first draft of my thesis

4 files changed, 935 insertions, 0 deletions

diff --git a/.gitignore b/.gitignore
index 7445e76..25e9f86 100644
--- a/.gitignore
+++ b/.gitignore
@@ -17,3 +17,8 @@ test-find-peaks
 *.xzdab
 *.tar.gz
 *.hdf5
+calculate_limits
+*.aux
+*.log
+*.pdf
+*.out
diff --git a/Makefile b/Makefile
index 27445de..96fe01c 100644
--- a/Makefile
+++ b/Makefile
@@ -7,3 +7,14 @@ default: all
 	cd src && "$(MAKE)" $@
 	cd utils && "$(MAKE)" $@
 
+all: sddm.pdf
+
+calculate_limits: calculate_limits.c
+
+sddm.pdf: sddm.tex
+	pdflatex $^
+
+clean:
+	rm calculate_limits
+
+.PHONY: clean sddm.pdf
diff --git a/calculate_limits.py b/calculate_limits.py
new file mode 100755
index 0000000..f297a69
--- /dev/null
+++ b/calculate_limits.py
@@ -0,0 +1,419 @@
+#!/usr/bin/env python3
+import numpy as np
+from scipy.integrate import quad
+from scipy.stats import expon
+from scipy.special import spherical_jn, erf
+from numpy import pi
+from functools import lru_cache
+
+# on retina screens, the default plots are way too small
+# by using Qt5 and setting QT_AUTO_SCREEN_SCALE_FACTOR=1
+# Qt5 will scale everything using the dpi in ~/.Xresources
+import matplotlib
+matplotlib.use("Qt5Agg")
+
+# speed of light (exact)
+SPEED_OF_LIGHT = 299792458 # m/s
+
+# depth of the SNO detector (m)
+# currently just converted 6800 feet -> meters
+h = 2072.0
+
+# radius of earth (m)
+# from the "Earth Fact Sheet" at https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html
+# we use the volumetric mean radius here
+R = 6.371e6
+
+# Approximate dark matter velocity in m/s. The true distribution is expected to
+# be a Maxwell Boltzmann distribution which is modulated annually by the
+# earth's rotation around the sun, but we just assume a single constant
+# velocity here. From Lewin and Smith Appendix B
+DM_VELOCITY = 230e3
+
+# Approximate earth velocity in the galactic rest frame (?)
+# from Lewin and Smith Equation 3.6
+EARTH_VELOCITY = 244e3
+
+# Approximate dark matter density in GeV/m^3. From Tom Caldwell's thesis.
+# from Lewin and Smith page 91
+DM_DENSITY = 0.4e6
+
+# Number density of scatterers in the Earth.
+#
+# FIXME: Currently just set to the number density of atoms in water. Need to
+# update this for rock, and in fact this will change near the detector since
+# there is water outside the AV.
+DENSITY_WATER = 1e3 # In kg/m^3
+
+# mean density of norite rock which is the rock surrounding SNO
+# probably conservative
+NORITE_DENSITY = 3e3 # In kg/m^3
+
+# atomic masses for various elements
+# from https://www.angelo.edu/faculty/kboudrea/periodic/structure_mass.htm
+element_mass = {
+  'H':1.0,
+  'C':12.011,
+  'O':15.9994,
+  'Na':22.98977,
+  'Mg':24.305,
+  'Al':26.98154,
+  'Si':28.0855,
+  'K':39.0983,
+  'Ca':40.08,
+  'Mn':54.9380,
+  'Fe':55.847,
+  'Ti':47.90
+}
+
+# composition and mean density of norite rock which is the rock surrounding SNO
+# the composition is from Table 3.2 in the SNOLAB User's handbook
+water = {'composition':
+ {'H':20,
+  'O':80},
+  'density':1e3}
+
+# composition and mean density of the mantle
+# from www.knowledgedoor.com/2/elements_handbook/element_abundances_in_the_earth_s_mantle.html
+# density from hyperphysics.phy-astr.gsu.edu/hbase/Geophys/earthstruct.html
+mantle = {'composition':
+ {'O':44.33,
+  'Mg':22.17,
+  'Si':21.22,
+  'Fe':6.3},
+  'density':4.4e3}
+
+# composition and mean density of norite rock which is the rock surrounding SNO
+# the composition is from Table 3.2 in the SNOLAB User's handbook
+norite = {'composition':
+ {'H':0.15,
+  'C':0.04,
+  'O':46.0,
+  'Na':2.2,
+  'Mg':3.3,
+  'Al':9.0,
+  'Si':26.2,
+  'K': 1.2,
+  'Ca':5.2,
+  'Mn':0.1,
+  'Fe':6.2,
+  'Ti':0.5},
+  'density':3e3}
+
+# Fiducial volume (m)
+FIDUCIAL_RADIUS = 5
+
+# Fiducial volume (m^3)
+FIDUCIAL_VOLUME = 4*pi*FIDUCIAL_RADIUS**3/3
+
+# proton mass from the PDG (2018)
+PROTON_MASS = 0.938 # GeV
+
+# proton mass from the PDG (2018)
+ATOMIC_MASS_UNIT = 0.931 # GeV
+
+# mass of Xenon in atomic units
+XENON_MASS = 131.293
+
+# mass of Neon in atomic units
+NEON_MASS = 20.18
+
+# mass of argon in atomic units
+ARGON_MASS = 39.948
+
+# mass of germanium in atomic units
+GERMANIUM_MASS = 72.64
+
+# mass of tungsten in atomic units
+TUNGSTEN_MASS = 183.84
+
+# mass of oxygen in atomic units
+OXYGEN_MASS = 15.999
+
+# mass of silicon in atomic units
+SILICON_MASS = 28.0855
+
+# mass of iron in atomic units
+IRON_MASS = 55.845
+
+# mass of magnesium in atomic units
+MAGNESIUM_MASS = 24.305
+
+# galactic escape velocity (m/s)
+# from Tom Caldwell's thesis page 25
+ESCAPE_VELOCITY = 244e3
+
+# conversion constant from PDG
+HBARC = 197.326978812e-3 # GeV fm
+
+# Avogadros number (kg^-1)
+N0 = 6.02214085774e26
+
+def get_probability(r, l):
+    """
+    Returns the probability of a dark photon decaying in the SNO detector from
+    a dark matter decay distributed uniformly in the Earth. Assumes that the
+    depth of SNO is much larger than the dimensions of the SNO detector.
+    """
+    if r <= h:
+        theta_min = 0
+    elif r <= 2*R - h:
+        theta_min = pi - np.arccos((h**2 + r**2 - 2*R*h)/(2*r*(R-h)))
+    else:
+        return 0
+    return (1 + np.cos(theta_min))*np.exp(-r/l)/(2*l)
+
+@lru_cache()
+def get_probability2(l):
+    p, err = quad(get_probability, 0, min(2*R-h,10*l), args=(l,), epsabs=0, epsrel=1e-7, limit=1000)
+
+    return p
+
+def get_event_rate(m, cs0, l, A):
+    """
+    Returns the event rate of leptons produced from self-destructing dark
+    matter in the SNO detector for a given dark matter mass m, a cross section
+    cs, and a mediator decay length l.
