectrum [69]. The discovery of the CMB and the confirmaon of its thermal spectrum in the ’70s established the radiaon’s cosmic source and therefore the hot big bang model of the Universe [70]. Small fluctuaons of the CMB’s temperature were expected, because they should trace gravitaonal fluctuaons necessary to grow and evolve into the cosmological structure of galaxy and clusters. The primary anisotropies originated in the baryon- photon plasma around the me of recombinaon. They result from the opposing processes of gravitaonal clustering, which forms regions of higher density, and radiaon pressure of the photons, which erases baryon over-densies. The resulng acousc oscillaons leave a characterisc mark in the CMB. The CMB temperature fluctuaons’ direconal dependence in the sky is usually expressed in terms of spherical harmonics,

∞ l X X δT (θ, φ) = almYlm(θ, φ) . (2.1.8) l=2 m=−l

7Today’s best measurement of the Cosmic Microwave Background (CMB) temperature is TCMB =(2.72548 ± 0.00057)K [67].

12 2.1. HISTORY AND EVIDENCE OF DARK MATTER IN THE UNIVERSE

angular scale 90° 45° 18° 9° 1° 0.2° 0.1°

6000

5000 ] 2 K 4000 π [ μ 2 / l 3000 C ) 1 + l (

l 2000

1000

0 2 3 4 5 10 20 30 500 1000 1500 2000 2500 multipole moment l

Figure 2.4: The power spectrum of the CMB, both the data and the best ﬁt of the cosmological ΛCDM model.

The CMB’s so-called power spectrum is nothing but the variance of the coeﬃcients,

l 1 X C = |a |2 , l 2l + 1 lm (2.1.9) m=−l which are directly related to the two-point correlaon funcon of the temperature fluctuaons. The acousc oscillaons’ signature is a number of peaks in the power spectrum, which encode the Universe’s geometry and content. In parcular, the acousc peaks depend crically on the density of baryonic and dark maer, since only the baryonic maer experiences the photon’s radiaon pressure, and dark maer starts to cluster even before recombinaon. For a long me, only the dipole (l = 2) was observed, which is due to Earth’s moon relave to the cosmic rest frame. In 1992, the satellite-borne Cosmic Back- ground Explorer (COBE) first reported the observaon of ny CMB anisotropies −5 of order δT/T ∼ 10 [71]. Following COBE, a number of ground and balloon based experiments were performed to extend the measurements to smaller angular scales, i.e. higher mulpoles l [72–75]. Another great milestone was the Wilkinson Microwave Anisotropy Probe (WMAP), a spacecra located at the Lagrange point L2, which measured the first three peaks in 2003 [76, 77]. The most recent measurement of the CMB power spectrum was obtained by WMAP’s successor Planck in 2013 [78] and is shown in figure 2.4[79] 8. It also shows 8The data used for this plot was taken from the Planck Legacy Archive [80]. For l ≤ 30 the scale

13 CHAPTER 2. DARK MATTER

the best fit of the ΛCDM model, the standard model of cosmology, which describes the Universe as a homogenous, isotropic, and flat spaceme, whose total energy density consists of ordinary maer, dark maer, and dark energy in the form of a cosmological constant Λ. In terms of the parameters Ωi ≡ ρi/ρc, hence the relave contribuon of the different constuents to the crical density, the most recent fit of the Planck power spectrum yielded [79]

ΩM = 0.315 ± 0.007 , Ωb = 0.0493 ± 0.0002 , (2.1.10)

where ΩM is the relave amount of maer and Ωb accounts for the baryonic mass only. In order to explain the power spectrum, baryonic maer can make up ∼15% of the total maer in the Universe only, and the majority of maer must be dark. But as opposed to the interpretaons during the first half of the 20th century, ‘dark maer’ does not just refer to unobserved maer, but is inherently different from baryonic maer, as it does not interact with photons at all. The fact that baryons contribute to the cosmological density by only such a small amount was confirmed by the independent predicons of Big Bang Nucle- osynthesis (BBN), the producon of light nuclei during the early Universe, first described by Alpher and Gamow in the infamous Alpher-Bethe-Gamow paper [81] and later refined e.g. in [82, 83]. In order to account for the observed abundance of light nuclei, the baryonic density is determined to lie between 0.046 and 0.055 [84], in perfect agreement with WMAP’s or Planck’s finding. In combinaon with observaons of high red shi type Ia supernovae, which also suggest a flat P universe with i Ωi = 1 [85, 86], we have a completely independent confirmaon of the CMB results.

This draws a consistent cosmological picture, which cannot work without large amounts of non-baryonic DM, in perfect agreement with evidence from smaller scales. Yet, this is not the last cosmological argument in favour of DM. It turned out that DM also played a crical role in the evoluon of the Universe’s structure.

Structure formaon During the evoluon of the cosmos, the inial over-densies, whose seeds we can observe in the CMB grew under the inﬂuence of gravity into large-scale structures of galaxies and clusters. The main approach to study cosmo-

is logarithmic, otherwise the scale is linear. For l > 30 the data is binned with a bin width of 30. The unbinned data is visible in light gray.

14 2.1. HISTORY AND EVIDENCE OF DARK MATTER IN THE UNIVERSE

(a) Observaons vs. simulaons of (b) An example of the simulated DM distribuon galaxy distribuons [94]. forming the cosmic web [95].

Figure 2.5: N-body simulaon of structure formaon. logical structure formaon are numerical N-body simulaons of many gravitang parcles, which describe their clustering under various assumpons and compare this to observaons of large structures. The pioneer of these simulaons was arguably Erik Holmberg, a Swedish astronomer who studied the gravitaonal interacon of galaxies in 1941 [87]. He set up an array of 37 light bulbs and used pho- 2 tocells to measure the “gravitaonal force”, employing the ∼ 1/r behaviour of the light intensity to mimic the gravitaonal force. The first numerical simulaons for cosmological scales were performed in the ’70s modelling galaxies as a gas of self-gravitang parcles [88]. Almost from the start, it was clear that, without DM, galaxies would have formed much too late, as baryonic maer starts to cluster later due to its dissipave non- gravitaonal interacons. In addion, it turned out that the formaon of smaller structures depends crically on the velocity of the DM and that fast thermal moon of the DM would wash out and suppress the formaon of small structures [89, 90]. Therefore, gravitaonal clustering of ‘hot’ DM would inially create large structures. In contrast, non-relavisc or ‘cold’ DM can collapse into low mass halos early on. In this case, cosmological structure also builds up hierarchical, but boom up, from stars to stellar clusters, from galaxies to clusters and super clusters. The observaon of small sub-structures in the first 3D surveys of galaxies [91] confirmed this hierarchical structure evoluon and lead to the cold DM paradigm [92, 93]. Newer simulaons with beer resoluons beaufully show how the DM and galaxies cluster and form the cosmic web, huge filaments surrounding enormous voids, most famously the Millennium simulaons from 2005 [94, 96]. The compar-

15 CHAPTER 2. DARK MATTER

ison to galaxy surveys [97, 98], as shown in ﬁgure 2.5a, show excellent agreement with the observaon on large scales. While the ‘DM-only’ type of simulaons succeeded in explaining the largest scales, they fell short on galacc scales in some respects. The major small-scale problems were coined the ‘too-big-to-fail’ [99], ‘cusp vs. core’ [100, 101], ‘diversity’ [102], and ‘missing satellite’ problem [103]. However, it is debated how severe these problems really are, as the inclusion of baryonic feedback e.g. from supernovae into the simulaons and the observaon of faint Milky Way satellites reduces the tension between observaons and predicons [104–106]. Especially the missing satellite problem seems to have disappeared over the last years.

During this chapter, dark maer was treated as an almost purely astrophysical ﬁeld. Yet the meaning of the term ‘dark maer’ shied signiﬁcantly in the second half of the 20th century. The term evolved slowly from referring to ordinary, but dim and unobserved maer, to something else enrely: an unknown form of maer, which seems to interact exclusively via gravity. Starng arguably with Gershtein and Zeldovich in 1966 [107], the quest for dark maer would merge cosmology, astrophysics, and parcle physics.

2.2 Parcle Dark Maer

In the previous chapter, we reviewed the compelling set of astrophysical evidence for the existence of DM in the Universe. Yet, very lile is known about its nature. During the ’70s, parcle physicists naturally started to speculate about the identy of DM, as discovering a new parcle was not uncommon at this point. Before exoc new parcles would be considered as the source of DM, the possibility that the galacc halos consist of ordinary maer needed to be explored. This can be regarded as the connuaon of earlier interpretaons of the ‘missing mass’ problem. For a while, it seemed possible that DM was comprised of Massive Com- pact Halo Objects (MACHOs)9, small, massive, and dark objects of baryonic maer, such as dim stellar remnants, black holes and neutron stars, rogue planets, rocks, and brown dwarfs, which dri unbound through the interstellar space and evade direct observaon. An early study by Hegyi and Olive in 1985 argued against baryonic halos [108],

9The term MACHO was coined by Kim Griest, as contrast to the WIMP, which we will discuss later.

16 2.2. PARTICLE DARK MATTER where they in parcular pointed out the incompability of the hypothesis of large amount of baryonic MACHOs and the central result of BBN, which strongly suggested that baryonic maer constutes only a small fracon of the cosmological energy density. This was also veriﬁed by CMB observaons as discussed in the previous chapter. During the same year, a new technique of looking for MACHOs, regardless of their composion, was presented by the Polish astronomer Bohdan Paczynski.´ He suggested to look for temporary ampliﬁcaon of the brightness of a large number of background stars, caused by gravitaonal lensing of a massive, transient object [109]. Paczynski´ called this process microlensing. This method does not require of the massive object to emit light itself, hence it is ideal to search for all kinds of galacc MACHOs, not just baryonic ones. Large microlensing surveys searched for dark stellar bodies, most notably the MACHO project [110] and EROS-2[111, 112]. While early results seemed to show an excess of microlensing events, eventually these surveys ruled out MACHOs as the primary source of DM in the galaxy.

While the 1985 paper by Hegyi and Olive le the possibility of black hole MA- CHOs open, it turned out that stellar remnant black holes did not have enough me to form and populate the halos. Yet, already 11 years earlier, the Brish theore- cians Bernard J. Carr and Stephen W. Hawking noted that density ﬂuctuaons in the early Universe, necessary for the formaon of structure, could collapse gravitaonally and form so-called Primordial Black Holes (PBHs) [113]. These non-stellar black hole relics could have survived to this day while accreng more mass and possibly form the galacc halo, without the need to invoke new forms of maer. However, there are severe constraints on PBHs as DM. These constraints have been re-evaluated more recently, aer the discovery of gravitaonal waves from black hole mergers by the Laser Interferometer Gravitaonal-Wave Observatory (LIGO) in 2016 [114]. While some conclude PBHs to be excluded by microlensing and dynamical constraints, at least as the only contributor to DM [115, 116], others ﬁnd that they remain a viable opon [117, 118].

In conclusion, the aempts to explain the observaons with ordinary maer alone failed, and the non-baryonic nature of the galacc halo seemed unavoidable [119].

17 CHAPTER 2. DARK MATTER

2.2.1 When astronomy and parcle physics merged

With ordinary baryonic maer being excluded as the soluon to the ‘missing mass’ problem, the next queson we should ask is, what condions a parcle would have to sasfy to act as DM. It should be noted that there is no reason to assume that all observaons of DM can be aributed to a single parcles. It could also be a dark sector with mulple parcles and interacons. A DM parcle should 1. obviously be non-luminous, i.e. not reﬂect, absorb or emit light,

2. be non-relavisc and therefore able to drive structure formaon,

3. be stable, at least on the me scale of the age of the Universe,

4. act collision-less and non-dissipave. This implies that the DM parcle should have no or only weak interacons with ordinary maer apart from gravity, and ﬁnally

5. get produced in the right amount during the early Universe via some mechanism. Naturally, the first approach is to check, if any of the known parcles can sasfy these criteria. Indeed, the neutrinos, originally predicted by Wolfgang Pauli in 1930 to explain the connuous energy spectrum of β-decay [120] and observed 26 years later by the Cowen-Reines neutrino experiment [121], seemed to fit the bill. The Russian physicists Semyon S. Gershtein and Yakov B. Zeldovich, two pioneers of what would become the field of astroparcle physics, discussed the cosmological implicaons of massive neutrinos in 1966 [107]. In analogy to the recently discovered CMB, they computed the thermal relic abundance of electron and muon neutrinos and derived upper limits on neutrino masses based on the expansion history. Even though they did not connect their findings to DM, their work can be considered as essenal groundwork for the idea of parcle DM and in parcular WIMP DM. Soon, others drew the connecon and started to consider neutrinos as potenal DM [122–124]. While the idea that DM was nothing but neutrinos must have been very ap- pealing, there are two problems. For one, the relic density of relic neutrinos was determined by known physics and turned out too low. The relave contribuon to the cosmic energy density is given by [125] P m Ω h2 = i i , ν 93.14eV (2.2.1) 18 2.2. PARTICLE DARK MATTER

−1 −1 where h is the Hubble parameter in units of 100 km s Mpc . With an upper limit P of around i mi < 0.11 eV for the sum of neutrino masses [126], we ﬁnd that

−3 Ων < 2.6 · 10 . (2.2.2)

Comparing to eq. (2.1.10), neutrinos can only account for a small fracon of non- baryonic maer in the Universe. The second reason against neutrino DM is that they would be relavisc during structure formaon and behave as hot dark matter. As such, it cannot reproduce the observed hierarchical formaon of cosmic structure [89, 90, 127]. In the end, the neutrino did not sasfy two of our ﬁve condions. But it can be regarded as the historic blueprint to many DM parcles proposed in the following years.

2.2.2 DM models and parcle candidates

When it became clear that the neutrino cannot account for the missing mass in the Universe, the full predicve power of parcle theorists was unleashed to come up with well-movated and ideally testable models of DM. Oen, new models proposed for independent reasons turned out to contain new parcles, which could act like dark maer. In this chapter, we list a exemplary selecon of the most common candidates for DM parcles in diﬀerent contexts. Naturally, this list is not exhaus- ve.

Sterile Neutrinos The neutrinos of theSM are the only fermions, which appear exclusively with le handed chirality, yet nothing forbids adding neutrinos with right handed chirality. These leptons would be gauge singlets and not interact with the other ﬁelds, which is why they are also called ‘sterile neutrinos’. Introducing heavy sterile neutrinos could also explain the small, non-vanishing neutrino masses via the seesaw mechanism [128]. If these heavy fermions are of keV scale mass, they would get produced in the early Universe via a non-thermal mechanism and act as DM [129, 130]. For a recent review on the observaonal status and producon mechanisms of sterile neutrino DM, we refer to [131].

Supersymmetry In the beginning of the ’70s, a fundamentally new kind of symmetry of quantum ﬁeld theories was discovered, which is called Supersymmetry

19 CHAPTER 2. DARK MATTER

(SUSY)10 [133–138]. SUSY relates the fermionic and bosonic degrees of freedom and therefore predicts that the fermions (bosons) of a parcle model are accompa- nied by a bosonic (fermionic) partner. Supersymmetric ﬁeld theories remain theorecally and phenomenologically well-movated and gain a great deal of aenon to this day. In addion, supersymmetric models have arguably been the most generous providers of DM parcle candidates. The introducon of SUSY roughly doubles the number of parcles, and the lightest of the supersymmetric parcles is typically regarded a candidate for DM, provided that it is stabilized by R-parity [139]. Due to its deep connecon to the Poincare´ group, supersymmetry as a local symmetry enforces that gravity has to be included. This is why local supersymmetry is usually called supergravity. In this context, the ﬁrst supersymmetric DM candidate was the gravino, the spin-3/2 partner of the graviton [140, 141]. With the formulaon of the Minimal Supersymmetric Standard Model (MSSM) in the ’70s and ’80s [142–146], the neutralino became one of the most studied potenal DM parcles [147–149]. The neutralino is a mixture of the fermionic partners of the neutral bosons of theSM. It became the archetype of a Weakly Interacng Massive Par- cle (WIMP), a much larger, more general class of DM parcle candidates with weak, but observable interacons with the SM. Other, less popular supersymmetric candidates include the scalar partner of the neutrino, the sneutrino [150] and the axino [151].

Extra dimensions One example for a non-supersymmetric WIMP arises from the consideraon of compacfied extra dimensions. The idea goes back to the Ger- man physicists Theodor Kaluza and Oskar Klein. In 1921, Kaluza tried to unify elec- tromagnesm with Einstein’s theory of gravity by postulang a fih spaal dimension [152], whereas Klein provided an interpretaon of this extra dimension as being microscopic and periodic [153]. Around 60 years later, Edward W. Kolb and Richard Slansky realized that compact extra dimensions could be associated with addional stable, heavy parcles [154]. These parcles arise due to the conservaon of the quanzed momentum along the compact extra dimension. The high- momentum states build up a tower of so-called Kaluza-Klein states, which appear −1 in the four large dimensions as parcles with increasing mass of scale MKK ∼ R , where R is the compacficaon scale. In models of Universal Extra Dimensions (UED) all SM fields are allowed to propagate through the extra dimension [155]. If the lightest of the Kaluza-Klein states

10For a comprehensive review of SUSY and the MSSM we recommend the ‘Supersymmetric Primer’ by Stephen P. Marn [132].

20 2.2. PARTICLE DARK MATTER is stabilized by some symmetry and cannot decay into the ground state, which corresponds to the usual SM parcle, this parcle would be a qualiﬁed DM parcle. As such it could get produced thermally and be observed e.g. via direct detecon [156–158].

Axions In 1977, Roberto D. Peccei and Helen R. Quinn proposed a soluon to the strong CP problem of Quantum Chromodynamics (QCD)[159, 160], which predicted a new parcle, as was quickly noted by the American theorists Stephen Weinberg and Frank Wilczek [161, 162]. The symmetries of QCD allow for the CP violang term θ h i L ⊃ Tr GµνG˜ , QCD 32π2 µν (2.2.3)

˜ where Gµν (Gµν) is the (dual) ﬁeld strength tensor of QCD and the trace runs over the SU(3) color indices. This term introduces CP violang interacons into QCD. However, from upper limits of the neutron’s electric dipole moment [163], we know that

−10 θ . 10 . (2.2.4)

The seemingly fine-tuned suppression of an otherwise allowed term is the strong CP problem. In their soluon, Peccei and Quinn explain the parameter’s suppression by introducing a new global, anomalous U(1) symmetry, which is spontaneously broken by a complex scalar field. In their model, the effecve θ parameter depends on this field and vanishes dynamically once the field aains its vacuum expectaon value. Furthermore, an addional light parcle arises naturally as the pseudo-Nambu-Goldstone boson. This parcle was called the axion. The standard Weinberg-Wilczek axion was excluded by prompt experimental searches, and more general realizaons such as the Invisible Axion were developed [164], culminang in a large class of Axion-Like Parcles (ALPs). The axion might not just be the by-product of the Peccei-Quinn mechanism, it could also be another candidate for DM, solving two problems at once [165, 166]. These light scalar parcles could get produced non-thermally and non-relaviscally during the early universe and act as a collision-less fluid sasfying all five DM condions. Consequently, a large number of experimental searches were performed aiming at the discovery of the axion [167, 168].

21 CHAPTER 2. DARK MATTER Y ( x ) Y ( 1 ) 1 σA=σ0

σA=10σ0 0.01 σA=100σ0

10-4

-6 moving number density 10 - Yeq(x) co 1 5 10 50 100 500 time variable x= mχ T

Figure 2.6: Thermal freeze-out of WIMPs.

2.2.3 Origin of DM in the early Universe

An essenal aspect of DM phenomenology is to explain how it was produced in the right amount during the early Universe. A lot of the proposed DM parcles share a common producon mechanism similar to the neutrinos’ and get produced thermally. They make up the large class of the WIMPs [169]. A WIMP is characterized by

• a mass of MeV-TeV scale,

• its non-gravitaonal interacons with ordinary maer. This interacon should not be stronger than the weak interacon of the SM.

• its thermal producon via ‘freeze-out’.

WIMPs are in thermal equilibrium with the SM bath during the early Universe, connuously created by and annihilang into lighter parcles. As the Universe kept expanding and cooling, the light parcles lost energy and with it the ability to generate new DM parcles. At this point, the DM populaon got depleted due to their ongoing annihilaons. But even the annihilaons stopped being eﬀecve, when the number density has dropped so low that the annihilaon rate dropped below the expansion rate, and parcles and an-parcles no longer came into contact. Af- terwards, a constant number of DM parcles remained in the Universe as a thermal relic, not unlike the photons of the CMB or the neutrino background. This producon mechanism is called ‘thermal freeze-out’. The relic abundance of a parcle can be computed using the Boltzmann equaon [139, 170], which determines the evoluon of the DM number density nχ in

22 2.2. PARTICLE DARK MATTER an expanding Universe,

dnχ 2 eq 2 + 3H(t)nχ = − hσAvi nχ − (nχ ) . (2.2.5) dt | {z } | {z } Hubble diluon creaon and annihilaon

Here, σA is the total annihilaon cross secon, v is the relave velocity, and h·i denotes the thermal average. In order to scale out the diluon due to expansion, 3 we can use the conservaon of the entropy density s, i.e. sR = const. We deﬁne nχ dY dnχ the DM density in a co-moving volume Y ≡ s , such that dt = dt + 3H(t)nχ. mχ Usually the me parameter is replaced by x ≡ T . Hence, the Boltzmann equaon becomes

dY xshσAvi 2 2 = − Y − Yeq . (2.2.6) dx H(T = mχ)

The freeze-out occurs during the radiaon dominated epoch of the Universe, which −2 fixes the Hubble parameter H(x) ∼ x . The entropy density per co-moving vol- −3 ume stays constant, s(x) ∼ x . Some example soluons for Y (x) with different annihilaon cross secons are shown in figure 2.6 and illustrate the thermal producon of a non-relavisc relic. While the equilibrium number density falls ex- −x ponenally Yeq(x) ∼ e , at some point the DM decouples and freezes out to a constant co-moving volume number density. The higher the annihilaon cross secon the longer the DM keeps annihilang in thermal equilibrium and the lower the final density. An approximate expression for the present WIMP density in units of the crical density is given by

σ v −1 Ω h2 ' A . χ pb 0.1c (2.2.7)

This procedure to compute the thermal relic of WIMPs gives a good esmate, but is not very precise. The calculaon has been greatly reﬁned to yield more precise predicons [171–174]. However, it was quickly noted that a weak scale cross secon would lead to Ωχ = O(1) , naturally explaining the origin of the right amount of DM. This rough agreement between the WIMP relic density and the observed DM density was called the ‘WIMP miracle’. It movated a great number of strategies and experiments in the following decades, aiming to detect WIMP DM. Unfor- tunately, these eﬀorts have not been successful so far, and the WIMP paradigm is geng constrained more and more [175]. Nonetheless, it has not been excluded altogether [176].

23 CHAPTER 2. DARK MATTER

direct detection

DM particle particle DM collider

interaction SM particle particle

indirect detection SM

Figure 2.7: The three main search strategies for non-gravitaonal DM interacons.

The DM parcles in the Universe do not need to be a thermal relic, many other producon mechanism have been proposed which do not rely on DM being in thermal equilibrium. Examples include the ‘freeze-in’ mechanism of very weakly interacng DM [177, 178] or DM producon from heavy parcle decays [179, 180]. For more informaon on non-thermal DM producon we refer to [181, 182].

2.2.4 Detecon strategies

Although the evidence for DM is almost conclusive, it is unfortunately purely gravitaonal. There is lile hope to directly observe or produce DM parcles, if the only DM-maer interacon is via gravity. However, there are good reasons that the dark and the bright sector share some other interacon. The DM parcle might not be a gauge singlet under theSM gauge groups and parcipate in some of the known forces. Alternavely, there might exist some interacve portal between the light and dark sector. The field acng as that portal might be known, such as the Z or the Higgs portal, or be a new field, e.g. a dark photon. There are three major strategies to search for non-gravitaonal effects of DM, illustrated in figure 2.7.

Indirect detecon If DM parcles could annihilate or decay into SM model parcles, these parcles could be observed with cosmic ray and neutrino telescopes. The approach to look for observaonal excesses of SM parcle ﬂuxes originang from regions of high DM density is called indirect detecon [183, 184]. These high- density regions could be the galacc center of the Milky Way, neighbouring dwarf galaxies, or galaxy clusters. In the case of the WIMP, the annihilaon cross secon determined the relic density. The indirect observaon of such annihilaons could thereby probe not just the existence of DM parcles, but also its origin.

24 2.2. PARTICLE DARK MATTER

Possible detecon channels are DM annihilaons into gamma-rays [124, 185], an-protons and positrons [186–189], or neutrinos [190]. The clearest and most conclusive indirect detecon of DM annihilaon would be a monochromac feature in the cosmic-ray spectrum. Another possible discovery channel of indirect detecon are neutrinos from the Sun. DM parcles could get gravitaonally captured by the Sun, aggregate in the solar core and annihilate resulng in an addional neutrino flux from the Sun [191–193]. Due to their weak interacons with maer, neutrinos could escape even those dense regions. The main challenge of indirect detecon is the disncon between the flux of the annihilaon products and background from astrophysical sources. Another challenge arises for the observaons of charged parcles which get reflected and diffused by galacc magnec fields or scaerings, impeding the assignment of an observaon to a parcular source [194, 195]. The interpretaon of an observed excess is difficult and drawing a definive conclusion of a DM discovery even more so. A number of experiments have indeed measured excesses. One example is an observed rise of the cosmic-ray positron fracon with energy, measured by the satellite-borne cosmic-ray observatories PAMELA and AMS-02 [196, 197]. The excess could in principle be interpreted as the product of DM annihilaons [198, 199].

Direct producon If DM couples to maer, it could be possible to produce DM parcles in high energy parcle collisions such as the Large Hadron Collider (LHC)[200– 203]. Aer their creaon, these parcles would leave the detector without a trace. Hence, the main signature of DM in colliders is Missing Transverse Energy (MET)11, and neutrino producon will be an important background. At hadron colliders, the typical signature event for the producon of a pair of DM parcles is a single jets or photons from inial state radiaon, plus MET. There are two draw-backs of collider searches of DM. For one, the search for new physics with a collider is always to some degree model-dependent, since a model is necessary to make predicons. This problem can be eased by a more general and standardized formulaon of DM-maer interacons using EFTs for contact interacons and simplified models in the presence of light mediators [205, 206]. More crically, colliders alone would not be able to confirm a newly discovered weakly interacng parcle as the source of DM. They could e.g. not test the stabil- ity of the parcle and only find weak lower bounds on its lifeme. The discovery

11A DM parcle would not be the ﬁrst parcle to be discovered through an MET signature, as the W boson was discovered through this technique in 1983 [204].

25 CHAPTER 2. DARK MATTER

of the same parcle by complementary astrophysical experiments is necessary to conclusively draw the connecon between the galacc halo and the observaon at a collider [207].

Direct detecon The basic idea of direct detecon is to search for recoiling nuclei or electrons in a terrestrial detector resulng from an elasc collision between an incoming DM parcle from the halo and a parcle of the detector’s target mass. Direct detecon is the main topic of this thesis and will be discussed in detail in the next chapter.

26 Chapter 3

Direct Detecon of Dark Maer

If the galacc halo is comprised of one or more new parcles which interact not just gravitaonally with ordinary maer, there is hope to detect these parcles here on Earth. These parcles would connuously pass through our planet in great numbers. The basic idea of the direct detecon of DM is to search for nuclear recoils resulng from an elasc collision between an incoming DM parcle from the halo and a nucleus inside a terrestrial detector. In this chapter, we review the development of the ﬁeld of direct DM searches, as well as the basic relaons and tools to make predicons for direct detecon experiments. We will treat both the standard nuclear recoil experiments and newer techniques to search for sub-GeV DM based on inelasc DM-electron interacons. The necessary model of the galacc halo, the scaering kinemacs, the descripon of DM-maer interacons, as well as the computaon of event rates for any detecon experiment are presented in detail, before we conclude with a short summary to direct detecon stascs and limits.

27 CHAPTER 3. DIRECT DETECTION OF DARK MATTER 3.1 The Historical Evoluon of Direct DM Searches

3.1.1 Standard WIMP detecon

The fundamental technique of direct detecon experiments was originally proposed by Andrzej Drukier and Leo Stodolsky in 1983 as a way to detect neutrinos via coherent elasc scaerings on nuclei [208]. Soon aer, three groups, one including Drukier himself, independently pointed out that this method might also serve as a possible detecon technique for DM parcles [209–211]. Especially the WIMP scenario can be probed by this strategy since GeV scale DM parcle would cause observable rates of keV scale nuclear recoils, and it did not take long unl the first direct detecon experiments took place [212, 213], seng the first direct detecon constraints on DM-maer interacons. In their 1986 paper, Drukier, Freese, and Spergel proposed to ulize the expected annual signal modulaon to disnguish between a potenal DM signal and background from other sources [211]. This modulaon occurs due to the orbital moon of the Earth around the Sun, which leads to a small variaon of the DM flux and event rate in a detector. About twenty years ago, the DAMA experiment first reported the observaon of such a modulaon [214]. The observed events in their sodium-iodide (NaI) target crystal showed not just an annual rate modulaon, the rate also peaked at the expected me of the year. DAMA/NaI and the upgrade DAMA/LIBRA connued to confirm their observaons over the years [215–217]. In the latest DAMA/LIBRA phase 2 run, they finished the observaon of the 20th annual cycle. Remarkably, they report a modulaon period of (0.999±0.001)yr and phase of (145±5) days, just as expected for DM1. The reason, why the DAMA claim is met with great scepcism, is the fact that no other experiment was able to confirm the observaon. Even worse, the findings of many other DM searches are in serious tension with the DAMA results and repeatedly exclude the parameter space favoured by DAMA. Since these experiments use different target materials, it is not impossible that some unexpected type of interacon would show up in a NaI crystal, but not e.g. in liquid xenon experiments. The comparisons rely on standard assumpons, which could of course be inaccurate. Unl an experiment of the same target material confirms or refutes the discovery claim, the DAMA signal’s origin remains unsolved. A number of direct detecon experiments with sodium-iodide crystals are being planned [218–221]. The first re-

1The theorecally predicted event rate under standard assumpons peaks around June 2nd, corresponding to a phase of 152 days.

28 3.1. THE HISTORICAL EVOLUTION OF DIRECT DM SEARCHES sult by DM-Ice17 [222] and COSINE-100 [223] did not show evidence for an annual modulaon and seem to indicate that the DAMA/LIBRA modulaon is indeed not due to parcles from the DM halo. Over the next few years, these experiments will ﬁnd a deﬁnive answer.

Direct detecon experiments can be classiﬁed in two broad categories of two- channel detectors, using either crystal or liquid noble targets. In both cases, the discriminaon between background events and a potenal DM signal is realized by spling the signal into two components, which are detected independently. The historically ﬁrst category are detectors with cryogenic solid-state targets. One two-channel technique of background rejecon is to simultaneously measure both ionizaon and heat signals [224, 225]. This method was applied to pure germanium targets by the EDELWEISS experiments [226–228] and silicon/germanium semiconductors by the CDMS detectors located in the Soudan Mine [229–231]. The newest generaon of SuperCDMS will be taking data from SNOLAB [232]. Another similar method is to simultaneously detect scinllaon photons and heat phonons, as done by the three CRESST experiments. Where CRESST-I used a sapphire target [233, 234], the other two generaons employed CaWO4 crystals [235–237]. Finally, a new generaon of solid-state detecon experiments looking for light DM, namely DAMIC [238, 239] and SENSEI [240], uses silicon CCDs as target, which read out ionized charges in each pixel, potenally caused by a DM-atom interacon in the silicon crystal. In these experiments, background events are rejected due to their characterisc spaal signal correlaon in neighbouring pixels, as opposed to point-like energy deposions by a WIMP. So far, the results of these experiments have been a series of null results with a few excepons. The CoGENT experiment, a germanium detector in the Soudan Underground Laboratory (SUL), has reported evidence for annual signal modulaons [241–243], however a re-analysis of the data found an underesmaon of background due to surface events [244, 245]. Furthermore, both CRESST-II (phase 1) and CDMS II reported an observed signal excess [235, 246]. For CRESST, these signals could also be aributed to background sources, and a DM interpretaon was eventually excluded [247]. The CDMS signal is in tension with constraints from other experiments, and a DM origin seems disfavoured as well [248]. These anoma- lies and their interpretaon, including the DAMA observaon, have been discussed in greater detail in [249]. In order to increase the sensivity of detectors to weaker interacons, larger

29 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

targets were needed. However, the upscaling of solid state detectors to sizes beyond the kg scale is costly. In the year 2000, a new detecon technology was proposed, the use of a two-phase noble target [250, 251]. Xenon has been the most common noble target, as it is easily purified, radio-pure, chemically inacve, has large ionizaon and scinllaon yield, combined with a heavy nucleus ideal to probe coherent spin-independent interacons. In addion, a xenon target can be realized even in ton scales. Just as for crystal targets, the signal is split in two parts to reject background. An incoming DM parcle would scaer on a xenon nucleus in the liquid phase, and the nuclear recoil causes the first scinllaon signal (S1) followed by a me-retarded second scinllaon signal (S2) in the detector’s gas phase. Both signals are observed with Photomulplier Tubes (PMTs). The S2-signal is caused by electrons which get ionized in the original scaering and dri towards the gas phase due to an external electric field. This ingenious idea to use the me- separated combinaon of scinllaon and ionizaon became the blueprint for a series of experiments with increasing target sizes all around the world.

While single-phase liquid xenon detectors have already been proposed and performed in the ’90s [252–254] and early ’00s [255, 256], the ﬁrst two-phase xenon detector was the ZEPLIN-II experiment at the Boulby Underground Laboratory in England, which reported the ﬁrst results in 2007 [257]. ZEPLIN-II was followed by a third generaon in 2009 [258]. Within Europe, they competed with the XENON experiments located at the Laboratori Nazionali del Gran Sasso (LNGS), with their three generaons, XENON10 [259, 260], XENON100 [261, 262], and XENON1T [263, 264]. The similar LUX experiment, installed at the Sanford Underground Research Facility (SURF) in the Homestake Mine, set leading constraints on WIMPDM[265, 266]. In the last few years, two more direct DM searches were performed by the PandaX collaboraon with xenon detectors at the China Jinping Underground Lab- oratory (CJPL) [267–270]. The Japanese XMASS-I experiment started to use single phase liquid noble targets again [271], while others switched from xenon to an argon target, namely the DarkSide-50 dual-phase detector at Gran Sasso [272, 273] and the single-phase detector DEAP-3600 at SNOLAB [274]. Planned future experiments include LUX’s successor, the next-generaon LUX-ZEPLIN (LZ) experiment, which is expected to take data with a 7 ton dual phase xenon target by 2020 [275], as well as proposals for XENONnT in Gran Sasso [276, 277], and a mul-ton dual phase xenon experiment DARWIN [278].

The connuing null results of these enormous experimental eﬀorts were certainly a disappointment for convenonal direct DM searches. While larger and

30 3.1. THE HISTORICAL EVOLUTION OF DIRECT DM SEARCHES larger detectors connue to constrain and rule out weaker and weaker WIMP interacons, no conclusive evidence for non-gravitaonal DM interacons in terrestrial detectors has ever been found. This caused a shi away from the standard WIMP paradigm. One way, to loosen the usual assumpons is to search for low-mass DM parcles.

3.1.2 Low-mass DM searches

The results presented in this thesis do not consider a parcular parcle physics model which contains a DM candidate. Instead it focuses on light, mostly sub-GeV, DM parcles and their phenomenology in direct detecon experiments, while stay- ing mostly agnosc about the parcle’s origin. Nonetheless, it should be noted that light DM arises in a number of well-movated parcle models. The standard WIMP picture can be modified in order to circumvent the Lee-Weinberg limit [123] such that DM of masses below ∼2 GeV can be produced as a thermal relic. This was shown to work for scalar DM [279], in supersymmetric models [280–282], e.g. as light axinos [283, 284], for Strongly Interacng Massive Parcles (SIMPs), meaning DM with strong self interacons [285, 286]2, for elascally decoupling DM [287], secluded sector DM [288], or light DM parcles which annihilate into heavier parcles during the early Universe [289]. Sub-GeV DM can also be produced non- thermally as asymmetric DM [290–294] or via freeze-in [178]. Direct detecon experiments only probe parcles down to some minimal mass. The energy deposits of even lighter DM fall below the experiment’s threshold and are insufficient to trigger the detector. Convenonal nuclear recoil experiments typically have a recoil threshold of the order of ∼keV and therefore probe DM parcles with masses above a few GeV. It is a great challenge to probe masses below the GeV scale using nuclear recoils. The experiments of the CRESST collaboraon are leading in this field and have pushed the limits of low-mass DM searches. They realized recoil thresholds of the order of O(10)eV for a gram scale detector, seng constraints on DM of masses down to ∼140 MeV [295]. The next generaon CRESST-III experiment is expected to reduce the threshold further [237]. There are a number of ideas, which do not require a new generaon of experiments. One possibility is to consider signatures of non-standard interacons at convenonal large-scale detectors. A sub-GeV DM parcle could cause an otherwise unobservable nuclear recoil, where the recoiling nucleus emits Bremsstrahlung.

2The abbreviaon SIMP is not used consistently in the literature, where some refer to strong self- interacons and others to strong couplings to ordinary maer. This is why we refrain from using it.

31 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

The emied photons are able to trigger the detector [296, 297]. Another idea is to employ the Migdal effect [298] and observe electrons dissociated from the atom through the nuclear scaering. As the nucleus recoils, the atomic electrons do not immediately follow and might therefore get excited or ionized, leading to a signal [299–303]. Both techniques of extending a detector’s sensivity to lower masses were recently applied by the LUX collaboraon to derive constraints on masses down to 400 MeV [304]. Another idea is to look for processes which could accelerate DM parcles as faster parcles are able to deposit larger recoil energies in a detector. If sub-GeV DM parcles scaer on hot constuents of the Sun, they could gain energy and make up a highly energec solar DM flux. This was shown to increase detectors’ sensivity for DM-electron interacons [305] and independently by us for DM-nucleus scaerings in Paper III. A similar idea is to apply the same argument to relavisc cosmic rays, which could also transfer energy to halo parcles, which could then be observed in terrestrial detector, even though they might have been undetectable before the scaering due to their low mass [306]. The solar reflecon of sub-GeV DM is the topic of chapter5.

Arguable the most promising idea to sub-GeV DM searches is to consider DM- electron scaerings [307–309]. MeV scale DM parcles can transfer almost their enre kinec energy to bound electrons and are hence able to ionize atoms. In parcular, the scaering on electrons in xenon experiments such as XENON10 [260] and XENON100 [310] have been invesgated. These experiments set strong constraints on DM-electron scaerings for DM masses as low as a few MeV [311, 312]. With DarkSide-50, the ﬁrst argon target detector has probed DM-electron scaerings in 2018 [313]. Despite their smaller exposures, semiconductor targets are even more promising due to their small band gaps of order O(1) eV [314–317]. The ﬁrst experiments to test electron scaerings were SENSEI (2018) [240, 318] and Super- CDMS (2018) [319], both using a silicon semiconductor targets. In the near future, the DAMIC-M collaboraon is planning to install a detector with a remarkably large semiconductor target mass of kg scale [320].

These are not the only new proposals for experimental techniques and search strategies aimed at light DM over the last few years. Others have suggested the use of scinllang materials [321], two-dimensional targets such as graphene for direconal detecon [322], superﬂuid helium targets [323–326], molecule dissocia- on [327], super conductors [328–330], and many other eﬀects and techniques [331– 336]. Many of these new ideas have been reviewed in [337].

