Acceleration factor of a Variance Reduction Technique on
Monte Carlo simulations for double gamma interactions in
a liquid xenon detector
Mehmet Arkin June 22, 2020
Report Bachelor Project Physics and Astronomy
Conducted between 30-03-2020 and 20-06-2020STUDENTNUMBER 11604808 INSTITUTE Nikhef
FACULTIES AND UNIVERSITIES Faculty of Science FNWI UvA SUPERVISOR prof. dr. A.P. Colijn
SECOND EXAMINER prof. dr. S. Bentvelsen VERSION First Version
Abstract
Dark matter exhibits almost no interaction with ordinary matter, making it difficult to detect. Liquid xenon (LXe) detectors like the Xenon1T experiment aim to detect individual dark matter particles by looking for interactions with xenon. The majority of interactions are from background radiation which have to be filtered out to distinguish the dark mat-ter signal from the noise. The prediction of the background signal requires a tremendous amount of particles to be simulated to achieve physical credibility as detectors become larger. A lack of computational power would render this task challenging however. To solve this problem a technique has been proposed [13] to reduce the variance of the simulations which cuts down the required sample size while maintaining the same confidence.
Most of the interactions occur in the outer layers of the xenon, which functions as a natural shield for the inner volume. This thesis is focused on how much the variance reduction technique (VRT) accelerates background simulations for double gamma scatters inside of this defined volume (fiducial volume). VRT already proved useful for single scatter simulations which were accelerated by a factor of α = 1.6 · 103 [13] for 1.5 MeV gamma particles. To examine double scatters both interactions have to take place inside of this fiducial volume. Considering 1.5 MeVγ-particles, the VRT achieved an α of 13.2 · 103for double scatters as can be seen in figure 9. Figures 8 and 10 showα values of 8.3 · 103and 16.3 · 103for 1 MeV and 2 MeV simulations respectively.
Popular summary (Dutch)
Donkere materie vertoont nauwelijks interactie met zijn omgeving waardoor het moeilijk te detecteren is. Het Xenon1T experiment probeert donkere materie te detecteren met een tank gevuld met xenon in vloeibare- en gasvorm. Het grootste deel van het signaal zal echter van achtergrondstraling zijn. Dit moet eruit gefilterd worden zodat het signaal van donkere materie beter te onderscheiden is. Om de achtergrondstraling goed te voor-spellen moeten er veel simulaties gedaan worden wat veel computerkracht vereist. Er is een techniek bedacht [13] die het benodigde aantal deeltjes verlaagt en dus de benodigde computerkracht vermindert door de variantie in de data te verlagen.
Dit verslag is gefocust op hoeveel deze techniek simulaties versnelt waarbij een ges-imuleerd deeltje twee interacties vertoont. Deze versnellingsfactor noemen wij α. Voor
deeltjes met een energie van 1.5 MeV was al een factor van 1.6 · 103 berekend bij één in-teractie. Voor twee interacties is deze factor 13.2 · 103. Verder is er gekeken naar simu-laties met deeltjes van 1 MeV en 2 MeV. Deze hebben versnellingsfactoren van 8.3 · 103en 16.3 · 103.