+    """
+    # For now we assume the event rate is constant throughout the earth, so we
+    # are implicitly assuming that the cross section is pretty small.
+    flux = DM_VELOCITY*DM_DENSITY/m
+
+    #p, err = quad(get_probability, 0, min(2*R-h,10*l), args=(l,), epsabs=0, epsrel=1e-7, limit=1000)
+    p = get_probability2(l)
+
+    cs = get_cross_section(cs0, m, A)
+
+    # FIXME: factor of 2 because the DM particle decays into two mediators?
+    return p*cs*(N0/A)*flux*FIDUCIAL_VOLUME
+
+def get_event_rate_sno(m, cs0, l, composition):
+    """
+    Returns the event rate of leptons produced from self-destructing dark
+    matter in the SNO detector for a given dark matter mass m, a cross section
+    cs, and a mediator decay length l.
+    """
+    rate = 0.0
+
+    for element, mass_fraction in composition['composition'].items():
+        rate += mass_fraction/100*get_event_rate(m,cs0,l,element_mass[element])
+
+    return rate*composition['density']
+
+def get_nuclear_form_factor(A, e):
+    """
+    Returns the nuclear form factor for a WIMP-nucleus interaction.
+
+    From Tom Caldwell's thesis page 24.
+
+    Also used Mark Pepin's thesis page 50
+    """
+    # mass of xenon nucleus
+    mn = A*ATOMIC_MASS_UNIT
+
+    # calculate approximate size of radius
+    s = 0.9 # fm
+    a = 0.52 # fm
+    c = 1.23*A**(1/3) - 0.60 # fm
+    r1 = np.sqrt(c**2 + (7/3)*pi**2*a**2 - 5*s**2)
+
+    q = np.sqrt(2*mn*e)
+
+    if q*r1/HBARC < 1e-10:
+        return 1.0
+
+    # Helm form factor
+    # from Mark Pepin's thesis page 50
+    f = 3*spherical_jn(1,q*r1/HBARC)*np.exp(-(q*s/HBARC)**2/2)/(q*r1/HBARC)
+
+    return f
+
+def get_cross_section(cs0, m, A):
+    """
+    Returns the WIMP cross section from the target-independent WIMP-nucleon
+    cross section cs0 at zero momentum transfer.
+
+    From Tom Caldwell's thesis page 21.
+    """
+    # mass of xenon nucleus
+    mn = A*ATOMIC_MASS_UNIT
+
+    # reduced mass of the nucleus and the WIMP
+    mr = (m*mn)/(m + mn)
+
+    # reduced mass of the proton and the WIMP
+    mp = (m*PROTON_MASS)/(m + PROTON_MASS)
+
+    return cs0*A**2*mr**2/mp**2
+
+def get_differential_event_rate_xenon(e, m, cs0, A):
+    """
+    Returns the event rate of WIMP scattering in the Xenon 100T detector.
+    """
+    # mass of nucleus
+    mn = A*ATOMIC_MASS_UNIT
+
+    # reduced mass of the nucleus and the WIMP
+    mr = (m*mn)/(m + mn)
+
+    cs = get_cross_section(cs0, m, A)
+
+    v0 = DM_VELOCITY
+
+    vesc = ESCAPE_VELOCITY
+
+    # earth's velocity through the galaxy
+    ve = EARTH_VELOCITY
+
+    # minimum wimp velocity needed to produce a recoil of energy e
+    vmin = np.sqrt(mn*e/2)*(mn+m)*SPEED_OF_LIGHT/(mn*m)
+
+    f = get_nuclear_form_factor(A, e)
+
+    x = vmin/v0
+    y = ve/v0
+    z = vesc/v0
+
+    # Equation 3.49 in Mark Pepin's thesis
+    k0 = (pi*v0**2)**(3/2)
+
+    # Equation 3.49 in Mark Pepin's thesis
+    k1 = k0*(erf(z)-(2/np.sqrt(pi))*z*np.exp(-z**2))
+
+    # From Mark Pepin's CDMS thesis page 59
+    if x <= z - y:
+        I = (k0/k1)*(DM_DENSITY/(2*ve))*(erf(x+y) - erf(x-y) - (4/np.sqrt(pi))*y*np.exp(-z**2))
+    elif x <= y + z:
+        I = (k0/k1)*(DM_DENSITY/(2*ve))*(erf(z) - erf(x-y) - (2/np.sqrt(pi))*(y+z-x)*np.exp(-z**2))
+    else:
+        return 0
+
+    return (N0/A)*(mn/(2*m*mr**2))*cs*f**2*I*SPEED_OF_LIGHT**2
+
+def get_event_rate_xenon(m, cs, A, threshold):
+    """
+    Returns the event rate of WIMP scattering in a dark matter detector using
+    an element with atomic mass A. Rate is in Hz kg^-1.
+    """
+    # mass of nucleus
+    mn = A*ATOMIC_MASS_UNIT
+
+    # reduced mass of the nucleus and the WIMP
+    mr = (m*mn)/(m + mn)
+
+    vesc = ESCAPE_VELOCITY
+
+    # earth's velocity through the galaxy
+    ve = EARTH_VELOCITY
+
+    # max recoil they looked for was 40 keV
+    emax = 2*mr**2*((ve+vesc)/SPEED_OF_LIGHT)**2/mn
+
+    rate, err = quad(get_differential_event_rate_xenon, threshold, emax, args=(m,cs,A), epsabs=0, epsrel=1e-2, limit=1000)
+
+    return rate
+
+if __name__ == '__main__':
+    import matplotlib.pyplot as plt
+
+    ls = np.logspace(-1,8,1000)
+
+    cs = 1e-50 # cm^2
+
+    # FIXME: should use water density for L < 1 m, silicon density for L ~ 1
+    # km, and iron for L >> 1 km
+    rate = np.array([get_event_rate_sno(1, cs*1e-4, l, water) for l in ls])
+
+    colors = plt.rcParams['axes.prop_cycle'].by_key()['color']
+
+    plt.figure()
+    plt.subplot(111)
+    plt.plot(ls, rate/np.max(rate),color=colors[0])
+    plt.xlabel("Mediator Decay Length (m)")
+    plt.ylabel("Event Rate (arbitrary units)")
+    plt.axvline(x=FIDUCIAL_RADIUS, color=colors[1], ls='--', label="SNO radius")
+    plt.axvline(x=h, ls='--', color=colors[2], label="Depth of SNO")
+    plt.axvline(x=R, ls='--', color=colors[3], label="Earth radius")
+    plt.gca().set_xscale('log')
+    plt.gca().set_xlim((ls[0], ls[-1]))
+    plt.title("Event Rate of Self-Destructing Dark Matter in SNO")
+    plt.legend()
+
+    # threshold is ~5 keVr
+    xenon_100t_threshold = 5e-6
+
+    # threshold is ~1 keVr
+    cdms_threshold = 1e-6
+
+    # threshold is ~100 eV
+    # FIXME: is this correct?
