Studies in Dark Matter, Dark Energy, and Gravitational Wave Sources

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Opening New Windows Onto the Universe: Studies in Dark Matter, Dark Energy, and Gravitational Wave Sources

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By Matthew C. Digman, M.S. Graduate Program in Physics

The Ohio State University 2020

Dissertation Committee: Christopher M. Hirata, Advisor

John F. Beacom

Eric Braaten

Klaus Honscheid c Copyright by

Matthew C. Digman

2020 Abstract

The past two centuries have seen tremendous expansion in human understanding of the structure and contents of the universe. Yet, less than 5% of the mass of the universe is contained within particles predicted and explained by the modern Standard Model of particle physics. An even smaller fraction, less than 0.5%, of the mass of the universe is in familiar, tangible objects like planets and stars. With so much of the universe yet obscure to use, we must develop new techniques to observe what our eyes (and telescopes) cannot see directly. In this thesis, I present three papers I have written exploring new windows for observing the hidden aspects of the universe. In the ﬁrst, presented in Chapter2,I explore theoretical constraints on models of strongly interacting dark matter. In the second, presented in Chapter3, I describe a formalism and code I developed to forecast the impact of long-wavelength density ﬂuctuations on future weak lensing surveys. In the third, presented in Chapter4, I present a forecast for the number of white dwarf binaries that will be detected by both LISA and the Nancy Grace Roman Space Telescope.

ii Acknowledgments

I would like to thank the many people who have assisted me on the journey to my PhD. My high school physics teacher, Marshall Pauli, who helped to spark my interest in physics. My professors in the introductory physics curriculum at Dartmouth who helped motivate me to pursue physics as a major, including Marcello Gleiser, Lorenza Viola, and Robyn Millan. My undergraduate thesis advisor, Robert Caldwell, who ushered me into the field of cosmology, gravitational waves, and theoretical physics generally. My time at The Ohio State University and the Center for Cosmology and AstroParticle Physics has been a valuable period of growth for me as an individual and as a scientist. My advisor, Chris Hirata, has been a great influence and has helped me to grow immensely as a researcher and scientist. I will miss our conversations. Joe McEwen was a positive influence as a first year mentor and helped me get started in CCAPP. The older members of our research group, including Xiao Fang, Paulo Montero, and Daniel Martens were positive and supportive role models and sparked many valuable conversations. My past and current office mates, including Su-Jeong Lee, Benjamin Buck- man, Bianca Davis, Jenna Freudenberg, and Makana Silva have been the source of many fruitful and entertaining conversations. Chris Cappiello’s close collaboration was an invaluable experience on our paper together. I have had many valuable conversations with the other graduate students in CCAPP, including Jahmour Givans, Gabriel Vasquez, Pawan Dhakal, Bei Zhou, Brian Clark, Jorge Torres Espinosa, and Kevin Fanning. The postdocs who passed through CCAPP during my time here have also been invaluable for conversation and mentorship, including Ashley Ross, Ami Choi, Niall MacCrann, Heidi Wu, Tim Linden, Jack Elvin-Poole, Johnny Greco, Katie Auchettl, and Anna Nierenberg. I am also grateful to my friends in physics at OSU outside of CCAPP, including Bryan Cardwell, Derek Everett, Daniel Heligman, Alexander Klepinger, Estefany Nunez, Andrew Piper, Bryan Reynolds, Srividya Suresh. Finally, thank you to Annika Peter and John Beacom who provided invaluable guidance and education both through our paper together and my time at CCAPP generally. I’d like to thank my family for their invaluable support and inspiration throughout my education, career, and life, including my father Karl, my mother Laura, and my siblings

iii Finn and Emily. I am also grateful to Emily Burland, who has been an invaluable source of support during my PhD. To everyone I mentioned above and others I surely forgot, you have helped me greatly through my PhD at OSU. This work would not have been possible without all of you.

iv Vita

2015 ...... B.A., Dartmouth College

2017 ...... M.S., The Ohio State University

Publications

Forecasting Super-Sample Covariance in Future Weak Lensing Surveys with SuperSCRAM Digman, M. C., McEwen, J. E., Hirata, C. M. 2019, JCAP 10, 004 [arXiv:1904.12071]

Not as Big as a Barn: Upper Bounds on Dark Matter-Nucleus Cross Sections Digman, M. C., Cappiello, C. V., Beacom, J. F., Hirata, C. M., Peter, A.G.H 2019, PRD 100, 063013 [arXiv:1907.10618]

LISA Galactic Sources in the WFIRST Microlensing Survey Digman, M. C., Hirata, C. M. 2019, In Prep

COSMIC Variance in Binary Population Synthesis Breivik, K., Coughlin, S. C., Zevin, M., Rodriguez, C. L., Kremer, K., Ye, C. S., Andrews, J. J., Kurkowski, M., Digman, M. C., Larson, S. L., Rasio, F. A. 2019, ApJ, Accepted [arXiv:1911.00903]

Fields of Study

Major Field: Physics

v Table of Contents

Page Abstract...... ii Acknowledgments...... iii Vita...... v List of Figures ...... ix List of Tables ...... xiii

Chapters

1 Introduction1 1.1 Preamble ...... 1 1.2 Dark Matter...... 1 1.3 Dark Energy ...... 2 1.4 Gravitational-Wave Detectors...... 4 1.4.1 Background...... 4 1.4.2 Detectors ...... 4 1.5 Gravitational-Wave Sources...... 9 1.6 Structure of Thesis...... 10

2 Not as Big as a Barn: Upper Bounds on Dark Matter-Nucleus Cross Sections 11 2.1 Introduction...... 12 2.2 Dark Matter Scattering Theory...... 14 2.2.1 Overview of basic assumptions ...... 15 2.2.2 Basic scattering theory...... 16 2.2.3 Derivation of ‘model-independent’ scaling ...... 17 2.3 Contact interactions ...... 20 2.3.1 Contact interaction with Born approximation...... 20 2.3.2 Contact interaction with partial waves...... 21 2.3.3 Attractive resonances ...... 23 2.3.4 Beyond contact interactions...... 24 2.4 Light Mediator ...... 25 2.4.1 Momentum-transfer cross section...... 26 2.4.2 Existing limits on light mediators...... 26 2.5 Composite Dark Matter ...... 27 2.6 Implications for Existing Constraints...... 27

vi 2.6.1 Scaling constraints...... 28 2.6.2 Detection ceilings...... 28 2.6.3 Dark matter-proton scattering constraints...... 29 2.6.4 Low-mass dark matter...... 29 2.7 Conclusions...... 30

3 Forecasting Super-Sample Covariance in Future Weak Lensing Surveys with SuperSCRAM 38 3.1 Introduction...... 39 3.2 Overview ...... 41 3.2.1 Notation and Deﬁnitions...... 41 3.2.2 Workﬂow ...... 41 3.2.3 Eigenvalue analysis...... 44 3.3 Methods and Code Organization ...... 45 3.3.1 Geometries ...... 45 3.3.2 Short-Wavelength Observables ...... 47 3.3.3 Long-Wavelength Observables...... 47 3.3.4 Cosmological Parametrizations ...... 49 3.3.5 Matter Power Spectrum...... 49 3.3.6 Fisher Matrix Manipulation...... 49 3.3.7 Long-Wavelength Basis ...... 51 3.4 Applications...... 51 3.4.1 Example Application: WFIRST+LSST ...... 51 3.4.2 Sensitivity to Survey Geometry...... 53 3.4.3 Other Applications...... 54 3.5 Conclusions...... 57

4 LISA Galactic Binaries in the Roman Microlensing Survey 59 4.1 Introduction...... 59 4.2 The surveys...... 60 4.2.1 Roman microlensing survey...... 60 4.2.2 LISA...... 61 4.3 Methods...... 62 4.3.1 Binary population synthesis...... 62 4.3.2 Tidal heating...... 63 4.3.3 White dwarf atmospheres ...... 63 4.3.4 Light curve generation...... 64 4.3.5 Dust extinction...... 64 4.3.6 Roman Exposures ...... 65 4.3.7 Survey Strategy...... 66 4.3.8 Chi-squared calculation ...... 66 4.3.9 Selecting binaries...... 67 4.3.10 Trials factor...... 67 4.3.11 Calculating detection eﬃciency...... 68 4.4 Known systems...... 68 4.5 Results...... 69

vii 4.6 Discussion...... 73 4.7 Conclusion ...... 73

5 Conclusion 75

Bibliography 76

Appendices

A Not as Big as a Barn: Upper Bounds on Dark Matter-Nucleus Cross Sections 101 A.1 Lippmann-Schwinger Equation ...... 101 A.2 Born Approximation...... 102 A.3 Partial wave analysis...... 103

B Forecasting Super-Sample Covariance in Future Weak Lensing Surveys with SuperSCRAM 105 B.1 Basis Decomposition...... 105 B.1.1 Response of Observables to Density Fluctuations ...... 107 B.2 Projected Power Spectra...... 108 B.3 Analytic Polygon Geometry...... 109 B.4 Galaxy Number Density...... 112 B.5 Growth Factor ...... 113 B.6 Code Tests ...... 114 B.7 Modifying the Matter Power Spectrum...... 115 B.8 Separate Universe Response...... 116

viii List of Figures

Figure Page

1.1 Example sensitivity curve [20] comparing the characteristic limiting strain sensitivities of advanced LIGO, LISA, and the International Pulsar Timing Array (IPTA) as a function of frequency, along with example sources for each detector...... 5 1.2 Schematic of Advanced LIGO’s optical layout from the Advanced LIGO Ref- erence Design [28]. This schematic does not include the squeezer, any of the optics in the pre-stabilized laser, or any auxiliary lasers...... 7

2.1 Claimed constraints on the spin-independent dark matter-nucleon cross section [86, 89–92]. Those from cosmology directly probe scattering with protons, but all others here are based on scattering with nuclei, and thus require the use of ‘model-independent’ scaling relations. Below, we show that assumptions used to derive these results are invalid over most of the plane...... 13 2.2 Top: Scaling with A for the contact interaction in Sec. 2.3 with |V0| = −5 −1 1/3 1.18 × 10 GeV, computed using k = 0.005A fm , RA = 1.2A fm. We include partial waves up to lmax = 8, which is suﬃcient to converge σχA to ∼ 10−16 precision. Attractive and repulsive interactions scale similarly, although the scaling deviates from A4 at high A due to form-factor suppression, accounted for here by including the contributions from higher partial waves. −3 Bottom: Same as above, but with |V0| = 1.18×10 GeV, which corresponds to the ‘scaling relations unreliable for A > 12’ line in Fig. 2.6. Repulsive and attractive interactions no longer scale the same way, and both saturate close 2 to 4πRA. The attractive potential shows resonances with A, which are sensitive to the speciﬁc choice of potential. For cross sections approaching the geometric cross section, any scaling with A is highly model-dependent. . . . . 32

ix 2.3 Top: Scaling of cross section with |V0| for A = 4 (helium), calculated using the contact interaction in Sec. 2.3. The cross sections are computed using the analytic partial wave results. For attractive potentials, once |V0| becomes large enough to support quasi-bound states, resonances can increase the cross section by several orders of magnitude, but only in a narrow range. Bottom: Same as above, but with A = 131 (xenon). A larger number of partial waves contribute due to the larger k ∝ A. There are many resonances, but they are not large enough to meaningfully increase the cross section above the geometric limit. Additionally, the resonances are not at the same values of |V0|, which prevents resonances from achieving a large cross section which scales predictably with A, as shown in Fig. 2.2...... 33 2.4 Scaling of the nuclear cross section with nucleon cross section for the repulsive −1 contact interaction of Sec. 2.3 at fixed kN = 0.005 fm . The contact interaction cannot achieve nucleon cross sections larger than the geometric cross section, denoted by the vertical red line. The cross section visibly deviates 4 −32 2 from A scaling at the O(1) level for heavy nuclei even for σχN ' 10 cm , 4 −28 2 and by the time scaling fails at the O(1) level for He at σχN ' 4×10 cm , the cross sections for heavy nuclei have completely saturated. The scaling could break down in different ways in other models...... 34 2.5 Top: Elastic scattering cross section contours as a function of mediator mass and coupling strength for the repulsive Yukawa potential in Eq. (2.33). We also show various constraints on the existence of such mediators from Ref. [134]. The largest cross sections achieved in unconstrained regions are −27 2 −9 σχN . 10 cm . For mφ < 10 GeV, fifth-force constraints become many orders of magnitude stronger and dominate other constraints [135]. Bot- tom: Same as above, but for the momentum-transfer cross section. The mt −32 2 largest cross sections achieved in unconstrained regions are σχN . 10 cm . 35 2.6 Summary of theoretically allowed regions for dark matter candidates. For 4 a contact interaction, A scaling breaks down for heavy nuclei for σχN & −32 2 −28 10 cm , and by σχN & 4 × 10 any scaling between different nuclei is model dependent. Here we define the failure of scaling as setting the LHS of Eq. (2.11) equal to 0.5. This choice approximately agrees with where scaling obviously fails in Fig. 2.4. The breakdown is purely on theoretical grounds. Also shown is the maximum allowed momentum-transfer cross section for −4 a mφ = 10 GeV light mediator using the constraints shown in Fig. 2.5, 4 coincidentally at a comparable scale. For mχ . 10 GeV we have applied 2 a conservative self interaction constraint σχχ/mχ < 10 cm /g [155]. For −25 2 σχN & 10 cm , no viable point-like dark matter candidates exist...... 36 2.7 Claimed constraints from Fig. 2.1, with the problematic regions identified in Fig. 2.6 highlighted. All existing detector ceiling calculations are deeply in the model-dependent regime, or entirely excluded for point-like dark matter. To the right of the dashed vertical line, the entire (small) direct-detection region must be reanalyzed...... 37

x 3.1 Conceptual overview of the basic workflow needed to obtain cosmological parameter forecasts. The specific modules implementing the structure are described in section 3.3 and shown in figure 3.2...... 42 3.2 Structure of the most important modules used by SuperSCRAM described in this paper to implement the workflow described in §3.1...... 46 3.3 Impact of super-sample covariance and mitigation on cosmological parameter constraints. Constraints in this plot use ∆χ2 = 2.3, with a set of forecast Planck constraints from the WMAP5 fiducial point as cosmological priors included [199]. Although it is apparent that super-sample covariance has an effect, the appearance of this plot is very sensitive to the choice of parametrization and the priors applied, as can be seen by comparison to figure 3.4. There- fore, it is not possible from this plot alone to accurately assess the magnitude of the super-sample effect. Figure 3.5 is more useful for isolating the effect of super-sample covariance...... 54 3.4 The same run as figure 3.3, but with an additional 1% prior applied to h. The apparent effect of super-sample covariance on cosmological parameter space 2 is now significantly different. Ωdeh is now well enough constrained that the super-sample effect is hardly noticeable, while w0 and wa now appear significantly contaminated by super-sample covariance, where they appeared much less contaminated previously. Because it is impossible to know at this stage the exact set of priors that an experiment like WFIRST will apply, this plot and figure 3.3 are both plausible forecasts. Because the conclusions about the impact of super-sample covariance drawn from such plots can vary significantly within the range of reasonable analysis choices, we recommend instead that the super-sample effect be assessed directly using our formalism, as in Table 3.1 and figure 3.5...... 55 3.5 The covariances CeSSC,g, CeSSC,SSC, CeSSC,mit, as defined in eq. (3.8) for the two directions in cosmological parameter space most affected by super-sample covariance. The Gaussian 1 − σ contour is a circle by construction, and the unmitigated super-sample component is diagonal, such that the axes of the ‘no mit’ ellipse align with the axes of the plot. The rotation of the ‘mit’ ellipse due to mitigation shows that the most contaminated direction has changed somewhat after mitigation, which is shown in Table 3.1. Nearly all of the super-sample effect is concentrated to these 2 combinations of parameters. Note that the magnitude of the contamination is not apparent from figure 3.3. Additionally, this plot is not drastically affected by changes to the parametrization, such as fixing wa = 0, while such changes affect figure 3.3 significantly...... 56

xi 3.6 Three example geometries tested, with results from a run of SuperSCRAM with each geometry. The black region is the LSST-like survey window, with white cutouts showing three different possible geometries for the WFIRST- like survey with identical angular areas 2098.2 deg2. All parameters other than the angular window are the same as the run summarized in Table 3.1. Without mitigation, the results distinctly favor more compact survey geometries. Mitigation may be more effective for extended geometries, such that the overall constraint with mitigation is slightly better for the "Strip" geometry, although it should be noted that our mitigation strategy is primarily for demonstration purposes and cannot be expected to precisely reflect the actual performance of a realistic mitigation strategy. The more significant result is that, provided WFIRST is completely embedded in an LSST-like survey window, the specific shape of the WFIRST survey window is largely a secondary consideration for the purposes of mitigating super-sample covariance. 57