Contents
1 Introduction 5
1.1 Dark Matter interaction . . . 5
1.1.1 Liquid Xenon detector . . . 7
1.2 Backgrounds . . . 7
1.3 Physics of gamma particles inside xenon . . . 8
2 Method 9 2.1 Standard simulation . . . 9
2.1.1 Monte Carlo standard deviation . . . 11
2.2 Variance Reduction Technique . . . 11
2.2.1 Technique . . . 12 2.3 Acceleration . . . 13 3 Results 14 4 Discussion 15 5 Conclusion 18 6 Acknowledgements 18
1
Introduction
Understanding the universe has led to many open questions like the nature and existence of dark matter. These unknown particles are believed to exhibit almost no interactions with their surroundings and yet make up most of the universe. The existence has been brought to light by many independent observations of the behaviour of astrophysical objects. These observations include the bending of light around massive objects like galaxies and galaxy clusters, rotation curves of spiral galaxies [1] and dispersion of velocity around elliptic galaxies [2]. The ellip-tic galaxies have a higher velocity dispersion around the centre of mass than expected from luminous matter, which increases as mass increases. All of these dynamics point to the ex-istence of extra non luminous matter to account for the deficiency of mass, which estimate a mass about five times higher than the matter that is visible. The Bullet Cluster (figure 1) is an example which was formed after two galaxy clusters collided. The Chandra X-ray Obser-vatory (CXO) detected two clumps [3] which contained most luminous baryonic matter (pink). Hot gas from one of the clusters moved through the hot gas from the other cluster when the collision happened. Gravitational lensing was used to determine that most of the mass was located in a spherical halo (blue) around the clumps and not at the intergalactic gas where all the baryonic matter was [4], which means non luminous matter was present. This proves that dark matter has noticeable gravitational interactions but only in bulk amounts. The gravita-tional strength of a single dark matter particle is too weak to provide an analysis so different techniques have to be used. It is hypothesised that these particles are (WIMPs) or Weakly Interacting Massive Particles [5]. Section 1.1.1 describes the detection method of dual phase xenon/noble element detectors and gives a brief explanation of the mechanisms of the Xenon1T detector. A description of the background signal of this detector is given in section 1.2. Un-derstanding the background is crucial so that it can be distinguished from WIMP interactions. A particle could have the same energy-signature as a WIMP, but when it scatters twice it is selected as background because a WIMP almost never interacts twice. Simulations of double scatters are therefore important so that particles mimicking WIMP interactions can be charac-terised as background. These simulations could become computationally heavy when detectors become larger, so in section 2 a variance reduction technique (VRT) is described which acceler-ates conventional Monte Carlo simulations, and finally the acceleration factor which the VRT provides.
1.1 Dark Matter interaction
When two particles interact they exchange a mediator or ’virtual particle’ [6]. The species of the virtual particle depends on which forces can couple to the interacting particles. An example
Figure 1: From [14]; The Bullet Cluster with luminous matter in pink, and non-luminous matter in blue. The highest mass concentration was found in the blue parts of the cluster where dark matter is believed to reside.
Figure 2: A WIMP (X) interacting with a quark (q) inside of a xenon nucleus through a Z-boson, Higgs or uknown mediator (φ).
would be a virtual photon which couples to two scattering electrons through the electromag-netic force. When a WIMP and nucleus scatter a mediator couples to one of the quarks inside of the nucleus and to the WIMP. Because dark matter has no electromagnetic interactions the virtual particle is either a Z-boson (spin dependent), Higgs (spin independent) or a new virtual particle. Figure 2 shows a simple illustration of a WIMP interacting with a single quark. Since the underlying physics of WIMP interaction is undetermined, the expected event rate in the detector is based on the assumption of the WIMP-nucleon cross section, the WIMP number density and the WIMP velocity distribution. A nuclear form factor is used to encapsulate the properties of the interaction [7](31-33).
Gravitational detection is considered as an indirect method to detect dark matter as it pro-vides no further analysis of an individual dark matter particle. Methods that employ different aspects of interaction are considered direct, like particle scattering, annihilation or collision.
Liquid xenon (LXe) detectors like XENONnT use particle scattering where a WIMP scatters elastically from a nucleus. The recoil energy is dissipated in the form of scintillation, ionization and heat. XENONnT uses ionization and scintillation which is described in section 1.1.1.