32 3.2. THE GALACTIC HALO 3.2 The Galacc Halo

To interpret the outcome of a direct detecon experiment, it is crucial to know how many and how energec DM parcles are expected to pass through the detector. It is necessary to esmate the local DM density and velocity distribuon, which are related through the gravitaonal potenal of the halo. The diﬀerenal number density of DM parcles close to the Sun’s orbit within the galacc halo with velocity 3 within (vχ, vχ + d v) can be wrien as

3 ρχ 3 dn = nχfhalo(v) d v = fhalo(v) d v , (3.2.1) mχ where fhalo(v) is the normalized velocity distribuon and nχ and ρχ are the local DM number and energy density respecvely. The DM density is assumed to remain constant along the Sun’s orbit around the galacc centrum throughout this thesis. However, this does not need to be true, which can have crical consequences for direct detecon [338]. Throughout this thesis, we use the canonical value of −3 ρχ = 0.3 GeV cm [339], even though newer evidence suggests a value closer −3 to ∼ 0.4 GeV cm [340]. The reason for this is the fact, that the former value is widely used in the literature and serves as a fiducial value. The velocity distribuon depends on the DM halo model. The convenonal choice is the Standard Halo Model (SHM), which models the DM of a galaxy as a self-gravitang gas of collision-less parcles in equilibrium, which form a spheri- −2 cal and isothermal halo [341]. As such, the density profile scales as ∼ r and the mass funcon as M(r) ∼ r, which explains the observed flatness of galac- c rotaon curves we discussed in secon 2.1.1. The velocies follow an isotropic Maxwell-Boltzmann distribuon, v2 f (v) ∼ exp − . halo 2 (3.2.2) 2σv

Even though this distribuon does not fit results from N-body simulaons very well [342–344], and a newer analysis of the stellar distribuon using the SDSS-Gaia sample shows that the halo shows more substructure [345], the SHM is by far the convenonal choice in DM detecon, since it does not introduces a crical error and simplifies the comparison between different results and experiments, similarly to the choice of the DM density3.

3We also menon that an update to the SHM was presented recently [346], which has only a small impact on non-direconal direct detecon experiments.

33 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

3.5 directional distribution 3.0

] 2.5 km / s

3 2.0 - winter summer 10 1.5 )[ v ( χ

f 1.0

0.5

0.0 0 200 400 600 800 1000 speed v [km/s]

Figure 3.1: The local speed distribuon of DM and its annual modulaon. A Mollweide projecon of the direconal distribuon with the characterisc dipole of the DM wind is shown as well.

The halo parcles’ speed will not exceed the galaxy’s escape velocity vesc, faster parcles would have le the galaxy a long me ago. Consequently, it is reasonable to truncate the distribuon, 1 1 v2 fhalo(v) = exp − Θ(vesc − v) , N 2 3/2 2σ2 (3.2.3a) esc (2πσv ) v

where Θ(x) is the Heaviside step funcon and the normalizaon constant Nesc is given by

! r 2 vesc 2 vesc vesc Nesc = erf − exp − . p 2 2 (3.2.3b) 2σv π σv 2σv

The galacc escape velocity typically chosen in the DM detecon literature is given v ≈ 544 km s−1 +54 km s−1 by esc , as obtained by the RAVE survey, 544-41 (90% conﬁ- dence) [347]. This value is sll the convenonal choice, even though it was updated +64 km s−1 σ to 533-46 (90%√ conﬁdence) more recently [348]. The velocity dispersion v −1 is set to σv = v0/ 2, where v0 ≈ 220 km s is the IAU value for the Sun’s circular velocity [349]. Finally, a Galilean transformaon into the Earth’s rest frame is necessary to obtain the local DM phase space fχ(v),

fχ(v) ≡ fhalo(v + v⊕) (3.2.4a) 1 1 (v + v )2 = exp − ⊕ Θ(v − |v + v |) . 3/2 3 2 esc ⊕ (3.2.4b) Nesc π v0 v0

34 3.2. THE GALACTIC HALO

Even though the original velocity distribuon was isotropic, this transformaon into the Earth’s rest frame breaks this isotropy and is the cause for the ‘DM wind’. More and faster parcles are expected to hit the Earth from its direcon of travel, which can be seen in the inset direconal distribuon in ﬁgure 3.1. The Earth’s velocity v⊕ is given in app. B.4. As the Earth orbits the Sun during a year, the Earth’s speed relave to the DM halo varies, which in turn causes a modulaon of the DM phase space and an annual signal modulaon of a direct detecon experiment [211]4. In many contexts, the direconal informaon of eq. (3.2.4) is irrelevant, and the marginal speed distribuon suﬃces. We obtain it by integrang out the velocity angles, Z 2 fχ(v) ≡ dΩ v f⊕(v) (3.2.5a) 2 2 1 v v + v⊕ vv⊕ = √ × 2 exp − 2 sinh 2 2 Nesc πv0v⊕ v0 v0 2 2 (v + v⊕) vesc + exp − 2 − exp − 2 Θ(|v + v⊕| − vesc) v0 v0 2 2 (v − v⊕) vesc − exp − 2 − exp − 2 Θ(|v − v⊕| − vesc) . (3.2.5b) v0 v0

The speed distribuon and its annual modulaon are ploed in figure 3.1. Provided with a local DM density and phase space distribuon, it is possible to compute the parcle flux of DM through a detector, the first necessary ingredient to predict scaering rates in a detector.

4A similar, smaller diurnal signal modulaon can be included, if, instead of v⊕, we use the laboratory velocity vlab, which addionally varies daily as the Earth rotates.

35 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

1.0

DM 0.8 particle particle DM 0.6 factor

γ 0.4

recoiling 0.2 target target 0.0 0.001 0.010 0.100 1 10 100 1000

mass ratio mχ/mT (a) Sketch of the scaering. (b) The γ factor.

Figure 3.2: Elasc scaering kinemacs between a DM and a target parcle.

3.3 Scaering Kinemacs

3.3.1 Elasc DM-nucleus collisions

In most WIMP searches, the fundamental processes which experiments look for are elasc coherent collisions between DM parcles and target nuclei in a detector. In this secon, we summarize the kinemac relaons required to describe direct detecon experiments [350]. A DM parcle of mass mχ and inial velocity vχ scaers on a target nucleus of mass mT , as illustrated in ﬁgure 3.2a. Energy and momentum conservaon alone determine the ﬁnal velocies of both parcles aer the elasc scaering,

0 mT |vχ − vT | mχvχ + mT vT vχ = n + , (3.3.1a) mT + mχ mT + mχ | {z } velocity of the CMS 0 mχ |vχ − vT | mχvχ + mT vT vT = − n + , (3.3.1b) mT + mχ mT + mχ

For terrestrial nuclei, the relave velocity is completely dominated by the DM speed, −3 |vχ − vT | ≈ vχ ∼ 10 . For example, in the atmosphere the thermal velocies of oxygen/nitrogen molecules are well below 1 km/s. The two momenta in the second term’s numerator can become comparable for very light DM. However, then the second term is negligible altogether. This leaves us with the approximaon of

0 mT vχn + mχvχ vχ ≈ (vT ≈ 0) , (3.3.2a) mT + mχ 0 −mχvχn + mχvχ vT ≈ (vT ≈ 0) . (3.3.2b) mT + mχ

We connue under the assumpon of resng targets, keeping in mind that it will

36 3.3. SCATTERING KINEMATICS not hold for hot nuclei in the core of the Sun.

The only unknown part is the unit vector n, which points into the direcon of the DM parcle’s final velocity in the Center of Mass System (CMS). This vector is determined by the scaering angle θ, defined as the angle between the inial and final velocity of the DM parcle in the CMS,

θ = ^(vχ, n) ∈ [0, π] . (3.3.3)

The distribuon of the scaering angle crically depends on the speciﬁc interacon between the DM and the nuclei and will be discussed in chapter 4.1.4. Once a scaering angle is speciﬁed, the momentum transfer q to the nucleus is given by

0 0 q = mT (vT − vT ) ≈ mT vT (vT ≈ 0) , (3.3.4a) 2 2 2 ⇒ q = 2µχT vχ (1 − cos θ) , (3.3.4b)

mimj µij ≡ where mi+mj is the reduced mass of a two body system. For an incoming speed the momentum transfer is bounded by

2 2 2 qmax = 4µχT vχ . (3.3.5)

The kinec quanty closer to the measured observable is the recoil energy ER, i.e. the kinec energy transferred to the nucleus in a collision,

2 0 q 1 02 ER ≡ ET − ET = ≈ mT vT (vT ≈ 0) 2mT 2 2 2 2 mT m v 1 − cos θ 4µ = χ χ (1 − cos θ) = γE γ ≡ χT . 2 χ with (3.3.6) (mT + mχ) 2 mχmT

For an incoming speed the nuclear recoil energy is bounded by

2 2 max 2µχT vχ ER = γEχ = . (3.3.7) mT

This determines the minimum speed vmin(ER), for which a DM parcle is capable to cause a nuclear recoil of energy ER, s ERmT vmin(ER) = 2 . (3.3.8) 2µχT

Furthermore, eq. (3.3.7) illustrates the relevance of the γ factor. It is the maximum

37 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

fracon of energy, the DM parcle can possibly transfer to the target. As shown in ﬁgure 3.2b, the factor γ is close to one, if the two parcles’ masses are similar and exactly one if they are degenerate. In this case, the DM parcle may lose all its kinec energy in a single scaering. This can also be seen from the deceleraon of the DM parcle,

0 r vχ 1 − cos θ = 1 − γ < 1 . (3.3.9) vχ 2

In summary, the energy transfer is governed by the rao of masses and the interacon type, which determines the distribuon of the scaering angle.

3.3.2 Inelasc DM-electron scaerings

For direct searches of sub-GeV DM parcles, it is possible to look for inelasc collisions between DM and an electron of an atom. The kinemacs are non-trivial since the bound electron’s momentum is not uniquely deﬁned, and there is no longer a direct relaon between the ﬁnal electron’s recoil energy and the momentum transfer, as eq. (3.3.6) for elasc nuclear scaerings [316]. The total energy transferred to the electron can be expressed as the energy lost by the DM parcle,

0 ∆Ee = −∆Eχ = Eχ − Eχ (3.3.10a) 2 mχ 2 |mχvχ − q| = vχ − (3.3.10b) 2 2mχ q2 = v · q − . (3.3.10c) 2mχ

Here, we neglected the fact that the atom also recoils as a whole. The maximum transferred energy ∆Ee with respect to q is therefore 1 ∆Emax = m v2 . e 2 χ χ (3.3.11)

As opposed to elasc nuclear scaerings, where the γ factor of eq. (3.3.7) yields the maximum kinemacally allowed relave energy transfer, a sub-GeV DM parcle is kinemacally allowed to lose its enre energy in a DM-electron scaering, and the kinec energy transfer is much more eﬃcient for electron than for nuclear targets. In analogy to eq. (3.3.8), the minimum DM speed kinemacally required for a

38 3.4. DESCRIBING DM-MATTER INTERACTIONS collision with momentum transfer q and transferred energy ∆Ee is

∆Ee q vmin(∆Ee, q) = + . (3.3.12) q 2mχ

In sub-GeV DM-electron scaerings, the electron is not just the lighter parcle. Nong that ve ∼ α ∼ 10-2, the electrons are also faster by one order of magnitude and hence dominate the relave velocity. Therefore, typical momentum transfers are of order q ∼ µχe|vχ − ve| ' meα ≈ 3.7 keV. The typical transferred energy is of the order of eV, suﬃcient to ionize and excite atoms.

3.4 Describing DM-Maer Interacons

At this point, all evidence in favour of the existence of DM is rooted in its gravitaonal influence on visible maer on astronomical scales, as discussed in chapter 2.1. In contrast, we know next to nothing about possible non-gravitaonal interacons, which are probed by direct detecon experiments. In the face of this ignorance, the queson arises, how we should model and describe these interacons. One aempt is to look for UV complete extensions of theSM, which are somehow well movated and contain a DM candidate parcle. The most prominent examples have been discussed briefly in chapter 2.2.2. Due to the large number of proposed models, these model driven and dependent descripons of DM-maer interacons hamper comparisons of different experiments. To avoid this, it is necessary to model the probed interacons in a more general framework. The use of an Effecve Field Theory (EFT) is a more suitable approach, since it is agnosc to parcular proposals for extending theSM[351]. It can be used to universally describe the low energy behavior of many proposed extensions of theSM and gives a general framework to model e.g. DM-quark interacons, independent on the underlying high energy degrees of freedom. Alternavely, it can make sense to formulate a simple prototype model with addional degrees of freedom and symmetries, so called simplified models, which can also be connected to more complete parcle models.

39 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

3.4.1 EFTs and DM-nucleus scaerings

The Lagrangian of theSM can be extended by a new, stable, massive ﬁeld χ, e.g. a massive Dirac fermion, which is the DM parcle. Its interacon with theSM quarks to leading order is wrien as Lorentz-invariant eﬀecve four-fermion operators, X Lint ⊃ αq (χΓχχ)(qΓqq) , (3.4.1) q

I µ 5 5 µ µν µν 5 where αq are the eﬀecve DM-quark couplings and Γi ∈ { , γ , γ γ γ , σ , σ γ } are the possible operators [139, 338, 352, 353]. The mediator for this interacon is assumed to be heavier than the momentum transfers of the scaerings and was integrated out yielding a contact interacon. This operator needs to be mapped to a nucleon operator as described in [139] in order to compute the relavisc matrix element for DM-nucleus scaerings. Aer taking the non-relavisc limit5, the resulng matrix element M enters the diﬀerenal cross secon, dσ 1 N = |M|2 . 2 2 (3.4.2) dER 32πmN mχvχ

Only the matrix element M depends on the speciﬁcs of the parcle physics model. The total scaering cross secon is obtained by integrang over all recoil energies,

max Z ER 2 2 dσN max 2µχN vχ σN = dER , with ER = . (3.4.3) 0 dER mN

Typically, the leading order operators are divided into two categories, depending on the cross secon’s dependence on the nuclear spin.

Spin-Independent (SI) interacons Spin independent interacons are mediated I µ by scalar or vector ﬁelds, with Γχ/q = or Γχ/q = γ respecvely,   SI X S V µ Lint = αq (χχ)(qq) + αq (χγµχ)(qγ q) , (3.4.4) q | {z } | {z } scalar vector

5Alternavely, the scaering cross secons can be described directly using Non-Relavisc Ef- fecve Field Theory (NREFT)[354, 355].

40 3.4. DESCRIBING DM-MATTER INTERACTIONS

The matrix element for the scaering between a DM parcle and a nucleus N in the non-relavisc limit turns out as6

SI M = 4mχmN (fpZ + fn(A − Z)) FN (q) , (3.4.5) and the diﬀerenal cross secon can be evaluated to be

SI dσN mN 2 SI 2 = [fpZ + fn(A − Z)] F (q) √ . 2 N q= 2mN ER (3.4.6) dER 2πvχ

We can use the total DM-proton scaering cross secon as a reference,

f 2µ2 σSI = p χp . p π (3.4.7)

Results of direct detecon experiments are usually presented in terms of this reference cross secon such that results from experiments with diﬀerent target nuclei SI can be compared directly. The diﬀerenal cross secon in terms of σp is then

SI SI 2 dσ mN σ f N = p Z + n (A − Z) |F (q)|2 √ . 2 2 N q= 2mN ER (3.4.8) dER 2µχpvχ fp

For isospin conserving interacons, the coupling to protons and neutrons is idencal (fp = fn) such that the diﬀerenal cross secon simpliﬁes to

f Z + n (A − Z) → A, fp and the DM parcle essenally couples to the nuclear mass with a coherence en- 2 hancement ∼ A . Isospin invariance is a standard assumpon in the context of direct detecon. However, it is not a requirement, see e.g. [356]. In most parts of this thesis, we will indeed assume that fp = fn. The excepon is the dark photon model, a simplified model which we will consider in the context of DM-electron scaering experiments, where fn = 0 such that the cross secon 2 is proporonal to Z . SI The nuclear form factor FN (q) describes the finite size of the nucleus and is defined as the Fourier transformed charge density. For large nuclei and large momentum transfers the DM parcle does not scaer coherently on the nucleus and starts to resolve the nuclear structure. The loss of coherence forSI interacons can

6For a more detailed derivaon, we refer to [139].

41 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

momentum transfer q in fm-1 0. 0.5 1. 1.5 2. 1 2 )

q 0.100 ( SI N F | 0.010

0.001

10-4 16O -5 10 131Xe nuclear form factor 10-6 0 100 200 300 400 momentum transfer q in MeV

Figure 3.3: The nuclear Helm form factor (solid line) and its exponenal approximaon (dashed line) for oxygen and xenon.

be described universally for all nuclei using the Helm form factor [357], sin(qr ) cos(qr ) q2s2 F SI(q) = 3 n − n exp − , N 3 2 (3.4.9a) (qrn) (qrn) 2

with

r 7 r = c2 + π2a2 − 5s2 , n 3 (3.4.9b) 1/3 c = 1.23A − 0.6 fm , (3.4.9c) a = 0.52 fm , s = 0.9 fm . (3.4.9d)

While there are more involved and accurate ways to determine the nuclear form factor, the Helm form factor agrees with these newer results to high accuracy for the range of momentum transfers relevant for direct detecons [358]. In cases, where the momentum transfer is small, we can also use an exponenal approximaon of the Helm form factor, r2 s2 F SI(q) ≈ exp − n + q2 . N 10 2 (3.4.10)

The Helm form factor and its approximaon are shown for two examples in ﬁg- ure 3.3.

For light DM with mχ mN , the maximum momentum transfer in a collision mχ −1 on a nucleus is qmax ≈ 5 1GeV MeV ≈ 0.03 fm . It is therefore a good approxi-

42 3.4. DESCRIBING DM-MATTER INTERACTIONS maon, to neglect the nuclear form factor and set FN (q) ≈ 1. In this limit, we can evaluate eq. (3.4.3) and obtain the total scaering cross secon, 2 2 SI SI µχN fn σN = σp Z + (A − Z) . (3.4.11) µχp fp

We can use the total cross secon and rephrase the diﬀerenal cross secon for light DM as

SI SI dσN σN = max , (3.4.12) dER ER which shows that the recoil energies follow a uniform distribuon. Since ER is connected to the scaering angle via eq. (3.3.6), we can already deduct that light DM scaers on nuclei isotropically in the CMS. We will discuss this in more detail in chapter 4.1.4.

Spin-Dependent (SD) interacons Even thoughSD interacons play no major role in this thesis, we brieﬂy introduce them at this point for completeness. TheSD interacon arises from an axial-vector coupling between DM and quarks,

SD X A 5 µ 5 Lint = αq χγ γ χ qγ γµq , (3.4.13) q

A with eﬀecve axial-vector DM-quark couplings αq . The diﬀerenal cross secon forSD DM-nucleus scaerings derived from this Lagrangian is [359],

SD dσN 2mN J + 1 2 SD 2 = (fphSpi + fnhSN i) F (q) √ . 2 N q= 2mN ER (3.4.14) dER πvχ J

2 2 SD 3µχpfp The total DM-proton cross secon, σp = π can serve as a reference cross secon, similar to theSI case,

SD SD 2 dσN 2mN σp J + 1 fn SD 2 = hSpi + hSN i F (q) √ . 2 2 N q= 2mN ER (3.4.15) dER 3µχpvχ J fp

Here, J is the nuclear spin, fn, fp are again effecve couplings to the nuclear spins, and hSpi,(hSN i) is the isotope-specific average spin contribuons of protons (neu- SD trons). Just as in the case ofSI interacons, a nuclear form factor FN (q) describes the nuclear structure. In this case, the spin structure depends on the specific isotope, and the nuclear form factor cannot be expressed universally [360, 361].

43 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

Compared toSI case, theSD interacon plays a subdominant role for direct 2 detecon. They lack the enhancement factor ∼ A , and the signal rate in a detector is consequently suppressed. Furthermore, most target isotopes have spin zero, as they include no unpaired nucleon, and the factors J, hSpi, and hSN i oen vanish.

3.4.2 Simpliﬁed models and DM-electron scaerings

The idea that DM consists of a single parcle, and there exists nothing else than theSM and that parcle, is certainly the minimal assumpon. But it could be argued that the existence of a “dark sector” with more parcles and interacons mediators is more likely, as it would mirror the diversity of the SM sector. In that case, DM could simply consist of the lightest stable of these dark parcles or even consist of different dark parcle species. The EFT approach used in the previous chapter applies best to the case, where the DM field is lighter than all other dark sector fields, including mediators, which can be integrated out. Then, the heavy fields would be kinemacally inaccessible in experiments, and their indirect effect on the lighter degrees of freedom can be absorbed into effecve operators. However, if there are fields lighter than the DM parcle which mediate the interacons to SM parcles, simplified models are typically a more suitable descripon of the new interacons [205, 206, 362]. Simplified models are prototype models which explicitly contain a small number of new degrees of freedom, in parcular the mediators. As such, the simplified model approach lies conceptually between a UV-complete extensions of theSM and EFTs. A good simplified model captures all the physical phenomena at the relevant energies and can be matched to more UV-complete models. The Lagrangian of such a simplified model of DM contains one or more stable DM candidates, a mediator coupling the visible to the invisible sector, and all renormalizable interacons consistent with the symmetries of the SM and dark sector. In the context of DM-electron scaerings, we are parcularly interested in the case, where the interacons are mediated by an ultralight field. We set up a simplified model, where theSM is extended by a dark sector of a spin-1/2 DM parcle χ of mass mχ and a new gauge group U(1)D. The new gauge boson, the so-called dark 0 photon A , can mix kinecally with the U(1) gauge bosons of theSM, which would act as a portal between the two sectors [363, 364]. The dark sector’s Lagrangian can be wrien as

µ 1 0 0µν 2 0 0µ 0µν LD =χ ¯(iγ Dµ − mχ)χ + F F + m 0 A A + εFµνF , 4 µν A µ (3.4.16a) 44 3.4. DESCRIBING DM-MATTER INTERACTIONS with the covariant derivave involving the new gauge coupling gD,

0 Dµ = ∂µ − igDAµ . (3.4.16b)

The dark photon is assumed to have acquired a mass mA0 , the new symmetry might e.g. be spontaneously broken by a Higgs mechanism. Furthermore, its kinec mixing is parametrized by .

DM-nucleus scaerings Similarly to the previous case, the mediator’s interacon to protons and neutrons can be expressed in terms of eﬀecve couplings fi [365]

0 µ µ Lint = eAµ (fppγ¯ p + fnnγ¯ n) , (3.4.17) which in turn can be related to the mixing parameters to the photon or Z boson,

2 1 − 4 sin θW 1 fp = γ + Z , fn = − Z . (3.4.18) 4 cos θW sin θW 4 sin θW cos θW

This is also where the weak mixing angle θW enters. The diﬀerenal cross secon for elasc DM-nucleus scaerings in terms of the momentum transfer q reads dσ 4παα 1 N = D (f Z + f (A − Z))2 |F (q)|2 . 2 2 2 2 2 p n N (3.4.19) dq (q + mA0 ) vχ

2 2 e gD Here, α ≡ 4π and αD ≡ 4π are the two fine structure constants, and FN (q) is the same nuclear form factor as forSI interacons. Under the assumpon that the dark photon mixes exclusively with the photon, i.e. ≡ γ 6= 0 and Z = 0, we find that the mediator naturally couples to charged parcles only (fn = 0). In analogy to eq. (3.4.7), we define a reference cross secon,

2 2 16πααD µ σ ≡ χp . p 2 2 2 (3.4.20) (qref + mA0 )

This cross secon depends on a reference momentum transfer qref which can be 2 2 chosen arbitrarily. For contact interacons, this dependence vanishes if mA0 q . With this reference cross secon, the diﬀerenal cross secon of eq. (3.4.19) takes the form

dσ σ N = p F (q)2 F (q)2 Z2 , 2 2 2 DM N (3.4.21) dq 4µχpvχ

45 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

where the q dependence is absorbed into the DM form factor

2 2 q + m 0 F (q) ≡ ref A , DM 2 2 (3.4.22) q + mA0

We will use the dark photon model to study sub-GeV DM parcles, therefore the nuclear form factor can be neglected via FN (q) ≈ 1. In addion, we focus on the two limits of ultraheavy and ultralight mediators, which can now be conveniently expressed in terms of the DM form factor,  2 2 1 , for mA0 qmax , FDM(q) = 2 (3.4.23) qref 2 2  q , for mA0 qmax .

These are the two form factors of contact and long-range interacons respecvely. We will also consider electric dipole interacons, characterized by q F (q) = ref . DM q (3.4.24)

This type of interacon does not arise in the dark photon model, instead it orig- 5 µν i inates from the operator χσ¯ µνγ χ F with σµν = 2 [γµ, γν] [366]. We will also consider results for electric dipole interacons in chapter 4.3.4.

DM-electron scaerings The diﬀerenal cross secon for DM-electron collisions is given by

2 dσ 4πααD 1 e = γ . 2 2 2 2 2 (3.4.25) dq (q + mφ) vχ

We introduce the reference DM-electron scaering cross secon,

2 2 16πααD µ σ ≡ χe e 2 2 2 (3.4.26) (qref + mA0 ) dσ σ ⇒ e = e F (q)2 , 2 2 2 DM (3.4.27) dq 4µχev

The reference cross secon is convenonally set to the typical momentum transfer of DM-electron scaerings, hence qref = αme. Despite this arbitrariness, the

46 3.4. DESCRIBING DM-MATTER INTERACTIONS

momentum transfer q [αme] 0.5 1 2 4 8 16 100 2 ) q ( -1 A 10

10-2 14Si -3 10 18Ar

32Ge 10-4

atomic form factor F 54Xe 10-5 1 5 10 50 100 momentum transfer q [keV]

Figure 3.4: The atomic form factor, given in eq. (3.4.30), which describes the screening of the nuclear electric charge for silicon, argon, germanium, and xenon nuclei. reference momentum transfer cancels in the rao of the two cross secons, 2 σp µχp = . (3.4.28) σe µχe

This rao shows a hierarchy of cross secons for DM masses above a few MeV. In the dark photon model with kinec mixing, DM-electron scaerings go hand in hand with much stronger DM-nucleus interacons as σp σe. Terrestrial eﬀects due to elasc scaerings on underground nuclei can have strong implicaons for direct detecon experiments, even if the nuclear recoils are undetectable and the experiment is probing DM-electron interacons. This has been studied e.g. in [315], as well as in Paper I and Paper V.

Charge screening The cross secons in eq. (3.4.21) and (3.4.27) apply to free nuclei and electrons with their charges in isolaon. In solids however, the electric charges are screened by the surrounding on larger distances. In the end, a solid is electrically neutral. For DM-nucleus scaerings, we can re-scale the nuclear charge to the eﬀecve charge using an atomic form factor,

Z → Zeﬀ = FA(q) × Z, (3.4.29)

lim FA(q) = 1 FA(0) = 0 with q→∞ and . The atomic form factor decreases the eﬀecve nuclear charge on longer distances, hence for low momentum transfers. One par-

47 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

cularly compact and simple form factor, which approximates the more elaborate Thomas-Fermi elasc form factor, is given in [367, 368] and reads a2q2 F (q) = , A 1 + a2q2 (3.4.30)

where a is the Thomas-Fermi radius,

1 9π2 1/3 0.89 a = a ≈ a , 4 2Z 0 Z1/3 0 (3.4.31)

a ≡ 1 ≈ 5.29 × 10−11 m wrien in terms of the Bohr radius 0 meα . This corresponds Ze −r/a to a screened Coulomb potenal r e . In turn, the electron charge is screened by the nuclei. Following [368], the corresponding atomic form factor takes the same 0 5.28 form as eq. (3.4.30), but we have to use a ≈ Z2/3 a0 instead of the Thomas-Fermi radius a.

2 Total cross secon For ultralight mediators and a DM form factor FDM ∼ 1/q , the Rutherford type differenal cross secon diverges in the IR, and the total cross secon can not simply be obtained by integrang eq. (3.4.21). However, the charge screening removes these divergences, and the atomic form factor drives the differenal cross secon to zero for q → 0 as shown for some examples in figure 3.4. Indeed, the total cross secon

2 qmax Z dσ σ = dq2 N |F (q)|2 , N dq2 A (3.4.32) 0

is ﬁnite and can be evaluated analycally, 2 µχN 2 σN = σp Z µχp

 1 2 2 2  1 + 2 2 − 2 2 log(1 + a q ) , FDM(q) = 1 ,  1+a qmax a qmax max for    q2 2 2 × ref 2 2 a qmax 1 q2 log(1 + a qmax) − 1+a2q2 , for FDM(q) ∼ q , (3.4.33)  max max   4 4  a qref 1  2 2 , FDM(q) ∼ 2 . (1+a qmax) for q

The ﬁrst line is the usual result for contact interacons of low-mass DM without charge screening. The case-speciﬁc factors depend on the screening length a. For

48 3.5. RECOIL SPECTRA AND SIGNAL RATES contact interacon it is of order one for mχ > 100 MeV, in other words charge screening is an irrelevant eﬀect for heavier DM. Note that the arbitrary reference momentum transfer, appearing in the other two cases, cancels out due to the arbitrary deﬁnion of the reference cross secon in eq. (3.4.20).

3.5 Recoil Spectra and Signal Rates

3.5.1 Nuclear recoil experiments

Once the DM flux and DM-maer interacon is specified, we can make predicons for direct detecon experiments. The differenal scaering rate between halo DM parcles and target nuclei of mass mN per unit me and mass is given by [369]

1 dR = × σN × v dnχ , (3.5.1a) mN |{z} | {z } |{z} scaering cross secon diﬀerenal DM ﬂux number of targets per unit mass substung eq. (3.2.1) and (3.2.4) results in

1 ρχ 3 = σN vfχ(v) d v . (3.5.1b) mN mχ

The recoil spectrum, i.e. the distribuon of events over the recoil energies ER, requires the diﬀerenal cross secon and integraon over velocies for which a DM parcle is kinemacally able to cause that recoil, ZZZ dR 1 ρχ 3 dσN = d v vfχ(v) Θ(v − vmin(ER)) . (3.5.2) dER mN mχ dER

For a given recoil energy, the step funcon limits the integral to the kinemacally allowed velocies. The minimum speed vmin is given by the kinemac relaon of (3.3.8). If we are not interested in direcon detecon, we can integrate out the direconal informaon of the distribuon and compute the recoil spectrum with the marginal speed distribuon, Z dR 1 ρχ dσN = dv vfχ(v) . (3.5.3) dER mN mχ v>vmin(ER) dER

If the cross secon σN does not explicitly depend on the DM speed v, then the

49 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

0.004

0.003 ] km / s )) [ R 0.002 DM mass E ( 1 GeV min v

η ( 10 GeV 0.001 100 GeV 1000 GeV

0.000 10-4 10-3 10-2 10-1 100 101 102 103

recoil energy ER [keV]

Figure 3.5: Examples of η(vmin(ER)) with varying DM masses for a 131Xe target.

dσN 1 ∼ 2 diﬀerenal cross secon scales as dER v and ZZZ dR 3 fχ(v) ∼ η(vmin) ≡ d v Θ(v − vmin) (3.5.4a) dER v Z f (v) = dv χ Θ(v − v ) . v min (3.5.4b)

Taking the distribuon of the SHM from (3.2.4), the funcon η(vmin) can be integrated analycally [370],   1 z < y , x < |y − z| ,  v0y for   1 4 −z2  erf (x + y) − erf (x − y) − √ ye  2Nescv0y π    for z > y , x < |y − z| , η(vmin) = 2 (3.5.5a)  1 erf (z) − erf (x − y) − √2 (y + z − x) e−z  2Nescv0y π    for |y − z| < x < y + z ,   0 for x > y + z ,

with

vmin v⊕ vesc x ≡ , y ≡ , z ≡ . (3.5.5b) v0 v0 v0

Some examples for a xenon target and diﬀerent DM masses are shown in ﬁgure 3.5. The η funcon also indicates the fact that a direct detecon experiment cannot probe arbitrarily low DM masses. Every experiment comes with a recoil threshold thr ER , below which recoil energies are too low to be detected. If even the fastest

50 3.5. RECOIL SPECTRA AND SIGNAL RATES halo parcles are kinecally incapable to cause a nuclear recoil above threshold, the experiment loses sensivity to this mass. Using eq. (3.3.7), the minimal probed DM mass is given by

min mN mχ = , q 2mN (3.5.6) thr vmax − 1 ER where the maximal speed of the galacc halo is the sum of the galacc escape velocity and the observer’s relave speed, vmax = vesc + v⊕. To extend the experiments sensivity towards lighter DM, one needs to either lower the threshold or use lighter targets. In chapter5, we will come back to this equaon and present a third opon concerning vmax, which involves the Sun. For an experiment with mulple target species of mass fracons fi we can simply add the spectra. The total scaering event rate R for a given threshold is then

Emax ZR,i X dRi R = fi dER . (3.5.7) dER i E thr

Lastly, we have to mulply the total signal rate by the exposure E in order to obtain the expected total number of events N. The exposure can be understood as the product of the target mass and the run me of the experiment, since the signal rate is given in events per unit mass and me.

N = E × R. (3.5.8)

Experimental observaons can be interpreted stascally by comparison to the spectra and number of events for a given DM mass and cross secon, as we will see in chapter 3.6.

Detector effects The previous expressions are the theorecal spectra and rates. Real detectors however have a finite energy resoluon, which might depend on the recoil energy, finite detecon efficiencies, and do not directly measure recoil energies but e.g. the number of photons in PMTs. The next step is to relate a theorecal spectrum to an observable spectrum. Underlying eq. (3.5.7) is the assumpon that the deposited energy corresponds exactly to the nuclear recoil energy. Instead a fracon of the recoil energy will be 0 lost into phonons, and the deposited energy E is less than the true recoil energy

51 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

0 ER, which can be expressed by the quenching factor Q ≡ E /ER. In pracce, this factor can oen be taken as a constant over the relevant energy ranges. Further- more, obtain a more realisc spectrum in terms of the actually detected energy ED by including the detector’s energy resoluon σE(ED) through convoluon with a Gaussian and detecon eﬃciency (ED),

∞ Z dR 0 0 X dRi = dE K(ED,E ) fi . (3.5.9a) dED dER E =E0/Q 0 i R i

with the detector response funcon

0 0 K(ED,E ) ≡ (ED) Gauss(ED|E , σE(ED)) . (3.5.9b)

Alternavely, the detector might use PMTs, which measure small discrete photon counts n, which are Poisson distributed. The spectrum in this case can be related to the theorecal recoil spectrum via

E ZRmax dR X dRi = f dE Poiss (n|ν(E )) , dn i R R dE (3.5.10) i R Ethr

where ν(ER) is the expected number of signals for a given recoil energy. In the end, we can sum over all n to get the total signal rate,

∞ X dR R = . dn (3.5.11) n=1

3.5.2 Detecon of sub-GeV DM with electron scaerings

In chapter 3.1.2, we discussed DM-electron scaerings as a promising detecon channel for sub-GeV DM. An incoming DM parcle could ionize or excite an electron bound in an atom causing a detecon signal. In this secon, we will review how to compute spectra and signal rates for these events for liquid noble gas [311–313] and semiconductor targets [315, 316]. Compared to nuclear recoils, the computaon of the DM-electron scaering cross secon is complicated by the quantum nature of the electrons bound in atoms. The electron’s momentum is indeterminate, and there is no one-to-one relaon between the momentum transfer and the deposited energy. As we showed in chapter 3.3.2, the kinemacs of this scaering allows the DM parcle to deposit its

52 3.5. RECOIL SPECTRA AND SIGNAL RATES enre kinec energy in a single interacon. With ∆Ee ∼ O(10-100)eV for MeV scale DM, there is an overlap of energy scales with the atomic scales. The electronic structure details of the inial bound state and the ﬁnal outgoing states can be absorbed into an ionizaon/excitaon form factor, which then enters the signal rate. The computaon of these form factors is typically involved. Fortunately, they do not depend on the speciﬁc DM model and have been tabulated and published for xenon and semiconductor targets [371].

Liquid noble targets We discussed two-phase noble target detectors in chapter 3.1.1, where the two me separaon of the two scinllaon signals (‘S1’ and ‘S2’) can disnguish DM-nucleus collisions from background. However, it turns out that experiments such as XENON or DarkSide-50 can also probe DM-electron interacons. Aer being ionized by a sub-GeV DM parcle, the primary electron dris towards the gas-phase of the target. It may scaer on other atoms and cause the ionizaon of secondary electrons. Upon reaching the gas-phase, these electrons cause a scinllaon signal (S2). The signature of DM-electron scaerings in two- phase xenon or argon detectors are therefore ‘S2 only’ events, where no inial S1 scinllaon photons were detected. The diﬀerenal ionizaon rate for atoms of mass mN with an ionized electron 02 of ﬁnal kinec energy Ee = k /(2me) is given [308, 309, 311, 316] by

nl dRion 1 ρχ X dhσionvi = , (3.5.12a) dEe mN mχ dEe nl with the diﬀerenal thermally averaged ionizaon cross secon,

nl dhσ vi σ Z 2 ion = e dq q |F (q)|2 f nl (k0, q) η (v (∆E , q)) , 2 DM ion min e (3.5.12b) dEe 8µχeEe

nl where ∆Ee = Ee+|EB |. The minimum speed vmin(∆Ee, q) is given by eq. (3.3.12). The sum runs over the quantum numbers (n, l) idenfying the atomic shells with nl nl 0 binding energy EB . The ionizaon form factor fion(k , q) is essenally the overlap of the inial and ﬁnal electron wave funcons,

02 Z 2 nl 0 2 2k X X 3 ∗ iq·x |f (k , q)| = d x ψ˜ 0 0 0 (x)ψ (x)e . ion 3 k l m i (3.5.13) (2π) 0 0 occupied states l m

The inial state of the target electron is approximated as a single parcle state of an isolated atom, the wave funcon ψi(x) can be evaluated numerically via tabu-

53 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

lated Roothaan-Hartree-Fock (RHF) bound wave funcons [372]. The first sum runs ˜ over all degenerate occupied inial states. The final state wave funcon ψk0l0m0 (x) is obtained by solving the Schrodinger¨ equaon for the hydrogen-type potenal of the new ion. The second sum runs over the final state angular momentum quan- 0 0 tum numbers l and m . The details of the ionizaon form factors’ derivaon are beyond the scope of this thesis and can be found in [309, 311, 313].

The next step is to compute the number of electrons, which reach the gas phase and cause the S2 signal. The inial electron can ionize more atoms and cause the release of addional ‘primary quanta’, E n(1) = e , W (3.5.14)

where W = 13.8 eV is the average energy to produce a single quanta for xenon [373]. Furthermore, the ionizaon of an inner shell electron brings about the emission of a photon and even more ‘secondary quanta’. For an quantum leap between two atomic shells with binding energies Ei and Ej, the number of secondary quanta from de-excitaon is E − E n(2) = i j . W (3.5.15)

The number of secondary quanta depends on the atomic shell. In table C.2, we list the number of secondary quanta for the diﬀerent shells of xenon.