4.1 Roman detection probability versus period for our fiducial run with temperatures from cosmic, with LISA S/N shown in color. The higher mass of the He-CO binaries gives them higher LISA S/N at fixed period, while the electromagnetic luminosity of the CO white dwarfs is generally smaller than He white dwarfs, resulting in the apparent inverse correlation between LISA S/N and Roman detection efficiency at fixed period. The small population of binaries separated from the overall trend in the top middle is a population of systems with very young (age< 5 Myr), hot secondaries...... 70 4.2 Roman detection probability versus period for our fiducial run with tides enhanced to match the temperatures of the J0651 system, with LISA S/N shown in color. For a given period, approximately 95% of He-He binaries fall above the dashed red line. In this model, all He-He binaries with Pf . 680 s have a > 0.3% chance of being detected in a single season of the nominal Roman microlensing survey. For He-CO binaries, all systems with Pf . 475 s have a > 0.3% chance of being detectable by Roman...... 71 4.3 Roman detection probability versus period for our fiducial run with tides enhanced to match the temperatures of the J0651 system, with LISA S/N shown in color. For a given period, approximately 95% of He-He binaries fall above the dashed red line. In this model, all He-He binaries with Pf . 950 s have a > 0.3% chance of being detected in a single season of the nominal Roman microlensing survey. For He-CO binaries, all systems with Pf . 475 s have a > 0.3% chance of being detectable by Roman...... 72

B.1 Applying Euler rotations to rotate an arc of angle β, highlighted in red, to a coordinate system where the side is along the equator, highlighted in yellow. 0 The contribution ∆alm,n is simple to calculate along the equator. Then the contribution to the geometry on the sky ∆alm,n may be obtained by applying 0 the rotations to the matrix ∆alm,n...... 111

xii List of Tables

Table Page

3.1 Results from a well converged run of SuperSCRAM with our example −1 WFIRST and LSST survey footprints. kcut = 0.0814 Mpc and zmax = 3, as described in Appendix B.1. This choice of kcut requires 10, 286, 527 basis modes, approximately the maximum number of basis modes for Super- SCRAM running with 1 TB of available RAM. For the short-wavelength observables, we use 20 logarithmically-spaced l bins 30 ≤ l < 5000. The product of eigenvalues, eigenvalue in most contaminated direction, and the coeﬃcients describing the most contaminated directions are as described in (i) subsection 3.2.3. The convergence of Π(i)λ with respect to kcut is . O(1%). The mitigation reduces the volume of parameter space contaminated by SSC by approximately a factor of 5. The most contaminated direction does not change drastically after mitigation...... 53

4.1 Expectation values for number of binaries detected with Porb < 3000 s, computed based on 20 Monte Carlo realizations of the 7 microlensing ﬁelds. Quoted 1 − σ errors are purely statistical uncertainties on the expectation values, and are far smaller than the inherent systematic uncertainties in the formation, evolution, and modeling of these systems...... 69

xiii Chapter 1 Introduction

1.1 Preamble

The Standard Model of particle physics explains less than 5% of the mass of the universe, and less than 0.5% is in the familiar stars in the night sky. In this thesis, I present three papers I have written exploring new windows for observing the aspects of the universe that we cannot yet see directly. In the first paper, presented in Chapter2, I explore previously under-appreciated constraints affecting many models of strongly interacting dark matter. In the second, Chapter3, I describe superSCRAM, a formalism and code I developed to forecast and guide mitigation of the impact of long-wavelength density fluctuations on future weak lensing surveys. In the third, presented in Chapter4, I provide forecasts of the number of white dwarf multi-messenger science targets obtained from combining data from LISA with data from the Nancy Grace Roman Space Telescope.

1.2 Dark Matter

For nearly a century, there has been compelling observational evidence that baryonic matter, the portion of the universe directly observable with electromagnetic telescopes, presents a highly incomplete accounting of the full contents of the universe. In 1933, Fritz Zwicky first postulated the existence of an unseen ’dark matter’ to explain the Coma cluster’s unexpected rotational velocity profile [1]. After decades of further observations and theoretical study, it has become generally (though not universally [2]) accepted that dark matter exists and makes up the bulk of the matter content of the universe. In addition to dynamical mass estimates such as rotational velocities [3,4], there exist multiple different types of evidence that the universe contains an unidentified matter component. Observations of gravitational lensing by galaxies show consistent agreement with dynamical mass estimates, indicating that galaxies contain substantially more mass than accounted for by visible stars and gas [5].

1 Images the Bullet cluster, formed from a high-speed collision between two smaller galaxy clusters, are often regarded as some of the most direct evidence for dark matter [6]. While the hot gas in both clusters was slowed by drag during the collision, the bulk of the matter of the cluster, as determined by gravitational lensing, continued apparently unimpeded by drag forces, suggesting that the majority of the matter in the cluster is an unknown weakly interacting substance. Observations of baryon acoustic oscillations combined with observations of the cosmic microwave background [7] provide good measurements of the fraction of baryonic matter in the universe, and, consistent with all other observations, show that known baryonic forms of matter make up a small fraction of the matter in the universe. Overall, the evidence for the existence of dark matter is excellent. Any alternative theory explaining the constellation of observations in favor of dark matter would almost certainly require a much more fundamental shift in our understanding of the physics of the universe than the existence of an as-yet-undiscovered particle [8–11]. However, its physical nature remains a mystery. Because it is abundant, it must have been produced in considerable quantity in the early universe, yet not interact with ordinary matter in ways that could disrupt big bang nucleosynthesis [12]. The so-called ’WIMP Miracle’ provides an explanation that could produce the correct abundances with a particle whose couplings could be weak enough that it would remain undiscovered. The search for WIMPs has motivated the construction of the largest dark matter direct detection experiments yet built, which search for scatterings of dark matter in the Milky Way halo with ordinary matter in human-made detectors. Such detectors have not yet entirely ruled out the miraculous WIMP scenario [13]. However, as such detectors become larger and more expensive, it becomes valuable to consider whether alternative dark matter models not explored by current direct detection technology could exist. If so, inexpensive smaller searches or tests using preexisting data from other experiments not designed for dark matter detection may provide valuable constraints. Some non-WIMP models of dark matter, such as axion-like particles, will be naturally tested by future gravitational-wave observatories [14]. Others can be tested by examining cosmological observables, or careful examination of large scale and galactic structure. Data provided by future survey telescopes missions intended to study dark energy will provide valuable constraints on such models. Some models of composite dark matter can be tested by examining terrestrial, lunar, or martian geology.

1.3 Dark Energy

By the 1990s, it appeared to many in the astrophysics community that the particle nature of dark matter was one of the last major mysteries in our understanding of the cosmology of

2 the universe. In 1970, it had famously been remarked that cosmology had become "a search for two numbers" [15] characterizing the expansion of the universe: the Hubble constant,

H0, and the deceleration parameter, q0, and nothing had yet come to disrupt this simple

picture. However, in 1998, it was discovered by two independent groups that q0, the so-called ’deceleration parameter’, was negative, at ∼ 3.9σ confidence [16, 17]. The discovery that the expansion of the universe is accelerating reinvigorated the field of cosmology. It presented compelling evidence that a second mysterious component, called dark energy, must make up an even larger fraction of the universe than dark matter. Einstein’s equations for general relativity allow a constant energy density to be added to all the other matter contents of the universe, called a cosmological constant. A properly tuned cosmological constant is the simplest way to reproduce the observed accelerating expansion rate of the universe. Consequently, the ΛCDM model, in which the evolution of the universe is dominated by a cosmological constant and cold dark matter, presents a good fit to most observational data and has become the default model in cosmological analyses. There are several tensions between various observables of this model. Investigating and resolving such tensions is a crucial driver of the field of precision cosmology. However, no alternative theoretical model yet provides strong evidence of a better fit to observational data. If the default explanation of a cosmological constant is correct, the physical principles underlying the constant’s numerical value is a tantalizing mystery. On the largest observable scales, our universe has no curvature (k = 0) within current experimental limits. Achieving a flat universe with k = 0 which requires that the energy densities produced by dark energy,

ΩΛ, dark matter ΩCDM, and ordinary baryonic matter Ωb add ΩΛ + ΩCDM + Ωb ' 1 to within ∼ 0.2% [7]. Achieving such agreement appears to require some fine-tuning, which the theory of inflation seeks to explain. Even if such agreement can be achieved, the value of the cosmological constant still appears to be extremely fine-tuned. The zero-point energy of the vacuum would seem a logical energy scale for the cosmological constant. However, it famously overpredicts the observed cosmological constant by a factor of 10122 [18]. Consequently, the idea that the zero-point energy of the vacuum should set the scale for the cosmological constant has been referred to as "the worst prediction in the history of science." If the two energy scales are, in fact, related, understanding the mechanism that causes them to differ so dramatically could be critical to a unified understanding of fundamental physics. If dark energy is not a cosmological constant, understanding what it is and how it arises is a promising avenue for discovering new physics beyond the Standard Model of particle physics and general relativity. In the absence of a robust theoretical basis for a particular physical model of dark energy, we must examine its imprint on the growth and structure of the universe. Current and planned survey missions observe galaxies and structures at

3 cosmological distances to constrain models of dark energy. Correctly translating observed survey data to constraints on models of the structure of the universe requires a proper understanding of the sources of error that impact the accuracy and constraining power of a cosmological survey. Regardless of its physical nature, the origin of the dark energy is an important theoretical question and a promising avenue for groundbreaking fundamental physics discoveries.

1.4 Gravitational-Wave Detectors

1.4.1 Background

In the weak ﬁeld limit, the metric tensor, which describes the structure of spacetime, can be decomposed

gµν = ηµν + hµν, (1.1)

where η = diag(−1, 1, 1, 1) is the Minkowski metric and a perturbation |hµν| 1 is induced 2 µ ν by a passing gravitational wave. The metric gives the line element ds = gµνdx dx which is used to measure the proper length between masses in free fall. Assuming the transverse traceless gauge, the distance between two test masses separated by a coordinate distance ∆x along the x axis with a gravitational wave perturbation propagating along the z axis is given [19]: Z ∆x Z ∆x q √ TT L = gxxdx = 1 + hxx (t, z = 0)dx, (1.2) 0 0 such that the fractional change in length of the arm over time is given 1 δL(t)/L ' hTT (t, z = 0). (1.3) 2 xx The amplitude of the fractional change in length h = |δL(t)/L| is the gravitational- wave strain. The characteristic strain amplitude is the primary observable for gravitational- wave detectors. Modern interferometric detectors use laser interferometry to observe the variations in the relative length of detector arms due to gravitational-wave strain. The minimum strain a detector is sensitive to as a function of frequency is often used to make plots comparing the characteristic sensitivities of diﬀerent detectors. An example of such a sensitivity curve is shown in Figure 1.1.

1.4.2 Detectors

The existence of gravitational waves within the framework of general relativity was proposed by Einstein in 1916 [21], shortly after the original development of the framework of general relativity [22].

1An interactive version of the plot is available at http://gwplotter.com/ and described in [20].

4 Figure 1.1: Example sensitivity curve [20]1 comparing the characteristic limiting strain sensitivities of aLIGO, LISA, and the International Pulsar Timing Array (IPTA) as a function of frequency, along with example sources for each detector.

Measurements of the decaying orbital period of the binary pulsar PSR 1913+16 [23] in 1982 closely matched the general relativity prediction for the orbital period derivative due to gravitational radiation, providing compelling but indirect evidence for the existence of gravitational waves. In addition, the detection of PSR 1913+16 was direct proof that systems capable of radiating any appreciable amount of gravitational waves can form in nature, which was not a settled question at the time. Proof of the existence of such systems helped pave the way for the substantial decades-long investments in gravitational wave detection technology leading up to the ﬁrst detection of gravitational waves by LIGO in 2015 [24]. Joseph Weber pioneered attempts to use aluminum bars as resonant-mass detectors to detect gravitational waves directly. However, his early eﬀorts were marred by repeated false claims of gravitational-wave detections at amplitudes that were not considered theoretically

5 plausible [25]. Laser interferometers have since eclipsed the capabilities of resonant-mass detectors. It is conceivable that variations on the resonant-mass concept may yet find practical gravitational-wave applications, such as detection of faint continuous-wave kHz gravitational-wave emission from known rotating neutron stars [26]. However, resonant-mass detectors are not currently the primary detection method of major planned gravitational- wave observatories. Laser interferometers are the state of the art gravitational-wave detectors. Develop- ing laser systems with noise and stability properties sufficient for gravitational-wave detection required significant advances in laser and control technology, resulting in technological advances which have spread to fields far removed from astrophysics. For example, the Pound-Drever-Hall technique [27], a widely used laser frequency stabilization technique, was originally developed for gravitational-wave detectors. Currently-operational ground-based detectors such as Advanced LIGO and Virgo are highly advanced variations on a traditional Michelson interferometer. A Michelson interferometer operates by splitting a beam of light down perpendicular paths at a beam splitter, sending it down perpendicular paths to be reflected by a distant mirror. The two beams are then recombined at the output port of the interferometer. Any changes in the relative length of the two arms will result in a changing interference pattern at the output port. Current generation detectors then use controls to position the mirrors such that the output photodiode is kept slightly offset from the center of a dark fringe of the interference pattern, minimizing the noise introduced by fluctuations in the input laser power. When gravitational-waves change the relative length of the arms, the output light intensity oscil- lates in time as the interference pattern changes. The first modification to the basic Michelson design was turning each arm into Fabry-Perot cavities, amplifying the circulating laser power in the arms and signal-to-noise of gravitational-wave proportionally. Additionally, in the initial LIGO design, a power- recycling mirror was added to prevent reflected laser power from being wasted. Advanced LIGO incorporates an additional signal recycling mirror at the output port and injects quantum states of squeezed vacuum into the detector, allowing readout at signal-to-noise beyond the classical shot-noise limit. In addition to these core elements of the interferometer design, a series of input and output mode cleaners are critical to removing noise from higher-order electromagnetic modes in the laser. Advanced LIGO is a very complex instrument. A schematic of the primary optical elements is shown in Fig. 1.2. For each interferometer, locking just the main cavity in observing mode requires controlling over 250 degrees of freedom, and a data acquisition system capable of processing ∼ 10, 000 "fast" (512 Hz − 16 kHz) DAQ channels in real time [28]. All of this enables LIGO’s 1064 nm (2.818 × 1014 Hz) light in the cavity to be the most stable monochromatic light source ever constructed, with a linewidth (full width at

6 Figure 1.2: Schematic of Advanced LIGO’s optical layout from the Advanced LIGO Ref- erence Design [28]. This schematic does not include the squeezer, any of the optics in the pre-stabilized laser, or any auxiliary lasers.