+    cresst_threshold = 1e-7
+
+    ms = np.logspace(-2,3,200)
+    cs0s = np.logspace(-50,-40,200)
+
+    mm, cs0cs0 = np.meshgrid(ms, cs0s)
+
+    rate1 = np.empty(mm.shape)
+    rate2 = np.empty(mm.shape)
+    rate3 = np.empty(mm.shape)
+    rate4 = np.empty(mm.shape)
+    rate5 = np.empty(mm.shape)
+    rate6 = np.empty(mm.shape)
+    for i in range(mm.shape[0]):
+        print("\r%i/%i" % (i+1,mm.shape[0]),end='')
+        for j in range(mm.shape[1]):
+            rate1[i,j] = get_event_rate_xenon(mm[i,j], cs0cs0[i,j]*1e-4, XENON_MASS, xenon_100t_threshold)
+            rate2[i,j] = get_event_rate_sno(mm[i,j], cs0cs0[i,j]*1e-4, 1.0, water)
+            rate3[i,j] = get_event_rate_xenon(mm[i,j], cs0cs0[i,j]*1e-4, GERMANIUM_MASS, cdms_threshold)
+            rate4[i,j] = get_event_rate_xenon(mm[i,j], cs0cs0[i,j]*1e-4, TUNGSTEN_MASS, cresst_threshold)
+            rate5[i,j] = get_event_rate_sno(mm[i,j], cs0cs0[i,j]*1e-4, 1e3, norite)
+            rate6[i,j] = get_event_rate_sno(mm[i,j], cs0cs0[i,j]*1e-4, 1e6, mantle)
+    print()
+
+    # Fiducial volume of the Xenon1T detector is 1042 +/- 12 kg
+    # from arxiv:1705.06655
+    xenon_100t_fiducial_volume = 1042 # kg
+
+    # Livetime of the Xenon1T results
+    # from arxiv:1705.06655
+    xenon_100t_livetime = 34.2 # days
+
+    plt.figure()
+    plt.subplot(111)
+    plt.gca().set_xscale('log')
+    plt.gca().set_yscale('log')
+    CS1 = plt.contour(mm,cs0cs0,rate1*3600*24*xenon_100t_fiducial_volume*xenon_100t_livetime,[10.0], colors=[colors[0]])
+    CS2 = plt.contour(mm,cs0cs0,rate2*3600*24*668.8,[10.0],colors=[colors[1]])
+    CS3 = plt.contour(mm,cs0cs0,rate3*3600*24*70.10,[10.0],colors=[colors[2]])
+    # FIXME: I used 2.39 kg day because that's what CRESST-3 reports in their paper
+    # but! I only use Tungsten here so do I need to multiply by the mass fraction of tungsten?
+    CS4 = plt.contour(mm,cs0cs0,rate4*3600*24*2.39,[10.0],colors=[colors[3]])
+    CS5 = plt.contour(mm,cs0cs0,rate5*3600*24*668.8,[10.0],colors=[colors[1]], linestyles=['dashed'])
+    CS6 = plt.contour(mm,cs0cs0,rate6*3600*24*668.8,[10.0],colors=[colors[1]], linestyles=['dotted'])
+    plt.clabel(CS1, inline=1, fmt="XENON1T", fontsize=10, use_clabeltext=True)
+    plt.clabel(CS2, inline=1, fmt=r"SNO ($\mathrm{L}_V$ = 1 m)", fontsize=10)
+    plt.clabel(CS3, inline=1, fmt="CDMSLite", fontsize=10)
+    plt.clabel(CS4, inline=1, fmt="CRESST-3", fontsize=10)
+    plt.clabel(CS5, inline=1, fmt=r"SNO ($\mathrm{L}_V$ = 1 km)", fontsize=10)
+    plt.clabel(CS6, inline=1, fmt=r"SNO ($\mathrm{L}_V$ = 1000 km)", fontsize=10)
+    plt.xlabel(r"$m_\chi$ (GeV)")
+    plt.ylabel(r"WIMP-nucleon scattering cross section ($\mathrm{cm}^2$)")
+    plt.title("Self-Destructing Dark Matter Limits")
+
+    x = np.linspace(0,300e-6,1000)
+    
+    # reproducing Figure 2.1 in Tom Caldwell's thesis
+    plt.figure()
+    plt.plot(x*1e6, list(map(lambda x: get_nuclear_form_factor(XENON_MASS, x)**2,x)), label="Xe")
+    plt.plot(x*1e6, list(map(lambda x: get_nuclear_form_factor(NEON_MASS, x)**2,x)), label="Ne")
+    plt.plot(x*1e6, list(map(lambda x: get_nuclear_form_factor(ARGON_MASS, x)**2,x)), label="Ar")
+    plt.plot(x*1e6, list(map(lambda x: get_nuclear_form_factor(GERMANIUM_MASS, x)**2,x)), label="Ge")
+    plt.xlabel("Recoil Energy (keV)")
+    plt.ylabel(r"$F^2(E)$")
+    plt.gca().set_yscale('log')
+    plt.legend()
+    plt.gca().set_ylim((1e-5,1))
+    plt.gca().set_xlim((0,300))
+    plt.title("Nuclear Form Factors for various elements")
+    plt.show()
diff --git a/sddm.tex b/sddm.tex
new file mode 100644
index 0000000..c7da970
--- /dev/null
+++ b/sddm.tex
@@ -0,0 +1,500 @@
+\documentclass{article}
+\usepackage{amsmath} % for \text command
+\usepackage{fullpage}
+\usepackage{tikz}
+\usepackage{hyperref}
+\usepackage{amsfonts}
+\newcommand*\diff{\mathrm{d}}
+\usetikzlibrary{shapes}
+\author{Anthony LaTorre}
+\date{\today}
+\title{Searching for Dark Matter with the Sudbury Neutrino Observatory}
+\begin{document}
+\maketitle
+\section{Introduction}
+\section{Estimating the Event rate in the SNO detector}
+The event rate of self destructing dark matter events, $R$, in the SNO detector is given by first integrating over the detector.
+\begin{equation}
+R = \int_\mathrm{SNO} \mathrm{d}^3r \, R(r) 
+\end{equation}
+Next, we integrate over the earth where the dark matter annihilates:
+\begin{equation}
+R = \int_\mathrm{SNO} \mathrm{d}^3r \, \int_{r'} \mathrm{d}^3r' R(r') \mathrm{P}(\text{detect at r} | \text{DM scatters at r'})
+\end{equation}
+where we have assumed above that the dark matter annihilates immediately after
+scattering. The event rate for scattering at a position $r'$ in the earth is:
+\begin{equation}
+R(r') = \Phi(r') \eta(r') \sigma(r')
+\end{equation}
+where $\Phi(r')$ is the flux at $r'$, $\eta(r')$ is the number density of
+scatterers at $r'$, and $\sigma$ is the cross section for the dark matter to
+scatter and annihilate. In general this will be a sum over the elemental
+composition of the earth at $r'$, but for notational simplicity we will assume
+a single cross section. We will also assume that the cross section is small
+enough that the flux is essentially constant over the whole earth so that the
+rate may be written as:
+\begin{equation}
+R(r') = \Phi \eta(r') \sigma(r').
