The Search for Dark Matter in Xenon: Innovative Calibration Strategies and Novel Search Channels Shayne Edward Reichard Purdue University

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12-2016 The search for dark matter in xenon: Innovative calibration strategies and novel search channels Shayne Edward Reichard Purdue University

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Recommended Citation Reichard, Shayne Edward, "The es arch for dark matter in xenon: Innovative calibration strategies and novel search channels" (2016). Open Access Dissertations. 994. https://docs.lib.purdue.edu/open_access_dissertations/994

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PURDUE UNIVERSITY GRADUATE SCHOOL Thesis/Dissertation Acceptance

This is to certify that the thesis/dissertation prepared

By Shayne Edward Reichard

Entitled THE SEARCH FOR DARK MATTER IN XENON: INNOVATIVE CALIBRATION STRATEGIES AND NOVEL SEARCH CHANNELS

For the degree of Doctor of Philosophy

Is approved by the final examining committee:

Rafael F. Lang Chair Dimitrios Giannios

Matthew Jones

Matthew L. Lister

To the best of my knowledge and as understood by the student in the Thesis/Dissertation Agreement, Publication Delay, and Certification Disclaimer (Graduate School Form 32), this thesis/dissertation adheres to the provisions of Purdue University’s “Policy of Integrity in Research” and the use of copyright material.

Approved by Major Professor(s): Rafael F. Lang

Approved by: John P. Finley 11/21/2016 Head of the Departmental Graduate Program Date

THE SEARCH FOR DARK MATTER IN XENON: INNOVATIVE CALIBRATION

STRATEGIES AND NOVEL SEARCH CHANNELS

A Dissertation

Submitted to the Faculty

Purdue University

Shayne Edward Reichard

In Partial Fulﬁllment of the

Requirements for the Degree

Doctor of Philosophy

December 2016

Purdue University

West Lafayette, Indiana ii

TABLE OF CONTENTS

Page LIST OF TABLES ...... v LIST OF FIGURES ...... vii ABSTRACT ...... xiv 1 DARK MATTER ...... 1 1.1 The Friedmann Equation ...... 2 1.2 Big Bang Nucleosynthesis ...... 6 1.3 The Cosmic Microwave Background ...... 9 1.3.1 The Curvature of the Universe ...... 13 1.3.2 Baryon Density ...... 13 1.3.3 Dark Matter Density ...... 13 1.4 Galaxies ...... 14 1.4.1 Milky Way Galaxy ...... 14 1.4.2 Other Spiral Galaxies ...... 15 1.4.3 Elliptical Galaxies ...... 16 1.5 Galaxy Clusters ...... 17 1.5.1 Orbits of Galaxies ...... 18 1.5.2 Hot Gas ...... 18 1.5.3 Gravitational Lensing ...... 19 1.6 Structure Formation ...... 21 1.7 Large-Scale Structures ...... 23 1.8 Dark Matter Candidates ...... 25 1.8.1 Weakly Interacting Massive Particle ...... 26 1.8.2 Axion ...... 28 2 DETECTION OF DARK MATTER ...... 31 2.1 Collider Experiments ...... 31 2.2 Indirect Detection ...... 33 2.3 Direct Detection ...... 34 2.3.1 Diﬀerential Energy Spectrum ...... 35 2.3.2 Dark Matter Halo Density ...... 36 2.3.3 Velocity Distribution ...... 37 2.3.4 Nuclear Form Factor ...... 38 2.3.5 Interaction Factor ...... 41 2.3.6 Energy Detection Eﬃciency ...... 42 iii

Page 2.3.7 Energy Resolution and Threshold ...... 43 2.3.8 Target Mass Fraction ...... 44 2.3.9 Annual Modulation ...... 44 2.3.10 WIMP Event Rate ...... 45 2.4 Low-Background Requirements ...... 48 3 SCATTERING EXPERIMENTS ...... 51 3.1 Liquid Noble Element Detectors ...... 52 3.1.1 WIMP Recoil Sensitivity ...... 52 3.1.2 Scintillation Signal ...... 53 3.1.3 Ionization Signal ...... 55 3.1.4 Electroluminescence ...... 56 3.1.5 Radiopurity ...... 57 3.1.6 Self-Shielding ...... 57 3.1.7 Scalability ...... 59 3.1.8 Discrimination ...... 59 3.2 Cryogenic Detectors ...... 60 3.2.1 Low-Temperature Phonon Signal ...... 60 3.2.2 Ionization Collection ...... 62 3.3 Solid Scintillators ...... 63 4 XENON ...... 65 4.1 The Principle of a Dual-Phase Time Projection Chamber ...... 65 4.2 Calibration ...... 67 4.3 Light Collection Eﬃciency ...... 68 4.4 Energy Scales ...... 70 4.4.1 Electronic Recoil Equivalent Energy ...... 70 4.4.2 Nuclear Recoil Equivalent Energy ...... 70 4.4.3 Combined Energy Scale ...... 71 5 RESULTS FROM A CALIBRATION OF XENON100 USING A DISSOLVED RADON-220 SOURCE ...... 73 5.1 The XENON100 Detector in Run 14 ...... 74 5.2 220Rn Decay Chain and Observed Time Evolution ...... 75 5.3 Contamination from the Veto ...... 78 5.4 Combined Leakage Rate ...... 80 5.5 Alpha Spectroscopy ...... 81 5.6 220Rn-216Po Coincidence and Convection ...... 83 5.7 The half-life of 212Po ...... 87 5.8 Low-Energy Calibration ...... 90 5.9 Conclusions ...... 94 6 DARK MATTER SCATTERING INELASTICALLY OFF XENON NUCLEI 95 6.1 Nuclear Structure and WIMP-Nucleon Currents ...... 96 iv

Page 6.2 Structure Factors ...... 97 6.3 Kinematics of Inelastic Scattering ...... 102 6.4 Signatures of Inelastic Dark Matter Scattering ...... 105 6.5 Implications for Dark Matter Searches ...... 111 6.6 Conclusions ...... 114 7 SUPERNOVA NEUTRINO PHYSICS WITH XENON DARK MATTER DE- TECTORS: A TIMELY PERSPECTIVE ...... 115 7.1 Supernova Neutrino Emission ...... 117 7.1.1 Supernova Neutrino Emission Properties ...... 118 7.1.2 Neutrino Flavor Conversion ...... 121 7.2 Supernova Neutrino Scattering with Dual-Phase Xenon Detectors .. 122 7.2.1 Scattering Rates in Terms of Recoil Energy ...... 124 7.2.2 Generation of the Observable S1 and S2 Signals ...... 127 7.2.3 Observable Scattering Rates ...... 130 7.3 S2-Only Analysis ...... 133 7.4 Supernova Neutrino Detection ...... 135 7.4.1 Detection Signiﬁcance ...... 136 7.4.2 Distinguishing Between Supernova Progenitors ...... 138 7.4.3 Reconstructing the Supernova Neutrino Light Curve ...... 138 7.4.4 Neutrino Diﬀerential Flux ...... 141 7.4.5 Total Energy Emitted into Neutrinos ...... 144 7.4.6 Comparison with Dedicated Neutrino Detectors ...... 147 7.4.7 SNEWS & False Alarms ...... 148 7.5 Experimental Factors ...... 150 7.5.1 Signal Uncertainty from Qy ...... 150 7.5.2 Sources of Increased Background Rates ...... 153 7.5.3 Sensitivity Limitation from Event Pile-Up ...... 154 7.6 Conclusions ...... 155 A Calculations of Velocity Distribution ...... 159 B Calculations of Event Rates ...... 161 C Convection Parameter ...... 163 LIST OF REFERENCES ...... 165 VITA ...... 180 v

LIST OF TABLES

Table Page 5.1 The results of a GEANT4 Monte Carlo simulation of leakage events from long-lived isotopes in the veto (buffer) region. The intensities, deposition probabilities, and overall contribution per mille are given for each gamma. The values correspond to the entire active region...... 79 5.2 The total leakage contribution from 208Tl is determined according to the different combinations of de-excitation gammas. The intensities, deposition probabilities, and overall contribution per mille are given for each decay path. The final leakage value accounts for the 36% branching of 208Tl. .. 79 5.3 Q-values, scintillation values (means and widths at 400 V/cm), and calculated light yields of each alpha decay in Figure 5.4...... 83 5.4 The results of a GEANT4 Monte Carlo simulation of leakage events from long-lived isotopes in the veto (buffer) region. The intensities, deposition probabilities, and overall contribution per mille are given for each gamma. The values correspond to the entire active region...... 93 5.5 The total leakage contribution from 208Tl is determined according to the different combinations of de-excitation gammas. The intensities, deposition probabilities, and overall contribution per mille are given for each decay path. The final leakage value accounts for the 36% branching of 208Tl. . 93 6.1 Minimal and maximal recoil energies, in keV, between which inelastic scattering is allowed (see Eq. (6.13)) for the two xenon isotopes and various WIMP masses...... 103

6.2 Minimum energy Evis in keV above which the observed inelastic spectrum for 129Xe, 131Xe and for the total spectrum starts to dominate the elastic one for various WIMP masses...... 113 vi

Table Page

7.1 Expected number of SN neutrino events per tonne of xenon target above various S1 and S2 thresholds. The SN burst occurs at 10 kpc from Earth and the neutrino flux has been integrated over the first 7 s after the core bounce. The light and charge yields, Ly and Qy, respectively, have been set to zero below recoil energies of 0.7keV. The number of events, for the case in which the threshold includes 0 PE (‘≥ 0’) and when it does not (‘> 0’), have been separated to show that many of the events have an S1 or S2 signal that is exactly zero. The symbol (?) indicates the most likely threshold values (see discussion in Sections 7.3 and 7.5 for details). An S2-only search for CEνNS from SN neutrinos is optimal as it results in a higher number of detected events...... 132 7.2 The typical uncertainty of the reconstructed total energy emitted in neutrinos over the first 7s, assuming our four SN progenitors situated 10 kpc from Earth, for different current and upcoming detectors...... 146 7.3 The number thresholds for supernovae signals are determined for different time intervals and background rates...... 149 7.4 The expected number of neutrino events per tonne for various S2 thresholds under different assumptions for Qy. We compare the Lindhard and Bezrukov models and assume that Qy = 0 for energies below Qy,min. The results are for the 27 M LS220 EoS progenitor at 10 kpc and integrated over the first 7 s. Similar results hold for other progenitor models. The signal uncertainty in each row is (S2max − S2min)/(S2max + S2min). The Lindhard model with Qy,min = 0.7 keV gives the smallest number of events per tonne and is the benchmark assumption that we have made in this paper...... 152 vii

LIST OF FIGURES

Figure Page 1.1 The abundances of D, 3He, and 7Li (relative to H) are predicted by Standard Big Bang Nucleosynthesis in terms of the abundance parameter, η10. The band widths represent uncertainties in nuclear and weak interactions. Ob- served primordial abundances indicate that the baryon density is Ωb ' 0.05. Plot based on data from [15]...... 8 1.2 The intensity of the Cosmic Microwave Background versus frequency. The curve represents the theoretical spectrum of a blackbody with T = 2.73 K. The data [21] match the curve well enough that the width of the curve hides the uncertainties of each point. Figure from [22]...... 10 1.3 False color images of the sky observed in microwave frequencies [24,25]. (1) The temperature is uniform at a large scale. (2) The same map is shown with temperatures between 2.721 K and 2.729 K. The dipole anisotropy results from the Earth’s motion relative to the CMB. (3) The subtraction of the dipole anisotropy eliminates most fluctuations in the map. The ones that remain are thirty times smaller. Hot regions are 0.0002 K hotter than the cold regions. (4) The anisotropies observed by Planck represent the seeds of all future structure formation in the Universe...... 11 1.4 The angular power spectrum of the Cosmic Microwave Background radiation from the Planck satellite. The standard cosmological model discussed here is able to reproduce the observed features to high accuracy, as shown by the curve. Figure provided by the Planck Collaboration [28]...... 12 1.5 A model rotation curve of the Milky Way galaxy is compared with data from several models. The red curves represent contributions from the bulge, disk and rings, and the dark halo. Figure from [29]...... 15 1.6 An exponential disk and halo are fitted to the observed rotation curve of the spiral galaxy NGC 3198. Figure from [30]...... 17 1.7 Distant galaxies are seen through Abell 1689, one of the most massive galaxy clusters. The effects of gravitational lensing may be used to determine the mass in the cluster. The representative color image is a composite of visible and near-infrared exposures [36]...... 20 viii

Figure Page 1.8 Composite image [37] of the Bullet Cluster with X-ray imaging from the Chandra X-ray Observator [38] and visible light from the Hubble Space Telescope [39]. The pink clumps are hot gas of normal baryonic matter in two clusters that are colliding. The blue clumps represent the concentrations of mass in the clusters based on gravitational lensing. The two regions are distinctly separate, demonstrating that most of the mass in the cluster is composed of dark matter...... 21 1.9 The Great Wall and the Sloan Great Wall presented at the same scale. They represent two of the largest coherent structures observed in the Universe. The Great Wall is a planar concentration of galaxies 500 million light years long. The Sloan Great Wall is an amalgamation of myriad clusters and galaxies. At 1.83 billion light years long, it is more than three time larger than the Great Wall. Image from [41]...... 24 1.10 Number density of WIMPs in the early Universe. Equilibrium abundances are given by the solid curve. The true abundance is represented by the dashed curves. Plot from [46]...... 27 2.1 The Feynman diagram of a dark matter particle and a standard model particle can be rotated to represent annihilation and production...... 32 2.2 Integrated recoil spectra as a function of energy threshold are shown for three targets, xenon, argon, and germanium, assuming values for the WIMP mass and WIMP-nucleon cross section...... 47 3.1 Light, charge, and phonons are used to detect dark matter via WIMP- nucleus scattering...... 51 3.2 Excitation and recombination of ion-electron pairs contribute to xenon’s scintillation signal. Bi-excitation quenches the scintillation signal for sufficiently high ionization densities. Figure from [74]...... 54 3.3 Compton scattering, photoelectric absorption, and pair production affect the attenuation of gammas in xenon...... 58 4.1 The principle of XENON’s dual-phase TPC is illustrated. An incident particle ionizes a xenon nucleus. Some of the free electrons recombine to produce a prompt scintillation (S1) signal; others drift along the field to produce a proportional scintillation signal (S2). The ratio of these two signals characterizes the type of recoil...... 66 ix

Figure Page 4.2 The benchmark WIMP region is defined by calibration sources. The electronic recoil events (blue) define a 99.75% rejection line. The nuclear recoil events (red) indicate where WIMP-induced recoils should occur in log(S2b/S1)- vs-energy parameter space. A minimal recoil energy is given by the detector’s threshold. Figure from [86]...... 68 4.3 The measured light signal depends on position. A light collection efficiency map was determined with the 662-keV gamma of 137Cs...... 69 5.1 The decay chain of 220Rn, following its emanation from a 228Th source. The three main analyses presented in this paper are labeled. Q-values are taken from [107], and half-lives and branching ratios from [108]...... 75 5.2 Temporal evolution of the various isotopes in the 220Rn decay chain as well as the 222Rn background. The times at which the source was opened and closed are indicated by the solid (green) and the dotted (red) line, respectively. The opening of the source is defined as time t = 0. Binning for 220Rn and 216Po is adjusted for times of low activity...... 76 5.3 Light collection map of XENON100 for high-energy alpha events, generated from 220Rn and 216Po decays. Events at low (high) Z and low (high) radius R have the highest (lowest) light collection and thus a correction factor less (greater) than unity...... 81 5.4 The alpha spectrum of the 220Rn decay chain is shown integrated over times before (40.3 hours, red) and after (148.7 hours, blue) the source is closed. The constant background of 222Rn (Figure 5.2) is visible only over the longer time period...... 82 5.5 Top: Number density of 220Rn216Po pairs across the TPC. Due to the short half-lives at the begining of the decay chain, these events are still concentrated near the entry point to the TPC. Bottom: Delayed coincidence of these pairs is used to map the average z-component of atomic velocity as a function of position. Atomic motion in XENON100 is primarily a result of a single convection cell, viewed here along the direction of its angular momentum vector...... 86 5.6 The spatial distribution of low-energy 212BiPo decays is shown. The events permeate the entire active region of the TPC...... 87 5.7 (Top) Time difference of BiPo coincidence events together with a fit of the half-life of 212Po. The lower bound of the fit is set at 720 ns to avoid peak- finding artefacts. (Bottom) The residuals of the fit in the top panel. ... 88 x

Figure Page 5.8 The fitted half-life of 212Po as a function of the minimum time difference that is considered. The lower bound of the half-life measurement is set at 720 ns. Based on the range [0.64, 0.90] µs, I estimate a 0.5-ns contribution to the total systematic uncertainty that results from the lower threshold of the time difference...... 89 5.9 The spatial distribution of low-energy decays is shown in two different perspectives. Since some clustering near the electrodes is apparent (top), events within 5 mm of either electrode have been rejected to display the XY - distribution (bottom). The distribution of BiPo events in Figure 5.6 shows that the beta emission of 212Pb permeates the entire active region of the TPC, confirming the suitability of this calibration source for low-energy detectors...... 92

129 6.1 Inelastic structure factors Sp (solid lines) and Sn (dashed) for Xe (top panel) and 131Xe (bottom panel) as a function of u = p2b2/2. The harmonic- oscillator length is b = 2.2853 fm for 129Xe and b = 2.2905 fm for 131Xe. Results are shown at the one-body (1b) current level and including two- body (2b) currents. The estimated theoretical uncertainty in the 2b currents is given by the red (Sp(u)) and blue (Sn(u)) bands...... 100 6.2 Comparison of the inelastic structure factors from Figure 6.1 to elastic counterparts from [126,127]. Both Sn(u) and Sp(u) are signiﬁcantly smaller than their elastic counterparts at u = 0. A maximum occurs for both isotopes in the elastic structure factors. Data for these structure factors are available in [121]...... 101

6.3 g(vmin)/g(0) as a function of vmin for several WIMP velocity distributions. 129 The lowest values of vmin are shown for elastic Xe (magenta), inelastic Xe 131 (cyan), and inelastic Xe (yellow) scattering. In the elastic case, vmin is calculated at XENON100’s nuclear recoil energy threshold of ∼ 7 keV (neglecting the diﬀerence in the nuclear mass number A)[109]...... 105

6.4 Diﬀerential nuclear recoil spectra dR/dER as a function of recoil energy ER for scattering oﬀ 129Xe and 131Xe, assuming “neutron-only” couplings of a −40 2 100 GeV WIMP with a WIMP-nucleon cross section σnucleon = 10 cm . Both elastic and inelastic recoil spectra are shown for comparison, including chiral 1b+2b currents, where the bands include the uncertainties due to WIMP-nucleon currents...... 106 xi

Figure Page

6.5 Differential energy spectra as a function of Evis for inelastic scattering off 129Xe (top) and 131Xe (bottom). Shown is the nuclear recoil spectrum (blue), the de-excitation gamma (green), and their sum (red), assuming realistic detector resolution and quenching of the nuclear recoil signal. As in Figure 6.4, results are shown with chiral 1b+2b currents, where the bands include the uncertainties due to WIMP-nucleon currents, and I have assumed “neutron- −40 2 only” couplings of a 100 GeV WIMP with σnucleon = 10 cm ...... 108 6.6 Integrated energy spectra of xenon for elastic and inelastic, spin-dependent scattering for “neutron-only” couplings and a 100 GeV WIMP with σnucleon = 10−40 cm2. The differential spectra are integrated from a given threshold value Evis to infinity. The inelastic contributions dominate over the elastic ones for moderate energy thresholds. As in Figure 6.4, results are shown with chiral 1b+2b currents, where the bands include the uncertainties due to WIMP-nucleon currents...... 109 6.7 Integrated total energy spectra of xenon as in Figure 6.6 for various WIMP masses from 25 − 500 GeV. The solid black curve is for a 100 GeV WIMP as in Figure 6.6. In all cases, minimum energies in Evis exist, above which the inelastic channel dominates the elastic one. The spectrum for a 500- GeV WIMP is lower than that of a 250- GeV WIMP because the particle number density in the galaxy decreases with increasing mass...... 110

7.1 The upper and lower panels show the neutrino luminosity Lνβ and mean

energy hEνβ i, respectively, as a function of the post-bounce time tpb for the 11 M (in blue) and 27 M (in red) SN progenitors with the LS220 EoS for νe (continuous lines),ν ¯e (dashed lines) and νx (dot-dashed lines). The panels on the left show the neutrino properties during the neutronization burst phase, the middle panes refer to the accretion phase, and panels on the right describe the Kelvin-Helmholtz cooling phase. The differences in the neutrino properties from different progenitors during the neutronization burst are small, but they become considerable at later times. The variation of the neutrino properties owing to a different nuclear EoS are smaller than the differences from a different progenitor mass, so for clarity, the progenitors with the Shen EoS are not shown here...... 119 xii

Figure Page

7.2 The upper panel shows the expected differential recoil spectrum dR/dER as a function of the recoil energy ER. The differential rate dR/dtpb as a function of the post-bounce time tpb is plotted in the middle panel. The lower panel represents the number of observable events R as a function of the detector’s energy threshold Eth. All panels show results for 11 M and 27 M progenitors with LS220 and Shen EoSs for a SN at 10 kpc. In the upper and lower panels, the neutrino flux is integrated over [0, 7] s after the core bounce, while the middle panel assumes Eth = 0 keV. All panels show that the event rate is larger for the 27 M SN progenitors while the LS220 EoS results in an O(25%) larger rate than the Shen EoS. Note that these rates are not directly observable since it is S1 and S2 that is measured, rather than ER...... 125 7.3 The left and right panels show the differential rates of the four SN progenitors in terms of the observable S1 and S2 signals, respectively. We have integrated the neutrino flux over the first 7 s after the core bounce and assumed that the SN burst occurs 10 kpc from Earth. Note that the axes in the right panel have units of 100 PE compared to PE in the left panel meaning that the S2 signal is generally larger than the S1 signal. The light and charge yields, Ly and Qy, respectively, have been set to zero below recoil energies of 0.7 keV. Integrated rates are given in Table 7.1...... 129 7.4 The detection significance is given as a function of the SN distance for a 27 M progenitor with LS220 EoS. The SN signal has been integrated over [0, 7] s. The different bands refer to XENON1T (red), XENONnT and LZ (blue), and DARWIN (green). The band width reflects uncertainties from our estimates for the background rate, discussed in Section 7.3. The vertical dotted lines mark the centre and edge of the Milky Way as well as the Large and Small Magellanic Clouds (LMC and SMC, respectively). For this SN progenitor, XENONnT/LZ could make at least a 5σ discovery of the neutrinos from a SN explosion anywhere in the Milky Way. DARWIN extends the sensitivity beyond the SMC...... 137 7.5 Event rate from an S2-only analysis as a function of the post-bounce time for a SN burst at 10 kpc. The event rate is shown for a 27M SN progenitor with LS220 EoS for three target masses: 2, 7, 40 tonnes in red, blue, and green respectively. The left panel covers the full time evolution with 500 ms time bins, including the Kelvin-Helmholtz cooling phase. The right panel shows the early evolution with 100 ms time bins and focuses on the neutronization and accretion phases...... 139 xiii

Figure Page

7.6 The left panel shows the reconstructed average neutrino energy hET i and amplitude parameter AT (green dot-circle) compared to the true value for the 27 M LS220 EoS progenitor at 10 kpc from Earth integrated between 0.1 s and 1 s (black triangle). Also shown are the 1σ contours from 2-, 7- and 40-tonne mock experiments following our ML analysis. The right panel shows the reconstructed neutrino flux as a function of the neutrino energy. The dashed green line represents the differential flux obtained with the best fit ML estimators and is compared with the true flux, shown by the dashed black line. Also shown are the 1σ intervals from our 2-, 7- and 40-tonne mock experiments...... 142 7.7 The reconstructed 1σ band of the total energy emitted into neutrinos in 30 mock experiments for each of XENON1T (red), XENONnT/LZ (blue) and DARWIN (green). The true value for the 27 M LS220 EoS progenitor integrated over the total time of the SN burst (taken as the first 7 s) is shown by the dashed vertical line...... 145 7.8 Variations in the dR/dS2 differential spectrum under different assumptions for Qy for the 27 M LS220 EoS progenitor at 10 kpc and integrated over the first 7 s. The quantity Qy,min is the energy below which Qy = 0 and the solid, dashed and dotted lines correspond to Qy,min values of 0.1 keV, 0.4 keV and 0.7 keV. The inset shows the Lindhard and Bezrukov Qy models together with the LUX measurements. The differences in dR/dS2 between the Lindhard and Bezrukov models are reasonably small compared to the larger differences from varying Qy,min...... 151 xiv

ABSTRACT

Reichard, Shayne Ph.D., Purdue University, December 2016. The Search for Dark Mat- ter in XENON: Innovative Calibration Strategies and Novel Search Channels. Major Professor: Rafael F. Lang.

The direct detection dark matter experiment XENON1T became operational in early 2016, heralding the era of tonne-scale dark matter detectors. Direct detection experiments typically search for elastic scatters of dark matter particles off target nuclei. XENON1T’s larger xenon target provides the advantage of stronger dark matter signals and lower background rates compared to its predecessors, XENON10 and XENON100; but, at the same time, calibration of the detector’s response to backgrounds with traditional external sources becomes exceedingly more difficult. A 220Rn source is deployed on the XENON100 dark matter detector in order to address the challenges in calibration of tonne-scale liquid noble element detectors. I show that the subsequent 212Pb beta emission can be used for low-energy electronic recoil calibration in searches for dark matter. The isotope spreads throughout the entire active region of the detector, and its activity naturally decays below background level within a week after the source is closed. I find no increase in the activity of the troublesome 222Rn background after calibration. Alpha emitters are also distributed throughout the detector and facilitate calibration of its response to 222Rn. Using the delayed coincidence of 220Rn-216Po, I map for the first time the convective motion of particles in the XENON100 detector. Additionally, I make a competitive measurement

212 of the half-life of Po, t1/2 = (293.9 ± (1.0)stat ± (0.6)sys) ns. In contrast to the elastic scattering of dark matter particles off nuclei, I explore inelastic scattering where the nucleus is excited to a low-lying state of 10 − 100 keV, with a subsequent prompt de-excitation. I use the inelastic structure factors for the odd- xv mass xenon isotopes based on state-of-the-art large-scale shell-model calculations with chiral effective field theory WIMP-nucleon currents, finding that the inelastic channel is comparable to or can dominate the elastic channel for momentum transfers around 150 MeV. I calculate the inelastic recoil spectra in the standard halo model, compare these to the elastic case, and discuss the expected signatures in a xenon detector, along with implications for existing and future experiments. The combined information from elastic and inelastic scattering will allow for the determination of the dominant interaction channel within one experiment. In addition, the two channels probe different regions of the dark matter velocity distribution and can provide insight into the dark halo structure. The allowed recoil energy domain and the recoil energy at which the integrated inelastic rates start to dominate the elastic channel depend on the mass of the dark matter particle, thus providing a potential handle to constrain its mass. Similarly, now that liquid xenon detectors have reached the tonne scale, they have sensitivity to all flavors of supernova neutrinos via coherent elastic neutrino-nucleus scattering. I consider for the first time a realistic detector model to simulate the expected supernova neutrino signal for different progenitor masses and nuclear equations of state in existing and upcoming dual-phase liquid xenon experiments. I show that the proportional scintillation signal (S2) of a dual-phase detector allows for a clear observation of the neutrino signal and guarantees a particularly low energy threshold, while the backgrounds are rendered negligible during the supernova burst. XENON1T (XENONnT and LZ; DARWIN) experiments will be sensitive to a supernova burst up to 25 (35; 65) kpc from Earth at a significance of more than 5σ, observing approximately 35 (123; 704) events from a 27 M supernova progenitor at 10 kpc. Moreover, it will be possible to measure the average neutrino energy of all flavors, to constrain the total explosion energy, and to reconstruct the supernova neutrino light curve. My results suggest that a large xenon detector such as DARWIN will be competitive with dedicated neutrino telescopes, while providing complementary information that is not otherwise accessible. xvi 1

1. DARK MATTER

If one were to ask an astronomer or a physicist what the universe is made of, the answer would be “We don’t know.” Admittedly, a task as daunting as knowing the intricacies of the Universe should give scientists difficulty, but surely we are not so ignorant. Astronomers can ascertain the chemical composition of stars and galaxies, and physicists can infer characteristics of the early Universe. The problem, however, is that we tacitly restrict ourselves to matter that is composed of atoms. Familiar objects, like humans and stars, may be made of atoms, but that does not necessarily hold true for the Universe as a whole. On the contrary, a significant body of evidence would indicate that the majority of matter in the Universe is not composed of atoms, at all. Instead, the Universe possesses a large amount of unknown mass, called dark matter, and unknown energy, called dark energy. Observations yield precise percentages of the Universe that contain dark matter, dark energy, and ordinary matter, but still the nature of dark matter remains enigmatic. “Dark matter” is the name given to the unknown mass in the Universe precisely because it does not interact electromagnetically (and, I suppose, because it reflects our ignorance about it). Its existence is surmised based on gravitational effects observed in ordinary matter, such as stars and luminous gas clouds. To exhibit these gravitational influences, the Universe must contain more mass than that which emits visible light. In order to fully understand from where dark matter comes, as well as its implications, one must first understand the geometry of the Universe. 2

1.1 The Friedmann Equation

The observed homogeneity and isotropy allow one to describe the geometry and evolution of the Universe. In fact, it is straightforward to propose that the universe is spatially homogeneous and isotropic but evolves over time [1]. This idea means that, in general relativity, the universe can be decomposed into spacelike slices such that each three-dimensional slice is maximally symmetric. The spacetime metric takes the form

ds2 = −dt2 + R2(t)dσ2, (1.1) where t is the timelike coordinate, R(t) is a scale factor, and dσ2 is the metric of a three-manifold Σ that can be expressed as

2 i j dσ = γij(u)du du , (1.2) with the coordinates on Σ, (u1, u2, u3), and a maximally symmetric three-dimensional metric, γij. These coordinates are known as co-moving coordinates–they are free of mixed terms dtdui, and there is no dependence between the coeﬃcient of dt and any of the ui. Only a co-moving observer will see the universe as isotropic. An observer on Earth will actually see a dipole anisotropy in the cosmic microwave background from the conventional Doppler shift that arises because Earth is not quite a co-moving reference frame. The spacetime metric is one such maximally-symmetric manifold that evolves over time and can be written as

dr¯2 ds2 = −dt2 + R2(t) +r ¯2dΩ2 , (1.3) 1 − kr¯2 with spacetime curvature k. This equation is known as the Robertson-Walker (RW) metric. As it is now, the scale factor, R(t), has units of distance, and the radial 3 coordinater ¯ is dimensionless. Fortunately, given that Eq. 1.3 is invariant under the transformation

R → λ−1R

r¯ → λr¯

k → λ−2k, (1.4) a convenient normalization may be chosen at will. Hence, one can deﬁne a dimensionless scale factor R(t) a(t) = , (1.5) R0 a coordinate with dimensions of distance

r = R0r,¯ (1.6) and a curvature parameter with dimensions of (length)−2

k κ = 2 . (1.7) R0 The Robertson-Walker metric transforms into

dr2 ds2 = −dt2 + a2(t) + r2dΩ2 . (1.8) 1 − κr2

The relationship between the scale factor and the energy-momentum of the universe is determined by Einstein’s equation

1 R − Rg − Λg = 8πGT , (1.9) µν 2 µν µν µν with the Ricci tensor Rµν, the Ricci scalar R, the space-time metric gµν deﬁned by Eq. 1.8, the cosmological constant Λ, Newton’s constant G, and the energy-momentum tensor

Tµν = −pgµν + (p + ρ)uµuν, (1.10) which describes the energy density, ρ, and the isotropic pressure, p, of the Universe under the assumption that matter is a perfect fluid; u = (1, 0, 0, 0) is the velocity 4 vector for the isotropic fluid in co-moving coordinates. Gliner [2] and Zel’dovich [3] have adopted an interpretation of Λ as an effective energy-momentum tensor for the vacuum of Λgµν/8πG. To calculate the Ricci tensor, it is necessary to first find the Christoffel symbols via the relation

1 Γλ = gλσ(∂ g + ∂ g − ∂ g ). (1.11) µν 2 µ νσ ν σµ σ µν

One ﬁnds that the Christoﬀel symbols are

aa˙ κr Γt = Γr = rr 1 − κr2 rr 1 − κr2 t 2 t 2 2 Γθθ = aar˙ Γφφ = aar˙ sin θ

r 2 r 2 2 Γθθ = −r(1 − κr )Γφφ = −r(1 − κr ) sin θ a˙ 1 Γr = Γθ = Γφ = Γθ = Γφ = tr tθ tφ a rθ rφ r φ θ Γθφ = cot θ Γφφ = − sin θ cos θ, (1.12) or related by symmetry. In turn, the nonzero components of the Ricci tensor, calculated by

α λ α α λ α λ Rµν = R µαν = ∂αΓνµ − ∂νΓαµ + ΓαλΓνµ − ΓνλΓαµ, (1.13) are

a¨ R = −3 00 a aa¨ + 2˙a2 + 2κ R = 11 1 − κr2 2 2 R22 = r (aa¨ + 2˙a + 2κ)

2 2 2 R33 = r (aa¨ + 2˙a + 2κ)sin θ, (1.14) and the Ricci scalar,

µ µν R = R µ = g Rνµ, (1.15) is a¨ a˙ 2 κ R = 6 + + . (1.16) a a a2 5

Finally, solving for the time-time component of Eq. 1.9( µν = 00) yields Friedmann’s equation a˙ 2 8πG κ 1 = ρ − + Λ. (1.17) a 3 a2 3 If one interprets the −κ/a2 term as the total energy, then the evolution of the Universe is dictated by the opposition of the potential energy, 8πGρ/3, and the kinetic energy (˙a/a)2. If the dependence of ρ on a(t) is known, Friedmann’s equation is suﬃcient to determine a(t). The rate of expansion is determined by the Hubble parameter

a˙ H = . (1.18) a

The contemporary value of the Hubble parameter is the Hubble constant, H0. Mea- surements demonstrate that the Hubble constant is

H0 = 100h km/s/Mpc, (1.19) where h ≈ 0.68 [4,5]. Customarily, the Friedmann equation is divided by H to obtain

8πG κ Λ ρ − + = 1. (1.20) 3H2 a2H2 3H2

This equation can be used to deﬁne a critical density such that κ = 0 when Λ = 0,

3H2 ρ = . (1.21) crit 8πG

The cosmological density parameter Ωtot is deﬁned as the density relative to the critical density, ρ Ωtot = . (1.22) ρcrit The critical density, which changes over time, is named such because the Friedmann equation may be expressed as

Λ κ Ω + − 1 = . (1.23) 3H2 a2H2 6

Hence, the sign of κ depends which of the following relations ρ satisﬁes:

Ω < 1 ⇔ κ < 0 ⇔ open

Ω = 1 ⇔ κ = 0 ⇔ ﬂat

Ω > 1 ⇔ κ > 0 ⇔ closed. (1.24)

The density parameter reveals which of the three RW geometries describes the Universe in which we live. Measurements of the cosmic microwave background anisotropy lead us to believe that Ω is very close to unity [6–8]. In cosmology, a common practice is to express the density of a species as a fraction

ρi of the critical density, Ωi = . It follows that Friedmann’s equation can be written ρcrit as

Ωmatter − Ωκ + ΩΛ = 1. (1.25) P In principle, the division of species can be arbitrarily speciﬁc as long as i Ωi = 1.