Assuming that the ionized xenon atom never recombines aer ionizaon, the electron fracon of the primary quanta is fe = 0.83 [311]. Under the same as- 0 00 0 sumpon, the number of electrons is ne = n + n , where n = 1 is the primary 00 (1) (2) electron, and n follows a binomial distribuon with n + n trials and success probability fe. Therefore the spectrum in terms of electrons is Z dR dR (1) (2) = dEe Binomial n + n , fe|ne − 1 . (3.5.16) dne dEe

The electrons are not detected directly, instead their scinllaon light is observed by PMTs, which count the Photoelectrons (PEs). We have to perform a last conversion into thePE spectrum of the actual S2 signal. One electron causes a normal distributed number of PEs. The Gaussian is determined by its mean neµPE and

54 3.5. RECOIL SPECTRA AND SIGNAL RATES

√ width neσPE. Hence,the ﬁnalPE spectrum reads

∞ dR X dR √ = Gauss(µPEne, neσPE|nPE) . (3.5.17) dnPE dne ne=1

The values for µPE and σPE for the diﬀerent experiments a listed in table C.4 of app. C.2.1. Next, the number of events for a given number of PEs is nothing but dN dR = (nPE) × E × , (3.5.18) dnPE dnPE where (nPE) is the overall detecon eﬃciency, and E is again the exposure. This expression can be used to make predicons at two-phase noble target experiments and derive constraints from experimental data. For the conversions from electron number ne to PEs, we followed [311, 312]. The signal rates derived in this chapter might be a slight underesmate, as we neglected the band structure of liquid noble targets, which could lower the ionizaon energy gap [374]. A very recent work pointed out the importance of atomic many-body physics, which are also neglected here [375].

Semiconductor targets Although semiconductor targets can not be scaled up as easily as liquid noble experiments, experiments with silicon or germanium crystals can probe even lower DM masses thanks to their small band gaps. While the binding energy of atoms is of order O(10)eV, the band gap to excite an electron to the conducng band of a semiconductor is as low as O(1)eV. This enables these experiments to search for DM parcles with masses / 1 MeV [314–316]. A semiconductor is a mul-body system, and the computaon of the ionizaon form factors of periodic crystal laces requires methods from condensed maer physics. The QEdark-module of the QuantumEspresso quantum simulaon code is a publicly available tool to compute these form factors [376], which have also been tabulated for silicon and germanium [316, 371]. The event rate was derived −1 −1 −1 in app. A of [316] and is given in kg s keV by

2 dRcrystal ρχ 1 σeαme = 2 dEe mχ Mcell µχe Z 1 × dq η(v (E , q)) |F (q)|2 |f (E , q)|2 . q2 min e DM crystal e (3.5.19a)

Here, Ee is the total deposited energy and |fcrystal(q, Ee)| is the crystal form factor encapsulang the electronic band structure of the target lace. The numbers of

55 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

unit cells per unit mass is 1/Mcell with  2 × mSi ≈ 52 GeV , for silicon, Mcell = (3.5.19b) 2 × mGe ≈ 135 GeV , for germanium.

Similarly to the previous case, we have to convert the spectrum to the observable quanty. Here, this signal is the ionizaon Q, the number of electron-hole pairs created per event. The conversion from deposited energy to Q can be approximated linearly via E − E Q(E ) = 1 + e gap , e (3.5.20)

where the average energy per electron-hole pair and energy gap Egap are

  3.6 eV , 1.11 eV , for silicon, = Egap = (3.5.21) 2.9 eV , 0.67 eV , for germanium.

Therefore, an energy deposion Ee ∈ [Egap + (Q − 1), Egap + Q) is assumed to generate Q electron-hole pairs. Furthermore, any semiconductor experiment has a ionizaon threshold Qthr, corresponding to the energy threshold

thr Ee = (Qthr − 1) + Egap . (3.5.22)

Obviously, the opmal threshold is Qthr = 1, such that the band gap energy is also the energy threshold. Assuming that DM can deposit all its kinec energy, 1 min the lowest theorecally observable DM mass is obtained by solving 2 mχ (vesc + 2 thr v⊕) = Ee . Hence,  2 ((Q − 1) + E )  , mmin = thr gap ≈ 0.3 (1.4) MeV for silicon, χ (v + v )2 esc ⊕ 0.2 (1.0) MeV , for germanium, (3.5.23)

where we substuted the ionizaon threshold Qthr = 1 (Qthr = 2). Remarkably, even sub-MeV DM masses are within reach of semiconductor experiments with single electron-hole pair thresholds.

56 3.6. STATISTICS AND EXCLUSION LIMITS 3.6 Stascs and Exclusion Limits

Apart from the controversial claims by the DAMA collaboraon, no direct detecon experiments succeeded to observe a clear signal of halo DM. In this situaon, the queson is how to interpret the null results and how to derive exclusion limits on physical parameters. One of the largest experimental challenges is the under- standing of the background. There are a number of known processes which can trigger the detector, and a great deal of eﬀort goes into the disncon between background and a potenal signal. Given a reliable background model, one can aribute a number of observed signals to the background, a process called background subtracon. Let us assume some experimental run of a generic direct detecon experiment, where either the background subtracon has already been performed or was con- servavely omied. Furthermore, we assume that this experiment observed N signals with energies {E1,...,EN }. Then the null result can be translated into exclusion limits on the physical parameters, here the DM mass mχ and the interacon cross secon σ. Roughly speaking, if a point in parameter space predicts more events than observed, the point is excluded. We will discuss this in greater detail, parcularly the most straight forward approach using Poisson stascs [126] and Yellin’s maximum gap method for cases with unknown backgrounds [377].

Poisson stascs The detector essenally counts events and measures their deposited energies. The probability to observe n events for an expected value of µ is given by the Probability Mass Funcon (PMF) of the Poisson distribuon, µn P (n|µ) = e−µ . n! (3.6.1)

If a set of parameters predicts that an experiments should have observed more events than it actually did with a certain Conﬁdence Level (CL), then this point is excluded by that CL.

∞ X µn = P (n > N|µ) = e−µ = 1 − (N|µ) CL n! CDF (3.6.2) n=N+1

For a given DM mass, we ﬁnd the cross secon σ corresponding to the value of µCL determined by eq. (3.6.2). The easiest way is to choose an arbitrary reference cross secon σref , compute the number of events Nref and ﬁnd the upper bound at con-

57 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

N µ90% µ95% µ99% 0 2.30 2.99 4.61 1 3.89 4.74 6.64 2 5.32 6.30 8.41 3 6.68 7.75 10.05 10 15.41 16.96 20.14 100 114.08 118.08 125.84

Table 3.1: Some soluons of eq. (3.6.2) for diﬀerent conﬁdence levels and event numbers.

ﬁdence level CL via

µCL σ < σref . (3.6.3) Nref

This means that we need to solve CDF(N|µCL) = (1 − CL) for µCL at a givenCL and number of observed events N. With the excepon of N = 0, the Poisson Cumulave Distribuon Funcon (CDF) can not simply be inverted. Instead we can exploit its close connecon to incomplete gamma funcons,

N X µn Γ(N + 1, µ) (N|µ) = e−µ = ≡ Q(N + 1, µ) , CDF n! N! (3.6.4) n=0

where we introduced the upper incomplete gamma funcon,

Z ∞ s−1 −t Γ(s, x) ≡ dt t e , (3.6.5) x

such that Γ(s, 0) = Γ(s). The corresponding regularized gamma funcon is deﬁned as

Γ(s, x) Q(s, x) ≡ . Γ(s) (3.6.6)

The regularized upper incomplete gamma funcon can be numerically inverted with methods described e.g. in [378]. Some relevant soluons of eq. 3.6.2 are listed in table 3.1. A set of generic direct detecon constraints is depicted in ﬁgure 3.6. The limit curve is characterized by a sharp loss of sensivity for masses below the minimum value, see eq. (3.5.6), caused by the truncated tail of the DM speed distribuon. For larger masses the upper limit increases, because the DM energy density is ﬁxed

58 3.6. STATISTICS AND EXCLUSION LIMITS

target lower

lighter

threshold p

σ background

more scattering cross section exposure

higher

90% CL 95% CL 99% CL

DM mass mχ

Figure 3.6: Generic exclusion limits from direct detecon and their scaling. and heavier DM parcles are more diluted. The figure also illustrates the effect of adjusng experimental parameters and how the constraints scale under change of threshold, target, background, and exposure. These steps to find an upper confidence limit on σ can be applied to either the total number of observed events or to a number of energy bins, independently for each bin. In the laer case, the bin with the lowest value for the upper cross secon bound sets the overall limit. It should be noted at this point that, strictly speaking, this procedure actually yields an exclusion at a lowerCL[379]. This method assumes no knowledge about the expected spectrum or the background and interprets all observed events as DM signal of equal significance. As such, it is the most conservave method to obtain limits. It assigns an observed energy of low energy the same importance as a high energy signal, even though the expected spectrum is exponenally decreasing. Yellin proposed several alternave criterions for deciding if the expected signal is too high compared to observaons, which take the expected energy spectrum into account.

Maximum Gap Method The Maximum Gap method is Yellin’s simplest method to obtain exclusion limits, if the data is contaminated with background events of an unknown source, which cannot be subtracted [377]. It avoids event binning and takes the funconal shape of the spectrum into account.

59 CHAPTER 3. DIRECT DETECTION OF DARK MATTER dE dN

expected event spectrum xi

Ei Ei+1 energy E

Figure 3.7: Yellin’s Maximum Gap method for ﬁnding upper limits.

For an observaon of N events of energies {E1,...,EN } and including the threshold Ethr and the maximum energy Emax, there are N + 1 gaps without ob- dN served events. Using the expected event spectrum dE , we can compute the expected number of events in each gap. This number is called the gap size,

Ei+1 Z dN x = dE . i dE (3.6.7) Ei

The size of the largest gap is denoted with xmax, which can be regarded as a random variable on its own. For a given expectaon value of the total number of events µ, the probability that the maximum gap is smaller than x is

µ b x c X (nx − µ)n n C (x, µ) ≡ P (x < x|µ) = e−nx 1 + . 0 max n! µ − kx (3.6.8) n=0

This CDF remarkably depends only on µ and not on the specific shape of the spectrum. The more signals are expected, the smaller is the expected size of the maximum gap, as we expect more and denser signals with smaller gaps. If a point in parameter space predicts a value for µ such that the corresponding maximum gap should most likely be smaller than the observed xobs, the point can be excluded with confidence level of that probability. Hence, we need to solve C0(xobs, µ) = CL for µ and find the bound on the cross secon σ on this way. The Maximum Gap Method can be generalized in several ways. For one, it can be extended to the Maximum Patch Method for direconal detecon experiments [380]. Furthermore, it can be modified to not only consider gaps of zero

60 3.6. STATISTICS AND EXCLUSION LIMITS events, but also intervals containing 1,2,... events. This is called the Opmum In- terval Method [377, 381]. A drawback of this method is the addional complicaon that the interval’s CDFs no longer have a closed analyc form and require to be tabulated usingMC simulaons.

All these constraints rely on a set of standard assumpons, especially concerning the DM halo model. Despite its shortcomings, the use of the SHM simpliﬁes the comparison of results. However, it is also possible to set halo-independent exclusion limits, see e.g. [382–385].

61 CHAPTER 3. DIRECT DETECTION OF DARK MATTER

62 Chapter 4

Terrestrial Eﬀects on Dark Maer Detecon

The convenonal detecon strategy for direct DM searches is to look for recoils of nuclei which collided with an incoming DM parcle of the halo. Given the basic assumpon that this fundamental process can occur in a detector, nothing prevents the DM parcle to also scaer undetected on other terrestrial nuclei. For a moderately high cross secon, it is then not unlikely that the parcle could scaer twice, once underground and subsequently inside a detector. If the underground scaering rate is significant, pre-detecon scaerings will alter the halo parcles’ density and distribuon locally at the experiment’s site and therefore also the expected DM signal. This is parcularly relevant for searches for low-mass DM, where new detecon channels aside from nuclear recoils have been proposed as discussed in secon 3.1.2. Elasc DM-nucleus collisions of light DM might not be observable, but they sll happen and affect experiments indirectly by deforming the DM phase space. They can amplify or reduce the local DM parcle flux through the detector. Over the course of this chapter, we will invesgate two phenomenological consequences of these scaerings usingMC simulaons, namely diurnal modulaons of the detecon signal rate and experiments becoming insensive to DM itself due to a flux aenuaon by the overburden.

1. Diurnal Modulaons: Assuming a significant probability for a DM parcle to scaer on a nucleus of the Earth’s core or mantle, the underground distribuon of DM parcles, both spaal and energec, will be distorted by the decelerang and deflecng collisions. Since the incoming parcles arrive pre- dominantly from a specific direcon due to the Earth’s velocity in the galacc frame, the underground distance a halo parcle has to travel to reach the de-

63 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

tector changes throughout a sidereal day due to the Earth’s rotaon. There- fore, the probability to scaer before being detected varies with the same frequency, and the signal rate in the detector should show a diurnal modulaon [386, 387].

2. Loss of sensivity to strongly interacng DM: With some excepons from the more recent past, most DM detectors are placed deep underground for the purpose of background reducon. But above a certain crical cross secon, the typical ∼ 1 km of rock overburden would start to shield oﬀ even DM parcles. Therefore, underground detectors would be severely limited in their ability to probe strongly interacng DM [209]. The constraints on the DM- proton cross secon extend up to this crical value only, and the parameter space above opens up and might be viable [388].

Each individual scaering decelerates and deflects the DM parcle on its path through the planet’s interior. While the effect of a single scaering can be described analycally [389], for mulple scaerings the effect of a series of deflecons is best treated with numerical simulaons of parcle trajectories. We developed and applied two scienficMC simulaon codes, the Dark Maer Simulaon Code for Underground Scaerings (DaMaSCUS)[7] and DaMaSCUS-CRUST[6], to quanfy diurnal modulaons and constraints on strongly interacng DM respecvely. Both codes are publicly available.

This chapter is structured as follows. In the ﬁrst part, we introduce the fundamentals of theMC simulaons of underground trajectories of DM parcles. The central outcome is the general formulaon of the simulaon algorithm. In chapter 4.2, we apply this algorithm to the whole Earth to perform aMC study of diurnal signal modulaons. The results have been published in Paper II. In the third secon, we determine the constraints on strongly interacng DM by simulang parcle trajectories inside the experiments’ overburden. The results are divided into two parts, where chapter 4.3.3 contains the constraints on low-mass DM based on nuclear recoil experiments originally published in Paper IV, and chapter 4.3.4 extends these results to light mediators and DM searches based on inelasc electron interacons. The treatment of DM-electron scaering experiments was published in Paper I and Paper V.

64 4.1. MONTE CARLO SIMULATIONS OF UNDERGROUND DM TRAJECTORIES 4.1 Monte Carlo Simulaons of Underground DM Tra- jectories

There are three fundamental random variables at the core of the DM parcle transport simulaon, which are required to get sampled repeatedly for every trajectory [390, 391].

1. The free path length: How far does a DM parcle propagate freely through the medium before it interacts with one of the constuent nuclei?

2. The target nucleus: Once a scaering takes place, of all the diﬀerent nuclei present, what isotope is involved in the collision?

3. The scaering angle: What is the scaering angle in the CMS of the DM-target system?

Aer a brief review of diﬀerentMC sampling methods, we will describe the distribuon and sampling of each of these quanes. In the last part of this chapter, we combine them into a generalMC simulaon algorithm for underground trajectories of DM parcles.

4.1.1 Monte Carlo sampling methods

It is usually not a problem to generate uniformly distributed (pseudo) random numbers. However,MC simulaons always include the generaon of some non-uniform random numbers, where the underlying distribuon is known and determined by the simulated physical process. This generaon is also called sampling. In this secon, we will describe how to sample random numbers for any underlying distribuon [126]. Assuming a connuous random variable X deﬁned on the domain [a, b], the Probability Density Funcon (PDF) fX (x) dx is deﬁned as the probability of X to take a value between x and x + dx,

fX (x) dx = P (x < X < x + dx) . (4.1.1)

As such it is a posive funcon, normalized on its domain,

b Z dx fX (x) = 1 . (4.1.2) a

65 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

Next, the Cumulave Distribuon Funcon (CDF) FX (x) is deﬁned as the probability that X assumes a value below x,

x Z 0 0 FX (x) = P (X < x) = dx fX (x ) . (4.1.3) a

The CDF is a non-decreasing posive funcon [a, b] → [0, 1] sasfying F (a) = 0 and F (b) = 1.

Inverse transform sampling The CDF is the central funcon for transforming the sample of a uniform random number into a sample of another, non-trivial but known PDF fX (x). Interpreng the CDF as a random variable on its own, Y = FX (X) ∈ [0, 1], we can compute its CDF,

FY (y) = P (Y < y)

= P (FX (X) < y) = P (X < F −1(y)) −1 = FX (FX (y)) = y . (4.1.4)

This is nothing but the CDF of a uniform random variable of domain [0, 1], and the CDF of any random variable is itself uniform, Y = U[0,1]. We can show that −1 the reverse is true as well. If Y = U[0,1], then the random variable Z ≡ FX (Y ) is idencal to X itself, since their CDFs are idencal,

FZ (z) = P (Z < z) −1 = P (FX (Y ) < z)

= P (Y < FX (z))

= FY (FX (z))

= FX (z) . (4.1.5)

In conclusion, we can use this fact to transform a sample ξ of U[0,1] into a sample x of any random variable X by solving

FX (x) = ξ . (4.1.6)

This is called inverse transform sampling and is a very eﬃcient way of sampling if −1 the CDF can be inverted explicitly, such that x = FX (ξ) can be computed directly.

66 4.1. MONTE CARLO SIMULATIONS OF UNDERGROUND DM TRAJECTORIES

1.0 fmax 0.4 X 0.8 ξ=0.67 0.3 rejected ) ) x 0.6 x ( ( X X 0.2

0.4 PDF f CDF F

0.1 0.2 accepted x=5.44 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 x x

Figure 4.1: Examples of inverse transform sampling (le) and rejecon sampling (right) for a normal distribuon with mean µ = 5 and standard deviaon σ = 1.

It is illustrated in the le panel of ﬁgure 4.1 for the example of a normal distribuon. If such an inversion is not possible, we can also solve eq. (D.1.1) numerically with a root ﬁnding algorithm. However, in most cases there is a beer alternave.

Rejecon sampling In many cases, the PDF is a complicated expression, and the CDF cannot be inverted analycally. Then, the acceptance-rejecon method, or simply rejecon sampling, provides an alternave efficient procedure to transform uniformly distributed random numbers into random numbers of some more complicated distribuon [392]. The only condion for this method is that the PDF can be evaluated efficiently. We already established that the CDF or in other words, the area under the PDF can be used to generate a sample of any distribuon from uniform random numbers. Rejecon sampling uses this fact and is closely related toMC integraon. Again, we assume a random variable X with a given PDF fX (x), defined on its max support [a, b]. Furthermore, the distribuon is bounded, fX (x) ≤ fX for all x ∈ [a, b]. We sample two random numbers ξ1, ξ2 of U[0,1] and find a random posion in the domain, x = a + ξ1(b − a). We accept x as a value of X, if

max ξ2fX ≤ fX (x) (4.1.7) turns out to be true. Otherwise we start over with two new random numbers ξ1, ξ2. The procedure is visualized for the example of a normal distribuon in the right panel of ﬁgure 4.1. By sampling uniformly distributed points (x, y) in a plane and only accepng points with y < fX (x), we obtain random numbers distributed according to fX (x). The more values for x get rejected before ﬁnding an acceptable sample, the

67 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

less efficient the method is. The efficiency can be quanfied as the rao of the two areas,

b R dx fX (x) a 1 = max = max . (4.1.8) (b − a)fX (b − a)fX

Hence, if the probability is concentrated in a small region of the support, rejecon sampling will involve a large number of rejecons and is therefore ineﬃcient. Luckily, rejecon sampling, as described above, can be generalized.

In fact, rejecon sampling allows to generate samples of X by sampling any other random variable, not just U[a,b]. Instead, we can choose any random variable Y , whose domain contains the domain of X and which we know how to sample e.g. by inverse transform sampling. Next we have to ﬁnd a constant M large enough that

fX (x) ≤ MfY (x) for all x ∈ [a, b] , (4.1.9)

and MfY (x) envelopes fX (x). Given two generated random values y of Y and ξ of U[0,1], y is accepted as a sample of X, if

ξMfY (y) ≤ fX (y) . (4.1.10)

1 The eﬃciency in this case is simply = M , and it takes on average M trials before a value is accepted.

We can convince ourselves of the validity of this procedure by considering the probability to accept a value y below x, fX (y) P (y < x is accepted) = P y < x ξ ≤ MfY (y) P y < x, ξ ≤ fX (y) MfY (y) = P ξ ≤ fX (y) MfY (y)

R x R fX (y)/(MfY (y)) dy fY (y) dξ fU (ξ) = a 0 [0,1] , R b R fX (y)/(MfY (y)) a dy fY (y) 0 dξ fU[0,1] (ξ)

68 4.1. MONTE CARLO SIMULATIONS OF UNDERGROUND DM TRAJECTORIES

f (ξ) = 1 and using U[0,1] , we obtain

1 R x dy f (y) = M a X 1 R b M a dy fX (y) = FX (x) . (4.1.11)

The accepted values of this method share the CDF with X and therefore provide an accurate sample of X. Aer this review of generalMC sampling methods, we come back to the concrete random variables for the DM trajectory simulaons.

4.1.2 Free path length

Distribuon of the free path length In order to determine the locaon of the next collision of a DM parcle on its path through a medium, it is necessary to know the stascal distribuon of the free path length. The probability to scaer depends on the type and strength of the interacon, as well as the properes of the medium, such as the density and composion. In general, the inﬁnitesimal probability dPscat to scaer within an inﬁnitesimal distance between x and x+ dx along the parcle’s path is simply proporonal to the distance and can be wrien in terms of the local interacon probability per unit length Σ(x, v),

dPscat(x) ≡ Σ(x, v) dx . (4.1.12)

Furthermore, the cumulave probability to scaer within a ﬁnite distance x is denoted as P (x). For two distances x1 < x2, we can relate the corresponding probabilies intuively via

P (x2) = P (x1) + (1 − P (x1)) Pscat(x1 < x ≤ x2) . (4.1.13) | {z } prob. to scaer between x1 and x2

For a small distance ∆x, this can be rewrien using eq. (4.1.12),

P (x + ∆x) − P (x) = (1 − P (x))Σ(x, v) + O(∆x)2 , ∆x (4.1.14) dP (x) ⇒ = (1 − P (x))Σ(x, v) , dx (4.1.15)

69 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

and ﬁnally integrated to the resulng probability to scaer within the following distance L (using P (0) = 0),

Z L P (L) = 1 − exp − dx Σ(x, v) . (4.1.16) 0

It should be noted at this point that eq. (4.1.16) is the CDF of the random variable L. Hence, the derivave yields the corresponding PDF, dP (x) p(x) dx = dx dx (4.1.17a) Z x 0 0 = exp − dx Σ(x , v) × Σ(x, v) dx . (4.1.17b) 0 | {z } | {z } = dPscat(x) =1−P (x)

The ﬁrst factor is the probability to reach posion x freely without scaering, whereas the second part is nothing but the probability to scaer between x and x + dx.

Mean free path With the PDF for the free path at hand, it is straight forward to compute the mean free path, Z ∞ λ ≡ hxi = dx x p(x) . (4.1.18) 0

However, to evaluate the integral, we would need to know the evoluon of Σ(x, v) along the enre path. For the case of an inﬁnite and homogenous medium, the expression simpliﬁes thanks to Σ(x, v) = Σ(v),

Z ∞ −1 λ = Σ(v) dx x exp (−Σ(v)x) = Σ(v) . (4.1.19) 0

This movates to deﬁne the local mean free path, also for inhomogeneous media, as

−1 λ(x, v) = Σ(x, v) , (4.1.20)

such that

1 Z x dx0 p(x) = exp − , 0 (4.1.21) λ(x, v) 0 λ(x , v) Z L dx P (L) = 1 − exp − . (4.1.22) 0 λ(x, v)

70 4.1. MONTE CARLO SIMULATIONS OF UNDERGROUND DM TRAJECTORIES

The local mean free path at a given locaon depends on the density and composion of the medium at that locaon, as well as the interacon strength as quanﬁed by the total scaering cross secon,

−1 X −1 X λ(x, v) = λi(x, v) ≡ ni(x)σχi . (4.1.23) i i

The index i runs over all targets present at x with number density ni. The total cross secon was introduced in chapter 3.4. Furthermore, we treated the targets as essenally resng relave to the DM parcle. The medium is usually characterized by the mass density ρ(x) and the mass fracons fi of the diﬀerent target species of mass mi such that the number densies are simply

fiρ(x) ni(x) = . (4.1.24) mi

Sampling of free path lengths As described in secon 4.1.1, we can determine the free path length between two scaerings for a parcular trajectory via inverse transform sampling and solve

0 P (L) = ξ , (4.1.25)

0 for L, where ξ is a sample of U[0,1]. This is equivalent to ﬁnding the soluon of

Z L dx 0 Λ(L) ≡ = − log ξ , with ξ = 1 − ξ . (4.1.26) 0 λ(x, v)

This is especially easy for an inﬁnite, homogenous medium, for which the mean free path does not depend on the posion,

L = − log ξ λ(v) . (4.1.27)

Passing sharp region boundaries Unfortunately, the situaon is usually more complicated. The medium might change either gradually, as for example in the Earth’s outer core, where the density increases connuously towards the center or have sharp boundaries, where the medium properes change abruptly. The laer occurs e.g. in transions between the Earth core and the mantle or between the Earth crust and a lead shielding layer, where both composion and density, and therefore also the mean free path, change disconnuously along the trajectory. If a DM parcle passes one or several of such region boundaries, the le hand side

71 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

of eq. (4.1.26) takes the form

l−1 l−1 X X Λ(L) = Λi(Li) + Λs(Ls) , and L = Li + Ls . (4.1.28) i=1 | {z } i=1 | {z } layer of scaering layers passed freely

Here, a parcle passes (l − 1) regions without interacons, where Li is the distance to the next boundary inside region i. In the last layer l, the parcle freely propagates a distance Ls before finally scaering. Hence, to find the soluon L of Λ(L) = − log ξ, we have to determine the number of freely passed layers (l−1), as well as the distance Ls inside the final layer l.

We start by sampling a uniformly distributed random number ξ ∈ (0, 1) and determine the distance L1 to the next region boundary along the parcle’s path. The next step is to compare − log ξ with Λ1(L1). If

− log ξ < Λ1(L1) , (4.1.29)

the parcle scaers within this layer. The exact locaon of this scaering is found by solving Λ1(L) = − log ξ for L. On the other hand, if

− log ξ > Λ1(L1) , (4.1.30)

the parcle passes through this region without interacng at all. We compute the distance L2 between the next two region boundaries along the trajectory and compare − log ξ − Λ1(L1) with Λ2(L2) in the same way. If, at this point,

− log ξ − Λ1(L1) < Λ2(L2) (4.1.31)

holds, then the parcle scaers in region 2 aer having travelled freely for L1 +Ls, where Ls is the soluon of − log ξ − Λ1(L1) = Λ2(Ls). Otherwise, we have to repeat these steps again for the next layer and connue these comparisons unl we reach the layer l of scaering, such that

l−1 X − log ξ − Λi(Li) < Λl(Ll) . (4.1.32) i=1

72 4.1. MONTE CARLO SIMULATIONS OF UNDERGROUND DM TRAJECTORIES

START

Sample ξ ∈ (0, 1). Find the distance Ll to No Does the parcle escape the Yes Start region: l = 1. the next region boundary. simulaon volume?

No No interacon in layer l. Check if eq. (4.1.32) holds. Move to next layer l → l + 1. Yes

Scaering in layer l. Final free path length: Pl−1 Solve eq. (4.1.34) for Ls. L = i Li + Ls.

STOP

Figure 4.2: Recursive algorithm to sample the free path length and therefore the locaon of the next scaering with diﬀerent regions and layers of sharp boundaries.

At last, the overall free path length is given by

l−1 X L = Li + Ls , (4.1.33) i=1 with Ls being the soluon of

l−1 X − log ξ − Λl(Ll) = Λl(Ls) . (4.1.34) i=1

If the DM parcle scaers before leaving the simulaon volume, this procedure will determine where exactly this collision occurs. However, if the parcle exits the simulaon volume, as the DM parcle passes e.g. the Earth’s surface towards space, then the trajectory’s simulaon is ﬁnished. These steps can be implemented as a recursive algorithm, depicted in ﬁgure 4.2.

4.1.3 Target parcle

Once the locaon x of the next collision is known, the identy of the target among the N diﬀerent parcle species present at x needs to be determined. The probability is given by its relave ‘target size’, which is proporonal to its number density

73 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

at x and the total scaering cross secon. n (x)σ (v ) P ( T ) = T χT χ . scaering on target species PN (4.1.35a) i=1 ni(x)σχi(vχ)

Using the deﬁnion of the mean free path in eq. (4.1.23), this simpliﬁes to

λ (x, v )−1 = T χ ≡ P (x, v ) . −1 j χ (4.1.35b) λ(x, vχ)

With these probabilies, sampling the target species is straight forward. The DM parcle collides with a nucleus isotope T found via

( n )

X T = min n ∈ {1,...,N} Pi(x, vχ) > ξ , (4.1.36) i=1

where a random number ξ of U[0,1] needs to be sampled.

4.1.4 Scaering angle

The distribuon of the scaering angle θ follows from the diﬀerenal cross sec- dσN on dER and is therefore a consequence of the chosen DM interacon model. The recoil energy is related to the scaering angle via eq. (3.3.6). Hence, the PDF of the random variable cos θ can in general be obtained as

max 1 dσN ER dσN fθ(cos θ) = = . (4.1.37) σN d cos θ 2σN dER

Together with the mass rao of the DM parcle and the nuclei, this distribuon determines the kinemacs of an elasc scaering. As we will see, it will diﬀer dras- cally between contact and long range interacons.

SI interacons In the limit of light DM the PDF forSI contact interacons is very simple. With eq. (3.4.12), we ﬁnd 1 f (cos θ) = . θ 2 (4.1.38)

Hence, cos θ is a uniform random variable, U[−1,1], and the scaering is completely isotropic in the CMS. Inverse transform sampling for cos θ is trivial, we sample a random number ξ of U[0,1] as usual and set cos θ = 2ξ − 1. When simulang heavier DM parcles, we cannot neglect the loss of coherence

74 4.1. MONTE CARLO SIMULATIONS OF UNDERGROUND DM TRAJECTORIES

1.4

1.2 1 GeV 10 GeV 1.0 50 GeV

θ ) 0.8 100 GeV 1 TeV cos (

θ 0.6 f 0.4

0.2

0.0 -1.0 -0.5 0.0 0.5 1.0 cos θ

Figure 4.3: Distribuon of the scaering angle cos θ for SI interacons for diﬀerent DM masses. and have to include the nuclear form factor. As the loss of coherence suppresses large momentum transfers, this means that forward scaering will be favoured. The PDF of the scaering angle becomes

SI 2 FN (q(cos θ)) fθ(cos θ) = , R +1 SI 2 (4.1.39) −1 d cos θ FN (q(cos θ))

q (1−cos θ) with q(cos θ) = qmax 2 and qmax = 2µχN vχ. The distribuon is shown for different masses in figure 4.3. The CDF cannot be inverted directly, and inverse transform sampling of the scattering angle is not an opon whenever the full Helm form factor is taken into account. Instead, rejecon sampling is a suitable way to generate scaering angles. It should be noted however that the sampling efficiency behaves as ∼ 1/fθ(1). For very heavy DM and large nuclei, fθ(1) increases, and rejecon sampling could become inefficient. It could be a good idea to solve eq. (4.1.6) numerically using a root-finding algorithm in that case.

Dark Photon In the dark photon model, the DM parcle couples to electric charge. Both the screening of charge and the presence of an ultralight mediator alter the scaering kinemacs signiﬁcantly, as they introduce a dependence on the momentum transfer q into the diﬀerenal cross secon. Starng from eq. (3.4.21), the PDF for the scaering angle can be obtained via 1 dσ 1 q2 dσ f (cos α) = N |F (q)|2 = max N |F (q)|2 . θ A 2 A (4.1.40) σN d cos α 2 σN dq

75 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

1.4 FDM=1 1 1 FDM∼ FDM∼ 2 1.2 q q 1.0 θ ) 0.8 cos (

θ 0.6 f 0.4 0.2 0.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 cos θ cos θ cos θ

km km vχ=50 sec vχ=300 sec vχ=vesc+v⊕ mχ = 1 MeV mχ = 10 MeV mχ = 100 MeV mχ = 1000 MeV

Figure 4.4: Distribuons of the scaering angle for contact, electric dipole, and long range interacons.

Here, we again express the momentum transfer in terms of the scaering angle using eq. (3.3.4b). The q dependence, and therefore the PDF, is determined by the DM and the atomic form factors, FDM(q) and FA(q). The three PDFs for contact, electric dipole, and long range interacons are

fθ(cos θ) =

 x3 1+x (1−cos θ)2 x , F (q) = 1 ,  4 x(2+x)−2(1+x) log(1+x) (1+ (1−cos θ))2 for DM  2  1  2 × x2 1+x (1−cos θ) 1 2 (1+x) log(1+x)−x (1+ x (1−cos θ))2 , FDM(q) ∼ q , (4.1.41) 2 2 for    1+x 1  x 2 , FDM(q) ∼ 2 , (1+ 2 (1−cos θ)) for q

x ≡ a2q2 ≈ × Z−2/3 mχ 2 v 2 where max 2255 100 MeV 10-3 is an auxiliary parameter. The distribuons are depicted in figure 4.4 and demonstrate the versale kinemacs of this model. For heavier DM masses, the charge screening has lile effect on a DM-nucleus collision. For GeV scale masses and contact interacons, the scattering is virtually isotropic with fθ ≈ 1/2, where the other two interacon types heavily favour forward scaering due to the suppression of large momentum transfers from the DM form factor of the cross secon. For lower masses, the typical momentum transfer decreases, and the effect of the charge screening on larger scales suppresses small momentum transfers. Hence, screened contact interacon favour backward scaering more and more, whereas long range interacons occur more and more isotropically. Here, the two effects of charge screening and the DM form 2 factor FDM ∼ 1/q cancel. The electric dipole interacon behaves in an intermediate way, favouring forward or backward scaering depending on the DM mass and speed. We can see that for mχ =10 MeV, slow parcles tend to scaer backwards, whereas faster halo parcles scaer more into the forward direcon. The

76 4.1. MONTE CARLO SIMULATIONS OF UNDERGROUND DM TRAJECTORIES behaviours of contact and long range interacons is enrely opposite. DM parcles of lower (higher) mass and lower (higher) speed scaer more isotropically if the interacon is mediated by a ultralight (ultraheavy) mediator. Simulang trajectories with the dark photon model, we sample scaering angles cos θ using inverse transform sampling, i.e. solving eq. (4.1.6). While we can evaluate the CDFs directly, there are no closed form soluons to Fθ(cos θ) = ξ for a ξ of U[0,1], and we have to ﬁnd the soluon numerically. Rejecon sampling is extremely ineﬃcient for these PDFs.

4.1.5 Trajectory simulaon

In this secon, we will introduce the general algorithm to simulate the trajectory of a DM parcle, as it traverse through a medium. In the context of diurnal modulaons in chapter 4.2, this medium will be the whole Earth. For the purposes of secon 4.3, it will be the atmosphere, the Earth crust, and possibly some addional shielding layers made of e.g. copper or lead. Nonetheless, the fundamental algorithm will be the same, and the ground work is already done. The three central random variables introduced in the previous chapter will come together to form theMC simulaon algorithm. As the ﬁrst step, we have to generate a set of inial condions (t0, x0, v0) for a DM parcle from the galacc halo in space. The details of the inial condions are more speciﬁc and will be discussed in the respecve context later on. The inial condions should be chosen in a way that the simulated parcle actually enters the simulaon volume. It would be a waste of computaonal me, if e.g. half the parcles miss the Earth. The point of entry (t1, x1, v1), where the parcle enters the simulaon volume, is determined by the distance d to its boundary along the parcle’s path. As long as a parcle moves freely inside the Earth, we assume that it moves along a straight path, and we neglect the gravitaonal force of the Earth’s mass acng on the DM parcle, which is a good approximaon for this case. Hence, d t1 = t0 + , x1 = x0 + d · vˆ0 , v1 = v0 , (4.1.42) v0

v where we introduced the notaon for the unit vector vˆ ≡ v . Now that the parcle is inside the simulaon volume, the locaon of the ﬁrst nuclear collision can be obtained by sampling the free path length L as discussed in chapter 4.1.2. The

77 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

START

Generate inial condions: Will the parcle enter No (t0, x0, v0). the simulaon volume? Yes

Point of entry: (t1, x1, v1).

Sample the next free path length Li using eq. (4.1.33). No

Parcle propagates freely: Li Yes Does the new speed fall ti+1 = ti + vi , below the cutoﬀ vmin? xi+1 = xi + Li · vˆi.

Parcle scaers. Find the target Yes Is the parcle sll inside via eq. (4.1.36) and the new the simulaon volume? velocity vi+1 via eq. (4.1.47). No

STOP

Figure 4.5: Flow chart for the trajectory simulaon algorithm.

me and place of the scaerings are L t2 = t1 + , x2 = x1 + L1 · vˆ1 . (4.1.43) v1

Assuming that x2 is sll well inside the simulaon volume, the parcle will scaer on a nucleus of type T with mass mT , sampled via eq. (4.1.36). Since the two parcles are assumed to scaer elascally, we can use the same kinemac relaons as in the context of direct detecon via nuclear recoils, discussed in chapter 3.3.1. In parcular, we obtain the new velocity of the DM parcle using eq. (3.3.2a),

mT v1n + mχv1 v2 = . (4.1.44) mT + mχ

The unit vector n points towards the ﬁnal velocity of the DM parcle in the CMS. For its determinaon, we sample the scaering angle θ or rather its cosine as described in secon 4.1.4. The vector n is deﬁned only up to the azimuth angle φ, which is distributed uniformly, U[0,2π). Having sampled a value for both angles, the vector

78 4.1. MONTE CARLO SIMULATIONS OF UNDERGROUND DM TRAJECTORIES reads

 sin θ  √ (e2 sin φ − e3e1 cos φ) + e1 cos θ 1−e2  3  sin θ n = √ (−e1 sin φ − e3e2 cos φ) + e2 cos θ ,  1−e2  (4.1.45)  3  p 2 sin θ 1 − e3 cos φ + e3 cos θ

T with vˆ1 ≡ (e1, e2, e3) and a parcular but arbitrary origin of the azimuth angle. It can easily be verified that n × vˆ1 = cos α and |n| = 1. Aer having simulated the scaering, which deflects and decelerates the DM parcle, the procedure repeats: We sample the free path length Li, find the locaon of the next scaering,

Li ti+1 = ti + , xi+1 = xi + Li · vˆi , (4.1.46) vi idenfy the target T of the collision, sample the scaering angles and obtain the new velocity,

mT vin + mχvi vi+1 = , (4.1.47) mT + mχ and so on. The repeon connues unl either the parcle exits the simulaon volume or its speed falls below a minimal speed vmin, which we defined in order to avoid wasng me on parcles too slow to contribute to our desired data sample. The simulaon algorithm is illustrated as a flow chart in figure 4.5.