1.1.1 Liquid Xenon detector
A dual-phase noble gas detector is a system which contains both liquid and gaseous xenon [7] (LXe and GXe). In this thesis the geometry of the system is taken to be cylindrical like XENONnT. First a particle enters the cylinder as can be seen in figure 3. An interaction occurs when a particle recoils on a xenon nucleus. The xenon recoil causes neighbouring xenon nuclei to be excited and ionized. After being excited, the nucleus emits photons (scintillation). The photons are detected by photomultiplier tubes at the bottom and top of the cylinder which is the first flash S1. The ionization of the nuclei creates free electrons which are drawn to the GXe
by the electric field (yellow arrow). Here the electrons are accelerated even more by another electric field which causes additional xenon nuclei to be excited. These excited nuclei create a second flash by scintillation S2. The time between these flashes is used to calculate the location of the interaction in the cylinder. The red stars represent the first interaction in the cylinder and the blue stars represent a possible second interaction if one occurred. To achieve a distinguishable signal from dark matter the volume has to be shielded from background radiation and this radiation has to be simulated to predict and filter the effects. The outer layers of LXe (≈ 10 cm thick) naturally shield the inner volume and stop a lot of background radiation. This background signal depends on materials in the LXe detector and external γ -radiation sources.
1.2 Backgrounds
The background signal of Xenon1T comes from nuclear recoils (NR), and electronic recoils (ER) of γ- andβ- particles. Cosmic muons may create neutrons outside the detector, which could cause WIMP like signals. The xenon detector is placed inside of a water tank with 84 PMTs which detect Cherenkov radiaton coming from muons which have a higher velocity than the phase velocity of light inside the water. The detector then ignores interactions which come shortly before or after the neutron interaction. This thesis is focused on ER backgrounds since it the most predominant background signal. For XENONnT the dominant contribution to the ER background is β-decay of214Pb (1) [9]. This is the eighth or ninth decay of238U (Uranium Series) depending on the branch.
238 92 U99KU-daughters99K21482 Pb β− −−−−−−→ 26.8 min 214 83 Bi (1)
Figure 3: Adapted from [8]: Schematic illustration of the LXe detector which uses LXe as a target for nuclear recoils of xenon. The green box depicts the inner volume or fiducial volume which is shielded from most of the background radiation. The scintillation of xenon causes lightflashes which are observed by the photomultiplier tubes (PMTs) at the top and bottom.
Theβ-particles of the214Pb decay can have varying energies which creates a fairly uniform ER energy spectrum. Theβ-decay of85Kr has a contribution to the background as well which has been reduced by distillation.
Theβ-particles interact through excitation and ionization with the orbital electrons. This thesis is focused on γ-particles from external γ-sources which interact with orbital electrons through Compton scattering and the Photoelectric effect. Compton scatters may look like WIMP interactions so it is important that these events can be selected in background pre-dictions.
1.3 Physics of gamma particles inside xenon
Aγ-particle may interact with xenon through a well known effect called the Photoelectric ef-fect, where the photon is completely absorbed by an inner shell electron without re-emitting a scattered photon. Note however that the electron is knocked out of the inner shell which is now ionized. The γ-particle could also interact with xenon through Compton scattering [10]. The photon enters a xenon atom and its energy is partially absorbed by an outer shell electron with a low binding energy. The electron is then knocked out of its orbit (Compton electron). The resulting photon energy is re-emitted with an angle as a scattered photon with a longer wavelength, because it now has less energy. Another possibility is pair production which is negligible for energies below 1.5 MeV as can be seen in figure 4. Below this energy the highest contribution to the attenuation are Compton scattering and the Photoelectric effect. The mean
free path of a photon is inversely related to the attenuation coefficient and density of xenon and for 1 MeVγ-particles it is about 3.5 cm [11], which is inside of the shielding layers.
The Compton effect is described by equation 2,
λout−λin=
h
mec(1 − cosθ) (2)
whereλout is the scattered photon wavelength andθ the scattering angle. With the relation
E = h f =λc this can be rewritten as Eout=
Ein
1 + ( Ein
mec2)(1 − cosθ)
(3)
to calculate the outgoing photon energy and angle. The energy deposited is then Ed e p= Ein− Eout= Ein−
Ein
1 + ( Ein
mec2)(1 − cosθ)
(4)
by a single scatter, which can be reapplied to calculate two or more scatters. When the scatter-ing angle and deposited energy are small, and the particle leaves the detector without havscatter-ing a second interaction, the signal could mimic WIMP interaction energies. This is because the probability that a WIMP interacts twice within the detector volume is negligible. The code has to choose between these two interaction processes when simulating an interacting particle. The choice is based on the relative cross sections of Compton scattering and the Photoelectric effect of a particle with energy E, extracted from the NIST database [12]. Common sources of
γ-particles are naturally occurring radioisotopes or cosmic rays which interact with particles in the atmosphere. The uranium series inside of the detector materials cause γ-interactions, even though the materials are chosen deliberately to reduce such activity. The water shield stops some γ-background and making the detector bigger also suppressesγ-noise. When the detector becomes larger, theγ-particles have to travel a longer distance through the LXe. This increases the chance of a second Compton scatter inside of the detector so that the event can be selected as background instead of a WIMP interaction.