1.2 Big Bang Nucleosynthesis

Big-Bang nucleosynthesis (BBN) is a critical part of the standard cosmological model. The synthesis of light elements is sensitive to physical conditions of the early Universe. Predictions of the abundances of D, 3He, 4He, and 7L, synthesized within the ﬁrst three minutes, validate the standard Big-Bang model. BBN provides strict constraints on deviations from standard cosmology and on new physics beyond the Standard Model [9–11]. In the beginning, protons and neutrons would collide continuously to fuse into deuterium, but the ambient radiation would cause the deuterium to photodissociate. After

4 minutes (T ≈ 80 keV/kB), photodissociation no longer occurred at an appreciable rate. Consequently, light elements—such as deuterium, tritium, helium and lithium— began to form and accumulate. When the universe was 24 minutes old (T ≈ 30 keV/kB), the population of neutrons was depleted, leaving only charged nuclei. At this point, the Coulomb barrier suppressed any additional fusion. Given that helium is energetically 7 favorable, nearly all of the neutrons end up in 4He. The fraction of baryons that end up in helium, or the helium mass fraction, is sensitive to the neutron-to-proton ratio, n/p (but not the overall baryon density). At the higher temperatures in the early Universe, weak interactions were in thermal equilibrium, ﬁxing the neutron-to-proton ratio to n/p = e−Q/T , where Q = 1.293 MeV is the neutron-proton mass diﬀerence. As the Universe cooled, the neutron-proton conversion rate, Γ ∼ T 5, fell more rapidly than the Hubble expansion rate, H ∼ T 2.

Hence, the neutrons and protons departed from chemical equilibrium at Tfr ' 1 MeV.

The neutron fraction at this point, n/p = e−Q/Tfr ' 1/6, was inﬂuenced by every known physical interaction. Q is determined by both the strong and electromagnetic forces; and Tfr depends on the weak and gravitational interactions. After freeze-out, neutrons were free to β decay, further reducing the neutron fraction to n/p ' 1/7 before nuclear reactions began. A simple analytic model of freeze-out yields the n/p ratio to an accuracy of ∼1% [12,13]. For n/p ' 1/7, one helium nucleus forms for about every twelve hydrogen nuclei, meaning we expect a helium mass fraction of ∼25%, with the remainder almost entirely in the form of hydrogen. Weak interactions compete with the expansion of the early Universe. The higher the baryon density, the sooner the deuterium bottleneck is breached. Then, the deuterium can form at a higher temperature, which corresponds to a higher neutron fraction. This eﬀect would allow heavier nuclei to form in higher abundances during BBN. Thus, a higher baryon density leads to a lower deuterium abundance after nucleosynthesis concludes. Deuterium is an excellent tool by which to measure the baryon density at the beginning of nucleosynthesis. Deuterium does not experience any sustained production processes [14], and, due to its large binding energy (2.2 MeV), deuterium is destroyed only in the evolution of the Universe. One can ascertain the chemical evolution of stars, and, wherever no metallicity has accrued, the observed deuterium abundance should be close to the primordial deuterium abundance. 8

As the abundances of deuterium and other light elements depend almost solely on the baryon-to-photon ratio, η, they can be used to probe the early Universe. Figure 1.1 displays the helium mass fraction and the abundances of D, 3He and 7Li relative to H,

10 3 −5 as a function of η10 ≡ 10 η [15]. The observed fractions of D and He are 10 , and the fraction of 7Li is 10−10 [13, 16]. Predictions from BBN can be compared to these observed abundances of light elements, which yield

5.1 < η10 < 6.5. (1.26)

Ω b 10-2 10-1 1

10-1 4He 10-2 (D/H) 10-3 P

10-4

-5 10 (3He/H) P -6

Abundance 10

10-7

10-8 7 -9 ( Li/H) 10 P

10-10

1 η 10 10

Figure 1.1. The abundances of D, 3He, and 7Li (relative to H) are predicted by Standard Big Bang Nucleosynthesis in terms of the abundance parameter, η10. The band widths represent uncertainties in nuclear and weak interactions. Observed primordial abundances indicate that the baryon density is Ωb ' 0.05. Plot based on data from [15]. 9

η10 relates to the fraction of baryonic matter, Ωb, by

2 η10 = 274Ωbh . (1.27)

2 The Wilkinson Microwave Anisotropy Probe (WMAP) obtained Ωbh = 0.0225 ±

0.0006 [17], which corresponds to η10 = 6.16 ± 0.16; the Planck Collaboration measured 2 Ωbh = 0.02205 ± 0.00028 [4], corresponding to η10 = 6.042 ± 0.077. These values “predict” the abundances of light elements to compare to observation [18]. The resultant D/H abundance agrees well with what is found in quasar absorption systems, and that of 3He agrees reasonably well with abundances in extra-galactic HII regions. However, there is poor agreement with the Li abundances observed in the atmospheres of halo dwarf stars [19].

3 The estimates of baryon density derived from deuterium and He are Ωb = 0.047 ± +0.017 0.003 and Ωb = 0.044−0.011, respectively [15]. If we were to trust these calculations, we would have to conclude that ordinary matter constitutes only ∼5% of the energy content in the Universe. One has to wonder what the other ∼95% of the Universe holds.

1.3 The Cosmic Microwave Background

The second key piece of evidence for our cosmological model is the presence of relic background radiation—the Cosmic Microwave Background (CMB)—discovered by Penzias and Wilson in 1964 [20]. The CMB arose from the heat of the Universe itself, and, as such, should possess a perfect thermal radiation spectrum. When the Universe was 380,000 years old, the temperature had fallen to 3,000 K. This temperature was sufficiently low to permit the recombination of charged nuclei and free electrons. Without electrons floating around, the Universe became transparent to the photons, and they were able to escape the hot plasma of hydrogen, helium, and electrons that existed since the end of BBN. At 3,000 K, the CMB spectrum originally peaked in visible light with wavelengths of a few hundred nanometers. Since then, the Universe has expanded by a factor of 103, stretching these photons by the same amount. Thus, their 10 wavelengths have shifted to about a millimeter, right in the middle of the microwave portion of the spectrum, corresponding to a temperature of a few Kelvin. In 1989, the NASA satellite Cosmic Background Explorer (COBE) was launched to investigate the CMB. The black-body spectrum of the CMB, shown in Figure 1.2, fits quite well to the theoretically calculated thermal radiation spectrum of 2.73 K [21–23]. It was already known that the CMB is uniform. The state of the early Universe must have been extremely uniform to produce such a smooth radiation spectrum. At first, this feature was regarded as evidence against the Big Bang because the early Universe must have had regions of enhanced density in order to form galaxies and clusters. However, COBE measurements demonstrated that the CMB was not perfectly uniform. Rather, the temperture varies by a few parts in 105 from one point to another.

Figure 1.2. The intensity of the Cosmic Microwave Background versus frequency. The curve represents the theoretical spectrum of a blackbody with T = 2.73 K. The data [21] match the curve well enough that the width of the curve hides the uncertainties of each point. Figure from [22]. 11

Figure 1.3. False color images of the sky observed in microwave frequencies [24, 25]. (1) The temperature is uniform at a large scale. (2) The same map is shown with temperatures between 2.721 K and 2.729 K. The dipole anisotropy results from the Earth’s motion relative to the CMB. (3) The subtraction of the dipole anisotropy eliminates most ﬂuctuations in the map. The ones that remain are thirty times smaller. Hot regions are 0.0002 K hotter than the cold regions. (4) The anisotropies observed by Planck represent the seeds of all future structure formation in the Universe. 12

More recently, NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) and ESA’s Planck satellite have provided conﬁrmation of the minute temperature ﬂuctuations [26, 27]. A close inspection of the CMB, reconstructed at various stages in Figure 1.3, reveals the distribution of matter in the early Universe. The variations in temperature verify that the density of the Universe varied from place to place. The “seeds” of structure formation were present at the time of nucleosynthesis.

Figure 1.4. The angular power spectrum of the Cosmic Microwave Back- ground radiation from the Planck satellite. The standard cosmological model discussed here is able to reproduce the observed features to high accuracy, as shown by the curve. Figure provided by the Planck Collabora- tion [28].

The CMB temperature and polization power spectra provide a wealth of information about the physical composition of the Universe. Given an angular scale, the temperature power spectrum separates the intensities of hot and cold spots. Figure 1.4 illustrates the angular power spectrum of the CMB from which we can obtain measurements of the following parameters: 13

1.3.1 The Curvature of the Universe

The angular size of a peak depends both on distance and the spatial geometry of the Universe. In Euclidean geometry, the angular size is given by

λ θ = , (1.28) d where λ is the wavelength of the peak and d is the sonic horizon. However, in hyperbolic geometry, the angular scale would be smaller, and, in spherical symmetry, the scale

−0.0065 would be larger. Planck constrains the curvature to Ωκ = −0.0010+0.0062, indicating that the Universe is (very nearly) ﬂat [4].

1.3.2 Baryon Density

A larger baryon density supplies a stronger gravitational force to resist radiation pressure. The peaks that correspond to compression of sound waves (the odd ones) are higher, and the peaks that correspond to rarefaction (the even ones) are lower. A second eﬀect of baryon density occurs in damping. A larger density reduces the diﬀusion length, and, thus, damping moves to a smaller angular scale. Thirdly, the speed of sound will diminish as the baryon density increases because baryons are heavy. Consequently, the frequency of oscillation will decrease, shifting the peaks and making the space between peaks larger. Recent measurements place the baryon density at

Ωb = 0.0479 ± 0.0007 [4].

1.3.3 Dark Matter Density

Oscillations that began in a radiation-dominated Universe should differ from oscillations that began in a matter-dominated Universe. Radiation drives oscillation, but matter does not. Radiation dominated the Universe only in its earliest epochs because the radiation density decreases more quickly than matter density as the Universe expands. As oscillations of smaller wavelengths begin first, only the peaks at smaller 14 angular scales are affected by this driving force. Therefore, the sizes of the peaks at lower multipoles clearly indicate the ratio of dark matter density to radiation density.

The CMB tells us that the dark matter density is Ωχ = 0.2565 ± 0.0067 [4].

1.4 Galaxies

The search for dark matter begins in our own Milky Way Galaxy. Then, the scope is extended to other galaxies and galaxy clusters.

1.4.1 Milky Way Galaxy

A celestial body’s orbital velocity relates directly to the mass enclosed within its orbit. If we approximate the the object’s orbit as circular, then

mv2 GM(r)m rot = (1.29) r r2 r GM(r) v = , (1.30) rot r for Z r M(r) = 4π ρ(r0)r02dr0. (1.31) 0 This relationship, in principle, provides the means by which to ascertain a complete distribution of mass in a galaxy by observing the velocities of stars at every unique distance from the galactic center. In practice, though, interstellar dust obscures our view of stars that are more than a few thousand light years away, making it difficult to measure stellar velocities. Fortunately, radio waves penetrate the dust, and clouds of atomic hydrogen gas emit a 21-cm spectral line. The Doppler shift of this 21-cm line reveals an object’s (orbital) velocity toward or away from Earth. For a rigid object, all points rotate in circular paths about the center of mass in the same rotational period. Points farther from the center of mass move in larger circles and, as a result, move faster. The orbital speed increases linearly with distance. In contrast, the orbital speed of the planets falls off with distance from the Sun because 15 the Sun possesses virtually all of the mass in the solar system. As the force wanes, the orbital speed diminishes. For the Milky Way, the orbital velocity flattens around 200 km/s [29], shown in Figure 1.5, which is distinctly different from the sharp decline in the rotation curve of the solar system. It follows that the mass is not concentrated at the center of the galaxy. On the contrary, the orbits of each subsequently distant body must encompass more mass. The flatness of the Milky Way’s rotation curve proves that the majority of our galaxy’s mass lies beyond the Sun, tens of thousands of light years from the galactic center.

250

200 V km/s

150

100 0 5 10 15 20 R kpc

Figure 1.5. A model rotation curve of the Milky Way galaxy is compared with data from several models. The red curves represent contributions from the bulge, disk and rings, and the dark halo. Figure from [29].

1.4.2 Other Spiral Galaxies

Other galaxies contain substantial amounts of dark matter, as well. It is possible to determine the amount of dark matter in a galaxy by comparing the galaxy’s mass to its luminosity. First, the galaxy’s luminosity is used to estimate the amount of mass 16 that the galaxy contains in the form of stars. Then, the total mass is calculated from the observed orbital velocities of stars and gas clouds. If the total mass is larger than the mass attributed to stars, one concludes that the excess is due to dark matter. To measure a galaxy’s luminosity, one gazes through a telescope and measures its apparent magnitude. The magnitude and its known distance from Earth provide the luminosity. A measurement of the galaxy’s total mass requires measurements of the orbital speed of stars and gas clouds far from the galaxy’s center. Atomic hydrogen gas clouds are found in spiral galaxies farther from the center than stars. Most data come from radio observation of the 21-cm line of these clouds. Doppler shifts of the 21-cm line indicates how quickly a cloud moves toward or away from Earth. The observations of several clouds at different orbital distances are used to construct the galaxy’s rotation curve. The rotation curves of most spiral galaxies, such as NGC 3198 whose velocity curve is plotted in Figure 1.6, end up being flat. Similar to the case of the Milky Way Galaxy, the flat rotation curves demonstrate that a significant portion of the matter lies far out in the halo of the galaxy. The composition of a typical spiral galaxy is 90% or more dark matter and 10% or less ordinary matter.

1.4.3 Elliptical Galaxies

One must employ a diﬀerent technique to evaluate elliptical galaxies because they are not supported by rotation (even if they were, they contain only trace amounts of atomic hydrogen gas). The motion of stars in elliptical galaxies are sporadic and thus cannot be synthesized into a meaningful rotation curve. Even so, their velocities still depend on the mass in the orbits. Every star has a unique Doppler shift, with some stars moving toward Earth and others away. The spectrum of the galaxy reﬂects the combination of all these Doppler shifts, yielding a broad line over a range of wavelengths rather than a narrow line. The larger the broadening, the faster the stars move relative to each other. The spectral lines 17

Figure 1.6. An exponential disk and halo are ﬁtted to the observed rotation curve of the spiral galaxy NGC 3198. Figure from [30].

from diﬀerent parts of the elliptical galaxy show that a star’s speed remains constant far from the center. As in spiral galaxies, elliptical galaxies contain matter beyond the center.

1.5 Galaxy Clusters

As stated previously, stars comprise up to 10% of the mass in a galaxy—the rest is dark matter. Observations of galaxy clusters imply that the fraction of dark matter is even greater. The mass in clusters can be up to 50 times the mass of stars. The evidence for dark matter in clusters arises from three methods of measuring cluster masses: the speeds of galaxies orbiting the center of the cluster, X-ray emissions of hot gas, and gravitational lensing. 18

1.5.1 Orbits of Galaxies

In the 1930s, while observing the Coma Cluster, Zwicky proposed that galaxy clusters are large clumps of galaxies bound together gravitationally, just like stars in a star cluster. As such, it is only natural that one could utilize galactic motion to measure a cluster’s mass. So, he commissioned an 18-in Schmidt telescope that had the ability to capture large numbers of galaxies in a single wide-angle photograph. Zwicky surveyed all galaxies in the Coma Cluster and used their measured redshifts to calculate their speeds relative to Earth. He determined the recession speed of the cluster—the speed at which expansion carries the cluster away from Earth—by averaging the speeds of the individual galaxies. Then, Zwicky found each individual galaxy’s speed relative to the cluster’s center. Finally, by application of the virial theorem, he calculated the total mass in the cluster. When Zwicky compared the Coma Cluster’s mass to its luminosity, he discovered that there is substantially more mass than one can attribute to the luminous matter. Accordingly, he named this excess “dunkle Materie” (dark matter) [31].

1.5.2 Hot Gas

A second technique to measure a cluster’s mass is based on the observation of X rays from the hot gas between galaxies [32–34]. The hot gas can help one to probe the mass in a cluster because its temperature depends on its mass. The gas is suﬃciently hot, typically tens of millions of degrees, to be (nearly) in a state of hydrostatic equilibrium— that is, the gas’s outward pressure balances the inward pull of gravity:

dP GM(r)ρ (r) = − gas (1.32) dr r2

Then, the ideal gas law can be written as

ρ (r)kT P = gas , (1.33) µmp 19 where µ is the mean molecular weight of the gas. And, combing these expressions, we ﬁnd the mass enclosed within a spherical volume of radius r,

kT r2 d M(r) = − (lnρgas(r)) (1.34) Gµmp dr

In this state, the average kinetic energy of particles is determined by the strength of gravity. Therefore, given that the temperature represents the average kinetic energy of particles, the temperature that is measured via X rays can be used to ascertain the mass content in the cluster. The total mass can be as much as 15 times larger than the luminous mass [35].

1.5.3 Gravitational Lensing

Up to this point, the evidence for dark matter has been based on Newton’s laws of gravity. In many different ways, Newton’s law continues to say that more matter exists in the Universe than we can see. It is reasonable to question whether this excess matter actually exists or whether there is a flaw in our comprehension of Newton’s laws. An alternative method derived from gravitational lensing provides valuable insight into this matter. Phenomenologically, gravitational lensing is the consequence of spatial distortion. A massive body bends space, thereby allowing it to alter the path light takes. The angle at which light bends around a massive body depends on that body’s mass; meaning, one can measure the body’s mass by observing the distortion. In Figure 1.7, countless distant galaxies are seen through Abell 1689, one of the most massive galaxy clusters. Abell 1689 acts as the lens that bends and magnifies the light from galaxies behind it. The degree to which the image is distorted far exceeds what one would expect based solely on the total mass of luminous matter. In Figure 1.8, we see a high-velocity merger of two subclusters in the Bullet Cluster. During this collision, the hot gas (rendered in pink) lags behind the subcluster galaxies (rendered in blue), which is illustrated by gravitational lensing. This observation is 20

Figure 1.7. Distant galaxies are seen through Abell 1689, one of the most massive galaxy clusters. The eﬀects of gravitational lensing may be used to determine the mass in the cluster. The representative color image is a composite of visible and near-infrared exposures [36].

explained by the fact that hot gas experiences collisions, but the subcluster galaxies do not. Furthermore, this interaction provides a limit on the self-interaction cross section of the invisible matter. A careful analysis of the distortion for which a cluster is responsible enables one to assess the cluster’s mass. Instead of Newton’s laws, Einstein’s general theory of relativity yields information about how massive the cluster must be to engender such a distortion. Reassuringly, this method produces results that generally agree with those of orbital velocities and hot-gas X-ray emissions. That these three diﬀerent methods demonstrate that clusters of galaxies carry vast amounts of excess mass strongly sup- ports the hypothesis of dark matter. 21

Figure 1.8. Composite image [37] of the Bullet Cluster with X-ray imaging from the Chandra X-ray Observator [38] and visible light from the Hubble Space Telescope [39]. The pink clumps are hot gas of normal baryonic matter in two clusters that are colliding. The blue clumps represent the concentrations of mass in the clusters based on gravitational lensing. The two regions are distinctly separate, demonstrating that most of the mass in the cluster is composed of dark matter.

1.6 Structure Formation

Stars, galaxies and clusters are gravitationally bound systems. Within these systems, gravity completely dominates the expansion of the Universe. Meaning, as the rest of the Universe expands, space is not expanding within our solar system, our galaxy, or in the vicinity of our galaxy. Theoretically, galaxies form from pockets of slightly higher density in the early Universe. After the Big Bang, the Universe expanded everywhere, and, gradually, these high-density pockets accreted ambient matter until the region stopped expanding. 22

Meanwhile, the Universe as a whole continues to expand. This accreted mass is known as a protogalactic cloud. If the mass in galaxies primarily takes the form of dark matter, then dark matter must be responsible for the creation of the protogalactic clouds. Hydrogen and helium gas within the clouds collapsed into a disc and transformed into stars, while dark matter remained on the periphery due to its inability to radiate its energy. Consequently, in the present day, luminous matter in a galaxy resides within a shell of dark matter that served as the catalyst for the galaxy’s formation. Clusters of galaxies manifest in the same manner. Initially, galaxies ﬂy apart with the expansion of the Universe. Then, at some point, the gravity from the galaxies’ dark matter will counteract and overpower the expansion of the Universe. The galaxies’ trajectories reverse, and they start to orbit each other. Some clusters seemingly have not completed their formation because they still at- tract new galaxies with their tremendous gravity. The Virgo Cluster, for example, continues to rein in the Milky Way and other galaxies. Hubble’s law tells us that, given the Virgo Cluster’s distance, the Milky Way should be drifting away from the Virgo Cluster as a result of universal expansion. On the other hand, the measured speed is 400 km/s slower than that predicted by Hubble’s law. Hence, one may reasonably conclude that the Virgo Cluster’s gravity pulls us against expansion. More explicitly, while it is true that the Milky Way and other galaxies still move away from the Virgo Cluster, their speeds are waning over time. Eventually, the galaxies may stop moving away entirely; at which point, the Virgo Cluster will draw them in closer until they merge. Clusters appear to yank at each other, as well, introducing the possibility of an even larger scale of gravitationally bound structures, called superclusters, in the Universe that are still in the early stages of formation. 23

1.7 Large-Scale Structures

Beyond 300 million light years from Earth, deviations from Hubble’s law resulting from gravitational pulls are negligible compared to universal expansion; so, Hubble’s law serves as the primary tool to measure galactic distances. Astronomers map the distribution of galaxies to reveal large-scale structures vaster than galaxy clusters. Structure formation commences with stars. Stars aggregate into galaxies, which group into clusters, which form superclusters. Beyond superclusters, lie sheets, walls and filaments, separated by immense voids. Before 1989, many scientists thought that galaxy clusters were the largest structures. However, at this time, Geller and Huchra discovered the “Great Wall,” a sheet of galaxies 500 million light years long, 200 million wide, and 15 million deep [40]. The map illuminated the complex structure of our part of the Universe. It demonstrated that galaxies are not scattered randomly throughout space, but they are arranged in large chains and sheets that span millions of light years. Clusters of galaxies are found at the intersections of these chains. Among these chains and sheets lie huge empty regions, or voids. This sheet eluded detection in the past because technology was not yet mature enough to handle the multitude of information necessary to properly map the three-dimensional postions of galaxies, requiring redshift measurements. In 2003, the Sloan Digital Sky Survey announced the discovery of the Sloan Great Wall, shown in Figure 1.9, which is a structure formed by a galaxy filament (a wall of galaxies observed among the millions of galaxies measured—spread out over one- fourth of the sky [41]). It extends 1.38 billion light years, or 1/60 the diameter of the observable universe. The Sloan Great Wall remained the largest known galaxy filament until 2013, when the Hercules-Corona Borealis Great Wall was discovered in a map of gamma-ray bursts based on data from the Swift Gamma-Ray Burst Mission and the Fermi Gamma-Ray Space Telescope [42, 43]. Immense structures like these have not yet collapsed into gravitationally bound systems. 24

Figure 1.9. The Great Wall and the Sloan Great Wall presented at the same scale. They represent two of the largest coherent structures observed in the Universe. The Great Wall is a planar concentration of galaxies 500 million light years long. The Sloan Great Wall is an amalgamation of myriad clusters and galaxies. At 1.83 billion light years long, it is more than three time larger than the Great Wall. Image from [41].

While the Universe may continue to grow at larger scales, there is a clear limit to its development. Beyond one billion light years, the distribution of galaxies is quite uniform, meaning that the Universe appears to be the same everywhere as suggested by the Copernican principle. Just as galaxies were born from regions of elevated density in the early Universe, these larger structures were also regions of elevated density. Galaxies, clusters, superclusters, the Great Wall, and the Sloan Great Wall very likely started as moderately-dense regions of diﬀerent sizes. The voids, conversely, likely started as low-density regions. 25

If this description of structure formation is correct, then the structures we see today must mirror the distribution of dark matter density in the early Universe. Supercom- puter models of structure formation are able to simulate the growth of galaxies, clusters, and larger structures from smaller points of elevated densities. These models showcase how we would expect to ﬁnd dark matter distributed in the observable universe [44].

1.8 Dark Matter Candidates

Plausibly, some of the dark matter particles could be ordinary, made of protons, neutrons and electrons, but with the quality that they do not emit detectable radiation. On the other hand, it is also likely that some of the dark matter particles are extraordinary, formed by particles that have yet to be discovered and understood. Given these distinctions, it is common to refer to ordinary and extraordinary dark matter as baryonic and non-baryonic matter, respectively. Matter does not necessarily need to be extraordinary to be dark. Any massive body that is too dim for modern telescopes to detect is classified as dark matter. Planets, brown dwarfs, and faint red main-sequence stars, for example, are too dim to be detected at present. Thus, in principle, it seems possible that many brown dwarfs, faint red stars, and Jupiter-size objects that are remnants of the Milky Way’s formation could still float around galaxy to this day, accounting for a large portion of the galaxy’s mass. These objects are named massive compact halo objects (MACHOs). By definition, MACHOs are too faint to observe directly, but there are other means by which to search for them. If trillions of dim faint stars and the like loiter around the halo, there are rare occasions when one drifts across our line of sight to a distant star. As the massive body lies between us and the star, its gravity focuses a portion of the star’s light toward Earth. Meaning, the distant star will appear to be (slightly) brighter for as long as the massive body traverses it. The duration of the lensing indicates the body’s mass. These observations show that such dim objects do, in fact, 26 exist; however, they are not numerous enough to account for all of the dark matter in the Milky Way [45]. Something else remains unseen. A more alluring prospect is that the dark matter in galaxies and clusters is not composed of baryonic matter at all. Neutrinos, for example, are dark by their very nature because the have no electrical charge and cannot emit electromagnetic radiation in any form. They are never found bound to charged particles in the sense that neutrons and protons are bound in atomic nuclei, so one is unable to discern their presence via particles that emit light. Particles like neutrinos interact with other particles only through two of the four (known) forces, gravity and the weak force. For this reason, they are called weakly interacting particles. The dark matter observed in galaxies cannot be composed of neutrinos because they traverse the universe at enormous speeds and, as such, easily escape the galaxy’s gravitational pull. Instead, one should seek a particle similar to the neutrino, but significantly heavier. These hypothetical particles, known as weakly interacting massive particles, would evade direct detection, and they would move more slowly as a consequence of their mutual gravitational influences, which holds them together. WIMPs may compose the majority of the dark matter in galaxies or clusters and, therefore, the majority of matter in the Universe. The nonrelativistic, nonelectromagnetic characteristics of dark matter form the basis of the Cold Dark Matter (CDM) paradigm.

1.8.1 Weakly Interacting Massive Particle

In the early Universe, Weakly Interacting Massive Particles (WIMPs) existed in thermal equilibrium with other particles. As the Universe expanded and cooled to a temperature below the WIMP mass, the WIMPs were too massive to remain in equilibrium. So, they began to self-annihilate, and their number density, n, fell exponentially at a rate Γ = hσvin, (1.35) 27 where hσvi is the thermally averaged cross section at a relative velocity v. In order to avoid extinction, there must be some type of asymmetry or the cross section must be suﬃciently low to leave a thermal relic. Once annihilation ceases, or “freezes out,” the particles that remain will move with a constant co-moving density. After a complete thermodynamical calculation, one ﬁnds that this thermal relic density is independent of the WIMP mass [46]: 3 · 10−27 cm3s−1 Ω h2 ' (1.36) χ hσvi The equilibrium and true abundances in a comoving volume are shown in Figure 1.10 as a function of x ≡ mχ/T . Larger annihilation cross sections lead to lower relic abundances.

Figure 1.10. Number density of WIMPs in the early Universe. Equilibrium abundances are given by the solid curve. The true abundance is represented by the dashed curves. Plot from [46].

A new weak-scale particle would have an annihilation cross section given by

hσvi ∼ α2(100 GeV)−2 ∼ 10−25 cm3s−1 (1.37) 28 for α ∼ 10−2. This value lies within a few orders of magnitude of the value required to explain the dark matter in the Universe. There is no obvious reason why a weak-scale parameter should fall this close to a cosmological parameter. The discovery of density enhancements further solidifies the idea that some of the dark matter consists of WIMPs and that their gravitation effects are responsible for structure formation. To explain that galaxies formed within a few billion years, the higher densities in the early Universe must have been considerably larger than that for which the temperature variations of a few parts in 105 could account. WIMPs by definition do not interact electromagnetically and, thus, would not directly affect the temperature of the CMB. However, WIMPs use gravity to accumulate clumps of ordinary matter, which does interact with photons. It follows that the small density enhancements detected by microwave telescopes echo a larger density enhancement composed of WIMPs. The WIMP hypothesis would account for spiral-galaxy halos rather than flattened disks as we observe for visible matter. Galaxies formed as gravity brought together matter of slightly larger densities in the early Universe. The matter would have con- tained mostly dark matter with bits of hydrogen and helium gas. As the gas particles collide, they convert their orbital energy to radiation that escapes the galaxy as photons. Consequently, the gas would collapse into a disk. On the other hand, WIMPs, which rarely interact or exchange energy with other particles, remain in their orbitals far from the galactic center. The faintest galaxies have the densest dark halos, which can be up to 103 denser than halos in the brightest parts of galaxies. The low-mass halos emerged first from the Universe as it expanded.

1.8.2 Axion

Initial speculation of the nature of dark matter resulted in two main ideas: that dark matter might be composed of WIMPs or, alternatively, that dark matter might be composed of light bosons, called axions [47]. The axion was proposed by Peccei-Quinn 29 theory as a solution to the strong CP problem [48]. The original model is ruled out, but axions that arise from higher symmetry-breaking energy scales are still allowed [49]. If axions exist they would have been created at the QCD phase transition. When Sikivie demonstrated that axions could be seen through coupling to two photons by conversion to a microwave photon in a magnetic ﬁeld [50], experiments diverged and specialized. Axion searches focused on improved cavity microwave experiments for dark matter, as in the Axion Dark Matter experiment (ADMX) [51], while WIMP searches concentrated on nuclear recoil detection through ionization, scintillation and vibrational methods using increasingly massive targets and lower background. The substantial improvements in background, which result from lower levels of radioactivity and self-shielding of the outer target region, have made it possible to use WIMP detectors to detect or set limits upon axions and other new types of hypothesized particles that may compose galactic dark matter. Astronomical observation is the most sensitive technique to detect axions. The Sun generates many of the observable axions through Bremsstrahlung, Compton scattering, axio-recombination, and axio-deexcitation. Axions may induce observable signals by coupling to photons, electrons or nuclei. Coupling to electrons may be tested by scattering oﬀ the electron cloud of a target, such as liquid xenon [52]. Axions, which are non-relativistic, would be entirely converted into electron recoil energy, yielding a monochromatic signal. 30 31

2. DETECTION OF DARK MATTER

To this day, we still do not know the nature of the most common substance in the Universe. This invisible matter neither emits nor absorbs electromagnetic radiation of any kind; we are cognizant of its existence only through its gravitational effects, which dominate on galactic scales. Evidence for dark matter exists on various scales, in fact, and it is important to acknowledge that the dark matter that binds a spiral galaxy may differ from the dark matter that collects and secures galaxies into clusters. This point is especially relevant for terrestrial experiments whose primary concern is the dark matter in the Milky Way halo and not necessarily the dark matter that lies in deepest recesses of space. The methods of dark matter detection, illustrated in Figure 2.1, fall under one of three categories: direct, indirect, and production. In direct detection, a dark matter particle and target nucleus collide; in indirect detection, dark matter annihilates or decays; and, in production, two standard model particles collide in a laboratory to produce dark matter particles. Each method of detection has its advantages and disadvantages, but all of the experimental endeavors are based upon the fundamental idea to continually identify, reduce, and reject background events in order to observe a rare dark matter event. A claim of discovery will have to be verified with at least two different methods.