79 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION 4.2 Diurnal Modulaons

Due to the Earth’s velocity in the galacc rest frame, there is a preferred direcon from which more DM parcles hit the Earth with higher energies, for details see chapter 3.2. Consequenally, the underground distance travelled by an average DM parcle from the halo before reaching a detector varies as the Earth rotates. This variaon translates into a modulaon of the probability to scaer prior to reaching a detector over a sidereal day. Assuming a moderately strong interacon between DM and terrestrial nuclei, underground scaerings redistribute and decelerate the DM parcles inside the Earth, periodically modifying the local density and velocity distribuon at any laboratory’s locaon on Earth. If e.g. the average DM parcle has to travel through the bulk mass of the Earth crust and core before it can cause a detecon signal, the chance of being deflected or decelerated below the energy threshold is much higher compared to when the DM wind hits the laboratory from above. Especially experiments in the southern hemisphere are sensive to this ‘Earth shadowing’ effect. The phenomenological signature in a detector is a diurnal modulaon of the signal rate, for which the amplitude depends on the latude and the phase on the longitude of the laboratory. In this chapter, we will describe and study this modulaon for light DM. Already in the 90s, diurnal modulaons of direct detecon rates due to underground scaerings have been invesgated and quanfied with earlyMC simulaons in the context of the classic WIMP[386, 387, 393, 394]. Later on, similar modulaons have been studied in the context of hidden sector DM [395–398], strong DM- nucleus interacons [399], and DM-electron scaering experiments [315]. Mul- ple experiments have searched for diurnal modulaons, including the COSME-II experiment in the early 90s [393], the DAMA/LIBRA experiment [400, 401], and LUX [402]. So far, no evidence for daily signal modulaons has been reported. The future experiment SABRE, designed to test the discovery claim by the DAMA collaboraon [216], is potenally relevant for diurnal modulaons, since it is located at the Stawell Underground Physics Laboratory in Australia [403]. Being in the southern hemisphere increases the sensivity to diurnal modulaons. The phenomenological signature of underground scaerings was studied with analyc methods in [389]. The authors quanfied the lab-frame DM distribuon modificaons due to a single Earth scaering and determined the diurnal modulaons for different effecve operators of the NREFT framework. As such, the procedure does not apply beyond the single scaering regime. For higher cross secons, where mulple scaerings before reaching the detector are expected, numerical

80 4.2. DIURNAL MODULATIONS methods such asMC simulaons are necessary. As a tool to study the eﬀect of mulple underground scaerings, we developed and published the Dark Maer Sim- ulaon Code for Underground Scaerings (DaMaSCUS), which complements and generalizes the EarthShadow code by Kavanagh et al. [404]. The analyc method will serve as a crucial consistency check for the newMC results. The simulaon of billions of parcles provides the stascal properes of DM inside the Earth and allows to compute the expected distoron of the underground DM density and velocity distribuon taking into account the Earth’s orientaon in the galacc frame, its composion and internal structure, as well as its me dependent velocity. Based on the simulaons, we can predict the diurnal modulaon of the signal rate, both amplitude and phase, based on an experiment’s setup, locaon and underground depth. We already covered the basics forMC simulaons of parcle trajectory through a medium in the previous chapter. In this secon of the thesis, we will apply them to simulate DM parcle trajectory propagang through the whole Earth undergoing mulple scaerings. Aer a brief descripon of the inial condions, we present the procedure to sample the free path length inside the Earth’s mantle and core, where the density is changing gradually. The next step will be to connect the trajectory simulaons with predicons for signal rates in a DM detector. Finally, we present and discuss results for a CRESST-II type experiment and diﬀerent benchmark points.

4.2.1 Inial condions

The inial condions consists of a starng me, locaon outside the Earth, and velocity. Naturally, the correct choice of inial condions for the simulated DM parcles is crucial for the accuracy of the ﬁnal results.

Inial me The inial me can be chosen arbitrarily, we can set t0 to a random value or just start at zero.

Inial velocity The inial velocity consists of two components, an isotropically distributed velocity sampled from the halo distribuon in the galacc rest frame and the Galilean boost into the Earth’s rest frame,

v0 = vhalo − v⊕(t) . (4.2.1)

81 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

The halo component is sampled from the velocity distribuon in eq. (3.2.3). The Earth’s velocity is given in eq. (B.4.2) of app. B.4.

Inial posion To find the inial posion of a DM parcle is slightly more deli- cate. The parcles of the halo should effecvely be distributed uniformly in space, however we would like the parcle to be on a collision course with the Earth. In previous works, the inial posions were chosen on top of the Earth’s surface [387, 393, 394]. This choice is arbitrary and does not correspond to uniformly distributed inial posions outside the planet. Starng all trajectories from the surface creates a finite volume bias at shallow underground depth. Too many parcles are being sent into the Earth with narrow angle, creang an over-density of DM parcles close to the surface, i.e. exactly where direct detecon experiments are located. This can be confirmed both analycally and with simulaons for a transparent Earth.

Given an inial velocity v0, all inial posions, which result in a collision with Earth, are located inside a cylinder oriented parallel to v0. We can sample a random point within this cylinder, which is equivalent to picking a random point on a circular disk of radius R⊕ at distance R > R⊕ from the center of the Earth, as depicted in ﬁgure 4.6, p x0 = Rez + ξR⊕ (cos φ ex + sin φ ey) , (4.2.2)

where ξ and φ are sampled values of U[0,1] and U[0,2π] respecvely, and the unit vectors ex and ey span the disk. This choice results in an eﬀecvely uniform distribuon of parcles, which we conﬁrmed for the case of a transparent Earth using simulaons.

4.2.2 Free path length inside the Earth

In chapter 4.1.2, we discussed how to sample the free path length of a DM parcle inside maer by solving eq. (4.1.26). We also presented how to handle sharp region boundaries. For the Earth, there is another complicaon in this context, as the mass density increases smoothly towards the planetary core within each layer, see eq. (B.5.1). With this polynomial parametrizaon of the Earth’s density proﬁle, we

82 4.2. DIURNAL MODULATIONS

R⊕

Figure 4.6: Sketch of the inial posion.

can compute Λi(Li) of eq. (4.1.28) analycally,

L /v Zi −1 Λi(Li) = dt vλ (x(t), v) 0 Li/v " # Z |x(t)| |x(t)|2 |x(t)|3 = vgl dt al + bl + cl + dl , r⊕ r⊕ r⊕ 0 | {z } =ρ⊕(|x(t)|) gl = 2 (C1al + C2bl + C3cl + C4dl) , (4.2.3a) r⊕ where we parametrized the parcle’s path through layer i as x(t) = x0 + tv and λ−1(x,v) gl ≡ use ρ⊕(x) . These factors only depend on the layer’s composion through the mass fracons fi, not on the locaon within that layer. Furthermore, if the interacon cross secon does not depend explicitly on the DM speed, which is the case if we neglect the nuclear form factor, they only have to be computed once for each mechanical layer. Otherwise they depend on the DM speed and have to be re-computed aer each scaering. The coeﬃcients Ci are

2 C1 = Lir⊕ , (4.2.3b) " r C = ⊕ L˜(L + x cos α) − x2 cos α 2 2 i 0 0 ˜ !# 2 2 Li + L + x0 cos α + x0 sin α log , (4.2.3c) x0(1 + cos α)

83 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

Earth axis

ΔΘ

v⊕ Θ

R⊕-d

Figure 4.7: Illustraon of the isodetecon angle and rings and their projecon onto the Earth surface at 0:00 and 12:00 with ∆Θ = 5°.

1 C = L x2 + x L cos α + L2 , 3 i 0 0 i 3 i (4.2.3d) " 1 2 ˜ 3 C4 = (5 − 3 cos α)(L − x0)x0 cos α 8r⊕ 2 ˜ ˜ 2 2 + 2Li L(Li + 3x0 cos α) + LiLx0(5 + cos α) ˜ !# 4 4 Li + L + x0 cos α + 3x0 sin α log . (4.2.3e) x0(1 + cos α)

Here, we used x · v q 0 ˜ 2 2 cos α ≡ , L ≡ Li + x0 + 2Lix0 cos α (4.2.3f) x0v

With this closed form for Λi(Li) inside a layer, the free path length inside the Earth can be found using the algorithm in ﬁgure 4.2.

4.2.3 Collecng data

The main objecve of theMC simulaons is to obtain a good esmate of the DM distribuon inside the Earth at a given underground depth. Even though the boost from the galacc to the Earth’s rest frame breaks the isotropy of the halo distribuon, the system preserves a residual rotaonal symmetry around the axis parallel to the Earth’s velocity v⊕. The polar angle Θ to this axis is called the isodetecon angle1, as the DM distribuon and direct detecon rates will always be constant ◦ 1An equivalent angle γ is being used in [389] and the EarthShadow code, deﬁned as 180 − Θ.

84 4.2. DIURNAL MODULATIONS

180

150 Θ [°] 120 LNGS 90 SUPL

60 INO SURF isodetection angle 30

0 0 12 24 36 48 60 72 time after 15.02.2016, 00:00UT [h]

Figure 4.8: Time evoluon of the isodetecon angle for different laboratories around the globe over the duraon of three days. along constant Θ. This symmetry is used in the simulaon, to define small isodetecon rings of equal finite angular size ∆Θ, as proposed originally in [387, 393], placed at an underground depth d (e.g. 1400 m for the LNGS), which are shown in figure 4.7. In this figure, we set ∆Θ = 5° for illustrave purposes. In the actual ◦ simulaons, we aim to have good resoluon and choose ∆Θ = 1 . This should be sufficiently small that the local DM distribuons will not vary over a single ring. As the Earth rotates, a laboratory travels through the different isodetecon rings, and its posion in terms of the isodetecon ring is

" (gal) # v⊕(t) · x (t) Θ(t) = arccos lab , (4.2.4) v⊕(t)(r⊕ − d) where we have to substute eq. (B.4.2) for the velocity of the Earth and eq. (B.4.5) for the posion of the experiment in the galacc frame. A given experiment will cover a certain range of Θ during a sidereal day depending on its latude, as shown in ﬁgure 4.8 for the LNGS( 45.454° N, 13.576° E), SUPL( 37.07° S, 142.81° E), INO (9.967° N, 77.267° E) and SURF( 44.352° N, 103.751° W). The plot shows how deeply a given experiment penetrates the Earth’s ‘shadow’ during its revoluon around the Earth axis. Already at this point, we can conclude that experiments in the southern hemisphere are more sensive to diurnal modulaons due to underground scaerings.

The 180 rings are deﬁned by their isodetecon angle Θk = k∆Θ and labeled by k ∈ [0,179]. By choosing a constant ∆Θ, their surface areas will diﬀer, and the area of ring k is given by

2 Ak = 2π(r⊕ − d) [cos(Θk) − cos(Θk + ∆Θ)] . (4.2.5)

85 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

For each of these rings, we have to find aMC esmate of the local DM velocity distribuon and density based on individually simulated parcle trajectories. Whenever a simulated parcle crosses an isodetecon ring, we record its velocity vector. By simulang a large number of trajectories, we gather a stascal velocity sample for each ring. The corresponding speed histograms will serve as a esmate of the PDF and contain all the informaon about how nuclear scaerings decelerate the halo parcles depending on the distance of underground propagaon. Since the rings vary in surface area, we have to ensure that the velocity sample sizes are sizeable even for the smaller rings close to Θ = 0° and Θ = 180°. In order to extract the redistribuon of parcles due to deflecons, we simply count the number of parcles crossing a given isodetecon ring. We will discuss the details of the data analysis in the next chapter. We should point out a number of differences to previous simulaons of this kind [387, 393, 394]. Collar et al. focused on the classic WIMP scenario with masses above 50 GeV and their detecon via nuclear recoil experiments, whereas we study sub-GeV DM and have new detecon strategies such as DM-electron scaerings in mind. Furthermore, there are substanal differences in the implementaon of the simulaons in the DaMaSCUS code, which include the corrected generaon of inial condions, which avoids to over-esmate the DM density close to the surface, as well as the more refined Preliminary Reference Earth Model (PREM). By posi- oning the isodetecon rings underground, the code is also able to simulate the shielding effect of the overburden for strongly interacng DM. However, it turns out to be much more efficient to set up a dedicated simulaon code for this problem, as we will discuss in chapter 4.3.

4.2.4 From MC simulaons to local DM distribuons and detecon rates

The DM parcle transport simulaons in the Earth generate local velocity data for all isodetecon rings. For each of these rings, an independent data analysis is performed to esmate the local DM density, velocity distribuon, and direct detecon event rate. The technical details of this analysis are presented in this secon.

Local DM Speed Distribuon Since we do not study direconal detecon, it suf- k ﬁces to esmate the DM speed distribuon fχ (v) for a given isodetecon ring k. The simplest, non-parametric way to esmate the distribuon underlying a data

86 4.2. DIURNAL MODULATIONS set is the histogram. However, the data must be weighted. We menoned earlier, that velocity data is recorded, once a DM parcle crosses an isodetecon ring, i.e. a ficous two-dimensional surface. If enough parcles crossed a small surface, we can esmate the distribuon in close proximity to the surface, but it is important to keep in mind that parcles crossing a surface define a parcle flux Φχ, which is related to the PDF via

3 3 Φχ(v) d v = nχfχ(v)v cos γ d v , (4.2.6) where γ is the angle between the velocity and the normal vector of the surface at the point of crossing [405]. Instead of connuously sending in a stream of DM parcles and me-tag each surface crossing, we effecvely simulate a single burst of incoming parcles and wait unl even the slowest parcle has le the Earth again. Hence, we do not actually track the flux Φχ, but instead Φχ/v. Consequently, given a parcle crossing at x with velocity v, we have to weigh this data point by the reciprocal cosine of γ, 1 x · v w = , cos γ ≡ , | cos γ| where xv (4.2.7) in order to really obtain an esmate of the distribuon funcon in proximity of the isodetecon ring’s surface. An alternave approach to understand this weight factor is to assume a DM parcle passing through a small patch of area dA. The effecve area for the parcle to pass the patch is dA cos γ, and a parcle moving almost parallel to the surface is less likely to cross, but should contribute to the distribuon just the same as a parcle crossing the surface perpendicular to the surface. Suppose we collected a MC data sample of Nsample weighted data points (vi, wi). The histogram’s domain is (vmin, vesc + v⊕), and the bin width ∆v is set by Sco’s normal reference rule [406], 3.5σ v √0 ∆v = 1/3 , with σ = , (4.2.8) Nsample 2

vesc+v⊕−vmin such that the histogram consists of Nbins = d ∆v e bins, B1 = [vmin, vmin + ∆v), B2 = [vmin + ∆v, vmin + 2∆v),.... The bin height of bin i is then

Nsample X Wi = wj 1Bi (vj) , (4.2.9) j=1

87 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

where we introduced the indicator funcon deﬁned by  1 if x ∈ A, 1A(x) ≡ (4.2.10) 0 otherwise.

Finally, the weighted histogram esmaon of the speed distribuon fχ(v) is simply

N 1 Xbins fˆ (v) = W 1 (v) , χ N i Bi (4.2.11) i=1

N Pbins where the denominator N = ∆v Wi ensures that the histogram is normalized, i=1 ˆ and fχ(v) can be regarded as a PDF. The variance of the bin height, and therefore the error of the PDF esmate, can be inferred from Poisson stascs,

Nsample 1 X σ2 ' w2 1 (v ) . Wi N 2 j Bi j (4.2.12) j=1

The average speed for some isodetecon angle is just the weighted mean,

Nsample Nsample 1 X X hvi = w v , w ≡ w , w i i with tot i (4.2.13) tot i=1 i=1

and for the associated standard error we can use the approximaon by Cochran [407],

Nsample " Nsample X 2 ( )2 ' (w v − hwihvi) SE (N − 1)w2 i i sample tot i=1 # 2 2 − 2hvi(wi − hwi)(wivi − hwihvi) + hvi (wi − hwi) . (4.2.14)

With an esmate of the DM velocity distribuon at hand, the next queson is to determine the local density distorons.

Local DM Density In the absence of DM interacons, the Earth becomes transpar- (0) −3 ent, and the DM density is constant throughout space with ρχ = 0.3 GeV cm . This can be ulized to esmate the local DM density by comparing simulaons with and without underground scaerings. Furthermore, the local number density in some given isodetecon ring is proporonal to the (weighted) number of passing parcles. By inially running a number of free trajectories without nuclear colli-

88 4.2. DIURNAL MODULATIONS

(0) sions, we obtain a reference value wtot for parcles passing the isodetecon ring, which can be related to its local DM density ρχ including Earth scaerings,

ρˆχ wtot (0) ∼ (0) . (4.2.15) ρχ wtot

The number of simulated parcles in the inial scaer-less simulaon run and the (0) main simulaons, Ntot and Ntot, generally diﬀers, which can simply be taken into account resulng in the density esmate

N (0) w ρˆ = tot tot ρ(0) . χ N (0) χ (4.2.16) tot wtot

The standard deviaon of the density is obtained via error propagaon,

" 2 # N 2 σ (0) tot 2 σwtot wtot 2 2 X 2 σρ = + ρχ , σw = wj . χ w2 (0) 2 where tot (4.2.17) tot (wtot) j=1

All these steps are performed for each isodetecon ring independently.

Direct Detecon Rates The standard nuclear recoil spectrum for a direct detecon experiment is given in eq. (3.5.2). ForSI interacons, it can also be wrien as

SI s dR ρχ σN ERmN = 2 η(vmin) , with vmin = 2 . (4.2.18) dER mχ 2µχN 2µχN

The DM density is already well esmated by eq. (4.2.16). The η funcon has been deﬁned in eq. (3.5.4), the analyc expression for the Maxwell-Boltzmann distribuon is given by eq. (3.5.5). If we can esmate this funcon on the basis of the MC data, we have a fullMC esmate of recoil spectra, event rates, and signal numbers. The integral of the speed distribuon is parcularly simple in our case as we use a histogram esmator. Therefore, we have to simply sum up the bins’ areas in order to obtain a histogram esmator of the η funcon. The bin height of bin i is

Z fˆ (v) H = dv χ i v v>(i−1)∆v

89 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

Nbins ˆ Nbins X fχ ((j − 1/2)∆v) 1 X Wj = ∆v = . (j − 1/2)∆v N (j − 1/2) (4.2.19) j=i j=i

The bin height Wj was deﬁned in (4.2.9). Hence, the resulng histogram esmate is nothing but

N Xbins ηˆ(vmin) = Hi 1Bi (vmin) . (4.2.20) i=1

In the end, we can substute the local DM density (4.2.16) and the histogram esmate of η(vmin) into eq. (4.2.18) and obtain theMC esmate of the nuclear recoil spectrum for any given experiment. The residual steps to compute total signal numbers, likelihoods, to include detector resoluons, mulple targets, efficiencies, as described in chapter 3.5, do not differ from the usual analyc computaons. The distorons due to underground scaerings enter only into the density and velocity distribuon. It has been verified as a consistency check that simulaons of the transparent Earth, i.e. without underground scaerings, reproduce the correct Standard Halo DM distribuon, as well as a spaally constant DM density. The detecon events agree perfectly to the standard analyc computaon in this case.

4.2.5 Results

We invesgate the effect of underground DM-nucleus scaerings for light DM and large enough cross secons that DM parcles passing through the Earth will scaer on one or more terrestrial nuclei. The first part of our results focus on the single scaering regime, where the average scaering probability is around 10%. In this case, the effect of mulple scaerings is of the order of 1%, andMC simulaons would not really be necessary, since this regime can be described analycally with the EarthShadow code [389]. Nonetheless, the analyc results give us the great opportunity to compare and check some simulaon results directly. Once we demonstrated that ourMC results agree with the independent EarthShadow results, we can be confident in the accuracy of our simulaon code and concentrate on its actual purpose, to study the effect of mulple Earth scaerings on direct detecon experiments. For every parameter set of DM mass and DM-proton scaering cross secon new simulaons have to be performed, therefore it makes sense to present the characterisc results for a set of benchmark points. For all of these points, we

90 4.2. DIURNAL MODULATIONS

SI Simulaon ID mχ [MeV] σp [pb] hNsci NSample 7 ‘SS’ 500 0.5206 0.12 10 7 ‘MS1’ 500 4.255 1.0 10 6 ‘MS10’ 500 42.45 10.0 2 × 10 6 ‘MS50’ 500 297.5 &50.0 10

Table 4.1: The four benchmark points to study the impact of DM-nucleus scaerings and diurnal modulaons at direct detecon experiments.

3.5 1.10 ] Free

km 3. /

s DaMaSCUS(Θ=0°)

3 1.05 - 2.5 DaMaSCUS(Θ=180°) 10 ( 0 ) )[ EarthShadow(Θ=0°) R v 1.00 ( / χ 2. EarthShadow(Θ=180°) 0.95 1.5

0.90 Free

1. event rate R DaMaSCUS 0.5 0.85

speed distribution f EarthShadow 0. 0.80 0 100 200 300 400 500 600 700 800 0 60 120 180 speed v [km/s] Isodetection angle Θ [°]

Figure 4.9: Comparison of the analyc EarthShadow and the MC results of DaMaSCUS. choose a DM mass of 500 MeV, which marks the minimum mass the CRESST-II detector was able to probe, and four increasing values for the cross secon. The lowest cross secon is chosen to correspond to the single scaering regime. The three higher cross secon are tuned to yield 1, 10, or 50 underground scaerings on average. The benchmark points are summarized in table 4.1. Here, hNsci is the average number of scaerings and NSample the sample size, i.e. the amount of velocity data points recorded per isodetecon ring. The collecon of such large data samples is computaonally expensive. All simulaons were performed on the Abacus 2.0, a 14.016 core supercomputer of the DeIC Naonal HPC Center at the University of Southern Denmark. They involved simulaons of up to 1011 individual DM trajectories.

The single scaering regime For the inial consistency check, we perform simulaons with a cross secon corresponding to an average scaering probability of 10% and generate the equivalent result using the EarthShadow tool. The DM mass and DM-nucleon scaering cross secon were set to be 500 MeV and ∼0.5 pb respecvely. This cross secon was determined using one of EarthShadow’s analyc rounes. The simulaons yielded a relave amount of freely passing parcles of ∼ 90%, the remaining 10% of the simulated parcles scaered at least once, the

91 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

average number of scaerings added up to 0.12. Therefore, the first comparison shows good agreement, as ∼ 1% of the parcles in theMC simulaons are expected to scaer twice or more, especially if they pass the denser Earth core. This should result in small deviaons between the analyc and theMC approach, which affect regions deeper in the Earth shadow most. There, the parcles have to travel longer distances underground and the scaering probability is largest. The central queson is, how the elasc collisions on terrestrial nuclei distorts both the DM speed distribuon as well as the spaal distribuon inside the planet. Both modificaons can be seen from the speed distribuon, if we spulate that the (0) −3 normalized distribuon corresponds to a DM energy density of ρχ = 0.3 GeV cm . R Hence, the local speed distribuon in the Earth’s shadow will sasfy dvf(v) < 1, (0) since ρχ < ρχ . On the other hand, if an experiment faces the DM wind, the local R DM density can also exceed the local halo density, such that dvf(v) > 1. In this case, many parcles which were originally not moving towards this region get deflected. A poron of the underground DM populaon gets redistributed from large to small values of Θ. The distribuons are ploed in the le panel of figure 4.9. Even for a relavely low scaering probability, the distribuons show this behaviour very well. Parcles heading to regions in the Earth’s shadow are likely to get scaered off their original path, and the local parcle density there is decreased. The shape of the distribuons of DaMaSCUS and EarthShadow agree remarkably well. Both show that the major modificaon of the DM distribuon is due to deflecons and the resulng redistribuon of the DM parcles. Deceleraon plays only a minor role, because a single scaering is not capable to cause a significant energy loss for a relavely light parcle. Given this excellent agreement, it is unsurprising that it propagates to the direct detecon event rates. In the right panel of figure 4.9, we show the event rate for a CRESST-II type detector2 for the different isodetecon rings and compare it to the equivalent calculaons based on the analyc approach. Here, the analyc result was taken from figure 7 of [389]. The Earth’s shadow is clearly visible, the event (0) rate drops significantly for 120° . Θ ≤ 180° down to ∼ 85% of the rate R , which does not takes underground scaerings into account. It is however important to note that an experiment in the northern hemisphere, e.g. located at the LNGS in Italy, is not sensive to this effect. The event rate is slightly increased, but next to completely flat for 0° ≤ Θ . 90°, and the isodetecon angles covered by such an experiment during a sidereal day never reach

2See app.C for a descripon of the CRESST-II experiment.

92 4.2. DIURNAL MODULATIONS

] 3.5 ] 4. km km / /

s s 3.5 Θ[°] 3 3. SS MS1 - - 3

10 10 3. 2.5 180 )[ )[ v v

( ( 2.5 χ 2. χ 2. 1.5 1.5 150 1. 1. 0.5 0.5 120

speed distribution f 0. speed distribution f 0. 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 speed v [km/s] speed v [km/s]

] 6 ] 7 90 km km / / s MS10 s 6 MS50

- 3 5 - 3

10 10 5 )[ 4 )[ 60 v v ( ( χ χ 4 3 3 2 30 2

1 1

speed distribution f 0 speed distribution f 0 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 0 speed v [km/s] speed v [km/s]

Figure 4.10: DM speed distribuons across the globe for our four benchmark −3 points. Note that they are normalized to 0.3 GeV cm . The black dashed line shows the speed distribuon of free DM. higher values as we saw in ﬁgure 4.8. In contrast, the same experiment located at the Stawell Underground Physics Laboratory (SUPL) in Australia would observe a diurnal modulaon of ∼ 15%.

Mulple Scaerings Having established the correspondence between analyc and simulaon results for the single scaering regime, we can direct our focus on the actual purpose of ourMC approach. We connue to explore the effect of mulple scaerings in the diffusion regime and increase the cross secon, such that we can expect 1, 10 or more than 50 scaerings on average. The values of the cross secon for the three benchmark points ‘MS1’, ‘MS10’ and ‘MS50’ have been fine-tuned to yield the respecve result for hNsci. The local distorons of the DM speed distribuon for all four benchmark points are shown in figure 4.10. As one might expect, the underground DM parcles get decelerated more severely for higher cross secons. As opposed to the single scattering regime, the peak of the speed distribuon shis significantly towards slower speeds as Θ increases. Similarly, the depleon of the DM populaon at large isodetecon angles occurs to a much higher degree, as the benchmark point ‘MS50’ demonstrates best. Locaons on Earth facing the DM wind show a vastly increased DM density caused by mulple scaerings. The effect is essenally the same as in the single scaering regime, but is pronounced much stronger. For very high cross

93 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

0.7 350 ] ] 3

0.6 s / cm / 0.5 km 300 〉 [ v GeV 〈 [ 0.4 χ ρ 0.3 250

0.2

0.1 average speed 200 DM density

0.0 0 60 120 180 0 60 120 180 Isodetection angle Θ [°] Isodetection angle Θ [°]

1.5 1.5 ( 0 ) ( 0 ) R R / / 1.0 1.0

0.5 0.5 event rate R event rate R LNGS SUPL 0.0 0.0 0 30 60 90 120 150 180 0 4 8 12 16 20 24 Isodetection angle Θ [°] local sidereal time [h] SS MS1 MS10 MS50

Figure 4.11: Results of the MC simulaons: The ﬁrst three plots depict the local DM density, average speed and event rate respecvely as a funcon of Θ. The last panel shows the corresponding diurnal modulaon in the local sidereal me for experiments in the northern and southern hemisphere.

secons, the distribuons show a new feature, a second peak close to zero speed, populated by very slow parcles. Current experiments are not sensive to such low-energec DM, but future experiments could potenally observe this peak as an increase in low recoil events. Furthermore, these parcles could parally get captured gravitaonally, leading to a similar effect [408, 409]. These distribuons are the primary output of the DaMaSCUS simulaons. They entail the local DM density and average speed and how they evolve with Θ. The result are ploed in the first two panels of figure 4.11. Depending on the cross secon the density can increase by more than a factor of 2 for locaons facing the −3 DM wind and decrease down to below 0.05 GeV cm in the Earth’s shadow. Simi- larly the deceleraon of DM parcles is most severe for larger isodetecon angles as expected. This results in an increase (decrease) of the event rate for direct detecon experiments located at small (large) isodetecon angle, shown in the third panel of figure 4.11. Again, we consider a CRESST-II type experiment as an example. Using eq (4.2.4) in connecon with eq. (B.4.2) and (B.4.5), we can relate this result to a diurnal modulaon specific to the laboratory’s fixed locaon on Earth.

94 4.2. DIURNAL MODULATIONS

100 SUPL LNGS Free SS

80 MS1 MS10 δ [%] MS50

60 Equator

Diurnal Modulation 20

0 -90 -60 -30 0 30 60 90 Latitude [°]

Figure 4.12: The percenle signal modulaon as a funcon of latude.

The last panel of figure 4.11 depicts this daily change of the event rate over the course of one sidereal day for an experiment both at the SUPL( 37.07°,142.81°E) in the southern hemisphere and at the LNGS( 45.454° N, 13.576° E) in Italy. Note that the phases only coincide, because we plot the signal rate as a funcon of the Local Apparent Sidereal Time (LAST). By transforming the modulaon in e.g. Universal Time (UT), we can obtain the modulaons’ phases as well. In the single scaering regime, experiments in the northern hemisphere are insensive to diurnal modulaons caused by Earth scaerings. However, they are no longer reserved for the southern hemisphere at larger cross secons. While it is sll generally true that experiments in e.g. Australia maximize the daily modulaon, we find that northern experiments can expect a sizeable effect as well, at least in the diffusion regime. To study the modulaon amplitude’s general dependence on the laboratory’s latude, we define the percenle signal modulaon,

Rmax − Rmin δ(Φ) = 100 % , (4.2.21) Rmax where Φ is the laboratory’s latude. The modulaons δ for the benchmark points are shown in ﬁgure 4.12. With the excepon of the poles’ neighbourhood, diurnal modulaons can be signiﬁcant almost all over the globe, if incoming DM parcles scaer mulple mes on terrestrial nuclei. For the benchmark points ‘MS1’,‘MS10’, and ‘MS50’, an experiment at LNGS could expect to observe modulaons of 10, 45, and almost 60% respecvely. The corresponding modulaons at the SUPL naturally exceed these values with about 18, 65 and more than 90%. For even higher cross secons we can expect underground scaerings to occur even in the overburden of an experiment, for example in the crust or atmo-

95 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

sphere. Above some crical value these layers shield oﬀ not only background but also DM parcles. Terrestrial experiments are not able to probe DM above some crical cross secon. Finding this cross secon for diﬀerent scenarios is the main goal of the next chapter.

4.3 Constraints on Strongly Interacng DM

The diurnal modulaons of the signal rate of DM parcles with moderate DM- nucleon interacon were the consequence of the shadowing or shielding effect of the Earth’s bulk. The planet’s mass reduces the DM flux at the detector, especially if the DM wind has to pass large parts of the Earth to reach the laboratory. For even higher cross secons experiments might lose sensivity altogether. The vast majority of direct detecon experiments are located deep underground, typically ∼ 1 km beneath the surface, in order to reduce the background such as atmospheric muons or highly energec cosmic rays. Above a certain crical DM-nucleon cross secon, the rocky overburden of the Earth crust does not just shield off the detector of this undesired background, but also the actual DM signal it is set up to observe. The DM not only scaers in the detector, but also on terrestrial nuclei in crust and atmosphere. These scaerings deflect and decelerate incoming parcles and reduce the detectable DM flux at the experiment. Consequently, cross secon above a crical value are not probed, and only a band of cross secons can be excluded. Already menoned in the original direct detecon by Goodman and Wien [209], this issue has been studied in various contexts. The central queson has been, whether or not this effect reduces constraints on strongly interacng DM in a way that an ‘open window’ in parameter space emerges [388]. The excluded band is bounded by underground direct detecon experiments from below and astrophysical constraints from above. Strong DM-baryon interacons would have an observable impact on various astronomical observaons. Strong interacons of this kind would lead to momentum transfer between the visible and dark sector and affect the CMB’s anisotropy [410–412], affect primordial nucleosynthesis [413], or introduce a collisional damping effect during cosmological structure formaon [414]. Furthermore, collisions of DM parcles and cosmic rays would lead to the producon of neutral pions and hence observable gamma-rays [413, 415, 416], or modify the cosmic ray spectrum via elasc scaerings [417]. Non-astrophysical constraints on strongly interacng DM are set by the satellite experiment IMP7/8 [418, 419], experiments on the Skylab space staon [419, 420], the X-ray Quantum Calorime-

96 4.3. CONSTRAINTS ON STRONGLY INTERACTING DM ter (XQC) placed in a rocket [421], early balloon-based experiments [422], and searches for new nuclear forces [336]. During the last 30 years, several allowed regions in parameter space were found for both light and heavy DM masses [388, 423]. These windows have since then been closed by arguments based on DM capture by the Earth and heat flow anoma- lies [424], observaons from the IceCube experiment [425] and the Fermi Gamma- ray Space Telescope [415]. Another window in the low mass region was closed more recently by a re-analysis of old data from DAMIC and XQC [426] and new data from the CRESST 2017 surface run [427]. Other works focused on super-heavy DM, where an allowed window was closed by a re-analysis of CDMS-I data [428]. The exact exclusion limits for direct detecon experiment taking the terrestrial effects into account have so far been determined mostly based on the analyc stopping power [366, 388, 398, 399, 416, 427–430], but also by usingMC simulaons [423, 426, 431, 432], which are best suited for this problem. The analyc approach fails to describe deflecons of DM parcles and will in most cases either under- or overesmate the overburden’s stopping power depending on the DM mass and interacon type. In this secon, we present a new systemacMC treatment of constraints on strongly interacng DM both for DM-nucleus and DM-electron scaering experiments. The main objecve of these simulaons is to systemacally find the crical cross secon, above which any given experiment becomes blind to DM itself. For DM-electron experiments, we do not limit our analysis to contact interacons as before and consider also electric dipole and long range interacons, mediated by ultralight dark photons. TheMC simulaons loosely resemble the ones from in the last chapter, but differ in a number of important points, as we will discuss in chapter 4.3.2. Versions of the DaMaSCUS-CRUST code developed for this purpose were released with Paper III and Paper V and are publicly available [6].

4.3.1 Analyc methods using the DM stopping power

There are mulple channels for a DM parcle to lose energy while traversing a medium. 1. Nuclear stopping: Elasc collisions with terrestrial nuclei.

2. Electronic stopping: Inelasc scaerings with bound electrons, leading to ionizaon and excitaon.

3. Atomic stopping: Elasc and inelasc collisions with bound electrons, where the electron remains bound and the whole atom recoils. This process is also

97 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

relevant for very light DM parcles, whose kinec energies fall below the atomic binding energy scales O(10)eV.

In the cases relevant for us, the nuclear stopping dominates, which is why our simulaons only include elasc DM-nucleus interacons [399]. Electronic and atomic stopping can however become relevant in certain models, where DM-quark interacons are not allowed or heavily suppressed [308]. The quantave descripon of the electronic stopping power is a great challenge requiring methods from condensed maer and geophysics. In parcular, the computaon is complicated by the

electronic structure of the chemical compounds found in rocks, such as e.g. SiO2,

Al2O3, or FeO. In Paper V, we discuss the electronic and atomic stopping power in greater detail. We also derive an esmate for the DM parcles’ energy loss due to ionizaon and atomic scaerings, potenally relevant for leptophilic models and use these esmates to set approximate constraints on leptophilic DM models. Naturally, the higher the average number of scaerings prior to reaching the detector, the lower the detectable DM ﬂux at the underground detector. We discussed in chapter 4.1.2 that the scaering probability is given in terms of the mean free path λ. However, the actual energy loss does not just depend on the number of interacons, but also on the relave energy loss in each interacon. This in turn depends crically on the DM mass mχ relave to the target mass and the interacon type. In a contact interacon between a DM parcle and target nucleus of similar mass, the DM parcle can lose a signiﬁcant fracon of its kinec energy. If however, the same scaering is mediated by an ultralight dark photon, forward scaering is heavily favoured, and the DM parcle’s deceleraon is marginal. The average relave energy loss in a single scaering is given by

1 Z ER ER(cos θ) = d cos θ fθ(cos θ) , (4.3.1) Eχ Eχ −1

where the scaering angle’s PDF is given in eq. (4.1.37). A 10 GeV DM parcle with a speed of 10-3 scaering on an oxygen nucleus illustrates the importance of the interacon type. For contact interacons, it loses 48% of its kinec energy −4 on average. However, if the mediator is ultralight, it loses only ∼ 6 × 10 % in the same scaering. It will take a lot more scaerings and a much shorter mean free path in the second case to effecvely aenuate the detectable DM flux in the overburden. In conclusion, the mean free path is generally not a good measure to quanfy the overburden’s ability to weaken the DM flux. Instead, the actual local

98 4.3. CONSTRAINTS ON STRONGLY INTERACTING DM quanty of interest should be the stopping power, the average loss of energy per distance travelled in the medium.

Nuclear stopping power As a DM parcle from the halo with kinec energy Eχ passes through a medium, it will lose energy due to elasc scaerings with nuclei. The average energy loss per travelled distance, the nuclear stopping power, is a local quanty and can be computed via [388, 399],

dhEχi X S (x,E ) ≡ − = n (x)σ hE i n χ dx i i R (4.3.2a) i Emax ZR X dσi = ni(x) dER ER . (4.3.2b) dER i 0

Note that the average recoil energy hERi depends on Eχ, and this is a first order differenal equaon. The soluon yields the energy (or speed) as a funcon of distance d travelled in the medium hEχi(d) (or hvχi(d)) for some specified inial condions.

In the case ofSI interacons of light DM, where the nuclear form factor can be neglected and the diﬀerenal cross secon simpliﬁes to eq. (3.4.12), the average recoil energy of isotropic contact interacons is simply

max 2 ER γ 2µχT hERi = = Eχ = Eχ , (4.3.3) 2 2 mχmT and the corresponding stopping power takes the form

2 SI X 2µχiσi SSI(x,E ) = n (x) E ≡ 2∆E . n χ i m m χ χ (4.3.4) i i χ

Here, we cleaned up the notaon and introduced the factor

2 SI X µχiσi ∆ ≡ n (x) . i m m (4.3.5) i i χ

For a homogenous medium, inside which the density and composion remains constant, it is possible to ﬁnd explicit soluons. Aer travelling a distance d in this (0) medium, the average energy of parcles with inial energy Eχ decreased expo-

99 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

nenally,

(0) hEχi(d) = Eχ exp [−2∆d] , (4.3.6a)

or in terms of the average speed,

(0) hvχi(d) = vχ exp [−∆d] . (4.3.6b)

We conﬁrmed the accuracy of this expression for the average speed withMC simulaons in Paper I.

In the dark photon model, the situaon is complicated by the charge screening, which can not be neglected for low-mass DM and an ultralight dark photon mediator. A closed form soluon for long range interacons in analogy to eq. (4.3.6) cannot be found. However, we can evaluate the integral in eq. (4.3.2),

4 2 σpq mχ 1 X ni(x)Z xi S (x,E ) = ref i log (1 + x ) − , n χ 16µ2 E m i 1 + x (4.3.7) χp χ i i i

8µ a2 x ≡ a q2 = χi i E where i i max,i mχ χ depends on the DM parcle’s kinec energy. Assuming that the nuclear composion, i.e. the isotopes’ mass fracons fi, do not change along the DM parcle’s path, we can separate the dependences on x and Eχ,

4 2 σpq mχ 1 X fiZ xi S (x,E ) = ρ(x) ref i log (1 + x ) − , n χ 16µ2 E m2 i 1 + x (4.3.8) χp χ i i i | {z } ≡f(Eχ)

factoring out the mass density ρ(x) as the only posion-dependent quanty. For a DM parcle moving a distance d in the medium, we can then write the implicit soluon as

hE i(d) Zχ Z dEχ = − dx ρ(x) . (4.3.9) f(Eχ) (0) Eχ

Further steps, e.g. solving for hEχi(d), must be performed numerically. Most proposed analyc methods to ﬁnd the crical cross secons are indeed based on the stopping power. We will review two of these methods and discuss the

100 4.3. CONSTRAINTS ON STRONGLY INTERACTING DM reasons of their inferiority compared to aMC approach in most scenarios.