+\end{equation}
+The rate may then be written as:
+\begin{equation}
+R = \Phi \int_\mathrm{SNO} \mathrm{d}^3r \, \int_{r'} \mathrm{d}^3r' \, \eta(r') \sigma(r') \mathrm{P}(\text{detect at r} | \text{DM scatters at r'})
+\end{equation}
+If we assume that the probability of detecting the dark matter is uniform
+across the SNO detector we may write it as:
+\begin{equation}
+R = \Phi \int_\mathrm{SNO} \mathrm{d}^3r \, \int_{r'} \mathrm{d}^3r' \, \eta(r') \sigma(r') \mathrm{P}(\text{detect at SNO} | \text{DM scatters at r'})
+\end{equation}
+This assumption is pretty well motivated since for most values of the mediator
+decay length the probability will be uniform across the detector. The only
+value for which it might not be a good approximation is if the mediator decay
+length is on the order of the detector radius in which case DM scattering in
+the rock of the cavity walls might have a higher event rate at the edge of the
+detector. Since the integrand no longer depends on $r$, we may write it as:
+\begin{equation}
+R = \Phi V_\text{fiducial} \int_{r'} \mathrm{d}^3r' \, \eta(r') \sigma(r') \mathrm{P}(\text{detect at SNO} | \text{DM scatters at r'})
+\end{equation}
+where $V_\text{fiducial}$ is the fiducial volume of the detector. The
+probability that the mediator $V$ is emitted in a direction $\theta$ and
+travels a distance $r$ in a spherical coordinate system centered on $r'$ may be
+written as:
+\begin{equation}
+f(r,\theta) = \frac{\sin\theta}{4\pi}\frac{e^{-r/L_V}}{L_V}
+\end{equation}
+To transform this probability distribution to the coordinate system centered on the SNO detector we first transform it to a cartesian coordinate system:
+\begin{align}
+f(r,\theta) &= \frac{\sin\theta}{4\pi}\frac{e^{-r/L_V}}{L_V}\frac{1}{r^2sin\theta} \\
+&= \frac{1}{4\pi r^2}\frac{e^{-r/L_V}}{L_V}
+\end{align}
+Then, the distribution is translated to the center of the detector, which
+doesn't change the form since the radial coordinate $r$ is the same in both
+coordinate systems. Finally, we switch back into spherical coordinates:
+\begin{align}
+f(r,\theta') &= \frac{1}{4\pi r^2}\frac{e^{-r/L_V}}{L_V}r^2\sin\theta' \\
+&= \frac{1}{4\pi}\frac{e^{-r/L_V}}{L_V}\sin\theta'
+\end{align}
+where $\theta'$ is the polar angle in the SNO coordinate system. We can now write the rate as:
+\begin{align}
+R &= \Phi V_\text{fiducial} \int_r \mathrm{d}r \, \int_\theta \mathrm{d}\theta \, \int_\phi \mathrm{d}\phi \, \eta(r,\theta,\phi) \sigma(r,\theta,\phi) \frac{1}{4\pi}\frac{e^{-r/L_V}}{L_V}\sin\theta
+\end{align}
+We now assume that the number density of scatterers $\eta$ and the cross
+section $\sigma$ are independent of the position in the earth. This is a good
+approximation for certain values of $L_V$ since the integral will be dominated
+by a single material. For example, if the mediator decay length $L_V$ is
+approximately 1 meter, then the vast majority of the events in the detector
+will be caused by dark matter scattering off of water. Similarly if the
+mediator decay length is approximately 1 km then the majority of the events in
+the detector will be caused by the dark matter scattering off of the norite
+rock surrounding the detector. With this approximation, the rate may be
+written:
+\begin{align}
+R &= \Phi V_\text{fiducial} \eta \sigma \int_r \mathrm{d}r \, \int_\theta \mathrm{d}\theta \, \int_\phi \mathrm{d}\phi \, \frac{1}{4\pi}\frac{e^{-r/L_V}}{L_V}\sin\theta \\
+&= \Phi V_\text{fiducial} \eta \sigma \int_r \mathrm{d}r \, \frac{1}{2}\frac{e^{-r/L_V}}{L_V} \int_\theta \mathrm{d}\theta \, \sin\theta
+\end{align}
+The $\theta$ integral goes from $\theta_\text{min}$ to $\pi$:
+\begin{align}
+R &= \Phi V_\text{fiducial} \eta \sigma \int_r \mathrm{d}r \, \frac{1}{2}\frac{e^{-r/L_V}}{L_V} \int_{\theta_\text{min}}^\pi \mathrm{d}\theta \, \sin\theta
+\end{align}
+where $\theta_\text{min}$ is equal to:
+\begin{equation}
+\theta_\text{min} =%
+\begin{cases}
+0 & \text{if } r < \text{depth} \\
+\pi - \arccos\left(\frac{\text{depth}^2 + r^2 - 2R\text{depth}}{2r(R-\text{depth})}\right) & \text{if } \text{depth} < r < 2R-\text{depth} \\
+\end{cases}
+\end{equation}
+where $R$ is the radius of the earth and $\text{depth}$ is the distance from
+the surface of the earth to the SNO detector.
+
+\section{Cross Section}
+In \cite{grossman2017} the differential scattering cross section for dark
+matter off a nucleus is calculated as
+\begin{equation}
+\frac{\diff \sigma_\text{scatter}}{\diff q^2} = \frac{g_V^2 \epsilon^2 e^2}{4\pi v^2 (q^2 + m_V^2)^2} |F_D(q^2)|^2 Z^2 F^2(q),
+\end{equation}
+where $q$ is the momentum transferred, $g_V$ and $\epsilon$ are coupling
+constants (FIXME: is this true?), $v$ is the velocity of the dark matter
+particle, $m_V$ is the mass of the mediator, $F_D(q^2)$ is a form factor for
+the dark matter to transition from a high angular momentum state to a lower
+angular momentum state, $Z$ is the atomic number of the nucleus, and $F^2(q)$
+is a nuclear form factor.
+
+In the limit of low momentum transfer, the cross section is approximately
+\begin{equation}
+\frac{\diff \sigma_\text{scatter}}{\diff q^2} \simeq \frac{g_V^2 \epsilon^2 e^2}{4\pi v^2 m_V^4} |F_D(q^2)|^2 Z^2 F^2(q).
+\end{equation}
+
+For existing direct detection dark matter experiments, the relevant cross
+section is (FIXME: is this true?)
+\begin{equation}
+\frac{\diff \sigma_\text{scatter}}{\diff q^2} \simeq \frac{g_V^2 \epsilon^2 e^2}{4\pi v^2 m_V^4} Z^2 F^2(q).
+\end{equation}
+
+A standard cross section can be defined as the total cross section in the zero
+momentum limit\cite{pepin2016}
+\begin{align}
+\sigma_0 &= \int_0^{4\mu_T^2 v^2} \frac{\diff \sigma_\text{scatter}}{\diff q^2}\bigg\rvert_{q \rightarrow 0} \diff q^2 \\
+&= \frac{\mu_T^2 g_V^2 \epsilon^2 e^2}{\pi m_V^4} Z^2,
+\end{align}
+where $\mu_T$ is the reduced mass of the WIMP and target nucleus.
+
+Since different experiments use different detector targets, it is also useful
+to define a standard cross section, $\sigma_p$ which is independent of the
+nuclear target:
+\begin{equation}
+\sigma_p = \left(\frac{\mu_p}{\mu_T}\frac{1}{Z}\right)^2 \sigma_0.