2.1 Collider Experiments

In science, there are known knowns, and there are known unknowns. Neither of these is necessarily a problem. What is certainly a problem, however, are the unknown unknowns. Both direct and indirect searches are, in principle, limited by the things we don’t know that we don’t know. Particle production at colliders, on the other hand, 32

DM DM Annihilation

Production SM SM Scattering

Figure 2.1. The Feynman diagram of a dark matter particle and a standard model particle can be rotated to represent annihilation and production.

are able to make matter regardless of whether we know anything at all about what we produce. Of course, there must be some type of interaction within the Standard Model that connects exotic particles to ordinary ones; otherwise, there would be no hope of detection. We know that dark matter interacts gravitationally, and that it does not interact electromagnetically or via the strong nuclear force. What we aren’t sure about is whether dark matter interacts through the weak nuclear force. A collider experiment can use an intermediate heavy particle to create dark matter via direct weak production. Basically, in direct weak production, the ﬁnal state of a collision consists of two WIMPs that may be accompanied by jets, a photon, a Z or W boson, a lepton, or even nothing at all. These ﬁnal states’ signatures will then be these products in addition to missing transverse energy (MET). One tries to remain agnostic about what occurs on the inside of these production mechanisms, which are generally based on quark (qq → χχ) or gluon (gg → χχ) collisions. 33

A beneﬁt of colliders is that they possess the unique possibility to probe the quark and gluon couplings to dark matter. For a given model and mass scale, a prediction can be made about the rates in direct- and indirect-detection experiments, such as XENON and FermiLAT. While a collider experiment could demonstrate that dark matter can be created, a direct-detection experiment would still be left to demonstrate that dark matter exists in large enough quantities to explain astrophysical observations.

2.2 Indirect Detection

WIMP annihilation or decay provides a second method by which to search for dark matter. If WIMPs are Majorana particles, they can self-annihilate to produce gamma rays or Standard Model particles; or, if WIMPs are unstable, they will decay into Standard Model particles. Indirect messengers come in the form of gamma rays [53], neutrinos [54], charged particles [55], and multiwavelength emissions. A prediction of indirect signals requires consideration for both the particle nature of dark matter and its astrophysical state. Hence, the diﬀerential intensity is represented by a particle term K and an astrophysical term J:

Φ = KJ. (2.1)

In the case of annihilation, dN hσvi K = 2 (2.2) dE 2mχ and 1 Z J = ρ2(s, φ)ds. (2.3) 4π Meanwhile, for decay, dN 1 K = (2.4) dE mχτχ and 1 Z J = ρ(s, φ)ds. (2.5) 4π dN The spectrum of particles produced in the interaction is given by dE , τχ is the dark matter particle’s lifetime, ρ is the dark matter density, s is the distance along the 34 line of sight, and θ is the angle from the galactic center. The annihilation astrophysical intensity depends on ρ2 because two dark matter particles are needed for the interaction. The shape of the spectrum encodes information about the particle’s properties. Soft channels produce continuous spectra of gamma rays. Hard channels include final state radiation of charged leptons in their final states. Also, there are line emissions from γγ, γZ and hγ channels. These particles are produced at energies up to the dark matter particle’s mass (or half the mass for decay). To observe gamma rays below 10 GeV, one must use a telescope in space, such as the Large Area Telescope. For higher energies, one needs a ground-based imaging telescope with a large effective area, like VERITAS. Indirect detection is the only way to determine the particle nature of dark matter in an astrophysical context. It is necessary to verify the identification of cosmological dark matter candidates detected in a collider or lab and its stability on cosmological timescales.

2.3 Direct Detection

Presently, several collaborations are investigating various hypotheses about the nature of dark matter via scattering. The most favored hypothesis holds that this uniden- tified non-luminous portion of our universe is composed of weakly interacting massive particles (WIMPs). These experiments aim to detect or set limits on WIMP-nucleus scattering cross section, based on the nuclear recoils that arise from collisions between WIMPs and target nuclei. The differential energy spectrum of nuclear recoils is expected to be a featureless, smoothly decreasing exponential. Once experimental observation has placed an upper limit on the differential rate for a particular recoil energy, there is a well-defined upper limit to the total rate of the dark matter signal, for a certain particle mass. Then, for assumed values of the dark matter density and dark matter flux, the limit on the total rate translates into a limit on the particles’ cross section. 35

For practical purposes, one must account for several potentially problematic features: (1) a form factor correction (<1) is required to represent the finite size of the target nucleus and depends mainly on the nuclear radius and momentum transfer, (2) the limits will differ between spin-dependent and spin-independent interactions, (3) the detection efficiency for nuclear recoils is typically different than that of electronic recoils, and the observed recoil energy will differ from the actual recoil energy, (4) instrumental and threshold effects may (will) be present, (5) the target might be composed of more than one element, resulting in a separate limit for each one, and (6) the detector is located on Earth, which orbits the Sun, which moves through the galaxy.

2.3.1 Diﬀerential Energy Spectrum

The theoretical diﬀerential energy spectrum of WIMP-nucleus scattering takes the form of a falling exponential: dR R0 − ER = e E0r . (2.6) dER E0r

R is the event rate per unit mass per unit time, ER is the recoil energy, R0 is the total

4MχMN rate, E0 is the most probable kinetic energy of the incident WIMP, and r = 2 (Mχ+MN ) is a kinematic factor that depends on the WIMP’s mass, Mχ, and the target’s mass,

MN . The expected value of the recoil energy is

Z ∞ dR hERi = ER dER = rE0. (2.7) 0 dER In practice, however, one must incorporate a few correction factors to represent the observed recoil spectrum:

dR 2 = R0S(ER)F (ER)I. (2.8) dER obs

S(ER) is a spectral function that includes information about the particles’ masses and 2 kinematics, F (ER) is the form factor that represents the ﬁnite size of the target nucleus, and I is the interaction factor. 36

2.3.2 Dark Matter Halo Density

The two quantities crucial for both direct and indirect detection are the local dark matter halo density, ρ0, and the velocity distribution, f(v). The rotation curve is the most important observable in determining ρ and f(v) because it measures the change in density and sets the scale for the galactic potential well. The mass distribution in the galactic bulge, the stellar disk, and perhaps a dark matter halo are also required to ﬁnd the local halo density. The rotation curve yields the total gravitational potential, including the contributions from the dissipationless dark matter halo and all other components. For example, the total rotational velocity that results from a stellar disk and a dark matter halo is given by

2 2 2 vtot(r) = vd(r) + vh(r), (2.9) where vd and vh are the disk and halo contributions, respectively, and r is the distance to the galactic center. If the density of dark matter particles depends only on vh, then

ρ0 = ρ(r0) can be approximated using only the local value and the slope of the rotation curve: 1 d ρ (r) = [rv2(r)]. (2.10) 0 4πGr2 dr h

The distance between the Sun and the center of the galaxy is r0 ≈ 8.5 kpc. In practice, though, it is useful to sample several points of the rotation curve and incorporate other dynamical information. The different galactic components are typically fitted to the rotation curve to obtain the local halo density. Hence, the uncertainty in ρ arises from (i) the uncertainty in the measured rotation curve, (ii) the uncertainty in the particular dark halo model, and (iii) the uncertainties in the other components’ contributions, which must be subtracted. Based on these principles, several estimates of the dark matter halo density, 0.2 GeV/cm3 ≤ ρ ≤ 0.6 GeV/cm3, have been calculated by [56–59]. One will often see the dark matter halo density modeled as the Navarro-Frenk-White profile [60], ρ(r) δ = , (2.11) ρcritical (r/a)(1 + r/a) 37 where δ is a (unitless) characteristic density, and a is a scale radius. The theoretical dark matter halos in computer simulations fit the Einasto profile best [61]:

−βrn ρ(r) = ρ0e . (2.12)

2.3.3 Velocity Distribution

The dark matter velocity distribution is the second factor that is relevant to both direct and indirect detection. The distribution, f(v), represents how dark matter moves in the galactic halo, and detectors have a velocity threshold below which they are insensitive to WIMP-induced nuclear recoils. The Earth’s motion through the galaxy should be incorporated into f(v), as well. The Sun’s potential field, in principle, affects the local velocity distribution, but the effect is small and typically not considered.

Given a dark matter velocity distribution, f(v, vE), the diﬀerential particle density can be represented by n dn = 0 f(v, v )d3v, (2.13) k E where k is the normalization constant such that

Z vesc dn ≡ n0. (2.14) 0

ρD The mean dark matter particle number density is given by n0 (= ), v is the partlcle’s MD velocity, vE is Earth’s velocity relative to the dark matter distribution, vesc is the local galactic escape velocity, and dn is the particle density of dark matter at velocities that are within d3v of v.

A Maxwellian dark matter velocity distribution, characterized by v0, is assumed for WIMPs: −→ −→ 2 −( v + v E ) v 2 f(v, vE) = e 0 . (2.15) 38

From Equation 2.13, k is found in the limit vesc → ∞,

∞ −→ −→ 2 Z −( v + v E ) 2 3 k = e v0 d v 0 √ 3 = ( πv0)

≡ k0

−→ −→ 2 −→2 Actual observation, however, is bound by the constraint ( v + v E) ≤ v esc, so

2π (cos θ) v −→ −→ 2 Z Z max Z max ( v + v E ) 2 − 2 k = dφ d(cos θ) v e v0 dv 0 −1 0 2 − vesc vesc 2 vesc v2 = k0 erf − √ e 0 v0 π v0

≡ k1, where   1, 0 ≤ v ≤ vesc − vE (cos θ)max = 2 2 . vesc−vE −v  , vesc − vE ≤ v ≤ vesc + vE 2vE v The full derivation of these constants can be found in AppendixA. Unsurprisingly, the normalization constant, k, is independent of vE. Typical values for the parameters,

k0 v0 = 230 km/s and vesc = 544 km/s, lead to = 0.997 [62]. The velocity distribution k1 is one of the primary sources of astrophysical uncertainty.

2.3.4 Nuclear Form Factor

When the momentum transfer q is such that the wavelength ~/q is no longer large compared to the nuclear radius, the eﬀective cross section falls with increasing q.A form factor, F , is suﬃcient to represent this phenomenon. To calculate the form factor, one must evaluate the matrix elements of the nucleon spin operators for a nuclear state. At zero momentum transfer, this is simply a calculation of the average spin of protons and neutrons in the nucleus; however, at nonzero momentum transfer, a form factor suppression that is based on nuclear wave functions must be included. The form factor 39

qrn is a function of the dimensionless quantity , where rn is the eﬀective nuclear radius. ~

With rn given by 1/3 rn = anA + bn (2.16) and

2 1/2 q = (2 × 0.932[GeV/c ]AER[keV]) , (2.17) and using ~ = 197.3 MeV · fm,

−3 1/2 1/2 1/3 qrn = 6.92 · 10 A ER (anA + bn), (2.18)

where ER is given in keV, and an and bn are given in fm. The cross section may take the form

2 σ(qrn) = σ0F (qrn), (2.19) where σ0 is the cross section at zero momentum transfer. The first factor, σ0, represents the strength of the specific interaction, while the second, F (qrn), includes the effects of momentum transfer. In the first Born approximation, the form factor is the Fourier transform of ρ(r), the density distribution of the “scattering centers”:

Z −→ −→ F (q) = ρ(r)ei q · r d3r (2.20)

Z 2π Z Z +1 = dφ r2ρ(r) eiqr cos θd(cos θ)dr 0 r −1 4π Z ∞ = r sin(qr)ρ(r)dr. q 0 At the most basic level, two models are considered to calculate the form factors: (a) a thin shell to approximate a single outer-shell nucleon for the case of a spin-dependent interaction,

F (qrn) = j0(qrn) = sin(qrn)/qrn; (2.21) and (b) a solid sphere to approximate a spin-independent interaction with the whole nucleus,

3 F (qrn) = 3j1(qrn)/qrn = 3[sin(qrn) − qrn cos(qrn)]/(qrn) . (2.22) 40

However, the form factor proposed by Helm [63], 2 2 j1(qrn) 2 F (qrn) = 3 × exp[−(qs) ], (2.23) qrn more accurately represents the nuclear structure in a spin-independent interaction; s is a measure of the nuclear skin thickness. The parameters in Eq. 2.23 can be determined from experimental estimates of rrms after observing that the skin thickness is practically constant [62]. For a uniform sphere of radius rn, 3 r2 = r2 ; rms 5 n and for Eq. 2.23, 3 r2 = r2 + 3s2. (2.24) rms 5 n Using a two-parameter least-squares ﬁt to the Fricke et al. [64] compilation of c

c ' 1.23A1/3 − 0.60 fm (2.25)

in conjunction with Eq. 2.24, rn for Eq. 2.23 is found to be 7 r2 = c2 + π2a2 − 5s2. (2.26) n 3 The parameters in the equations above are taken to be a ' 0.52 fm and s=0.9 fm. The form factor, F 2(q) = S(q)/S(0), may be modeled in terms of a structure funtion, S(q), which has three parts to represent either the interactions with protons, neutrons and mixing, or the isoscalar (p+n), isovector (p-n) and interference terms:

2 2 S(q) = a0S00(q) + a1S11(q) + a0a1S01(q). (2.27)

In order to fully represent the momentum dependence of the form factor, each part of

1 2 2 the form factor is ﬁtted with a polynomial in terms of the variable u = 4 b q ,

X (k) k Sij = cij u , (2.28) k where b = (1 fm) · A1/6. The form factor will vary from nucleus to nucleus. Precise calculations have been carried out for 73Ge [65], 27Al [66], 39K [66] and 29Si [67]. 41

2.3.5 Interaction Factor

Dark matter interactions with nuclei can be characterized in terms of scalar (spin- independent) and axial-vector (spin-dependent) couplings. Each type has its own interaction factor that scales the total interaction cross section: σ ∝ I. For interactions that are independent of spin, there are A scattering amplitudes that, at suﬃciently low momentum transfers (qrn 1), add in phase to yield a coherent cross section proportional to A2. Thus, the interaction cross section of a single nucleon is simply multiplied by this coherent interaction factor to obtain the total interaction cross section:

2 Ic = A . (2.29)

However, it is important to note that coherence is lost for qrn & 1 because scattering amplitudes no longer add in phase. On the other hand, for interactions that depend on spin, the scattering amplitudes of paired nucleons cancel because their signs switch with spin. Only targets with an odd number of nucleons may contribute. The interaction factor for spin-dependence is described by

2 2 Is = C λ J(J + 1), (2.30) where J is the total angular momentum, C is a factor that accounts for the quarks’ spin contribution to a nucleon,

X 3 C = Tq ∆q, (q = u, d, s), (2.31) q

3 ∆q is the fractional quark spin contributed to nucleon spin by quark species q, Tu,d,s = 1 1 1 2 , − 2 , − 2 is the third component of isospin, and λ is given by the relation [J(J + 1) + s(s + 1) + l(l + 1)]2 λ2 = . (2.32) 4[J(J + 1)]2 The full treatment from Engel et al. [68] incorporates couplings to both protons and neutrons: J I = [C hS i + C hS i]2 . (2.33) s p p n n J + 1 42

hSp(n)i is the expected nuclear spin contribution from protons (neutrons), calculated based on the shell model. The spin-independent term will increase relative to the spin-dependent term according to A; spin-independent interactions dominate around

A & 30 [62].

2.3.6 Energy Detection Eﬃciency

For scintillation and ionization detectors calibrated with γ sources, the observed nuclear recoil energy is less than the actual value. The ratio fn, dubbed the “relative eﬃciency,” is determined through neutron-scattering measurements. Experimentalists tend to work with γ-calibrated energies for easy identiﬁcation of γ background. In the rate expressions and energy spectra, the recoil energy ER is often replaced in favor of

Ev, the “visible” energy, via the relations ER = Ev/fn and dR dfn dR = fn + ER . (2.34) dER dER dEv

The preferred model for calculating the ionization eﬃciency, , of an element is that of

Lindhard et al. [69], who represent fn by

kg() f = , (2.35) n 1 + kg() where, for a nucleus of atomic number Z,

−7/3 = 11.5ER(keV)Z ,

k = 0.133Z2/3A1/2, and g() is well ﬁtted to: g() = 30.15 + 0.70.6 + . 43

2.3.7 Energy Resolution and Threshold

The ﬁnite energy resolution of a detector means that N recoils at a single energy E0 would be observed in a spectrum distributed in an approximate Gaussian shape:

dN(E) N 0 2 2 = √ e−(E−E ) /2∆E , dE 2π∆E resulting in dR 1 Z 1 dR 0 2 2 √ −(Ev−Ev) /2∆E 0 = 0 e dEv. (2.36) dEv 2π ∆E dEv In general, ∆E is energy dependent. For detectors with a linear response, the statistical √ ﬂuctuations alone are given by ∆E(E) ∝ E, although additional terms occur in practice. The energy resolution is traditionally presented as the ratio of the full width √ 0 at half maximum to the mean energy, ∆EFWHM /E , with ∆EFWHM = 8ln2∆E. The detector signal may exist as a discrete quantity, such as the number of count from a photomultiplier, and at low energies this number may be suﬃciently small such that the Gaussian would lead to inaccurate count losses due to negative energy. The statistical component of the resolution can be correctly represented through a Poisson process rather than a Gaussian one:

0 dR 1 Z dR E 0 v −Ev/ 0 = 0 e dEv (2.37) dEv n! dEv with Ev = n. A threshold must be set in this type of detector in order to reduce intrinsic rates, resulting in a reduced detection eﬃciency at low energies. Consider two PMTs running in coincidence with identical thresholds. If the PMTs are balanced (i.e. produce the same mean number of photoelectrons), then, for an event that produces n photoelectrons, the estimate of the probability that m ≤ n arrive at one PMT (and n − m at the other) is

p(m) = Ae−n/2(n/2)m/m!, (2.38)

Pn where A is a normalization factor yielding m=0 p(m) ≡ 1; hence, (n/2)m/m! p(m) = . (2.39) Pn k k=0(n/2) /k! 44

So, for a threshold NT measured in photoelectrons, only events for which NT ≤ m ≤ n − NT would be accepted. Meaning, the counting eﬃciency is

Pn−NT (n/2)m/m! η(n, N ) = m=NT . (2.40) T Pm=n m m=0 (n/2) /m!

2.3.8 Target Mass Fraction

In the case of a compound target, it is useful to set a limit on each component. The diﬀerential rate is deﬁned per unit mass of the target. It follows that, if events are attributed to an element that contributes a fraction of the target mass, f, then the observed rate for that element is

dR 1 dR = . dE element f dE More generally, if there are n constituent parts, the total rate can be expressed as

n dR X 1 dRi = . dE f dE i=1 i

2.3.9 Annual Modulation

Realistically, one should incorporate the Sun’s motion as well as the Earth’s. Over time, the Earth’s motion around the Sun moves in and out of alignment with the Sun’s motion in the Galaxy. Consequently, the dark matter event rate modulates. Since a detector’s velocity relative the dark matter halo depends on the time of year, the event rate will exhibit sinusoidal behavior with a period of one year [62].

dR (E, t) ≈ S (E) + S (E) cos[2π(t − t )/year], (2.41) dE 0 m 0 where S0(E) is the time-averaged rate, and Sm(E) is the modulation rate. The dark matter flux is highest in June, giving the phase t0, and lowest in December. The event rate is ∼2% larger when the Earth’s motion is aligned with the Sun’s than when they are anti-aligned. As modulation is a small effect, a large number of events would be 45 required to claim significant detection. However, modulation can serve as a powerful method by which to distinguish signal from backgrounds, such as ambient radioactivity, that are not expected to display any dependence on time.

2.3.10 WIMP Event Rate

The event rate per unit mass from a target of atomic mass A AMU with cross section per nucleus σ is N dR = 0 σvdn, (2.42) A where N0 is Avagadro’s number. In the following, the total event rates and energy spectra are given without corrections. So, σ0 is the “zero momentum transfer” cross section. Then

N Z R = 0 σ vdn A 0 N Z n = 0 σ 0 vf(v, v )d3v A 0 k E N ≡ 0 σ n hvi. (2.43) A 0 0

√2 R0 is deﬁned as the event rate per unit mass for vE = 0 and vesc = ∞ (then, hvi = π v0):

2 N0 ρD R0 = √ σ0v0, (2.44) π A Mχ such that √ π hvi R = R0 2 v0 Z R k0 1 3 = 4 vf(v, vE)d v (2.45) R0 k 2πv0 2 2 vE 2 2 vesc 1 k0 − 2 √ 1 v0 vE vE vesc 1 vE − 2 v0 v0 = e + π + erf − 2 2 + 2 + 1 e . (2.46) 2 k 2 vE v0 v0 v0 3 v0

2 R0, normalized to ρD = 0.4 GeV/c and v0 = 230 km/s, becomes

540 σ ρ v R = 0 D 0 (2.47) 0 −2 −3 −1 AMχ 1 pb 0.4GeVc cm 230 kms 46

503 σ ρ v R = 0 D 0 (2.48) 0 −2 −3 −1 MχMT 1 pb 0.4 GeVc cm 230 kms 2 where Mχ and MT are in GeV/c . The recoil energy of a nucleus colliding with a dark 1 2 matter particle of kinetic energy E = 2 MDv , scattered at an angle θ is:

ER = Er(1 − cos θ)/2. (2.49)

Assuming that scattering is isotropic (i.e. uniform in cos θ), the recoil energies are uniformly distributed over the range 0 ≤ ER ≤ Er; so, dR Z Emax dR(E) = (2.50) dER Emin Er

Z vmax 2 dR 1 v0 = 2 dR(v), (2.51) dER E0r vmin v where Emin = ER/r is the smallest incident kinetic energy of the WIMP that can induce 1 2 a recoil of energy ER, E0 = 2 MDv0, and 1/2 1/2 2Emin ER vmin = = v0. (2.52) Mχ E0r Therefore, using Eq. 2.45, one ﬁnds,

Z vmax dR R0 k0 1 1 3 = 2 f(v, vE)d v. (2.53) dER E0r k 2πv0 vmin v

Here, the vmin scale is deﬁned:

Z vmax k0 1 1 3 T (vmin) = 2 f(v, vE)d v. (2.54) k 2πv0 vmin v

For 0 < vmin < vesc − vE,

v +v 2 v −v 2 v2 v +v Z esc E (v−vE ) Z esc E (v+vE ) esc Z esc E k 1 − − − 2 0 v 2 v 2 v T (vmin) = e 0 dv − e 0 dv − e 0 dv k 2vE vmin vmin vesc−vE √ 2 vesc k0 π v0 vmin + vE vmin − vE − 2 = erf − erf − e v0 . k 4 vE v0 v0

However, if vesc − vE < vmin < vesc + vE,

v +v 2 v2 v +v Z esc E (v−vE ) esc Z esc E k 1 − − 2 0 v 2 v T (vmin) = e 0 dv − e 0 dv k 2vE vmin vmin √ 2 − vesc k0 π v0 vesc vmin − vE 1 v2 = erf − erf − vesc + vE − vmin e 0 . k 4 vE v0 v0 2vE 47

Full derivations of all previous calculations are found in AppendixB. The expected event rates of WIMP-nucleus spin-independent scattering are shown in Figure 2.2 for three target elements. These rates incorporate the corrections imposed by the finite range of the velocity distribution as well as the nuclear form factors for the corresponding target. Firstly, one can see the benefits of coherence in xenon’s spectrum at nuclear-recoil energies below ∼35 keV. Even though xenon loses coherence at higher momentum transfers, xenon is the optimal for WIMP detection because one is generally more interested in the lower-energy recoils (≤40 keV). Lighter targets have an advantage in searches for light dark matter. Secondly, the sensitivity of the total event rate to threshold effects is clearly visible. The falling exponential form of the differential rate means that every improvement in a detector’s threshold yields a subsequently larger return in the total event rate.

-45 2 Mχ=50 GeV, σ=10 cm 10-3

10-4

10-5 Xenon (A=131)

Event Rate [count/day/kg] Argon (A=40) Germanium (A=73)

-6 10 0 5 10 15 20 25 30 35 40 45 50 Energy Threshold [keV]

Figure 2.2. Integrated recoil spectra as a function of energy threshold are shown for three targets, xenon, argon, and germanium, assuming values for the WIMP mass and WIMP-nucleon cross section. 48

2.4 Low-Background Requirements

The strategy of direct detection (i.e. scattering experiments) fundamentally reduces to two tactics: scale and background reduction. Scalability is straightforward for liquid noble element detectors; a larger target mass will lead directly to a larger exposure. One can continually increase the size of the detector as long as the integrity of the detector’s design remains intact and additional target material is available. Background reduction, on the other hand, is usually trickier. Backgrounds arise externally from cosmic rays, detector materials, and environmental sources, as well as internally from the inherent radioactive decays of the target medium. Several different methods must be employed to suppress the background rate as much as possible. The first order of business is to choose a suitable location for a dark matter detector. On the Earth’s surface, cosmic radiation quickly overwhelms the detector. So, it becomes necessary to position the detector deep underground. Once shielded by the Earth, the cosmogenic background is reduced by several orders of magnitude. Then, the contaminants in and around the active region of the detector are the dominant sources of background. The second step is a careful selection of detector materials. All detectors contain instrumentation and a containing vessel that provides either pressure or a vacuum. These components must be composed of materials with low levels of radioactivity. Common choices to satisfy this requirement are stainless steel, titanium and copper. Third, one must shield the detector’s sensitive region from external sources of background. Traditionally, detectors use a combination of lead, which shields from external gamma radiation, and some type of hydrogenated material, like polyethylene, which shields from external neutrons. However, as detector scales increase, experiments are shifting to water-based shielding, which is easily thick enough (a few meters) to protect from both gamma and neutron backgrounds. These materials constitute the passive shield. Additionally, the outer part of a sufficiently dense target volume can provide 49 self-shielding to further suppress the external background. This phenomenon is called the active shield. With external backgrounds under control, consideration must be given to internal backgrounds. An ideal target is one that has no long-lived radioactive isotopes and would not interact in some way that contributes to the background. Naturally, one turns to the noble elements, which are inert by nature. 50 51

3. SCATTERING EXPERIMENTS

Direct searches for dark matter base themselves on collisions between dark matter particles and target nuclei. Various technologies exist to detect these nuclear recoils: liquid noble element detectors, cryogenic detectors, and solid scintillators. The majority of experiments capitalize on some combination of ionization, scintillation, and low- temperature phonon techniques. All share the same basic theoretical interpretation.

Light

Liquid Xe, Ar

NaI, CsI, CaWO4

Phonon Charge

Ge, Si Xe, Ar, Ne

CaWO4 Ge, Si

Figure 3.1. Light, charge, and phonons are used to detect dark matter via WIMP-nucleus scattering. 52

3.1 Liquid Noble Element Detectors

Xenon and argon are the primary targets in direct detection of dark matter. Certain properties of xenon make it better suited to find these rare dark matter events. Xenon is the heaviest non-radioactive noble gas. The high mass number increases the rate of coherent scattering with dark matter. Xenon’s large density shields the innermost volume from many external sources of background. None of the xenon isotopes is long-lived; the longest is 127Xe with a half-life of 36.3 days. The absence of any intrinsic radioactivity means that there also will be almost no internal sources of background. Of the nine stable isotopes, two have nonzero nuclear spin, making them sensitive to spin-dependent interactions. A detection medium must produce a measurable signal from the energy it absorbs. Liquid xenon emits both scintillation photons and ionization electrons directly as a result of ionizing radiation. In fact, xenon is a highly efficient scintillator, yielding ∼45 photons per keV. Additionally, xenon is transparent to its own scintillation light. The high mobility of ionization electrons in liquid xenon allows for the extraction of electrons via an externally applied electric field. One can procure xenon as a byproduct of liquefaction of air as it separates into oxygen and nitrogen. The resultant liquid oxygen contains both xenon and krypton, which may be removed with fractional distillation. Xenon is extracted from the xenon- krypton mixture via distillation.

3.1.1 WIMP Recoil Sensitivity

The WIMP sensitivity of xenon exceeds the sensitivities of germanium and argon for spin-independent interactions at suﬃciently low energy thresholds by taking advantage of the A2-term in the coherent spin-independent cross section. The size of the xenon nucleus causes scattering to lose coherence at high energies; however, for low energy thresholds, xenon is the most sensitive target. The threshold is determined by the 53 eﬃciency in detecting scintillation photons. Low-energy nuclear recoils typically yield on the order of ten scintillation photons per keV; thresholds can go as low as ∼5 keV. Finally, xenon is sensitive to spin-dependent interactions that result from coupling to neutrons. The natural odd-neutron isotopes of xenon, 129Xe and 131Xe, occur with abundances of 26.4% and 21.2%, respectively.

3.1.2 Scintillation Signal

An incident particle that recoils in liquid xenon will deposit energy along a track. Some of these xenon atoms get excited (excitons), and others get ionized (ions). Within a few picoseconds, excitons (Xe∗) collide with neighboring xenon atoms to form an

∗ excited molecular state (Xe2), known as an excimer. Ionized atoms also form excimers through the processes

+ + Xe + Xe → Xe2 , (3.1)

+ − ∗∗ Xe2 + e → Xe + Xe, (3.2) Xe∗∗ → Xe∗ + heat, (3.3)

∗ ∗ Xe + Xe → Xe2. (3.4)

Subsequently, the excimer decays to the dissociative ground state and produces scintillation light: ∗ Xe2 → 2Xe + hν. (3.5)

The scintillation photons from liquid xenon fall within the vacuum ultraviolet range (VUV), with a wavelength of 178 nm (7 eV) and a width of 13 nm [70]. The scintillation light receives contributions from both the singlet and triplet states of the excimers. For relativistic electrons under an applied electric ﬁeld, the singlet and triplet states have decay times of 2.2 and 27 ns [71], respectively; consequently, xenon is one of the fastest scintillators. However, if no electric ﬁeld is applied, the recombination time dominates, and a single decay of 45 ns is observed [72]. The ratio of singlets to triplets is higher at larger ionization densities. 54

Another mechanism enters the picture for high ionization densities. Even though the scintillation decay time does not depend on the ionization density, quenching may occur before the creation of excitons. Hitachi [73] introduced the “bi-exitonic” quenching mechanism,

∗ ∗ ∗∗ + − Xe + Xe → Xe2 → Xe + Xe + e , (3.6) in which two excitons collide to produce an electron-ion pair. Two exitons, which typically each produce one free electron for recombination, yield only a single free electron. Figure 3.2 shows a schematic of these processes.

Figure 3.2. Excitation and recombination of ion-electron pairs contribute to xenon’s scintillation signal. Bi-excitation quenches the scintillation signal for suﬃciently high ionization densities. Figure from [74].

Several effects can reduce the number of scintillation photons created by ionizing radiation in liquid xenon. An electric field reduces scintillation by reducing the fraction 55 of recombination. Tracks with low ionization density, such as electronic recoils, have a smaller scintillation yield due to escaped electrons. Nuclear quenching reduces the scintillation yield of nuclear recoils because of the energy lost to atomic motion. And tracks with a high excitation density suffer from bi-excitonic quenching. Similarly, a dark matter particle induces both excitation and ionization of argon atoms. The excimers and ions form molecular states and recombine through the same processes as xenon to produce a scintillation signal (see Figure 3.2). The transition of the singlet and triplet states of argon to the dissociative ground state produces an emission spectrum with a narrow peak at 128 nm and a width of 20 nm [75]. The decay times of the states are 5 ns and 1600 ns, respectively.

3.1.3 Ionization Signal

The average energy required to produce an electron-ion pair is 15.6 eV [76], which makes liquid xenon the liquid noble element with the largest ionization yield. Electrons that are excited from the valence band to the conduction band by ionizing radiation can be drifted over long distances. However, in order to measure the ionization signal, an electric ﬁeld must be applied to prevent electrons to recombine, and the electron must drift without attachment to impurities.