2
Method
2.1 Standard simulation
To simulate background events a code was developed [13] in python which uses Monte Carlo simulations (MC). An MC simulation repeatedly traces gamma rays with random starting pa-rameters through the detector, in this case particles with random starting positions and trav-eling directions. The code generates a particle on the surface of the cylinder in figure 5 with starting coordinates X0 and directional vector ~X0 which represents the path of the particle. It
Figure 4: Mass attenuation of xenon (µρ) extracted from NIST [12]. Higher attenuation corresponds to a lower material transparency toγ-particles which means a shorter mean free path through the xenon. Pair production does not play a significant role below 1.5 MeV yet.
throws away any particle that does not intersect with the inner volume (fiducial volume). The energy of the particle is used to interpolate the attenuation length (λ) in xenon (NIST) so that the path length (S) inside of the LXe can be calculated with
P = 1 − e−Sλ . (5)
P is the probability that a particle interacts within a distance S. Only interactions inside of the cylinder are simulated. Based on a few requirements the adequate interaction is chosen which is either a Compton scatter or the Photoelectric effect. One of the parameters of the generated particle is an energy-cut which limits the amount of energy that can be deposited. This is used to simulate WIMP interactions within a region of interest below 250 keV. If the energy-cut is lower than the total energy of the particle the Photoelectric effect is prohibited and only Comp-ton scattering can occur. The cut is turned off when simulating backgrounds of high energy
γ-particles of 1-2 MeV because they can deposit more than WIMPs. The choice of interaction is then based on the relative cross sections of Compton scattering and the Photoelectric effect. If the Photoelectric effect is chosen, theγ-particle is completely absorbed locally (Xs1in figure 5)
and a second scatter is not possible. The coordinates and particle properties before absorption are saved and a new particle is generated on the surface of the cylinder. If Compton scatter-ing is chosen, an arbitrary azimuthal angle (φ) is generated and an inclination angle (θ1) is
up-dated and the particle path now leads to the exit point (X2) where it exits the fiducial volume and cylinder if only single scatters are simulated. For double scatters the particle is further tracked by calculating a second possible interaction point ~Xs2 on the line ~X2− ~Xs1. The new
particle energy is now E0= E − Ed e pwhich is used to calculate the second angleθ2. A new path
now leads to the exit and a new particle is generated.
2.1.1 Monte Carlo standard deviation
From the MC simulation, the probability that aγ-particle will cause a double scatter inside the xenon fiducial volume can be calculated with
Pi=Ni
Nγ, (6)
where Ni is the number of double scatters inside of the cylinder and Nγthe number of sim-ulated γ-particles. A reference simulation is performed to converge the MC simulation to a deterministic probability of interaction
Pref=
Nref
Nγ , (7)
which requires around Nγ= 109. Increasing this number might yield a slightly more deter-ministic probability at the cost of computational power and time. In this thesis this number is sufficient, because the simulation with the highest number of particles is done with 100x less particles. The correct standard deviation of an arbitrary MC simulation which is Nγdependent can now be calculated with
σmc(Nγ) =pNN i
ref =
pPiNγ
PrefNγ (8)
assuming that the reference simulation does not have any uncertainty. Each simulation has a different Pi so we take the average of n simulations to get a more accurate value;
¯ σmc= Pnσ mc n . (9) ¯
σmcis calculated for Nγ= 102, 103...106, each having n=100 samples to take the average from. 2.2 Variance Reduction Technique
The variance of a Monte Carlo simulation gives an analysis of how future simulations might behave around the mean. When the variance is high, it is hard to predict simulations which means that the sample size or the number of samples has to be increased to generate a higher confidence prediction. The generated standard deviations follow a Gaussian distribution around a mean, so the variance has the relation σ=pVar(Ni). The variance increases when
Figure 5: Schematic illustration of a possible particle path through the LXe. A particle is generated on the cylinder surface at X0and enters the fiducial volume (Xentry). An interaction point Xs2is calculated
and saved if it interacts inside the cylinder. The path is updated depending on the interaction (Compton scattering or Photoelectric effect), and a second interaction point is calculated.