+\end{equation}
+
+The direct detection cross section is then:
+\begin{equation}
+\frac{\diff \sigma_\text{scatter}}{\diff q^2} \simeq \frac{1}{4 \mu_p^2 v^2} \sigma_p Z^2 F^2(q).
+\end{equation}
+and the cross section for the dark matter to annihilate is:
+\begin{equation}
+\frac{\diff \sigma_\text{scatter}}{\diff q^2} \simeq \frac{1}{4 \mu_p^2 v^2} \sigma_p |F_D(q)|^2 Z^2 F^2(q).
+\end{equation}
+
+\subsection{Nuclear Form Factor}
+The nuclear form factor, $F(q)$, characterizes the loss of coherence as the de
+Broglie wavelength of the WIMP approaches the radius of the
+nucleus\cite{caldwell2015}. The most commonly used form factor calculation used
+in the direct detection community is that of Helm which is given by:
+\begin{equation}
+F(q) = 3\frac{j_1(q r_1)}{q r_1} e^{-\frac{(q s)^2}{2}},
+\end{equation}
+where $j_1$ is the spherical bessel function of the first order, $s$ is a
+measure of the nuclear skin thickness, and $r_1$ is a measure of the nuclear
+radius\cite{pepin2016}. The values used for these constants were
+\begin{align}
+s &= 0.9 \text{ fm} \\
+a &= 0.52 \text{ fm} \\
+c &= 1.23 A^\frac{1}{3} - 0.60 \text{ fm} \\
+r_1 &= \sqrt{c^2 + \frac{7}{3}\pi^2 a^2 - 5 s^2}
+\end{align}
+
+\begin{figure}
+\centering
+\begin{tikzpicture}[scale=0.1]
+% earth
+\draw [thick,domain=120:150] plot[smooth] ({200*cos(\x)},{200*sin(\x)});
+\begin{scope}[shift={(-100,100)},rotate=45]
+% interaction
+\node[star,star points=9,draw] at (20,20){};
+% acrylic vessel
+\draw [thick,domain={90+asin(75/600)}:{360+90-asin(75/600)}] plot[smooth] ({6*cos(\x)},{6*sin(\x)});
+\draw [thick] ({6*cos(90+asin(75/600))},{6*sin(90+asin(75/600))}) -- ({6*cos(90+asin(75/600))},{6*sin(90+asin(75/600))+7.5}) --
+      ({6*cos(90+asin(75/600))+2*0.75},{6*sin(90+asin(75/600))+7.5}) --
+      ({6*cos(90+asin(75/600))+2*0.75},{6*sin(90+asin(75/600))});
+% PSUP
+\draw [domain=0:360] plot ({8.89*cos(\x)},{8.89*sin(\x)});
+% cavity
+\draw (-9.5,-10.5) --
+      (-10.6,-10.5+5.6) --
+      (-10.6,-10.5+14.93) --
+      (-9.5,-10.5+30) --
+      (9.5,-10.5+30) --
+      (10.6,-10.5+14.93) --
+      (10.6,-10.5+5.6) --
+      (9.5,-10.5) --
+      (-9.5,-10.5);
+\draw[->,ultra thick] (-25,0) -- (25,0) node[right]{$x$};
+\draw[->,ultra thick] (0,-25) -- (0,25) node[right]{$y$};
+\end{scope}
+\end{tikzpicture}
+\end{figure}
+
+\section{Event Reconstruction}
+In order to reconstruct the physical parameters associated with an event we
+compute a likelihood for that event given a proposed energy, position,
+direction, and initial time. The likelihood may be written as:
+\begin{equation}
+\label{likelihood}
+\mathcal{L}(E, \vec{x}, \vec{v}, t_0) = P(\vec{q}, \vec{t} | E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+where $E$, $\vec{x}$, $\vec{v}$ represent the initial particle's kinetic
+energy, position, and direction respectively, $t_0$ represents the initial time
+of the event, $\vec{q}$ is the charge seen by each PMT, and $\vec{t}$ is the
+time recorded by each PMT.
+
+In general the right hand side of Equation~\ref{likelihood} is not factorable
+since for particle tracks which scatter there will be correlations between the
+PMT hits. However, to make the problem analytically tractable, we assume that
+the probability of each PMT being hit is approximately independent of the
+others. With this assumption we can factor the right hand side of the
+likelihood as:
+\begin{equation}
+\mathcal{L}(E, \vec{x}, \vec{v}, t_0) = \prod_i P(\text{not hit} | E, \vec{x}, \vec{v}, t_0) \prod_j P(\text{hit}, q_j, t_j | E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+where the first product is over all PMTs which weren't hit and the second
+product is over all of the hit PMTs.
+
+If we introduce the variable $n$ which represents the number of photoelectrons detected we can write the likelihood as:
+\begin{equation}
+\mathcal{L}(E, \vec{x}, \vec{v}, t_0) = \prod_i P(n = 0 | E, \vec{x}, \vec{v}, t_0) \prod_j \sum_{n = 1}^{\infty} P(n, q_j, t_j | E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+
+We can factor the right hand side of the likelihood as:
+\begin{equation}
+\mathcal{L}(E, \vec{x}, \vec{v}, t_0) = \prod_i P(n = 0 | E, \vec{x}, \vec{v}, t_0) \prod_j \sum_{n = 1}^{\infty} P(q_j, t_j | n, E, \vec{x}, \vec{v}, t_0) P(n | E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+
+If we now assume that the charge and time observed at a given PMT are
+independent we can write the likelihood as:
+\begin{equation}
+\mathcal{L}(E, \vec{x}, \vec{v}, t_0) = \prod_i P(n = 0 | E, \vec{x}, \vec{v}, t_0) \prod_j \sum_{n = 1}^{\infty} P(q_j | n, E, \vec{x}, \vec{v}, t_0) P(t_j | n, E, \vec{x}, \vec{v}, t_0) P(n | E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+
+Since there are many photons produced in each event each of which has a small
+probability to hit a given PMT, we will assume that the probability of
+detecting $n$ photons at a given PMT is poisson distributed, i.e.
+\begin{equation}
+P(n | E, \vec{x}, \vec{v}, t_0) = e^{-\mu} \frac{\mu^n}{n!}
+\end{equation}
+
+We can therefore write the likelihood as:
+\begin{equation}
+\mathcal{L}(E, \vec{x}, \vec{v}, t_0) = \prod_i e^{-\mu_i} \prod_j \sum_{n = 1}^{\infty} P(q_j | n, E, \vec{x}, \vec{v}, t_0) P(t_j | n, E, \vec{x}, \vec{v}, t_0) e^{-\mu_j} \frac{\mu_j^n}{n!}
+\end{equation}
+where $\mu_i$ is the expected number of photoelectrons detected at the ith PMT
+(given an initial particle's energy, position, and direction).
+
+First, we'll calculate the expected number of photoelectrons for a single non-showering track which undergoes multiple scattering through small angles. In this case, we can calculate the expected number of photoelectrons as:
+\begin{equation}
+\mu_i = \int_x \diff x \int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} P(\text{detected} | E, x, v)
+\end{equation}
+where $x$ is the position along the track and $\lambda$ is the wavelength of
+the light.