At lower values of the applied field, the drift velocity vd is proportional to the applied field; the constant of proportionality is called the electron mobility, µ. At larger field strengths, the drift velocity reaches a maximal value. Ions travel at significantly lower drift velocities than electrons. When electrons drift in liquid xenon and collide with electronegative impurities, they attach to these impurities. The ionization signal diminishes because the ions drift at a lower speed. Attachment is arguably the most critical factor that affects the performance of large-scale liquid xenon detectors that measure ionization signals of recoiling particles. The average time an electron survives before it attaches to an electronegative impurity is known as the electron lifetime, τe. For uniformly distributed 56 impurities, the number of electrons that survive after traveling a distance z in a detector from an initial number of electrons N0 is given by

z N(z) = N0exp − . (3.7) vdτe

O2 is the most prevalent culprit responsible for attachment in liquid xenon. More generally, the electron lifetime depends on the concentrations of different electromagnetic impurities, ni, and their rate constants for attachment, ki, by τe = −1 (Σkini) . Since the probability to capture an electron is a function of the electron’s energy, the applied electric field has an effect on the rate constants for electron attachment.

3.1.4 Electroluminescence

It is possible to extract ionization electrons from liquid xenon into gaseous xenon. The larger dielectric constant of liquid xenon relative to gaseous xenon creates a repul- sive image potential at the liquid-gas interface. An electric ﬁeld can drift these electrons to overcome the potential barrier. As long as an electron’s momentum perpendicular to the interface exceeds the potential, they may be extracted into the gaseous phase. In the gaseous phase, the electrons continue to drift, and they gain enough energy to excite xenon atom to produce scintillation light. This proportional scintillation signal, or electroluminescence, yields a number of photons per unit length per electron

dN ph = α(E − βp), (3.8) dx g

−1 −1 where α = 70 photons/keV, β = 1kVcm atm , and Eg and p are the the electric ﬁeld strength and pressure in the gaseous phase [77]. This process allows one to amplify ionization signals that may otherwise be too small to detect or measure. 57

3.1.5 Radiopurity

As xenon is a noble element, purification is a relatively simple process. Commercial xenon is manufactured by distillation and purified in the lab by heated getters that remove non-noble contaminants at ≤ 1 ppb. Contamination with other noble elements may be resolved with gas chromatography. The absence of natural long-lived radio-isotopes makes it possible to achieve extremely low background levels. The few backgrounds that remain are typically electron recoils, which may be distinguished by the charge-to-light ratio. The primary intrinsic background comes from krypton, which occurs at the level of 5 ppb in the best commercially available xenon. 85Kr decays with a half-life of 10.77 years to the stable 85Rb. At the WIMP recoil energies, the beta spectrum yields 2.2 · 10−3 beta particles per keV per decay [78]. Esti- mating the relative abundance of 85Kr, one ppb of krypton contamination corresponds to ∼0.05 events/keV/kg/day [79]. Krypton is removed from xenon either through distillation or chromatography. In chromatography, a portion of contaminated xenon is sent through a charcoal column by a carrier gas (helium). The greater adsorption of xenon onto charcoal separates the gases within the column. The krypton emerges first and is diverted into a trap.

3.1.6 Self-Shielding

Liquid xenon’s high density (∼3 g/cm3) and the absence of any long-lived radio- isotopes allow the detector mass itself to be a reliable source of shielding from external electromagnetic background. Since xenon’s signals come from electronic excitation, the stopping power of liquid xenon is relevant. The stopping power for electrons increases as energy decrease below 1 MeV; thus, the tracks of electronic recoils will exhibit higher ionization and excitation densities near the end of the track. Conversely, nuclear recoils will exhibit higher ionization and 58 excitation densities at the beginning of the track. In the energy range of interest for dark matter searches, nuclear recoils have a higher ionization density than electronic recoils. The probability of recombination increases with ionization density. Hence, nuclear recoils will experience a smaller reduction in scintillation than electronic recoils in the presence of an electric field. The large stopping power of liquid xenon, due to its high density, allows neutron and gammma events to be rejected effectively, and an inner (fiducial) volume with a very low interaction rate may be selected. Interaction lengths of neutron and gammas in liquid xenon are 15 cm and 10 cm, respectively, for even the most penetrative gamma rays. The rate of single scatters due to neutrons and gammas at a particular point in the

−d/x0 detector is proportional to e , where x0 is the interaction length of the particle, and d is the distance the particle must travel. For any path to the center of the detector, a particle must traverse the entire active region of xenon, making a single scatter unlikely to occur.

104 10-4 103 Total 10-3 -1

g 2

2 10 10-2 10 10-1 1 1 10-1 Compton 10 -2 Attenuation Length (cm) 10 Photoelectric Absorption Attenuation Coefficient cm 102 10-3 Pair Production 103 10-4 10-3 10-2 10-1 1 10 Energy (MeV)

Figure 3.3. Compton scattering, photoelectric absorption, and pair production aﬀect the attenuation of gammas in xenon. 59

X rays and γ rays interact through the photoelectric absorption, Compton scattering, or pair production. In the case of Compton scattering, a γ ray continues along its path to produce other energetic electrons in subsequent interactions. Figure 3.3 shows attenuation for these three eﬀects and their sum as a function of energy. We conclude that the main external background that produces electronic recoils in the energy region of interest for dark matter searches arises from low-energy single Compton scatters. Most low-energy γ rays are absorbed in the outer layers of the detection medium.

3.1.7 Scalability

WIMP sensitivity scales with detector mass. Current limits already demonstrate the need for target masses of several tonnes. While instrumentation scales with area (of the PMT arrays), the sensitivity scales with volume. Meaning, large-scale detectors are economical in design. The only limitations on size are those set by high voltage constraints and the ability to see scintillation light and drifted charge over a considerable distance. In XENON10, the drift length of charge was measured to be 4 meters [80,81].

3.1.8 Discrimination

The availability of both light and charge signals in xenon allows for discrimination between nuclear and electronic recoils. The ionization density of nuclear recoils is larger than that of electronic recoils. Consequently, more recombination occurs, and nuclear recoils have a smaller charge-to-light ratio. More than 99% of electronic recoils can be rejected. Additionally, as a result of the significantly different decay times in argon, Pulse Shape Discrimination (PSD) is possible between electronic and nuclear recoils. Elec- tromagnetic radiation preferentially excites argon’s triplet state, while nuclear recoils excite the singlet state. The shorter lifetime of the singlet state means that the signal intensity in the beginning of a scintillation pulse from a singlet will be noticeably larger 60 than the intensity of a triplet. A PSD parameter is defined by the ratio of the light in the first portion of a waveform to the total amount of light in the signal. The employment of PSD is complementary to the charge-to-light ratio for a dual-phase detector and necessary for a single-phase detector.

3.2 Cryogenic Detectors

Cryogenic detectors operate at very low temperatures, typically less than 100 mK. They detect dark matter through vibrational modes in a crystal lattice, known as a phonons. In addition to phonon signals, cryogenic detectors measure charge signals based on the creation of electron-hole pairs.

3.2.1 Low-Temperature Phonon Signal

A phonon is a quantized normal-mode vibrational excitation of a periodic lattice. Geometrical anisotropies in the lattice cause the phonon phase velocity to depend on the direction of propagation and the polarization mode. The phonon dispersion relation is 2 X X ρω µ = cµσντ kσkν τ , (3.9) τ σν where ρ is the crystal’s mass, ω is the phonon frequency, µ is a component of the polarization vector, cµσντ is the stiﬀness tensor and kσ is a component of the phase velocity. All indices run over x, y, z. Three mechanisms govern the generation of phonons: primary phonons, recombination phonons, and Luke phonons [82]. When a particle interacts with the crystal electromagnetically or through a nuclear recoil, the particle imparts energy to an electron or the nucleus in the substrate. As the exited electron or nucleus returns to equilibrium, it deposits its energy as a phonon.

Accounting for the energy already spent to excite NQ electrons, the energy diverted into the phonon is

Eprimary = Erecoil − NQEg, (3.10) 61

where Eg is the energy band gap of the crystal. Once electron-hole pairs are created, they will recombine unless a suﬃciently strong external electric ﬁeld is applied to drift them apart. If they recombine, they release the corresponding energy back to the phonon. Even if the electron and the hole do not recombine with each other, they may still recombine with other holes and electrons as they drift to their respective electrodes. Hence, recombination phonons have an energy

Erecomb = NQEg. (3.11)

Luke phonons are the crystalline analog of Cherenkov radiation. Luke phonons contribute additional energy to the crystal and is given by the work done on NQ electron- hole pairs by the external electric ﬁeld:

2NQ Z X ~ ELuke = qi E · d~x. (3.12) i path If the electric ﬁeld is uniform, then

X di E = eV , (3.13) Luke a i

th where V is the applied voltage, a is the size of the crystal, and di is the distance the i charge travels. In the case of complete charge collection, where the sum of the distances traveled by each part of a pair is precisely the size of the crystal, one obtains

ELuke = eV NQ. (3.14)

An energy Eehp of 3 eV (3.83 eV) is required to produce an electron-hole pair in a germanium (silicon) crystal. Then, the ionization energy is EQ = NQEehp. At particular bias voltages (-4 V for germanium; -3 V for silicon), the Luke phonon energy is equivalent to the ionization energy. The production of Luke phonons allows for increased sensitivity to low-energy recoils. However, an oversized electric ﬁeld would undermine the contribution from the recoil phonon, which, in turn, would compromise the detec- 62 tors ability to discriminate based on the ionization-to-recoil-energy ratio. The total energy is

Etotal = Eprimary + Erecomb + ELuke

= Erecoil + EQ (3.15)

Measurements of the total energy and the ionization energy can yield a calculation of the recoil energy. The lower atomic mass of silicon typically makes it a less sensitive target for spin-independent interactions due to coherent enhancement of the scattering cross section with atomic mass. However, its lower mass is beneﬁcial for low-mass WIMP searches.

3.2.2 Ionization Collection

Thermal or electromagnetic agitation can excite electrons from the valance band to the conducting band in a semiconductor. At low temperatures, their energy is no longer adequate to induce an excitation over the bandgap. If one removes the substrate from the presence of high-energy radiation, then only penetrative particles could possibly create electron-hole pairs. Incident electromagnetic radiation bestows energy onto electrons, and uncharged particles collide with nuclei, which then transfer energy to electrons. In either scenario, the excited electrons give energy to other electrons, and thus a track of electon-hole pairs is formed along the particle’s recoil. An external electric field is applied to prevent recombination of electron-hole pairs. Bias voltages are applied to electrodes on one face of the germanium or silicon substrate; the other face is grounded. Due to the anisotropic nature of an electron’s motion, it is preferred to measure holes, and hence the voltage vias is negative. The electrode senses the electric current induced by a nearby charge because as the charge moves it changes the number of electric field lines that terminate at the end of the elctrode. The final charge signal is proportional to the number of charge carriers that were created. 63

Assuming the charges do not get trapped, the ionization signal is roughly independent of the depth in the crystal.

3.3 Solid Scintillators

Solid scintillators are some of the oldest radiation detectors. Light output is converted into voltage pulses that are then digitized in the same way as pulses from semiconductor detectors and liquid noble element detectors. The point of a scintillator is to produce a large light output in the visible range. Two types of scintillators exist: organic and inorganic. Inorganic crystal scintillators are relevant to dark matter detection. The scintillation of an inorganic crystal depends on the structure of the crystalline lattice. A pure inorganic crystal, such as sodium iodide, permits electrons to occupy only certain energy bands. External sources of energy can excite electrons from the valence band to the conduction band, leaving a gap in the valence band. However, de- excitation of these electrons via photonic emission is too ineﬃcient to be useful; energy is dissipated through other mechanisms, such as Coulomb interactions or Bremsstrahlung radiation. Even if this process were eﬃcient, the band gap is too large to produce a photon within the visible range. In order to make inorganic crystal scintillators useful, trace impurities are incorporated into the crystal lattice. These impurities are called activators. They alter the band gap structure within the lattice. New energy states are created within what would otherwise be the forbidden band. Then, the electrons can descend to the valence band through these new energy levels. The smaller transitions in energy levels correspond to emissions with larger wavelengths that fall within the visible region. An incident particle will create many electron-hole pairs in the crystal. The holes drift to activator sites, which are preferentially ionized because of their lower ionization energy. The electron, free to move throughout the conduction band, will do so until it encounters an ionized activation site. The electron then falls into the impurity and 64 quickly de-excites, emitting a photon. The activator is chosen such that this photon is in the visible spectrum. The half-lives of the activators are typically on the order of hundreds of nanoseconds, and electron migration is even shorter. Inorganic scintillators include NaI(Tl), CsI(Tl), CsI(Na), and LiI(Ei). 65

4. XENON

As we saw in Chapter1, cold dark matter comprises roughly 25% of the Universe. Of the many dark matter candidates that exist outside the Standard Model, the WIMP is the most favored. One method by which we can search for WIMPs, as discussed in Chapter2, is based on nuclear scattering. A terrestrial experiment uses one of the various detector technologies to obtain sensitivity to the low-energy nuclear recoil that results from WIMP-nucleus scattering. The XENON experiment, stationed underground at the Laboratori Nazionali del Gran Sasso (LNGS), uses liquid xenon to conduct these searches for WIMPs. The detector is highly sensitive to recoils because of its large active volume, its ultra-low background, and its ability reconstruct the energy and three-dimensional position of recoils on an event-by-event basis. All of these properties are crucial to overcome the challenge posed by the low interaction rate of WIMPs. The XENON detector employs a dual-phase (liquid-gas) time projection chamber (TPC) that simultaneously detects scintillation light of a few keV and ionization at the single-electron level. The active region is a cylinder 30.6 cm × 30.5 cm, containing 62 kg of xenon. Event localization to millimeter precision and the self-shielding properties of liquid xenon facilitate background reduction by selection of a ﬁducial volume. XENON100 has ten times more mass than XENON10 and a background 100 times lower [83,84]. Full detector speciﬁcations are presented in [85].

4.1 The Principle of a Dual-Phase Time Projection Chamber

A schematic of XENON’s dual-phase TPC is shown in Figure 4.1. An incident particle scatters oﬀ a xenon nucleus to produce scintillation light and ionization electrons. An electric ﬁeld is applied between the cathode, which is on negative voltage, and the 66

PMT Array Gas S2 Anode

Gate Grid e-

Liquid

S1 Drift Field Cathode

PMT Array

Figure 4.1. The principle of XENON’s dual-phase TPC is illustrated. An incident particle ionizes a xenon nucleus. Some of the free electrons recombine to produce a prompt scintillation (S1) signal; others drift along the ﬁeld to produce a proportional scintillation signal (S2). The ratio of these two signals characterizes the type of recoil.

grounded gate grid in order to drift the ionization electrons to the liquid-gas interface. A stronger electric field applied between the gate grid and the positive anode to induce proportional scintillation. The 178-nm scintillation light from both the prompt (S1) and proportional signals (S2) are seen by two arrays of PMTs at the top and bottom of the vessel. The PMTs are optimized for vacuum ultraviolet (VUV) emissions. The uniform drift field establishes a constant electron drift velocity of approximately 1.7 mm/µs, depending on the detector’s operation conditions. The time difference between the S1 and S2, ∆t = tS2 − tS1, then provides an event’s z-position, which lies along the cylindrical chamber’s central axis. The z-position resolution is 0.3 mm. The 67

finite width of an S2 means that two S2 pulses will only be resolved separately if their z-coordinates differ by more than 3 mm. The (x, y) position of an interaction vertex is ascertained from the hit pattern of the S2 in the top PMT array, based on a localized spot just above the liquid-gas interface. Three independent position-reconstruction algorithms are used to obtain (x, y) from a comparison with a hit pattern generated by Monte Carlo simulation: χ2, support vector machine (SVN), and neural network (NN). The NN algorithm gives the most homogenous response and the best agreement with Monte Carlo. Event-by-event reconstruction allows for position-dependent signal corrections and the selection of a fiducial volume to suppress external sources of background. The x- and y-position resolutions are 3 mm.

4.2 Calibration

To search for WIMP scatters, one must know the detector’s response to both electronic and nuclear recoils very well. Electronic calibrations are performed with 137Cs, 57Co, 60Co, and 232Th. The electronic recoil background of XENON100 is dominated by the decay chains of radioactive contaminants 238U, 232U, 60Co, and 40K. The response to single-scatter nuclear recoils, the expected WIMP signature, is determined with 241AmBe, an (α,n) source. A 10-cm lead shield suppresses the contribution from its high-energy gamma rays (4.4 MeV). In addition, 241AmBe provides gammas from inelastic scattering of xenon and ﬂuorine at 40 keV (129Xe), 80 keV (131Xe), 110 keV (19F), 164 keV (131mXe), 197 keV (19F), and 236 keV (129mXe). Figure 4.2 illustrates the electronic recoil (blue) and nucear recoil (red) bands. These bands deﬁne the upper and lower boundaries of the benchmark WIMP region. A lower bound in energy is determined by the detector’s energy threshold. 68

Figure 4.2. The benchmark WIMP region is deﬁned by calibration sources. The electronic recoil events (blue) deﬁne a 99.75% rejection line. The nuclear recoil events (red) indicate where WIMP-induced recoils should occur in log(S2b/S1)-vs-energy parameter space. A minimal recoil energy is given by the detector’s threshold. Figure from [86].

4.3 Light Collection Eﬃciency

When energy is deposited in the TPC, the amount of energy that is actually measured depends on where this deposition occurs. This position dependence is attributed to several factors including Rayleigh scattering, solid angle effects, reflectivity and mesh transmission. It is necessary to create a three-dimensional correction map of light collection efficiency (LCE) that converts the observed energy values to the actual energy values. In XENON100, three sources are used to infer the correction map: (1) 137Cs, which emits a photon at 662 keV; (2) the 40-keV line of neutrons scattering inelastically off 129Xe; and (3) the 164-keV line of neutron-activated 131mXe. The map is shown in Figure 4.3 as a function of (r2, z). One finds that the largest LCE efficiency occurs at the 69 center of the bottom of the TPC, and the smallest LCE occurs along the circumference of the top. Furthermore, the light yield at 122 keV is needed to determine the equivalent nuclear recoil energy scale of S1 signals. Since gammas of this energy cannot penetrate the TPC to reach the fiducial volume, the light yield cannot be measured directly. Instead, the volume-averaged scintillation yields of several peaks are used to interpolate the light yield: 40 keV (129Xe), 80 keV (131Xe), 164 keV (131mXe), and 662 keV (137Cs). The calculated value of light yield at 122 keVee is 2.20±0.09 PE/keV for a drift field strength of 0.53 kV/cm [85].

0 1.8 z (mm) -50 1.6

1.4 -100 1.2

-150 1 0.8 -200 0.6

0.4 -250 0.2 × 3 -300 10 0 0 2 4 6 8 10 12 14 16 18 20 22 r2 (mm2)

Figure 4.3. The measured light signal depends on position. A light collection eﬃciency map was determined with the 662-keV gamma of 137Cs.

Additionally, two independent effects distort the size of an S2. The first is a result of electron absorption over the drift length, requiring a z-dependent correction. The second is due to various reductions in the S2 light collection efficiency. The fraction of the solid angle covered by the PMT arrays decreases the closer an S2 is to the walls of the TPC. Non-functional PMTs cause spatial variations in the signal. Differences 70 in quantum efficiencies of adjacent PMTs and non-uniformity in the proportional scintillation gap result in asymmetric light collection. The correction map was found in the same way as S1 light collection efficiency, using 137Cs and the 40-keV and 164-keV gammas from inelastic scattering and neutron activation. These effects necessitate an (x, y)-dependent correction.

4.4 Energy Scales

The signals created by recoiling particles can be used to reconstruct a particle’s energy and perhaps to determine which types of particles they are. A detector, however, does not inherently know the type of interaction that occurred within the detector’s sensitive volume. The signal it observes could correspond to a nuclear recoil of energy

Enr or, equivalently, an electronic recoil of energy Eee. In order to assign a particular measure of the energy to the observed signals, one must calibrate the detector. Three diﬀerent energy scales for these detector technologies are discussed in this section.

4.4.1 Electronic Recoil Equivalent Energy

The electronic recoil equivalent energy scale is the scale in which a γ source is used to ascertain the conversion between energy and the detector’s response. γ sources are even used to calibrate detectors that are meant to measure nuclear recoils because the electron-equivalent energies facilitate the identiﬁcation of lines from radioactive isotopes in background spectra. This scale is denoted with the unit “keVee”, or keV electron-equivalent.

4.4.2 Nuclear Recoil Equivalent Energy

The scintillation yield depends on the incident particle and the amount of energy it deposits. The absolute scintillation yield is difficult to measure; instead, the relative scintillation efficiency of nuclear recoils, Leff, is used to convert scintillation signals into 71

nuclear recoil energies. Leff is an energy-dependent quantity that is defined as the ratio of the scintillation yield of nuclear recoils to that of electronic recoils from 122-keV γ rays of 57Co at zero drift field:

Ly(Enr) Leﬀ(Enr) = . (4.1) Ly(122 keV)

In a detector designed to search for nuclear recoils, such as a dual-phase liquid xenon TPC, the energy scale is often based on the scintillation of nuclear recoils in the detection medium. The recoil energy in liquid xenon is found directly from the scintillation signal, S1: S1 1 Ser Enr = , (4.2) Ly Leff Snr where Ly is the light yield of 122-keV γ rays, Leff is the relative scintillation efficiency of nuclear recoils in liquid xenon, and Ser and Snr are the scintillation field quenching factor for electronic and nuclear recoils, respectively. This energy scale is denoted with the unit “keVnr”, or keV nuclear recoil equivalent.

4.4.3 Combined Energy Scale

Due to the anticorrelation betwen scintillation and ionization, one can create a linear energy scale in which the recombination ﬂuctuations are canceled by adding the two signals with appropriate factors [87]. Several γ sources are used to construct this scale, which assumes units of keVee. The combined energy scale is useful for comparison between background spectra and Monte Carlo simulations because it aids in the separation of lines from various radioisotope decays. 72 73

5. RESULTS FROM A CALIBRATION OF XENON100 USING A DISSOLVED RADON-220 SOURCE

Signiﬁcant experimental progress in particle physics continues to be made in searches for rare events such as neutrinoless double-beta decay [88] or scattering of dark matter particles [89]. Experiments that use the liquid noble elements xenon or argon are at the forefront of these searches [90–95]. As these detectors are scaled up to improve their sensitivity, the self-shielding of external radioactive sources yields lower backgrounds and thus further improvement in their performance. However, this feature renders calibration with external sources impractical and necessitates the development of novel strategies. Radioactive calibration sources that can be directly mixed into the liquid target promise to provide an alternate method to calibrate current and future detectors. A dissolved 83mKr source was proposed in [96–98] and employed in the LUX dark matter detector [92,99] for calibration of the electronic recoil energy scale at low energies. Tritiated methane was also employed in LUX [100] in order to exploit the beta decays that fall within the low energy range of interest for dark matter investigations. However, since the activity of tritiated methane must be proactively extracted from the detector with a hot zirconium getter in a xenon recirculation loop, the time required to fully purify the detector after a tritiated methane calibration increases with the volume of the detector. Here, the XENON100 dark matter experiment [85] is used to characterize a dissolved 220Rn source [101] for use in current and future low-background experiments. The source is well suited to calibrate the low-energy electronic recoil background [102]. In addition, the isotopes in the 220Rn decay chain provide alpha and beta radiation that improve our understanding of intrinsic 222Rn [103, 104], which is a dominant source of background [105,106]. Finally, given the short decay time of the whole decay chain, all 74 introduced activity decays within one week, independent of the size of the detector or the speed of recirculation.

5.1 The XENON100 Detector in Run 14

The XENON100 detector, described in detail in [85], is a cylindrical liquid/gas time projection chamber (TPC) that is 30 cm in height and diameter and uses 62 kg of high-purity liquid xenon as a dark matter target and detection medium. An energy deposition in the TPC produces scintillation photons and ionization electrons. The photons provide the prompt scintillation signal (S1). A cathode, which is biased at -12 kV and positioned at the bottom of the TPC, is combined with a grounded gate and an anode mesh near the liquid-gas interface to define an electric field of 400 V/cm. This field drifts the electrons from the interaction site into the gas phase, where a second signal (S2) is generated through proportional scintillation. Both S1 and S2, measured in photoelectrons (PE), are observed by two arrays of photo-multiplier tubes (PMTs), one in liquid xenon at the bottom of the TPC below the cathode and the other in the gaseous xenon above the anode. The TPC is surrounded by a buffer volume containing 99 kg of liquid xenon. A diving bell is used to keep the liquid level constant between the gate and anode meshes. Gaseous xenon is continuously recirculated at 2.6 standard liters per minute through a purification loop. The returning gas pressurizes the diving bell to approximately 2.05 atm, with most of the gas pressure being relieved into the buffer volume through a pipe that is used to define the height of the liquid level. In a separate loop, gaseous xenon is liquefied and returned to the top of the TPC. This open design results in a vertical temperature gradient of 0.8 K over the height of the TPC. 75

5.2 220Rn Decay Chain and Observed Time Evolution

I present the results from a calibration campaign using a 33.6-kBq 220Rn source. The suitability of this source for its employment under low-background conditions was previously reported in [101]. The source contains 228Th electrolytically deposited on a stainless steel disc 30 mm in diameter and housed in a standard vacuum vessel that is flanged onto the xenon gas recirculation system using 1/4” VCR piping. The 220Rn atoms are emanated from the source and flushed into the TPC through the xenon gas stream. With 20 m of piping and a (40 ± 10)% source emanation efficiency, 1.8 × 109 220Rn atoms are acquired in the TPC while the source is open for 1.7 days. A total of 1.7 × 107 decays are observed in the active region during the full calibration run.

220 RnPo α-decays: convection & ions 220Rn 212Pb β--decay: low-energy calibration 55.6s 212BiPo decay: half-life measurement α 6405keV 212Po 216Po 299ns 0.145s α 8954keV 212 β- (64%) Bi α 2252keV 60.6m 6906keV 208 212 Pb α (36%) β- Pb stable 6207keV 569.8keV 10.6h β- 208 4999keV Tl 3.05m

Figure 5.1. The decay chain of 220Rn, following its emanation from a 228Th source. The three main analyses presented in this paper are labeled. Q- values are taken from [107], and half-lives and branching ratios from [108]. 76

102 220 10 Rn 1 10-1 10-2 Activity [mBq/kg] 10-3 2 10 0 1 2 Elapsed3 Time [days]4 5 6 7 216 10 Po 1 10-1 10-2 Activity [mBq/kg] 10-3

10 0 1 2 Elapsed3 Time [days]4 5 6 7

1 Low Energy (2-30) keV 10-1

10-2 Activity [mBq/kg] 10-3

2 10 0 1 2 Elapsed3 Time [days]4 5 6 7 212 10 BiTl

10-1 Activity [mBq/kg] 10-2

2 10 0 1 2 Elapsed3 Time [days]4 5 6 7 212 10 BiPo 1 10-1 10-2 Activity [mBq/kg] 10-3

0 1 2 Elapsed3 Time [days]4 5 6 7 222Rn 10-1

Activity [mBq/kg] 10-2 0 1 2 3 4 5 6 7 Time Since Source Opening [days]

Figure 5.2. Temporal evolution of the various isotopes in the 220Rn decay chain as well as the 222Rn background. The times at which the source was opened and closed are indicated by the solid (green) and the dotted (red) line, respectively. The opening of the source is deﬁned as time t = 0. Binning for 220Rn and 216Po is adjusted for times of low activity. 77

The relevant portion of the 220Rn decay chain is shown in Figure 5.1. I first address the overall viability of this calibration source and defer the description of the respective event selection criteria to the following sections, which also detail the physics that can be extracted from each of the steps in the decay chain. Figure 5.2 shows the isotopic temporal evolution as observed in XENON100. These rates are corrected for deadtime effects, which arise from significant DAQ saturation due to a high trigger rate of O(100) Hz while the source is open. The short-lived isotopes, 220Rn (55.6 s) and 216Po (0.145 s), grow into the active region of the detector within minutes after opening the source, and they quickly decay once the source is closed. Delayed coincidence of 220Rn and 216Po provide the means to detail fluid dynamics within the detector volume. Particle flow is of particular interest because it has the potential to improve the efficiency of purification systems and may inspire new methods to reduce radon backgrounds using self-veto cuts. The primary utility of a 220Rn calibration source comes from the ground-state beta decay of 212Pb with a Q-value of 569.8 keV and a branching ratio of 11.9%. This decay results in low-energy electronic recoils that can be used for background calibration in dark matter searches. As I show here, the isotope’s long half-life, 10.6 hours, gives it ample time to spread throughout the entire detector volume while being sufficiently short to allow the activity to decay within a week. I find 300 low-energy 212Pb decays within the central 34-kg fiducial region used in [109]. Meaning, 3 in every 104 decays are useful for low-energy electronic recoil calibration. For comparison, this ratio is two orders of magnitude higher than that of typical external Compton sources in XENON100, and it is expected to remain constant as detectors become larger. The beta decays of 212Bi may be selected with high efficiency and purity due to their delayed coincidence with the alpha decays of 212Po, occuring within the same digitized trace. The BiPo events provide a standard candle to measure the total activity within the detector. Consequently, they yield the most accurate measurement of the introduced activity’s dissipation time of ∼ 7 days, as seen in Figure 5.2. 78

Further utility of this source comes from the 2.6-MeV gamma decay of 208Tl, which is close to the Q-value of the 136Xe double-beta decay. Due to other low-energy gammas that accompany it, multiple steps are created in the energy spectrum and can be exploited in calibration. Additionally, the alpha decays of 220Rn, 216Po, and 212Bi can be used to calibrate position-dependent light and charge collection eﬃciencies.

5.3 Contamination from the Veto

Figure 5.2 also shows the expected behavior of the various isotopes based on a simple calculation of the exponential decay chain. This treatment effectively assumes instantaneous and complete mixing of all isotopes. As can be seen, this model provides a good fit to the observed time evolution. However, a comparison of the short- and long-lived portions of the decay chain suggest that there is either a deficit of short-lived isotopes or an excess of long-lived isotopes. It has been shown that a significant fraction of the injected radon is diverted into the veto region of the detector. Due to their high energies, some of the gammas that are emitted by 212Pb, 212Bi and 208Tl, will travel into the active region from the veto and interact. Hence, the observed rate of long-lived isotopes is inflated. This effect is modeled with a simple GEANT4 Monte Carlo simulation of the 30- cm diameter active region, surrounded by PTFE 6 mm in thickness and a 4-cm veto region on the outside. The relevant gamma rays are produced in the veto region with isotropic emission, and the fraction that successfully penetrates the active region in determined. Table 5.1 lists all of the gammas that were considered, the decay mode that produces them, their intensities, the probability of depositing energy in the active region given a decay has occurred, and the total contribution. For example, 43.6% of 212Pb decays emit a 238-keV gamma. Of these, 7.8% deposit energy in the active region. Meaning, 212Pb decays in the veto have a leakage intensity of 3.4%. Similarly, there are two contributions from 212Bi, 727 keV and 1.620 MeV, which have leakage intensities of 1.5% and 0.36%, respectively. Finally, there are five decay paths of 208Tl, all of 79

Table 5.1 The results of a GEANT4 Monte Carlo simulation of leakage events from long-lived isotopes in the veto (buﬀer) region. The intensities, deposition probabilities, and overall contribution per mille are given for each gamma. The values correspond to the entire active region.

Isotope Energy [keV] Intensity [%] Deposition [%] Count [per 103]

212Pb 238 43.6 7.8 34.2 212Bi 727 6.7 21.7 14.5 1620 1.5 24.2 3.6 208Tl 277 6.6 10.2 6.7 510 22.6 18.8 42.5 583 85.0 20.4 173 763 1.8 22.0 3.9 860 12.5 22.6 28.3 2614 99.8 25.5 254

Table 5.2 The total leakage contribution from 208Tl is determined according to the diﬀerent combinations of de-excitation gammas. The intensities, deposition probabilities, and overall contribution per mille are given for each decay path. The ﬁnal leakage value accounts for the 36% branching of 208Tl.

Path Gammas [keV] Intensity [%] P(n≥1) [%] Count [per 103]

A 763+583 3.2 17.6 5.6 B 510+583 24.2 20.8 50.4 C 277+583 6.6 17.9 11.8 D 860 12.5 2.8 3.5 E 583 49.1 17.3 85.0 Total 156.3 Leakage 133.4 80 which emit multiple gammas. These paths are listed in Table 5.2 with their intensities, leakage probabilities, and total contribution. Each path terminates with the emission of a 2.614-MeV gamma. Path A occurs in 3% of 208Tl decays, with a leakage probability of 17.6%, resulting in a 0.56% leakage intensity. The combined leakage intensity of all ﬁve paths is 15.6%. Ultimately, accounting for the 1.620-MeV gamma, the probability that one observes at least one event in the active region induced by a gamma from any decay path of 208Tl in the veto is 37.1%.