mean. A background event is counted when a particle goes through and has exactly two scat-ters inside of the fiducial volume. An analysis of the background is only possible in this region of interest, which becomes rare when the interaction probability goes down. Forcing the simu-lation to only generate these preferred events would lower the variance, because one does not have to wait for the simulation to go through all possible events until it finally reaches the desired events. This is not physical however, so a technique must be used which accounts for the skipped undesirable event space.
2.2.1 Technique
Just like the standard method in 2.1 the simulation with the variance reduction technique (VRT) generates a particle on the surface. A directional vector is assigned and the particle, again, is thrown away if it does not lead through the fiducial volume. This time however, before calculating an interaction, the particle is transported to the edge of the fiducial volume Xentry. Normally there would be a chance of the particle interacting in the path ~Xentry− ~X0,
but this event space is now thrown away. To account for the disposed event space between |~Xentry− ~X0| without having to explore it all, a distribution is created using the weight [13];
whereρ= 3.5 g
cm3 is the density of LXe andµ(Ein) the mass attenuation coefficient. Inside the
fiducial volume, the particle is forced to have an interaction within the path |~X1− ~Xentry|. Now
the code has to account for an interaction occuring, or;
w2= 1 − e−ρµ(Ein)|~X1−~Xentry|. (11)
Just like the standard MC, an angleθ1is calculated using the particle energy and the amount
of energy deposited, and another interaction is forced within the path |~X2− ~Xs1| using the same logic;
w3= 1 − e−ρµ(Ein)|~X2−~Xs1|. (12)
The particle now has a weight W = w1w2w3. Note that we have not taken into account the
restriction that an energy-cut brings to the Photoelectric effect and the limited possible angles which follow from the Klein-Nishina equation. If the particle energy is larger than the energy-cut, the Photoelectric effect is not possible. For this a fourth weight is assigned;
w4=
σCompton σtotal
, (13)
which is the chance of a Compton scatter occurring instead of the Photoelectric effect. Due to the restriction on the maximum recoil energy, Emax, not all scattering angles are allowed. A
fifth weight accounts for this by integrating the cross section over all possible Compton angles; w5= 1 σCompton Z Emax 0 dσCompton dE dE. (14)
When no energy-cut is applied the maximum allowed deposit energy would have an angle ofπ. The particle exits the cylinder with W = w1w2w3w4w5 and scattering angleθ2. The standard
deviation can be calculated with
σVRT= q 1 n Pn(N i− ¯Ni)2 ¯ Ni = pVar(Ni) ¯ Ni , (15)
where Niis not a whole integer like in the standard MC, but has the weight W assigned. Each
sample size in Nγ= 102, 103...106again has n=100 samples taken.