half maximum) of only ∼ 1 Hz[29]. The technical sophistication of Advanced LIGO’s mirror assemblies and vibration isolation systems is necessary due to the noisy seismic, acoustic, gravitational, and electromagnetic environment on the surface of the earth. Such environmental constraints are greatly reduced in space, allowing a substantially less complex space-based interferometer to detect astrophysical gravitational-wave sources. Reduced technical complexity is also necessary for a space-based instrument, which must be light and compact enough to ﬁt in a launch vehicle yet robust enough to survive the substantial launch forces and vibration without damage to precise alignments. Space-based experiments must also be able to operate for years with no preventative or corrective maintenance possible. In contrast, Advanced LIGO’s complex systems require weekly preventative maintenance and frequent corrective maintenance. Future and proposed ground-based LIGO successors, such as A+, Voyager, Cosmic Ex-

7 plorer, and Einstein Telescope, will adopt various further advanced technologies, including frequency-dependent squeezing, cryogenic cooling, adding a third arm, and may even advance beyond the Michelson interferometer-based design concept. Ultimately, future sophisticated ground-based interferometers may reach a limiting astrophysical stochastic gravitational-wave background between 10 − 100 Hz. Ground-based detection technology is fundamentally limited at low frequencies by gravity-gradient noise induced by the movement of both seismic waves in the earth’s crust and the movement of objects on the earth’s surface, which induces motions in the test masses indistinguishable from those induced by gravitational waves [30]. Proposed space-based detectors take advantage of the much longer arm lengths practical in space to achieve sensitivity to lower-frequency astrophysical gravitational-wave sources. The Laser Interferometer Space Antenna (LISA), a gravitational-wave observatory scheduled to launch in 2034, will have ∼ 2.5 million km long arms, and proposed successor missions could have & 5 au arms [31]. The longer arms shifts the natural frequency the detectors are most sensitive to; for LIGO, the best sensitivity is achieved at ∼ 100 Hz, while LISA’s sensitivity will be best at ∼ 10 mHz. The lower-frequency sensitivity is the primary advantage of operating a gravitational-wave detector in space because it enables the detection of entirely different classes of sources. Additionally, sources in this frequency range evolve much more slowly than at ∼ 100 Hz, allowing some sources to be observed for the entire duration of the mission, accumulating high signal-to-noise and generally better source localizations than are achievable from the ground. Long observation windows and comparatively good localization make collaboration with electromagnetic observatories easier, facilitating powerful multi-messenger science applications. Reaching even lower gravitational-wave frequencies requires moving beyond the stability of human-made solar-system based reference sources and observing times beyond the en- durance of typical spacecraft missions. Pulsar timing arrays seek to achieve this through using ground-based radio telescopes to observe pulsars, rotating neutron stars that emit pulses with extremely stable timing. Gravitational waves passing through the solar system shift the relative phase of these pulses in different areas on the sky, potentially allowing sensitivity to gravitational waves with periods of years or even decades. Unlike laser interferometer-based detectors, the advances required to achieve better pulsar timing measurements of gravitational waves are not exclusively technological. Rather, they are a combination of improved modeling of both the pulsars themselves and the intergalactic medium, the discovery of more usable pulsars, and longer observation times. Such efforts are ongoing and may yield the first detection of low-frequency gravitational waves in the next few years.

8 1.5 Gravitational-Wave Sources

General relativity predicts that any non-spherically-symmetric acceleration of matter radi- ates gravitational waves, just as accelerating electric charges produce Bremsstrahlung radiation in electrodynamics [32]. However, gravity’s coupling to matter is extremely weak. Con- sequently, only extraordinarily energetic astrophysical events radiate any significant power in gravitational waves, and once emitted the travel through the universe virtually unimpeded by ordinary matter, requiring extremely sensitive detectors to observe. The most powerful gravitational-wave sources currently known are the inspirals and mergers of compact object binaries containing black holes, neutron stars, or white dwarfs. Other sources that may emit gravitational-waves detectable by manmade observatories include supernovae, rotating neutron stars, the early universe around the time of inflation, and other exotic sources that could revolutionize our understanding of fundamental physics, such as cosmic strings. Ground-based gravitational wave detectors have primarily focused on detecting the late stages of inspiral, merger, and ringdown of black hole and neutron star binaries. Searches for continuous wave sources such as rotating neutron stars are also being conducted. They have already placed non-trivial constraints on the geometry of neutron stars, but have yet to yield a detection. Because the inspiral of compact objects accelerates close to merger, merging binaries always appear as transient sources, only visible to current-generation ground-based detectors for seconds to a few minutes. The LIGO and Virgo collaborations publicly confirmed ten binary black hole mergers and one binary neutron star merger in their first two observing runs, O1 and O2 [33]. During O3, 56 candidate events were publicly alerted and remain unretracted, although the final O3 catalog paper has not yet been published as of this writing [34]. As analysis techniques improve and the LIGO and Virgo detectors’ behavior becomes better understood, public catalogs will become increasingly probabilistic and include a spectrum of marginal event classifications, rather than a single definitive classification of an event as a specific source class, noise, or probable glitch. Future detectors such as Einstein telescope are expected to achieve observing volumes several orders of magnitude larger, such that multiple inspirals will always be observable in the detector at any given time. In just a few years, ground-based gravitational-wave observatories have moved from exciting laboratories to test a fundamental prediction of general relativity to telescopes for the rapidly growing field of multi-messenger astronomy. Understanding the mass function and spin distribution of black holes can help answer fundamental questions about the growth of black holes and the structure of galaxies, while neutron star mergers provide windows into nuclear physics, the origin of the elements that make up the solar system, and tests of cosmological expansion history. Higher signal-to-noise and better-calibrated observations of

9 the ringdowns from black hole mergers provide powerful tests of whether general relativity is the correct theory of gravity. However, observations by ground-based observatories are limited to the classes of sources that reach frequencies & 1 Hz. Many types of sources will not reach such high frequencies 5 7 before merger. Supermassive black hole binaries with masses 10 − 10 m merge while their gravitational-wave emission is still primarily in the mHz, and observing such mergers is a vital way to study the formation mechanisms of the supermassive black holes at the centers of galaxies. Such mergers may also be multi-messenger targets if either black hole maintains an accretion disk. Extreme mass ratio inspirals (EMRIs) are mergers where a stellar-mass black hole merges with a much larger supermassive black hole, with a mass ratio & 1000 : 1. Observing an EMRI will allow some of the most stringent possible tests of deviations from general relativity, as well as potentially provide evidence for or rule out a variety of dark matter models. Another test of general relativity will be achieved by comparing the predicted merger time from space-based observations to the observed merger time in ground-based detectors. Any diﬀerence between the predicted and actual observed time of merger would be evidence for deviations from general relativity. Galactic white dwarf binaries are excellent multi-messenger targets, as they are both strong mHz gravitational-wave sources and visible with a wide variety of electromagnetic telescopes. White dwarf binaries merge or are otherwise disrupted before they reach ground- based detectors sensitivity bands, making them prime targets for space-based detectors.

1.6 Structure of Thesis

This thesis is divided into three primary sections, illustrating three major projects I have led during my graduate studies. In the ﬁrst, Chapter2, I describe a study exploring theoretical constraints on strongly interacting models of dark matter. In the second, Chapter3,I describe a formalism and code I developed to forecast the impact of long-wavelength density ﬂuctuations on future weak lensing surveys. Finally, in Chapter4, I present a forecast for the number of galactic binary systems that may be detectable by both Roman Space Telescope’s microlensing survey and LISA.

10 Chapter 2 Not as Big as a Barn: Upper Bounds on Dark Matter-Nucleus Cross Sections

Matthew C. Digman, Christopher V. Cappiello, John F. Beacom, Christopher M. Hirata, Annika H. G. Peter

Here I present my paper exploring theoretical limits on strongly interacting dark matter [35].

Abstract

Critical probes of dark matter come from tests of its elastic scattering with nuclei. The results are typically assumed to be model-independent, meaning that the form of the potential need not be specified and that the cross sections on different nuclear targets can be simply related to the cross section on nucleons. For point-like spin-independent scatter- 2 2 4 2 ing, the assumed scaling relation is σχA ∝ A µAσχN ∝ A σχN , where the A comes from 2 2 2 coherence and the µA ' A mN from kinematics for mχ mA. Here we calculate where model independence ends, i.e., where the cross section becomes so large that it violates its defining assumptions. We show that the assumed scaling relations generically fail for dark −32 −27 2 matter-nucleus cross sections σχA ∼ 10 − 10 cm , significantly below the geometric sizes of nuclei, and well within the regime probed by underground detectors. Last, we show on theoretical grounds, and in light of existing limits on light mediators, that point-like dark −25 2 matter cannot have σχN & 10 cm , above which many claimed constraints originate from cosmology and astrophysics. The most viable way to have such large cross sections is composite dark matter, which introduces significant additional model dependence through the −32 2 choice of form factor. All prior limits on dark matter with cross sections σχN > 10 cm with mχ & 1 GeV must therefore be re-evaluated and reinterpreted.

11 2.1 Introduction

The nature of dark matter is one of the most pressing problems in both fundamental physics and cosmology. Decades of observations indicate that dark matter makes up the vast majority of matter in our universe, yet increasingly advanced experiments have yet to determine its physical nature. Once discovered, the particle properties of dark matter will be a guidepost to physics beyond the Standard Model as well as to an improved understanding of galaxies and cosmic structure [8–11, 36–38]. Progress depends on accurately assessing the regions of dark matter parameter space that remain viable. One of the best motivated dark-matter candidates is a single weakly interacting massive particle: a WIMP. There are several ways to search for WIMPs. First, through missing transverse momentum searches at colliders [39–46]. Second, through searches for WIMP self-annihilation products and decay [13, 47–58]. Third, by energy transfer through elastic scattering with nuclei and electrons. Laboratory direct-detection experiments [59–69] provide the tightest bounds on dark matter-nucleus elastic scattering cross sections, with other constraints provided by cosmology and astrophysics [70–88]. While there are no robust signals yet, progress is rapid. For these searches, the two most common benchmarks for the performance of dark matter detection experiments are the dark matter self-annihilation cross section and the spin- independent dark matter-nucleon elastic scattering cross section, the simplest case (for more general treatments, see, e.g., Refs. [93–99]). These benchmarks allow constraints set by different experiments to be scaled to each other. Here, we focus on spin-independent elastic dark matter-nucleus scattering for dark matter with mχ & 1 GeV. For generality, we do not require that dark matter be a thermal relic. Most direct-detection searches focus on pushing sensitivity to small cross sections, but it is also important to consider constraints on large cross sections [89–92, 100–109]. Direct- detection experiments are typically located beneath the atmosphere, rock, and detector shielding, such that dark matter with too large of a cross section loses too much energy above the detector. Energy loss in the detector overburden may open a window where strongly interacting dark matter is allowed [100]. Figure 2.1 summarizes prior claimed constraints. The ‘IMP+IMAX+SKYLAB’ region is based on atmospheric and space-based detectors and is dashed because the results are commonly cited but are not based on detailed analyses in peer-reviewed papers [89]. The X-ray Quantum Calorimeter (XQC) experiment is rocket-based [110]. There are several similar proposed XQC regions [89, 90, 92]; we adopt that of Ref. [92]. The ‘Underground Detectors’ region is taken directly from the summary plot in Ref. [91]. For the ‘Cosmology’ region, we plot the strongest constraint that depends only on dark matter-proton scattering [86] (including helium would make the constraints somewhat stronger [78]). The details of

12 10 10 10 13 Cosmology 10 16 10 19 10 22 XQC IMP + IMAX + SKYLAB ] 25 2 10 m

c 28

[ 10

N 10 31 Underground Detectors 10 34 10 37 10 40 10 43 10 46 1 103 106 109 1012 1015 1018 m [GeV]

Figure 2.1: Claimed constraints on the spin-independent dark matter-nucleon cross section [86, 89–92]. Those from cosmology directly probe scattering with protons, but all others here are based on scattering with nuclei, and thus require the use of ‘model-independent’ scaling relations. Below, we show that assumptions used to derive these results are invalid over most of the plane.

13 which constraints are plotted do not affect our conclusions. Direct-detection searches for spin-independent interactions benefit from an essentially 2 2 model-independent A coherent enhancement, as well as a kinematic factor of µA, such that σχA is related to the dark matter-nucleon elastic scattering cross section σχN by σχA ∝ 2 2 A µAσχN . For mχ mA, the dark matter-nucleus reduced mass µA ' AmN , such that 4 the scaling becomes σχA ∝ A σχN . This straightforward scaling allows constraints on dark matter-nucleus scattering to be related to each other and to the cross section on nucleons. This scaling is ‘model-independent’ in the sense that it is independent of the detailed shape of the potential. In Fig. 2.1, all constraints except the one labeled ‘Cosmology’ deal with nuclear targets with A > 1, and hence assume this scaling relation. How large of cross sections are allowed before the defining assumptions are violated? Here we systematically calculate the theoretical upper limits on dark matter-nucleon cross sections. We show that most of the parameter space of Fig. 2.1 is beyond the point where the simple scaling relations above are valid, or where point-like dark matter is even allowed. Our results are based first on generic considerations of theoretically allowed cross sections for short-range interactions with nuclei, and second on classes of models where we consider light mediators as a mechanism to obtain large cross sections. As far as we are aware, this is the first systematic exploration of these issues for dark matter-nucleus scattering (for related considerations in strongly self-interacting dark matter sectors, see, e.g., Ref. [111]). Our results will require the reinterpretation of a large and varied body of work, e.g. Refs. [73, 76, 78, 79, 89–92, 100–109, 112–122]. In Sec. 2.2, we review the nonrelativistic scattering theory used to obtain the model- independent scaling relations. In Sec. 2.3, we examine the various ways that scaling relations can break down for contact interactions. In Sec. 2.4, we examine the possibility of achieving a larger cross section with a light mediator in light of present constraints on light mediators. In Sec. 2.5, we briefly discuss the possibility that dark matter itself could have a nonzero physical extent. In Sec. 2.6, we discuss the implications for existing constraints and future experiments. Finally, we summarize our results and the outlook for future work in Sec. 2.7.

2.2 Dark Matter Scattering Theory

We brieﬂy review the basic nonrelativistic scattering theory required to derive the ‘model- independent’ scaling relation for the spin-independent elastic scattering cross section. We also discuss how some of the key assumptions may break down. Throughout, we set ~ = c = 1.

14 2.2.1 Overview of basic assumptions

1. Single particle: Dark matter is primarily a single unknown particle. The number density of dark matter is then determined only by its mass and the local dark matter density.

2. Point-like: Dark matter is a point-like particle with no excitation spectrum.

3. Electrically neutral: It is typically assumed that dark matter is electrically neutral. Millicharged dark matter has diﬀerent dynamics and is too strongly constrained to produce large cross sections.

4. Equal coupling to all nucleons: For simplicity, we assume that dark matter has equal coupling to both protons and neutrons, although this assumption is not essential to any of our conclusions. D E 0 0 3 0 5. Local: The interaction is assumed to be local, x Vb x = V (x )δ (x − x). 6. Energy-independent potential: The potential for the interaction is assumed to be energy-independent, such that the cross section for the interaction is also energy- independent up to a form factor. For a spin-independent interaction, the potential must also be independent of the incident angular momentum l.

7. Elastic: For laboratory experiments, dark matter-nucleus scattering is assumed to occur at typical Milky Way virial velocities, v ∼ 10−3 c. Typical recoil energies of O(1 keV) are not suﬃcient to produce Standard-Model particles, or to excite internal degrees of freedom of nuclei. Therefore, elastic scattering is the dominant interaction channel. In any case, all physical scattering processes have at least some elastic component [123].

8. Coherence: Closely related to the assumption of purely elastic scattering is the assumption of coherence. For coherence to hold, it must be a good approximation to treat the dark matter as interacting with the nucleus as a whole, rather than with individual nucleons. Coherence is typically a good approximation provided the momentum transfer q is insuﬃcient to excite internal degrees of freedom in a nucleus,

which is true provided 1/q is large compared to the characteristic nuclear radius rA,

qrA 1 [124]. The breakdown of coherence can be parametrized by including a momentum-dependent form factor in the diﬀerential cross section.