5.4 Combined Leakage Rate

For every three 212Pb decays, one expects three 212Bi and one 208Tl decay. Any excess activity observed in the active region due to decays in the veto is thus split in the ratio Pb : Bi : Tl = 0.096 : 0.058 : 0.379. Meaning, 0.534 of every 13 decays (4%) in the veto yield an event observed in the active region. A single leakage parameter η is introduced in the analytic model seen in Figure 5.2. This parameter quantifies the excess required to explain the observed activity. For every 4.36 decays of isotopes in the active region, there are η decays (of long-lived isotopes) in the veto that contribute to leakage. In Run 14 of XENON100, I find η = 180, implying that for every true 212Pb decay in the active region there are 42.9 contaminant 212Pb decays in the veto. Similarly, the ratios for 212Bi and 208Tl are 1 : 26.5 and 1 : 471. It follows that the total activity in the veto is 180/4.36/0.04 = 1000 times larger than that in the active region. I consider BiPo events to estimate the true activity inside the active region. A BiPo event can be selected efficiently with high purity due to its time structure. Incorporating livetime corrections, one concludes that there are 1.60 × 105 BiPo events. This count indicates that there are 1.04×106 decays in total within the active region during the full calibration campaign, explaining only 6% of the observed activity. Furthermore, by this second method, I find that the activity in the veto is (0.94×1.7×107/0.04/4.36)/(1.04× 106) = 350 times larger than in the active region, which agrees within a factor of three 81

0 1.6 1.5 -50 1.4 1.3 -100 1.2 -150 1.1

Z [mm] 1 -200

0.9 Correction Factor

-250 0.8 0.7 -300 0.6 0 30 60 90 120 150 2 R /R0 [mm]

Figure 5.3. Light collection map of XENON100 for high-energy alpha events, generated from 220Rn and 216Po decays. Events at low (high) Z and low (high) radius R have the highest (lowest) light collection and thus a correction factor less (greater) than unity.

with the previous ratio. Therefore, in XENON1T, which does not have the open design of XENON100, one can expect a total activity that is O(102 − 103) times higher.

5.5 Alpha Spectroscopy

The interactions of alpha particles in liquid xenon are easily identiﬁable because of their tracks’ large ionization density, which results in small S2’s and large S1’s. This distinction is due to a higher probability of recombination and allows the alphas to be selected based on their S1 yield, thereby rejecting backgrounds from beta or gamma sources. I derive a correction factor to account for the light collection eﬃciency varying across the TPC [85]. As the PMT bases in XENON100 have been optimized for low- energy events in the search for dark matter, a dedicated correction map is required 82

Energy [MeV] 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 1200 212Bi 220Rn 1000

800

600 216Po 400 Count per 100 PE 200 222Rn 0 19 20 21 22 23 24 25 26 27 S1 [1000 PE]

Figure 5.4. The alpha spectrum of the 220Rn decay chain is shown integrated over times before (40.3 hours, red) and after (148.7 hours, blue) the source is closed. The constant background of 222Rn (Figure 5.2) is visible only over the longer time period.

to account for nonlinearity in the response of PMTs to high-energy events. For each spatial bin, the observed mean scintillation value of the 6.4- and 6.9-MeV alpha decays of 220Rn and 216Po is calculated. Then, these mean scintillation values are scaled by the volume-averaged value to obtain a relative correction factor, shown in Figure 5.3. The

2 radial parameter R /R0 is defined according to the detector’s radius, R0 = 150 mm, and has the advantage of being within the interval [0,R0] while preserving uniformity in the transformation from Cartesian to cylindrical coordinates. Events at low (high) Z and low (high) R have the highest (lowest) light collection and thus a correction factor less (greater) than unity. The energy spectrum of the alpha decays of 222Rn, 212Bi, 220Rn, and 216Po is shown in Figure 5.4, after applying the alpha light correction map. The population of alphas is 83 split into periods before (red) and after (blue) the source was closed. These events have been selected in the fiducial volume defined by R ≤ 100 mm and −200 ≤ Z ≤ −5 mm in order to optimize the identification of different isotopes. The energy, mean scintillation value, and light yield (LY) are listed for each isotope in Table 5.3. The light yield is constant in this energy range to within 0.3%. The ability to identify and characterize the various alpha particles is the foundation of the multiple modes of delayed coincidence that are discussed in subsequent sections.

Isotope Q [MeV] S1 [PE] σ [PE] LY [PE/keV] 222Rn 5.590 20231 ± 17 315 3.62 212Bi 6.207 22451 ± 3 296 3.62 220Rn 6.405 23306 ± 3 291 3.64 216Po 6.906 25152 ± 4 316 3.64

Table 5.3 Q-values, scintillation values (means and widths at 400 V/cm), and calculated light yields of each alpha decay in Figure 5.4.

5.6 220Rn-216Po Coincidence and Convection

The combination of spatial and temporal information permits one to match 216Po with its parent 220Rn. As a result, I measure the high-energy position resolution, map the ﬂuid dynamics, and calculate a lower limit on the drift speed of 216Po ions in the XENON100 TPC. I select RnPo pairs from within 3σ of their respective scintillation peaks (see Fig- ure 5.4) with the requirement that a candidate 216Po daughter occur within 1 s and 8 mm of a candidate 220Rn parent. Under these conditions, less than 0.3% of the 220Rn candidates have more than one possible 216Po partner and consequently are removed 84 from the analysis. A total of 45441 RnPo pairs are found in multiple calibrations between June and November 2015. Each RnPo pair provides diﬀerential position values in the vertical (∆Z) and the horizontal (p(∆X)2 + (∆Y )2) directions. The resultant distributions yield measurements of the position resolutions at high energies: σZ = 0.2 mm and σXY = 0.7 mm.

These resolutions are better than those reported at low energies in [85](σZ = 0.3 mm,

σXY = 3 mm) thanks to the larger signals. Moreover, I study ﬂuid dynamics of the liquid xenon using RnPo pairs. I consider a view of the cylindrical TPC through the lateral surface. To this end, I introduce the parameter Y˜ , derived in the AppendixC, which preserves number density in a projection of the cylindrical TPC onto a plane containing its central axis:

2 1 Y˜ = R θ − sin(2θ) − 1 , (5.1) 0 π 2 where Y 0 θ = arccos − , (5.2) R0 and R0 = 150 mm is the radius of the TPC. For reasons that will shortly become apparent, I consider a view at the azimuthal angle φ = −45◦ and calculate the rotated coordinate Y 0 = X sin(−45◦) + Y cos(−45◦). In Figure 5.5 (top), the number density of RnPo pairs is shown. The pairs tend to concentrate along the outer surface near Y˜ = 150 mm beneath the point at which they enter the TPC from the xenon gas system. Large variations in the number density exist because the half-lives of both 220Rn and 216Po are much shorter than the time it takes an atom to fully traverse the TPC. Moreover, I show in Figure 5.5 (bottom) the average z-velocities of the 216Po daugh-

216 ter, vz = ∆Z/∆t, as a function of position, where ∆t is the time the Po atom takes to decay. For Y˜ > 0 (Y˜ < 0), particles move downward (upward) at speeds up to 7.2 mm/s (4.8 mm/s). Ergo, the TPC of XENON100 is a single convection cell whose net angular momentum lies along φ ≈ 135◦. Such convective motion has significant implications 85 for the deployment of calibration sources and for the development of techniques for background mitigation in future experiments. The 214Pb daughter is the main contributor to the low-energy electronic recoil background that arises from the 222Rn decay chain. A known convection pattern can be used to track this 214Pb daughter from the site of its parent 218Po’s alpha decay. In XENON1T, for example, 214Pb traverses at most 0.7% of the TPC during its 27.1-min half-life. Consequently, a low-energy 214Pb decay can be tagged by its parent 218Po, effectively reducing the electronic recoil background. Finally, due to the difference between the minimum and maximum velocities in Figure 5.5, I conclude that there is a subdominant contribution to the total particle motion that results from the applied electric field, 400 V/cm. Figure 5.5 (bottom) shows a bias toward downward velocities because more pairs move downward than upward (as a result of their quick decay). I therefore require |Y˜ | > 100 mm to calculate the variance- weighted mean velocities on either side,v ¯up andv ¯down, for each of ten equal Z bins in the range [−200, −100] mm. The two velocities must be determined separately because of their unequal number densities. Then, the velocity offset within each Z bin is found by averaging the two components: voffset = (¯vup +v ¯down)/2. Consequently, the mean offset over all ten bins is found to bev ¯offset = (0.9 ± 0.3) mm/s toward the cathode. This offset constitutes a lower limit on the ion drift speed of 216Po, corresponding to a completely ionized population. This limit is consistent with the ion drift speeds presented by EXO-200 for 218Po at 380 V/cm [103]. Delayed coincidence of 212Bi and 208Tl (BiTl) is also attempted following a method- ology similar to that of RnPo. With atomic speeds up to ∼ 7 mm/s in XENON100, the 208Tl atom can travel up to ∼ 140 cm during its 3-minute half-life. This distance is much larger than the 30-cm dimension of XENON100, making it impossible to accurately match BiTl pairs at this stage. However, BiTl coincidence may be a useful component of this calibration source for meter-scale detectors such as XENON1T or single-phase detectors such as EXO-200. 86

0 800

-50 700

600 -100 500

-150 400 Count Z [mm] -200 300 200 -250 100

-300 0 -150 -100 -50 ~ 0 50 100 150 Y [mm] 0 8

6 -50 4 -100 2

-150 0 [mm/s] z Z [mm] -2 v -200 -4 -250 -6

-300 -8 -150 -100 -50 ~ 0 50 100 150 Y [mm]

Figure 5.5. Top: Number density of 220Rn216Po pairs across the TPC. Due to the short half-lives at the begining of the decay chain, these events are still concentrated near the entry point to the TPC. Bottom: Delayed coincidence of these pairs is used to map the average z-component of atomic velocity as a function of position. Atomic motion in XENON100 is primarily a result of a single convection cell, viewed here along the direction of its angular momentum vector. 87

5.7 The half-life of 212Po

The beta decay of 212Bi (Q = 2.2 MeV) and the alpha decay of 212Po are easy to identify because they occur in quick succession (τ1/2 = 299 ns) within a single, continuously recorded digitizer trace. The selection of BiPo events is made within the energy range 20 PE < S1β < 7000 PE, 10000 PE < S1α < 55000 PE, and with the requirement that the S1β appear in the waveform before the S1α. Figure 5.6 shows that the BiPo events are distributed throughout the active region. Furthermore, the drift time of the ionization electrons from the beta decay of 212Bi must exceed the decay time of the 212Po alpha decay. Consequently, I exclude events within 5 mm of the anode. Figure 5.7 shows the resulting distribution of time diﬀerences of the selected alpha and beta decays. One can infer from this distribution that this BiPo sample is > 99.75% pure.

0 103 -50

-100 102 -150 Count Z [mm] -200 10 -250

-300 1 0 30 60 90 120 150 2 R /R0 [mm]

Figure 5.6. The spatial distribution of low-energy 212BiPo decays is shown. The events permeate the entire active region of the TPC. 88

4 τ =(293.9±(1.0) ±(0.6) ) ns 10 1/2 stat sys χ2/NDF=143.6/113

s 10 µ

102 Count per 0.8 10

30 100 200 300 400 500 600 700 800 900 1000 ∆ τ (samples) ] 2 σ 1 0 -1 -2 Residuals [ -3 -4 0 1 2 3 4 5 6 7 8 9 10 ∆ t [µs]

Figure 5.7. (Top) Time difference of BiPo coincidence events together with a fit of the half-life of 212Po. The lower bound of the fit is set at 720 ns to avoid peak-finding artefacts. (Bottom) The residuals of the fit in the top panel.

The distribution of 212Po decay times is ﬁtted with an exponential model of the

−∆t/τ event rate, N(∆t) = N0e +B, where N0 is the event rate at small time diﬀerences, and B is the background rate. Fitting over the range [0.72, 10] µs, I ﬁnd a half-life

τ = (293.9 ± (1.0)stat) ns. The residuals of this ﬁt are shown in the bottom panel of Figure 5.7. 89

310

305

300

295 Fitted half-life [ns]

290

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Lower Threshold of the Fit [µs]

Figure 5.8. The fitted half-life of 212Po as a function of the minimum time difference that is considered. The lower bound of the half-life measurement is set at 720 ns. Based on the range [0.64, 0.90] µs, I estimate a 0.5-ns contribution to the total systematic uncertainty that results from the lower threshold of the time difference.

Various systematic effects are considered for the uncertainty of this half-life measurement. The minimum time difference considered for the fit is chosen to minimize the combined statistical and systematic uncertainty. Figure 5.8 shows the fitted half-life as a function of this minimum time difference. The RMS value of the four points in the range [0.64, 0.90] µs, 0.5 ns, is taken as the systematic uncertainty resulting from the fit range. The clock of the digitizer contributes less than 0.3 ns to the systematic uncertainty. I repeat the measurement with different lower energy thresholds of the selection of S1α and S1β, estimating a 0.1 ns contribution to the uncertainty. I find that the result is independent of temporal resolution and the choice of binning. Peak- finding jitter, S1 scintillation properties, and other effects that modify the time stamp 90 of individual S1 peaks average out. Ultimately, I measure the half-life of 212Po to be

τ = (293.9 ± (1.0)stat ± (0.6)sys) ns.

5.8 Low-Energy Calibration

The decay of 212Pb (Q = 569.8 keV) emits beta particles that may be useful for the calibration of liquid noble gas detectors in the search for dark matter. The direct decay of 212Pb to the ground state of 212Bi, occurring with a branching ratio of 11.9%, is the relevant decay for low-energy electronic recoil calibration. Specifically, ∼10% of these direct decays fall within the low-energy range of interest, 2 − 30 keV. The source is viable in this capacity as long as the activity of 212Pb spreads throughout the entire active region of the TPC. In order to identify these low-energy beta decays with high purity, three types of selection cuts are applied following previous XENON100 dark matter analyses of electronic recoils [52, 110, 111]. The first type includes cuts that remove excessively noisy events, ensuring that one chooses actual particle interactions. The second type checks for consistency between the drift time and the width of the S2. Finally, I select the relevant low-energy beta decays by applying a cut on the scintillation signal, 3 PE < S1 < 60 PE, which corresponds to the energy range 2 − 30 keV [52, 110]. However, as discussed in Sections 5.3 and 5.4, many of these low-energy events are induced by gammas that travel into the TPC from the buffer volume. The spatial distribution of these low-energy events is shown in Figure 5.9. Clus- tering near the electrodes is apparent for 212Pb events. For the XY -distribution, an additional cut was thus applied to exclude events within 5 mm of either the cathode or the anode. As can be seen, the low-energy activity is present throughout the TPC. Despite the fact that low-energy 212Pb events cannot be disentangled from the buffer events, the distribution of BiPo events in Figure 5.6 demonstrates that the low-energy 212Pb events reach the center of the active region. The convection pattern observed with the short-lived alpha decays does not have a measurable effect on the distribu- 91 tion. This measurement validates the 220Rn source as a low-energy electronic recoil calibration source for noble element detectors. The same Monte Carlo simulation presented in Section 5.3 is used to determine the fraction of leakage events that fall within the inner 34-kg fiducial volume applied to the low-energy population in 5.2. Table 5.4 lists the emission intensity, deposition probability, contribution per gamma, and leakage fractions with and without relevant branching ratios for each energy consider in the simulation. Additionally, the contributions from the multiple decay paths of 208Tl are detailed in Table 5.5. Safely assuming that the activity in the buffer volume is 103 times larger than the activity in the active region, I find 0.23 low-energy events induced by decays in the buffer for every 212Pb atom in the fiducial volume. Equivalently, there are 19 low-energy events from buffer decays for every low-energy 212Pb decay that is truly within the 34-kg fiducial volume. Therefore, 300 of the 5300 events observed in the 34-kg fiducial volume are actually the low-energy 212Pb beta events that one seeks, and about 3 in every 104 events are useful for calibration in XENON1T or future dark matter experiments. This source efficiency agrees well with the original proposal, being approximately two orders of magnitude better than that of the Compton sources used in the past. Since the full decay chain has a collective decay time of less than 12 hours, the introduced activity decays within a week. Unlike calibration with tritiated methane, this time scale is independent of the recirculation speed or efficiency, making this source useful for the largest detectors envisioned. I observe no activity attributable to either 224Ra (the parent of 220Rn) or 222Rn after the source is closed and thus place limits on the inadvertent introduction of these isotopes. In the case of 224Ra, the rate of 220Rn events is determined for periods both before and after the source is deployed, resulting in a limit on the introduced 224Ra activity of 1.0 µBq/kg at 90% confidence. The bottom panel of Figure 5.2 shows that the rate of 222Rn remains constant over the period during which the calibration source is deployed, resulting in a limit on the introduced 222Rn activity of 13 µBq/kg at 90% confidence. 92

-50

-100 102

-150 Count Z [mm] -200 10

-250

-300 1 0 30 60 90 120 150 2 R /R0 [mm] 150

100

50 102

0 Count Y [mm] Y -50 10

-100

-150 1 -150 -100 -50 0 50 100 150 X [mm]

Figure 5.9. The spatial distribution of low-energy decays is shown in two diﬀerent perspectives. Since some clustering near the electrodes is apparent (top), events within 5 mm of either electrode have been rejected to display the XY -distribution (bottom). The distribution of BiPo events in Figure 5.6 shows that the beta emission of 212Pb permeates the entire active region of the TPC, conﬁrming the suitability of this calibration source for low-energy detectors. 93

Table 5.4 The results of a GEANT4 Monte Carlo simulation of leakage events from long-lived isotopes in the veto (buﬀer) region. The intensities, deposition probabilities, and overall contribution per mille are given for each gamma. The values correspond to the entire active region.

Isotope Energy [keV] Intensity [%] Deposition [%] Count [per 103]

212Pb 238 43.6 0.001 0.006 212Bi 727 6.7 0.035 0.024 1620 1.5 0.038 0.006 208Tl 277 6.6 0.004 0.002 510 22.6 0.026 0.058 583 85.0 0.029 0.251 763 1.8 0.037 0.007 860 12.5 0.040 0.050 2614 99.8 0.033 0.326

Table 5.5 The total leakage contribution from 208Tl is determined according to the diﬀerent combinations of de-excitation gammas. The intensities, deposition probabilities, and overall contribution per mille are given for each decay path. The ﬁnal leakage value accounts for the 36% branching of 208Tl.

Path Gammas [keV] Intensity [%] P(n≥1) [%] Count [per 103]

A 763+583 3.2 0.026 0.008 B 510+583 24.2 0.031 0.075 C 277+583 6.6 0.025 0.017 D 860 12.5 0.005 0.006 E 583 49.1 0.025 0.123 Total 0.555 Leakage 0.200 94

5.9 Conclusions

I have presented a novel calibration method for liquid noble element detectors using a dissolved 220Rn source. The 220Rn decay chain provides several isotopes that allow for a variety of different calibrations, including the response to low-energy beta decays, high-energy gamma lines, and the important 222Rn background. The activity enters the active volume as soon as the source is opened to the gas recirculation system. No contamination is observed from long-lived isotopes, and the introduced activity naturally decays within a week after the source is closed. Since this dissipation time is independent of the size of the detector, calibration with 220Rn is particularly appealing for large-scale detectors, such as XENON1T, XENONnT, LZ, DARWIN, DEAP-3600, and nEXO. The primary utility of the source is the beta decay of 212Pb, which can be employed to calibrate a detector’s response to low-energy electronic recoil backgrounds in the search for dark matter. The 212Pb atoms permeate the entire active region, including the center which is beyond the reach of traditional calibrations with external Compton sources. Furthermore, the high-energy alpha decays of 220Rn and 216Po provide the means by which to map atomic motion. One observed a single convection cell in XENON100 at speeds up to ∼ 7 mm/s as well as subdominant ion drift in the electric field of the TPC. Such an improved understanding of fluid dynamics within a detector promises to motivate analytic techniques for background mitigation. Beyond the development of calibration techniques, I have used the beta decay of 212Bi and the alpha decay of 212Po to make a high-purity, high-statistics measurement

212 of the half-life of Po of τ = (293.9 ± (1.0)stat ± (0.6)sys) ns. 95

6. DARK MATTER SCATTERING INELASTICALLY OFF XENON NUCLEI

Astrophysical evidence indicates that the Milky Way disc is embedded into a non- baryonic dark matter halo [112], which could be made of weakly interacting massive particles (WIMPs) [113]. The WIMP dark matter hypothesis is testable, as WIMPs may be detected directly by scattering oﬀ nuclei in low-background underground detectors, indirectly by observing annihilation products above astrophysical backgrounds, and by producing them at the LHC [114]. In direct detection experiments, only the elastic scattering channel, with an exponential nuclear recoil energy spectrum, is usually exploited [115]. Another avenue to direct detection is to observe inelastic WIMP-nucleus scattering, inducing transitions to low-lying excited states [116]. The experimental signature is a nuclear recoil together with the prompt de-excitation photon. In odd-mass nuclei with low-lying excited states, a galactic WIMP with suﬃcient kinetic energy can induce inelastic excitations. Xenon is an excellent target material because natural xenon contains the odd 129Xe and 131Xe isotopes with abundances of 26.4% and 21.2%, respectively. The excitation energies of the lowest-lying states are reasonably low, with a

+ + 129 + 3/2 state at 39.6 keV above the 1/2 ground state in Xe and a 1/2 state at 80.2 keV

+ 131 above the 3/2 ground state in Xe. The nuclear decays are electromagnetic M1 and E2 with half-lives of 0.97 ns and 0.48 ns, respectively. Searches for inelastic scattering have been performed in the past [117–120]. Here, I show that inelastic WIMP-nucleus scattering in xenon is complementary to elastic scattering for spin-dependent interactions, as the inelastic channel dominates the integrated spectra above ∼ 10 keV energy deposition, depending on the WIMP mass [121]. This aspect not only provides a consistency check for a xenon experiment, 96 but in the case of dark matter detection via this channel, it would oﬀer a clear indication for the spin-dependent nature of the fundamental interaction.

6.1 Nuclear Structure and WIMP-Nucleon Currents

The calculation of inelastic WIMP scattering off nuclei requires a reliable description of the structure of the initial and final nuclear states. Considering that low-lying transitions occur in odd-mass isotopes with different spins between the ground and excited states, spin-dependent WIMP scattering generally dominates in the inelastic channel. I reference [122] for a study of the spin-independent case. Thus, in what follows, only spin-dependence is considered. In collaboration with Achim Schwenk, Javier Menendez, and Philipp Klos, I used the shell-model code ANTOINE [123] and the GCN5082 [124, 125] interaction in the

0g7/2, 1d5/2, 1d3/2, 2s1/2 and 0h11/2 valence spaces to perform state-of-the-art large-scale nuclear structure calculations for 129Xe and 131Xe. As long as WIMPs interact with quarks at the momentum scales relevant to nuclei and WIMP-nucleus scattering, the relevant degrees of freedom are nucleons and spin. Chiral eﬀective ﬁeld theory (EFT) systematically expands WIMP-nucleon coupling into powers of momentum Q based on symmetries of QCD. At leading orders Q0 and Q2, chiral EFT predicts one-body (1b) currents given by [126,127]

A A 2 2 X X 1 gA(p ) gP (p ) J3 = a σ + a τ 3 σ − (p · σ ) p , (6.1) i,1b 2 0 i 1 i g i 2mg i i=1 i=1 A A where the summation is over all A nucleons, and spin-1/2 WIMPs are considered. a0 3 and a1 are the isoscalar and isovector WIMP-nucleon couplings; σi and τi are the spin and isospin matrices; p = pi − pf represents the momentum transfer from nucleons 2 2 2 to WIMPs; gA(p ) and gP (p ) are the axial and pseudoscalar couplings including Q corrections given in [126]; gA = gA(0); and m is the nucleon mass. In addition to coupling through 1b currents, chiral EFT predicts two-body (2b) currents at order Q3, by which WIMPs couple to two nucleons [126]. Two-body currents 97 are quantitatively important in medium-mass and heavy nuclei because the momentum involved is sensitive to both momentum transfer and the typical momentum of a nucleon in a nucleus. The inclusion of long-range 2b currents results in additional contributions to the isovector axial and pseudoscalar currents. Combining the 1b and 2b currents to order Q3 yields [127]

2 2 P 2 3 1 3 gA(p ) gP (p ) δa1 (p ) Ji,1b+2b = a1τi + δa1(p) σi + − + 2 (p · σi) p , (6.2) 2 gA 2mgA p with momentum- and density-dependent renormalizations of the axial-vector and pseu-

P doscalar components, δa1(p) and δa1 (p), respectively, ρ 1 1 h σ σ i δa1(p) = − 2 c4 + 3I2 (ρ, p) − I1 (ρ, p) Fπ 3 4m 1 1 1 +c ˆ + −c + Iσ(ρ, p) − 6 I (ρ, p) , (6.3) 3 3 4m 1 12m c6 2 P ρ −2c3p c3 + c4 P 1 +c ˆ6 δa1 (p) = 2 2 2 + I (ρ, p) − Ic6(ρ, p) , (6.4) Fπ mπ + p 3 12m with density ρ, pion mass and decay constant mπ, Fπ, and chiral EFT low-energy σ σ couplings c3, c4,c ˆ6. The values for the couplings and the functions I1 (ρ, p), I2 (ρ, p), P I (ρ, p), and Ic6(ρ, p) (due to integrals in the exchange terms) are given and explained in detail in [127]. The uncertainties in chiral 2b currents are dominated by the uncertainties of the c3 and c4 couplings. A weak dependence on p leads to δa1(0) = −(0.14 − 0.32) [127], such that 2b currents reduce the axial part of the WIMP-nucleon currents. Conversely,

P δa1 (mπ) = 0.23−0.54 [127] enhances the pseudoscalar part. This enhancement vanishes at p = 0, due to the pseudoscalar nature, but increases with momentum transfer.

6.2 Structure Factors

The structure factors of inelastic, spin-dependent WIMP-nucleus scattering are calculated based on the initial and ﬁnal nuclear states. The structure factor SA(p) receives 98 contributions from longitudinal (L5), transverse electric (T el5) and transverse magnetic (T mag5) multipoles

X 5 2 X el5 2 mag5 2 SA(p) = hJf ||LL||Jii + hJf ||TL ||Jii + hJf ||TL ||Jii , (6.5) L>0 L>1 where the detailed expressions for the multipoles in terms of the 1b+2b currents of Eq. (6.2) are given in [127]. As a result of parity conservation, only odd (even) L multipoles contribute to the electric (magnetic) multipoles for an inelastic transition between nuclear states with the same parity. While magnetic multipoles do not contribute to elastic scattering because of time-reversal symmetry, they do contribute to inelastic scattering. The inelastic structure factors for WIMP scattering off 129Xe and 131Xe are shown in Figure 6.1. These structure factors are given in terms of “proton-only” (a0 = a1 = 1) and “neutron-only” (a0 = −a1 = 1), Sn(u) and Sp(u), which are functions of the dimensionless momentum transfer u = p2b2/2 with harmonic oscillator length b. These couplings are more sensitive to neutrons and protons, respectively. The fact the xenon possesses an even number of protons implies that (practically) all the spin, and, hence, the nuclear spin response, is drawn from neutrons. Fits of both elastic [126] and inelastic [127] structure functions are available in [121]. In Figure 6.2, the elastic and inelastic structure factors are compared in the experimentally relevant region. At p = 0, both inelastic Sn(u) and Sp(u) are significantly smaller than their elastic counterparts, one order for 129Xe and more than two orders for 131Xe. For both isotopes, a maximum occurs in the inelastic structure factor at low momentum transfer; this feature does not exist in elastic scattering, where the maximum always occurs at p = 0 followed by a steep decrease. Therefore, for u = 1 − 2 (p = 125−175 MeV), the inelastic channel is comparable to the elastic channel. The significance is that this region falls within the range of allowed momentum transfers. The inelastic structure factors even dominate their elastic counterparts in the case of 129Xe. For 131Xe, the inelastic structure functions are slightly smaller but still comparable to the elastic ones. 99

Similar results of a comparison of elastic and inelastic scattering were found in [128], although the structure factors presented there diﬀer greatly from those presented here. For example, the inelastic structure factor of 129Xe in [128] is monotonically decreasing; in the case of 131Xe, the suppression of the inelastic response at p = 0 is an order of magnitude smaller than in the case presented here. It should also be noted that elastic and inelastic structure factor alike for 131Xe are larger a u ∼ 1 in [128]. The diﬀerences are the consequences of the well-tested interactions and the smaller truncations in the present calculations. To explain the behavior at p = 0 and the maxima in the inelastic structure factors,

129 + one must consider the spin and parity of the ground state of Xe, 1/2 , and its ﬁrst

+ 131 + + excited state, 3/2 . The situation is reversed for Xe, with a 3/2 ground and 1/2 first excited state. Only L = 1, 2 multipoles contribute to inelastic scattering. For elastic scattering, in which the magnetic multipoles do not contribute, the allowed multipoles are L = 1 for 129Xe and L = 1, 3 for 131Xe. At p = 0, only the L = 1 multipole contributes to scattering, an only orbitals of the initial and final states with the same orbital angular momentum l can be connected. Because these contributions are naturally maximal in elastic scattering–andsmaller when nucleons are excited into different l orbitals–a noticeable reduction in the inelastic structure factors occurs at p = 0 in the inelastic case. On the other hand, at p > 0, all multipoles contribute, and inelastic scattering is no longer suppressed. Since magnetic multipoles with L = 2 only contribute to inelastic scattering the maximum of the inelastic structure factors is enhanced with respect to the elastic ones. The full effects of the suppression at p = 0 and the maximum of the inelastic structure factor depend on the nuclear structure details of the states involved and on the relative contribution of the magnetic multipoles. 100

0.1 S (u) 1b currents 129 p 0.01 Xe Sn(u) 1b currents Sp(u) 1b + 2b currents

0.001 Sn(u) 1b + 2b currents (u)

i -4

S 10

-5 10

-6 10

-7 10 1 2 3 4 5 6 7 8 9 0.1 S (u) 1b currents 131 p 0.01 Xe Sn(u) 1b currents Sp(u) 1b + 2b currents

0.001 Sn(u) 1b + 2b currents (u)

i -4

S 10

-5 10

-6 10

-7 10 0 1 2 3 4 5 6 7 8 9 10 u

Figure 6.1. Inelastic structure factors Sp (solid lines) and Sn (dashed) for 129Xe (top panel) and 131Xe (bottom panel) as a function of u = p2b2/2. The harmonic-oscillator length is b = 2.2853 fm for 129Xe and b = 2.2905 fm for 131Xe. Results are shown at the one-body (1b) current level and including two-body (2b) currents. The estimated theoretical uncertainty in the 2b currents is given by the red (Sp(u)) and blue (Sn(u)) bands. 101

0.1 S (u) 1b + 2b inelastic 129 p Xe Sn(u) 1b + 2b inelastic Sp(u) 1b + 2b elastic

0.01 Sn(u) 1b + 2b elastic (u) i S 0.001

-4 10

0.5 1 1.5 0.1 131 Xe

0.01 (u) i S 0.001

-4 10

0 0.5 1 1.5 2 u

Figure 6.2. Comparison of the inelastic structure factors from Figure 6.1 to elastic counterparts from [126,127]. Both Sn(u) and Sp(u) are signiﬁcantly smaller than their elastic counterparts at u = 0. A maximum occurs for both isotopes in the elastic structure factors. Data for these structure factors are available in [121]. 102

6.3 Kinematics of Inelastic Scattering

Given the promising results for the inelastic structure factors, I calculate the expected recoil spectra for inelastic dark matter scattering. By ﬁrst principles, the WIMP- nucleus scattering must conserve momentum and energy:

qi = qf + q , (6.6) 2 2 qi qf ∗ = + ER + E , (6.7) 2mχ 2mχ where qi and qf are the initial and ﬁnal WIMP momenta, q = qi − qf (= −p) is ∗ the momentum transfer, mχ is the WIMP mass, and E the excitation energy of the nucleus. The nuclear recoil energy is given by

q2 ER = , (6.8) 2mA with the mass of the nucleus mA. Eliminating qf in Eq. (6.7), this leads to a quadratic equation for q,

2 ∗ q − (2µvi cos β)q + 2µE = 0 , (6.9) with the reduced mass µ = mAmχ/(mA + mχ), initial WIMP velocity vi = qi/mχ, and where β is the angle between qi and q. This equation has two solutions: s ! 2E∗ q± = µvi cos β 1 ± 1 − 2 2 . (6.10) µvi cos β

This leads to two constraints:

1 1 E∗ ≤ µv2 cos2 β ≤ µv2 , (6.11) 2 i 2 i and 1 q E∗ q E∗ v = + ≥ + ≡ v , (6.12) i cos β 2µ q 2µ q min where the allowed range is cos β ≥ 0. The upper limit in Eq. (6.11) on the excitation energy shows that the inelastic excitation is more sensitive to WIMPs with high velocities in the tail of the dark matter halo distribution. Because q− (q+) is monotonically 103

decreasing (increasing) with respect to vi cos β, the absolute minimal (maximal) momentum transfer occurs at vi cos β = vi for cos β = 1. This deﬁnes the minimal and maximal recoil energies:

2 2 s ∗ ! (µvi) 2E ER,min/max = 1 ∓ 1 − 2 . (6.13) 2mA µvi

In the Earth’s rest frame, the maximal vi is given by vesc + vEarth, where vesc and vEarth denote the galactic escape velocity and Earth’s speed in the galaxy, respectively. I assume vesc = 544 km/s and an average vEarth = 232 km/s in subsequent calculations. 1 2 ∗ As 2 µvi → E , the domain of recoil energies over which the recoil spectrum is deﬁned 2 ∗ shrinks, converging to the value (µvi) /2mA = µE /mA. Table 6.1 gives the minimal and maximal recoil energies for the two xenon isotopes and various WIMP masses.