2.3 Acceleration
The acceleration of the variance reduction on the MC simulation is defined as the factor be-tween the number of particles Nγ simulated between VRT and nonVRT, when their relative standard deviations are equal;
σVRT(NγVRT) =σnonVRT(NγnonVRT) →α=
NγnonVRT
which means their variance is the same. A way to approach this is by plotting theσVRT and
σnonVRTvalues in aσ(Nγ) graph. Both methods can be fit with y(x) =pq
x+ c (17)
and the horizontal distance between both lines will be the acceleration factorα;
α=NγnonVRT NγVRT = ( qγ nonVRT qγ VRT )2. (18)
3
Results
Three different γ-energies have been simulated which are 1 MeV, 1.5 MeV and 2 MeV. The results are illustrated in five figures. Figure 6 shows the probability of the second interaction in the fiducuial volume, which converges to about 8 · 10−2. The first interaction has a reference probability of 3.1 ·10−5, which is significantly lower. The product of these two is the probability of a double scatter event inside the xenon fiducial volume, which determines the acceleration factor. Figure 7 shows the energy spectrum of 1 MeVγ-particles in a cylindrical volume (A) and a fiducial volume (B). This spectrum is the total amount of energy deposited after two scatters. The VRT has better results than the nonVRT when there are less interactions (counts) in total which can be seen in B. The VRT has a lower variance than the nonVRT which has multiple peaks. This difference in variance is shown in figures 8, 9 and 10 where the relative error σ¯x is shown as a function of N, the amount of particles simulated. Each figure has a logarithmic scale on both axis to represent the fit functions (17) as straight lines. The horizontal distance between these lines is the acceleration factorα. The acceleration factor for 1 MeVγ-particles in a fiducial volume isα1M eV = 8.3 · 103 which is shown in figure 8. The VRT achieved higher
acceleration factors for 1.5- and 2 MeVγ-particles which areα1.5M eV= 13.2 · 103 andα2M eV=
Figure 6: Probability of a particle having a second interaction inside the fiducial volume. The value converges to about 8 %.
4
Discussion
We have analyzed the acceleration factor of VRT on MC simulations for doubleγ-scatters inside of a fiducial LXe volume. The particles have an energy-cut of 250 keV to simulate the region of interest for WIMPs. As provided by [13], the acceleration for single scatters with 1.5 MeV
γ-particles in this region of interest is 1.6 · 103. The variance is directly correlated with Prefas
can be seen in equation 8. Lowering the probability for an event increases the variance and consequently the acceleration factor as well. Since double scatter events have a lower proba-bility than single scatter events inside of a fiducial volume, it is expected that the acceleration factor increases. The acceleration now depends on the probability of the second interaction. If the second interaction had the same probability as the first interaction, the acceleration fac-tor would increase quadratically. However, after interacting once inside the fiducial volume, the particle has already crossed the shielding layers and is not hindered by it anymore, which means the probability of an interaction inside of this volume is higher (figure 6). The results support this expectation because the acceleration is about 10 times higher and is not increased quadratically. Increasing theγ-particle energy also increases the acceleration. This is because higher energy particles have a lower chance of depositing at ≤ 250 keV, making the event more rare.
Figure 7: The energy spectrum of 1 MeVγ-particles in a large fiducial volume (r = 64 cm, h = 149 cm) (A) and a smaller fiducial volume (r = 57 cm, h = 134) (B) where the total cylinder has dimensions r = 65 and h = 150. The fiducial volume shows less interactions (counts) as expected since most interactions happen in the outer layer. The nonVRT simulation has a higher variance as can be seen in B with multiple peaks while the VRT has a more spread distribution. In black is a simulation of particles with less than 250 keV to represent possible dark matter signals.
Figure 8: The acceleration factorα = 8.3 · 103for 1 MeVγ-particles, calculated by fitting the standard simulation and the VRT simulation with pq
x+ c independently, and determining the horizontal distance
between both functions asα = (qmc
qvrt)
2. A logarithmic scale has been used to depict the fits as straight
lines. The value is lower than the 1.5 MeV and 2 MeV simulations.
Figure 10: An acceleration factor ofα = 16.3 · 103for 2 MeVγ-particles.
5
Conclusion
The VRT provides a significant acceleration on Monte Carlo simulations which increases when preferred events from simulations become more rare and have a higher variance. This is very suitable when searching for rare events like double scatters and WIMP interactions. Increasing theγ-energy also increases the acceleration factor. The acceleration does not increase quadrat-ically from single scatters because the second interaction has a much higher probability than the first interaction.
6
Acknowledgements
First and foremost, I would like to thank my supervisor prof. dr. Auke-Pieter Colijn for provid-ing an excellent project, where his continuous help and support enabled me to finish this thesis with utmost content.