9. No bulk effects: The scattering should be well approximated as being with a single nucleus, such that initial and final state effects in the bulk medium can be ignored. This approximation is good as long as the characteristic momentum transfer q is large compared to the characteristic inter-atomic spacing, which is typical. 15 The rest of this paper deals with the failure of the following additional assumptions:

10. S-wave scattering For s-wave (l = 0) scattering, the scattering is isotropic in the center of momentum frame. As shown in Sec. 2.2.3, assuming l = 0 is required to 2 2 derive the model-independent A µA scaling relation. However, real interactions may deviate signiﬁcantly from isotropic scattering and we do not require l = 0 in this analysis.

2 2 11. Weak Interaction: For A µA scaling to hold, the interaction must be weak enough for the Born approximation to hold. We discuss this assumption in Sec. 2.2.3.

2.2.2 Basic scattering theory

Here we provide a brief review of the scattering theory formalism [123–130] used in later sections. To be detectable, dark matter must have some kind of interaction with ordinary matter in a detector, written here as a potential V (r). We specialize to spin-independent interactions, and restrict our analysis to spherically symmetric potentials V (r) = V (r) that fall oﬀ faster than r−1 as r → ∞. In the center of momentum frame, the time-independent Schrödinger equation giving the evolution of a nonrelativistic two-particle system with wavefunction ψ(r) and reduced mass µ is 1 − ∇2 + V (r) ψ(r) = Eψ(r). (2.1) 2µ r As shown in Appendix A.1, far from the potential the solution of Eq. (2.1) may be written:

ikr r→∞ µe Z 0 ψ(r) −−−→ψ (r) − V (r0)ψ(r0)e−ikf ·r d3r0 0 2πr eikr =ψ (r) + (2π)−3/2 f (k , k ) , (2.2) 0 r i f

−3/2 ik ·r where f (ki, kf ) = f(k, θ) is the scattering amplitude, ψ0(r) ≡ (2π) e i , and ki ≡ kˆz and kf are the initial and ﬁnal dark matter momenta, respectively. From the scattering amplitude, we obtain the diﬀerential cross section: dσ = |f(k, θ)|2, (2.3) dΩ and the total elastic scattering cross section: Z dσ σ = dΩ. (2.4) χA dΩ

If the scattering is isotropic, f(k, θ) = f(k), Eq. (2.4) is proportional to the rate of detectable scattering events in a detector. However, to be detectable, a collision must deposit suﬃcient energy into the detector. If the scattering angle is peaked close to θ = 0, 16 very little momentum is transferred, and hence insuﬃcient energy deposited in the detector. Therefore, it is sometimes more useful to weight the integral in Eq. (2.4) by the momentum transfer to obtain the momentum-transfer cross section, Z dσ σmt = (1 − cos θ) dΩ. (2.5) χA dΩ

mt For isotropic scattering, σχA = σχA. For a potential with characteristic radius rA, isotropic scattering is generically a good approximation at low energies, krA 1, as discussed further in Sec. 2.2.3. Forward scattering is a major concern for light mediators

(Sec. 2.4); in the Coulomb scattering limit where the mediator mass mφ → 0, then σχA → ∞, mt while σχA remains ﬁnite.

2.2.3 Derivation of ‘model-independent’ scaling

Now we discuss approximation methods for f(k, θ). The two approaches we consider here are the Born approximation and the partial wave expansion. Both allow us to derive the 2 2 2 σχA = A µA/µN σχN scaling with nuclear mass number A. The reduced masses are deﬁned µA ≡ mAmχ/(mA + mχ), µN ≡ mN mχ/(mN + mχ), where the mass of a nucleus

with mass number A is related to the mass of a single nucleon mN using mA = AmN . We begin with the Born approximation because it is the simple and familiar derivation. Because the partial wave expansion is valid even when the Born approximation fails, it allows us to more concretely show the behavior at large scattering cross sections.

Born approximation

Inspecting Eq. (2.2), a natural ﬁrst approach to obtaining f(k, θ) is to solve for ψ(r) by iteration, which is the Born approximation, as demonstrated in Appendix A.2. The ﬁrst Born approximation to f(k, θ) is simply the Fourier transform of the potential:

2µ Z ∞ f (1) (k, θ) = f (1) (q) = − A V (r0) sin(qr0)r0dr0, (2.6) q 0 where q = |q| = 2k sin θ/2 is the momentum transfer.

Now, assume that the potential has some maximum radius rA, and we have low energy 0 0 scattering, krA 1. Then we can approximate sin(qr ) ≈ qr and integrate only up to the

maximum radius rA: Z rA (1) 0 02 0 f (k, θ) ≈ −2µA V (r )r dr . (2.7) 0 Eq. (2.7) is a remarkable result. Provided the required approximations are valid, f (1) (k, θ) depends only on the volume integral of the potential; it contains no information at all about the shape. Provided the volume integral of the potential is proportional to the

17 nuclear mass number A, we have the scaling:

(1) f (k, θ) ∝ AµA. (2.8)

Plugging into Eq. (2.4): (1) 2 2 σχA ∝ A µA, (2.9)

which can be recast more precisely in terms of the dark matter-nucleon reduced mass µN (1) and scattering cross section σχN :

2 (1) 2 µA (1) σχA = A 2 σχN . (2.10) µN Eq. (2.10) is the famous ‘model-independent’ scaling relation for the spin-independent elastic scattering cross section. Provided the potential falls off faster than 1/r, this scaling relation is generally a good approximation at sufficiently low energies, so long as the first Born approximation reasonably approximates f(k, θ). However, we must examine when the first Born approximation fails. We discuss the validity of the Born approximation in Appendix A.2. A useful condition for the validity of the first Born approximation is [129]:

Z ∞ µA 0 2ikr0 0 V (r ) e − 1 dr 1. (2.11) k 0

We can simplify Eq. (2.11) using our assumption of a maximum range rA and krA 1:

Z rA 0 0 0 2µA V (r )r dr 1. (2.12) 0 Eq. (2.12) is equivalent to the statement that the potential is much too weak to form a bound state even if V (r) was purely attractive [131]. While Eq. (2.7) is a volume integral, Eq. (2.12) is an area integral of the potential. Therefore, the first Born approximation is valid when some potential-weighted effective area is small. The effective area in question is, in fact, the elastic scattering cross section, as shown for a contact interaction in Sec. 2.3.

Partial wave expansion

To investigate what happens when the Born approximation fails, the ﬁrst step is to expand the scattered wave function in terms of Legendre polynomials and calculate the phase shift of each contribution. The phase shifts may be found by numerically integrating the Schrödinger equation, as described in Appendix A.3. The elastic scattering cross section may be written

18 in terms of the phase shifts δl(k):

∞ 4π X σ = (2l + 1) sin2 (δ ). (2.13) χA k2 l l=0 The momentum-transfer cross section in Eq. (2.5) may also be written in terms of partial wave phase shifts:

∞ 4π X σmt = (l + 1) sin2(δ − δ ). (2.14) χA k2 l+1 l l=0 The mathematical decomposition in terms of partial waves is valid even beyond interactions that can be described in nonrelativistic potential scattering theory. However, when the number of partial waves becomes too large, it may be impractical to compute the phase shifts individually, and semiclassical approximations become useful [123, 128]. Physically, the sum over partial waves in Eq. (2.13) is equivalent to the classical operation of averaging over all possible impact factors b = pl(l + 1)/k [123, 130]. Classically, for a potential with maximum range rA, there would be no collisions for b > rA. Therefore, a useful approximate upper limit on the highest partial wave that can meaningfully contribute to the sum in nonrelativistic quantum scattering is lmax ≈ krA, and contributions from higher l > lmax fall oﬀ quickly [123]. Our derivation of the ‘model-independent’ scaling of

Eq. (2.10) in the Born approximation assumes krA 1, which is equivalent to saying only the l = 0 (s-wave) term contributes a nonvanishing phase shift. Using the same iterative procedure as for the Born approximation, we can obtain the ‘model-independent’ form of the s-wave phase shift [130]:

Z rA 0 02 0 δ0(k) ≈ −2µAk V (r )r dr , (2.15) 0

where the required approximation is |δ0(k)| 1. Plugging Eq. (2.15) into Eq. (2.13) and

again using |δ0(k)| 1, we obtain precisely the same expression for the scattering amplitude we obtained for the Born approximation in Eq. (2.7), as expected. Again, the requirement

|δ0(k)| 1 places an upper bound on the maximum σχA where the relation can apply.

If the cross section were instead the maximum allowed by unitarity, δ0(k) = π/2, we would obtain 4π σ = , (2.16) χA k2 2 which decreases as 1/A with increasing A, assuming k = µAv ∝ A, rather than increasing as A4. 2l+1 Higher partial waves necessarily scale diﬀerently with k [123], δl(k) ∝ k for krA 1.

Higher δl(k) also contain information about the shape of the potential. Therefore, we do not expect any special ‘model-independent’ scaling when higher partial waves contribute.

19 2.3 Contact interactions

In this section, we consider the limits on cross sections that can be obtained through a contact interaction, and how the scaling relations break down. A contact interaction is useful as an illustrative case because we do not need to consider the speciﬁc mechanism that produces the interaction.

2.3.1 Contact interaction with Born approximation

As a simple case, we consider a contact interaction with a nucleus, as could be produced by a heavy mediator. We roughly approximate the nuclear charge density as having a top hat shape with radius r : A  V0 r < rA V (r) = (2.17) 0 otherwise. We assume the maximum charge density is roughly independent of atomic mass number A, 1/3 such that rA ≈ A rN , where rN ' 1.2 fm. We use this toy model with a sharp cutoﬀ because both the Born approximation and the

partial wave phase shift δl(k) can be found analytically. The eﬀect of using a more realistic charge distribution is discussed in Sec. 2.3.3. Fourier transforming Eq. (2.17) using Eq. (2.6) gives:

2µ V f (1)(q) = A 0 [qr cos(qr ) − sin(qr )] . (2.18) q3 A A A The total elastic scattering cross section in the ﬁrst Born approximation is then:

πµ2 V 2 σ(1) = A 0 4kr sin(4kr ) + cos(4kr ) + 32k4r4 − 8k2r2 − 1. (2.19) χA 16k6 A A A A A

In the limit krA 1, Eq. (2.19) becomes: 16π σ(1) ≈ µ2 r6 V 2. (2.20) χA 9 A A 0

(1) 16π 2 6 2 For scattering with a nucleon, Eq. (2.20) would become σχN ≈ 9 µN rN V0 . Substituting 1/3 rA ≈ A rN , we recover the required scaling relation of Eq. (2.10):

2 (1) 2 µA (1) σχA ≈ A 2 σχN . (2.21) µN

In the krA 1 limit, the condition for validity of the ﬁrst Born approximation in Eq. (2.12) is simply 2 µArAV0 1. (2.22)

20 Comparing to Eq. (2.20), we can rewrite the condition Eq. (2.22) as: 16 σ(1) πr2 . (2.23) χA 9 A Eq. (2.23) has a simple physical interpretation. The ﬁrst Born approximation is only appli- cable for elastic scattering cross sections much smaller than the geometric cross section of the nucleus. Using rN ≈ 1.2 fm in Eq. (2.23), the Born approximation result only applies (1) −25 2 for σχN 10 cm . Going to higher orders in the Born approximation does not unlock cross sections signif- (1) 16 2 icantly exceeding the geometric limit of the potential. For σχA > 9 πrA, the Born series is not even guaranteed to converge for all energies [123]. However, it may still be possible to obtain a meaningful cross section in regimes where the Born approximation fails using partial wave analysis. We explore this technique below.

2.3.2 Contact interaction with partial waves

For r < rA in Eq. (2.17), the radial wave function decomposed in partial waves has an 0 0 2 1/2 analytic solution in terms of partial waves, ul(r) = Clrjl(k r), where k ≡ k − 2µAV0 could be either pure real or pure imaginary. First, we consider the s-wave cross section, with 2 2 l = 0. Expanding in the limit where V0 and k are small, (krA) 1, |V0| 1/(2µArA), we recover (see Appendix A.3):

2µ kr3 V 8µ2 kr5 V 2 δ (k) ≈ − A A 0 + A A 0 + O |V |3 . (2.24) 0 3 15 0 The corresponding s-wave cross section is 16π 128π σl=0 ≈ µ2 r6 V 2 − µ3 r8 V 3 + O |V |4 , (2.25) χA 9 A A 0 45 A A 0 0 which is identical to Eq. (2.20) to lowest order in |V0|, as anticipated in Sec. 2.2.3. Now we can see how the ‘model-independent’ A scaling fails as the coupling strength 3 3 8/3 gets stronger. The second-order term in Eq. (2.25) scales ∝ µA/µN A , and either reduces the cross section for a repulsive potential V0 > 0 or increases it for an attractive one. For

V0 large enough for the second-order and higher corrections to matter, there is, therefore, l=0 no model-independent scaling with A, because σχA depends on the details of the potential. The breakdown of the scaling as a function of A is shown in Fig. 2.2. Once details of the potential begin to matter, corrections from a more realistic charge distribution would also become important, as discussed in Sec. 2.3.3. 2 To illustrate further, if we instead considered the strong coupling limit, |V0| 1/2µArA,

21 for V0 > 0 we would obtain:

δ0(k) = −krA, (2.26) and 4π σl=0 = sin2 (−kr ) (2.27) χA k2 A 2 ' 4πrA, (2.28)

so the repulsive cross section completely saturates at four times the geometric cross section.

The saturation is plotted as a function of |V0| in Fig. 2.3. Physically, the well has simply become an impenetrable hard sphere with a ﬁxed radius. Therefore, we have obtained a physical maximum cross section for the repulsive contact interaction at small k which is only 9/4 times larger than the maximum we obtained using the ﬁrst Born approximation.

Higher partial waves

While the s-wave cross section is limited to the geometric cross section, it is natural to wonder if the contributions from higher partial waves could allow a larger total cross section. For

krA 1, we can approximate semi-classically lmax ≈ krA, as described in Sec. 2.2.3. Of course, in the quantum case, it is possible for higher partial waves to contribute, but their

contributions fall oﬀ rapidly for l > lmax. Assuming a sharp cutoﬀ is adequate for deriving an approximate maximum physically achievable cross section. The maximum possible cross

section is achieved by saturating partial wave unitarity, i.e., taking δl≤lmax (k) = π/2:

l 4π Xmax σ = (2l + 1) (2.29) χA k2 l=0 4π = (1 + l )2 (2.30) k2 max 2 ≈ 4πrA. (2.31)

Now, we see that the saturation at approximately the geometric cross section, found for

krA 1 in Eq. (2.28), also holds for krA 1.

In fact, for a very strong repulsive contact interaction, the phase shifts for l ≤ lmax 2 approach δl≤lmax (k) ≈ lπ/2 − krA [123], such that σχA ≈ 2πrA. Therefore, a repulsive hard sphere almost saturates the unitarity limit of Eq. (2.31). Including higher partial waves is therefore not a useful way of increasing the cross section,

because the potential remains limited by the characteristic radius rA. Figure 2.4 shows the breakdown of the A4 scaling for several example nuclei, fully taking into account contributions from higher partial waves. Direct detection constraints

22 for underground detectors are aﬀected by the breakdown of scaling at the O(1) level for −32 2 σχN ' 10 cm .

2.3.3 Attractive resonances

A final possible approach would be to saturate unitarity at δl = π/2 while krA 1, such that 4π σmax = (2l + 1) (2.32) χA k2 can become large. The limit δl = π/2 at krA . 1 can be achieved through resonances, which occur when an attractive potential becomes strong enough to support a bound state. In reality, the resonant scattering cross section would achieve large values only for a narrow range of k relative to the incident dark matter velocity distribution [123, 132]. Be- cause k = µAv, the resonances are generically at different incident dark matter velocities for different elements, which guarantees that there are not any useful model-independent scaling relations relating the observed cross sections for strongly attractive potentials between different target materials.