129Xe 131Xe

Mass [GeV] ER,min ER,max ER,min ER,max 10 − − − − 25 1.5 31 − − 50 1.2 110 6.8 81 100 1.1 285 5.4 244 250 1.1 659 4.9 601 500 1.1 954 4.7 885

Table 6.1 Minimal and maximal recoil energies, in keV, between which inelastic scattering is allowed (see Eq. (6.13)) for the two xenon isotopes and various WIMP masses.

I calculate the nuclear recoil spectra for elastic and inelastic, spin-dependent WIMP scattering oﬀ 129Xe and 131Xe following [129]: √ dR πv0 R0 g(vmin) σ ρ0 v0 = −36 2 −3 −1 , (6.14) dER 2 mχmA E0 r 10 cm 0.3 GeVcm 220 km s 104 where the WIMP-nucleus cross section σ is given by [68]: 4 π µ 2 σ = SA(q) σnucleon . (6.15) 3 2Ji + 1 µnucleon

R0 = 361 events/(kg d) is the total event rate per unit mass for the Earth being at rest and an inﬁnite escape velocity, E0 is the most probable kinetic energy of an incident WIMP (given in terms of the the characteristic parameter of the Maxwell-Boltzmann distribution v0), ρ0 is the local WIMP density in our galaxy [130], and r is a kinematic factor r = 4µ/(mA + mχ). Ji is the nuclear spin in the initial state, µnucleon is the

WIMP-nucleon reduced mass, and σnucleon is the zero momentum transfer cross section for the nucleon. g(vmin) is the integral containing information about the WIMP velocity distribution f(v + vEarth)[131]: Z ∞ f(v + vEarth) 3 g(vmin) = d v , (6.16) vmin v where the integral is from vmin, and the velocity distribution f(v + vEarth) is truncated at vesc.

Figure 6.3 shows the normalized velocity integral, g(vmin)/g(0), for three WIMP velocity distributions: the Standard Halo Model [132], the Double Power Law proﬁle [133], and the Tsallis model [134,135]. The diﬀerent types of interactions are sensitive to different velocities. While for elastic scatters, the lowest vmin is determined by a detector’s

p ∗ energy threshold, in the inelastic case the lowest vmin occurs at 2E /µ (or recoil en- ∗ ergy ER = µE /mA, see the previous discussion). Figure 6.4 shows the differential recoil spectra for scattering off 129Xe and 131Xe for “neutron-only” couplings and for the structure factors that include chiral 1b+2b currents. The elastic structure factors are taken from [127]. The widths of the bands reflect the theoretical uncertainties in the WIMP-nucleon currents. The elastic spin- dependent spectra are shown for comparison. For illustration, all spectra are shown over a wide range of expected rates; but, it should be noted that, for the interpretation of future experimental results, only the highest-rate parts will be relevant. As expected, the inelastic spectra fall to zero at the bounds of the recoil energy, and the domain for 131Xe is smaller because of the higher excitation energy compared to 129Xe. 105

1 Standard Halo Model Double Power Law Tsallis Model 0.1

)/g(0) 0.01 min g(v Xe Xe 129 0.001 131 Elastic Inelastic Inelastic -4 10 0 100 200 300 400 500 600 700 800

vmin (km/s)

Figure 6.3. g(vmin)/g(0) as a function of vmin for several WIMP velocity distributions. The lowest values of vmin are shown for elastic Xe (magenta), inelastic 129Xe (cyan), and inelastic 131Xe (yellow) scattering. In the elastic case, vmin is calculated at XENON100’s nuclear recoil energy threshold of ∼ 7 keV (neglecting the diﬀerence in the nuclear mass number A)[109].

6.4 Signatures of Inelastic Dark Matter Scattering

Dark matter detectors that use liquid xenon can be expected among those most sensitive to inelastic, spin-dependent WIMP scattering [91]. In these detectors, the nuclear recoil and subsequent gamma emission will occur at the same space-time coordinates: the half-lives of the lowest excited states in 129Xe and 131Xe are 0.97 ns and 0.48 ns, respectively, and the mean free paths of gammas with energies of 39.6 keV and 80.2 keV in liquid xenon are ∼ 0.15 mm and 0.92 mm, whereas experimental resolutions are typically of order 10 ns and 3 mm [85]. Therefore, experiments can search for a total energy deposition,

∗ Evis = f(ER) × ER + E , (6.17) 106

-4 10 129 -5 10 Xe elastic 131

) -6 Xe elastic

-1 129 10 Xe inelastic -7 131 10 Xe inelastic keV

-1 -8 d 10 -1 -9 10 (kg

R -10 10

-11 10 dR/dE

-12 10

-13 10 0 50 100 150 200 250 300

ER (keV)

Figure 6.4. Diﬀerential nuclear recoil spectra dR/dER as a function of 129 131 recoil energy ER for scattering oﬀ Xe and Xe, assuming “neutron- only” couplings of a 100 GeV WIMP with a WIMP-nucleon cross section −40 2 σnucleon = 10 cm . Both elastic and inelastic recoil spectra are shown for comparison, including chiral 1b+2b currents, where the bands include the uncertainties due to WIMP-nucleon currents.

in the detector, where Evis is the observed energy and f(ER) is an energy-dependent quenching factor for nuclear recoils [129]. Only a fraction f(ER) of the nuclear recoil energy will be transferred to electronic excitations. These can be observed as scintillation light in a single-phase detector or as prompt scintillation and delayed charge signals in liquid xenon time projection chambers [136, 137]; the rest is transferred to heat and remains undetected. Figure 6.5 shows the differential nuclear recoil spectra for inelastic scattering off 129Xe and 131Xe, the differential energy spectra of the de-excitation gammas, and the sum of these two contributions. The energy scale is based on the total number of quanta detected in a liquid xenon dark matter experiment. For nuclear recoils, I assume the 107

Lindhard theory [69] with a conservative choice of the proportionality constant between the electronic stopping power and the velocity of the recoiling xenon atom, k = 0.110, as shown in Figure 1 of [138]. For the gamma lines, I assume an energy resolution as obtained in the XENON100 detector at these energies in inelastic neutron-xenon scatters [85], using a linear combination of the primary scintillation and proportional scintillation signals [139]: σ/E = 9% at 40 keV and σ/E = 6.5% at 80 keV. Due to the prompt gamma from the nuclear de-excitation, the region of interest for the dark matter search is shifted to higher energies compared to elastic scattering. Thus, experiments can search in their observed differential energy spectra for clear structures around 40 keV (129Xe) and 80 keV (131Xe), although the higher energy region is kinematically suppressed. This suppression is particularly pronounced for WIMP masses below ∼ 100 GeV (see Figure 6.7). In near-future detectors, observed event rates are expected to be small. While a precise measurement of the differential energy spectrum will thus be statistically pro- hibited, integrated rates can provide valuable insight. Figure 6.6 shows the integrated spectra for elastic and inelastic scattering of a 100 GeV WIMP as well as their sum, while Figure 6.7 displays these for a range of WIMP masses. These spectra are shown for natural xenon, thus the isotopic contributions of each isotope, given by Eq. (6.14), are weighted by the aforementioned abundances. The inelastic channel is favored over the elastic one at momentum transfers above 36 MeV, 45 MeV, 54 MeV, 68 MeV and 76 MeV (corresponding to nuclear recoil energies of 5 keV, 8 keV, 12 keV, 19 keV and 24 keV) for WIMP masses of 25, 50, 100, 250 and 500 GeV, respectively. Should both elastic and inelastic signatures be observed by a current or future xenon experiment, it would offer a strong case for the spin-dependent nature of WIMP-nucleon interactions. On the other hand, if signal-like events are observed with no excess signal in the spin-dependent inelastic channels, this would indicate a spin-independent nature of WIMP-nucleon interactions. 108

-5 10 129 Nuclear Recoil -6 Xe EM Deexcitation ) 10 Total -1

-7

keV 10 -1 d

-1 -8 10 (kg

vis -9 10

dR/dE -10 10

-11 10 0 20 40 60 80 100

Evis (keV) -5 10 Nuclear Recoil -6 131 EM Deexcitation ) 10 Xe Total -1

-7

keV 10 -1 d

-1 -8 10 (kg

vis -9 10

dR/dE -10 10

-11 10 0 20 40 60 80 100 120 140

Evis (keV)

Figure 6.5. Diﬀerential energy spectra as a function of Evis for inelastic scattering oﬀ 129Xe (top) and 131Xe (bottom). Shown is the nuclear recoil spectrum (blue), the de-excitation gamma (green), and their sum (red), assuming realistic detector resolution and quenching of the nuclear recoil signal. As in Figure 6.4, results are shown with chiral 1b+2b currents, where the bands include the uncertainties due to WIMP-nucleon currents, and I have assumed “neutron-only” couplings of a 100 GeV WIMP with −40 2 σnucleon = 10 cm . 109

129 Xe inelastic 131 -4 Xe inelastic 10 Total inelastic 129 Xe elastic

) 131

-1 Xe elastic d -5 Total elastic -1 10 Total R (kg

-6 10

-7 10 0 20 40 60 80 100

Evis (keV)

Figure 6.6. Integrated energy spectra of xenon for elastic and inelastic, spin- dependent scattering for “neutron-only” couplings and a 100 GeV WIMP −40 2 with σnucleon = 10 cm . The diﬀerential spectra are integrated from a given threshold value Evis to inﬁnity. The inelastic contributions dominate over the elastic ones for moderate energy thresholds. As in Figure 6.4, results are shown with chiral 1b+2b currents, where the bands include the uncertainties due to WIMP-nucleon currents. 110

25 GeV 50 GeV -4 10 100 GeV 250 GeV 500 GeV ) -1

d -5

-1 10 R (kg

-6 10

-7 10 0 20 40 60 80 100 120

Evis (keV)

Figure 6.7. Integrated total energy spectra of xenon as in Figure 6.6 for various WIMP masses from 25 − 500 GeV. The solid black curve is for a 100 GeV WIMP as in Figure 6.6. In all cases, minimum energies in Evis exist, above which the inelastic channel dominates the elastic one. The spectrum for a 500- GeV WIMP is lower than that of a 250- GeV WIMP because the particle number density in the galaxy decreases with increasing mass. 111

6.5 Implications for Dark Matter Searches

Current xenon-based dark matter experiments are either single-phase (liquid) detectors or dual-phase (liquid/gas) time projection chambers (TPCs) [136, 137]. In single- phase detectors, a large liquid xenon volume is instrumented with photodetectors, and the prompt scintillation light signal is observed. In a TPC, the prompt light signal is observed with two arrays of photosensors, and, in addition, free electrons are drifted away from the interaction site toward the vapor phase above the liquid and detected via an amplified, proportional scintillation light signal. In both cases, background reduction is achieved by self-shielding and fiducialization, namely the selection of an inner, low-background liquid xenon volume based on the reconstructed vertex of each event. For TPCs, an additional tool to distinguish background from signal events is the charge- to-light ratio [140], which depends on the electronic stopping power dE/dx, and hence on the type of particle interaction. Operational liquid xenon dark matter experiments such as XENON100 [85] and XMASS [141] have reached overall background levels of ∼ 0.01 events/(kg d keV) [142] and ∼ 0.7 events/(kg d keV) [143], respectively, in the energy region < 200 keV. This implies sensitivities to inelastic, spin-dependent WIMP-nucleon cross sections of 3.8×10−36 cm2 (XENON100) and 1.3×10−33 cm2 (XMASS) for a WIMP with mass 130 GeV. These sensitivities are four (XENON100) and seven (XMASS) orders of magnitude worse than the current best limit for spin-dependent interactions from XENON100 at this WIMP mass, but could be improved by a dedicated analysis. A temporal or spatial coincidence between the nuclear recoil and the de-excitation gamma cannot be employed to further suppress the background for aforementioned reasons related to detector resolution. However, the observed background spectra are flat in this energy range, and no peak-like structures are expected from internal (such as 222Rn and 85Kr) or external sources. Fast neutrons can in principle inelastically scatter off xenon nuclei, and, in fact, interactions from 241AmBe neutrons or similar sources are used as calibration sources for both elastic and inelastic interactions [85]. However, current dark matter experi- 112 ments are designed such that a background level of 1 elastic nuclear recoil per given exposure is expected from neutrons, implying an even lower rate for inelastic neutron scatters. Hence, a dedicated analysis can search for a peak-like enhancement in an otherwise flat background spectrum. Additionally, dual-phase detectors can discriminate between the purely electronic recoil background and the WIMP-induced signal, which contains an additional nuclear recoil component based on the charge-to-light ratio. The LUX experiment [144], which started a first physics run in 2013, is expected to reach a background level that is a factor 10 below the one of XENON100. Future experiments such as XMASS-5t [145], XENON1T [146], LZ [147], and DARWIN [148,149] are to lower their backgrounds by yet another factor of 100−1000, compared to XENON100. This reduction can be expected to continue until the overall electronic-recoil background will be dominated by irreducible neutrino-electron scatters from solar neutrinos at the level of 8×10−6 events/(kg d keV) [148], at which point further improvements in sensitivity to this interaction channel will be inhibited. A particular experiment, using the same detector, target, and isotopic abundances, can measure both the elastic and inelastic recoil spectra. Because, for a given halo model, the ratio of the cross sections for elastic and inelastic scattering is fixed, these measured spectra would provide insight into the characteristics of the dark matter halo. For instance, the relative contributions of the elastic and inelastic scattering to the spin-dependent channel may be compared in order to inspect the WIMP velocity distribution. The fraction of the event rate that each channel contributes is necessarily connected to the integral of the velocity distribution given in Eq. (6.14) and shown in Figure 6.3, such that any deviation from the expected contributions would reflect a deviation in the expected WIMP velocity distribution. The mass of the dark matter particle is uniquely connected to both the recoil energy domain and to the point at which the inelastic channel dominates the elastic one, thereby providing two methods by which to use the observed spectra to probe the WIMP mass. Table 6.2 lists the energies at which the inelastic spectrum starts to dominate the elastic one for the WIMP masses considered in Section 6.4. Due to the 113 kinematic constraints discussed in Section 6.3, inelastic scattering is not possible for 129Xe at WIMP masses below 14 GeV or for 131Xe at masses below 31 GeV.

Mass [GeV] 129Xe 131Xe Total 10 − − − 25 5 − 5 50 7 17 9 100 7 24 12 250 9 32 19 500 11 35 24

Table 6.2 Minimum energy Evis in keV above which the observed inelastic spectrum for 129Xe, 131Xe and for the total spectrum starts to dominate the elastic one for various WIMP masses.

Once a potential dark matter signal is observed in a xenon-based experiment, one could use a target enriched or depleted in 129Xe and 131Xe. Thus, one would control the ratio of odd-even isotopes to enhance or deplete certain types of interactions. While the observed rate in the spin-independent channel would remain roughly unchanged, the rate in the spin-dependent channel, both elastic and inelastic, would increase or decrease with the fraction of odd xenon isotopes in the detector. Inelastic scattering of dark matter can also be studied in other isotopes considered for searches of spin-dependent WIMP scattering. The nucleus 73Ge would be especially promising [150] as it has a very low-lying state at 13 keV, plus two other low-lying excited states at 67 and 69 keV. In addition, 127I[151] has a low-lying state at 58 keV. The ﬁrst excited states of lighter isotopes relevant to experimental searches (19F and 23Na) have higher excitation energies > 100 keV. Combined with the smaller reduced WIMP-nucleus mass, these lighter nuclei are therefore not sensitive to inelastic scattering. 114

6.6 Conclusions

In conclusion, I have shown that for spin-dependent WIMP-nucleus interactions, inelastic scattering dominates over elastic scattering if the momentum transfer is above a value that is still kinematically accessible for galactic dark matter to scatter oﬀ a terrestrial xenon target. To this end, I performed detailed calculations of the inelastic, spin-dependent nuclear structure factors for 129Xe and 131Xe, which include estimates of the theoretical uncertainties in WIMP-nucleon currents. If the observed energy in a given detector is limited to a value above 4−16 keV, depending on the WIMP mass, the sensitivity to spin-dependent WIMP-nucleon interactions will be given by the inelastic channel alone. This can have important consequences for the appropriate analysis strategy. In addition, it is conceivable that detectors that are optimized for a higher energy threshold but lower radioactive background levels [152] can obtain competitive spin-dependent WIMP sensitivity using this channel alone, provided their target contains an isotope with an exited state that is kinematically accessible. Finally, this additional detection channel can provide useful information to disentangle a potential nuclear recoil signal from other nuclear recoil backgrounds. If signal events are observed in the elastic channel, simultaneous detection of signal events in the inelastic channel will point towards a spin-dependent nature of the interaction. With suﬃcient statistics, data from a single detector can be analyzed to extract information about the particle mass and the dark matter halo in a fully complementary way, breaking degeneracies otherwise inherent to analyses focusing on the elastic channel alone. 115

7. SUPERNOVA NEUTRINO PHYSICS WITH XENON DARK MATTER DETECTORS: A TIMELY PERSPECTIVE

Core collapse supernovae (SNe) are among the most energetic transients that occur in the universe, originating from the death of very massive stars [153, 154]. Despite the remarkable progress we have seen in our understanding of the core-collapse physics over the last decade, we are still far from fully grasping the physical processes that underlie the SN engine and, in particular, the role that neutrinos play in powering it [153, 155, 156]. A high-statistics detection of neutrinos from the next Galactic SN explosion in detectors that operate with different technologies will shed light on both the stellar engine and the properties of neutrinos. Neutrino flavor discrimination will be crucial to investigate neutrino oscillation physics and scenarios with non-standard neutrino properties [155,157–161]. On the other hand, the detection of all six neutrino flavors will be essential to reconstruct global emission properties, such as the total explosion energy emitted into neutrinos [162,163]. At present, several neutrino detectors are ready for the next Galactic SN explosion, while others are under construction or being planned [155, 164]. Among these experiments, the most-promising technologies include Cherenkov telescopes and liquid scintillators, as used in or proposed for IceCube [165], Super-Kamiokande [166, 167], IceCube-Gen2 [168], Hyper-Kamiokande [169], LVD [170], Borexino [171], JUNO [172], RENO-50 [173], and KamLAND [174]. Both of these technologies will be able to probe

ν¯e neutrinos with high accuracy. In contrast, the planned liquid argon detector within the DUNE facility [175] will accurately probe the νe channel. There are also proposals to study the νe properties with Cherenkov telescopes or liquid scintillators [176–178] or with experiments that use lead or iron targets [179–181]. Together, these experiments will accurately measure theν ¯e and νe ﬂuxes from the next Galactic SN explosion [164]. 116

The elastic scattering of neutrinos on protons [182,183] and on nuclei [162] are alternative tools to detect astrophysical neutrinos. Neutrino-nucleus scattering is especially attractive because, at low energies, the scattering cross section is coherently enhanced by the square of the nucleus’s neutron-number [184]. Supernova neutrinos with energies of O(10) MeV induce O(1) keV nuclear recoils through coherent elastic neutrino-nucleus scattering (CEνNS). Although recoils in this energy range are too small to be detected by conventional neutrino detectors, it is precisely this energy range for which direct detection dark matter experiments are optimized [89]. The primary purpose of these experiments is to search for nuclear recoils induced by Galactic dark matter particles.

Yet, sufficiently large experiments (& tonne of target material) are also sensitive to CEνNS from SN neutrinos [163,185–187]. Mediated by Z-boson exchange, CEνNS is especially intriguing because it is equally sensitive to all neutrino flavors. Detectors that observe CEνNS are therefore sensitive to theν ¯µ, νµ,ν ¯τ and ντ (otherwise dubbed νx) neutrinos within their main detection channel, in addition to theν ¯e and νe neutrinos [162]. This feature carries numerous implications for experiments that detect CEνNS. For instance, the neutrino light curve could be reconstructed without the uncertainties that arise from neutrino oscillation in the stellar envelope [188], the total energy emitted into all neutrino species could be measured, or, by assuming adequate reconstruction of theν ¯e and νe emission properties with other detectors, CEνNS detectors provide a way to reconstruct the νx emission properties. In this paper [189], we revisit the possibility of detecting CEνNS from SN neutrinos in the context of XENON1T [105] and larger forthcoming direct detection dark matter experiments that employ a xenon target, such as XENONnT [105], LZ [106], and DARWIN [190]. Among the various technologies used in direct detection experiments, dual-phase xenon experiments have many advantages: the large neutron-number of the xenon nucleus enhances the CEνNS rate compared to nuclei used in other direct detection experiments; they are sensitive to sub-keV nuclear recoils; the deployment of XENON1T heralds the era of tonne-scale experiments, which are relatively straight- 117 forward to scale to even larger masses; despite their large size, the background rates are very low; and, finally, they have excellent timing resolution, O(100) µs, in the data analysis mode discussed here. As we will demonstrate in Section 7.4, these factors mean that XENON1T is already able to detect neutrinos from a SN up to 25 kpc at a significance of more than 5σ. To forecast the signal that is expected from SN neutrinos in forthcoming xenon detectors, we adopt inputs of four hydrodynamical SN simulations from the Garching group [155, 191] that differ in the progenitor’s mass and nuclear equation of state in such a way as to provide a reasonable estimate of the signal band. The neutrino properties for the adopted progenitor models are introduced in Section 7.1. In Section 7.2, for the first time, we accurately simulate the expected signal in terms of the measured quantities in dual-phase xenon experiments (scintillation photons and ionization electrons). In Section 7.3, we discuss the advantages of a dual-phase xenon detector in the observation of SN neutrinos (the low-energy sensitivity of the proportional-scintillation- signal analysis mode) as well as the expected backgrounds and achievable threshold. Section 7.4 contains our main physics results. We discuss the detection significance of SN neutrinos with future-generation xenon detectors, the reconstruction of the SN neutrino light curve as well as the average neutrino energy and the total explosion energy. Uncertainties related to the detector modeling are outlined in Section 7.5. Finally, in Section 7.6, we present our conclusions.

7.1 Supernova Neutrino Emission

In order to forecast the expected recoil signal in a xenon detector, we require the differential flux of the νe,ν ¯e and νx neutrinos as a function of time and energy. The differential flux for each neutrino flavor νβ at a post-bounce time tpb for a SN at a distance d is parametrized by

Lν (tpb) ϕν (E, tpb) f 0 (E, t ) = β β , (7.1) νβ pb 2 4πd hEνβ (tpb)i 118

where Lνβ (tpb) is the νβ luminosity, hEνβ (tpb)i is the mean energy, and ϕνβ (E, tpb) is the neutrino energy distribution. The neutrino energy distribution is deﬁned in [192, 193] as:

αβ (tpb) E (αβ(tpb) + 1)E ϕνβ (E, tpb) = ξβ(tpb) exp − . (7.2) hEνβ (tpb)i hEνβ (tpb)i

The ﬁt parameter αβ(tpb) satisﬁes the relation

2 hEνβ (tpb) i 2 + αβ(tpb) 2 = , (7.3) hEνβ (tpb)i 1 + αβ(tpb) R while ξβ(tpb) is a normalization factor deﬁned such that dE ϕνβ (E, tpb) = 1. In the following, we show results for a benchmark distance of d = 10 kpc for the Galactic SN.

7.1.1 Supernova Neutrino Emission Properties

The neutrino emission properties that we adopt are from the one-dimensional (1D) spherically symmetric SN hydrodynamical simulations by the Garching group [155, 191, 194]. More recent 3D SN simulations exhibit hydrodynamical instabilities such as large-scale convective overturns and the standing accretion shock instability (SASI) that are responsible for characteristic modulations in the neutrino signal not observable in 1D SN simulations [153,195–198]. However, as in this paper we are interested in the general qualitative behavior of the SN neutrino event rate in a xenon detector, we can safely neglect these eﬀects and adopt the outputs from 1D spherically symmetric SN simulations. To investigate the variability of the expected signal as a function of the progenitor mass, we use the neutrino data from two SN progenitors with masses of 11.2 M and

27 M . We also consider the dependence of the expected event rates on the nuclear equation of state (EoS) by adopting, for each SN progenitor, simulations obtained from the Lattimer and Swesty EoS [199] with a nuclear incompressibility modulus of K = 220MeV (LS220 EoS) and the Shen EoS [200]. These four progenitors will provide a gauge of the astrophysical variability of the expected recoil signal. 119

400 80 12 ν 350 e 70 27 M , LS220 EoS Sun 10 ν 300 e 60 11 MSun, LS220 EoS erg/s]

51 ν 8 250 x 50 200 40 6 150 30 4 100 20

Luminosity [10 2 50 10 0 0 0 18 18 18 X axis

16 16 16

14 14 14

12 12 12

10 10 10

Mean [MeV] Energy 8 8 8

6 6 6 0 0.02 0.04 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 Post-Bounce Time [s] Post-Bounce Time [s] Post-Bounce Time [s]

Figure 7.1. The upper and lower panels show the neutrino luminosity Lνβ

and mean energy hEνβ i, respectively, as a function of the post-bounce time tpb for the 11M (in blue) and 27M (in red) SN progenitors with the LS220 EoS for νe (continuous lines),ν ¯e (dashed lines) and νx (dot-dashed lines). The panels on the left show the neutrino properties during the neutronization burst phase, the middle panes refer to the accretion phase, and panels on the right describe the Kelvin-Helmholtz cooling phase. The differences in the neutrino properties from different progenitors during the neutronization burst are small, but they become considerable at later times. The variation of the neutrino properties owing to a different nuclear EoS are smaller than the differences from a different progenitor mass, so for clarity, the progenitors with the Shen EoS are not shown here. 120

Figure 7.1 displays the neutrino luminosities (top panels) and mean energies (bottom panels) of all neutrino flavors (νe,ν ¯e and νx) as a function of the post-bounce time in the observer frame for the 27 M and 11 M SN progenitors with the LS220 EoS. The variation of the neutrino properties owing to a different nuclear EoS are smaller than the differences shown here from the different progenitor mass. The neutrino signal emitted from a SN explosion lasts for more than 10 s but, as Figure 7.1 demonstrates, the luminosity drops considerably after a few seconds. Therefore, in this paper, we focus on the initial 7 s of the neutrino signal after the core bounce. The left, middle, and right panels of Figure 7.1 show the three main phases of the SN neutrino signal: the neutronization burst, the accretion phase, and the Kelvin- Helmholtz cooling phase, respectively. The neutronization burst originates while the shock wave is moving outward through the iron core. Free protons and neutrons are released as the shock wave dissociates iron nuclei. Consequently, rapid electron capture by nuclei and free protons produces a large νe burst. As evident from the top left panel in Figure 7.1, the width and amplitude of the νe luminosity during the neutronization burst are approximately independent of the SN progenitor mass and EoS [201, 202].

Generally, the νx luminosity rises more quickly than that ofν ¯e during the ﬁrst 10–

20 ms of the signal due to the high abundance of νe and electrons, which suppress the rapid production ofν ¯e. The accretion phase is shown in the middle panels of Figure 7.1. During this phase, the SN shock loses energy while moving outward and dissociating iron nuclei until it stalls at a radius of about 100–200km. According to the delayed-neutrino SN explosion mechanism [203,204], neutrinos provide additional energy to the shock to revive it after tens to hundreds of milliseconds and finally trigger the explosion. As the in-falling material accretes onto the core, it is heated, and the subsequent e+e− annihilation produces neutrinos of all flavors. Due to the high abundance of νe during the neutronization burst, the production ofν ¯e and νx is initially suppressed. The production ofν ¯e increases as the capture of electrons and positrons on free nucleons starts to become 121 more efficient. The non-electron neutrinos remain less abundant as they can only be produced via neutral-current interactions. The explosion of 1D SN simulations may require an artificial initiation, especially for more massive progenitors. In the simulation shown in Figure 7.1, the explosion was triggered at tpb ' 0.5 s. After this, the Kelvin-Helmholtz cooling phase of the newly-born neutron star begins. As shown in the right panel of Figure 7.1, the neutrino luminosities gradually decrease as the proto-neutron star cools and de-leptonizes. As the explosion is artificially triggered in these simulations, the exact transition time from the accretion to the Kelvin-Helmholtz cooling phase should be taken with caution. The neutrino signal during this phase is sensitive to the progenitor mass and the EoS. In fact, while the differences among the neutrino properties from different progenitors during the neutronization burst are small, at later times they become considerable.

7.1.2 Neutrino Flavor Conversion

The neutrino transport in SN hydrodynamical simulations is solved within the weak- interaction basis for all three neutrino flavors. Neutrinos oscillate while they are propagating through the stellar envelope as well as on their way to Earth. This affects the neutrino flavor distribution detected on Earth. In particular, neutrinos undergo the Mikheev-Smirnov-Wolfenstein (MSW) effect [205,206], which affects the survival probability of each neutrino flavor according to the adiabaticity of the matter profile. The MSW effect could be modified by turbulence or significant stochastic fluctuations in the stellar matter density (see e.g. [207–210]). In addition, neutrino–neutrino interactions are believed to be important and can affect the neutrino flavor evolution and therefore the expected energy distribution [155,156,211]. For our purpose, however, the details of the oscillation physics are not important. This is because CEνNS is sensitive to all neutrino flavors and the total neutrino flux is conserved. Hence, the same total flux produced at the SN core will reach the detector on Earth. 122

Non-standard physics may lead to situations where the total flux is not conserved, such as a scenario with light sterile neutrinos [157,158,212,213], non-standard neutrino interactions [159, 160, 214], or light dark matter particles [161, 215, 216]. All of these cases affect the heating of the star, implying that the total neutrino flux reaching the Earth could be different from the total neutrino flux at the neutrinosphere. In this paper, we will not consider these scenarios further but focus on the Standard Model scenario.

7.2 Supernova Neutrino Scattering with Dual-Phase Xenon Detectors

With the launch of the XENON1T experiment [105], which contains two tonnes of instrumented xenon, direct detection dark matter searches have entered the era of tonne-scale targets. The detection principle of this experiment is similar to smaller predecessors, including LUX [144], PandaX [217], XENON100 [85], XENON10 [218], and the three ZEPLIN experiments [219–221]. Future experiments using the same technology include XENONnT [222] and LZ [106] with each planning for approximately seven tonnes of instrumented xenon. The DARWIN consortium [148,190,223] is investigating an even larger experiment to succeed XENONnT and LZ with approximately 40 tonnes of instrumented xenon. In principle, the technology can be extended to even larger detectors at comparatively modest cost. These experiments consist of a dual-phase cylindrical time projection chamber (TPC) filled primarily with liquid xenon and a gaseous xenon phase on top. The energy deposited by an incident particle in the instrumented volume produces two measurable signals, called the S1 and S2 signals, respectively, from which the energy deposition can be reconstructed. An energy deposition in the liquid xenon creates excited and ionized xenon atoms, and the prompt de-excitation of excited molecular states yields the S1 (or prompt scintillation) signal. An electric drift field of size O(1) kV/cm draws the ionization electrons to the liquid-gas interface. A second electric field of size O(10) kV/cm extracts the ionization electrons from the liquid to the gas. Within the gas phase, 123 these extracted electrons collide with xenon atoms to produce the S2 (or proportional scintillation) signal. The S1 and S2 signals are observed with two arrays of photomultiplier tubes (PMTs) situated at the top and bottom of the TPC. A measurement of both the S1 and S2 signals allows for a full 3D reconstruction of the position of the energy deposition in the TPC. In typical dark matter searches, only an inner volume of the xenon target is used to search for dark matter (the “fiducial volume”), but the background rate for the duration of the SN signal is sufficiently small that all of the instrumented xenon can be used to search for SN neutrino scattering (see Section 7.3 for further discussion). In the following, we will thus always refer to the instrumented volume. The general expression for the differential scattering rate dR in terms of the observable S1 and S2 signals for a perfectly efficient detector is

d2R Z d2R = dtpbdER pdf (S1, S2|ER) . (7.4) dS1dS2 dERdtpb

The differential rate is an integral over the time-period of the SN neutrino signal, expressed in terms of the post-bounce time tpb, and an integral over the recoil energy ER of the xenon nucleus. The differential scattering rate in terms of ER is convolved with the probability density function (pdf) to obtain S1 and S2 signals for a given energy deposition ER. In Subsections 7.2.1 and 7.2.2, we describe the procedure to calcu- 2 late d R/dERdtpb and pdf(S1, S2|ER) respectively. Subsequently, in Subsection 7.2.3, we present the expected neutrino-induced scattering rates in terms of the S1 and S2 observable quantities. Before moving on, we briefly comment on single-phase xenon experiments, such as XMASS [141], which only have the liquid phase. The absence of the gas-phase implies that there is no S2 signal, so any generated ionization only adds to the S1 signal. Thus, the instrument is more sensitive in S1 but lacks the inherent amplification of the S2 signal using proportional scintillation. Ultimately, due to quantum efficiencies of photon detection and some sources of background, single-phase detectors have a higher energy threshold compared to dual-phase detectors. As we demonstrate in the 124 next subsection, the recoil spectrum increases rapidly at low energies; so, dual-phase experiments are significantly more sensitive to SN neutrinos. For this reason, we do not consider single-phase detectors and refer the reader to the literature for further discussion [187].