+
+If the particle undergoes many small angle Coulomb scatters, the net
+angular displacement of the particle after a distance $x$ will be a Gaussian
+distribution by the central limit theorem\cite{pdg2017}. The distribution of
+the net angular displacement at a distance $x$ along the track is then given
+by\footnote{This distribution will be correlated between different points along the track.}:
+\begin{equation}
+f(\theta,\phi) = \frac{\theta}{2\pi\theta_0^2}e^{-\frac{\theta^2}{2\theta_0^2}}
+\end{equation}
+where
+\begin{equation}
+\theta_0 = \frac{13.6 \text{ MeV}}{\beta c p}z\sqrt{\frac{x}{X_0}}\left[1 + 0.038\ln\left(\frac{x z^2}{X_0 \beta^2}\right)\right]
+\end{equation}
+where $p$, $\beta c$, and $z$ are the momentum, velocity, and charge of the
+particle, and $X_0$ is the radiation length of the particle\cite{pdg2017}.
+
+Now, we integrate over the angular displacement of the track around the original velocity:
+\begin{align}
+\mu_i &= \int_x \diff x \int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} \int_\theta \diff \theta \int_\phi \diff \phi P(\text{detected} | \theta, \phi, E, x, v) P(\theta, \phi | E, x, v) \\
+\mu_i &= \int_x \diff x \int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} \int_\theta \diff \theta \int_\phi \diff \phi P(\text{detected} | \theta, \phi, E, x, v) f(\theta,\phi)
+\end{align}
+The probability of being detected can be factored into several different compontents:
+\begin{align}
+\mu_i &= \int_x \diff x \int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} \int_\theta \diff \theta \int_\phi \diff \phi P(\text{emitted towards PMT i} | \theta, \phi, E, x, v) f(\theta,\phi) P(\text{not scattered or absorbed} | \lambda, E, x, v) \epsilon(\eta) \mathrm{QE}(\lambda) \\
+\label{eq:mui}
+\mu_i &= \int_x \diff x \int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} P(\text{not scattered or absorbed} | \lambda, E, x, v) \epsilon(\eta) \mathrm{QE}(\lambda) \int_\theta \diff \theta \int_\phi \diff \phi P(\text{emitted towards PMT i} | \theta, \phi, E, x, v) f(\theta,\phi)
+\end{align}
+where $\eta$ is the angle between the vector connecting the track position $x$
+to the PMT position and the normal vector to the PMT, $\epsilon(\eta)$ is the
+collection efficiency, and $\mathrm{QE}(\lambda)$ is the quantum efficiency of
+the PMT.
+
+The probability that a photon is emitted directly towards a PMT is given by a
+delta function (we make the assumption here that the probability is uniform
+across the face of the PMT):
+\begin{equation}
+P(\text{emitted towards PMT i} | \theta, \phi, E, x, v) = \delta\left(\frac{1}{n(\lambda)\beta} - \cos\theta'(\theta,\phi,x)\right) \frac{\Omega(x)}{4\pi}
+\end{equation}
+where $\theta'$ is the angle between the track and the PMT and $\Omega(x)$ is the solid angle subtended by the PMT.
+
+In a coordinate system with the z axis aligned along the original particle velocity and with the PMT in the x-z plane, the angle $\theta'$ is defined by:
+\begin{equation}
+\cos\theta' = \sin\theta\cos\phi\sin\theta_1 + \cos\theta\cos\theta_1
+\end{equation}
+where $\theta_1$ is the angle between the PMT and the original particle velocity.
+
+We can now solve the integral on the right hand side of Equation~\ref{eq:mui} as:
+\begin{align}
+P(\text{emitted towards PMT i}) &= \int_\theta \diff \theta \int_\phi \diff \phi \delta\left(\frac{1}{n(\lambda)\beta} - \cos\theta'(\theta,\phi,x)\right) \frac{\Omega(x)}{4\pi} \frac{\theta}{2\pi\theta_0^2}e^{-\frac{\theta^2}{2\theta_0^2}} \\
+P(\text{emitted towards PMT i}) &= \frac{\Omega(x)}{4\pi} \frac{1}{2\pi\theta_0^2}\int_\theta \diff \theta \int_\phi \diff \phi \delta\left(\frac{1}{n(\lambda)\beta} - \cos\theta'(\theta,\phi,x)\right) \theta e^{-\frac{\theta^2}{2\theta_0^2}} \\
+P(\text{emitted towards PMT i}) &= \frac{\Omega(x)}{4\pi} \frac{1}{2\pi\theta_0^2}\int_\theta \diff \theta \int_\phi \diff \phi \delta\left(\frac{1}{n(\lambda)\beta} - \sin\theta\cos\phi\sin\theta_1 - \cos\theta\cos\theta_1\right) \theta e^{-\frac{\theta^2}{2\theta_0^2}}
+\end{align}
+
+We now assume $\theta$ is small (which should be valid for small angle scatters), so that we can rewrite the delta function as:
+\begin{align}
+P(\text{emitted towards PMT i}) &= \frac{\Omega(x)}{4\pi} \frac{1}{2\pi\theta_0^2}\int_\theta \diff \theta \int_\phi \diff \phi \delta\left(\frac{1}{n(\lambda)\beta} - \theta\cos\phi\sin\theta_1 - \cos\theta_1\right) \theta e^{-\frac{\theta^2}{2\theta_0^2}}
+\end{align}
+
+We can rewrite the delta function and solve the integral as:
+\begin{align}
+P(\text{emitted towards PMT i}) &= \frac{\Omega(x)}{4\pi} \frac{1}{2\pi\theta_0^2}\int_\theta \diff \theta \int_\phi \diff \phi \frac{1}{\left|\cos\phi\sin\theta_1\right|}\delta\left(\theta - \frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\cos\phi\sin\theta_1}\right) \theta e^{-\frac{\theta^2}{2\theta_0^2}} \\
+&= \frac{\Omega(x)}{4\pi} \frac{1}{2\pi\theta_0^2} \frac{1}{\left|\sin\theta_1\right|} \int_\phi \diff \phi \frac{1}{\left|\cos\phi\right|} \int_\theta \diff \theta \delta\left(\theta - \frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\cos\phi\sin\theta_1}\right) \theta e^{-\frac{\theta^2}{2\theta_0^2}} \\
+&= \frac{\Omega(x)}{4\pi} \frac{1}{2\pi\theta_0^2} \frac{1}{\left|\sin\theta_1\right|} \int_\phi \diff \phi \frac{1}{\left|\cos\phi\right|}\frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\cos\phi\sin\theta_1}H\left(\frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\cos\phi\sin\theta_1}\right)e^{-\frac{1}{2\theta_0^2}\left(\frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\cos\phi\sin\theta_1}\right)^2} \\
+&= \frac{\Omega(x)}{4\pi} \frac{1}{2\pi\theta_0^2} \frac{1}{\left|\sin\theta_1\right|}\sqrt{2\pi}\theta_0 e^{-\frac{1}{2\theta_0^2}\left(\frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\sin\theta_1}\right)^2} \\
+&= \frac{\Omega(x)}{4\pi} \frac{1}{\sqrt{2\pi}\theta_0} \frac{1}{\left|\sin\theta_1\right|} e^{-\frac{1}{2\theta_0^2}\left(\frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\sin\theta_1}\right)^2}.