In Fig. 2.2, we show the behavior as a function of A for two different values of V0. When resonances are possible, the scaling with A need not be monotonic. The behavior is fairly complex for even the simple rectangular well of Eq. (2.17). Realistic scaling is likely to be even more complicated, because the nuclear charge distribution changes as a function of A. Even two different nuclei with the same A but different atomic numbers could have different charge distributions, and hence different resonant cross sections. Scaling with A for strong attractive couplings is therefore highly model-dependent. In Fig. 2.3, we show the saturation of the s-wave cross section as a function of the coupling strength, as well as the resonant behavior which occurs once the potential becomes strong enough to support a quasi-bound state. The height and shape of resonances is a strong function of the assumed velocity distribution of the incident dark matter particles; here we assume typical Milky −3 Way halo thermal velocities of v ∼ 10 c, where k = µAv. The resonances for A = 4 are fairly narrow low energy s-wave resonances, as for a nucleon the scattering is still well approximated in the low-k limit. However, for A = 131, the low-k limit is no longer a good approximation, and resonances are broadened to the point that they do not significantly increase the scattering cross section. Additionally, there are many more resonances, from multiple partial waves. Note we have not implemented any velocity dispersion for this plot; the spreading of resonances is entirely due to broadening of peaks and overlapping contributions from multiple partial waves at finite k. Applying a realistic dark matter velocity distribution would smooth the peaks. For heavy nuclei with multiple naturally occurring isotopes (e.g. xenon), averaging over a distribution of isotopes would smooth the peaks even further.

23 Overall, even if a carefully tuned resonance could achieve a large cross section for a single light nucleus, other nuclei would not necessarily have correspondingly large cross sections. Scaling relations between specific nuclear cross sections would also be highly model- dependent, such that constraints from different types of nuclei would be difficult to compare because the full resonance structure would not be known. For example, using more realistic charge distributions, such as an exponential potential for A = 4 and a Woods-Saxon potential for A = 131 [133] would shift the positions of the resonances somewhat.

More realistic charge distributions

The rectangular barrier potential in Eq. (2.17) is a toy model. Realistic nuclear charge distributions have a smooth cutoff, and an exponential tail to larger radii [133], as in a Woods-Saxon potential. Because there is not a sharp cutoff, allowing the interaction strength to be arbitrarily large would cause the potential to grow logarithmically with |V0|. However, −1 there are limits to how strong |V0| can be. For |V0| & 10 GeV, QCD corrections break the simple nonrelativistic contact interaction picture. For |V0| & 2 GeV, the interaction may be strong enough to pull proton-antiproton pairs out of the vacuum. We have verified by numerically computing partial-wave amplitudes that for |V0| . −1 10 GeV using a Woods-Saxon potential increases the maximum σχA by a factor of . 10. By definition, an increase in the computed cross section due to a different potential can only appear for |V0| strong enough that the ‘model-independent’ form of the cross section has already significantly broken down. Therefore, using a more realistic charge distribution cannot significantly change our conclusions about contact interactions. Realistic potentials are also not perfectly spherically symmetric, but the same basic picture of geometrical limitations still applies, and resonances are still possible.

2.3.4 Beyond contact interactions

We have established that a contact interaction with a nucleus at typical Milky Way halo thermal velocities cannot achieve cross sections much larger than the geometrical cross section of the nucleus. The case where large cross sections might be achieved, a strongly attractive potential, produces resonances that are sensitive to the detailed structure of the potential and is far too model-dependent to possess any simple scaling relation relating the cross sections at diﬀerent A. To circumvent these problems, we need an interaction with a larger characteristic range. One possible way to achieve a larger characteristic range is to insert a light mediator for the interaction, as discussed in Sec. 2.4. Another possibility is composite dark matter with an intrinsic radius, discussed in Sec. 2.5.

24 2.4 Light Mediator

A simple approach to achieving a larger characteristic radius is to insert a light mediator, of mass mφ = 1/rφ, which generically results in a potential of the form

λ λ e−r/rφ V (r) = A χ , (2.33) 4π r

where λχ and λA = AλN are the coupling strengths of the particle φ to the dark matter and nucleus respectively. To achieve cross sections much larger than a nucleus, we should have

rφ 1 fm. The dark matter and target nucleus are distinguishable particles, so the Yukawa potential can be either attractive or repulsive. We assume the mediator is a scalar, although the general form of the potential would be similar for other light mediator candidates. The scattering amplitude of Eq. (2.33) is easily calculated using Eq. (2.6):

(1) µAλχλA f (q) = − 2 2 , (2.34) 2π(q + 1/rφ) which gives the total elastic scattering cross section:

2 2 2 4 (1) µAλχλArφ σχA = 2 2 . (2.35) π(1 + 4k rφ)

Because the characteristic radius is larger than the geometric radius of a nucleon, Eq. (2.35) can in principle achieve larger cross sections within the domain of validity of the Born

approximation than a contact interaction could. Because k ∝ µA, the scaling with A is now 2 2 more complicated due to the k rφ term in the denominator. Assuming mχ mA such that µA ≈ AmN , we have two limits:  4 (1) A σ kArφ 1 σ(1) ≈ χN (2.36) χA 2 (1) A σχN kArφ 1. −1 −3 For direct-detection, kN ∼ 0.005 fm is set by Milky Way halo velocities v ' 10 c 131 −1 and the mass of a single nucleon, kA ' AkN . For Xe, kA ≈ 0.7 fm , such that kArφ > 1 4 occurs for rφ & 1.4 fm. Therefore, A scaling is at best marginal for heavy nuclei for any rφ −25 2 that could conceivably produce a cross section σχA & 10 cm . Additionally, as established for a general potential with a characteristic radius rφ in Sec. 2.2.3, increasing the coupling 4 strengths λχ or λN at ﬁxed rφ causes the Born approximation, and therefore the A scaling, to fail before cross sections larger than the geometric cross section are achieved. Therefore, 4 (1) −25 2 4 for a light mediator, the A scaling is only possible if σχA 10 cm . The failure of A scaling occurs without even considering the constraints on the existence of light mediators discussed in Subsection 2.4.2. The A4 scaling is preserved to somewhat larger cross sections

25 than for the contact interaction shown in Fig. 2.2.

2.4.1 Momentum-transfer cross section

In fact, even the A2 scaling is too optimistic for the detectable momentum transfer in a detector with high A. Inspection of Eq. (2.34) shows that for krφ 1, the scattering becomes strongly peaked at θ = 0. Therefore, it is more useful to consider the momentum- transfer cross section, Eq. (2.5). Using the Born approximation, we can calculate Eq. (2.5) analytically for the Yukawa potential:

2 2 2 mt,(1) µAλχλA 2 2 2 2 2 2 σχA = 4 2 2 (1 + 4k rφ) log(1 + 4k rφ) − 4k rφ . (2.37) 8πk (1 + 4k rφ)

mt,(1) (1) For krφ 1, Eq. (2.37) simpliﬁes to σχA ≈ σχA as expected for isotropic scattering. However, for krφ 1, we have:

µ2 λ2 λ2 σmt,(1) ≈ A χ A log(4k2r2) − 1 . (2.38) χA 8πk4 φ

Eq. (2.38) grows only ∝ log(A) for k = µAv ∝ A, such that, for a ﬁxed total detector mass, the total energy deposited in the detector would be larger for nuclei with smaller A. Direct detection experiments that focus on protons and other light nuclei, such as Refs. [62, 136– 138], may therefore be eﬀective ways of constraining the landscape for model-dependent direct detection.

2.4.2 Existing limits on light mediators

mt,(1) −25 2 If there were no other constraints on rφ or λA, Eq. (2.38) would allow σχA 10 cm , albeit with a less useful scaling relation between different nuclei. However, because the light mediator couples to the Standard Model directly, other experiments already place mt constraints on such a particle. Figure 2.5 shows the maximum achievable σχN and σχN for a repulsive Yukawa potential, conservatively using the perturbativity limit [134] λχ ' 4π, −1 −4 µN ≈ mp, and k = 0.005 fm . When Eq. (2.12) is > 10 , Fig. 2.5 uses the results from a numerical partial wave expansion with adaptive lmax, rather than the Born approximation. In practice, the Born approximation is adequate in the entire unconstrained region. −27 2 mt −32 2 Including all such constraints, we have σχN . 10 cm and σχN . 10 cm . Con- straints that rely on lower relative velocities, such as the cosmological constraints discussed in Sec. 2.6.3, could achieve larger cross sections, but their constraints would need to be scaled correctly to compare them to direct detection constraints. The momentum-transfer mt −25 2 −6 cross section is restricted to be σχN . 10 cm even for velocities as low as 10 c. It is also possible to produce the light mediator in a collision [139]. Particle production is an inelastic scattering process and beyond the scope of this paper, but it could be another 26 avenue to transfer momentum between dark matter and a detector. The detailed constraints in Fig. 2.5 could be different for different types of mediators. For example, for a vector mediator, the BBN constraints would be stronger [134]. Other con- −27 2 straints might be weaker. However, σχN . 10 cm is already smaller than the geometric cross section of the nucleus, and circumventing individual constraints is unlikely to drastically change the overall conclusion that light mediators do not appear to be a promising approach to achieving large cross sections.

2.5 Composite Dark Matter

Another mechanism for achieving a larger characteristic interaction radius is dark matter that is not a point particle, but instead has a finite physical extent [120, 140–154]. Such dark matter could take the form of a composite particle. Because such dark matter would likely require an entire dark sector, any conclusions about the largest possible cross section with composite dark matter would be intrinsically model-dependent. Because the largest physical scale in the problem is no longer related to a property of the target nucleus, the cross section need not scale with A at all. The actual scaling with A could only be determined by examining the particular model of composite dark matter. Additionally, achieving cross sections significantly larger than a nucleus with composite dark matter will always require krdm & 1 for typical Milky Way virial velocities, so constraints on composite dark matter will need to be computed with a specific dark matter form factor in mind. See Sec. 2.6.3 for discussion of limits at the lower velocities relevant to cosmological limits. Analyses setting constraints on specific form factors at large cross sections should consider whether their specific choice of form factor can be achieved at the cross sections they are constraining in a physically realistic model. Therefore, limits on composite dark matter need to be calculated in specific models. Calculation of constraints on specific models of composite dark matter is left to future work.

2.6 Implications for Existing Constraints

Figure 2.6 summarizes the approximate limits for the repulsive contact-interaction cross sections discussed in Sec. 2.3. In the colored regions, the Born approximation begins to break down when the proton cross section is scaled to heavier nuclei, ultimately failing even for light nuclei. For point-like dark matter with a repulsive contact interaction, cross sections much larger than the geometric cross section are completely forbidden. As discussed in Sec. 2.4, the limits for a light mediator are similarly below the geometric cross section. For 16 mχ & 10 GeV, the entire (small) exclusion region for underground detectors is aﬀected by the failure of scaling relations. Future improvements to constraints could change the region

27 where the entire exclusion region would fail. Additionally, all detectors’ computed ceilings are aﬀected by the breakdown of scaling relations.

2.6.1 Scaling constraints

In the regime where scaling relations are unreliable, it becomes more difficult to compare constraints between experiments. When the scaling relations fail, scaling constraints from different nuclei to the dark matter-nucleon cross section using the A4 is no longer meaningful. For both contact interactions and light mediators, as the cross section begins to saturate, the momentum-transfer cross section scales less than linearly with A. Therefore, at fixed total detector mass, there is more detectable momentum transfer into the detector for lighter target nuclei. The failure of the scaling relations also occurs at larger cross sections for smaller A. For example, a 12C-based detector would be able to use the Born approximation, and therefore the scaling relations, up to about 3000 times larger dark matter-nucleon cross section than a 131Xe-based detector. Therefore, robustly covering the large cross section regime may be best accomplished by detectors using light nuclei, e.g. [62, 136–138]. One option is to simply not scale constraints at large cross sections. While with resonances it could be possible for heavy nuclei to have smaller cross sections than a single nucleon, broadening by the dark matter velocity dispersion may limit the effect of narrow resonances on the overall detectable signature. Therefore, a relatively conservative approach could be to plot the actual momentum-transfer cross section constraints obtained from different nuclei on the same scale. In fact, if composite dark matter as discussed in Sec. 2.5 is indeed the most plausible strongly interacting dark matter candidate, disregarding scaling with A may be the most correct way of plotting constraints.

2.6.2 Detection ceilings

Now we brieﬂy consider if the detection ceilings (i.e., the largest cross sections that can be probed by a given detector based on the detector’s overburden) shown in Fig. 2.1 are preserved. In our simple model in Sec. 2.3, cross sections simply saturate at four times the geometric cross section for heavier nuclei. Even if all nuclei in the detector overburden have an elastic scattering cross section equal to their geometric cross section, dark matter cannot be stopped by the overburden above some mχ [112]. Because all currently computed detector ceilings exist at cross sections where the breakdown of the A4 scaling is severe, correctly calculated detector ceilings must be specialized to a speciﬁc model. For basic energy-independent cross section scaling, the weakened ceilings 16 likely lead to stronger direct detection constraints for mχ . 10 GeV. For such models, direct detection may even have exhausted the parameter space for cross sections up to the largest cross sections achievable with point-like dark matter. For other dark matter form

28 factors, the behavior around the ceiling could be more complicated. Further work is required to make detailed adjustments to existing constraint contours to determine what dark matter parameter space has been constrained at large cross sections.

2.6.3 Dark matter-proton scattering constraints

Constraints that rely only on dark matter scattering directly with protons are not directly affected by the breakdown of scaling relations with A. These are primarily constraints from cosmology and astrophysics, although at least one laboratory experiment uses proton targets [138]. Astrophysics constraints (e.g. disk stability, stars, cosmic ray interactions, gas clouds, etc) are typically assumed to occur at galactic virial velocities, as for direct detection. Cosmology constraints, such as CMB and structure formation constraints, typically assume collisions occur at smaller relative velocities. As shown in Fig. 2.6, the cross sections of interest for cosmological/astrophysical constraints are too large to be point-like dark matter. Therefore, they should be reinterpreted as constraints on specific models of composite dark matter with a specified form factor, as discussed in Sec. 2.5. For cosmology constraints set at lower relative velocities, the suppression of the cross section by the form factor of dark matter is not as severe. One consequence is that it is possible to achieve somewhat larger cross sections for point-like dark matter than those shown in Fig. 2.5, either with a light mediator or an attractive contact interaction with low- energy s-wave resonances. Even for velocities as low as v ' 10−6 c, existing constraints on mt −25 2 light mediators would still require σχN . 10 cm . However, invoking such models would require additional caution, as direct detection constraints would not be scaled correctly relative to the cosmology constraints, such that it would no longer be appropriate to plot cosmology and direct detection constraints on the same axes, as done in Fig. 2.1.

Cosmological and astrophysical constraints set at masses mχ < 1 GeV, discussed in Sec. 2.6.4, are at lower cross sections, and may still be meaningful constraints on point- like dark matter. However, analyses at lower masses should either directly investigate how high their limits can be extrapolated, or make it much clearer that there are caveats in extrapolating their results to much larger masses.

2.6.4 Low-mass dark matter

Because for mχ 1 GeV, µA ' mχ, low-mass dark matter constraints beneﬁt only from a single factor of A2 from coherence. Therefore, the loss of the A2 scaling at large cross sections will be orders of magnitude less severe than the impact from the loss of A4 scaling at larger masses. The momentum transfer is also smaller, so the loss of coherence due to an assumed form factor for the dark matter would be less severe. Contact interactions are still

29 limited by the geometric size of the nucleus, but constraints on light mediators will become a function of mχ [134]. We leave a detailed assessment of the impact of our considerations at low mass to future work. However, we reiterate our caution that constraints set at low masses should carefully state the limitations on extrapolating their constraints to mχ & 1 GeV.