7.2.1 Scattering Rates in Terms of Recoil Energy

The interaction of a SN neutrino with a xenon nucleus through CEνNS causes the nucleus to recoil with energy ER. The differential scattering rate in terms of ER is given by d2R X Z dσ = N dE f 0 (E , t ) , (7.5) Xe ν νβ ν pb dERdtpb Emin dER νβ ν 27 where the sum is over all six neutrino flavors, NXe ' 4.60 × 10 is the number of min p xenon nuclei per tonne of liquid xenon, Eν ' mNER/2 is the minimum neutrino energy required to induce a xenon recoil with energy ER, mN is the mass of the xenon nucleus, and f 0 (E , t ) is as defined in Eq. (7.1). Finally, dσ/dE is the coherent νβ ν pb R elastic neutrino-nucleus scattering cross section [162]:

2 dσ GF mN 2 mNER 2 = QW 1 − 2 F (ER) , (7.6) dER 4π 2Eν

2 where GF is the Fermi constant, QW = N − (1 − 4 sin θW )Z is the weak nuclear 2 hypercharge of a nucleus with N neutrons and Z protons, sin θW ' 0.2386 is the weak mixing angle at small momentum transfer [224], and F (ER) is the nuclear form factor. For xenon, the Helm form factor provides an excellent parametrization for the small values of ER induced by CEνNS with which we are concerned [225]:

2 3j1(qrn) (qs) F (ER) = exp − , (7.7) qrn 2

2 where q = 2mNER is the squared momentum transfer, s = 0.9 fm is the nuclear skin 2 2 7 2 2 2 1/3 thickness, rn = c + 3 π a −5s is the nuclear radius parameter, c = 1.23A −0.60fm, a = 0.52 fm, A is the atomic number of xenon, and j1(qrn) is the spherical Bessel function. 125

16 14 0

25 0

5 Integrated Rate [count/tonne]

0 0 1 2 3 4 5 6 7 8 9 10 Energy Threshold [keV]

Figure 7.2. The upper panel shows the expected differential recoil spectrum dR/dER as a function of the recoil energy ER. The differential rate dR/dtpb as a function of the post-bounce time tpb is plotted in the middle panel. The lower panel represents the number of observable events R as a function of the detector’s energy threshold Eth. All panels show results for 11 M and 27 M progenitors with LS220 and Shen EoSs for a SN at 10 kpc. In the upper and lower panels, the neutrino flux is integrated over [0, 7] s after the core bounce, while the middle panel assumes Eth = 0 keV. All panels show that the event rate is larger for the 27 M SN progenitors while the LS220 EoS results in an O(25%) larger rate than the Shen EoS. Note that these rates are not directly observable since it is S1 and S2 that is measured, rather than ER. 126

The differential scattering rates dR/dER as a function of the xenon recoil energy ER for the four progenitor models are shown in the upper panel of Figure 7.2 for tpb integrated over [0, 7] s. As evident in all of this figure’s panels, the event rate is larger for the 27 M SN progenitors, while there is a smaller difference owing to the different equations of state, with the LS220 EoS resulting in a slightly larger predicted event rate.

The middle panel of Figure 7.2 shows the differential scattering rate dR/dtpb as a function of the post-bounce time ttb for the four progenitor models. In this figure, we have integrated over the recoil energy, assuming an idealistic threshold energy Eth = 0 keV. The qualitative behavior is similar for other threshold energies although the rate is smaller. The differential scattering rates among the different SN progenitors

−2 are comparable for tpb . 10 s, reflecting the similarities of the neutrino emission properties during the neutronization burst (cf. left panels of Figure 7.1). As the post- bounce time increases, the differences among the differential rates become larger. Most of the scattering events occur for tpb . 1 s. The total number of events observed by an experiment is determined by integrating the differential scattering rate above a given energy threshold Eth over the full time period of the SN burst. These integrated spectra are shown in the lower panel of Figure 7.2. Again, we see that the number of signal events is about twice as large for the

27 M SN progenitors, while there is O(25%) difference owing to the different equation of state. The total number of events drops quickly as Eth increases, demonstrating the importance of pushing Eth as low as possible. However, as ER is not directly measurable, these rates are not directly observable. Therefore, a careful treatment is needed to discuss the rates in terms of S1 and S2 signals instead. Besides scattering off xenon nuclei, neutrinos can also scatter off electrons in the xenon atom. We neglect the latter interaction as the rate of electron recoils is very small, approximately 10−5 counts/tonne, compared to the rate of approximately 10 counts/tonne for recoils with a xenon nucleus. 127

7.2.2 Generation of the Observable S1 and S2 Signals

To convert the nuclear recoil energy ER induced by a SN neutrino into the S1 and S2 signals, we perform a Monte Carlo simulation of a xenon TPC following the method employed by the XENON1T collaboration [105]. In this subsection, we discuss the technical details of our Monte Carlo simulation. The S1 and S2 signals are directly proportional to the number of scintillation photons

Nph and ionization electrons Nel, respectively. The mean numbers of photons and electrons are modeled as

hNphi = ER Ly(ER) , (7.8)

hNeli = ER Qy(ER) , (7.9)

where both the photon yield, Ly, and electron yield, Qy, are functions of ER. We use the emission model developed by the LUX collaboration with data from an in situ nuclear recoil calibration [92]. We ignore the small effects that may arise from having different drift fields [226–228] across the various detectors. The quantities Qy and Ly have been directly measured down to an energy of 0.7 keV and 1.1 keV, respectively [92]. Unless otherwise stated, we assume that Qy and Ly are zero below 0.7 keV. Hence, our rate predictions tend to be conservative, and we discuss the impact of this assumption in Section 7.5. In a realistic detector, we must account for quantum and statistical fluctuations. Part of the nuclear recoil energy is lost to heat dissipation. Therefore, the number of scintillation photons and ionization electrons that are produced by the xenon target,

NR NQ = Nph + Nel, is only a fraction of the total number of quanta produced by an electronic recoil at the same energy. In electronic recoils, the energy lost to heat is negligible and hNQi = ER/13.7 eV is the total number of quanta available [229, 230]. NR We model the intrinsic ﬂuctuation in NQ with a Binomial distribution characterized NR by a trial factor hNQi and probability fNR = hNQ i/hNQi:

NR NQ = Binomial(hNQi, fNR) . (7.10) 128

NR In addition to the intrinsic ﬂuctuation in NQ , the fraction of quanta that are emitted as scintillation photons also ﬂuctuates. We model this with a second Binomial distribution

NR NR with trial factor NQ and probability fph = hNphi/hNQ i:

NR Nph = Binomial(NQ , fph) . (7.11)

By conservation of the number of quanta, we have that the number of electrons is simply

NR Nel = NQ − Nph. Next, we consider detector-speciﬁc ﬂuctuations as we convert the number of generated scintillation photons and ionization electrons into observed S1 and S2 signals. Both S1 and S2 are measured in photoelectrons (PE). For the S1 signal, the number of detected photoelectrons is

NPE = Binomial(Nph, fPE) , (7.12) where fPE is the photon detection efficiency (also referred to as g1 or 1 in other studies [92, 223]). For LUX and XENON1T, fPE ' 0.12 [92, 105] and we assume this value for all other detectors, as well. This efficiency may be optimistic for detectors that are much larger than XENON1T since the geometry of larger detectors generally means that fPE decreases. However, as we discuss in Section 7.3, the S1 signal is less important than the S2 signal such that this assumption does not affect our conclusions. Finally, for the S1 signal, we must account for the response of a PMT, which is modeled with a Gaussian distribution

p S1 = Gauss(NPE, 0.4 NPE) . (7.13)

To obtain the S2 signal, we must account for the loss of ionization electrons due to electronegative impurities in the liquid as they drift toward the liquid-gas interface. This attenuation is represented by the electron survival probability ∆z psur = exp − , (7.14) vdτ where ∆z is the distance an electron traverses, τ is the so-called electron lifetime in the liquid and vd is the electron drift velocity. The value measured in XENON100 was 129

6 0

1 Differential Rate [count/tonne/PE] 0 0 1 2 3 4 5 6 S1 [PE] 6 0

Differential Rate [count/tonne/100PE] 0 0 1 2 3 4 5 6 S2 [100 PE]

Figure 7.3. The left and right panels show the differential rates of the four SN progenitors in terms of the observable S1 and S2 signals, respectively. We have integrated the neutrino flux over the first 7 s after the core bounce and assumed that the SN burst occurs 10 kpc from Earth. Note that the axes in the right panel have units of 100 PE compared to PE in the left panel meaning that the S2 signal is generally larger than the S1 signal. The light and charge yields, Ly and Qy, respectively, have been set to zero below recoil energies of 0.7 keV. Integrated rates are given in Table 7.1. 130

vd ' 1.7 mm/µs [231] and we assume this value for all other detectors (cf. [232]). For simplicity, we assume that ∆z/τ is distributed uniformly over [0, 2/3] mm/µs and that it holds for all of the detectors sizes that we consider. This condition means that, as detectors become larger, their purity increases proportionally to meet or exceed this requirement. For XENON1T, the maximum drift length ∆zmax ' 967 mm implies that the electron lifetime is at least τ ' 1450 µs, which is straightforward to achieve. The number of ionization electrons that reach the liquid-gas interface is

˜ Nel = Binomial(Nel, psur) , (7.15) and we assume that the extraction eﬃciency from the liquid into gas is 100%. Finally, the extracted electrons are accelerated by a strong electric ﬁeld in the gas phase and induce an S2 signal that is modeled with a Gaussian distribution: q ˜ ˜ S2 = Gauss(20 Nel, 7 Nel) . (7.16)

We have conservatively assumed that the average yield for each extracted electron is 20 PE, but the yield could be higher. For example, the LZ design goal is 50 PE per electron [106].

7.2.3 Observable Scattering Rates

Now that we have given expressions for dR/dER and detailed our procedure for generating the S1 and S2 signals, it is straightforward to use Eq. (7.4) to calculate the differential rates dR/dS1 and dR/dS2. These rates are shown in the left and right panels of Figure 7.3, respectively, where we have integrated tpb over [0, 7]s and assumed that Ly and Qy are zero below 0.7 keV, as discussed in the previous subsection. In the left panel, we have integrated over all S2 values by assuming an S2 threshold of zero, while the S1 signal has been integrated with an S1 threshold of zero in the right panel. Similar to the previous figures, the differential rates are highest for the 27 M SN progenitors and for the LS220 EoS. Comparing the two panels, it is apparent that 131 the S2 signal is generally larger than the S1 signal (note that the axes in the right panel have units 100 PE compared to PE in the left panel). The left panel shows that the differential rate has peaks at integer multiples of 1 PE. A similar behavior is also present in the right panel, although the effect is smaller and the peaks appear at multiples of 20 PE, the average number of photoelectrons generated for each extracted electron, cf. Eq. (7.16) (see also Figure 7.8 where the effect is more apparent). The roll- off in dR/dS2 below approximately 100 PE (right panel) is a result of the assumption that Qy is zero below 0.7 keV. In Section 7.5, we show that this assumption does not have a significant impact on our results. Table 7.1 lists the total number of expected events per tonne of xenon target for various values of the S1 and S2 thresholds. For the listed S1 thresholds, we have integrated over all S2 values, and vice versa for the listed S2 thresholds. The S2 thresholds are given as multiples of 20 PE, the average number of detected photoelectrons for each extracted electron. The number of events is reported for our four SN progenitor models located 10 kpc from Earth and for tpb integrated in the range [0, 7] s. We have separated the number of events for the case in which the threshold includes 0 PE and when it does not to show that approximately 50% and 30% of the events have an S1 or S2 signal that is exactly zero, respectively, and are therefore not observable even in an ideal detector. Generally, the number of S2 events is much higher than the number of S1 events, and the event rate drops more slowly as the S2 threshold is increased, compared to an increase in the S1 threshold. This trend reflects the fact that the S2 signal from low-energy depositions is easier to detect in a dual-phase xenon TPC due to the amplification that is inherent to the process of proportional scintillation. For example, the mean S1 signal of a 1 keV energy deposition is hS1i ' 0.5 PE, while the mean number of electrons and mean S2 signal are hNeli ' 7.4 and hS2i ' 150 PE, respectively. Since dual-phase xenon detectors are sensitive to single electrons [233,234], even very small energy depositions result in detectable S2 signals. On the basis of these preliminary results, we show in the next section that an S2- only analysis is the optimal channel for detecting CEνNS from SN neutrinos. We will 132

Table 7.1 Expected number of SN neutrino events per tonne of xenon target above various S1 and S2 thresholds. The SN burst occurs at 10kpc from Earth and the neutrino ﬂux has been integrated over the ﬁrst 7s after the core bounce. The light and charge yields, Ly and Qy, respectively, have been set to zero below recoil energies of 0.7keV. The number of events, for the case in which the threshold includes 0 PE (‘≥ 0’) and when it does not (‘> 0’), have been separated to show that many of the events have an S1 or S2 signal that is exactly zero. The symbol (?) indicates the most likely threshold values (see discussion in Sections 7.3 and 7.5 for details). An S2-only search for CEνNS from SN neutrinos is optimal as it results in a higher number of detected events.

27 M 11 M LS220 EoS Shen EoS LS220 EoS Shen EoS

S1th [PE] ≥ 0 26.9 21.4 15.1 12.3 > 0 13.3 9.8 6.9 5.2 1 11.0 8.0 5.6 4.1 2 7.3 5.1 3.6 2.6 3 (?) 5.2 3.5 2.4 1.7

S2th [PE] ≥ 0 26.9 21.4 15.1 12.3 > 0 18.5 14.0 9.9 7.6 20 18.4 14.0 9.8 7.6 40 18.1 13.7 9.7 7.4 60 (?) 17.6 13.3 9.4 7.2 80 17.0 12.8 9.0 6.9 100 16.3 12.2 8.6 6.5 133 discuss realistic values of the S2 threshold and show that an S2-only search is not limited by background events. In Section 7.5, we also show the signal uncertainty is not a limitation.

7.3 S2-Only Analysis

The canonical dark matter search in a dual-phase xenon experiment requires the presence of both an S1 and an S2 signal. This stipulation reduces the background rate by two primary means. Firstly, measuring both S1 and S2 enables discrimination between the dominant electronic recoil backgrounds and the expected nuclear recoil signal, based on the ratio S2/S1 at a given value of S1. Secondly, the S1 and S2 signals allow for a 3D reconstruction of the interaction vertex, based on the time difference between the S1 and S2 signal events and the PMT hit pattern. The latter means that events can be selected from the central region of the detector, where the background rate is lowest. In these canonical dark matter searches, which utilize data collected over O(100) days, the S1 threshold is typically 2 PE or 3 PE, while the S2 threshold is typically ∼ 150 PE (see e.g. [92,109,235]). For SN neutrinos though, the O(10) s burst of the signal requires less stringent discrimination capabilities to reduce the background signal. Although the requirement of detecting both an S1 and an S2 signal has the effect of reducing the background rate, it also significantly reduces the signal rate, especially for processes such as SN neutrino scattering where the nuclear recoil energy is small [236–241]. For example, for

S2th = 60 PE and any value of S1 (including no S1 signal), the number of SN neutrino events for the 27M SN progenitor with the LS220 EoS is 17.6 events/tonne. However, when additionally requiring an S1 signal with S1th = 2 PE, the number of events drops to only 7.2 events/tonne. Requiring both an S1 and an S2 signal therefore significantly reduces the rate of CEνNS compared to an S2-only analysis. We now show that for a SN burst, the background rate is small enough that an S2-only analysis does not require the additional discrimination capabilities otherwise 134 afforded by the S1 signal. Although the low-energy S2 background in dual-phase xenon experiments is not yet fully understood, the dominant contribution is believed to arise from photoionization of impurities in the liquid xenon and the metal surfaces in the TPC [234], caused by the relatively high energy of the 7eV xenon scintillation photons. Another background contribution may be from delayed extraction of electrons from the liquid to gas-phase [233]. Such processes create clusters of single-electron S2 signals and, occasionally, these single-electron signals overlap and appear as a single S2 signal from multiple electrons. The resultant low-energy background S2 signals are very similar to those expected in the case of a SN neutrino interaction. The background rate for these events has been characterized by XENON10 [237, 241] and XENON100 [242], which found background rates of approximately 2.3 × 10−2 and 1.4 × 10−2 events/tonne/s, respectively. These rates are consistent with the general expectation that the S2-only background rate is independent of the detector size. Based on these measurements, we therefore assume that the average background rate in XENON1T and future detectors will lie in the range (1.4−2.3)×10−2 events/tonne/s. This background rate corresponds to 0.1−0.2 events/tonne during the initial 7s of the SN signal, which is at least a factor of 40 smaller than the signal rate from the 11M with Shen EoS progenitor, the smallest rate in Table 7.1 (assuming S2th = 60 PE). Additionally, it is worth recalling that the background signal grows linearly in time, whereas the SN neutrino signal does not, resulting in an even better signal-to-background ratio in the early times of the SN burst. Finally, we motivate an appropriate choice of the S2 threshold. This threshold is largely determined by two factors. The first is the ‘trigger-efficiency’ for an experiment to detect an S2 signal. For XENON10, the trigger-efficiency was 50% for S2 ' 20 PE and reached 100% for S2 ' 30 PE [241], while for XENON100, it was 50% for S2 ' 60 PE and reached 100% for S2 ' 140 PE [231]. Values have not yet been reported for LUX. The trigger system for XENON1T has been significantly upgraded relative to XENON100 and is expected to lead to an improvement in the trigger-efficiency. Therefore, while the trigger-efficiency does vary between different experiments, here 135

we assume a benchmark value of S2th = 60 PE and make the simplifying assumption that the trigger-eﬃciency is 100% above this value. This benchmark value is consistent with the threshold in the sensitivity studies of LZ, where it was assumed that the S2- only threshold is 2.5 extracted electrons [106], corresponding to S2th = 50 PE with an average of 20 PE per extracted electron.

The second consideration when deciding S2th is the signal uncertainty induced by the choice of the electron yield Qy (cf. Eq. (7.9) for where it enters our analysis). We postpone a full discussion of this uncertainty until Section 7.5 and, for now, simply state that the signal uncertainty from Qy is smaller than 10% when S2th = 60 PE. This is appreciably smaller than the ∼ 25% variation for the LS220 and Shen EoS for the same progenitor mass as well as the approximate factor-of-two difference for different progenitor masses; so, this uncertainty should only have a small effect on our results. For all of the reasons outlined above, our main results (presented in Section 7.4) have been obtained by adopting an S2-only analysis with S2th = 60 PE. Since the background rate is significantly smaller than the signal rate, we ignore it unless stated otherwise.

7.4 Supernova Neutrino Detection

In this section, we calculate the discovery potential of an S2-only search for SN neutrinos as a function of the SN distance and discuss the discrimination power of xenon detectors with respect to the SN progenitor. We then show that it is possible to reconstruct the SN neutrino light curve and, therefore, to discriminate among the different phases of the neutrino signal. Furthermore, we demonstrate that xenon detectors can reconstruct both the neutrino differential spectrum and the total energy emitted by the SN into all flavors of neutrinos. Finally, we present a concise comparison of the performance of xenon detectors with dedicated neutrino detectors. 136

7.4.1 Detection Signiﬁcance

We first investigate the sensitivity of present and upcoming xenon detectors to a SN burst as a function of the SN distance from Earth. Figure 7.4 shows the SN burst detection significance as a function of the SN distance from Earth for the 27 M progenitor with LS220 EoS. We see that XENON1T will be able to detect this SN burst at more than 5σ significance up to 25 kpc from Earth, while XENONnT and LZ will make at least a 5σ discovery anywhere in the Milky Way. DARWIN’s much larger target mass will extend the sensitivity to a 5σ discovery past the Large Magellanic Cloud (LMC) and the Small Magellanic Cloud (SMC). In this figure, the SN signal has been integrated over the first 7 seconds after the core bounce. We calculate the detection significance following the likelihood-based test for the discovery of a positive signal described in [243]. Our null hypothesis is that the observed events are only due to the background processes described in Section 7.3, while our alternative hypothesis is that the observed events are due to both the background processes and from SN neutrino scattering. A detection significance of 5σ means that we reject the background-only hypothesis at this significance, which we therefore regard as a 5σ discovery of the SN neutrino signal. The bands in Figure 7.4 show the detection significance for a background rate spanning the range (1.4−2.3)×10−2 events/tonne/s, our assumption for the background rate discussed in Section 7.3, based on the measured rates in XENON10 and XENON100.

Figure 7.4 shows the detection signiﬁcance for the 27 M LS220 EoS progenitor, which gives the highest event rate among the four progenitors that we consider. How- ever, from this ﬁgure and Table 7.1, it is straightforward to calculate the detection significance for the other progenitors. The expected number of events simply scales with the inverse square of the SN distance, which implies that the distance d2|nσ for an nσ detection of an alternative SN progenitor is related to the distance d27,LS220|nσ for an nσ detec- p tion of the 27M LS220 EoS progenitor by d2|nσ = d27,LS220|nσ (events27,LS220)/events2.

Here ‘events’ is simply the number of events calculated from the S2th = 60 PE row in 137

�� (��) ��/�� (��) ] �� (��) σ [ ��

�

� ��

� �� [��]

Figure 7.4. The detection significance is given as a function of the SN distance for a 27 M progenitor with LS220 EoS. The SN signal has been integrated over [0, 7] s. The different bands refer to XENON1T (red), XENONnT and LZ (blue), and DARWIN (green). The band width reflects uncertainties from our estimates for the background rate, discussed in Section 7.3. The vertical dotted lines mark the centre and edge of the Milky Way as well as the Large and Small Magellanic Clouds (LMC and SMC, respectively). For this SN progenitor, XENONnT/LZ could make at least a 5σ discovery of the neutrinos from a SN explosion anywhere in the Milky Way. DARWIN extends the sensitivity beyond the SMC. 138

Table 7.1 (which gives the number of events per tonne and thus must be multiplied by the detector size). With this formula, we estimate that the SN burst from the 11 M , Shen EoS progenitor can be detected at 5σ signiﬁcance at 16 kpc, 26 kpc and 44 kpc from Earth for XENON1T, XENONnT/LZ and DARWIN, respectively.

7.4.2 Distinguishing Between Supernova Progenitors

Besides spotting a SN burst, we are also interested in investigating whether dual- phase xenon detectors could help us to constrain the SN progenitor physics and the neutrino properties. Given the sensitivity of xenon detectors to SN neutrinos and the expected insignificant background, detection should allow the progenitor mass to be discerned. With the neutrino flux from only four progenitor models, we cannot perform a detailed study of the precision with which the progenitor mass could be reconstructed. However, we can make some general statements on the performance of the different xenon experiments. For a SN at 10 kpc, the expected numbers of events in XENON1T, XENONnT/LZ and DARWIN for the 27 M LS220 EoS progenitor are 35, 123 and 704, respectively, which are 3.8σ, 7.1σ and 16.9σ higher than the 11M LS220 EoS progenitor, where the expectations are 19, 66 and 376 events. This demonstrates that when the SN distance is well known, DARWIN will be able to discern between these progenitor masses with a high degree of certainty, while even XENON1T’s ability will be reasonably good.

7.4.3 Reconstructing the Supernova Neutrino Light Curve

We now discuss the reconstruction of the SN neutrino light curve from a Galactic SN burst. Figure 7.5 shows the neutrino event rate for the most optimistic of the four SN progenitors (27 M with LS220 EoS) as a function of the time after the core bounce. The rate has been obtained for a SN at 10 kpc from Earth by adopting an 139

�� ≤ �� ≤ �[�]� �� ] �� (��) ��/�� (��) �� / �� (��) �� [ ��

��

� � � � � � � � � ��-�� [�]

�� ≤ �� ≤ �[�]� ��

] �� (��) �� /�� (��) �� / �� (��) �� [

��

� �� -�� [�]

Figure 7.5. Event rate from an S2-only analysis as a function of the post- bounce time for a SN burst at 10 kpc. The event rate is shown for a 27 M SN progenitor with LS220 EoS for three target masses: 2, 7, 40 tonnes in red, blue, and green respectively. The left panel covers the full time evolution with 500 ms time bins, including the Kelvin-Helmholtz cooling phase. The right panel shows the early evolution with 100 ms time bins and focuses on the neutronization and accretion phases. 140

S2-only analysis with a benchmark threshold of 60 PE for XENON1T, XENONnT/LZ and DARWIN. In this analysis, we neglect the small background rate. The left panel of Figure 7.5 shows the light curve during the full time evolution of the SN burst with 500 ms bins. For a Galactic SN, a detector the size of DARWIN clearly shows the characteristic behavior of the Kelvin-Helmholtz cooling phase where the event rate slowly decreases between 1 to 7s, following the same neutrino luminosity trend (cf. Figs. 7.1 and 7.2). This behavior is also partially distinguishable with XENONnT/LZ, albeit with a smaller significance, and essentially unobservable with XENON1T where the number of events per bin is too low. The right panel of Figure 7.5 focuses on the early time evolution of the SN signal during the neutronization and the accretion phases of the burst. In this panel, 100 ms time binning is used. A detector the size of DARWIN will be able to discern the neutronization peak in the neutrino signal shown in Figure 7.1. However, the neutronization peak cannot be distinguished at a high level of significance in a seven-tonne or two- tonne detector for a SN at 10 kpc. We thus conclude that it will be necessary to have a xenon detector with O(40) tonnes of xenon in order to constrain the SN light curve with high precision. Such an experiment will be competitive with existing neutrino telescopes. We stress that the results shown here are for a SN exploding 10 kpc from Earth. The DARWIN results shown in Figure 7.5 for a SN at 10 kpc are equivalent to the XENON1T or XENONnT/LZ results for a SN at 2.2 kpc and 4.2 kpc respectively. As shown in Figs. 7.1 and 7.2, the neutrino signal is almost independent of the progenitor mass and the nuclear EoS for tpb . 10 ms, while, for later times, it depends on the progenitor properties. Therefore, an accurate measurement of the later-time light curve will tell us about properties of the SN progenitor, such as its mass and EoS. The single-flavor light curve, which depends on the neutrino mass ordering and on flavor oscillation physics [155], should be accurately reconstructed with traditional neutrino detectors. The fact that xenon-based direct detection dark matter experiments are flavor insensitive will allow for the possibility of combining the all-flavor light curve with results from detectors sensitive to a single-flavor light curve. This complementarity 141 between xenon detectors and traditional neutrino experiments should, therefore, allow for tests of oscillation physics as well as the possible existence of non-standard physics scenarios (e.g. [244]). We leave a detailed study of this feature for future work.

7.4.4 Neutrino Diﬀerential Flux

Up to this point, we have extracted information using the event rate integrated over the S2 range for a given threshold S2th. However, xenon detectors are also able to accurately measure the S2 value of an individual event. For the first time, we investigate the physics that can be extracted from this spectral information. In particular, in this subsection, we demonstrate that xenon detectors can reconstruct the all-flavor neutrino differential flux as a function of the energy and, in the next subsection, that the total SN energy emitted into all flavors of neutrinos can be reconstructed. The neutrino differential flux, f 0 (E , t ), defined in Eq. (7.1), enters the calculation νβ ν pb for the rate of events in a xenon detector in Eq. (7.5). From this equation, we see that it is the time-integrated differential flux summed over all neutrino flavors that determines the number of SN neutrino scattering events in a detector. This quantity may therefore be reconstructed. It depends on the SN progenitor and the time window of the observation, but we would like to reconstruct it by making as few assumptions as possible about the initial SN progenitor. We thus make the following ansatz for the time-integrated differential flux summed over all neutrino flavors:

X Z t2 dt f 0 (E , t ) pb νβ ν pb t1 νβ (7.17) αT Eν −(1 + αT )Eν ≡ AT ξT exp . hET i hET i With this ansatz, we assume that the time-integrated differential flux can be parametrized with three free parameters: an amplitude AT , an average energy hET i, and a shape parameter αT . Here, ξT is a normalization parameter defined such that

Z αT Eν −(1 + αT )Eν dEν ξT exp = 1 . (7.18) hET i hET i 142 ] � - �� ≤ ≤ [ ] ��

[ �σ �� (��) � ��/�� (��) �� (��) �

� ▽▽ ⊙⊙

� ▽ �� ⊙ �� 〈�� 〉 [��]

] �

� �� - �� ≤ �� ≤ ��[�] �σ �� - �� (��) ��/�� (��) �� (��) ��

�� [ � ��

� �� [��]

Figure 7.6. The left panel shows the reconstructed average neutrino energy hET i and amplitude parameter AT (green dot-circle) compared to the true value for the 27 M LS220 EoS progenitor at 10 kpc from Earth integrated between 0.1 s and 1 s (black triangle). Also shown are the 1σ contours from 2-, 7- and 40-tonne mock experiments following our ML analysis. The right panel shows the reconstructed neutrino flux as a function of the neutrino energy. The dashed green line represents the differential flux obtained with the best fit ML estimators and is compared with the true flux, shown by the dashed black line. Also shown are the 1σ intervals from our 2-, 7- and 40-tonne mock experiments. 143

−1 With these definitions, AT has units of area , αT is dimensionless, and hET i has units of energy. As suggested by our notation, hET i is the average neutrino energy of the time-integrated flux summed over all flavors.

In practice, however, the shape parameter αT is difficult to constrain experimentally since it is degenerate with hET i, which also controls the shape of the observed recoil spectrum i.e. dR/dS2. We therefore make a simplifying assumption motivated by the observation from SN simulations that the differential neutrino flux can be approximated by a Fermi-Dirac distribution with zero chemical potential [192, 193, 245]. For this distribution, the relation hE2i/hEi2 ' 1.3 holds [192], and from Eq. (7.3), implies

αT ' 2.3. This value of αT is fixed in the subsequent results. This should be a reasonably good approximation everywhere, except during the very short neutronization burst phase (tpb . 10 ms), where the spectrum is significantly pinched with respect to a Fermi-Dirac distribution [193]. We first consider the reconstruction of the time-integrated differential flux summed over all neutrino flavors in the time window [t1, t2] = [0.1, 1] s (cf. Eq. (7.17)) for the

27 M LS220 EoS progenitor at 10 kpc from Earth. This window corresponds to the accretion phase of the SN burst. To reconstruct AT and hET i, we perform a maximum likelihood (ML) analysis and maximize the extended likelihood function

N X ln L(AT , hET i) = N ln µ − µ + ln f (S2i; hET i) , (7.19) i=1 where µ = µ(AT , hET i) is the mean number of expected events, N is the observed number of events, and f(S2i; hET i) is the probability density function evaluated at the S2 value of the ith event.

Figure 7.6 shows the ML estimators for AT and hET i for XENON1T, XENONnT/LZ and DARWIN mock experiments, where, by construction, each mock experiment has the same ML estimators for AT and hET i. The N observed events are randomly drawn from the dR/dS2 spectrum of the 27M LS220 EoS progenitor at 10 kpc, integrated from 0.1s to 1 s. We consider all events in the S2 range from S2th = 60 PE to S2max = 2000 PE. For this progenitor and this time window, the mean number of expected events is 144

7.0 events/tonne, so N is drawn from Poisson distributions with means of 14, 49 and 280 events for XENON1T, XENONnT/LZ and DARWIN, respectively. The left panel of Figure 7.6 shows the best fit ML estimators (green dot-circle) together with the 1σ contours for XENON1T, XENONnT/LZ and DARWIN. These contours are obtained from the ML by ln L = ln Lmax − 2.3/2. The black triangle shows the values of these parameters from our input SN progenitor. The DARWIN reconstruction of the parameters is excellent, while the XENON1T reconstruction has a significantly larger uncertainty. The dashed green line in the right panel of Figure 7.6 shows the differential flux obtained with the best-fit ML estimators substituted into Eq. (7.17). This can be compared with the true flux from the 27 M LS220 EoS progenitor, which is shown by the dashed black line. The ML reconstruction is in very good agreement with the true flux. Also shown are the 1σ intervals. At each value of the neutrino energy, the intervals were obtained by propagating all points in the 1σ regions for AT and hET i through Eq. (7.17) and selecting the maximum and minimum values of the neutrino flux. The right panel demonstrates that a DARWIN-sized experiment will be capable of accurately reconstructing the neutrino flux. The errors from XENON1T however are substantial, owing to the fact that with XENON1T one would observe only 14 events during this time window, compared to 280 with DARWIN.