Analyc speed criterion Arguably the simplest method is to ﬁnd the cross secon for which even the fastest halo parcles get decelerated that they are no longer detectable. In other words, even the highest recoil they could cause in the detector falls below the recoil threshold Ethr. For a DM parcle to be detectable means to have kinec energy of at least

min Ethr mT mχ Eχ = = 2 Ethr . (4.3.10) γ 4µχT

Here, mT is the mass of the detector’s lightest target nuclei. To model the path of the DM parcles towards the detector, we can assume that the parcles take the direct path between Earth’s surface and the detector, i.e. a straight path of length corresponding to the detector’s underground depth d. The fastest parcles in the max 1 2 halo have kinec energy of Eχ = 2 mχ(vesc + v⊕) . If even these parcles are on average slowed down to non-detectabilility, the detector is assumed to have lost sensivity. As the Earth’s crust can be modelled as a homogenous medium, the situaon is parcularly easy forSI contact interacons, where we have the closed soluons SI given in eq. (4.3.6). Then the crical DM-nucleon scaering cross secon σp is the soluon of

" 2 SI # X 2µχiσi Emin = Emax exp − f ρ d , χ χ i m2m (4.3.11) i i χ −1 " 4 2 # max X 2µχiAi Eχ ⇒ σSI = f ρ d × log . p i µ2 m2m Emin (4.3.12) i χp i χ χ

This cross secon can serve as a first esmate of the crical cross secon. The analyc speed criterion was first formulated in [388] and applied e.g. in [399, 429], as well as in Paper I and Paper IV, which also contains a method comparison. Knowing the fullMC results, this method turns out to yield a reasonable esmate a posteri- ori, yet it comes with a list of deficiencies.

1. The crical cross secon obtained via eq. (4.3.12) depends solely on the detector’s threshold and is not found by compung expected numbers of events using e.g. Poisson stascs. The experiment’s details, in parcular the exposure, do not enter the esmate, and the two boundaries of the exclusion limits are not on equal foong.

101 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

2. Similarly, the knowledge of the halo DM parcles’ speed distribuon is not being used, except for the PDF’s hard cutoﬀ, where the distribuon is trun- (0) cated. Seng vχ = vesc + v⊕ might be conservave, but is also rather arbitrary.

3. The incoming parcles do not follow straight paths towards the detector. Deflecons prolong the underground trajectories, and the average distance travelled will be larger than the underground depth. Furthermore, deceleraon is not the only process, which aenuates the underground DM flux. Parcles can also get reflected back to space, which is not taken into account. However, there are some cases, where forward scaering is heavily favoured. This is typically the case for light mediators or very heavy DM. In these cases, this problem is less severe.

4. In the case of GeV scale DM and contact interacons, where a single scaering causes a significant relave loss of kinec energy, the analyc stopping equaon overesmates the reducon of the detectable DM flux, as pointed out by Mahdawi and Farrar [426, 431]. We emphasized that the stopping power describes the average energy loss. If the reducon of the flux occurs through only a hand full of scaerings, it fails to account for the small, but non-negligible number of rare parcles, which scaer fewer mes. Although their number is naturally suppressed, this is compensated by the large cross secon and probability to trigger the detector, once a parcle reaches the detector depth. Higher cross secons increase both the overburden’s shielding as well as the event rate in the detector. This problem is most severe, if the DM mass is close to the mass of terrestrial nuclei.

The last two problems can not be addressed by analyc means alone3. To solve these problems, parcle transport simulaons are the method of choice. However, the problems 1 and 2 can indeed be addressed by substuon of the stopping power into the computaon of the DM distribuon and therefore the detecon rates.

Analyc signal criterion The central DM property which enters the direct detecon rate in eq. (3.5.2) is the DM velocity distribuon. Typically, we use the halo distribuon of the SHM, assuming that the maer surrounding the experiment has

3Deﬂecons can in principle be described analycally, but this has only been done in the single- scaering regime [389].

102 4.3. CONSTRAINTS ON STRONGLY INTERACTING DM

0.010

] halo km / 0.008 d=100m sec

)[ d=500m χ v

( 0.006

d χ d=1000m

0.004

0.002 distribution f

0.000 0 100 200 300 400 500 600 700

speed vχ [km/sec]

Figure 4.13: The underground DM speed distribuon of eq. (4.3.14) SI -30 2 for mχ = 1 GeV and σp = 10 cm at different depths. no impact on the DM flux. It is however important to keep in mind that the correct distribuon is the one at the detector. It is possible to include the impact of the d overburden’s stopping power into the local underground speed distribuon fχ(vχ) at depth d, provided that we have a handle on the stopping soluon hvχi(d), as e.g. forSI contact interacons in eq. (4.3.6). Based on our assumpon that the parcles move along a straight path and lose energy connuously, no parcles get lost and the parcle flux is conserved,

d d d d (0) (0) (0) fχ(vχ)vχ dvχ = fχ(vχ )vχ dvχ . (4.3.13)

For contact interacons, we can use the soluons in eq. (4.3.6) and ﬁnd

d d d fχ(vχ) = exp[2∆d]fχ(exp[∆d]vχ) , (4.3.14) which is shown for a few examples of stopping in the Earth crust4 in ﬁgure 4.13. The local DM density increases as the parcles get slower, which is reﬂected by the fact that this distribuon is no longer normalized. For the recoil spectrum, we substute the result into eq. (3.5.3) and express it in terms of the halo distribuon,

Z SI dR 1 ρχ dσi −∆d (d) = dvχ vχfχ(vχ) (ER, e vχ) . (4.3.15) ∆d dER mT mχ vχ>e vmin(ER) dER

This spectrum can be used to compute signal rates and counts, constraints etc., and the overburden’s aenuang eﬀect is automacally taken into account. Above

4The Earth crust model, in parcular the density and nuclear composion, can be found in app. B.5.1.

103 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

a certain cross secon the signal rate starts to decrease with higher interacon strengths. Beyond the crical cross secon, the number of expected events falls so much that the parameter space is unconstrained. Comparing the analyc speed and signal criteria, we see that the crical cross secon obtained by the first method corresponds to the cross secon, for which eq. (4.3.15) predicts zero events. The crical cross secon based on the speed criterion will always be a slight over-esmaon when comparing to this method. This method can be improved by not just taking straight paths between the Earth surface and detector depth, but taking the direconality into account [427]. Around half of the parcles approach the detector from below, not above, yet for cross secons close to the crical cross secon, these will surely not contribute to the detectable DM flux. This would lead to addional aenuaon of roughly 50%. Since the signal rate drops very sharply, as we will see later on, the effect on the value of the crical cross secon is negligible. Finally, the nuclear form factor could be included to correctly compute constraints on super-heavy DM [428, 433].

For light mediators, the situaon is not that simple, as we have to numerically solve eq. (4.3.9) to obtain an incoming parcle’s ﬁnal speed at the detector and therefore the underground speed distribuon. More details can be found in Paper V.

4.3.2 MC simulaons of DM in the overburden

TheMC simulaon of the DM flux aenuaon due to an experiments overburden follows the principles introduced in chapter 4.1. They track individual parcles incoming from the halo, as they travel on straight paths unl they collide elascally on a terrestrial nucleus, changing direcon and losing energy in the process. The most important difference to the DaMaSCUS simulaons of chapter 4.2 is the simulaon volume. Instead of the whole planet, we reduce the volume to the shielding layers between the detector and space, which could include the rocky Earth crust, the atmosphere, and addional layers made of e.g. lead or copper. For cross secons close to the crical cross secons, virtually all detectable DM parcle reach the laboratory from above. The relevant scaerings occur in a km scale volume. On these scales the Earth surface’s curvature may be neglected, and we can model the overburden as a stack of parallel, planar layers as illustrated in figure 4.14. To check this approximaon’s accuracy, we can monitor the parcles’ horizontal dis-

104 4.3. CONSTRAINTS ON STRONGLY INTERACTING DM

Space (a) Atmosphere 1/3

84 km Atmosphere 2/3

Atmosphere 3/3

Earth Crust < 2 km ∼ cm (b)

Lead shield (c) Laboratory

Figure 4.14: Sketch of the MC simulaons of parallel shielding layers (not to scale). placements along their trajectories. If they stay within a few km at the crical cross secon, the inclusion of the Earth’s geometry is indeed not necessary. The simulaon code is opmized for a variable number of user-deﬁned constant density layers, using the algorithm of ﬁgure 4.2. A layer is characterized by its thickness, density, composion, and locaon relave to the other layers. For a layer of non-constant density, the atmosphere being the obvious example, we simply divide it into a larger number of sublayers, each of which is again of constant density. For more details on the atmospheric or the Earth crust model, we refer to app. B.5.1.

Trajectory simulaon The main objecve of the simulaons is to yield an accurate esmate of the underground distribuon of detectable parcles, which in turn can be used to compute local event rates for direct detecon experiments. A DM parcle is deﬁned as detectable, if is able to trigger the detector with a recoil energy above its threshold. It is by deﬁnion faster than s mT (Ethr − 3σE) vmin = 2 , (4.3.16) 2µχT where we specify an experiment’s threshold Ethr, its lightest target’s mass mT , and energy resoluon σE. For light DM searches the interval of interest [vmin, vesc + v⊕] typically falls into the halo distribuon’s high energy tail, and we would like to avoid wasng me by simulang undetectable parcles. Therefore, we only simulate parcles from this interval. Before any parcle is simulated, we need to specify how inial condions are being sampled. Since our setup of stacked, parallel, planar shielding layers is in-

105 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

variant under horizontal translaons. Therefore the underground DM distribuon solely depends on the depth, and we can pick any inial posion above the overburden. For the velocity, we average over the halo distribuon’s anisotropy and sample the inial speed v0 ∈ [vmin, vesc + v⊕) via rejecon sampling of the speed distribuon in eq. (3.2.5). This has the advantage that we do not have to specify, where and when the experiment took place. To determine the velocity’s direcon we define α as the angle between the inial velocity v0 and the vercal line. Since the inial posions are effecvely distributed uniformly in space, and we send off all parcles from a fixed altude, this angle is not isotropically distributed. In order not to overesmate the number of parcles with shallow incoming angle (i.e. α close to 90°), we have to use the correct PDF for this angle, fα(cos α) = 2 cos α, with cos α ∈ [0, 1). We transform a sample ξ of U[0,1] into a sample of cos α via inverse transform sampling, Z cos α 0 0 p d(cos α )fα(cos α ) = ξ ⇒ cos α = ξ . (4.3.17) 0

This leaves us with the inial condions     √  0 sin α 1 − ξ       t0 = 0, x0 = 0 , v0 = v0  0  = v0  0  . (4.3.18)      √  0 − cos α − ξ

Here, we picked a parcular horizontal direcon, which is allowed by the simulaon volume’s rotaonal symmetry. As illustrated in ﬁgure 4.14, there are three triggers terminang a trajectory simulaon. The DM parcle is simulated and tracked unterground, unl

(a) the parcle gets reﬂected back to space,

(b) the parcle’s speed falls below the threshold of eq. (4.3.16), or

In the last case, we record the parcle’s speed and stascal weight. Otherwise we simply count the failed parcle for the esmate of the aenuaon.

Rare event techniques The crical cross secon is typically very high and would naively predict a large number of events in a detector. However, the crical cross

106 4.3. CONSTRAINTS ON STRONGLY INTERACTING DM secon is defined as the value, where the detector loses sensivity to DM. To compensate signal rates of large magnitude, the underground DM flux has to be at- tenuated by the same amount. Therefore, the instance (c) of a detectable parcle reaching the detector without being reflected or decelerated below threshold is extremely rare. Yet, these are the parcles we are interested in. To esmate the detectable DM flux, we need to know how many and how fast DM parcles reach the detector despite the overburden. If the success probability for a single parcle is vanishingly small, the problem arises that the simulaons are computaonally extremely expensive, inefficient and wasteful. Rare-event simulaon is a well-studied challenge to parcle transport simulaons, going back to von Neu- mann and neutron transport simulaons for nuclear reactors. We implement the two most common rare event techniques, which increase the success probability in a controlled manner [391, 434, 435]. We should stress that these techniques are absolutely essenal forMC simulaons in this context to be applicable at all.

If certain stascal properes of the desired data set are known, one can introduce a bias into the simulaons’ PDFs which increases the success rate and amplifies sampling favourable values, while keeping track of this bias by a stascal weighing procedure. This method is called Importance Sampling (IS). Its general principles and specific applicaon for our purpose is reviewed in the app. E.1. It was first applied for this purpose by Mahdawi and Farrar [426, 431]. The method yields stable and fast results for GeV scale DM, where a single scaering can cause significant energy losses. However, if the DM flux gets aenuated only by hundreds or even thousands of scaerings, this method is not reliable.

Another standard rare event technique is Geometric Importance Spling (GIS), which was ﬁrst applied to DM simulaons in Paper V. This method supports the simulaon of successful parcles by idenfying ‘interesng’ parcles which get close to the detector and then spling these parcles in a number of idencal copy, each of which gets tracked independently. This in turn increases the chance that one of them reaches the detector. Furthermore, parcles which get less interest- ing by moving away from the detector have a certain chance to be terminated by what is typically called Russian Roulee. Compared toIS, the spling technique turned out to be more generally applicable and yielded stable and fast results for various cases including trajectories with large numbers of scaerings. One reason is that GIS leaves the simulaon’s underlying PDF’s untouched. In the DaMaSCUS- CRUST code, the spling of the parcle in copies of itself is realized by a recursive parcle simulaon funcon. We discuss the technical details in app. E.2.

107 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

Finding the crical cross secon TheMC simulaons provide knowledge about how the underground scaerings in the overburden redistribute and aenuate the incoming DM parcles. The generated data needs to be connected to a predicon of the direct detecon event rate. This is done by esmang the local speed d distribuon fχ(vχ) at depth d. Based on the speed data, we obtain an esmate of the normalized speed distribuon through Kernel Density Esmaon (KDE), a non-parametric probability density esmaon procedure, which returns a smooth funcon. The technique of KDE is reviewed in app. D.3. The second necessary part is the overall aenuaon factor ad, the fracon of incoming parcles which passed the distance d with enough energy to cause a signal. The distribuon is hence given by

ˆd ˆKDE fχ(vχ) = ad × f (vχ) . (4.3.19)

The aenuaon factor is simply the fracon of successful parcles. If the total number of incoming parcles is Ntot, and N trajectories ended at the laboratory depth with respecve weights wi, then

N 1 X a = w . d N i (4.3.20) tot i=1

Here, we used that both the averageIS weight and the inial GIS weight is equal PNtot to 1 such that i=1 wi = Ntot. We have to keep in mind that the total number of simulated parcles Nsim is not the same as the total number of incoming parcles, as we only picked inial condions from the interval of interest [vmin, (vesc + v⊕)] in order speed up computaons. However, we can easily relate the two using the inial parcles’ speed distribuon, i.e. eq. (3.2.5),

Nsim Ntot = v +v ≥ Nsim . (4.3.21) escR ⊕ dvχ fχ(vχ) vmin

The speed distribuon of eq. (4.3.19) is used to compute detecon signal rates with e.g. eq. (3.5.3) or (3.5.12) in the usual way. At this point, we are able to compute numbers of events and likelihoods for any kind of direct detecon experiment, based on either nucleus or electron scaerings. We summarize the procedure. For a point (mχ, σ) in parameter space, the simulaon of DM parcle trajectories through the overburden generates speed data {(v1, w1), ..., (vN , wN )} and the aenuaon factor ad. Using KDE, we obtain

108 4.3. CONSTRAINTS ON STRONGLY INTERACTING DM a smooth esmate of the local, aenuated speed distribuon, which in turn determines the recoil spectra, event rates, likelihoods, etc., depending on the experiment of interest. For a given mass, we start at the usual lower bound on the cross secon and systemacally increase the cross secon, repeang the procedure from above, to find the point where the predicted number of signals starts to decrease. Above that value, the overburden’s shielding power dominates the signal rate in the detector, and we carefully keep increasing the cross secon in smaller and smaller steps. It is crucial to avoid going beyond the crical cross secon, as this is typically the regime, where the necessary computaon me grows dramacally. Instead, we carefully approach the crical cross secon from below. Once the likelihood grows above (1-CL), for a specified certainty level CL, the experiment no longer sets constraints on the point in parameter space. The exact value of the crical cross secon is finally obtained by interpolaon of the likelihood as a funcon of the cross secon and solving it for (1-CL). There are a few effects we neglect in our simulaons, which affect the predicted number of events by order one.

• Around half the parcles approach the detector from below and get shielded, whereas we assume all parcles to reach the Earth from above the experiment. The exact addional aenuaon depends on the experiment’s locaon on Earth relave to the DM wind. As we do not specify this and average over the halo distribuon’s anisotropy, we neglect this aenuaon of order 1/2.

• A few DM parcles could pass the detector’s depth subsequently get reﬂected back up, which would increase the predicon for the event number.

• It was reported that the modelling of the atmosphere as a planar stack of parallel layers leads to an error in the predicted number of detectable parcles at the detector of less than a factor 2 when compared to a more accurate geometric setup [432].

It should be emphasized that these errors lead to order one changes in the predicted event numbers, which have only minimal eﬀects on the value of the crical cross secon. This is due to the fact that, as soon as the signal rates start to drop above some cross secon, it drops extremely steeply, as e.g. shown in ﬁgure 4.15. The Earth crust turns ‘opaque’ to DM rapidly. However, the quanty of interest is the crical cross secon, not the underground event rate. The simulaons are not meant nor able to produce precise predicons of detecon signal rates, which

109 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

1012 no shielding speed criterion

1010 signal criterion MC

1 108 Likelihood

106

104 0.5 total number of events 102

100

0.1 10-46 10-44 10-42 10-40 10-38 10-36 10-34 10-32 10-30 SI 2 cross section σp [cm ]

Figure 4.15: Comparison of the diﬀerent methods and the predicted number of events for XENON1T (solid lines) and DAMIC (2011) (dashed lines) for a DM mass of 10 GeV.

would require much more details, even if the three issues menoned above were accounted for.

4.3.3 Nuclear recoil experiments

In Paper IV, we focused onSI contact interacons and constraints from nuclear recoil experiments. This is an interesng case, as the fundamental process which is probed in the detector is also the source of the signal’s aenuaon. Elasc nuclear recoils in the overburden and detector act as antagonizers, where the stopping effects dominates above some interacon strength. In this secon, we compare the different methods of the previous chapters and derive the constraints onSI contact interacons for a number of direct detecon experiments. The three methods of obtaining the crical cross secon discussed in chapter 4.3.1 and 4.3.2 are compared for the example of DAMIC (2011) [238] and XENON1T [263] in figure 4.15. The plot shows the predicted number of events as a funcon of the DM-proton cross secon determined with the analyc stopping power in eq. (4.3.15), as well as the more accurateMC simulaons. It also includes the simple speed based esmate of eq. (4.3.12), which in the case of XENON1T underesmates the crical cross secon, while it slightly overesmates it for DAMIC. Without theMC results being available, a quality assessment of this simple criterion is not possible, but it can clearly serve as an easy-to-compute first guess on its own. For low cross secons, the overburden has naturally no impact on the local speed distribuon in the laboratory nor the predicted event numbers, since the

110 4.3. CONSTRAINTS ON STRONGLY INTERACTING DM mean free path is much longer than the underground depth. Hence, the signal numbers increase linearly with the cross secon. The recoil spectrum of eq. (4.3.15) reproduces the qualitave behaviour nicely, where the signal strength above a certain value of the cross secon starts to drop very steeply. However, compared to theMC based predicons, it overesmates the pace with which the signal number decreases and therefore underesmates the crical cross secon by up to an order of magnitude. By looking for the two values of the cross secons, where the likelihood crosses (1-CL), we obtain the excluded interval. For cross secons of order 100pb, we see that the shielding has no visible effect on the signal rates, which seems to be in conflict with the results of chapter 4.2, where we had signal modulaons of order 100%. We menon once more that some of the assumpons, e.g. that all parcles approach the detector from above, are only valid for cross secons close to the crical cross secon. For intermediate values, the parcles from below are addionally aenuated since they have to pass the enre planet before they pass the target material. This regime can only be studied by simulang the whole planet, as we did earlier. The advantages of theMC simulaon of trajectories start to emerge at this point. The analyc stopping descripon neglects the rare, stubborn parcles, which scaer fewer mes than expected and thereby overesmates the aenuaon of the DM flux. However, in other cases, where reflecon, not deceleraon, is the dominant process of DM aenuaon, it can also yield an underesmate. The simulaons will produce a more accurate, realisc, and consistent result in all cases, which might also be more constraining. It was claimed that the analyc stopping equaon should not be applied for the derivaon of the crical cross secon of strongly interacng DM [431]. The authors jusfy this statement with the discrepancy in the event numbers of many orders of magnitude. We would argue that, looking at figure 4.15, any conservave method of finding the crical cross secon, such as the speed criterion or a recently proposed similar, even more conservave esmate [416], will yield values, for which correspondingMC simulaons will predict huge event rates. The resulng exclusion bands may be conservave, but are sll valid. Using the DaMaSCUS-CRUST simulaon code, we derive exclusion limits based on CRESST-II [236], the CRESST 2017 surface run [295], DAMIC (2011) [238], and XENON1T [263], which are ploed together with constraints from XQC [421] and the CMB[412] in figure 4.16. For the more specific details of the different experiments, we refer to the respecve app. C.1. We also include the neutrino floor [436].

111 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

10-22

10-24 CMB 10-26

10-28 XQC

-30 10 CRESST

-32 2017 ] 10 2 surface cm [ 10-34 SI p DAMIC σ -36 10 (2011 CRESST ) 10-38 II

cross section 10-40

10-42

10-44 XENON1T 10-46 neutrino floor 10-48

10-1 100 101 102 103

DM mass mχ [GeV]

Figure 4.16: The constraints (90% CL) by various direct detecon experiments, together with limits from XQC and the CMB.

The direct detecon constraints cover DM masses between 100 MeV and 1 TeV, and cross secons between 10-46 and 10-27cm2 can be excluded depending on the experiment. For each DM mass, we obtain a clearly deﬁned, consistent exclusion interval, where both bounds are computed the same way. We included constraints of the DAMIC 2011 run, despite the fact that these constraints are fully covered by the two experiments by the CRESST collaboraon. These results are useful since they allow direct comparison to the results obtained with the DMATIS code [426], which serves as a independent evaluaon. The two results agree to a reasonable precision with deviaons of around 10%. Both CRESST-II and XENON1T were located deep underground at LNGS under 1400m of rock. They are therefore insensive to strongly interacng DM, with their respecve constraints reaching up to ∼ 10-30cm2 and ∼ 10-31cm2 in their low mass region. On the other side, the CRESST 2017 surface run of a prototype detector developed for the ν-cleus experiment is much more powerful in this context. It was performed in a surface laboratory with only a few cm of concrete and the atmosphere as shielding. Despite its ny exposure it is much more successful in probing high cross secons and extends the excluded intervals by around three orders of magnitude towards stronger interacon strengths. All allowed windows

112 4.3. CONSTRAINTS ON STRONGLY INTERACTING DM between underground experiments and the XQC rocket experiment can be closed this experiment by the CRESST collaboraon. Only a ny window between CMB constraints remains, which might could get narrowed by including constraints from cosmic rays [413] or even closed by consideraons of heat flow in the Earth [424] or newer limits due to gas cloud cooling [437]. The comparison between CRESST-II and XENON1T illustrates furthermore that the exposure has only a marginal effect on the crical cross secon. The exposure of XENON1T exceeds the one by CRESST-II by a factor of ∼700, yet the crical cross secon is only up to ∼10% higher. Compared to DAMIC and the CRESST surface run, it is clear that the underground depth is the dominant factor. In order to probe strongly interacng DM with a direct detecon experiment, larger exposure are not an efficient strategy. Instead the detectors must be placed above ground, prefer- ably at high altudes, such as mountains, balloons, rockets, or possibly satellites.

In this chapter, we systemacally determined the crical cross secon above which a nuclear recoil experiment loses sensivity to DM usingMC simulaons of trajectories in the crust and atmosphere. We presented constraints onSI contact interacons between DM parcles and nuclei coming from a number of experiments. The constraints reach from the interesng sub-GeV mass regime to 1 TeV. Next, we want to focus exclusively on sub-GeV DM and experiments using DM-electron scatterings as potenal discovery process.

4.3.4 Electronic recoil experiments

DM parcles of sub-GeV mass do not have enough kinec energy to trigger a convenonal detector, which is why DM-electron scaerings were proposed as a new search channel for low masses. Nonetheless, for large cross secons, undetectable DM-nucleus scaerings can indirectly affect DM-electron scaering experiments by redistribung the underground DM parcles. DM-electron scaerings in the crust or atmosphere might in principle do the same, but the stopping due to electron scaerings tends to be a subdominant effect, as discussed in chapter 4.3.1. In the dark photon model, DM couples to both electrons and protons with a hierarchy between the respecve cross secons. However, the electronic stopping of DM is much weaker than nuclear stopping in this model, as we discussed in greater detail in Paper V and plays a significant role only, if the model does not allow DM-quark interacons. The only truly leptophilic models, where DM-nucleus interacons are

113 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

not even generated on the loop level, include pseudo-scalar and axial vector interacons [308]. In Paper V, we also presented an analyc esmate for the constraints on strongly interacng DM in these class of models. A certain model dependence is unfortunately unavoidable, as we need a relaon between the two cross secons, since the stopping and detecon mechanisms are no longer idencal. The dark photon model relates the two cross secons in a simple way, ﬁxing the rao to the rao of reduced masses, see eq. (3.4.28). While the algorithmic procedure of ﬁnding the crical cross secon is the same as in the previous chapter, the DaMaSCUS-CRUST code [6] has been extended in the following ways.

1. More general interacons: We do not limit ourselves to contact interacons and consider light mediators and electric dipole interacons as well. This changes the scaering kinemacs crically, as we discussed in chapter 4.1.4.

2. New rare eventMC technique: Geometric Importance Spling was ﬁrst implemented in this context. The method’s details are reviewed in app. E.2.

3. New analyses: For DM-electron scaerings, the computaon of event rates and energy spectra are more involved and require the corresponding ionizaon form factors. The necessary relaons have been discussed in chapter 3.5.2.

In this chapter, we will study how scaerings in overburdens affect direct searches for light DM. For contact, electric dipole, and long range interacons, the detecon constraints on strongly interacng DM are determined viaMC simulaons, and the constraints’ general scaling behaviours under change of exposure or underground depth is studied. We re-visit the most recent experiments seng constraints on the DM-electron scaering cross secon and find the limits’ extend towards strong interacons, before the crust or atmosphere shields off the detector. Namely, we analyse the data of XENON10 [260, 312], XENON100 [310, 312], SEN- SEI (2018) [240], and SuperCDMS (2018) [319]. In addion, we present projecons for future runs of DAMIC and SENSEI at different underground laboratories. Finally, we explore the prospects of a high-altude run of a direct detecon experiment. A semiconductor target could be performed on either a balloon or in Earth orbit. For such an experiment, the strong orbital signal modulaon due to the Earth’s shadowing effect would be of great value to disnguish a potenal signal from an expectedly large background.

114 4.3. CONSTRAINTS ON STRONGLY INTERACTING DM

10-26 -24 10-28 10 -28 10 10-26 10-30 -30 -28

] ] 10

] 10 2 2 2 -32 -30 cm 10 cm cm [ [

[ 10

e -32 e e 10 σ σ σ 10-32 10-34 10-34 10-34 -36 yr 10 -36 g -36 g yr 10 0.1 10 0.1 g yr cross section cross section

0.1 cross section 1 g yr 1 g yr yr -38 1 g yr -38 g yr 10 -38 g 10 10 yr 10 10 yr 10 g g yr 100 g g yr 100 -40 100 10 -40 -40 2 10 FDM=1 10 FDM=αme/q FDM=(αme/q) 10-42 0 1 2 3 100 101 102 103 104 100 101 102 103 104 10 10 10 10

DM mass mχ[MeV] DM mass mχ[MeV] DM mass mχ[MeV]

Figure 4.17: Exposure scaling of the constraints (95% CL) for a generic semiconductor detector.

Exposure and depth scaling Increasing the exposure by running experiments of larger targets for longer mes allows to probe weaker DM interacons and extends the exclusion band towards lower cross secons. In the case of nuclear recoil experiments, we previously found that such an increase has only lile effect on the upper boundary of the exclusion band. The crical cross secon turned out insensive to the exposure. Here, we return to this queson to check if this conclusions holds also for DM-electron scaering experiments and, more importantly, light mediators. For that, we consider a generic silicon semiconductor experiment, where we fix the ionizaon threshold to 2 electron-hole pairs, the underground depth to 100m, and assume the absence of background events. The exposure is varied between 0.1 and 100 gram year. The resulng exclusions are shown in figure 4.17. The crical cross secon is again found extremely insensive to the exposure. The exposure increase of four orders of magnitude yielded an improvement of the limits by ∼60% or less. It seems to be a generic feature, that the overburden decreases the detecon signal rate very rapidly for a cross secon increase beyond some crical value. The shielding layers turn effecvely opaque to DM rather sud- denly, and the experimental parameters are not the dominant factors. The only advantage of larger exposures in this case concerns contact interacons, where the probed interval of DM masses can indeed be extended. Here, larger exposures increase the sensivity to heavier DM.

The exact dependence on the depth should be quanﬁed. We consider the same experiment as before, but ﬁx the exposure to 1 gram year. This me, we vary the underground depth between 1 km underground, to 40 km above ground, an altude a balloon experiment might reach. There the only shielding comes from

115 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

-22 10 10-20 10-24 -24 -22 10 10 +40km 10-26 +40km 10-24 +10km + -26 0m 40 10 +10km m ] ] 10 km ] - 2 -28 0m 2 -26 10 2 10 m + - -100 10 -28 10m cm cm

cm m [

10 [

km [ -1000 e -100 e -28 0m e m 10 σ -30 σ 10 - σ 10 -30 -1000m - m 10 100 10-30 -32 - m 10 1000 10-32 -32 m 10 -34 10 -34 -34 cross section cross section cross section 10 10

-36 -36 10 10 10-36 -38 10 2 10-38 FDM=1 10-38 FDM=αme/q FDM=(αme/q) 10-40 0 1 2 3 100 101 102 103 104 100 101 102 103 104 10 10 10 10

DM mass mχ[MeV] DM mass mχ[MeV] DM mass mχ[MeV]

Figure 4.18: Underground depth scaling of the constraints (95% CL) for a generic semiconductor detector.

the high, very thin layers of the atmosphere and the experimental setup itself. Here, we approximate the shielding layers of the apparatus as one layer of steel (2mm) and one of copper (1mm). The enre atmosphere’s stopping power corresponds to the stopping power of about ∼5m of rock or ∼1m of steel. The atmosphere can be neglected in the simulaons for 100m and 1000m. The resulng constraints (95% CL) are shown in ﬁgure 4.18 demonstrang the fact that the crical cross secon grows by about one order of magnitude for each order of magnitude the underground depth is decreased. Having a detector in a laboratory on a mountain will further improve the situaon by less than one order of magnitude. To gain more sensivity, it could be a good idea to run a small-scale balloon-borne experiment, since the exposure is not important at all. We will come back to this idea later in this chapter.

Constraints Direct detecon experiments, both with noble and semiconductor targets, have been performed to search for DM-electron interacons. In the absence of a posive result, various collaboraons have published constraints on the DM-electron scaering cross secon. We reanalyse their data on the basis of our MC simulaons, determining the extend of the excluded band in parameter space. Since it is nuclear scaerings which aenuate the DM ﬂux but electron scaerings which get detected, we are forced to assume some model, in order to have a relaon between the proton and electron cross secons. In Paper I, we presented ﬁrst esmates for the dark photon model based on a simple speed criterion. In Paper V and this thesis, we present the constraints for the same model, but based on a full data analysis such that lower and upper boundary of the exclusion band are truly on the same foong.

116 4.3. CONSTRAINTS ON STRONGLY INTERACTING DM

-24 10 10-22 10-24

-24 10-26 10 10-26

SENSEI SENSEI -26 ]

] ] 10

2 -28 -28 10 2 10 2 - - surface surface cm

cm cm SENSEI [ [ [

e -28 e e 10 σ -30 - 10 σ -30 σ surface SuperCDMS 10 SuperCDMS -30 -32 10 10 SuperCDMS 10-32 -32 10-34 10

cross section XENON100 cross section cross section XENON10 XENON100 -34 XENON100 10 XENON10 -34 XENON10 10-36 10

-38 -36 -36 2 10 FDM=1 10 FDM=αme/q 10 FDM=(αme/q)

0 1 2 3 0 1 2 3 10 10 10 10 100 101 102 103 104 10 10 10 10

DM mass mχ[MeV] DM mass mχ[MeV] DM mass mχ[MeV]

Figure 4.19: Direct detecon constraints (95% CL) by various DM-electron scaering experiments.

In parcular, we present updated constraints (95% CL) from the S2-only data of XENON10 [260, 312] and XENON100 [310, 312], as well as the surface runs of SENSEI (2018) [240], and SuperCDMS (2018) [319]. The details of the respecve analyses are presented in app.C. The constraint plot for contact, electric dipole, and long range interacons are shown in figure 4.19. The constraints for contact interacons in the le panel also include the constraints in the absence of charge screening as doed lines (FA(q) = 1). For very light DM, the charge screening effecvely decreases the DM-proton scaering cross secon, such that the overburden stopping is less effecve and the crical cross secon is higher. The dashed line in the right panel show an analyc esmate of the crical cross secon for light mediators based on eq. (4.3.9). For low masses, the analyc approach fails to account for deflecons or reflecons of DM parcles and yields an overesmate. For GeV masses however, we found that the DM parcle scaer sharply in a forward direcon and move on relavely straight lines connuously scaering on nuclei. This is well captured by the analyc descripon of nuclear stopping, and the two esmates converge around mχ ≈ 1 GeV. Above this mass,MC simulaons become more and more impraccal, because the number of scaerings increases extremely, where each individual scaering changes the DM speed only minimally. Then, the analyc method should yield reliable results. While the lower boundary of the exclusion band depends crucially on the experiments’ details, the crical cross secon is mostly determined by the overburden. The extend towards larger scaering cross secons is similar for SENSEI and SuperCDMS, since both experiments were performed on the surface. The same goes for XENON10 and XENON100 at the LNGS 1400m underground. For contact interacons, we observe a decrease of the crical cross secon for heavier

117 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

-24 10-26 10-26 10

-26 -28 10 10-28 10 -28 -30 10 ] -30 ] 10 ] 2 10 2 2 -30

cm cm cm 10

[ [ -32 [

e e 10 e 10-32 σ σ σ 10-32 -34 -34 10 10 10-34 10-36 10-36 10-36 at MINOS

cross section cross section -38 SENSEI cross section 10 SNOLAB -38 at MINOS -38 at MINOS at 10 SENSEI 10 SENSEI SENSEI -M at SNOLAB at SNOLAB -40 SENSEI DAMIC -40 -M SENSEI M 10 10 DAMIC 10-40 - DAMIC 2 FDM=1 10-42 FDM=αme/q 10-42 FDM=(αme/q) 10-42 10-1 100 101 102 103 104 100 101 102 103 104 100 101 102 103

DM mass mχ[MeV] DM mass mχ[MeV] DM mass mχ[MeV]

Figure 4.20: Projected constraints (95% CL) for the planned experiments DAMIC-M and SENSEI.

2 mχ σp/σe ≈ 2 DM masses, which is caused by the increasing rao in eq. (3.4.28), me . For electric dipole interacons and ultralight mediators, this eﬀect is counteracted by the DM form factor. In these two cases, the momentum transfer grows with the −2 −4 DM mass, while the diﬀerenal cross secon gets suppressed by ∼ q and ∼ q respecvely. For light mediators, the suppression of large momentum transfers is strong enough to dominate over the increasing cross secon rao, and the crical cross secon increases with DM mass.

Projecons and outlook There are two experimental collaboraons preparing a direct detecon experiment using CCDs, where the silicon semiconductor acts as the target, DAMIC [239] and SENSEI [318]. We derive constraint projecons for these next-generaon experiments and obtain their sensivity to strongly interacting DM. The DAMIC-M experiment is planned to run at the Laboratoire Souterrain de Modane in France at a depth of 1780m and will most likely have the largest exposure. SENSEI is planned to be located at SNOLAB slightly deeper at 2km underground. However, a smaller detector might also be used at the MINOS facility at Fer- milab at a relavely shallow depth of 107m. The projected exposures are 10, 100, and 1000 gram year for SENSEI at Minos, SENSEI at SNOLAB, and DAMIC at Modane respecvely. We assume the opmal ionizaon threshold of one electron-hole pair for all three detectors. Regarding the background, we assume to have observed 103, 104, and 105 events in the ne =1 bin. The results are shown in ﬁgure 4.20. Having the largest target mass, DAMIC-M would probe the smallest cross secon. Being at the shallowest site, a SENSEI run at MINOS would cover the strong interacon regime the furthest.

118 4.3. CONSTRAINTS ON STRONGLY INTERACTING DM

3. ] 2 1 Balloon ISS Balloon - no shadowing

2.5 sec ISS 1 ] - 1.5 gr 7 km

/ 2. halo s 10 [ - 3 1 10 1.5 )[ v (

χ 1. f 0.5 0.5 total event rate 0. 0 100 200 300 400 500 600 700 800 0 200 400 600 800 1000 1200 1400 DM speed v [km/s] UT [min]

Figure 4.21: Orbital modulaon of the DM speed distribuon and detecon rates for a balloon and a satellite experiment for mχ = 100 MeV, 2 2 σe = 10-23cm , and a light mediator (FDM ∼ 1/q ).

To probe cross secons above the reach of terrestrial experiments, we can entertain the idea to run a direct detecon experiment at great heights, either in the upper atmosphere in a balloon or in a lower Earth orbit onboard a satellite. The exposure of a balloon-borne experiment would be of order ∼gram hour, but shielded only by the upper atmospheric layers of low density. On a satellite, the situaon improves further, allowing longer exposures, while the shielding is minimized to the apparatus’ setup and self-shielding by the target. There is an intermediate regime of cross secons, where the Earth blocks off the observable DM flux enrely, while high altude experiments are sll sensive. The DM signal in these experiments would show a strong modulaon due to the Earth’s shadowing effect [386, 399], similarly to the diurnal modulaon we studied in chapter 4.2. For balloon experiments, we should expect a diurnal modulaon. For a space-borne experiment, the modulaon frequency would depend on the satellite’s orbit. We can esmate this modulaon without the need forMC simulaon by compung the local DM speed distribuon at the experiment’s locaon x via Z 2 f(x, v) = dΩ v f(v) × S(x, v) , (4.3.22) where we filter out parcles, which have to pass the planet’s mass, to reach the locaon. This is done with a simple ‘shadowing funcon’, defined as  0 , if |x + tv| = R⊕ has a soluon t < 0 , S(x, v) = (4.3.23) 1 , otherwise.

We discuss two examples of a silicon semiconductor target experiment.

119 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

1. A geostaonary detector running onboard a balloon, 30km over Pasadena, ◦ ◦ Ca (34.1478 N, 118.1445 W). The me evoluon of the posion vector in the galacc rest frame is given in eq. (B.4.5). Besides the upper 54 km of the thin, upper layers of the atmosphere, the target is assumed to be shielded by two addional layers, namely 1 mm of copper5 and 5 mm of mylar6.