2.7 Conclusions

How do dark matter particles interact with matter? One of the most commonly considered cases to probe is the spin-independent interactions of mχ > 1 GeV point-like dark matter with nuclei. In the literature, a vast array of constraints — based on astrophysical and cosmological tests, as well as direct-detection searches with a wide range of nuclei and overburdens — are all compared to each other in simple plots of the dark matter-nucleon cross section and dark-matter mass. Comparing searches in this way requires the assumption 4 of scaling relations, e.g., σχA ∝ A σχN for mχ mA, that are widely assumed to be model- independent. We systematically examine the validity of the assumptions used to derive these relations, calculating where model independence ends. Figure 2.7 summarizes our results. We ﬁnd:

−32 2 1. For small cross sections, σχN 10 cm , the usual scaling relations are valid, and multiple reasonable models can produce the same scaling relation.

−32 2 −25 2 4 2. For 10 cm . σχN . 10 cm , the assumed A scaling for a contact interaction progressively fails for all nuclear targets as cross sections for heavier nuclei begin to saturate at their geometric cross sections. Experimental constraints on the existence of light mediators prevent simple light mediator models from achieving cross sections in this range at all, such that constraints set in this range of cross sections should be specialized to a model.

−25 2 3. For σχN > 10 cm , dark matter with repulsive interactions cannot be point-like. For all nuclei heavier than helium, even attractive interactions are prevented by unitarity from achieving cross sections more than three order of magnitudes larger than 2 the geometric σχA ' 4πrA limit for a repulsive interaction at typical Milky Way halo thermal velocities v ' 10−3 c. No stable isotope of any atomic nucleus heavier than neon can achieve cross sections even one order of magnitude above the repulsive geometric limit. Simple light mediators are strongly ruled out. Dark matter with cross sections in this range must be composite or exhibit low-energy attractive resonances with light nuclei.

The failure of the scaling relations should influence the design of future dark matter searches. For interactions with cross sections that scale less than linearly with A, such as 30 some models of composite dark matter, dark matter detectors with lighter nuclei are more efficient per unit detector mass. As a result, future direct-detection searches for strongly interacting dark matter may benefit from constructing detectors with light nuclei. Constraints on dark matter parameter space are most useful if they can be compared between different experiments. Where the A4 scaling is not reliable, results need to be recast in terms of specific models. A comprehensive analysis should include clear statements about the mass ranges their results can reasonably be extrapolated to. Because constraints will −32 2 not be the same for different models, plots including cross sections σχN & 10 cm must specify a model, whether it involves a contact interaction, light mediator, composite dark matter, or something else.

Acknowledgements

We are grateful for useful discussions with Laura Baudis, Kimberly Boddy, Juan Collar, Adrienne Erickcek, Vera Gluscevic, Rafael Lang, Hitoshi Murayama, Ethan Nadler, and Juri Smirnov. MCD and CMH were supported by the Simons Foundation award 60052667, NASA award 15-WFIRST15-0008, and the US Department of Energy award DE-SC0019083. CVC and JFB are supported by NSF grant PHY-1714479 to JFB. AHGP is supported by NASA grant ATP 80NSSC18K1014 and NSF grant AST-1615838.

31 23 2 10 4 RA ] 2 10 26 m c [

A 10 29

10 32

10 23 ] 2 10 26 m c [

A 10 29 4 A N repulsive 10 32 attractive

0 20 40 60 80 100 120 140 A

Figure 2.2: Top: Scaling with A for the contact interaction in Sec. 2.3 with |V0| = 1.18 × −5 −1 1/3 10 GeV, computed using k = 0.005A fm , RA = 1.2A fm. We include partial waves −16 up to lmax = 8, which is suﬃcient to converge σχA to ∼ 10 precision. Attractive and repulsive interactions scale similarly, although the scaling deviates from A4 at high A due to form-factor suppression, accounted for here by including the contributions from higher −3 partial waves. Bottom: Same as above, but with |V0| = 1.18×10 GeV, which corresponds to the ‘scaling relations unreliable for A > 12’ line in Fig. 2.6. Repulsive and attractive 2 interactions no longer scale the same way, and both saturate close to 4πRA. The attractive potential shows resonances with A, which are sensitive to the speciﬁc choice of potential. For cross sections approaching the geometric cross section, any scaling with A is highly model-dependent.

32 10 21

2 ] 24 4 R 2 10 A m c [

A 10 27

10 30

10 21

] 24 2 10 m c [

A 27 10 2 V0 repulsive 10 30 attractive 10 6 10 5 10 4 10 3 10 2 10 1 |V0| [GeV]

Figure 2.3: Top: Scaling of cross section with |V0| for A = 4 (helium), calculated using the contact interaction in Sec. 2.3. The cross sections are computed using the analytic partial wave results. For attractive potentials, once |V0| becomes large enough to support quasi- bound states, resonances can increase the cross section by several orders of magnitude, but only in a narrow range. Bottom: Same as above, but with A = 131 (xenon). A larger number of partial waves contribute due to the larger k ∝ A. There are many resonances, but they are not large enough to meaningfully increase the cross section above the geometric limit. Additionally, the resonances are not at the same values of |V0|, which prevents resonances from achieving a large cross section which scales predictably with A, as shown in Fig. 2.2.

33 10 23 A=131

10 25

] A=40 2 m 2 N c r [ 27 A=12 10 A 4 A=4

29 10 A=1

10 31 10 33 10 31 10 29 10 27 10 25 2 N [cm ]

Figure 2.4: Scaling of the nuclear cross section with nucleon cross section for the repulsive −1 contact interaction of Sec. 2.3 at ﬁxed kN = 0.005 fm . The contact interaction cannot achieve nucleon cross sections larger than the geometric cross section, denoted by the vertical red line. The cross section visibly deviates from A4 scaling at the O(1) level for heavy nuclei −32 2 4 even for σχN ' 10 cm , and by the time scaling fails at the O(1) level for He at −28 2 σχN ' 4 × 10 cm , the cross sections for heavy nuclei have completely saturated. The scaling could break down in diﬀerent ways in other models.

34 10 3 10 4 Meson decay 10 5 6 10 2 cm 7 22 10 n-Xe scattering 10

N 8 10 BBN 10 9 2 2 10 cm 2 10 cm 30 26 cm 34 11 0 10 10 1 0 2 5th force 1 m Stellar cooling 8 c 10 12 3 0 1 10 13 1 4 10 m [GeV] Meson decay 10 5

6 2 10 10 26 cm 7 10 n-Xe scattering

N 8 30 2 10 10 cm BBN 10 9

10 2 10 10 34 cm 10 11 5th forceStellar cooling 12 38 2 10 10 cm 10 13 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 1 m [GeV]

Figure 2.5: Top: Elastic scattering cross section contours as a function of mediator mass and coupling strength for the repulsive Yukawa potential in Eq. (2.33). We also show various constraints on the existence of such mediators from Ref. [134]. The largest cross sections −27 2 −9 achieved in unconstrained regions are σχN . 10 cm . For mφ < 10 GeV, ﬁfth-force constraints become many orders of magnitude stronger and dominate other constraints [135]. Bottom: Same as above, but for the momentum-transfer cross section. The largest cross mt −32 2 sections achieved in unconstrained regions are σχN . 10 cm .

35 Cannot be Point-like DM 10 25

No Reliable Scaling For A > 4 10 28

] A > 12 2 m c

[ A > 40 10 31

N A > 131

10 34

37 10 Upper Limit for Light Mediator

1 103 106 109 1012 1015 1018 m [GeV]

Figure 2.6: Summary of theoretically allowed regions for dark matter candidates. For a 4 −32 2 contact interaction, A scaling breaks down for heavy nuclei for σχN & 10 cm , and −28 by σχN & 4 × 10 any scaling between diﬀerent nuclei is model dependent. Here we deﬁne the failure of scaling as setting the LHS of Eq. (2.11) equal to 0.5. This choice approximately agrees with where scaling obviously fails in Fig. 2.4. The breakdown is purely on theoretical grounds. Also shown is the maximum allowed momentum-transfer −4 cross section for a mφ = 10 GeV light mediator using the constraints shown in Fig. 2.5, 4 coincidentally at a comparable scale. For mχ . 10 GeV we have applied a conservative self 2 −25 2 interaction constraint σχχ/mχ < 10 cm /g [155]. For σχN & 10 cm , no viable point-like dark matter candidates exist.

36 10 10 10 13 Cosmology 10 16 Cannot Be Point-Like DM 10 19 10 22 XQC IMP + IMAX + SKYLAB ] 25 2 10 m

c 28

[ 10

Scaling Uncertain N 10 31 10 34 Underground Detectors 10 37 2 2 10 40 A A Scaling Ok 10 43 10 46 1 103 106 109 1012 1015 1018 m [GeV]

Figure 2.7: Claimed constraints from Fig. 2.1, with the problematic regions identiﬁed in Fig. 2.6 highlighted. All existing detector ceiling calculations are deeply in the model- dependent regime, or entirely excluded for point-like dark matter. To the right of the dashed vertical line, the entire (small) direct-detection region must be reanalyzed.

37 Chapter 3 Forecasting Super-Sample Covariance in Future Weak Lensing Surveys with SuperSCRAM

Matthew C. Digman, Joseph E. McEwen, Christopher M. Hirata

Here I present my paper describing a code and formalism designed to forecast the impact of super-sample covariance on future weak lensing surveys [156].

Abstract

The observable universe contains density perturbations on scales larger than any finite volume survey. Perturbations on scales larger than a survey can measure degrade its power to constrain cosmological parameters. The dependence of survey observables such as the weak lensing power spectrum on these long-wavelength modes results in super-sample covariance. Accurately forecasting parameter constraints for future surveys requires accurately accounting for the super-sample effects. If super-sample covariance is in fact a major component of the survey error budget, it may be necessary to investigate mitigation strategies that constrain the specific realization of the long-wavelength modes. We present a Fisher matrix based formalism for approximating the magnitude of super-sample covariance and the effectiveness of mitigation strategies for realistic survey geometries. We implement our formalism in the public code SuperSCRAM: Super-Sample Covariance Reduction and Mitigation. We illustrate SuperSCRAM with an example application, where the modes contributing to super-sample covariance in the WFIRST weak lensing survey are constrained by the low- redshift galaxy number counts in the wider LSST footprint. We find that super-sample covariance increases the volume of the error ellipsoid in 7D cosmological parameter space by 38 a factor of 4.5 relative to Gaussian statistical errors only, but our simple mitigation strategy more than halves the contamination, to a factor of 2.0.

3.1 Introduction

Observational cosmology has experienced a remarkable rebirth as a precision science in the last two decades. In 1998, two teams using Type Ia supernovae found that the expansion of the Universe is accelerating [16, 17]. Galaxy surveys such as the 2dF Galaxy Redshift Survey [157], SDSS [158], BOSS [159], and WiggleZ [160] have provided an independent means of exploring the recent expansion history of the Universe and have dramatically confirmed cosmic acceleration. Meanwhile, the high-redshift Universe has been probed to great precision using the large area surveys of the cosmic microwave background by WMAP [161, 162] Planck [7, 163], ACT [164], and SPT [165]. The cause of the accelerated expansion of the universe is tied to the origin and ultimate fate of the universe. Depending on how the acceleration changes over time, the universe could expand forever, re-collapse, rip itself apart, or perhaps do something else entirely [166, 167]. The most common parametrization for the cause of cosmic acceleration introduces a fluid with positive energy density but negative pressure, called dark energy. Dark energy models p are characterized by an equation of state parameter w ≡ ρ , which relates its pressure p and the energy density ρ [168]. In the simple case where w = −1, dark energy is a cosmological constant, which is the standard ΛCDM cosmological model. The most studied physical mechanisms which could produce a w 6= −1 would also allow w to vary as a function of time, w = w(z) [169]. Current cosmological observations do not give good constraints on w(z) because errors on w(z) at different times are highly correlated [170, 171]. Tomographic weak lensing surveys directly probe the statistical properties of the matter distribution in projection, using sources at a range of redshifts, and hence can be used to constrain w(z) [170, 172]. Several completed and ongoing experiments have measured or will measure weak lensing, including CFHTLens [173], KiDS [174], DES [175], and HSC [176]. Ideally, the precision of weak lensing measurements would be limited primarily by fundamental statistical uncertainties due to their limited samples of galaxies and survey volumes. Such statistical uncertainties will be greatly reduced by the large volumes of future surveys, such as LSST [177], Euclid [178], and WFIRST [179]. However, future weak lensing surveys will also suffer from many sources of observational and astrophysical systematic errors, which will need to be properly understood to avoid systematic biases in the results [180–183]. One source of statistical error for future experiments to consider is the coupling of measured results to matter density fluctuations outside the survey window, called super- sample covariance (SSC). The direct coupling between the amplitude of short-wavelength modes, which fit in the survey window, and long-wavelength modes, which do not, was

39 described as beat-coupling in refs. [184, 185] and found to be the dominant contribution to the matter power spectrum covariance on small scales. Further studies have considered other aspects of the effect and described formalism for analyzing it [186–191]. The SSC effect on lensing observations has been shown to depend strongly on the details of the survey geometry [192]. Survey geometry design is subject to many practical instrument constraints, such as observatory slew times, galactic plane avoidance, and calibration. Future experiments, such as WFIRST, should consider whether optimizing survey design to reduce super-sample covariance is a significant additional consideration. Additionally, missions should consider whether combining other probes of long-wavelength density fluctuations can be a useful means of mitigating their covariances [193, 194]. In this paper, we present a formalism for estimating the SSC contribution to the covariances of future experiments and investigating possible mitigation strategies in a realistic survey geometry. We implement our formalism in a public code, SuperSCRAM: Super- Sample Covariance Reduction and Mitigation.2 The code is intended to be modular and extensible to facilitate the addition of further observables, physical effects, and mitigation strategies. SuperSCRAM provides Fisher matrix forecasts of parameter constraints, which is useful for estimation of the relative magnitude of effects and effectiveness of mitigation strategies, but is not a substitute for a full Markov chain Monte Carlo likelihood code such as CosmoLike [193, 194]. We do not analyze other sources of non-Gaussian covariance, although SuperSCRAM could be extended to include them. Refs. [195–198] have applied a similar harmonic expansion formalism to calculate SSC effects on cluster counts including geometrical effects. This paper focuses on weak lensing observables, although the code’s structure is modular, so an extension to include cluster counts in the future would be straightforward. The structure of the paper is as follows. In section 3.2, we describe the formalism and overall analysis workflow used throughout the paper. In section 3.3, we describe the specific implementation of our formalism in SuperSCRAM. In subsection 3.4.1, we present a practical demonstration of a useful application of SuperSCRAM, in which we assess the effectiveness of embedding the WFIRST weak lensing survey program in the LSST field of view. In subsection 3.4.2 and subsection 3.4.3, we discuss other applications and extensions of SuperSCRAM. Finally, in section 3.5 we discuss our conclusions and the outlook for future work. 2SuperSCRAM is available at https://github.com/mcdigman/SuperSCRAM

40 3.2 Overview

3.2.1 Notation and Deﬁnitions

For our analysis, we define two types of observables, which must be handled in separate ways: short-wavelength (SW) observables, denoted {OI }, and long-wavelength (LW) observables, denoted {Oa}. The cosmological parameters are denoted {Θi}. Finally, the long- wavelength matter density modes are expanded in terms of a set of basis modes {ψα(r)}, P α δ(r) = α δ ψα(r). In this paper, we use indices IJK... for SW observables, abc... for LW observables, ijk... for cosmological parameters, and αβγ... for the density mode basis. Short-wavelength observables, discussed in subsection 3.3.2, are observables which can only directly probe fluctuations of shorter wavelength than the survey’s window, such as the shear-shear lensing power spectrum, or cluster counts. In this paper, they are the “primary” source of cosmological information, which a survey is designed to extract. Long-wavelength matter density fluctuations, with wavelengths larger than the survey window, add covariance to estimators of the short-wavelength modes measured within the window, which ultimately degrades cosmological parameter estimation. Long-wavelength observables, discussed in subsection 3.3.3, are chosen to provide information about the amplitude of the long-wavelength matter density fluctuations, so the degradation of estimates of short-wavelength observables by super-sample covariance can be mitigated. Long- wavelength observables need not originate from the same survey that measures the short- wavelength observables.