My sincere gratitude also goes to my daily supervisor Gijs Leguijt, who helped me with my daily questions and provided great feedback to my works. He always responded to my questions with all the information I needed.
Besides my supervisor and daily supervisor, I would like to thank Olivier Kesber who helped me understand the code of the simulation and some complicated uncertainty calcu-lations.
My thanks goes to Joran Angevaare as well, who helped me set up the software in order to run the code through the Nikhef computers.
I would also like to thank prof. dr. Stan Bentvelsen for taking the time to be my second examiner, and to read this report.
Last but not least I would like to thank the Dark Matter group; prof. dr. Patrick Decowski, Peter Gaemers, Stefan Bruenner, Lucas de Vries, Alvaro Loya, Isis Hobus and Leonora Verveld for asking informative questions during my short presentations where I got the group up to date.
References
[1] Massimo Persic, Paolo Salucci, Fulvio Stel, The universal rotation curve of spiral galax-ies — I. The dark matter connection, Monthly Notices of the Royal Astronomical Society, Volume 281, Issue 1, July 1996, Pages 27–47, https://doi.org/10.1093/mnras/278.1.27 [2] Hoeft, M., Mücket, J. P., & Gottlöber, S. (2004). Velocity dispersion profiles in dark matter
halos. The Astrophysical Journal, 602(1), 162.
[3] M. Markevitch; A. H. Gonzalez; D. Clowe; A. Vikhlinin; L. David; W. Forman; C. Jones; S. Murray & W. Tucker (2004). "Direct constraints on the dark matter self-interaction cross-section from the merging galaxy cluster 1E0657-56". Astrophys. J. 606 (2): 819–824. arXiv:astro-ph/0309303. Bibcode:2004ApJ...606..819M. doi:10.1086/383178
[4] Douglas Clowe et al. “A Direct Empirical Proof of the Existence of Dark Matter”. The As-trophysical Journal 648.2 (Sept. 10, 2006), pp. L109–L113. doi: 10.1086/508162. arXiv: astro-ph/0608407 (cit. on pp. 13, 15).
[5] Kamionkowski, M. (1997). WIMP and axion dark matter. arXiv preprint hep-ph/9710467. [6] Peskin, M.E., Schroeder, D.V. (1995). An Introduction to Quantum Field Theory, Westview
Press, ISBN 0-201-50397-2, p. 80.
[7] Aalbers, J. (2018). Dark matter search with XENON1T
[8] E. Aprile et al. “The XENON1T Dark Matter Experiment”. Eur.Phys.J. C77(Dec. 18, 2017), p. 881. doi: 10.1140/epjc/s10052-017-5326-3 (cit. on pp. 47, 48, 54, 97).
[9] Aprile, E, et al. “XENON1T Dark Matter Data Analysis: Signal & Background Models, and Statistical Inference.” ArXiv.org, 1 Mar. 2019, arxiv.org/abs/1902.11297.
[10] P. Christillin (1986). "Nuclear Compton scattering". J. Phys. G: Nucl. Phys. 12 (9): 837–851. Bibcode:1986JPhG...12..837C. doi:10.1088/0305-4616/12/9/008.
[11] Israelashvili, I., Cortesi, M., Vartsky, D., Arazi, L., Bar, D., Caspi, E. N., & Breskin, A. (2015). A comprehensive simulation study of a Liquid-Xe detector for contraband detection. Journal of Instrumentation, 10(03), P03030.
[12] “NIST Database.” NIST, 27 Apr. 2020, https://physics.nist.gov/PhysRefData/Xcom/html/xcom1.html [13] Bruenner, S., Colijn, A.P., Decowski, M.P., & Kesber, O.V., (2020). Variance reduction
[14] Chandra Collaboration. 1E 0657-56: NASA Finds Direct Proof of Dark Matter. Aug. 21, 2006. url: http://chandra.harvard.edu/photo/2006/1e0657/ (visited on 01/02/2018) (cit. on p. 13).