7.4.5 Total Energy Emitted into Neutrinos

Finally, we show that it is possible to reconstruct the total energy emitted by neutrinos, which is simply the luminosity integrated over the duration of the SN burst (here taken as the ﬁrst 7 s) and summed over all neutrino ﬂavors. This is related to the free parameters in our ansatz by (see Eqs. (7.1) and (7.17))

Z 7 s X 2 Etot = dtpb Lνβ (tpb) = 4πd AT hET i . (7.20) 0 s νβ 145

�� ≤ �� ≤ �[�] �σ ��

�� (��)

��/�� (��) ��

�� (��)

�� [�� ]

Figure 7.7. The reconstructed 1σ band of the total energy emitted into neutrinos in 30 mock experiments for each of XENON1T (red), XENONnT/LZ (blue) and DARWIN (green). The true value for the 27 M LS220 EoS progenitor integrated over the total time of the SN burst (taken as the ﬁrst 7 s) is shown by the dashed vertical line.

This relation follows from noting that

Z X Z A hE i = dE E dt f 0 (E , t ) , (7.21) T T ν ν pb νβ ν pb νβ

and using Eq. (7.1) to express f 0 (E , t ) in terms of L (t ). νβ ν pb νβ pb Figure 7.7 shows the 1σ range of the reconstructed total energy emitted into neutrinos in 30 mock experiments for each of XENON1T, XENONnT/LZ and DARWIN. 146

Table 7.2 The typical uncertainty of the reconstructed total energy emitted in neutrinos over the ﬁrst 7 s, assuming our four SN progenitors situated 10 kpc from Earth, for diﬀerent current and upcoming detectors.

27 M 11 M LS220 Shen LS220 Shen XENON1T (2t) 20% 25% 30% 36% XENONnT/LZ (7t) 11% 13% 16% 20% DARWIN (40t) 5% 6% 7% 9%

As in the previous subsection, we use the ML method to ﬁnd the estimators of the parameters AT and hET i for the signal integrated over the ﬁrst 7 s of a 27 M LS220

EoS progenitor at 10 kpc from Earth. Then, we calculate Etot from Eq. (7.20) and the 1σ range using the propagation of errors as described in [246]. We do not include any uncertainty on the distance d in our reconstruction. The dashed vertical line shows the total energy from the SN simulation of the

27 M LS220 EoS progenitor. As we would expect, each mock experiment results in a diﬀerent mean and variance with the property that the 1σ region covers the true value in approximately 68% of the mock experiments. The typical uncertainty on the reconstructed energy for all four SN progenitors is given in Table 7.2. This number is the average of the ratio of the 1σ error over the mean for 250 mock experiments.

The uncertainty is smallest for the 27 M LS220 EoS progenitor since it results in the highest number of events, and is largest for 11M Shen EoS progenitor, which gives the lowest number of events. Unsurprisingly, the errors decrease substantially as the target mass is increased from 2 tonnes in XENON1T to 40 tonnes in DARWIN. However, even XENON1T can give a reasonably precise estimate of the total energy emitted into neutrinos for a SN at 10 kpc. 147

7.4.6 Comparison with Dedicated Neutrino Detectors

We brieﬂy compare the expected number of events of the forthcoming xenon detectors with existing or future neutrino detectors (see also Table 1 of [155] for an overview). For a SN burst at 10kpc, XENON1T and XENONnT/LZ will measure approximately 35 and 120 events in total. This is approximately one order of magnitude less than DUNE,

3 which is expected to measure approximately O(10 ) events mostly in the νe channel with a 40 tonne liquid argon detector (see Figure 5.5 of [175]). In theν ¯e channel, a larger event rates are expected from IceCube which should see approximately 106 events (see Figure 52 of [169]), Hyper-Kamiokande which is expected to measure approximately 105 events (see Figure 54 of [169]), and JUNO which should detect about 6000 events (see Figs. 4-7 of [172]). The proposed DARWIN direct detection dark matter detector, with 40 tonnes of liquid xenon, will measure approximately 700 events for all six flavors, and is thus starting to be competitive in terms of the event rate with these dedicated neutrino detectors. Of course, the quoted numbers depend on different assumptions for the adopted SN model and therefore have to be viewed only as rough estimations of the expected number of events. For what concerns the reconstruction of the SN neutrino light curve, IceCube, Hyper-Kamiokande and JUNO will all measure many more events [O(104−105) events/s] compared to DARWIN, which will see approximately 330 events during the first second and 370 events in the remainder of the SN burst. Even though the number of events is smaller for DARWIN, it is important to remember that it is sensitive to all six neutrino flavors, while the existing and planned neutrino detectors are primarily sensitive to a single flavor. Moreover, as discussed in previous sections, dual-phase xenon detectors will provide us with all-flavor information about the energetics of the explosion that should be compared with the flavor-dependent energy spectra possibly reconstructed, e.g., in JUNO or Hyper-Kamiokande with high resolution. In this sense, in the event of a SN burst, a global analysis of the burst with events from all experiments will benefit 148 from the inclusion of DARWIN data to better constrain the properties of neutrinos and the SN progenitor.

7.4.7 SNEWS & False Alarms

Due to the high significance with which XENON1T can detect Galactic supernovae, XENON1T is able to join the Supernova Early Warning System (SNEWS) [247, 248]. A member of SNEWS is required to produce no more than one false alarm per 10 days. In this section, we determine the number of events n such that the probability that we observe n or more events in any given interval ∆t with a background λ during a data run of livetime T = 10 days more than once is less than 0.5%. In other words, we calculate the supernovae signal threshold values n = n0 such that the probability the background randomly fluctuates above these thresholds more than once every 10 days is less than 0.5%. First, let us consider that probability that n events coincide within ∆t. Given that the first event occurs at t = 0, the probability that n − 1 more events occur in the interval (0, ∆t] is Z P (n; λ, ∆t) = P (n − 2; λ, ∆t) · dP (∆t) (7.22) where (λ∆t)n−2 µn−2 P (n − 2; λ, ∆t) = e−λ∆t = e−µ, (7.23) (n − 2)! (n − 2)! is the probability of n − 2 events happening in the interval (0, ∆t), and

dP (∆t) = λ · dt = dµ (7.24) is the probability of the nth event happening at t = ∆t. Combining these expressions, we obtain Z λ∆t µn−2 P (n; µ) = e−µdµ, (7.25) 0 (n − 2)! and, with a change of variable a = n − 1, a partial gamma function

1 Z λ∆t P (n; µ) = µa−1e−µdµ. (7.26) Γ(a) 0 149

The probability calculated in Eq. 7.26 is conditional upon the ﬁrst event’s occur- rence. In order to generalize this probability, we must consider the trial factor, which is simply the number of events in the run

Ntrials = Nevents = λT. (7.27)

Then, the probability of obtaining a false alarm k times in the livetime T follows the binomial distribution with an individual probability p = P (n; µ), Ntrials P (k; N , p) = pk(1 − p)Ntrials−k. (7.28) trials k

The condition in which we are interested is

P (k > 1; Ntrials, p) < 0.5%. (7.29)

A complete computation of the aforementioned probabilities for various intervals ∆t and background rates λ provides the n0 threshold values listed in Table 7.3. If the background rate successfully reaches the level of 10−4 Hz, then we have only to observe two consecutive events within a few seconds to say that a supernovae has occurred.

n0 λ [Hz] ∆t [s] 10−1 10−2 10−3 10−4 1 6 4 3 2 2 7 4 3 2 5 8 5 3 2 10 10 5 3 2

Table 7.3 The number thresholds for supernovae signals are determined for diﬀerent time intervals and background rates. 150

7.5 Experimental Factors

In this section, we discuss the uncertainties related to Qy, the detector performance during calibration periods, and the eventual pile-up of events that could prevent a clean identiﬁcation of individual S2 signals if a SN burst occurred too close to the Earth.

7.5.1 Signal Uncertainty from Qy

An accurate prediction of the S2 signal relies on knowledge of Qy, the charge yield in liquid xenon, at sub-keV energies. The LUX collaboration has provided the most accurate measurement of Qy and has measured it down to a nuclear recoil energy of 0.7 keV. The LUX data points from [92] are reproduced in the inset of Figure 7.8. The Lindhard model [249] provides a good ﬁt to the measurements and, following the LUX collaboration, is the default parametrization that we have assumed for energies above

0.7 keV. For energies below this value, we have conservatively assumed that Qy = 0. In this subsection, we investigate the uncertainty that this assumption introduces on the number of events observed with a dual-phase xenon experiment.

In order to extrapolate Qy to the lowest energies, we use either the Lindhard model or the alternative model by Bezrukov et. al. [250]. As can be seen in the inset of

Figure 7.8, both the Lindhard and Bezrukov models fit the data well. We then set Qy to zero below various values Qy,min. The case of Qy,min = 0.7 keV can be seen as the minimum predicted signal. At approximately 0.1 keV or below, the Lindhard model is expected to break down due to atomic effects [251]. We thus also test Qy,min = 0.1 keV and 0.4 keV, as an intermediate example. The main panel of Figure 7.8 shows different realizations of the dR/dS2 spectrum for the 27M LS220 EoS progenitor at 10 kpc integrated over the first 7 s. The spectra are obtained for the Lindhard and Bezrukov models of Qy with three values of Qy,min.

As expected, the lower the assumed Qy,min value, the greater the number of signal electrons that can be detected from low-energy nuclear recoils. The diﬀerences between 151

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Figure 7.8. Variations in the dR/dS2 differential spectrum under different assumptions for Qy for the 27 M LS220 EoS progenitor at 10 kpc and integrated over the first 7 s. The quantity Qy,min is the energy below which Qy = 0 and the solid, dashed and dotted lines correspond to Qy,min values of 0.1 keV, 0.4 keV and 0.7 keV. The inset shows the Lindhard and Bezrukov Qy models together with the LUX measurements. The differences in dR/dS2 between the Lindhard and Bezrukov models are reasonably small compared to the larger differences from varying Qy,min. 152

the Lindhard and Bezrukov models for Qy are much smaller than the diﬀerences from varying Qy,min. For a given Qy,min, the Lindhard model gives a signal that is shifted to lower S2 values, which follows from the lower energy yield for given recoil energy, as seen in the insert of Figure 7.8.

Table 7.4 The expected number of neutrino events per tonne for various S2 thresholds under diﬀerent assumptions for Qy. We compare the Lindhard and Bezrukov models and assume that Qy = 0 for energies below Qy,min. The results are for the 27 M LS220 EoS progenitor at 10 kpc and integrated over the ﬁrst 7s. Similar results hold for other progenitor models. The signal uncertainty in each row is (S2max − S2min)/(S2max + S2min). The Lindhard model with Qy,min = 0.7 keV gives the smallest number of events per tonne and is the benchmark assumption that we have made in this paper.

27 M LS220 EoS

Lindhard Qy,min Bezrukov Qy,min Signal

S2th [PE] 0.1 keV 0.7 keV 0.1 keV 0.7 keV uncertainty 20 22.9 18.4 23.8 18.5 13% 40 21.0 18.1 22.2 18.3 10% 60 (?) 19.4 17.6 20.6 17.9 8% 80 18.1 17.0 19.2 17.5 6% 100 16.9 16.3 17.9 16.9 5%

Table 7.4 shows the total number of expected events per tonne of xenon target in the various Qy scenarios considered. The number of events corresponds to the 27 M LS220 EoS SN progenitor at 10 kpc and the neutrino signal is integrated over 7 s. The ﬁnal column in Table 7.4 gives an estimate of the signal uncertainty for each S2 threshold, calculated in each row as (S2max − S2min)/(S2max + S2min). In all cases, the minimum number of events per tonne is found for the Lindhard model with Qy,min = 0.7 keV, which is the benchmark assumption that we have made in all calculations reported in this paper. The highest number of events is found for the Bezrukov model with 153

Qy,min = 0.1 keV. For the 27 M LS220 EoS progenitor and our benchmark value

S2th = 60 PE, the uncertainty from the Qy parametrization is around 8%. The signal uncertainties with this S2 threshold for the other SN progenitors are similar, with an uncertainty of 9% (9%, 10%) for the 27 M Shen EoS (11 M LS220 EoS, 11 M Shen EoS) progenitor. The neutrino flux amplitude and mean energy reconstruction analyses in Section 7.4.4 may be more adversely affected by the uncertainty in the charge yield Qy, since they also take into account the shape of the recoil spectrum. The Qy modeling uncertainty could be straightforwardly incorporated into a ML analysis (as the Ly, Qy and Milky Way halo uncertainties are routinely incorporated into dark matter studies). Here, we simply test a higher S2 threshold, S2th = 120 PE, to reduce the Qy modeling uncertainty by repeating our analysis that led to Figure 7.6. In this case, we find similar results as in Figure 7.6. The number of events is reduced from 7.0 events/tonne to 6.3 events/tonne, which leads to an increase in the 1σ regions of the mean energy and amplitude by only 13% and 20% respectively. Thus, the quantitative conclusions drawn from this analysis are only slightly affected by the present uncertainty in Qy.

Clearly, it would be most desirable to further reduce the Qy uncertainty by having other low-energy measurements of this quantity.

7.5.2 Sources of Increased Background Rates

The low background rates discussed in Section 7.3 are applicable when the detector is in dark matter search mode. However, in contrast to dedicated SN neutrino detectors, direct detection dark matter detectors can in some cases spend half of their time taking calibration data [111]. Various calibration sources are utilized, from external Compton or neutron calibrations to radioactive isotopes that are dissolved directly in the liquid target. The particular background rate in the S2-only channel discussed previously can vary signiﬁcantly during calibration and may depend on the particular calibration source employed. However, even with an event rate during calibration two orders of 154 magnitude above the rates during a dark matter search, the background is still smaller than the expected signal rates from a Galactic SN. Another potential source of increased background to SN signals comes from photoionization on impurities in the liquid xenon. During the commissioning of a detector, the purity may be low, and thus the background rate may be increased. Furthermore, the diminished electron survival probability from their drift would in eﬀect raise the S2-based energy threshold, possibly rendering the detector blind to SN events. Since such initial commissioning times are supposed to be short, we do not discuss them further here.

7.5.3 Sensitivity Limitation from Event Pile-Up

In a xenon TPC, a single SN neutrino scattering event produces a number of ionization electrons that are drifted to the gas phase, where the S2 signal is produced from proportional scintillation. At a drift velocity of order 2 mm/µs [232], a 1 m high TPC is expected to smear the arrival times of the electrons by about 250 µs. This aspect limits the timing resolution of this detection channel.

To get a better estimate of the maximum number of SN neutrino events (Npileup) before pile-up becomes an issue, we perform a Monte Carlo simulation of the events in the TPC. Once the electrons are extracted from the liquid, the observed S2 signal is a pulse with a width of O(1) µs [231]. To resolve individual events without the need to use information from the PMT hit pattern, one S2 pulse should not overlap another S2 pulses. This will limit the sensitivity of the detector once pile-up becomes significant. Motivated by Figure 8 in [231], we define events to be well separated if the spacing from the start of one S2 pulse to the start of the next pulse is more than 10µs. We randomly distribute events in time according to the differential time distribution dR/dtpb. We test both the 27 M LS220 EoS and 11 M Shen EoS progenitors to get an idea about the impact of these models on our conclusions. As the event rate is highest at the start of the SN burst (cf. Figs. 7.1 and 7.2), we focus on the first second after the explosion. 155

We then distribute the events uniformly throughout the TPC and take into account the time delay as the ionization electrons drift from the interaction site to the liquid-gas interface, assuming the XENON100 drift velocity vd = 1.7 mm/µs [231]. In each mock (and real) experiment, the vertex sites, the number, and the time distribution of the events varies. We thus use a statistical procedure and deﬁne Npileup to be the number of events at which 90% of mock experiments observe at least 5% of events with a spacing of less than 10 µs. We have performed our calculation for three TPC sizes, 967, 1450 and 2600 mm, corresponding to the expected sizes of the XENON1T [105], XENONnT/LZ [106] and

DARWIN TPCs [223]. We ﬁnd that Npileup is approximately independent of these three

TPC sizes, varying by less than 0.2%. For the 27 M LS220 EoS and 11 M Shen EoS progenitors, Npileup = 4810 events and Npileup = 4780 events respectively, consistent with our simple estimate of approximately 250 µs for the timing resolution.

The maximum number of SN neutrino events, Npileup, can be converted into a minimum progenitor distance from Earth so that pile-up is not an issue. Given that 8.3 events per tonne and 4.1 events per tonne are expected during the ﬁrst second for the 27 M LS220 EoS and 11 M Shen EoS progenitors at 10 kpc, we ﬁnd minimum distances of {0.6, 1.1, 2.6} kpc and {0.4, 0.8, 1.8} kpc for {XENON1T, XENONnT/LZ,

DARWIN} for the 27 M LS220 EoS and 11 M Shen EoS progenitors, respectively. A SN explosion that is much closer than these distances will still be detected by a xenon detector, but precision studies of the SN neutrino light curve or neutrino ﬂux parameters will become degraded as it becomes diﬃcult to distinguish between individual events.

7.6 Conclusions

With the launch of XENON1T with 2 tonnes of xenon target, and given the plans for larger experiments employing the same technology such as XENONnT and LZ with 7 tonnes and DARWIN with 40 tonnes, we here revisited the possibility of detecting a 156

Galactic supernova (SN) through coherent elastic neutrino-nucleus scattering (CEνNS) with such dual-phase xenon direct detection dark matter experiments. In order to gauge the astrophysical variability of the expected signal, we studied the neutrino signal from four hydrodynamical SN simulations, differing in the progenitor mass and nuclear equation of state. For the first time, we have performed a realistic detector simulation of SN neutrino scattering, expressing the scattering rates in terms of the observed signals S1 (prompt scintillation) and S2 (proportional scintillation). We have shown that focusing on the S2 channel maximizes the number of events that can be detected, thanks to the lower energy threshold. We have discussed appropriate values of the S2 threshold and proved that the background rate is negligible compared to the expected signal. Hence, high-significance discoveries can be expected. As a concrete example, we have shown that for a 27 M SN progenitor, the XENON1T experiment will be able to detect a SN burst with more than 5σ significance up to 25 kpc from Earth. Furthermore, the XENONnT and LZ experiments will extend this sensitivity beyond the edge of the Milky Way, and the DARWIN experiment will be sensitive to SN bursts in the Large and Small Magellanic Clouds. Due to the low background rate, these experiments should even be able to actively contribute to the Supernova Early Warning System. For a SN burst at 10 kpc, features of the neutrino signal such as the neutronization burst, accretion phase, and Kelvin-Helmholtz cooling phase will be distinguishable with the DARWIN experiment. In addition, with DARWIN it will be possible to make a high- precision reconstruction of the average neutrino energy and differential neutrino flux. Since CEνNS is insensitive to the neutrino flavor, the signal in dual-phase xenon detectors is unaffected by uncertainties from neutrino oscillation physics. A high-precision measurement of CEνNS from SN neutrinos will therefore offer a unique way of test- ing our understanding of the SN explosion mechanism. The sensitivity to all neutrino flavors also means that it is straightforward to reconstruct the total energy emitted into neutrinos. We have shown that even XENON1T could provide a reasonably good reconstruction of this energy. 157

It has already been discussed that a large multi-tonne xenon detector such as DAR- WIN would be able to measure solar neutrino physics [252] and exploit novel dark matter signals [121, 223, 253]. Here, we have illustrated that DARWIN will also be able to reconstruct many properties of SN progenitors and their neutrinos with high precision. Large dual-phase xenon detectors are expected to be less expensive and more compact than future-generation dedicated neutrino telescopes, encouraging the construction of liquid xenon experiments as SN neutrino detectors. Specifically, DAR- WIN will allow for all-flavor event statistics that are competitive with next-generation liquid argon or scintillation neutrino detectors [172, 175], which are sensitive to only some of the neutrino flavors. At the same time, being flavor blind, dual-phase xenon detectors will provide complementary information on the SN neutrino signal that is not obtainable with existing or planned neutrino telescopes. 158 APPENDICES 159

A. Calculations of Velocity Distribution

In the limit vesc → ∞, the normalization is

∞ −→ −→ 2 Z −( v + v E ) v 2 3 k ≡ k0 = e 0 d v 0 2π ∞ 1 2 2 Z Z Z −(v +vE +2vvE cos θ) 2 2 = dφ v e v0 d(cos θ)dv 0 0 −1 ∞ 2 2 2 Z (v−vE ) (v+vE ) πv0 − 2 − 2 = v e v0 − e v0 dv vE 0 2 ∞ 2 ∞ 2 πv Z − x Z − x 0 v 2 v 2 = (x + vE)e 0 dx − (x − vE)e 0 dx vE −vE vE 2 vE 2 vE 2 ∞ 2 πv Z − x Z − x Z − x 0 v 2 v 2 v 2 = xe 0 dx + vE e 0 dx + 2vE e 0 dx vE −vE −vE vE 2 vE 2 ∞ 2 πv Z − x Z − x 0 v 2 v 2 = 0 + 2vE e 0 dx + 2vE e 0 dx vE 0 vE ∞ 2 Z − x 2 v 2 = 2πv0 e 0 dx 0 Z ∞ 2 v0 −1/2 −u = 2πv0 u e du 0 2 3 = πv0 · Γ(1/2) 3 √ = πv0 π

√ 3 = ( πv0)

−→ −→ 2 −→2 For a moving Earth, there is the constraint ( v + v E) ≤ v esc,

2π (cos θ) v −→ −→ 2 Z Z max Z max ( v + v E ) 2 − 2 k = dφ d(cos θ) v e v0 dv 0 −1 0 v 2 2 (cos θ) Z max v +vE Z max 2vvE cos θ 2 − 2 − 2 = 2π v e v0 e v0 d(cos θ)dv 0 −1 where   1, 0 ≤ v ≤ vesc − vE (cos θ)max = 2 2 ; vesc−vE −v  , vesc − vE ≤ v ≤ vesc + vE 2vE v 160 so,

2 vesc−v v2+v2 2v v 2v v Z E E E − E πv0 − 2 2 2 k = ve v0 e v0 − e v0 dv vE 0 2 2 2 v +v 2 2 2v v v −v −v Z esc E v +vE E − esc E − 2 2 2 + ve v0 e v0 − e v0 dv vesc−vE 2 2 vesc+v 2 vesc−v 2 v vesc+v Z E (v−vE ) Z E (v+vE ) − esc Z E πv0 − 2 − 2 2 = ve v0 dv − ve v0 dv − e v0 vdv . vE 0 0 vesc−vE

Using the substitutions u = v − vE and u = v + vE in the ﬁrst and second terms, respectively, one ﬁnds

2 2 Z vesc u2 Z vesc u2 vesc Z vesc+vE πv − − − 2 0 v 2 v 2 v k = (u + vE)e 0 du − (u − vE)e 0 du − e 0 vdv vE −vE vE vesc−vE 2 2 Z vE u2 Z vesc u2 vesc Z vesc+vE πv − − − 2 0 v 2 v 2 v = ue 0 du + 2vE e 0 du − e 0 vdv vE −vE 0 vesc−vE vesc 2 2 vesc Z v0 2 − πv0 t v2 = 0 + 2vEv0 e dt − 2vEvesce 0 vE 0 √ 2 − vesc 3 π vesc vesc v2 = 2πv0 erf( ) − e 0 2 v0 v0 2 − vesc vesc 2 vesc v2 = k0 erf( ) − √ e 0 v0 π v0

≡ k1. 161

B. Calculations of Event Rates

In the following, we expound upon the calculation of dark matter event rates in direct detection experiments.

√ π hvi R = R0 2 v0 Z R k0 1 3 = 4 vf(v, vE)d v (B.1) R0 k 2πv0 v +v 2 v −v 2 v2 v +v Z esc E (v−vE ) Z esc E (v+vE ) − esc Z esc E k0 1 2 − 2 2 − 2 2 2 v0 v0 v0 = 2 v e dv − v e dv − e v dv k 2v0vE 0 0 vesc−vE Z vesc u2 Z vesc u2 k0 1 2 2 − 2 2 2 − 2 v0 v0 = 2 (u + 2vEu + vE)e du − (u − 2vEu + vE)e du k 2v0vE −vE vE 2 vesc 2 1 2 − 2 − 2v v + v e v0 E esc 3 E

Z vE u2 Z vesc u2 Z vE u2 k0 1 1 2 − 2 − 2 − 2 v0 v0 v0 = 2 u e du + 2 ue du + vE e du k v0 vE 0 vE 0 2 vesc 2 1 2 − 2 − v + v e v0 esc 3 E 2 √ 2 2 vE vesc vE 1 k0 − 2 π v0 vE − 2 − 2 √ vE vE = − e v0 + erf − 2 e v0 − e v0 + π erf 2 k 2 vE v0 v0 v0 2 2 2 vesc vesc 1 vE − 2 v0 − 2 2 + 2 e v0 3 v0 2 2 vE 2 2 vesc 1 k0 − 2 √ 1 v0 vE vE vesc 1 vE − 2 v0 v0 = e + π + erf − 2 2 + 2 + 1 e . (B.2) 2 k 2 vE v0 v0 v0 3 v0

For 0 < vmin < vesc − vE,

v +v 2 v −v 2 v2 v +v Z esc E (v−vE ) Z esc E (v+vE ) esc Z esc E k 1 − − − 2 0 v 2 v 2 v T (vmin) = e 0 dv − e 0 dv − e 0 dv k 2vE vmin vmin vesc−vE 2 Z vesc w2 Z vesc w2 vesc k 1 − − − 2 0 v 2 v 2 v = e 0 dw − e 0 dw − 2vEe 0 k 2vE vmin−vE vmin+vesc v v esc esc v2 Z v Z v esc k0 1 v0 0 −t2 0 −t2 − 2 = e dt − e dt − e v0 k 2 vE vmin−vE vmin+vE v0 v0 162

√ 2 vesc k0 π v0 vmin + vE vmin − vE − 2 = erf − erf − e v0 . k 4 vE v0 v0

However, if vesc − vE < vmin < vesc + vE,

v +v 2 v2 v +v Z esc E (v−vE ) esc Z esc E k 1 − − 2 0 v 2 v T (vmin) = e 0 dv − e 0 dv k 2vE vmin vmin vesc 2 vesc Z v0 2 − k0 1 v0 −t v2 = e dt − (vesc + vE − vmin)e 0 k 2 vE vmin−vE v0 √ 2 − vesc k0 π v0 vesc vmin − vE 1 v2 = erf − erf − vesc + vE − vmin e 0 . k 4 vE v0 v0 2vE 163

C. Convection Parameter

The challenge we face when viewing the convection cell through the lateral surface of the cylindrical TPC is that we see larger subvolumes of the detector closer to the central axis. These unequal volumes could potentially introduce a bias in our measurement of atomic motion. A given subvolume is represented by its projection onto a circle of radius R0 and centered at the origin in the XY plane. The projection is at the position Y : q 2 2 dA = 2 R0 − Y dY. (C.1)

We aim to convert A(Y ) into a function A(Y˜ ) such that dA/dY˜ = constant. To this

2 2 end, we ﬁrst apply a substitution Y ≡ −R0 cos u, which yields dA = 2R0 sin u du. Then, we set dY˜ 1 = sin2 u = (1 − cos 2u) (C.2) du 2 which gives 1 1 Y˜ (u) = u − sin 2u + C. (C.3) 2 2 To preserve the symmetry around Y = 0, we require Y˜ (u(0)) = 0 and, thereby, deﬁne

π C = − 4 . Furthermore, we scale by 4R0/π, 2 1 Y˜ (u) = R u − sin 2u − 1 , (C.4) 0 π 2

˜ so that Y is deﬁned on [−R0,R0]. 164 LIST OF REFERENCES 165

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VITA

Education

2011 Bachelor of Science in Physics with Honors & Distinction, The Ohio State University, USA. 2011 Bachelor of Science in Mathematics with Honors & Distinction, The Ohio State University, USA. 2011 Bachelor of Science in Astronomy with Honors & Distinction, The Ohio State University, USA.

Research Experience

2011–today Graduate Researcher at Purdue University. Developed the phenomenology of supernova neutrino detection in dual-phase liquid xenon detectors. Prepared a dissolved 220Rn source for calibration of XENON1T and future dark matter experiments. Es- tablished a new channel, based on inelastic interactions, for the discovery and characterization of dark matter in liquid xenon detectors. Analyzed a source of background that arises from accidental coincidence of light and charge signals. Contributed to the exclusion of two models that were proposed as an explanation of DAMA’s claimed signal. 2010–2011 Undergraduate Researcher at The Ohio State University. Worked in the BaBar group to search for exotic decay modes of the Z boson. 181

Primary Publications

2016 E. Aprile et al. (XENON100), Results from a Calibration of XENON100 Using a Source of Dissolved Radon-220, (2016), arXiv:1611.03585. 2016 R. F. Lang et al., Supernova Neutrino Physics with Xenon Dark Matter Detectors: A Timely Perspective, (2016), doi: 10.1103/Phys- RevD.94.103009, arXiv:1606.09243. 2013 L. Baudis et al., Signatures of Dark Matter Scattering Inelastically Oﬀ Nuclei, Phys.Rev. D88 (2013), 115014, doi: 10.1103/Phys- RevD.88.115014, arXiv:1309.0825. 2016 E. Aprile et al. (XENON100), XENON100 Dark Matter Results from a Combination of 477 Live Days, (2016), arXiv:1609.06154. 2015 E. Aprile et al. (XENON100), Exclusion of Leptophilic Dark Mat- ter Models Using XENON100 Electronic Recoil Data, Science 2015 vol. 349 no. 6250 pp. 851-854, doi: 10.1126/science.aab2069, arXiv:1507.07747. 182

Secondary Publications

2016 E. Aprile et al. (XENON100), Search for Two-Neutrino Double Elec- tron Capture of 124Xe with XENON100, (2016), arXiv:1609.03354. 2016 J. Aalbers et al., DARWIN: Towards the Ultimate Dark Mat- ter Detector, (2016), doi: 10.1088/1475-7516/2016/11/017, arXiv:1606.07001. 2016 E. Aprile et al. (XENON100), Low-mass Dark Matter Search Us- ing Ionization Signals in XENON100, (2016), doi: 10.1103/Phys- RevD.94.092001,arXiv:1605.06262. 2015 E. Aprile et al. (XENON1T), Physics Reach of the XENON1T Dark Matter Experiment, JCAP04(2016)027, doi: 10.1088/1475- 7516/2016/04/027, arXiv:1512.07501. 2015 E. Aprile et al. (XENON100), Search for Event Rate Modulation in XENON100 Electronic Recoil Data, Phys. Rev. Lett. 115, 091302 (2015), doi: 10.1103/PhysRevLett.115.091302, arXiv:1507.07748. 2015 E. Aprile et al. (XENON1T), Lowering the Radioactivity of the Pho- tomultiplier Tubes for the XENON1T Dark Matter Experiment, doi: 10.1140/epjc/s10052-015-3657-5, arXiv:1503.07698. 2014 E. Aprile et al. (XENON1T), Conceptual Design and Simula- tion of a Water Cherenkov Muon Veto for the XENON1T Experi- ment, JINST 9 (2014), P11006, doi: 10.1088/1748-0221/9/11/P11006, arXiv:1406.2374. 2014 E. Aprile et al. (XENON100), First Axion Results from the XENON100 Experiment, Phys.Rev. D90 (2014), 062009, doi: 10.1103/PhysRevD.90.062009, arXiv:1404.1455. 183

Teaching Experience

2011–2016 Graduate Teaching Assistant at Purdue University. Astroparticle Physics, Electric and Magnetic Interactions, Electricity and Optics 2010–2011 Tutor of Physics at The Ohio State University. Introductory Level Honors Courses 2008–2011 Tutor of Mathematics at The Ohio State University. Algebra, Calculus, Ordinary Diﬀerential Equations

Awards

2016 Bilsland Dissertation Fellowship 2016 George W. Tautfest Award 2013–2014 Purdue Research Foundation Fellowship 2013 Ford Foundation Fellowship, Honorable Mentions 2010–2011 Mathematics & Physical Sciences Undergraduate Research Scholar- ship 2010 Edward R. Grilly Summer Undergraduate Research Scholarship 2010 Mathematics and Statistics Learning Center Tutoring Award 2006–2010 Prestige Scholarship, Honors Department at The Ohio State Univer- sity 2006–2010 The Ohio State University Academic Scholarship 2006–2010 The State of Ohio Academic Scholarship 184

Synergistic Activities

2016 Oral Presentation at TeV Particle Astrophysics Conference, CERN 2016 Poster Presentation at NEUTRINO Conference, Imperial College 2016 Graduate Student Seminar, Purdue University 2014–2015 Graduate Student Mentor 2014 Poster Presentation at SLAC Summer Institute 2013 Summer Undergraduate Research Fellowship Symposium, Judge/Moderator 2010–2011 Treasurer of Sigma Pi Sigma, Physics Honor Society

Memberships

2012–today XENON Collaboration 2012–today DARWIN Consortium 2012–today Bujinkan Budo Taijutsu Martial Arts Dojo at Purdue University 2014–2016 Physics Graduate Student Association, Committee Member

Languages

English Native French Fluent Japanese Fair

Computer Skills

C++, python, root, LATEX, Linux, Microsoft Windows, Microsoft Oﬃce