3.2.2 Workﬂow

A sketch of the workﬂow of our mitigation procedure is provided in ﬁgure 3.1. To provide cosmological parameter forecasts, we calculate the total Fisher information matrix for the cosmological parameters by combining prior constraints and the information obtained from the short-wavelength observables measured by the survey:

tot prior survey Fij = Fij + Fij , (3.1) where the indices i and j run over all possible combinations of cosmological parameters. For the purposes of this paper, we use the WMAP5 ﬁducial point with a set of forecast prior survey constraints from Planck for Fij , taken from ref. [199]. To calculate Fij , we use

∂OI ∂OJ F survey = F tot , (3.2) ij ∂Θi IJ ∂Θj

I tot IJ −1 where here O are the short-wavelength observable vectors. FIJ = Ctot is calculated

41 Calculate SW observables

Project SSC onto SW observables in speciﬁc geometry Input survey geometry

Combine SW covariance matrices Apply migaon Strategies

Get cosmological parameter forecasts Calculate SSC Fisher matrix in basis

Figure 3.1: Conceptual overview of the basic workflow needed to obtain cosmological parameter forecasts. The specific modules implementing the structure are described in section 3.3 and shown in figure 3.2.

as the inverse of the short-wavelength covariance matrix,

IJ IJ IJ IJ Ctot = Cg + Cng + CSSC. (3.3)

IJ The Gaussian covariance Cg is the lowest order contribution, and would be the only nonzero contribution if the ﬂuctuations in the underlying density ﬁeld were purely Gaussian IJ distributed. The non-Gaussian covariance Cng contains higher order non-Gaussian correlations between short-wavelength modes that occur even if the large scale mean overdensity IJ is zero. The CSSC term contains higher order correlations resulting from the dependence IJ of short-wavelength on long-wavelength modes. We do not consider Cng any further because other analyses have found it is subdominant [200–202], and the focus of this paper is assessing the impact of the SSC contribution. We decompose the long-wavelength modes into a basis as described in Appendix B.1, and calculate the super-sample covariance in our basis using perturbation theory, such that

42 we can write

∂OI ∂OJ CIJ = (F SSC)−1 . (3.4) SSC ∂δα αβ ∂δβ The response of the observable to a density fluctuation ∂OI /∂δα is used to propagate the covariance matrix of the density fluctuations into the covariance matrix for the short- wavelength observables, and encodes the dependence of the SSC term on the size and shape of the survey window function. The calculation of ∂OI /∂δα is described in subsection B.1.1. Other contributions to the SSC term, such as the tidal contributions [203, 204], would be added by incorporating the additional term into ∂OI /∂δα, since the large-scale density SSC modes also determine the large-scale tidal fields. Fαβ is the Fisher information matrix for long-wavelength matter density fluctuations decomposed into our basis, which includes the SSC contribution from perturbation theory given in eq. (B.8), plus information provided by any number of mitigation strategies, indexed {1..n}:

SSC+mit SSC X n Fαβ = Fαβ + Fαβ. (3.5) n The response to the long-wavelength fluctuations and the Fisher information matrices of long-wavelength observables contain explicit dependence on the geometry of the survey. Note that eq. (3.5) implicitly assumes that the long-wavelength observables are statistically independent on large scales. Statistical independence is generally a reasonable assumption if the two observables are coming from different types of measurements or non-overlapping surveys. However, care must be taken to avoid adding the same information twice if observables contain information from overlapping datasets. Note that SSC mitigation, as we consider it here, directly provides information on the specific realization of the large scale modes that are responsible for SSC. This is different from combining the data vectors OI from different surveys, since the latter usually include power spectra or correlation functions and do not provide information on the specific realizations of any modes. It is worth noting that the SSC term is not purely an observational limitation from IJ partial sky coverage, and CSSC would not vanish even for a full sky survey, because an important component of the SSC term is radial. In fact, the improvements to the Gaussian covariance from increasing the sky area may make the SSC term more important relative to the Gaussian covariance, although both terms decrease as the survey gets larger. The speed of decrease of the SSC versus Gaussian covariance can even depend on the survey geometry and scale (see Figure 1 of Ref. [189]). Therefore, the SSC term is an important component of the error budget even for large surveys [202]. The specific mitigation approach implemented in this paper — embedding a survey in a larger survey — would not work in this case, but

43 it is possible that other approaches would.

3.2.3 Eigenvalue analysis

Inspecting the individual values of the elements of the covariance matrix of the cosmolog- ij ical parameters Ctot can give a rough idea of the magnitude of the SSC contamination or effectiveness of a mitigation strategy. However, a more systematic method is required to quantitatively assess the overall impact. The direction of parameter space most contaminated by SSC or improved by mitigation is very difficult to determine by visual inspection of two-dimensional slices of many-dimensional ellipsoids, such as figure 3.3. To assess the overall impact, we use one covariance matrix, as a metric to identify the directions most changed in another perturbed covariance matrix. For our purposes, ij a useful metric is the purely Gaussian covariance in cosmological parameter space, Cg . For this section, we will refer to the covariance in cosmological parameter space with the ij super-sample contribution but no mitigation as CSSC, and the covariance with all mitigation ij applied as Cmit. We then consider the product matrix

i ik −1 M j = (CSSC)(Cg )kj, (3.6)

(i) i i j(k) (k) i(k) which has eigenvalues λ and eigenvectors v j such that M jv = λ v . For this section, Latin indices ijk... run over cosmological parameter space and must be raised or ij −1 lowered by applying the Cg or its inverse (Cg )ij. Indices in parentheses (i)(j)(k)... are ij labels for the eigenvalues and eigenvectors, and are not summed over here. If CSSC is a positive semidefinite perturbation to the metric, all of the eigenvalues satisfy λ(i) ≥ 1, and the largest eigenvalue corresponds to the direction in parameter space most contaminated by ij CSSC. The resulting eigenvalues are dimensionless, and can therefore be compared between i cosmological parametrizations. Note that the matrix M j is not symmetric in general, which is inconvenient for computing eigenvalues numerically, so we instead find the eigenvalues of the matrix −1 kl −1 Mfîˆj = (L )kîCSSC(L )lˆj, (3.7)

iˆj ij ikˆ jkˆ where L is the lower triangular Cholesky decomposition Cg ≡ L L . Mfîˆj is a symmetric matrix with the same eigenvalues λ(i) as Mi and eigenvectors u (j) related to the eigenvectors j î vi(j) of Mi by transforming u (j) = (L−1) vk(j). The transformed eigenvectors then exist in j î kî a new vector space where the metric is 1. Indices in this vector space, îˆjk...ˆ , may therefore be raised and lowered freely. Now, we would like to interpret the most contaminated directions in our physical (j) −1 k(j) (j) parameter space. We define the one-form v˜i ≡ (Cg )ikv . v˜i has the property (i) jk (l) ˆj(i) kˆ(l) (j) v˜j CSSCv˜k = u Mfˆjkû , such that v˜i gives the directions in the physical parameter space most contaminated by super-sample covariance relative to the Gaussian covariance.

44 We require the normalization ukˆ(i)u(j) ≡ 1(i)(j). kˆ The covariance matrix in the basis of directions most contaminated by super-sample covariance is given: (i)(j) (i) kl (j) CeSSC,mit ≡ v˜ k Cmitv˜l , (3.8) ij where Cmit is the covariance matrix with both super-sample covariance and mitigation ap- (i)(j) ij plied. The elements of CeSSC,mit give the covariance relative to Cg in the directions most contaminated without mitigation. If mitigation is perfect, Cij = Cij, we have C(i)(j) = 1(i)(j). mit g emit,Cg ij ij (i)(j) (i) (i)(j) If instead the mitigation had no effect, Cmit = CSSC, we have CeSSC,SSC = λ 1 . The product of the eigenvalues is the ratio of the volumes of the uncertainty ellipsoid with Q (i) and without contamination, (i) λ = det(CSSC)/det(Cg). Note that because changing ij ij Q (i) the cosmological parametrization affects both CSSC and Cg in the same way, (i) λ and the largest eigenvalue λtop are both relatively insensitive to the choice of parametrization, which makes them useful for examining the effect of super-sample covariance in isolation. The product of eigenvalues and top eigenvalue are two useful points of comparison between different survey and mitigation strategies. In subsection 3.4.1, we examine the eigenvalues most contaminated by the addition of (i) (i) the SSC term before and after mitigation, λSSC, and λmit.

3.3 Methods and Code Organization

In this section, we present a framework and code for forecasting super-sample eﬀects on future observations in a survey with a given geometry. SuperSCRAM is written and tested in Python 2.7 and uses the SciPy [205], NumPy [206], and Astropy [207, 208] libraries. Convergence tests were run on the CCAPP condos of the Pitzer and Ruby clusters at the Ohio Supercomputer Center [209]. The overall structure of the important modules in SuperSCRAM is sketched out in ﬁgure 3.2.

3.3.1 Geometries

The base class for survey geometries is the Geo class, which contains basic functionality for setting up the tomographic lensing bins and the resolution bins to be integrated over. Classes that extend Geo for a speciﬁc angular window function must provide a method for calculating the angular area in steradians, angular_area(), and the spherical harmonic coeﬃcients alm, a_lm(l,m). We provide several possible methods of implementing survey geometries. Pixelated geometries extend the PixelGeo class, which allows for an arbitrary pixelation of the sky, though all current geometries use HEALPix [210]. Our PolygonGeo class allows a survey geometry to be input as an arbitrary bounding spherical polygon, with segments of great

45 Input Setup Geometry Maer Power Spectrum Cosmology Details Geo PolygonGeo FASTPT HalofitPk CosmoPie PolygonPixelGeo MatterPowerSpectrum WMatcher LSSTGeo WFIRSTGeo

Observable Setup Basis SW Observables LW Observable SphBasisK ShearPower DNumberDensityObservable ShearShearLensingObservable LensingPowerBase

Obtain Results Parameter Forecasts SW Survey Covariance LW Survey SWSurvey SuperSurvey FisherMatrix LWSurvey MultiFisher SWCovMat

Figure 3.2: Structure of the most important modules used by SuperSCRAM described in this paper to implement the workﬂow described in §3.1.

circles as edges. The primary advantage of this class is that the spherical harmonic decomposition alm can be calculated analytically for such polygons, as described in Appendix B.3. For comparison with pixelated methods also provide the PolygonPixelGeo class, which implements spherical polygons using a HEALPix pixelation. We have implemented several utility geometries which perform useful operations on other geometries. The AlmDifferenceGeo implements the diﬀerence between any two Geo objects, which is necessary for cutting a mask out of a survey window. AlmRotationGeo uses the Euler rotation method described in Appendix B.3 to rotate any Geo to a speciﬁed position on the sky. PolygonUnionGeo and PolygonPixelUnionGeo calculate the union of PolygonGeo and PolygonPixelGeo objects respectively; we have not currently implemented a more general purpose class for calculating unions or intersections between two arbitrary geometries. We also provide a number of demonstration and testing geometries in premade_geos.py. For our demonstration we use the LSSTGeo and WFIRSTGeo, which implement approximations

46 of possible LSST and WFIRST window functions using PolygonGeo objects, combined using the various utility geometries. Various other geometries used for the demonstrations shown in ﬁgure 3.6 include StripeGeo, which describes a spherical rectangle, and CircleGeo, which approximates a circle as a many-sided spherical polygon. For testing purposes, we have implemented FullSkyGeo and HalfSkyGeo, which use the analytic alm for the full sky or an entire hemisphere respectively. Various utility functions, such as those needed to reconstruct the images of geometries in ﬁgure 3.6 from their alm representations, are provided in alm_utils.py, ylm_utils.py, and geo_display_utils.py.

3.3.2 Short-Wavelength Observables

The SWObservable class provides an interface which implementations of specific short- wavelength observables must provide. A short-wavelength observable must implement the method get_dO_I_ddelta_bar, which should calculate ∂OI /∂δ¯ and return it as a NumPy array. All short-wavelength observables currently implemented in SuperSCRAM are subclasses of LensingObservable, which are projected power spectra calculated by ShearPower, as described in Appendix B.2. We treat lensing between each pair of tomographic bins as separate LensingObservable objects. To avoid unnecessarily recomputing power spectra in different tomographic bins, each SWSurvey object creates a single shared LensingPowerBase object to store all the ShearPower objects which its LensingObservable objects will need. To create a specific lensing observable, a LensingObservable subclass, such as the shear- shear lensing power spectrum ShearShearLensingObservable, needs to be provided with two QWeight objects, in this case QShear, which are the functions qA(z) described in Ap- pendix B.2.

3.3.3 Long-Wavelength Observables

In this section, we describe a sample mitigation strategy of embedding a weak lensing survey, such as WFIRST, in a wider but shallower survey of galaxy number counts, such as LSST. The difference between the number density measured by the larger survey inside the lensing survey window, n1, and the number density outside, n2, provides information about long- wavelength density fluctuations. Our observable is therefore ∆n12 ≡ n1 − n2. The number density in a radial bin of the mitigating survey defined by the comoving radius rmin ≤ r ≤ rmax can be written to linear order in perturbation theory as

Z Z rmax 3 2 n1 = 3 3 dΩ r drn(r) [1 + b(r)δ(r, Ω)] (rmax − rmin)Ω1 Ω1 rmin " # 3 Z Z rmax X = dΩ r2drn(r) 1 + b(r) δαψ (r, Ω) , (3.9) (r3 − r3 )Ω α max min 1 Ω1 rmin α

47 where b(r) is the linear bias, n(r) is the expected number density of galaxies at comoving P α distance r, Ω1 is the angular area of the window, and δ(r, Ω) = α δ ψα(r, Ω) is a density ﬂuctuation expanded in our basis described in Appendix B.1. For the number density of galaxies as a function of redshift in the mitigation survey, we use the expected number densities for the LSST 1-year survey to a depth of i < 24 from ref. [211], which gives approximately 18 galaxies/arcmin2. We have deliberately chosen a very conservative use of LSST. We obtain the bias by abundance matching to the Sheth- Tormen halo mass function, as described in Appendix B.4. Assuming the wider survey has uniform sensitivity as a function of sky position, we have the response to long-wavelength density perturbations: ∂∆n ∂n ∂n 12 ≡ 1 − 2 (3.10) ∂δα ∂δα ∂δα Z rmax Z Z 3 2 1 1 = 3 3 r dr n(r)b(r) dΩψα(r, Ω) − dΩψα(r, Ω) . rmax − rmin rmin Ω1 Ω1 Ω2 Ω2 The “noise” in the long-wavelength observable is the shot noise, since for our application the clustering of the galaxies due to the underlying large-scale matter density ﬁeld is the signal of interest. The mean number density of galaxies in this slice is

Z rmax 3 2 hni = 3 3 n(r)r dr. (3.11) (rmax − rmin) rmin We can calculate the variance under the assumption that the shot noise is Poissonian:

1 1 N12 ≡ Var (n1 − n2) = hni + , (3.12) V1 V2

3 3 where Vi ≡ (rmax − rmin)Ωi/3. In real samples, the long-wavelength shot noise may deviate from the Poisson prediction due to non-linear clustering, exclusion, and satellite galaxies (see, e.g., ref. [212] for an extensive discussion). For our example application in section 3.4, 3 the mitigation performance is only weakly dependent on N12. Then, we can write the Fisher matrix for this mitigation strategy ∂∆n ∂∆n F mit = 12 (N )−1 12 . (3.13) αβ ∂δα 12 ∂δβ This long-wavelength observable is implemented in the DNumberDensityObservable class, which extends the general speciﬁcations for a long-wavelength observable, LWObservable. Subclasses of LWObservable should implement the method get_fisher, which DNumberDensityObservable obtains using eq. (3.13) by numerically integrating

3Increasing the galaxy number count survey’s limiting magnitude from i < 24.1 to i < 25.1 more than doubles the number of galaxies, but reduces the volume of the error ellipsoid in our 7D cosmological parameter space by < 1%.