+\end{align}
+
+To simplify this expression we can write
+\begin{equation}
+P(\text{emitted towards PMT i}) = \frac{\Omega(x)}{4\pi} \frac{1}{\sqrt{2\pi}\theta_0} \frac{1}{\left|\sin\theta_1\right|} e^{-\frac{\Delta^2(\lambda)}{2\theta_0^2}}
+\end{equation}
+where
+\begin{equation}
+\Delta(\lambda) = \frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\sin\theta_1}
+\end{equation}
+
+Plugging this back into Equation~\ref{eq:mui}
+\begin{align}
+\label{eq:mui-exact}
+\mu_i &= \frac{1}{\sqrt{2\pi}\theta_0} \int_x \diff x \frac{\Omega(x)}{4\pi} \frac{1}{\left|\sin\theta_1\right|} \epsilon(\eta) \int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} P(\text{not scattered or absorbed} | \lambda, E, x, v) \mathrm{QE}(\lambda) e^{-\frac{\Delta^2(\lambda)}{2\theta_0^2}}
+\end{align}
+
+Ideally we would just evaluate this double integral for each likelihood call,
+however the double integral is too computationally expensive to perform for
+every likelihood call (FIXME: is this true?). We therefore assume that the
+second integral will be dominated by the Bessel function which has a
+singularity when it's argument is zero, and rewrite Equation~\ref{eq:mui-exact}
+as:
+\begin{align}
+\mu_i &= 2 \frac{1}{\sqrt{2\pi}\theta_0} \int_x \diff x \Omega(x) \frac{1}{\left|\sin\theta_1\right|} \epsilon(\eta) P(\text{not scattered or absorbed} | \lambda_0, E, x, v) \mathrm{QE}(\lambda_0) e^{-\frac{\Delta^2(\lambda_0)}{4\theta_0^2}} \int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} K_0\left(\frac{\Delta^2(\lambda)}{4\theta_0^2}\right)
+\end{align}
+where $\lambda_0$ is the wavelength at which $\Delta(\lambda) = 0$.
+
+For small values of $\Delta$, the Bessel function may be approximated as:
+\begin{equation}
+K_0(x) \simeq -\log(x) + \log(2) - \gamma
+\end{equation}
+
+We may therefore approximate the expected charge as
+\begin{multline}
+\label{eq:mui-approx}
+\mu_i = 2 \frac{1}{\sqrt{2\pi}\theta_0} \int_x \diff x \Omega(x) \frac{1}{\left|\sin\theta_1\right|} \epsilon(\eta) P(\text{not scattered or absorbed} | \lambda_0, E, x, v) \mathrm{QE}(\lambda_0) e^{-\frac{\Delta^2(\lambda_0)}{4\theta_0^2}} \\
+\int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} \left(-\log\left(\frac{\Delta^2(\lambda)}{4\theta_0^2}\right) + \log(2) - \gamma\right)
+\end{multline}
+
+The number of Cerenkov photons produced per unit length and per unit wavelength
+is given by\cite{pdg2017}
+\begin{equation}
+\frac{\diff^2 N}{\diff x \diff \lambda} = \frac{2\pi\alpha z^2}{\lambda^2}\left(1 - \frac{1}{\beta^2 n^2(\lambda)}\right)
+\end{equation}
+where $\alpha$ is the fine-structure constant and $z$ is the charge of the
+particle in units of the electron charge.
+
+We can therefore write the second integral in Equation~\ref{eq:mui-approx} as
+\begin{align}
+\int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} \left(-\log\left(\frac{\Delta^2(\lambda)}{4\theta_0^2}\right) + \log(2) - \gamma\right) &=
+2\pi\alpha z^2 \int_\lambda \diff \lambda \frac{1}{\lambda^2}\left(1 - \frac{1}{\beta^2 n^2(\lambda)}\right) \left(-\log\left(\frac{\Delta^2(\lambda)}{4\theta_0^2}\right) + \log(2) - \gamma\right) \\
+\label{eq:lambda}
+&\simeq 2\pi\alpha z^2 \left(1 - \frac{1}{\beta^2 n^2(\lambda_0)}\right) \int_\lambda \diff \lambda \frac{1}{\lambda^2}\left(-\log\left(\frac{\Delta^2(\lambda)}{4\theta_0^2}\right) + \log(2) - \gamma\right)
+\end{align}
+
+Since the $\Delta$ function only depends on the wavelength through the index
+which depends weakly on the wavelength, we can approximate the index of
+refraction as:
+\begin{equation}
+n(\lambda) \simeq a + \frac{b}{\lambda^2}.
+\end{equation}
+
+The integral in Equation~\ref{eq:lambda} may then be solved analytically
+\begin{multline}
+\int_{\lambda_1}^{\lambda_2} \diff \lambda \frac{1}{\lambda^2}\left(-\log\left(\frac{\Delta^2(\lambda)}{4\theta_0^2}\right) + \log(2) - \gamma\right) =
+\left[\log(4\theta_0^2) + \log(\sin^2\theta_1) + \log(2) - \gamma\right]\left(\frac{1}{\lambda_1}-\frac{1}{\lambda_2}\right) + \\
+\left.\left(-4\sqrt{\frac{a}{b}}\arctan\left(\sqrt{\frac{a}{b}}\lambda\right) +
+4\sqrt{\frac{1+a\Delta_0}{b\Delta_0}}\arctan\left(\sqrt{\frac{1+a\Delta_0}{b\Delta_0}}\lambda\right) -
+\frac{1}{\lambda}\log\left[\left(\Delta_0+\frac{\lambda^2}{b+a\lambda^2}\right)^2\right]\right)\right|_{\lambda_1}^{\lambda_2}
+\end{multline}
+where $\lambda_1$ and $\lambda_2$ are chosen to cover the range where the
+quantum efficiency is non-zero, typically between 300 nm and 600 nm.