2. A small detector in low Earth orbit. We take the ISS orbit as an example, which orbits our planet at about 400km altude with an orbital period of around 90 ◦ minutes. The orbit has an inclinaon of around 50 , and the ISS transions between being exposed to or hidden from the DM wind. Here, the target is assumed to be shielded by 1 mm of mylar only.

The variaon of the locally aenuated speed distribuon of DM parcles is shown in the le panel of ﬁgure 4.21. The plot on the right side shows the corresponding orbital signal modulaons for a DM parcle with mχ = 100 MeV and σe = 10-23cm2, which interacts via an ultralight mediator. This parameter point is chosen as an example, which is sll allowed by terrestrial experiments. Especially the populaon of fast DM parcles gets depleted within the Earth’s shadow. The fraconal modulaon is deﬁned as

Rmax − Rmin Rmax − Rmin fmod ≡ = , (4.3.24) Rmin + Rmax 2hRi

in terms of the minimum and maximum signal rates Rmin and Rmax. In both cases, we find a significant modulaon. For the balloon, we obtain a smaller fraconal modulaon with ∼ 30%. The same experiment in the southern hemisphere would be more sensive to the diurnal modulaon. For the ISS orbit, we see a high frequency orbital modulaon with fmod ' 85%. The satellite moves deep into the Earth’s shadow where most parcles, in parcular the fast ones, are geng stopped. The modulaon is expected to increase for lower DM masses, as their detecon relies more on the high-energy tail of the DM distribuon. The event modulaon due to the shadow effect could be used to disnguish a DM signal from backgrounds. For experiments without major shielding, we have to expect large background event rates. We assume an experimental run with exposure E, observing B background events. The 5σ discovery reach of a modulang

3 5Copper: ρ = 8.96 gram/cm . 6 3 Mylar: C10H8O4 with ρ = 1.4 gram/cm , (62.5% carbon, 33.3% oxygen, and 4.2% hydrogen) [438].

120 4.3. CONSTRAINTS ON STRONGLY INTERACTING DM

-16 COLL 10 2 FDM=(αme/q) CMB N 10-18 eff satellite SLAC 10-20 balloon 10-22 10-24

] -26 2 10 BBN Neff

cm -28 [ 10 e detection σ direct -30 SN 10 ΩDM 0.01 Ωχ= -32 0.1 ΩDM 10 Ωχ= ΩDM Ωχ= 10-34 10-36 -38 10 freeze-in (Ωχ=ΩDM) 10-40 10-1 100 101 102 103 104

mχ[MeV]

Figure 4.22: Constraints on DM with an ultralight dark photon mediator. signal due to DM over this background can be determined by solving

fmodNtot √ = 5 , (4.3.25) Ntot + B for σe [309]. The cross secon enters this equaon via the total number of events Ntot ≡ EhRi. For the balloon (satellite), we assume a background of 106 (109) signals, such that the 5 σ discovery reach corresponds to Ntot ' 16000 (182000), where we substuted fmod = 0.31 (0.87). It should not go unmenoned that the shielding layers are sll modelled as planar, and the results should be regarded as a first, but reasonable esmate. The exact geometry of the experimental apparatus would need to be implemented for more precise esmates, and the simulaons would need to be extended to more complex simulaon geometries. The projected modulaon discovery reaches are depicted in figure 4.22. This plot also shows our results for the direct detecon constraints, constraints from supernova cooling [439], cosmological bounds on Neff from the CMB and BBN[440], and collider constraints from a search for milli-charged parcles by SLAC [441, 442]. The blue line shows the parameter space favoured by the freeze-in producon mechanism [309, 443]. The open window in parameter space in the top right corner is inconclusive. It

121 CHAPTER 4. TERRESTRIAL EFFECTS ON DARK MATTER DETECTION

might e.g. be narrowed by translang the convenonal direct detecon constraints from the CRESST collaboraon into bounds on the electron cross secon by using eq. (3.4.28). There are further constraints based on the requirement that DM has decoupled at recombinaon [444, 445]. Most of these constraints assume that the strongly interacng DM parcles are the only component of DM, i.e. Ωχ = ΩDM. Large parts of the parameter space open up for the scenario where a milli-charged DM parcle makes up only a fracon of the total DM amount. This possibility has aracted a fair amount of aenon in light of the 21cm anomaly observed by the EDGES collaboraon [446], see e.g. [432, 447, 448]. In addion, some authors claimed that milli-charged DM would get ejected from the galacc disk through its magnec field and supernova shock waves and could therefore never be found by direct searches [444]. In a recent paper, it was claimed to the contrary that milli- charged parcles from the halo connuously re-populate the disk with a highly energec DM flux which extends previous detecon bounds significantly [449]. The fate of this interesng segment of parameter space is not clear at this point. If region remains allowed by the experimental bounds, a ballon- or satellite-borne experiment would be able to probe large parts of it.

122 Chapter 5

Solar Reﬂecon of Dark Maer

Direct searches targeng low-mass DM are complicated by the fact that light DM parcles cause smaller energy deposits in the detector. If even the largest possible recoil energy is too so and falls below the experiment’s threshold, DM is too light to be probed by that experiment. The minimum DM mass a detector of recoil thresh- thr old ER and target mass mN can test is given by eq. (3.5.6). Usually, the maximum speed entering this relaon is set to the galacc escape velocity plus the Earth’s relave moon vmax = vesc + v⊕. Next to the two obvious approaches to extend the experimental search to lower masses by lowering the target mass or threshold, it is also an opon to consider mechanisms that might increase the maximum speed vmax. Indeed, there is a virtually halo model independent mechanism which generates a populaon of fast DM parcles in the solar system, provided that the DM mass falls below the GeV scale. A light DM parcle may gain energy by elasc collisions on hot and highly energec targets, reaching speeds far beyond the maximum of the SHM. In this chapter, we will consider the idea of DM parcles geng accelerated by the hot constuents of the Sun. This idea was first proposed in [305] and independently shortly aer in Paper IV. Recently, cosmic rays have also been proposed as a potenal DM accelerator [306]. A DM parcle entering the Sun could scaer on a nucleus in the solar core and get reflected with great speed. This mechanism is effecve, if the kinec energy of infalling DM falls below the thermal energy of the solar targets and therefore applies only to low-mass DM. Heavy WIMPs most likely lose energy by scaering in the Sun and might get gravitaonally captured permanently. To obtain a significant scaering rate in the Sun and thereby a potenally observable parcle flux from solar reflecon of DM, the respecve interacon must

123 CHAPTER 5. SOLAR REFLECTION OF DARK MATTER

be strong enough. However, if the cross secon is too large, the cooler outer layers of the Sun shield off the hot solar core. If the last scaering before leaving the star occurs on a colder target, the DM parcle would most likely have lost energy. The only hope to observe reflected DM relies on the reflecon spectrum extending beyond the halo parcles’ maximum speed. Below that value, the standard halo DM will always dominate. Aer falling into the gravitaonal potenal of the Sun, the DM velocies are completely dominated by the solar escape velocity. The halo’s original distribuon therefore has no significant effect on the final reflecon spectrum, and the dependence on the assumed halo model is very weak. The obligatory existence of an addional flux of very fast DM parcles in the solar system can be used to extend the detecon sensivity of low-threshold direct detecon experiments to lower masses. Depending on the experimental exposure, this might set a novel kind of constraints on sub-GeV DM. The general idea to look for DM, which gained energy inside the Sun was first proposed for parcle evaporaon aer geng gravitaonally captured [450]. While evaporaon relies on the captured DM being thermalized, which is only true down to a certain DM mass, the mechanism of solar reflecon does not require such assumpons.

In chapter 5.1 we present the computaon and results of Paper III. Therein, we extend the analyc formalism by Gould [451–453]. In parcular, the new treatment takes the Sun’s opacity fully into account, smoothly connecng the opaque and transparent regime. Finally, we find the spectrum of solar reflecon via a single scaering and study the implicaons for direct detecon. In the second part, chapter 5.3, we show first results for aMC treatment of solar reflecon including the effect of mulple scaerings and finish by discussing the prospects of this promising approach. The results are preliminary and have not been published at this point. Throughout this chapter, we use the Standard Solar Model (SSM) to model the Sun’s interior. The SSM is introduced in app. B.5.2.

124 5.1. DM SCATTERINGS IN THE SUN

] 3. km / s 3 - 2.5 10 )) [ 2. w,r ( u ( χ 1.5

0.5 speed distribution f 0. 0 200 400 600 800 1000 1200 speed w [km/s]

r→∞ r=100r☉ r=10r☉ r=r☉ r=0.1r☉

Figure 5.1: Gravitaonal acceleraon of DM close to the Sun 5.1 DM Scaerings in the Sun

Compared to the descripon of underground scaerings in the Earth, we have to extend the simulaon algorithms for DM trajectories in the Sun.

1. The Sun’s gravitaonal well is much deeper than the Earth’s and cannot be neglected. In fact, the DM speed distribuon inside the Sun will be dominated by the kinec energy gained through falling into the Sun.

2. With the target being a hot plasma, it is not a valid approximaon to assume resng nuclei. In fact, this is precisely the reason why DM parcles can get accelerated by an elasc collision in the ﬁrst place.

Asymptocally far away from the Sun, the DM speed follows the halo distribuon fχ(u) introduced in chapter 3.2. Note that in this chapter, the asymptoc velocity (speed) is denoted by u (u). As a halo parcle approaches the Sun, it gains kinec energy. Using energy conservaon, the speed w at radius r is given by

p 2 2 w(u, r) = u + vesc(r) . (5.1.1)

The Sun’s local escape velocity vesc(r) is part of the SSM[454] and can be found in eq. (B.5.3). The direcon of w is unknown at this point and depends on the full r and u vectors. The local speed distribuon of infalling DM at radius r results from Liouville’s theorem [453],

p 2 2 fχ(w, r) = fχ(u(w, r)) , with u(w, r) = w − vesc(r) . (5.1.2)

125 CHAPTER 5. SOLAR REFLECTION OF DARK MATTER

This ‘blue shi’ of the halo distribuon in the Sun’s neighborhood is shown in ﬁg- ure 5.1. The second necessary adjustment is due to the fact that the solar medium is a thermal plasma. The nuclear velocies follow an isotropic Maxwell-Boltzmann distribuon, 3 r 3 κ −κ2v2 3 mN fN (vN , r) d vN = √ e N d vN , with κ ≡ , (5.1.3) π 2T (r)

where the dependence on the radius r enters via the solar temperature T (r) given in app. B.5.2.

5.1.1 Scaering rate

To describe solar reflecon (or capture), we need to know the rate at which DM parcles scaer to a specified final speed v. We divide the Sun into spherical shells of infinitesimal thickness dr. The differenal rate of DM parcles colliding on solar targets within a given shell can be wrien as a product of three factors,

dS = dΓ × dPscat × Pshell . (5.1.4) |{z} | {z } |{z} free DM passing rate scaering probability probability to reach the shell

The first factor is the rate with which DM parcles in an infinitesimal phase space volume would pass a spherical shell, if the Sun was completely transparent. Natu- rally, parcles might scaer already before reaching that parcular shell. The real passing rate is therefore rather dΓ × Pshell, with Pshell being the probability to make it to the shell without colliding beforehand. Finally, dPscat is the probability to scaer while passing the spherical shell. This means that eq. (5.1.4) yields the rate of the first scaering aer a DM parcle enters the Sun. For the computaon of the first two factors, we use results obtained by Press, Spergel [193], and Gould [451, 452] in the late ’80s. The third factor generalizes their framework to include the Sun’s opacity. This way, we avoid that the final equaons apply only to a certain regime of the scaering cross secon.

DM passing rate inside the Sun We start by imagining a spherical surface of radius R R with the Sun at its center. Being asymptocally far away from the Sun, the DM follows the standard halo distribuon. The diﬀerenal DM ﬂux through this surface into the sphere, i.e. the number of parcles entering the sphere per unit

126 5.1. DM SCATTERINGS IN THE SUN area and me, is obtained to be [193] 1 π dΦ = n f (u)u du d(cos2 θ) , 0 < θ < . χ 4 χ χ with 2 (5.1.5)

The angle θ lies between the DM velocity and the surface normal. We transform 2 2 the variable cos θ to the orbital invariant J , the angular momentum (per mass),

J = uR sin θ , (5.1.6a) 1 dJ 2 ⇒ dΦ = n f (u) du , 0 < J 2 < R2u2 . χ 4 χ χ R2u2 with (5.1.6b)

The total rate of parcles entering the spherical volume is du dJ 2 dΓ = 4πR2 dΦ = πn f (u) . χ χ χ u (5.1.7)

Parcles entering a volume of radius R are not guaranteed to pass the Sun’s surface or a given spherical shell of radius r < R . Incoming DM parcles follow an unbound hyperbolic Kepler orbit towards the Sun, while the orbit inside the Sun is no longer Keplerian. The queson whether or not a parcle passes a shell of radius r is equivalent to the queson whether its orbit’s perihelion distance sasﬁes rp < r. We can use the fact that the angular momentum J is an invariant and that θ = π/2 holds at the perihelion. Thus, the condion for a parcle of asymptoc speed u to pass a spherical surface of radius r is

J < w(u, r)r . (5.1.8)

For example, the total rate of DM parcles entering the Sun can be computed to be

2 2 Z ∞ Z w(u,R ) R 2 fχ(u) Γ (mχ) = πnχ du dJ (5.1.9a) 0 0 u 2 2 −1 = πR nχ hui + vesc(R ) hu i (5.1.9b) −1 29 mχ −1 ≈ 8 · 10 s (5.1.9c) GeV

It should be noted that we used the DM speed distribuon of eq. (3.2.5), where the anisotropy of the full velocity distribuon due to the Sun’s velocity in the galac- c rest frame is averaged out. The rate in eq. (5.1.7) is to be understood as the average over the spherical surface.

127 CHAPTER 5. SOLAR REFLECTION OF DARK MATTER

Scaering probability in a spherical shell The scaering probability inside the Sun cannot be described in terms of eq. (4.1.12), which we applied for the Earth. The reason is that the medium consists of a plasma of energec targets instead of eﬀecvely resng nuclei. The relave velocity is not dominated by the DM velocity, and the thermal moon of nuclei inside the Sun cannot be neglected. Instead of a mean free path λ, the central quanty is the collision frequency or scaering rate Ω. This can be understood by considering a resng DM parcle. Using (4.1.12), this parcle would never scaer unless it moves relavely to the targets. In a hot medium however, one of the thermal nuclei would eventually scaer on even a resng DM parcle.

The probability for a DM parcle of velocity w to scaer inside a spherical shell of radius r can be expressed in terms of the collision frequency Ω, dl dPscat = × Ω(r, w) . (5.1.10) w | {z } |{z} scaering rate me spent in shell

The distance travelled inside the shell is denoted with dl, as illustrated on the right hand side of ﬁgure 5.2. The diﬀerenal collision frequency on target species i with inial velocity vi is

3 dΩ(r, w) = σi |w − vi| ni(r)fi(vi) d vi , (5.1.11)

where fi(vi) is the Maxwell-Boltzmann distribuon of the target, and σi is the total scaering cross secon. Therefore the total scaering rate is obtained by integrat- ing over the targets’ velocies and summing over all target species with number density ni, X Ω(r, w) = ni(r)hσi|w − vi|i . (5.1.12) i

The brackets h·i denotes the thermal average. If the cross secon does not explicitly depend on the velocity, the collision frequency simpliﬁes to X Ω(r, w) = ni(r)σih|w − vi|i . (5.1.13) i

We can express the thermal average of the relave speed explicitly using eq. (5.1.3),

128 5.1. DM SCATTERINGS IN THE SUN

Z 3 h|w − vi|i = d vi |w − vi|fi(vi) (5.1.14a) 2 2 1 + 2κ w 1 2 2 = erf (κw) + √ e−κ w . 2κ2w πκ (5.1.14b)

We note that this thermal average depends on the radius r implicitly via the temperature.

If we insist on deﬁning a mean free path such that dPscat = dl/λ, it has to be wrien as

Ω(r, w) X h|w − vi|i λ−1(r, w) = = n σ , w i i w (5.1.15) i provided that the DM parcle is in moon. For resng targets, |w−vi| ≈ w, and we re-obtain eq. (4.1.23). At this point, we should point out a common misconcepon P −1 in the literature, where the mean free path in the Sun is taken to be ( i niσi) P −1 (or ( i nihσii) ), instead of eq. (5.1.15)[305, 455–459]. This can result in a significant overesmaon of the mean free path, which could crically alter the results as the mean free path enters the scaering probability in the exponent. This is especially severe in the case of electron scaering.

Whether a parcle gets captured or reflected by the Sun, depends not only on the total scaering rate, but also on the final speed of the scaering w → v. Starng from eq. (5.1.11), we can derive an expression for the differenal scaering rate to final speed v. For a derivaon, we refer to the app. of [452].

2 dΩ± 2 v dv X µ (w → v) = √ + σ n (r) dv π w µ i i i h −µκ2(v2−w2) i × χ(±β−, β+)e + χ(±α−, α+) . (5.1.16a)

Here, we took over Gould’s notaon,

m µ ± 1 µ ≡ χ , µ ≡ , m ± 2 (5.1.16b) √ i π χ(a, b) ≡ [erf (b) − erf (a)] , 2 (5.1.16c) α± ≡ κ(µ+v ± µ−w) , β± ≡ κ(µ−v ± µ+w) . (5.1.16d)

129 CHAPTER 5. SOLAR REFLECTION OF DARK MATTER

1 2 rr p dl

θ dr

Figure 5.2: Sketch of a DM trajectory crossing a solar shell of radius r twice.

The plus sign corresponds to acceleraon (v > w), whereas the minus sign applies to deceleraon (v < w).

Probability to reach a spherical shell The probability to travel a given path freely without interacon can be inferred from (4.1.16), Z Psurv = exp − dPscat , (5.1.17) path

where the scaering probability can be found in eq. (5.1.10). We focus on a path starng at radius rA travelling to a larger radius rB > rA. Using the symmetry of the free DM orbits, we can construct all relevant paths in terms of Z dl Psurv(rA, rB) = exp − Ω(r, w(u, r)) (5.1.18a) rA→rB w(u, r) Z rB dl Ω(r, w(u, r)) = exp − dr (5.1.18b) rA dr w(u, r)

dl Looking at the right hand side of ﬁgure 5.2, we can express dr in terms of J = rw sin θ,

dl J 2 −1/2 = 1 − . dr w2(u, r)r2 (5.1.19)

As menoned previously, a DM parcle can only pass a spherical shell of radius r if J < w(u, r)r. Then, the parcle will pass the shell up to two mes. This is illustrated on the le hand side of ﬁgure 5.2. The probability to reach the shell the second me without scaering is naturally lower than for the ﬁrst me. We use the orbit’s symmetry around its perihelion at distance rp such that both probabilies

130 5.1. DM SCATTERINGS IN THE SUN add up to

Pshell(r) = Θ (w(u, rp)rp − J) | {z } shell passing condion   2 × Psurv(r, R ) + Psurv(r, R )Psurv(rp, r)  (5.1.20a) | {z } | {z } 1st passing 2nd passing 2 = Psurv(r, R ) 1 + Psurv(rp, r) Θ(w(u, rp)rp − J) . (5.1.20b)

Since this is the sum of two probabilies 0 ≤ Pshell < 2. Now all three factors in our original eq. (5.1.4) are in place, and we can write down the result for the diﬀerenal scaering rate inside the Sun.

Final result for DM scaering rate The differenal rate of DM parcles falling into the Sun to scaer for the first me in a spherical shell of radius r to final speed v is obtained by substung eqs. (5.1.7), (5.1.10), (5.1.16), and (5.1.20) into eq. (5.1.4),

dS Z dP = dΓ scat P (r) dr dv dr dv shell (5.1.21a) ∞ w(u,r)2r2 Z Z f (u) dΩ J 2 −1/2 = πn du dJ 2 χ (w(u, r) → v) w(u, r)2 − χ u dv r2 0 0 2 × Psurv(r, R ) 1 + Psurv(rp, r) (5.1.21b)

This equaon captures the ﬁrst scaerings regardless of the cross secon. It smoothly connects the opaque and transparent regime. In the laer, Psurv ≈ 1, and the in- 2 tegral over J can be evaluated analycally,

∞ dS Z f (u) dΩ ≈ 4πr2n du χ w(u, r) (w(u, r) → v) , dr dv χ u dv (5.1.22) 0 which reminds of Gould’s capture equaon (2.8) of [451]. In the opaque regime, we ﬁnd Ω(w) dl P (r , r) ≈ 0 , P (r, R ) ≈ δ(r − R ) , surv p and surv w dr (5.1.23) where the le hand side of the second equaon can be understood as the probability density in terms of r. In other words, in the extreme opaque regime, the

131 CHAPTER 5. SOLAR REFLECTION OF DARK MATTER

parcle scaers immediately upon arriving at the solar surface. This can be seen from the total scaering rate, which can be evaluated to be

2 2 −1 S ≈ πR hui + vesc(R ) hu i . (5.1.24)

Comparing to. (5.1.9), this is nothing but the total rate of parcles entering the Sun. Next, we have to quanfy how many of the scaered parcles get captured and how many manage to escape the star’s interior without scaering a second me.

5.1.2 Solar reﬂecon and capture

A DM parcle which scaers on a solar nucleus at radius r is kicked into a new orbit 0 0 with perihelion distance rp, angular momentum J , and speed v. If it loses kinec mχ 2 energy of at least 2 u , the ﬁnal speed falls below the local escape velocity, and the parcle gets captured. Otherwise, it gets reﬂected, if it survives back to the solar surface without re-scaering. The probability to get captured is simply

Pcapt(v, r) = Θ(vesc(r) − v) . (5.1.25)

On the other side, the chance of geng reﬂected is

Pleave(v, r) = Θ(v − vesc(r)) | {z } escape condion   1 × P (r, R ) + P (r, R )P (r0 , r)2 .  surv surv surv p  (5.1.26) 2 | {z } | {z } short path out long path out

This is an expression similar to Pshell of eq. (5.1.20). But here, we average over the two possible direcons a DM parcle could take to leave the star. These two probabilies can be used to filter out the captured or reflected parcles out of the scaering rate. The differenal capture and reflecon rates are dC dS = P (v, r) , dv dr dv dr capt (5.1.27) dR dS = hP (v, r)i 02 . dv dr dv dr leave J (5.1.28)

In the last line, we average over final angular momenta, again assuming isotropic scaerings. To get an idea about the parcles geng reflected or captured, we dS plot hPleave(v, r)iJ02 and overlay contour lines of dr dv in figure 5.3. It illustrates

132 5.1. DM SCATTERINGS IN THE SUN

Figure 5.3: Average probability hPleave(v, r)iJ02 to escape the Sun without re-scaering for mχ = 100 MeV and increasing cross secons. beaufully how an increase in the scaering cross secon shis the locaon of the first scaering towards the solar surface and renders the core more and more opaque. As the outer shells are cooler, the DM parcle’s final velocity decreases correspondingly, and the scaering rate peaks at lower values for v. About half of the parcles have a good chance of leaving the Sun without re-scaering, the other half either gets captured or scaers again. Strictly speaking, there is no single scaering regime, even for very weak interacon strengths, as half the DM parcles get captured, enter a bound orbit and will eventually scaer a second me. By integrang over the Sun’s volume and ‘red-shiing’ the reflected parcles’ speeds, we finally obtain the spectrum of reflected DM parcles on Earth, where they might pass through a detector,

Z R dR dR dv = dr √ . (5.1.29) du dv dr du 2 2 0 v= u +vesc(r)

Following from that expression, the differenal parcle flux is dΦ 1 dR R = , du 4π`2 du (5.1.30) where ` = 1 A.U. is the distance between the Sun and the Earth. This establishes a new populaon of potenally fast DM parcles in the solar system. Their flux through Earth is compared to the standard halo flux in figure 5.4. For speeds below the halo distribuon’s cutoff, the reflecon flux is naturally suppressed. However, the differenal reflecon flux has no speed cutoff and extends to much higher energies than present in the DM halo, provided that the DM mass is similar or lighter than the targets. By increasing the cross secon, scaering on solar targets becomes more and more likely. The flux of slow reflected parcles therefore increases as well. However, the hot core also gets shielded off more in this case, and very

133 CHAPTER 5. SOLAR REFLECTION OF DARK MATTER

1010 halo 8 SI ] 10 σp =0.1pb - 1 )

s SI / σp =1pb 106 km ( σSI=10pb s p 2 1

m 4 SI [ 10 σp =100pb ℛ du d Φ 102

DM flux 100

10-2

0 250 500 750 1000 1250 1500 1750 2000 speed u [km/s]

Figure 5.4: Differenal DM parcle flux for the halo and solar reflecon for DM masses of 10 GeV (solid), 1 GeV (dashed), and 100 MeV (doed).

fast DM parcles are less likely to escape, which explains the ﬂux’s decrease with increased cross secon for high energies. In all our computaon we include the four largest targets in terms of niσi, namely hydrogen 1H, helium 4He, oxygen 16O, and iron 56Fe. Smaller targets may be neglected. They might slightly increase the scaering rates, but also marginally shield oﬀ the hot core.

5.2 Direct Detecon of Reﬂected DM

The event rate in a direct detecon experiment is proporonal to the DM flux through the detector, as covered in chapter 3.5. Solar reflecon is the source of a new flux component, which can easily be incorporated into eq. (3.5.2),   ∞ dR 1 Z  ρ 1 dR  dσ = du  χ uf (u) +  N .  ⊕ 2  (5.2.1) dER mN mχ 4π` du  dER umin(ER) | {z } | {z } halo DM reflected DM

We present results for a detector of the CRESST-III type [237, 460, 461]. For these projecons we assume that phase 2 will have an exposure of 1 ton day and a recoil threshold of 100eV. The app. includes a summary of the experimental setup, see chapter C.1.2. The constraints (90% CL) for solar reﬂected DM are shown in ﬁg- ure 5.5 as red shaded regions and compared to the corresponding constraints from halo DM.

134 5.2. DIRECT DETECTION OF REFLECTED DM

10-32

10-34 ] 2 cm [ p σ

10-36 cross section

10-38

10-40 0.10 0.15 0.20 0.25 0.30 0.35 0.40

mass mχ[GeV]

Figure 5.5: The red (yellow) contours are projected constraints for a CRESST-III type (idealized sapphire) detector with exposures of 1, 10, and 100 ton days (10, 100, and 1000 kg days) in solid, dashed, and doed lines respecvely.

At this point, the reflecon constraints are sub-dominant, but they differ from the halo constraints in one crucial quality. For increased exposure, lower and lower masses are being probed, whereas the halo constraints never reach below the minimum mass given by eq. (3.5.6). For a detector of the CRESST-III type with exposures larger by a factor of 10 to 100 and no addional background, parameter space below the naive minimum mass becomes accessible. Above a certain exposure, the minimal probed DM mass starts to decrease for larger exposures. These constraints are widely insensive to the halo model and its velocity distribuon, as the gravitaonal acceleraon and subsequent scaering erase most of the inial distribuon’s informaon. The sensivity to reflected DM increases significantly for lower recoil thresholds. Then, solar reflecon could be used to extend the halo constraints even for smaller exposures. Figure 5.5 also contains projecons for an idealized detector made of sapphire with perfect resoluon, no background, and a threshold of 20 eV. A similar detector with such a low threshold was already realized by the CRESST collaboraon [295]. For exposures above around 10 kg days, the solar reflecon flux would already be the source of observable numbers of events. There are mulple ways to disnguish reflected sub-GeV DM from standard

135 CHAPTER 5. SOLAR REFLECTION OF DARK MATTER

halo DM of heavier mass.

1. The recoil spectrum would have a non-Maxwellian tail.

2. Direconal detectors would show the Sun as the signal’s source.

3. If scaerings in the Sun are common, they would also occur in the Earth. For a given cross secon, the Sun and the Earth are similarly opaque. In chapter 4.2, we already discussed in detail that this would result in a diurnal modulaon of the signal.

4. Due to the eccentricity of the Earth’s orbit and the associated variaon of the distance to the Sun, the reﬂecon ﬂux, and therefore the signal rate, will have a ∼ 7% annual modulaon peaking at the perihelion around January 3rd as opposed to the standard annual modulaon, which peaks in Summer.

The central result of this chapter is a demonstraon of the existence of an addional, highly energec DM populaon in the solar system and the possibility to detect those parcles in standard direct detecon experiments of low threshold. For their first results of the CRESST-III experiment, a threshold below 100eV was already achieved [237]. There are no addional assumpons underlying this result. For low thresholds and exposures above a crical value, these parcles set constraints on sub-GeV masses, where ordinary DM cannot, regardless of exposure. Furthermore, their spectrum is largely independent of the halo model. As a side product, we extended Gould’s expression for the DM capture rate in eq. (5.1.27) to account for the Sun’s temperature and opacity. The results of this chapter are conservave, as the computed flux contains single scaering reflecon only. Nonetheless, due to the correct treatment of the Sun’s opacity, the equaons apply to all masses and cross secons and smoothly connect the transparent and opaque limits. UsingMC simulaons will improve the results by accounng for mulple scaerings in the Sun, which increases the reflecon flux.

136 5.3. MC SIMULATIONS OF SOLAR REFLECTION 5.3 MC Simulaons of Solar Reﬂecon

The results of the previous chapter should be considered as conservave, as they only include the first scaering between DM and solar nuclei. Solar reflecon from the second, third or 100th scaering is not captured by our formulae, and the resulng DM flux is a conservave underesmate. In order to improve the result and include the effect of mulple scaerings inside the Sun,MC simulaons of DM trajectories inside the Sun are the appropriate tool. The simulaons are similar to what we presented in chapter 4.2. However, there are a few major adjustments to be implemented.

1. It is no longer acceptable to assume straight paths for the DM parcles trajectories in between scaerings. The eﬀect of the Sun’s gravitaonal potenal is crucial, both for the shape of the DM orbits an the speeds. This will involve the soluon of the equaons of moon with numerical methods.

2. As discussed in chapter 5.1, the desired eﬀect is an acceleraon of incoming DM parcles due to collisions on hot nuclei. The thermal moon of the targets can not be neglected.

3. Due to gravitaonal focusing and acceleraon, the generaon of inial condions in chapter 4.2.1 has to be improved.

4. Since DM parcles can be accelerated, it is no longer sensible to include a minimum speed cutoﬀ vmin into the algorithm, as done in the simulaon algorithm of ﬁgure 4.5.

This type ofMC simulaons to describe DM trajectories inside the Sun have been performed already in the ’80s[462] to verify Gould’s analyc formalism and more recently to study solar reflecon of DM via electron scaerings [305]. Therein, the effect of solar nuclei has been neglected. Even if the nuclei do not contribute to the reflecon of accelerated DM parcle, they also decrease the DM flux by shielding off the hot solar core, unless the considered model does not allow interacons with quarks.

5.3.1 Simulang DM orbits and scaerings inside the Sun

Equaons of moon Pung aside scaerings for the moment, the DM parcles’ trajectories are standard Keplerian orbits with the modiﬁcaon that they may penetrate the Sun’s interior. The parcles’ moon happens in a plane perpendicular to

137 CHAPTER 5. SOLAR REFLECTION OF DARK MATTER

the conserved angular momentum J. In terms of polar coordinates (r, φ) of this plane, the moon is described by the following Lagrangian,

∞ 1 Z G m M(r0) L = m r˙2 + r2φ˙2 + dr0 N χ . 2 χ r02 (5.3.1) r

The mass-radius relaon M(r) is part of the Standard Solar Model, see app. B.5.2. The corresponding Euler-Lagrange equaons are G M(r) r¨ − rφ˙2 + N = 0 , r2 (5.3.2a) 2 ˙ r φ ≡ J = const . (5.3.2b)

Using the conservaon of angular momentum, we will solve the equaons of moon in two dimensions and translate the soluons back to 3D. Given a point in conﬁguraon space (t, r, v), we can switch to the coordinate system spanned by r r × v xˆ = , zˆ = , yˆ = zˆ × xˆ , |r| |r × v| (5.3.3)

such that the polar coordinates in the orbital plane are

r = |r| , φ = 0 , and J = |r × v| . (5.3.4)

As before, theˆ· denotes vectors of unit length or the normalized version of a vector, ˆ V i.e. V ≡ |V| . Aer solving the equaons of moon up to a new locaon given by 0 0 (r , φ ) we can return to our original coordinate system,

0 0 0 0 r = r (cos φ xˆ + sin φ yˆ) , (5.3.5a) 0 0 0 ˙0 0 0 0 0 0 ˙0 0 v = r˙ cos φ − φ r sin φ xˆ + r˙ sin φ + r φ cos φ yˆ , (5.3.5b)

˙0 J with φ = r02 . Inside the Sun, the orbits are no longer Keplerian and we need to solve the equaons of moon numerically. We can write the Euler-Lagrange equaons (5.3.2) as the set of ﬁrst order ordinary diﬀerenal equaons, G M(r) J r˙ = v , v˙ = rφ˙2 − N , φ˙ = , r2 r2 (5.3.6)

and solve them numerically with the Runge-Kua-Fehlberg (RK45) method [463], which we introduce in app. D.4.

138 5.3. MC SIMULATIONS OF SOLAR REFLECTION

Scaerings on solar nuclei The probability for a DM parcle to scaer on a nucleus while travelling a distance L inside the Sun is given by eq. (5.1.10), which allowed us to deﬁne the speed dependent local mean free path λ(r, w) in eq. (5.1.15).

Hence, the CDF of the freely travelled distance l inside the Sun is idencal to the corresponding expression for the Earth in eq. (4.1.22), but in terms of the adjusted mean free path,

 L  Z dx P (L) = 1 − exp −  λ(r, w) (5.3.7a) 0  L  Z X h|w − vi|i = 1 − exp − dx n (r)σ .  i i w  (5.3.7b) 0 i

To ﬁnd the locaon of the next scaering, we use inverse transform sampling and 0 solve P (L) = ξ ∈ (0, 1) along an orbit,

L Z dx − log(ξ) = ξ ≡ 1 − ξ0 . λ(r, w) with (5.3.8) 0

While solving the equaon of moon (5.3.6) step by step with the RK45 method, we add up the step’s contribuon to the right hand side of eq. (5.3.8) unl it exceeds − log(ξ). In other words, we connue the trajectory unl

r 2 X ∆li X J = r˙2 + ∆t λ−1(r ) ≥ − log(ξ) . λ(r ) r2 i (5.3.9) i i i

Then the parcle scaers at the current posion. The identy of the target nucleus is sampled in the same way as before, using eqs. (4.1.35) and (4.1.36), with the adjustment to use the solar mean free path. The ﬁnal velocity of the DM parcle aer the elasc collision can be found in eq. (3.3.1a). The crucial diﬀerence to scatterings on terrestrial nuclei is the fact that the solar targets are hot. Their velocity follows an isotropic Maxwell-Boltzmann distribuon, given by eq. (5.1.3), with the temperature depending on the underground depth. This means that we not only have to sample the target’s identy and the scaering angle, but also the target’s velocity for each scaering. As the CDF of the Maxwell-Boltzmann distribuon cannot simply be inverted, we can either sample the target speed via inverse transform sampling inverng the CDF numerically or use rejecon sampling. The direcon of the target’s velocity can be chosen isotropically.

139 CHAPTER 5. SOLAR REFLECTION OF DARK MATTER

With these adjustments, we can extend the previous simulaon code to propagate DM parcles in the Sun, as they scaer, gain or lose energy, get captured and ﬁnally leave the Sun again as a reﬂected parcles aer having scaered any number of mes. The next step concerns the inial condions and how to sample them for theMC simulaons.

5.3.2 Inial condions

The objecve of this secon is to

(a) generate inial condions (t0, r0, v0) far away from the Sun, such that the parcles are guaranteed to hit the Sun and are eﬀecvely distributed homogeneously in space.

(b) to propagate the parcle to a radius r & R using the analyc soluon of the Kepler problem such that the DM parcle ends up in close proximity to the Sun with (t1, r1, v1) (with |r1| = Ri & R ), which are the inial condions for the numerical procedure.

Outside the Sun, the incoming, unbound DM parcles have posive total energy and follow hyperbolic Kepler orbit.

Hyperbolic Kepler orbits Starng at (t0, r0, v0), we want to compute (t1, r1, v1) without passing by the periapsis. The orbit is characterized and fully determined by the following parameters,

2 2 2 u = v0 − vesc(r0) (asymptoc speed) , (5.3.10a) G M a = − N , u2 (semimajor axis) (5.3.10b) J 2 p = (semilatus rectum) , (5.3.10c) GN M r p e = 1 − > 1 , a (eccentricity) (5.3.10d) q = a(1 − e) (periapsis) , (5.3.10e) 1 p cos θ = − 1 , e r (angle from periapsis) (5.3.10f) G M v2 = N 1 + e2 + 2e cos θ , p (speed) (5.3.10g)

140 5.3. MC SIMULATIONS OF SOLAR REFLECTION

e sin θ tan φ = 1 + e cos θ (angle of velocity to the perpendicular of the radial direcon) , (5.3.10h) e + cos θ cosh F = , 1 + e cos θ (eccentric anomaly) (5.3.10i) M = e sinh F − F (mean anomaly) , (5.3.10j) s (−a)3 t − tp = M (me from periapsis) . (5.3.10k) GN M

The second crucial ingredient is the orientaon of the axes in three dimensions, such that θ = 0 corresponds to the orbit’s periapsis.

r0 × v0 zˆ = , xˆ = cos θ0 ˆr0 + sin θ0 ˆr0 × zˆ, yˆ = zˆ × xˆ (5.3.11) |r0 × v0|

Using eqs. (5.3.10), the new posion and velocity can be expressed as s (−a)3 t1 = t0 + sign(r1 − r0) (M1 − M0) , (5.3.12a) GN M

r1 = r1 (cos θ1 xˆ + sin θ1 yˆ) , (5.3.12b) e sin θ ˆr + (1 + e cos θ ) zˆ × ˆr v = v 1√1 1 1 , 1 1 2 1 + e + 2e cos θ1 s G M = N (e sin θ ˆr + (1 + e cos θ ) zˆ × ˆr ) , p 1 1 1 1 (5.3.12c) with

1 p θ1 = sign(r1 − r0) arccos − 1 . (5.3.12d) e r1

Inial posion Let us assume, we sampled a inial velocity v0 and want to sample an inial posion r0, such that the parcle is guaranteed to penetrate the Sun’s surface. The queson, whether or not a given parcle hits the Sun, is equivalent to the queson if the orbit’s periapsis is smaller than the solar radius or if the angular momentum fulﬁls

J = |r0 × v0| = r0v0 sin α < w(u, R )R , (5.3.13)

141 CHAPTER 5. SOLAR REFLECTION OF DARK MATTER

x v 0 0 h

Rdisk

x1 d≫R☉ v1 α

R☉ Ri≳R☉

Figure 5.6: Sketch of the inial condions for DM parcles entering the Sun (not to scale).

p 2 2 where w(u, r) = u + vesc(r). Similar to the situaon in chapter 4.2.1, all incoming parcles on collision course will pass through a disk of radius Rdisk(u) > R 2 vesc(r0) at distance d R (such that u2 1), oriented perpendicular to u. Due to gravitaonal focussing the disk’s radius is larger than R and depends on u. It can sin α = h h be derived from eq. (5.3.13) using r0 . The distance to the disk’s center must fulﬁll r v2 (R ) v2 (r ) h < R (u) = 1 + esc R + O esc 0 . disk u2 u2 (5.3.14)

This is illustrated in ﬁgure 5.6. Choosing a random inial posion on this disk will correspond to the subset of homogeneously distributed parcles on collision course with the Sun, u x(t ) = dzˆ + pξ R (cos ϕ xˆ + sin ϕ yˆ) , zˆ ≡ − . 0 disk with u (5.3.15)

Here, xˆ and yˆ span the disk, while ξ and ϕ are samples of the random variables U[0,1] and U[0,2π] respecvely. To not waste me by solving the equaons of moon numerically for the infalling parcle outside the Sun, we perform an analyc Kepler shi to a radius R & R with eqs. (5.3.12). This leaves us with the ﬁnal recipe to sample inial condions of DM trajectory simulaons inside the Sun.