+
+For simplicity we will write this previous expression as $f(x)$
+\begin{multline}
+f(x) = \left[\log(4\theta_0^2) + \log(\sin^2\theta_1) + \log(2) - \gamma\right]\left(\frac{1}{\lambda_1}-\frac{1}{\lambda_2}\right) + \\
+\left.\left(-4\sqrt{\frac{a}{b}}\arctan\left(\sqrt{\frac{a}{b}}\lambda\right) +
+4\sqrt{\frac{1+a\Delta_0}{b\Delta_0}}\arctan\left(\sqrt{\frac{1+a\Delta_0}{b\Delta_0}}\lambda\right) -
+\frac{1}{\lambda}\log\left[\left(\Delta_0+\frac{\lambda^2}{b+a\lambda^2}\right)^2\right]\right)\right|_{\lambda_1}^{\lambda_2}
+\end{multline}
+
+We can now write Equation~\ref{eq:mui-approx} as
+\begin{equation}
+\mu_i = 2 \frac{1}{\sqrt{2\pi}\theta_0} 2\pi\alpha z^2 \int_x \diff x \Omega(x) \frac{1}{\left|\sin\theta_1\right|} \epsilon(\eta) P(\text{not scattered or absorbed} | \lambda_0, E, x, v) \mathrm{QE}(\lambda_0) e^{-\frac{\Delta^2(\lambda_0)}{4\theta_0^2}} \left(1 - \frac{1}{\beta^2 n^2(\lambda_0)}\right) f(x)
+\end{equation}
+
+The probability that a photon travels to the PMT without being scattered or absorbed can be calculated as follows
+\begin{align}
+P(\text{not scattered or absorbed} | \lambda, x) &=
+P(\text{not scattered} | \lambda, x) P(\text{not absorbed} | \lambda, x) \\
+&= \int_l^\infty\frac{1}{s(\lambda)}e^{-\frac{x}{s(\lambda)}}\int_l^\infty\frac{1}{a(\lambda)}e^{-\frac{x}{a(\lambda)}} \\
+&= e^{-\frac{l}{s(\lambda) + a(\lambda)}}
+\end{align}
+where $l$ is the distance to the PMT from the position $x$, $s(\lambda)$ is the
+scattering length, and $a(\lambda)$ is the absorption length.
+
+We can therefore write the expected charge as
+\begin{equation}
+\mu_i = 2 \frac{1}{\sqrt{2\pi}\theta_0} 2\pi\alpha z^2 \int_x \diff x \Omega(x) \frac{1}{\left|\sin\theta_1\right|} \epsilon(\eta) e^{-\frac{l(x)}{s(\lambda) + a(\lambda)}} \mathrm{QE}(\lambda_0) e^{-\frac{\Delta^2(\lambda_0)}{4\theta_0^2}} \left(1 - \frac{1}{\beta^2 n^2(\lambda_0)}\right) f(x)
+\end{equation}
+
+The last integral is calculated numerically when the likelihood is evaluated.
+
+We now return to the likelihood and calculate the probability of observing a
+given time. In principle, this depends on the number of photons hitting a PMT
+since the PMT hit will only register the \emph{first} photoelectron which
+crosses threshold. However, since this is expected to be a small effect, we
+assume that the probability of observing a given time is independent of the
+number of photons which hit the PMT, i.e.
+\begin{equation}
+P(t_j | n, E, \vec{x}, \vec{v}, t_0) \simeq P(t_j | n \geq 1, E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+
+We first condition on the \emph{true} time at which the photon hits the PMT
+\begin{equation}
+P(t_j | n \geq 1, E, \vec{x}, \vec{v}, t_0) = \int_{t_j'} \diff t P(t_j | t_j') P(t_j' | n \geq 1, E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+where we used the fact that the probability of a measured time only depends on the true PMT hit time.
+
+Now, we integrate over the track
+\begin{equation}
+P(t_j | n \geq 1, E, \vec{x}, \vec{v}, t_0) = \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x P(t_j', x | n \geq 1, E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+where $x$ here stands for the event that a photon emitted at a distance $x$ along the track makes it to the PMT.
+
+We now use Bayes theorem to rewrite the last probability
+\begin{align}
+P(t_j | n \geq 1, E, \vec{x}, \vec{v}, t_0) &= \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x P(t_j', x | n \geq 1, E, \vec{x}, \vec{v}, t_0) \\
+&= \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x P(t_j' | x, n \geq 1, E, \vec{x}, \vec{v}, t_0) P(x | n \geq 1, E, \vec{x}, \vec{v}, t_0) \\
+\end{align}
+The first term in the integral is just a delta function (up to slight differences due to dispersion) since we are assuming direct light
+\begin{align}
+P(t_j | n \geq 1, E, \vec{x}, \vec{v}, t_0)
+&= \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x \delta\left(\frac{l(x)n(\lambda_0)}{c}-t_j'\right) P(x | n \geq 1, E, \vec{x}, \vec{v}, t_0) \\
+\end{align}
+
+We now use Bayes theorem to rewrite the last term
+\begin{align}
+P(t_j | n \geq 1, E, \vec{x}, \vec{v}, t_0)
+&= \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x \delta\left(\frac{l(x)n(\lambda_0)}{c}-t_j'\right) \frac{P(n \geq 1 | x, E, \vec{x}, \vec{v}, t_0) P(x | E, \vec{x}, \vec{v}, t_0)}{P(n \geq 1 | E, \vec{x}, \vec{v}, t_0)} \\
+&= \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x \delta\left(\frac{l(x)n(\lambda_0)}{c}-t_j'\right) \frac{P(x | E, \vec{x}, \vec{v}, t_0)}{P(n \geq 1 | E, \vec{x}, \vec{v}, t_0)} \\
+&= \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x \delta\left(\frac{l(x)n(\lambda_0)}{c}-t_j'\right) \frac{P(x | E, \vec{x}, \vec{v}, t_0)}{1 - e^{-\mu_j}} \\
+&= \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x \delta\left(\frac{l(x)n(\lambda_0)}{c}-t_j'\right) \frac{\mu_j(x)}{1 - e^{-\mu_j}} \\
+&= \frac{1}{1 - e^{-\mu_j}} \int_x \diff x \mu_j(x) \int_{t_j'} \diff t_j' P(t_j | t_j') \delta\left(\frac{l(x)n(\lambda_0)}{c}-t_j'\right) \\
+\end{align}
+
+We assume the transit time spread is equal to a gaussian (we ignore the pre and late pulsing)
+\begin{align}
+P(t_j | n \geq 1, E, \vec{x}, \vec{v}, t_0)
+&= \frac{1}{1 - e^{-\mu_j}} \int_x \diff x \mu_j(x) \int_{t_j'} \diff t_j' \frac{1}{\sqrt{2\pi}\sigma_t} e^{-\frac{(t_j-t_j')^2}{2\sigma_t^2}} \delta\left(\frac{l(x)n(\lambda_0)}{c}-t_j'\right) \\
+&= \frac{1}{1 - e^{-\mu_j}} \frac{1}{\sqrt{2\pi}\sigma_t} \int_x \diff x \mu_j(x) e^{-\frac{(t_j-t_0(x))^2}{2\sigma_t^2}}
+\end{align}
+where in the last expression we define
+\begin{equation}
+t_0(x) \equiv \frac{l(x)n(\lambda_0)}{c}
+\end{equation}
+
+\begin{thebibliography}{9}
+\bibitem{grossman2017}
+    Grossman, et al. \textit{Self-Destructing Dark Matter}. \href{https://arxiv.org/abs/1712.00455}{{\tt arXiv:1712.00455}}. Dec 2017.
+\bibitem{pepin2016}
+    M. Pepin. \textit{Low-Mass Dark Matter Search Results and Radiogenic Backgrounds for the Cryogenic Dark Matter Search}. \url{http://hdl.handle.net/11299/185144}. Dec 2016.
+\bibitem{caldwell2015}
+    T. Caldwell. \textit{Searching for Dark Matter with Single Phase Liquid Argon}. \url{https://repository.upenn.edu/dissertations/1632}. 2015.
+\bibitem{pdg2017}
+    C. Patrignani et al. (Particle Data Group), Chin. Phys. C, 40, 100001 (2016) and 2017 update.
+\end{thebibliography}
+\end{document}