1. Sample a velocity v0 = u far away from the Sun from the halo distribuon fχ(u), given in eq. (3.2.4). Since we simulate the parcles in the Sun’s rest

142 5.3. MC SIMULATIONS OF SOLAR REFLECTION

frame, we replace v⊕ with the solar velocity v given in eq. (B.4.1).

2. Sample the inial posion via eq. (5.3.15).

3. Propagate the parcle analycally on its hyperbolic Kepler orbit using eqs. (5.3.12) to a locaon close to the Sun.

Finally, the resulng (t1, r1, v1) are the inial condions for the numerical RK45 method.

5.3.3 Data collecon

The simulaon of a low-mass DM parcle connues unl it leaves the Sun. As it propagates through the star’s interior and scaers on the nuclei, it loses and gains energy and might get temporarily captured. Eventually, it will gain enough energy to escape the Sun’s gravitaonal potenal. If it reaches the Sun’s surface without re-scaering, it counts as reflected. We analycally propagate the parcle to the Earth’s orbital radius with the eqs. (5.3.12) and save the final DM velocity as a data point. This is repeated unl the data sample is sufficiently large.

Anisotropy and isoreflecon rings The DM velocity distribuon in the boosted reference frame of the Sun is no longer isotropic. The anisotropy of the DM wind might leave a trace in the reflected DM flux. The detecon signal due to these parcles would consequently show a modulang behavior as the Earth orbits the Sun. This way, solar reflecon could be the source of a novel annual modulaon. If we want to study this effect withMC simulaons, we have to track the direconality of the reflected parcles. Again we can use the residual symmetry of the problem. The system sll has a rotaonal symmetry around v . This is in principle no different from the situaon in chapter 4.2.3, where we also exploited this symmetry to define isodetecon rings. Again, we use the polar angle Θ of the symmetry axis to define finite-sized rings of symmetry. For the data collecon with the terrestrial simulaons, we fixed the isodetecon rings’ sizes by a constant angle ∆Θ following [387, 394]. The area of these rings differed considerably, geng smaller towards the ‘poles’ following eq. (4.2.5). A good angular resoluon was necessary to accurately describe diurnal modulaon for any terrestrial detector on the globe. Unfortunately, the data collecon for the smaller rings takes much more me, as the probability for a parcle to hit a small area is of course low. The varying area of the isodetecon rings was an unavoidable

143 CHAPTER 5. SOLAR REFLECTION OF DARK MATTER

Figure 5.7: Comparing the equal-angle isodetecon rings for the Earth simulaons and the equal-area isoreﬂecon rings for the Sun simulaons.

boleneck of the Earth simulaons. For the Sun however, we do not require the same angular resoluon and define the isoreflecon rings to be of equal area. We call these equal-area rings isoreflecon rings. For N rings, the isoreflecon rings are defined via 2 Θ = arccos cos Θ − , θ = 0 i ∈ {1, ..., N} . i i−1 N with 0 and (5.3.16)

A comparison between the equal-angle isodetecon rings and the equal-area isore- ﬂecon rings is shown in ﬁgure 5.7. The loss of angular resoluon is most severe at the poles. However, the Earth never passes through these regions, and this is not a problem.

5.3.4 First results and outlook

In this chapter of the thesis, we will present some preliminary results for theMC treatment of solar reﬂecon. The simulaon of DM trajectory in the Sun can shed light on the impact of mulple scaerings. The analyc formalism presented in chapter 5.1 provides conservave results and only accounts for single scaering reﬂec- on. Similarly to the Earth simulaons in chapter 4.2.5, these analyc results can be used to test and verify the independent approaches.

Comparison of analyc and MC result The number of isoreﬂecon rings N is ad- justable in the simulaon code. In order to compare to the analyc results, which assumed isotropy, we can simply set it to one. Naturally, we include the same number of solar targets in the simulaon. For the comparison, we are only interested

144 5.3. MC SIMULATIONS OF SOLAR REFLECTION

1.0

σn=0.1 pb σn=1 pb σn=10 pb σn=100 pb 0.8

☉ 0.6 R /

0.4 radius r

0.2

0.0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 final speed v [km/s] final speed v [km/s] final speed v [km/s] final speed v [km/s]

Figure 5.8: Histogram of the radius and final DM speed of the first scaering in the dS Sun. The black contours trace the analyc scaering rate dv dr . in the first scaering, as this is the one described by the analyc equaons. For each simulated trajectory, we save the radius and the final speed and plot the data in a two-dimensional histogram. The differenal first scaering rate in a shell of radius r to a speed v was computed previously in eq. (5.1.21) and ploed as black contours in figure 5.3. In figure 5.8, we compare theMC histograms with the same contours. We can see that the simulaons yield the same locaon of the first scaering and the same final DM speed, which is a first indicator for the consistency of the two results.

Mulple scaerings There is no reason why a DM parcle cannot be reflected with high energy aer many collisions. In fact, since about half of the DM parcles are gravitaonally captured by their first scaering, they are bound to scaer at least one more me. As discussed earlier, this statement is independent of the assumed cross secon, and a ‘single scaering regime’ does not really exist. We consider two example trajectories for a DM parcle of 1 MeV and a DM- proton cross secon of 1 pb. The evoluon of their total energy Eχ(t) = T + V and radial coordinate r(t) are ploed in figure 5.9 as solid and dashed lines respecvely. The red example describes a parcle geng captured by the first collision. Even though the third scaering accelerate the DM parcle sufficiently to potenally escape, rendering its total energy posive, the parcle does not reach the solar surface before the next scaering. Only aer the seventh scaering does the parcle finally leave the star with an energy increased by a factor of around 7. The yellow example shows another parcle geng captured to a bound orbit parally outside the Sun. As it re-enters the hot, dense core it scaers a few more mes and gets eventually reflected with more than twelve mes its original energy aer

145 CHAPTER 5. SOLAR REFLECTION OF DARK MATTER

14 4 12 10 8 3

( 0 ) χ 6 E / ☉ χ

4 R / 2 2 0 radius r -2 DM energy E -4 1 -6 -8 -10 0 0 5000 10 000 15 000 20 000 25 000 30 000 35 000 time [s]

Figure 5.9: Time evoluon of the total DM energy (solid lines) and the radial coordinate (dashed lines) for two example trajectories.

a total of 13 scaerings. By counng the reflected DM parcles passing the isoreflecon rings and saving their speed, we can determine the differenal reflecon flux. TheMC result has to be re-scaled to yield the correct total reflecon flux,

dR nrefl ˆ = Γ (mχ) × Φ(u) . (5.3.17) du nsim |{z} | {z } normalized flux distribuon total reflecon flux

Here, we use eq.(5.1.9), the rate of halo parcles entering the Sun, which can be explicitly evaluated via

2 2 −1 Γ (mχ) = πR nχ hui + vesc(R ) hu i (5.3.18a) ρχ v0 2 = πR mχ Nesc " 2 2 1 v vesc(R ) v v0 v × √ exp − 2 + + + erf π v0 v0v v0 2v v0 # 2 v (R )2 v2 v2 v2 √ esc gal gal − 1 + 2 + 2 + 2 exp − 2 (5.3.18b) π v0 v0 3v0 v0

The galacc escape velocity is denoted as vgal, not vesc, to disnguish it from the solar escape velocity vesc(r). This expression can be used in connecon with the rao of the total number of simulated parcles nsim and the number of reflected parcles nrefl to scale up theMC numbers to the realisc values. Ulmately, we will ˆ esmate the normalized flux distribuon φ(u) using KDE (see app. D.3). However,

146 5.3. MC SIMULATIONS OF SOLAR REFLECTION ]

1 12 mχ=200 MeV, σp=1 pb - ) s / km

( 10 1 - s 26

10 8 [ ℛ du d 6

2 reflection spectrum

0 0 500 1000 1500 2000 2500 DM speed u [km/sec]

Figure 5.10: Contribuon of mulple scaerings to the solar reflecon spectrum. we will start by using histograms, as this enables to study and visualize the effect of mulple scaerings. In figure 5.10, we show the solar reflecon spectrum for mχ = 200 MeV and σp = 1 pb, using a histogram esmate. The bar’s alternang colors indicate the contribuon of reflecon by one scaering, two scaerings, three scaerings, etc. The black line corresponds to our analyc result of eq. (5.1.29) and again shows good agreement with the single scaering fracon. It is obvious that, while single scattering reflecon makes up a large fracon of the flux especially for lower speeds, the inclusion of mulple scaerings viaMC simulaons increases the reflecon flux significantly. Especially the high energy tail, which is essenal for DM detecon, is greatly enhanced. Whereas the individual contribuons of each number of scaerings above 2 is negligible, their fracons due to large number of collisions accumu- late to a sizeable flux amplificaon.

Outlook The results of the Sun simulaons presented in this secon are preliminary. The simulaons have not yet been applied to derive direct detecon constraints on sub-GeV DM. An efficient procedure to scan the two-dimensional mass- cross secon parameter space will be necessary, as each parameter point requires its own simulaon run. Once this is set up, we can expect the resulng constraints to far exceed the analyc results in figure 5.5. The final hope is to find cases, where detecon runs can employ solar reflecon to extend their sensivity. The detecon of DM accelerated by the Sun by mulple scaerings should be studied for nuclear recoil experiments, as well as DM-electron scaering experiments. Solar electrons would need to be considered as an addional target. Elec-

147 CHAPTER 5. SOLAR REFLECTION OF DARK MATTER

trons were already considered in the context of leptophilic DM, where the Sun’s nuclear targets may be neglected [305]. If we want to include both target species, a model-dependent relaon between the DM-nucleus and DM-electron cross secons would have to be assumed. Furthermore, the annually varying Sun-Earth distance would leave a modulaon signature in the detecon signal. This modulaon could potenally be modified, if the anisotropy of the inial condions due to the DM wind leaves a trace in the reflecon spectrum.MC simulaons would allow to probe the assumpon of isotropy. With the definion of the isoreflecon rings, we will be able to address this queson. The mechanism of solar reflecon of DM has so far only been invesgated for contact interacons, eitherSI DM-nuclear, or DM-electron collisions. The inclusion of more general interacons is one of the straight-forward extensions. It would be interesng to consider e.g. spin-dependent interacons or the general NREFT operators [354, 355], the relevant form factors for the solar targets have been computed in the context of DM capture [464]. Finally, implemenng interacons mediated by ultralight fields into the simulaons could be a promising direcon.

148 Chapter 6

Conclusions and Outlook

Our enre knowledge about DM rests upon its gravitaonal interplay with baryonic maer. The presence of large amounts of unseen maer manifests itself in astronomical observaons starng in the 1930s. Back then, Fritz Zwicky found large discrepancies between the gravitaonal and the luminous mass of a galaxy cluster. A few years later, Horace Babcock was the ﬁrst to point out a peculiar ﬂaening of a galacc rotaon curve. Furthermore, DM revealed itself by bending the light of background sources and by dominang the formaon of cosmological structures. Another crucial piece of evidence is the observaon of the anisotropies of the CMB. Here, DM le a unambiguous trace in the acousc peaks of the power spectrum during the early Universe. The history and evidence of DM was the main topic of chapter2. The minimal explanaon for these measurements would be a new form of matter interacng with the parcles of theSM exclusively via gravity. Gravitaonal interacons are unavoidable due to gravity’s universality, and this could indeed be the only portal coupling to visible maer. As such, DM would reside in a secluded sector, and it would hardly be possible to unveil its origin and properes in the fore- seeable future. Nonetheless, there is the well-movated hope that this is not the path that nature chose for our Universe. The ulmate objecve of all direct detecon experiments is to search for an addional interacons between the bright and dark sectors of maer by measuring the aermath of a DM collision in a detector. Such experiments were the focus of this thesis, and their fundamental principles were covered in chapter3. Provided that parcles of the DM halo can collide with ordinary parcles, these collisions are of course not restricted to take place inside a detector’s target. For high enough scaering probabilies, it is therefore illegimate to treat the detector

149 CHAPTER 6. CONCLUSIONS AND OUTLOOK

as an isolated object situated in the galacc halo. The surrounding medium of the Earth needs to be taken into account, since the incoming DM parcles have to pass through it to reach the detector and might scaer beforehand. The central subject of this thesis was to describe underground scaerings both in the Earth and the Sun and invesgate the implicaons for direct detecon experiments searching for low-mass, mostly sub-GeV, DM parcles. With the excepon of chapter 5.1, the results of this thesis were obtained by the use ofMC simulaons of individual DM parcle trajectories. By tracking large numbers of parcles, their stascal properes can be used to quanfy the eﬀect of mulple underground scaerings on direct DM searches. The simulaon algorithms were formulated from the ground up and implemented fully parallelized in C++. The resulng scienﬁc codes underwent extensive tesng and were released as open source code to the community together with the respecve publicaons. Most of the results of this thesis were obtained by high-performance computaons on the ABACUS 2.0 of the DeIC Naonal HPC Center, a supercomputer with 584 nodes of 24 cores each.

DM in the Earth Concerning underground scaerings inside the Earth, we started off by simulang DM parcles on their path through our planet’s core and mantle. These simulaons are relevant for DM with masses below mχ / O(500)MeV and SI proton scaering cross secon around σp ∼ O(1-100)pb. For such light DM, the constraints from direct detecon rapidly grow weaker as the nuclear recoils fall below the experimental thresholds. The results can also apply to the possibility that the strongly interacng DM makes up a subdominant component of the total amount of DM in the Universe. In this region of the parameter space, the probability of scaering before entering the detector is significant and must be taken into account. They distort the expected nuclear recoil spectrum, potenally increasing or decreasing the expected signal rate compared to unscaered halo parcles. We modelled the Earth in theMC simulaons on the basis of the PREM and improved the generaon of inial condions from similar works of the past. The resulng DaMaSCUS code returns a data-based esmate of the underground DM density and velocity distribuon for any locaon and me. For a specified direct detecon experiment, we analyze the generatedMC data and determine the me evoluon of the recoil spectrum and signal rates throughout a sidereal day, quanfying both the phase and amplitude of the diurnal modulaon.

150 For the single scaering regime, we showed that theMC results are in excellent agreement with previous analyc results. Therefore the simulaon code passed a crucial consistency check, and we connued to study larger cross secons and the impact of mulple scaerings. We computed the diurnal modulaon for a detector of CRESST-II type and a number of benchmark points. For the largest cross secon, we found a sizeable modulaon of almost 100% to be expected for an experiment in the southern hemisphere. The simulaons could be extended by the consideraons of more general interacons, such as long range forces or higher-order operators of the NREFT framework. On the side of the data analysis, it could be interesng to determine the modulaon of DM-electron scaering experiments due to underground DM-nucleus scaerings.

In the second part of chapter4, we systemacally determined how far the various direct detecon constraints extend towards stronger interacons. Due to scatterings in the overburden of the experiments, which are located deep underground in most cases, the Earth crust and atmosphere can aenuate the ﬂux of strongly interacng DM. Ulmately, the experiment is not able to probe cross secons above a crical value. We usedMC simulaons to calculate this crical cross secon for a variety of DM-nucleus and DM-electron recoil experiments. In the laer case, we shied our focus further to models with light mediators, which modify the scattering kinemacs substanally. These kind of simulaons are highly relevant for direct detecon experiments, as they reveal the limitaons of detectors’ sensivity and might point out open parameter space above the usual constraints. We demonstrated that direct detecon experiments need to be located in shallower sites, either on the surface or even at higher altudes, if it is supposed to probe strongly interacng DM. Since the relevant scaerings occur in the Earth’s crust and atmosphere above the laboratory, it was no longer necessary to simulate trajectories through the whole planet, and we could simplify the simulaon volume to parallel planar shielding layers. The ulmate goal of the simulaon is to determine the DM distribuon and density at a given underground depth. Here, the greatest challenge was the fact that close to the crical cross secon, only ny fracons of the incoming detectable DM ﬂux reach this depth. We solved this problem using rare-event simulaon techniques such asIS and GIS. The resulng DaMaSCUS-CRUST code is publicly available and can be extended to future experiments in a straight-forward fashion.

151 CHAPTER 6. CONCLUSIONS AND OUTLOOK

For nuclear recoil experiments, we found the that the CRESST 2017 surface run closes most open windows in parameter space, as shown in figure 4.16. The scenario of strongly interacng DM with spin-independent contact interacons is strictly constrained. The situaon is different in the presence of light mediators, which we studied for noble and semiconductor target detectors probing inelas- c DM-electron collisions. As shown in figure 4.22, the different bounds might leave some interesng parts of the parameter space unconstrained especially if the strongly interacng DM parcle are only a small component of the total DM. However, the situaon here is not clear at this point and requires further invesgaons. We presented some discovery prospects for balloon- and satellite-borne experiments targeng this region, which could become very relevant in the near future.

DM in the Sun The phenomenology of DM parcles scaering inside Sun was studied in the second main chapter of this thesis. Therein, we developed the new idea of solar reflecon, when DM parcles fall into the Sun, gain kinec energy via elasc scaerings on hot solar constuents and leave the Sun with speeds far above the maximum of the normal DM halo populaon. The central consequence of solar reflecon is the existence of an addional DM populaon in the solar system, whose spectrum is widely insensive to the choice of the halo model. By extending Gould’s analyc framework, we established the possibility that the resulng flux of reflected DM parcles can be used to set constraints in the low-mass parameter space, where the standard halo DM can not regardless of the experiment’s exposure. Low-threshold detectors could extend their sensivity to DM masses below their naive minimum. In contrast to standard halo DM constraints, the minimum testable mass due to solar reflecon depends on the experiment’s exposure gets lower for larger exposures. The constraints presented in figure 5.5 are conservave, as they only include solar reflecon by a single scaering. This is why we started to set up aMC descripon of solar reflecon, which accounts for the contribuon of mulple scaerings inside the Sun, where parcles could also get temporarily captured gravitaonally before they finally get reflected. The ground work for such simulaons is presented in chapter 5.3, but there is sll work to be done, before the simulaons can be used to derive robust constraints. New experiments like CRESST-III [237] have realized recoil thresholds below 100 eV. Direct detecon might be able to probe solar reflecon in the near future.

152 In conclusion of this thesis, we demonstrated the power ofMC simulaons to quanfy the effect of mulple underground scaerings for direct searches of light DM. The simulaons of DM parcle trajectories in the Earth or Sun can be applied to answer various quesons, and we applied them to study diurnal modulaons of detecon rates, constraints on strongly interacng DM, and the new idea of solar reflecon of highly energec DM parcles. In these contexts,MC simulaons are an invaluable tool to quanfy or extend the sensivity of detecon experiments here on Earth. Future experiments will hopefully succeed to reveal a portal between the visible and dark maer sectors. For low-mass DM with significant scaering rates, the results and tools developed in this thesis should be of great value for the discovery of DM and further invesgaons of its nature.

153 CHAPTER 6. CONCLUSIONS AND OUTLOOK

154 Appendix A

Constants and Units

A.1 Physical Constants

The physical parameter, couplings, and masses required for the computaons of this thesis are listed in SI units in table A.1.

Quanty Symbol SI-Value [126] Physical constants −1 speed of light in vacuum c 299 792 458 m s −34 Planck constant h 6.626 070 040(81) × 10 J s −19 electric charge e 1.602 176 620 8(98) × 10 C −11 3 −1 −2 Newton constant GN 6.674 08(31) × 10 m kg s −23 −1 Boltzmann constant kB 1.380 648 52(79) × 10 JK −7 −2 vacuum permeability µ0 4π × 10 NA 2 2 2 weak mixing angle sin θW (MS) 0.231 22(4) (at q = MZ ) Masses −31 electron mass me 9.109 383 56(11) × 10 kg −27 proton mass mp 1.672 621 898(21) × 10 kg −27 atomic mass unit mn 1.660 539 040(20) × 10 kg Derived quanes h −34 Planck constant (reduced) ~ ≡ 2π 1.054 571 800(13) × 10 J s 1 −12 −1 0 ≡ 2 8.854 187 817 × 10 F m vacuum permivity µ0c α ≡ e2 q2 = 0 ﬁne structure constant 4π0 c 1/137.035 999 139(31) (at ) ~ 2 4π0~ −10 a ≡ 2 0.529 177 210 67(12) × 10 m Bohr radius B mee

Table A.1: Physical constants and masses in SI units

155 APPENDIX A. CONSTANTS AND UNITS A.2 Natural Units

Instead of SI units, so-called natural units are used in many ﬁelds of physics for reasons of convenience and convenon. The set of natural units most common in high energy physics and used throughout this thesis, are deﬁned by seng the speed of light in vacuum, the reduced Planck constant, and the Boltzmann constant to 1,

c = 1 , (A.2.1a) ~ = 1 , (A.2.1b) kB = 1 . (A.2.1c)

Our electromagnec units are based on the raonalized Lorentz-Heaviside units,

0 = 1 , (A.2.2a) µ0 = 1 , (A.2.2b) √ ⇒ e = 4πα = 0.302 822 12 . (A.2.2c)

Consequenally, all unit dimensions can be expressed in powers of energy, and we can express physical quanes in terms of a single unit. Throughout the computaons for this thesis we use GeV as default unit. The conversion factors between SI and natural units are found in table A.2.

Dimension Unit Conversion factor SI-Value −10 energy 1 GeV ×1 = 1.602 18 × 10 J −1 −16 length 1 GeV ×~c = 1.973 27 × 10 m −1 −25 me 1 GeV ×~ = 6.582 12 × 10 s 1 −27 mass 1 GeV × c2 = 1.782 66 × 10 kg 1 GeV × 1 = 1.160 45 × 1013 K temperature kB eSI −19 Electric charge 1 × e = 5.290 82 × 10 C

Table A.2: Conversion between SI and natural units

156 Appendix B

Astronomical requirements

Various astronomical details of a more technical nature are necessary for the diﬀer- ent computaons and simulaons, ranging from astronomical coordinate systems to models of the Earth and the Sun. We present a review of these basics in this app. both for completeness and to serve as a reference.

B.1 Constants

The relevant astronomical parameters are listed in table B.1.

Quanty Symbol SI-Value [126] Units of length 15 light year ly 9.4607 × 10 m 16 parsec pc 3.085 677 581 49 × 10 m astronomical unit AU 149 597 870 700 m Solar parameters 30 Mass M 1.988 48(9) × 10 kg 8 Radius R 6.957 × 10 m 7 Core temperature T (0) 1.549 × 10 K Terrestrial parameters 24 Mass M⊕ 5.9724(3) × 10 kg 6 Radius R⊕ 6.371 × 10 m

Table B.1: Astrophysical quanes

157 APPENDIX B. ASTRONOMICAL REQUIREMENTS B.2 Sidereal Time

For a precise predicon of the diurnal signal modulaon’s phase, the Earth’s rotaon needs to be tracked precisely. The sidereal me is the appropriate me scale as it is based on that very rotaon. One sidereal day is deﬁned as the duraon of one rotaon relave to vernal equinox Υ and is about four minutes shorter than a mean solar day (23.934 469 9 h)[465]. It is oen given as an angle instead of a me unit. Furthermore, the LAST is deﬁned as the me since the local meridian of some posion on Earth passed vernal equinox Υ and is the primary measure for the Earth’s rotaonal phase. In order to calculate the LAST for any given me and locaon on Earth, all mes will be given relave to 01.01.2000 12:00 Terrestrial Time (TT), or short ‘J2000.0’, a commonly used reference me. The fraconal number of days relave to J2000.0 for a given date D.M.Y and me h : m : s (UT) is found by the following relaon [466],

˜ ˜ nJ2000 = b365.25Y c + b30.61(M + 1)c + D h m s + + + − 730563.5 , 24 24 × 60 24 × 602 (B.2.1a)

where  Y − 1 M = 1 2 , ˜  if or Y ≡ (B.2.1b) Y if M > 2 ,

and  M + 12 M = 1 2 , ˜  if or M ≡ (B.2.1c) M if M > 2 ,

and b·c is the ﬂoor funcon. As an example, the me of 31.01.2019 at 23:59UT epoch corresponds to nJ2000 = 6970.5. It will be useful to deﬁne the as n T ≡ J2000 . J2000 36525 (B.2.2)

The ﬁrst step towards the LAST is the determinaon of the Greenwich Mean Side- real Time (GMST), expressed in seconds via the formula [465]

GMST[s] =

158 B.3. COORDINATE SYSTEMS

86400 × 0.7790 5727 3264 + nJ2000mod 1 + 0.0027 3781 1911 35448 nJ2000 + 0.000 967 07 + 307.477 102 27 TJ2000 + 0.092 772 113 T 2 + O(T 3 ) . J2000 J2000 (B.2.3)

Secondly, the equaon of equinoxes is added to obtain the Greenwich Apparent Sidereal Time (GAST)

GAST = GMST + Ee(TJ2000) . (B.2.4a)

The equaon of equinoxes can be approximated as

Ee(TJ2000) ≈ ∆ψ cos A + 0.000176s sin Ω + 0.000004s sin 2Ω , (B.2.4b) where

∆ψ ≈ −1.1484s sin Ω − 0.0864s cos 2L, (B.2.4c) Ω = 125.0445 5501◦ − 0.0529 5376◦n + O(T 2 ) , J2000 J2000 (B.2.4d) L = 280.47◦ − 0.98565◦n + O(T 2 ) , J2000 J2000 (B.2.4e) = 23.4392 79444◦ − 0.01301021361◦T + O(T 2 ) . A J2000 J2000 (B.2.4f)

Finally, the LAST for a place with latude and longitude (Φ, λ) in seconds results by adding the longitude, λ (λ) = + 86 400 s . LAST GAST 360◦ (B.2.5)

Here, western longitudes are negave and LAST ∈ (0,86400)s. The errors of the approximaons are of the order of tens of milliseconds compared to the tables in [465].

B.3 Coordinate Systems

For the simulaon of the DM parcle trajectories through the Earth, the planet’s orientaon has to be speciﬁed. The simulaons are carried out in the galacc frame. We review all relevant coordinate systems in this chapter based on [465, 466], together with the transformaons. All of the coordinate systems listed in table B.2 are right handed and rectangular.

159 APPENDIX B. ASTRONOMICAL REQUIREMENTS

Frame Descripon galacc (gal) – heliocentric – x-axis points towards the galacc center – the z-axis points towards the galacc north pole – x- and y-axis span the galacc plane – see figure B.2a heliocentric, (hel-ecl) – heliocentric eclipc – x-axis points towards vernal equinox Υ – z-axis points towards the eclipc north pole – x- and y-axis span the eclipc plane –see figure B.2b geocentric, (geo-ecl) – geocentric eclipc – x-axis points towards vernal equinox Υ – z-axis points towards the eclipc north pole – x- and y-axis span the eclipc plane geocentric, (equat) – geocentric equatorial – x-axis points towards vernal equinox Υ – z-axis points towards the Earth north pole – x- and y-axis span the equatorial plane – see figure B.2c laboratory (lab) – laboratory-centric – x-axis points towards east – z-axis to the sky.

Table B.2: Coordinate systems

160 B.3. COORDINATE SYSTEMS

The transformaons into the galacc frame, where all computaons take place, require a number of me-dependent transformaon matrices.

(lab) ←→ (equat): The transformaon from the laboratory to the equatorial frame is done via   − sin φ − cos θ cos φ sin θ cos φ   x(equat) = N x(lab) , N =  cos φ − cos θ sin φ sin θ sin φ , with   (B.3.1) 0 sin θ cos θ

θ = π − Φ φ = 2π LAST(Φ,λ) (Φ, λ) where 2 , 86400s , and are the laboratory’s latude and longitude respecvely.

(hel-ecl)←→(geo-ecl): The two eclipc frames are connected by the trivial transformaon

x(geo-ecl) = −x(equat) . (B.3.2)

(geo-ecl.) ←→ (equat): A vector x(geo-ecl) is transformed to equatorial coordinates through the following rotaon,   1 0 0   x(equat) = Rx(geo-ecl) , R = 0 cos − sin  , with   (B.3.3) 0 sin cos

◦ ◦ where = 23.4393 − 0.0130 TJ2000 is the obliquity or axial lt of the eclipc.

(equat)←→(gal): The equatorial frame at J2000.0 can be related to the galacc frame,

x(gal) = Mx(equat)(J2000.0) , (B.3.4a) with

M11 = − sin lCP sin αGP − cos lCP cos αGP sin δGP , (B.3.4b)

M12 = sin lCP cos αGP − cos lCP sin αGP sin δGP , (B.3.4c)

M13 = cos lCP cos δGP , (B.3.4d)

161 APPENDIX B. ASTRONOMICAL REQUIREMENTS

M21 = cos lCP sin αGP − sin lCP cos αGP sin δGP , (B.3.4e)

M22 = − cos lCP cos αGP − sin lCP sin αGP sin δGP , (B.3.4f)

M23 = sin lCP cos δGP , (B.3.4g)

M31 = cos αGP cos δGP , (B.3.4h)

M32 = sin αGP cos δGP , (B.3.4i)

M33 = sin δGP . (B.3.4j)

The occuring angles, namely the J2000.0 right ascension of the north galacc pole

αGP, the J2000.0 declinaon of the north galacc pole δGP and the longitude of the

north celesal pole in J2000.0 galacc coordinates lCP, are

◦ ◦ ◦ αGP = 192.85948 , δGP = 27.12825 , lCP = 122.932 . (B.3.4k)

The rotaon in equatorial coordinates from J2000.0 to any me epoch TJ2000 is performed with the next matrix,

(equat) (equat) x (TJ2000) = Px (J2000.0) , (B.3.5a)

with

P11 = cos ζA cos θA cos zA − sin ζA sin zA , (B.3.5b) P12 = − sin ζA cos θA cos zA − cos ζA sin zA , (B.3.5c) P13 = − sin θA cos zA , (B.3.5d) P21 = cos ζA cos θA sin zA + sin ζA cos zA , (B.3.5e) P22 = − sin ζA cos θA sin zA + cos ζA cos zA , (B.3.5f) P23 = − sin θA sin zA , (B.3.5g) P31 = cos ζA sin θA , (B.3.5h) P32 = − sin ζA sin θA , (B.3.5i) P33 = cos θA . (B.3.5j)

Here the equatorial angles were used, which are me-depending and given by

ζ = 2306.08322700T + 0.29885000T 2 , A J2000 J2000 (B.3.5k) z = 2306.07718100T + 1.09273500T 2 , A J2000 J2000 (B.3.5l) θ = 2004.19190300T + 0.42949300T 2 . A J2000 J2000 (B.3.5m)

162 B.3. COORDINATE SYSTEMS

1 − R (equat) N (hel-ecl.) (geo-ecl.) (lab) at TJ2000

M (equat) (gal) at J2000.0

Figure B.1: Flowchart for transformaons between the coordinate systems.

Summary and examples Any transformaon matrix from one frame to any other can be expressed as a product of the respecve matrices. The ﬂow chart in ﬁg- ure B.1 is a small helper to connect two frames, where the inverse rotaon matrix is to be chosen if opposing the arrow’s direcon. To rotate e.g. from equatorial to galacc coordinates for any given epoch TJ2000, the transformaon reads

(gal) −1 (equat) x = MP x (TJ2000) , (B.3.6a) similarly, from heliocentric, eclipc coordinates to galacc coordinates,

−1 ( x(gal) = −MP Rx hel-ecl) . (B.3.6b)

T T This allows to express the axis vectors ex = (1, 0, 0) , ey = (0, 1, 0) and ez = T (0, 0, 1) of the equatorial and heliocentric-eclipc frame in terms of galacc coordinates,

e(gal) = MP−1e x,equat x     −0.0548763 0.0242316     2 =  0.494109  +  0.002688  T + O(T ) ,     J2000 J2000 (B.3.7a) −0.867666 −1.546 · 10−6

e(gal) = MP−1e y,equat y     −0.873436 −0.001227     2 = −0.444831 +  0.011049  T + O(T ) ,     J2000 J2000 (B.3.7b) −0.198076 −0.019401

e(gal) = MP−1e z,equat z

163 APPENDIX B. ASTRONOMICAL REQUIREMENTS

    −0.483836 −0.000533     2 =  0.746982  +  0.004801  T + O(T ) .     J2000 J2000 (B.3.7c) 0.455984 −0.008431

In the same way, the axis vectors of the heliocentric-eclipc frame can be evaluated to

(gal) −1 e = −MP Rex x,hel-ecl     0.054876 −0.024232     2 = −0.494109 +  −0.002689  T + O(T ) ,     J2000 J2000 (B.3.8a) 0.867666 1.546 × 10−6

(gal) −1 e = −MP Rey y,hel-ecl     0.993821 0.001316     2 = 0.110992 + −0.011851 T + O(T ) ,     J2000 J2000 (B.3.8b) 0.000352 0.021267

(gal) −1 e = −MP Rez z,hel-ecl     0.096478 0.000227     2 = −0.862286 + 0.000015 T + O(T ) .     J2000 J2000 (B.3.8c) −0.497147 0.000018

Some of these matrices will also be used in the computaon of the laboratory’s velocity in the galacc frame, which we will review next.

B.4 Velocity of the Sun, the Earth, and the Laboratory

For the computaon of direct detecon rates one has to specify the velocity of the laboratory in the galacc frame. It is composed of the Sun’s orbital velocity around the galacc center, the Sun’s peculiar moon relave to neighbouring stars, the Earth’s orbital velocity around the Sun, which is responsible for the annual signal modulaon [211] as illustrated in figure B.2a, and finally the laboratory’s rotaonal velocity around the Earth’s axis. The laer is causing a similar diurnal modulaon, but can be neglected in most cases. For a long me the standard reference for the Earth’s velocity in the context of direct DM searches has been the review by Smith and Lewin [369]. However, it turned out to contain an error in the first order correcon related to the Earth’s

164 B.4. VELOCITY OF THE SUN, THE EARTH, AND THE LABORATORY

galactic pole (gal) (hel-ecl) north pole (equat) aphelion

ecliptic pole june

DM 60° galactic plane wind equator

plane december

ecliptic

perihelion south pole

(a) Orbital velocity in (gal) (b) Orbital velocity in (c) Rotaonal velocity in (hel-ecl) (equat)

Figure B.2: The Earth’s orbital velocity in the galacc and heliocentric eclipc frame and the laboratory’s rotaonal velocity in the equatorial frame. orbital eccentricity, which was pointed out in [467] and conﬁrmed in [466]. This secon is based on the laer.

The Sun The Sun’s velocity v in the galacc rest frame, has two components,

v = vr + vs , (B.4.1a) given by the galacc rotaon of the Sun,   0   −1 vr = 220 km s ,   (B.4.1b) 0 the peculiar moon of the Sun [468],   11.1   −1 vs = 12.2 km s .   (B.4.1c) 7.3

The Earth For the Earth’s velocity, we add the orbital velocity relave to the Sun, shown in ﬁgure B.2b, is added,

v⊕(t) = v + vorbit(t) , (B.4.2a)

165 APPENDIX B. ASTRONOMICAL REQUIREMENTS

with (gal) vorbit(t) = −hv⊕i (sin L + e sin(2L − $)) e x,hel-ecl + (cos L + e cos(2L − $)) e(gal) , y,hel-ecl (B.4.2b)

e(gal) where the axis vectors of the heliocentric, eclipc frame, i,hel-ecl, which are listed in eqs. (B.3.8). The parameters are the mean orbital speed hv⊕i, the orbit’s eccentricity e, the mean longitude L, and the longitude of the perihelion $,

−1 hv⊕i = 29.79 km s , (B.4.2c) e = 0.01671 , (B.4.2d) L = 280.460° + 0.985 647 4°nJ2000 mod 360° , (B.4.2e) $ = 282.932° + 0.000 047 1°nJ2000 mod 360° . (B.4.2f)

The correcon due to the orbit’s eccentricity are mostly irrelevant for the simulaon result and are only include for completeness.

The laboratory The detector’s exact velocity in the galacc frame is obtained by adding the rotaonal velocity around the Earth axis,

vlab = v⊕(t) + vrot(t) . (B.4.3)

The first step is to find the spherical coordinate angles (θ, φ) of the detector’s posion in the geocentric equatorial coordinate system, such that the transformaon into the galacc frame is straight forward. The locaon of a detector on the Earth is defined via the latude and longitude (Φ, λ) and the underground depth d of the laboratory. As menoned in secon B.3, the x-axis of the equatorial coordinate system points towards the vernal equinox or March equinox. Hence, the spherical coordinate angles for a given LAST are π θ = − Φ , φ(t) = ω (Φ, λ) . 2 rotLAST (B.4.4)

2π The rotaon frequency is simply ωrot = 86 400 s . Note that these are sidereal seconds. Finally the posion vector can be transformed from the equatorial to the

166 B.5. MODELLING THE EARTH AND SUN

12 Ocean Crust I ]

3 10 Crust II cm / g 8 LID & LVZ ρ [ Transition Zone I 6 Transition Zone II Transition Zone III mass density 4 Lower Mantle 2 Outer Core ρPREM(r) Inner Core 0 0 1000 2000 3000 4000 5000 6000 radius r [km]

(a) Mechanical layers. (b) Mass density proﬁle.

Figure B.3: The Preliminary Reference Earth Model. galacc frame,   (r⊕ − d) sin θ cos φ (gal) −1   x = MP (r⊕ − d) sin θ sin φ . lab   (B.4.5) (r⊕ − d) cos θ

Lastly, the rotaonal velocity follows directly, 2π(r − d) ⊕ −1 (equat) vrot = cos ΦMP eφ (x) Td | {z } ≡veq = −v cos Φ sin(φ(t)) e(gal) − cos(φ(t)) e(gal) . eq x,equat y,equat (B.4.6)

−1 The rotaonal speed at the equator is veq ≈ 0.465 km s , and the axis vectors are listed in eq. (B.3.7).

B.5 Modelling the Earth and Sun

B.5.1 The Earth

Planetary model of the Earth In the study of diurnal modulaons, DM parcles are simulated as they traverse through the Earth and scaer on terrestrial nuclei. To model the Earth for this purpose, the planets interior structure needs to be specified, including the chemical composion and the mass density profile. We split the Earth split into two composional layers [469], namely the core with a radius of 3480 km and the mantle. The chemical abundances of the different nuclei are listed in table B.4.

167 APPENDIX B. ASTRONOMICAL REQUIREMENTS

l Layer Depth [km] al bl cl dl 0 Inner Core 0 – 1221.5 13.0885 0 -8.8381 0 1 Outer Core 1221.5 – 3480 12.5815 -1.2638 -3.6426 -5.5281 2 Lower Mantle 3480 – 5701 7.9565 -6.4761 5.5283 -3.0807 3 TZ I 5701 – 5771 5.3197 -1.4836 0 0 4 TZ II 5771 – 5971 11.2494 -8.0298 0 0 5 TZ III 5971 – 6151 7.1089 -3.8045 0 0 6 LVZ& LID 6151 – 6346.6 2.6910 0.6924 0 0 7 Crust I 6346.6 – 6356 2.9 0 0 0 8 Crust II 6356 – 6368 2.6 0 0 0 9 Ocean 6368 – 6371 1.020 0 0 0 10 Space > 6371 0 0 0 0 −3 Table B.3: The density proﬁle coeﬃcients in g cm and radii of the mechanical layers in the PREM.

1.2 US Standard Atmosphere

] 2 layers

3 1.0 m

/ 4 layers

kg 0.8 8 layers ρ [

0.6

0.4 mass density 0.2

0 5000 10 000 15 000 20 000 25 000 30 000 altitude [m]

Figure B.4: Density proﬁle of the US Standard Atmosphere and diﬀerent layer approximaons for the MC simulaons.

The mass density proﬁle of the Earth enters the computaon of e.g. the mean free path of DM. A polynomial parametrizaon in terms of the radius r is part of the PREM[470], where the Earth’s bulk is split into ten mechanical layers, as shown in ﬁgure B.3a. In each of these layers the density is given by

2 3 r ρ⊕(r) = al + blx + clx + dlx , where x ≡ . (B.5.1) r⊕

The ten layers are listed in table B.3, together with their dimension and density coeﬃcients. We plot the density in the right panel of ﬁgure B.3b.

Earth crust and atmosphere When describing the shielding eﬀect of an experiment’s overburden, the most eﬀecve DM stopping is caused by the Earth crust

168 B.5. MODELLING THE EARTH AND SUN

Element Core Mantle Crust Atmosphere [469][469][471][472] 1H 0.06 0.01 – – 12C 0.2 0.01 – 0.02 14N – – – 75.6 16O – 44 46.6 23.1 20Ne – – – 0.001 23Na – 0.27 2.6 – 24Mg – 22.8 2.1 – 27Al – 2.35 8.1 – 28Si 6 21 27.7 – 31P 0.2 0.009 – – 32S 1.9 0.03 – – 40Ar – – – 1.3 39K – – 2.8 – 40Ca – 2.53 3.6 – 52Cr 0.9 0.26 – – 55Mn 0.3 0.1 – – 56Fe 85.5 6.26 5.0 – 58Ni 5.2 0.2 – – Total 100.26 99.83 98.5 100

Table B.4: Relave element abundances in wt% of the Earth core, mantle,crust, and atmosphere.

169 APPENDIX B. ASTRONOMICAL REQUIREMENTS

for underground and the atmosphere for surface laboratories respecvely. In this context, where the interacon cross secon is very high, there is no need to model the enre planet. The simulated geometry an be simplified to a set of parallel layers above an experiment. Hence, the Earth crust is regarded as a layer of constant −3 mass density ρ = 2.7 g cm , consisng of nuclei, whose abundances are given in table B.4[471]. In order to account for the atmosphere’s shielding effect, the US Standard At- mosphere (1976) [472] is implemented, which extends to an altude of 86km. The composion is also listed in table B.4. The atmospheric density decreases with altude as shown in figure B.4. It is easier for theMC simulaons to simulate trajectories through layers of constant density. The atmosphere is therefore split into R a set of parallel layers of constant density, such that the integral ρ(x) dx yields the same value for each layer. This way, each layer has a similar stopping power, as visible in figure B.4 for different numbers of layers. Typically, a set of four layers is used to model the atmosphere. This discrezaon has been checked, the results are stable under variaon of the number of layers.

170 B.5. MODELLING THE EARTH AND SUN

1.0 1025 ]

0.8 - 3 cm )[

r 23

( 10 i 0.6

0.4 1021

0.2 number density n 1019

0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

radius r/R☉ radius r/R☉

1 4 3 12 16 56 temperature T(r)/T(0) mass M(r)/M☉ escape velocity vesc(r)/vesc(0) H He He C O Fe

(a) Solar mass-radius relaon, (b) Number density of the solar isotopes. temperature, and escape velocity.

Figure B.5: The solar model AGSS09.

B.5.2 The Sun

Similarly, the Sun needs to be modelled in order to describe DM parcles as they fall into the Sun’s gravitaonal well and possibly scaer on hot solar nuclei. The solar model used in this thesis is the Standard Solar Model (SSM) AGSS09 [454]. It provides the mass-radius relaon M(r), the temperature T (r), the mass density ρ(r), and the mass fracon fi(r), and number densies ni(r) of 29 solar isotopes,

fi(r)ρ(r) ni(r) = , (B.5.2) mi which are required for the calculaon of the local mean free path, or rather the collision frequency, inside the star. Furthermore, the local escape velocity is given by

 R  2G M R Z M(r0) v2 (r) = N 1 + dr0 , esc  02  (B.5.3) R M r r

2 2GN M which simplifies to vesc(r) = r outside the Sun. The solar mass-radius relaon, temperature, and local escape velocity are shown −1 in figure B.5a. The core escape velocity and temperature are vesc(0) ≈ 1385 km s , 7 and T (0) = 1.549 × 10 K respecvely. Furthermore figure B.5b shows the number density of the abundant isotopes.

171 APPENDIX B. ASTRONOMICAL REQUIREMENTS

172 Appendix C

Direct Detecon Experiments

In chapter4 and5 of this thesis, we compute detecon recoil spectra and constraints for a number of direct detecon experiments, probing both DM-nucleus and DM-electron interacons. In this appendix, we summarize the detectors’ details, required for various computaons. In addion, the essenal parameter for nuclear recoil experiments are listed in table C.1.

C.1 DM-Nucleus Scaering Experiments

C.1.1 CRESST-II

The CRESST-II [236] experiment was located at LNGS under 1400m of rock. Its phase 2 set leading constraints down to 500 MeV. The target of the ‘Lise’ module consisted of 300 g of CaWO4 crystals. Its recoil threshold was Ethr = 307 eV, and the energy resoluon was σE = 62 eV. With an exposure of 52.15 kg days, the predicted signal spectrum can be computed via eq.(3.5.9a), Z ∞ dR X dRi = dER i(ED)Gauss(ED|ER, σE) fi , (C.1.1) dED min dER ER i

min where i runs over (O, Ca, W) and ER = Ethr − 2σE. In the acceptance region of [Ethr, 40 keV] a number of 1949 events survived all cuts, and the energy data was released in [461] together with the cut survival probability i(ED) of each target nucleus. We compute the constraints using Yellin’s maximum gap method [377] which, despite the fact that the collaboraon used Yellin’s opmum interval method, reproduces the 90% CL constraints on low-mass DM to a very good accuracy.

173 APPENDIX C. DIRECT DETECTION EXPERIMENTS

C.1.2 CRESST-III

The successor of CRESST-II uses the same target at the same locaon. In the context of solar reflecon, we regarded a CRESST-III type detector. The CRESST-III phase 2 expects an exposure of 1 ton day according to [461]. We assume an energy threshold of Ethr = 100 eV, an energy resoluon of σE = Ethr/5, and efficiencies (ED) provided by the CRESST collaboraon. The computaon of the spectrum does not differ from CRESST-II. First results of CRESST-III’s phase 1 have been published in 2017 [237]. They presented new, leading exclusion limits on light DM based on nuclear recoils down to a mass of 350 MeV. The exposure was reported as 2.39 kg days, for which 33 events were detected in the acceptance region. We do not include results for CRESST-III in chapter 4.3, as the recoil data has not been published yet. Due to its similarity to CRESST-II, it will be straighorward to implement aer the data release.

C.1.3 CRESST surface run (2017)

The surface run of a prototype detector for the ν-cleus detector was used by the

CRESST collaboraon in 2017 [295]. It consisted of a small, 0.49g sapphire (Al2O3) target. Despite a ny net exposure of 0.046 g days, it was ideal for probing strongly interacng DM, because it was set up on the surface in a building of the Max Planck Instute in Munich, with only 30 cm of concrete ceiling as shield. The energy threshold and resoluon were remarkably low with Ethr ≈ 19.7 eV and σE ≈ 3.74 eV. Within the region of interest [Ethr, 600 eV], 511 events were observed. The cut efficiency was set to 1 as a conservave measure. The energy data was released as ancillary files on arXiv. Apart from the different targets, the computaon of the spectrum does not differ from CRESST-II or III, and the official constraints can be reproduced using Yellin’s maximum gap method with only small deviaons.

C.1.4 DAMIC

The DAMIC uses CCDs of silicon as a target to probe low-mass DM [238]. In 2011, an engineering run of a 0.5g target was used to derive ﬁrst constraints. These limits constrain strongly interacng DM more than later bigger runs, as it was located at a relavely shallow underground site at Fermilab with a underground depth of 350’(≈ 107 m). Addionally, a lead shield of 6”(≈ 15 cm) thickness was installed. The main reason to include DAMIC in our results, even though its constraints are covered by other experiments, is the fact that we can directly compare to DAMIC

174 C.2. DM-ELECTRON SCATTERING EXPERIMENTS

Experiment Target Shielding Ethr [eV] σE[eV] Exposure CRESST-II CaWO4 d=1400m 307 62 52.15 kg days [236, 461] CRESST-III CaWO4 d=1400m 100 20 ∼1 ton day [237, 461] CRESST 2017 Al2O3 atmosphere 19.7 3.74 0.046 g days surface run 30cm concrete [295] DAMIC Si CCDs d=106.7m 550 – 0.107 kg days [238] 15cm of lead XENON1T Xe d=1400m 5000 – 35.6 ton days [263]

Table C.1: Summary of direct detecon experiments based on nuclear recoils included in our analyses. constraints obtained by Mahdawi and Farrar using very similar simulaons [426]. This is also the reason, why we follow them in their analysis. DAMIC’s exposure was reported as 0.107 kg days, and its recoil threshold was Ethr = 0.04 keVee. Overall, 106 events were observed in the region of interest [Ethr, 2 keVee] (in terms of nuclear recoils [0.55,7]keV). In order to have comparable results with [426], we calculate the 90% CL constraints using Poisson likelihoods.

C.1.5 XENON1T

The first results by XENON1T, which is located at the LNGS, were presented in May 2017 [263]. The experiment with a liquid xenon target had reported exposure of 35.6 ton days, consistent with the background-only hypothesis. The threshold of XENON1T is Ethr = 5 keV, and the region of interest covers the interval [Ethr, 40 keV]. We used a simplified recoil spectrum to compute the constraints based on eq. (3.5.3) and neglected other detector effects apart from a flat 82% efficiency.

C.2 DM-Electron Scaering Experiments

C.2.1 XENON10 and XENON100

Both XENON10 and XENON100 have used the observaon of ‘S2-only’ events for the search of light DM [260, 310]. This data has been used to set constraints on DM- electron interacons [311, 312]. The equaons necessary to describe the ionizaon rate of xenon atoms due to DM parcles have been reviewed in the main body of

175 APPENDIX C. DIRECT DETECTION EXPERIMENTS

the thesis, in chapter 3.5.2. The necessary informaon about the atomic shells included in the analysis is summarized in table C.2. This table includes the number of secondary quanta for each shell, necessary to convert the spectrum from the primary electron’s energy to the total number of ionized electrons.

nl (2) atomic shell EB [eV] n 5p6 12.4 0 5s2 25.7 0 4d10 75.6 4 4p6 163.5 6-10 4s2 213.8 3-15 nl (2) Table C.2: Binding energy EB and number of secondary quanta n of xenon’s atomic shells. For the limits, the lowest number of secondary quanta is used.

The detector speciﬁc parameters, e.g. the exposure and detector resoluon, for XENON10 and XENON100 can be found in table C.3. These parameters are necessary to convert the spectra further in terms of the actual observable, the PEs, as described by eq. (3.5.17). Both experiments were located at the LNGS.

XENON10 XENON100 exposure 15 kg days 30 kg years µPE 27 19.7 σPE 6.7 6.2 min nPE 14 80 bin width 27 20 underground depth 1400m 1400m

Table C.3: Parameters for the analysis of XENON10 and XENON100.

For XENON10, we assume a flat cut efficiency of 92% [260], which needs to be mulplied by the trigger efficiency in order to obtain the total efficiency. The trigger efficiency is given in figure 1 of [311]. The corresponding acceptance and trigger efficiency for XENON100 can be found in figure 3 of [310]. Finally, the signal bins and event numbers for both experiments are listed in table C.4. For a given DM mass, we find the upper limit on the cross secon using Poisson stascs independently for each bin. The lowest of these values sets the overall limit. For XENON100 we only use the first three bins.

176 C.2. DM-ELECTRON SCATTERING EXPERIMENTS

XENON10 XENON100 bin [PE] events bin [PE] events [14,41) 126 [80,90) 794 [41,68) 60 [90,110) 1218 [68,95) 12 [110,130) 924 [95,122) 3 [130,150) 776 [122,149) 2 [150,170) 669 [149,176) 0 [170,190) 630 [176,203) 2 [190,210) 528

Table C.4: Binned data for XENON10 and XENON100.

C.2.2 SENSEI (2018) and SuperCDMS (2018)

The recent results by the SENSEI and SuperCDMS collaboraon have a number of things in common [240, 319]. They use a silicon semiconductor target, presented results from a surface run in 2018 and are sensive to single electronic excitaons. The formalism necessary to compute signal spectra and numbers can be found in chapter 3.5.2. SENSEI had the lower exposure with 0.07 gram×456 min ≈ 0.02 gram days. The observed events, taken from table I of [240], are also listed on the le side of table C.5. The SENSEI surface run took place at the Silicon Detector Facility at Fermilab, shielded only by a few cm of concrete, aluminium, wood and cooper. These layers can safely be neglected, and the atmosphere is included as the only overburden. SENSEI SuperCDMS ne eﬃciency events eﬃciency events 1 0.668 140302 0.88 ∼53000 2 0.41 4676 0.91 ∼400 3 0.32 131 0.91 ∼74 4 0.27 1 0.91 ∼18 5 0.24 0 0.91 ∼7 6 – – 0.91 ∼14

Table C.5: Eﬃciencies and observed signals in the electron bins for SENSEI (2018) and SuperCDMS (2018).

The surface run of SuperCDMS had an exposure of 0.487 gram days. The cut efficiency and event numbers can be found in figure 3 of [319]. The event numbers, listed on the right side in table C.5, were esmated from the histogram in the same figure. The SuperCDMS collaboraon obtained their limits using Yellin’s opmum interval method, where they removed the data more than 2σ away from

177 APPENDIX C. DIRECT DETECTION EXPERIMENTS

Element fi [wt%] 1H 0.33 16O 52.28 23Na 0.02 24Mg 0.10 27Al 0.33 28Si 40.85 32S 0.16 35Cl 0.01 39K 0.06 40Ca 5.59 56Fe 0.27

Table C.6: Nuclear composion of concrete in terms of the isotopes’ mass fracons fi.

the electron peaks with a charge resoluon of σ =0.07 electron-hole pairs. We simplify the analysis and compute the constraints with simple Poisson stascs for each bin. We also take the data removal procedure into account by adding a ﬂat eﬃciency factor of ≈ 0.9545, corresponding to the 2σ. Regarding the shielding, we take the 60cm of concrete above the detector into account in our simulaons. −3 We model the concrete as a layer of mass density ρ = 2.4 g cm , the composion is given in table C.6[473].

178 Appendix D

Numerical and Stascal Methods

In this appendix, we review some of the methods and rounes, which have been applied in the diﬀerent scienﬁc codes. The book Numerical Recipes by Press et al. has been a great resource in this context [378].

D.1 Adapve Simpson Integraon

In order to numerically compute the one-dimensional integral

b Z I(f; a, b) = dx f(x) , (D.1.1) a

Simpson’s rule approximates the funcon f(x) as a second order polynomial P (x) a+b a+b with f(a) = P (a), f(b) = P (b) and f( 2 ) = P ( 2 ) and integrates P (x) analyt- ically, b − a a + b I(f; a, b) ≈ S(a, b) ≡ f(a) + 4f + f(b) . 6 2 (D.1.2)

Apart from a few exceponal cases, this will not be very accurate. For that reason Simpon’s rule is promoted to an adapve method [474], that will compute the integral (D.1.1) up to any tolerance . If the error of S(a, b) exceeds this tolerance we a+b divide the interval (a, b) into subintervals (a, c) and (c, b) with c = 2 and apply eq. (D.1.2) to the sub-intervals. This repeats recursively unl

|S(a, c) + S(c, b) − S(a, b)| < 15 , (D.1.3)

179 APPENDIX D. NUMERICAL AND STATISTICAL METHODS

0.6

0.4

0.2 ) x

( 0.0 f

-0.2

-0.4

-0.6 0 5 10 15 20 x

Figure D.1: Example for Steﬀen interpolaon. Ten equidistant points (x, f(x)) with f(x) = cos(x) sin(x) are chosen and interpolated. Note that the extrema of the interpolaon funcon coincide with the data points.

as suggested by J.N. Lyness [475]. This method has been implemented in a recursive way, which avoids redundant evaluaons of the funcon f.

D.2 Interpolaon with Steﬀen Splines

Many methods have been proposed to interpolate in between a set of number pairs {(x1, y1), ..., (xN , yN )}. One parcular example for one dimensional funcons is the use of so-called Steﬀen splines [476], a smooth interpolaon method using third order polynomials as interpolaon funcons. This method guarantees connuous ﬁrst derivaves and monotonic behaviour of the interpolaon funcon and avoids relic oscillaons. Local extrema can only occur directly on a data point. In any given interval (Xi,Xi+1), the interpolaon funcon is given by

3 2 fi(x) = ai(x − xi) + bi(x − xi) + ci(x − xi) + di . (D.2.1a)

The coeﬃcients are given by

0 0 0 0 yi + yi+1 − 2si 3si − 2yi − yi+1 ai = 2 , bi = , (D.2.1b) hi hi 0 ci = yi , di = yi , (D.2.1c)

with

yi+1 − yi hi = (xi+1 − xi) , and si = . (D.2.1d) hi

180 D.2. INTERPOLATION WITH STEFFEN SPLINES

0 The remaining step is to ﬁnd yi. For i ∈ {2, ..., N − 1} it is given by  0 s s ≤ 0 ,  if i−1 1 0  yi = 2a min(|si−1|, |si|) if |pi| > 2|si−1| or |pi| > 2|si| , (D.2.2a)   |pi| otherwise.

Here, we used

si−1hi + sihi−1 a = sign(si−1) = sign(si) , pi = . (D.2.2b) hi−1 + hi

0 At the boundaries, we obtain y using  0 p s ≤ 0 ,  if 1 1 0  y1 = 2s1 if |p1| > 2|s1| , (D.2.3a)   p1 otherwise, and h1 h1 p1 = s1 1 + − s2 . (D.2.3b) h1 + h2 h1 + h2 on the le and  0 p s ≤ 0 ,  if N N−1 0  yN = 2sN−1 if |pN | > 2|sN−1| , (D.2.4a)   pN otherwise, and hN−1 hN−1 pN = sN−1 1 + − sN−2 . (D.2.4b) hN−1 + hN−2 hN−1 + hN−2 on the right boundary. For further explanaons and foundaons of the method, we refer to the original publicaon. An example can be found in ﬁgure D.1.

For the locaon of the argument in the tables during a funcon call, we implemented a combinaon of bisecon and a hunng algorithm, which also accounts for potenal correlaons of subsequent funcon calls [378].

181 APPENDIX D. NUMERICAL AND STATISTICAL METHODS D.3 Kernel Density Esmaon

TheMC simulaon of DM parcles typically provides a data set of velocies or speeds, which survived e.g. an energy cut or pass a certain locaon on Earth. The central problem of the following data treatment is to esmate the unknown PDF. More speciﬁcally, we assume a simulaon resulng in a data set {x1, ..., xN } with associated weights {w1, ..., wN } and the underlying PDF f(x) of domain I, i.e. x ∈ I, is unknown. The simplest non-parametric density esmate of f(x) is just the histogram of the data. Another, more sophiscated non-parametric approach ˆ is KDE[477, 478], which produces a connuous and smooth esmate f(x) of the true PDF, N 1 X x − xi fˆ (x) = w K . h h P w i h (D.3.1) i i i=1

The parameter h is called the bandwidth and plays a similar role as the bin width for histograms. The funcon K(x) is called a kernel and sasﬁes

K(x) ≥ 0 , for all x ∈ I, (D.3.2a) Z dx K(x) = 1 , (D.3.2b) I Z dx xK(x) = 0 . (D.3.2c) I

For praccal reasons, the scaled kernel is deﬁned as 1 x K (x) ≡ K , h h h (D.3.3)

such that

N 1 X fˆ (x) = w K (x − x ) . h P w i h i (D.3.4) i i i=1

The choice of the kernel is relavely insigniﬁcant, and a simple Gaussian kernel usually suﬃces, 1 −x2 K(x) = √ exp . 2π 2 (D.3.5)

182 D.3. KERNEL DENSITY ESTIMATION

True PDF 0.015 KDE

] KDE + C&H km /

s 0.010 )[ v ( PDF f 0.005

0.000 600 650 700 750 speed v [km/s]

Figure D.2: KDE for a bounded domain with and without the pseudo data method of Cowling and Hall (C&H).

The choice of the bandwidth h, being the only free parameter, is crucial, similarly to the bin width for histograms. A too large value for h might smooth over interesng features of the PDF, a value too small will resolve stascal ﬂuctuaons of the data. The bandwidth can be esmated based on the data via Silverman’s rule of thumb [479],

4 1/5 h = σˆ , 3N (D.3.6) which takes the sample’s size N and standard deviaon σˆ into account.

A well known and studied problem of KDE occurs if the domain or support is bounded. Especially using the Gaussian kernel can be problemac and introduce significant errors since the kernel’s domain is unbounded. In these cases, the KDE esmate of eq. (D.3.4) underesmates the true PDF in proximity of the domain’s boundaries. The kernel does not include any knowledge of the boundary and assigns weight to the region beyond, where no data points exist. The problem is most severe, if the true PDF does not vanish at the domain edges. TheMC simulaons in the context of secon 4.3 generate data samples with a hard speed cutoff, such that the PDFs to be esmated have a bounded domain. An example, using a data sample of 5000, generated with DaMaSCUS-CRUST, is shown in figure D.2, where the unmodified KDE drops below the true PDF close to the speed cutoff.

In order to remove this boundary bias, a plethora of methods of varying com- plexity has been developed and studied [480]. The most simple ﬁx is to reﬂect the data around the boundary. For a PDF f(x) of domain [xmin, xmax] with f(xmin) > 0 and a data sample {x1, ..., xN } with xmin ≤ xi ≤ xmax, the following adjustment

183 APPENDIX D. NUMERICAL AND STATISTICAL METHODS

of eq. (D.3.4) can reduce the bias considerably,

N 1 X fˆ (x) = w K (x − x ) + K x − xreﬂ , h P w i h i h i (D.3.7) i i i=1

refl with the reflected pseudo data point xi ≡ 2xmin − xi. The drawback of this ˆ0 easy to implement method is the fact that f (xmin) = 0, which introduces a new bias, unless the true PDF shares this property. But the general idea of generang pseudo data beyond the boundary is applicable for other PDFs as well. Cowling and Hall proposed such a pseudo data procedure [481]. They showed that certain linear combinaons of data points from within the domain into pseudo data points (x(−1), ..., x(−m)) beyond the boundary can correct the flawed edge behaviour of ˆ f(x). In parcular, they propose to use the modified KDE

" N m # 1 X X fˆ (x) = w K (x − x ) + w K (x − x ) , h P w i h i (−i) h (−i) (D.3.8a) i i i=1 i=1

where one possible combinaon is the three-point-rule

x(−i) = 4xmin − 6xi + 4x2i − x3i , (D.3.8b) w + w + w w = i 2i 3i , (−i) 3 (D.3.8c) N m = . 3 (D.3.8d)

As opposed to eq. (D.3.4), the normalizaon of (D.3.8a) on its support is no longer guaranteed, it is therefore a good idea to re-normalize the density esmate. As shown in ﬁgure D.2, the bias towards the domain boundary vanishes.

D.4 The Runge-Kua-Fehlberg Method

The Runge-Kua-Fehlberg method is a member of the Runge-Kua (RK) family to explicitly and adapvely solve ordinary differenal equaons (ODEs) of first order [463]. It requires the same number of funcon evaluaons as RK6, but is rather a combinaon of RK4 and RK5, such that the error of each step can be esmated. This esmate can be used to adjust the step size adapvely. For a 1st order ordinary differenal equaon with given inial condions,

y˙ = f(t, y) , y(t0) = y0 , (D.4.1)

184 D.4. THE RUNGE-KUTTA-FEHLBERG METHOD we can ﬁnd the soluon via the iteraon step 25 1408 2197 1 y = y + k + k + k − k , k+1 k 216 1 2565 3 4101 4 5 5 (D.4.2a) where

k1 = ∆t f (tk, yk) , (D.4.2b) 1 1 k = ∆t f t + ∆t, y + k , 2 k 4 k 4 1 (D.4.2c) 3 3 9 k = ∆t f t + ∆t, y + k + k , 3 k 8 k 32 1 32 2 (D.4.2d) 12 1932 7200 7296 k = ∆t f t + ∆t, y + k − k + k , 4 k 13 k 2197 1 2197 2 2197 3 (D.4.2e) 439 3680 845 k = ∆t f t + ∆t, y + k − 8k + k − k , 5 k k 216 1 2 513 3 4104 4 (D.4.2f) ∆t k = ∆t f t + , 6 k 2 8 3544 1859 11 y − k + 2k − k + k − k . k 27 1 2 2565 3 4104 4 40 5 (D.4.2g)

Interesngly enough, we can use the same coeﬃcients to get a 5th order esmate of the soluon, 16 6656 28561 9 2 y˜ = y + k + k + k − k + k , k+1 k 135 1 12825 3 56430 4 50 5 55 6 (D.4.3) giving us a direct measure |yk+1 − y˜k+1| of the 4th order method’s error. Specifying an error tolerance , we can adjust our step size adapvely,

1/4 ∆tk+1 = 0.84 ∆tk . (D.4.4) |yk+1 − y˜k+1|

The new step size is either used in the next step or to repeat the previous step, if the error exceeds the error tolerance.

185 APPENDIX D. NUMERICAL AND STATISTICAL METHODS

186 Appendix E

Rare Event Simulaons

To determine direct detecon constraints on strongly interacng DM withMC simulaons, we simulate trajectories through the shielding overburden above a laboratory, where most parcles fail to reach the detector. The desired data is composed of the rare events, whenever a parcle against all odds makes it to the detector, while sll being energec enough to cause a signal in the detector. These events can be so rare that millions of trajectories need to be simulated, in order to obtain a single data point. It is clear that this yields a challenge to ‘brute force’MC simulaons of rare events, which are in the best case just ineﬃcient, in the worst case would require such a tremendous amount of compung me, that they become praccally unapplicable. In this app., we present and review two advancedMC methods for variaon reducon, Importance Sampling (IS) and Geometric Importance Split- ng (GIS), which deal with this problem. Both methods are implemented in the DaMaSCUS-CRUST code and increase the probability of a successful simulaon run, therefore reducing computaonal me1.

E.1 Importance Sampling

Typically, certain events inMC simulaons are rare, if the region of interest of the involved probability densies is found in the distribuons’ tails. Naturally, values from these tails are rarely sampled. Importance Sampling (IS) is a standardMC technique and arﬁcially increases the probability to sample these more ‘important’ values [434]. It introduces a bias in the simulaon’s PDFs, ensuring to compensate this bias by a appropriately chosen weighng factor. Assuming aMC simulaon, which involves a single PDF f(x) and where a suc-

1For a more detailed introducon to rare event simulaons, we recommend [435] and [391].

187 APPENDIX E. RARE EVENT SIMULATIONS

cessful run is extremely unlikely. The final data set is the result of an aggressive filtering process, where most simulaon runs fail and do not contribute. The stascal properes of the successful runs consequently differ considerably from the ‘typical’ run. For example, a simulaon run might be more likely to succeed, if the random variable is repeatedly sampled from the suppressed region of interest of f(x). The distribuon of this random variable in successful simulaon differs from the distribuon f(x) of an average run and follows its own underlying PDF g(x) determined by the filter criteria. The difference in the PDFs between a rare successful and an average run can be exploited. If the simulaon no longer samples from the original f(x), but instead of some new distribuon gˆ(x) which approximates g(x), it would imitate the stascal behaviour of successful simulaon runs. Consequently, the desired events occur more frequently. The modificaon of the PDF introduces a bias which needs to be compensated by a data weight. Furthermore, if the biased distribuon funcon gˆ(x) is chosen poorly and does not approximate the true distribuon of the rare events, the method becomes unstable and does no longer produce reliable results. A comparison between the unmodified and theIS simulaon for some examples should always be performed as consistency check. To determine the weighng factor, assume a random variable X with PDF f(x) and the expectaon value of a quanty Y (X) on a given interval I, Z hY iI = dx Y (x)f(x) . (E.1.1a) I

This can trivially be wrien as

Z f(x) = dx Y (x) gˆ(x) . (E.1.1b) I gˆ(x)

The funcon gˆ(x) can be interpreted as a new distribuon funcon, whereas the f(x) factor gˆ(x) may be regarded as the weighng funcon. Importance Sampling means to sample from gˆ(x) instead of f(x) during the simulaons. Under the assumpon that gˆ(x) is chosen wisely in the sense described above, the rare events of interest occur with higher probability. This way,IS can make rare event simulaon praccally feasible, where ‘brute force’ simulaons fail due to ﬁnite me and resources. This method is of special interest for the simulaon of strongly interacng DM parcles in e.g. the Earth crust, where the parcles can lose a signiﬁcant fracon of their kinec energy in a single scaering on a nucleus. It is remarkably powerful for

188 E.1. IMPORTANCE SAMPLING

GeV scale DM and contact interacons, where its applicaon was ﬁrst proposed by Mahdawi and Farrar [426, 431]. They showed how the DM parcles, which reach the detector depth, deviate in their stascal properes in this case from the average parcles.

1. Successful parcles scaer fewer mes, or in other words, propagate freely for longer distances than the mean free path.

2. They also tend to scaer more in the forward direcon, as opposed to the typical parcle, whose scaerings are isotropic.

Obviously, both properes increase the chance of reaching the detector depth before they get reflected or lose too much energy to be detectable. It also shows, how the simulaon’s PDFs should be modified forIS, as will be shown in detail. The central random variables in the DM simulaons were introduced in chapter 4.1. The first is the distance L a parcle propagates freely before scaering on a shielding target, with its PDF 1 x f (x) = exp − , λ λ λ (E.1.2) where λ is the mean free path. A reasonableIS modificaon of this distribuon to address point 1 is to increase the mean free path, 1 x gλ(x) = exp − , (E.1.3) (1 + δλ)λ (1 + δλ)λ with δλ > 0. If a distance Li is sampled as described in chapter 4.1.1 by solving

L Z i dx gλ(x) = ξ , (E.1.4) 0 the accompanying weighng factor is f (l ) λ i δλ wλ,i = = (1 + δλ)(1 − ξ) . (E.1.5) gλ(li)

For a trajectory of nS scaering events, the total weight is

n YS wλ = wλ,i . (E.1.6) i=0

The random variable which can address point 2 is the scaering angle in the

189 APPENDIX E. RARE EVENT SIMULATIONS

1.0 0.10

0.8

] 0.08 δ=0 1 - θ ) m 0.6 δ=0.2

)[ 0.06 cos x ( ( θ λ δ=0.4 0.4 0.04 δ=0.6 PDF f PDF f

0.02 0.2 δ=0.8 δ=1 0.00 0.0 0 5 10 15 20 25 30 35 -1.0 -0.5 0.0 0.5 1.0 distance x [m] scattering angle cos θ

(a) Free propagaon distance. (b) Scaering angle.

Figure E.1: Importance Sampling modiﬁcaons of the simulaon’s PDFs.

center of mass frame θ. As discussed in more detail in chapter 4.1.4, its distribuon 2 is isotropic for contact interacons of light DM, as the nuclear form factor FN (q ) can be approximated by 1. Then cos θ is a uniform random quanty with PDF 1 f (cos θ) = , θ 2 (E.1.7)

and domain (−1, 1). TheIS-modiﬁed distribuon should favour forward scaerings lowering the reﬂecon probability and the energy loss in a scaering. Both support a successful outcome of the respecve trajectory. One possible implementaon of this idea is theIS biased probability density 1 + δ cos θ g (cos θ) = θ , δ ∈ [0, 1] . θ 2 with θ (E.1.8)

Whenever we determine a sample of cos θi by inverse transform sampling,

cos θ Z i d cos θ gθ(cos θ) = ξ , (E.1.9) −1 p 2 −1 + (1 − δθ) + 4δθξ ⇒ cos θi = . (E.1.10) δθ

the weighng factor is

fθ(cos θi) 1 wθ,i = = gθ(cos θi) 1 + δθ cos θi 2 −1/2 = (1 − δθ) + 4δθξ . (E.1.11)

In the non-IS limit, isotropic scaering is re-obtained, as lim cos θi = 2ξ − 1. The δθ→0 IS modiﬁcaons are shown in ﬁgure E.1. For the simulaon of heavier DM parcles, the loss of coherence needs to be

190 E.2. GEOMETRIC IMPORTANCE SPLITTING taken into account, and the nuclear form factor FN (q) can no longer be neglected. Instead it favours small momentum transfers and forward scaerings as we saw in secon 4.1.4, with the PDF given by eq. (4.1.39). Similarly to eq. (E.1.8), theIS PDF can be deﬁned as δ g (cos θ; v ) = f (cos θ; v ) + θ cos θ , θ χ θ χ 2 (E.1.12) in order to increase the chance of forward scaering even more. Yet it needs to be ensured that gθ(cos θ; vχ) > 0 for backwards scaering, which occurs if

δθ > 2fθ(−1; vχ) . (E.1.13)

The weight of a successful trajectory with nS scaerings is given by the product

n YS wθ = wθ,i . (E.1.14) i=1

At last, the total stascal weight of a data point obtained withIS of both involved random variables is just the product of the respecve weights, w = wλwθ.

E.2 Geometric Importance Spling

As menoned in the previous secon,IS yields stable results, only if the simulated DM parcle can lose a large fracon of its energy in very few scaerings. In other cases, such as for very heavy DM or interacons with light mediators, this is no longer the case. Instead even successful trajectories involve hundreds or thousands of scaerings, each of which cause a ny relave loss of energy. For these cases, we implemented an alternaveMC method for rare event simulaon, called Geomet- ric Importance Spling (GIS), which goes back to John von Neumann and Stanislav Ulam and was ﬁrst described by Kahn and Harris in the context of neutron transport [434].

E.2.1 Spling and Russian Roulee for rare event simulaon

UnlikeIS, the implementaon of GIS does not modify any the PDFs, and the sampling of the random variables is unchanged. Instead, ‘important’ parcles are being split into a number of copies, whose propagaon is simulated independently. In addion, the simulaon of ‘unimportant’ parcles has a chance of being terminated

191 APPENDIX E. RARE EVENT SIMULATIONS

prematurely. This stresses the physical intuion that we do not simulate fundamental parcles but packages of parcles, which can be split into smaller sub-packages. The size of such a package is quanﬁed by its stascal weight, which decreases when it gets split. This raises the queson, what it means for a parcle to be ‘important’. The ‘im- 3 portance’ of a parcle is deﬁned by an importance funcon I : R → R, which is the central object for simulaons with GIS. As a parcle approaches the detector depth, the importance funcon should increase. In fact, it is a monotonously increasing funcon of the parcle’s underground depth. The GIS algorithm is the following. Assuming a parcle of weight wi and importance Ii scaers at depth z, its previous importance is compared to the new value Ii+1 ≡ I(z). If

Ii+1 ν ≡ > 1 , (E.2.1) Ii

the parcle will be split into n copies of weight wi+1, where  ν , if ν ∈ N ,  n = bνc , if ν∈ / N ∧ ξ ≥ ∆ , (E.2.2)   bνc + 1 , if ν∈ / N ∧ ξ < ∆ ,

and

w w ≡ i . i+1 n (E.2.3)

Here ∆ ≡ ν − bνc is the non-integer part of ν and ξ is a sample of U[0,1]. The expectaon value of n for non-integer ν is nothing but ν, as hni = ∆(bνc + 1) + (1 − ∆)bνc = ν. As coined by von Neumann and Ulam, the opposite acon of parcle spling is called Russian Roulee. If a parcle’s importance is decreasing, i.e.

Ii+1 ν = < 1 , (E.2.4) Ii

the parcle’s simulaon is terminated with a probability of pkill = 1−ν. In the case of survival, its weight gets increased via

wi wi wi+1 = = > wi . (E.2.5) 1 − pkill ν

192 E.2. GEOMETRIC IMPORTANCE SPLITTING

surface 0 w=1 w=1w=1 w=1/4

-20 I=1 w=1/2

/2 =1 -40 w ]

4 m = / [ I 2

1 z 1/2 w= =

-60 w w= 1/4 -80 I=4 w=1/4 1/4 w=1/4 w=

-100 detector depth -100 -50 0 50 100 x[m]

splitting Russian Roulette survival Russian Roulette kill

Figure E.2: Illustraon of two DM trajectories using GIS with three importance domains of equal size. The parcle can split into either 2 or 4 copies. The weight development along the paths is shown as well. such that the expectaon value of the new weight is just the old weight, hwi+1i = pkill · 0 + (1 − pkill) · wi+1 = wi. Russian roulee can be regarded as a special case of eq. (E.2.2).

E.2.2 The Importance Funcon and Adapve GIS

The central challenge ofIS was to find an appropriate modificaon of the simulaon’s PDFs. Similarly, the central problem of GIS is the construcon of the importance funcon, which determines the parcle splings. For the purposes of this thesis, the shielding layers above a direct detecon experiment at depth d can simply be divided into NI importance domains, defined 0 > l > l > ... > l > d by a set of planar spling surfaces of depth 1 2 NI −1 . The importance of a parcle changes, if it passes any of these boundaries. Assigning k−1 domain k an importance of e.g. Ik = Nsplits, with Nsplits = 2, 3, ..., a parcle from domain k reaching k + 1 will be split into Nsplits copies. However, the importances do not need to be integers. The number of importance domains is to be determined adapvely. Choos- ing NI too small will not exploit the full benefits of GIS, choosing it too large can cause a single parcle to split into a large number of copies. If these reach the detector, the data set can be highly correlated, as a large fracon of the final data originates from one parcle. For example with NI =10 and Nsplits =3, a parcle which scaers in domain 9 for the first me will split into 38 =6561 copies. If hard scaerings with large relave energy losses dominate the simulaon, the

193 APPENDIX E. RARE EVENT SIMULATIONS

choice NI ∼ d/λ, with λ being the mean free path, would work well. However, it is a beer idea to determine NI based on the integrated stopping power, described in chapter 4.3.1, as this also works for the case of ultralight mediators,

d Z Nlayers X l h∆Ei ≡ dx Sn(mχ, σp, hvχi) = tlSn(mχ, σp, hvχi) . (E.2.6) 0 l=1

R vmax tl l hvχi = vf(v) Here, is the thickness of the physical shielding layer , and vcutoﬀ is the average inial speed of the simulated DM parcles. We can now determine NI adapvely via h∆Ei N = κ · , I mχ 2 2 (E.2.7) 2 (hvχi − vcutoﬀ )

where κ is a dimensionless parameter, which can be freely adjusted to inﬂuence the pace with which the number of importance domain is increasing. For the locaon

of the spling surfaces, l1, ..., lNI −1, it should be ensured, that the domains are of equal integrated stopping power, i.e. h∆Ei/NI . This way, we avoid placing a lot of the boundaries into regions of relavely weak stopping power. For example, if we simulate both the Earth crust and the atmosphere, most if not all domain boundaries should be located in the crust. The benefit of this method is a simulaon speed-up of up to two orders of magnitude. The GIS results have been compared to non-GIS results for different examples, to verify the method’s validity. All results are invariant under (de-)acvaon of the GIS method up to stascal fluctuaons.

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