48 eq. (3.10) as described in subsection B.1.1. Alternatively, to use the more eﬃcient numerical Fisher matrix manipulation method described in subsection 3.3.6, an LWObservable may instead implement a get_perturbing_vector method, which for DNumberDensityObservable ∂∆n12 −1 returns ∂δα and (N12) for the perturbing vector V and inverse variance K respectively. −1 For the current implementation (N12) is required to be diagonal. Which method should be used is toggled by setting the LWObservable object’s fisher_type variable to True to use get_fisher and False to use get_perturbing_vector.

3.3.4 Cosmological Parametrizations

In order to properly calculate the response of a set of short-wavelength observables {OI } to cosmological parameters {Θi}, ∂OI /∂Θi, we must select a cosmological parametrization. The parametrization is speciﬁed in the CosmoPie module, along with a set of rules for calculating derived parameters, so that the parametrization can easily be interchanged. Parameters relating to the dark energy equation of state w(z) are handled separately from 2 2 2 the rest. By default, SuperSCRAM uses {Ωmh , Ωbh , Ωdeh , ns, ln As} as its basic set of parameters, where Ωm, Ωb,, and Ωde are the total matter, baryon, dark energy densities respectively. This parametrization is chosen to be compatible with the parametrization in

[199]. For simplicity, we set Ωk = 0. If we changed the ﬁducial model to a curved Universe, then we would have to build the basis modes ψα(r) for a curved Universe, using the same ultraspherical Bessel functions that appear in CMB anisotropy calculations [213]. We also P ﬁx the neutrino masses to be mν = 0 in the present implementation.

3.3.5 Matter Power Spectrum

SuperSCRAM provides several interchangeable sample implementations of the matter power spectrum accessible using a MatterPowerSpectrum object, including Python implementations of the Takahashi [214] and Casarini [215] revisions of the Halofit [216] model, linear matter power spectra from CAMB [217], and the FAST-PT [218, 219] implementation of the one-loop power spectrum. Additionally, we have extended both Halofit and FAST-PT to facilitate an arbitrary w(z) using the same procedure used in refs. [215, 220] to extend the Halofit model as described in Appendix B.7, although we have not attempted to validate the resulting power spectra through simulations. In this paper, we report only results for the Halofit model.

3.3.6 Fisher Matrix Manipulation

Fisher and covariance matrix manipulations are primarily handled by the FisherMatrix class. The FisherMatrix class is optimized to eﬃciently perform the manipulations of Fisher matrices required by SuperSCRAM with sizes up to the memory limitations of the

49 machine it runs on. For memory efficiency, it can internally store either a Fisher matrix, covariance matrix, or their Cholesky decompositions. The representations can be exchanged in place internally, and all external class methods intended for use by other modules can be used regardless of the internal representation. For some functions, the choice of internal representation can affect run time significantly, especially for large matrices. The overall logic for managing the Fisher matrices for the short and long-wavelength observables and cosmological parameters is handled by the MultiFisher class. A MultiFisher object takes an LWBasis object, an SWSurvey object, and a LWSurvey object. From these objects, it extracts the appropriate projection matrices for converting between the space of long-wavelength and short-wavelength observables, and the space of short-wavelength observables and cosmological parameters, ∂OI /∂δα and ∂Oi/∂Θi respectively. It then performs the necessary projections and applies priors internally. As an example, if we have already created a Fisher matrix object called multi_f, we can use

f_set = multi_f.get_fisher_set(include_priors=False)

to obtain a set of FisherMatrix objects in cosmological parameter space containing just the Gaussian covariance and total covariance with and without mitigation. To then obtain the eigensystems with and without mitigation using the Gaussian covariance as a metric, as described in subsection 3.2.3, we can use eig_set = multi_f.get_eig_set(f_set)

Long-Wavelength Convergence

Obtaining well converged results for the super-sample covariance in our formalism may require a long-wavelength Fisher matrix larger than can be stored in memory. However, αβ because the long-wavelength covariance matrix CSSC as written in Appendix B.1 is block diagonal, it is possible to calculate the contribution to the short-wavelength covariance IJ matrix CSSC from each block individually, which requires individual matrices only as large 2 as (2lmax +1) . However, taking advantage of the block diagonal covariance matrix prevents directly perturbing the Fisher matrix with mitigating information as in eq. (3.5). Provided it is possible to decompose

mit T a b Fαβ = (V KV)αβ = V αKabV β, (3.14)

as in eq. (3.13),4 we may instead use the Woodbury matrix identity

αβ −1 T −1 αβ αβ T −1 T −1 αβ CSSC+mit = [(CSSC + V KV) ] = CSSC − [CSSCV (K + VCSSCV ) VCSSC] . (3.15)

4The implementation in SuperSCRAM has a diagonal K, but this is not essential to the formalism.

50 αβ Here CSSC+mit is not necessarily block diagonal, so storing it would require a full-size matrix. However, the matrix ∂OI W aI ≡ V a Cαβ (3.16) α SSC ∂δβ αβ aα a βα can be computed without storing the full CSSC, as can X ≡ V βLSSC, using the fact that αβ αβ the Cholesky decomposition LSSC of a block diagonal matrix CSSC is also block diagonal. We can then write ∂OI ∂OJ CIJ = Cαβ − W aI [(K−1 + XXT)−1] W bJ , (3.17) SSC+mit ∂δα SSC ∂δβ ab which can now be computed without ever storing a full-size matrix. In addition to allowing better-converged results due to the much larger number of practically usable basis elements, this procedure is also orders of magnitude faster and is more numerically stable than perturbing the Fisher matrix directly. This procedure is implemented in the SphBasisK module. To allow this procedure to be used, all LWObservable objects must set fisher_type=False and create a get_perturbing_vector method, which returns V and K, for all long-wavelength observables in the survey. We implement this functionality for our DNumberDensityObservable class.

3.3.7 Long-Wavelength Basis

A basis for long-wavelength observations can be speciﬁed as an extension of the LWBasis class. A long-wavelength basis must at minimum implement get_fisher(), which returns a FisherMatrix object, and get_dO_I_ddelta_alpha(geo,integrand), which calculates ∂OI /∂δα by integrating an observable input in a grid [∂OI /∂δ¯](z), where the grid should correspond to the ﬁne z grid of a given Geo object. The long-wavelength basis described in Appendix B.1 is provided by the SphBasisK class.

3.4 Applications

3.4.1 Example Application: WFIRST+LSST

As a demonstration, we calculate the importance of the super-sample covariance in a simulated WFIRST weak lensing survey footprint, with mitigation from a simulated LSST footprint. For the LSST footprint, we use a simpliﬁed survey window from +5◦ to −65◦ equatorial latitude spanning 360◦ in longitude, with a ±20◦ mask around the galactic plane, which gives about 13600 deg2 of usable survey area. For the WFIRST footprint, we use an approximately 2100 deg2 footprint, depicted along with the LSST footprint in the right panel of ﬁgure 3.6. Neither footprint is necessarily what will be implemented; for example, the LSST DESC has proposed an extended footprint going farther north [221].

51 For LSST, we use the forecast number density of galaxies for the LSST year 1 dataset with i < 24.1 from ref. [211], which gives a total of about 18 galaxies/arcmin2. We choose the i < 24.1 cutoff to give a conservative estimate of the number of galaxies with good photometric redshifts; by the time of the WFIRST analysis, the photometric redshifts for these bright objects should be very good. We use 4 evenly spaced redshift bins of width ∆z = 0.3 from z = 0 to z = 1.2, which should be very large compared to the photometric redshift uncertainties. For the lensing source galaxies, we use the WFIRST weak lensing sources, using typical High Latitude Survey parameters of 5 exposures of 140 s each. The source densities were computed using v15 of the WFIRST Exposure Time Calculator [222] and the Phase A throughput tables.5 The input galaxy catalog was from the CANDELS GOODS-S region [223], using the photometric redshifts from ref. [224]. Cuts were applied requiring detection S/N of the galaxy > 18, ellipticity error per component < 0.2, and resolution factor > 0.4 in the convention of ref. [225]. These cuts leave about 34 galaxies/arcmin2 for H band only. The actual source density will be larger in the combined images from all 4 filters, but, for simplicity, we focus only on the H band. The results are most sensitive to the number density chosen for the WFIRST lensing source galaxies because increasing the number of sources decreases the Gaussian covariance. These number densities should give an adequate approximation of the survey galaxy number densities for purposes of this demonstration. The intent of this demonstration is to give a rough idea of the impact of super-sample covariance alone on a realistically shaped survey footprint and to give a rough idea of how much improvement could be expected from a simple mitigation strategy. It is not intended to be a full forecast of the actual constraints WFIRST will be able to achieve. In particular, we do not include any systematic or non-Gaussian effects beyond the super-sample covariance. This is because our philosophy is to compare each added term in the error budget (such as SSC) to the irreducible statistical error bars, and not to the combination of all other uncertainties where we have mitigation efforts in progress. In Figures 3.3 and 3.4, we show traditional triangle plots of our constraints with different reasonable priors applied. It is apparent from these plots that super-sample covariance has an effect. However, the overall magnitude is difficult to read off, and different choices of priors change the apparent effect of super-sample covariance, which is undesirable for comparison purposes. Instead, we recommend comparing the magnitude of the effect in the most contaminated directions in parameter space, as described in subsection 3.2.3. The comparison is shown in figure 3.5, with the corresponding effect summarized numerically in Table 3.1. From figure 3.5, it is much more apparent that there exist directions significantly affected by super-sample covariance. Additionally, our formalism removes the dependence on the choice of priors by isolating the effect of super-sample covariance on the constrain-

5See https://wfirst.gsfc.nasa.gov/science/WFIRST_Reference_Information.html

52 ing power of the survey. Figure 3.5 and Table 3.1 show more clearly that a mitigation strategy can signiﬁcantly reduce the amount of constraining power lost due to super-sample covariance.

(i) top 2 2 2 Π(i)λ λ ncs Ω\mh Ω[bh Ω\deh ln\(As) wc0 wca CSSC 20.26 9.94 329.3 5051.1 -8425.4 -390.9 269.4 -206.3 -58.6 Cmit 4.02 2.93 271.0 4319.2 -7132.2 -346.2 229.2 -176.5 -48.9

Table 3.1: Results from a well converged run of SuperSCRAM with our example WFIRST −1 and LSST survey footprints. kcut = 0.0814 Mpc and zmax = 3, as described in Ap- pendix B.1. This choice of kcut requires 10, 286, 527 basis modes, approximately the maximum number of basis modes for SuperSCRAM running with 1 TB of available RAM. For the short-wavelength observables, we use 20 logarithmically-spaced l bins 30 ≤ l < 5000. The product of eigenvalues, eigenvalue in most contaminated direction, and the coeﬃcients describing the most contaminated directions are as described in subsection 3.2.3. The con- (i) vergence of Π(i)λ with respect to kcut is . O(1%). The mitigation reduces the volume of parameter space contaminated by SSC by approximately a factor of 5. The most contaminated direction does not change drastically after mitigation.

3.4.2 Sensitivity to Survey Geometry

One application of SuperSCRAM is to test the impact of varying the survey geometry. Previ- ous analytic studies [193] have for the most part considered only circular survey geometries although see, e.g., [192, 226]. Results comparing several different survey geometries are shown in figure 3.6. The magnitude of the effect varies at approximately the O(10%) level between different contiguous, reasonably compact high latitude survey geometries. Because more compact geometries produce a smaller super-sample covariance, estimates using only a circular geometry may tend to understate the magnitude of the effect. The differences between geometries are small enough that it is possible that including additional super- sample effects beyond the density contrast effect considered here, such as tidal effects and redshift-space distortions, could change the relative ordering of the effect on different survey geometries. Because the mitigation reduces the super-sample contamination far more than the differences between any reasonable geometries, the possibility of applying mitigation strategies appears to be a more important survey-design consideration than the detailed survey geometry for controlling super-sample covariance.

53 1.3e+02 ssc+mit+g

s ssc+g

n 7.6e+01 g

2.3e+01

4.6e-04 2 h

m 0.0e+00

-4.6e-04

8.8e-05 2 h

b 0.0e+00

-8.8e-05

3.8e-02 2 h

e 0.0e+00 d

-3.8e-02

1.2e-02 ) s A

( 0.0e+00 n l

-1.2e-02

1.4e-01 0 0.0e+00 w

-1.4e-01

2.6e-01 a 0.0e+00 w

-2.6e-01

-2.1e-03 0.0e+00 2.1e-03 -4.6e-04 0.0e+00 4.6e-04 -8.8e-05 0.0e+00 8.8e-05 -3.8e-02 0.0e+00 3.8e-02 -1.2e-02 0.0e+00 1.2e-02 -1.4e-01 0.0e+00 1.4e-01 -2.6e-01 0.0e+00 2.6e-01 2 2 2 ns mh bh deh ln(As) w0 wa

Figure 3.3: Impact of super-sample covariance and mitigation on cosmological parameter constraints. Constraints in this plot use ∆χ2 = 2.3, with a set of forecast Planck constraints from the WMAP5 fiducial point as cosmological priors included [199]. Although it is apparent that super-sample covariance has an effect, the appearance of this plot is very sensitive to the choice of parametrization and the priors applied, as can be seen by comparison to figure 3.4. Therefore, it is not possible from this plot alone to accurately assess the magnitude of the super-sample effect. Figure 3.5 is more useful for isolating the effect of super-sample covariance.

3.4.3 Other Applications

A variety of possible applications would be straightforward to implement but are not currently included. Additional short-wavelength observables, such as galaxy-galaxy lensing and galaxy power spectra, could be added to better understand the full constraining power of a weak lensing survey. Many more mitigation strategies are possible. The current mean number density observable is only a simple demonstration of the information that can be extracted from a larger survey and does not include information about galaxy number density as function of sky position. Additionally, CMB lensing could be useful to constrain over-densities at higher redshifts. The mean tangential shear could also provide additional information.

54 1.3e+02 ssc+mit+g

s ssc+g

n 7.7e+01 g

2.3e+01

4.5e-04 2 h

m 0.0e+00

-4.5e-04

8.8e-05 2 h

b 0.0e+00

-8.8e-05

1.0e-02 2 h

e 0.0e+00 d

-1.0e-02

1.2e-02 ) s A

( 0.0e+00 n l

-1.2e-02

5.3e-02 0 0.0e+00 w

-5.3e-02

1.4e-01 a 0.0e+00 w

-1.4e-01

-2.1e-03 0.0e+00 2.1e-03 -4.5e-04 0.0e+00 4.5e-04 -8.8e-05 0.0e+00 8.8e-05 -1.0e-02 0.0e+00 1.0e-02 -1.2e-02 0.0e+00 1.2e-02 -5.3e-02 0.0e+00 5.3e-02 -1.4e-01 0.0e+00 1.4e-01 2 2 2 ns mh bh deh ln(As) w0 wa

Figure 3.4: The same run as figure 3.3, but with an additional 1% prior applied to h. The apparent effect of super-sample covariance on cosmological parameter space is now 2 significantly different. Ωdeh is now well enough constrained that the super-sample effect is hardly noticeable, while w0 and wa now appear significantly contaminated by super-sample covariance, where they appeared much less contaminated previously. Because it is impossible to know at this stage the exact set of priors that an experiment like WFIRST will apply, this plot and figure 3.3 are both plausible forecasts. Because the conclusions about the impact of super-sample covariance drawn from such plots can vary significantly within the range of reasonable analysis choices, we recommend instead that the super-sample effect be assessed directly using our formalism, as in Table 3.1 and figure 3.5.

It would be useful to compare the results of our Fisher matrix method to a Monte Carlo code. To achieve more realistic forecasts, photometric redshift uncertainties and other systematic and non-Gaussian eﬀects would need to be implemented. SuperSCRAM can be used to investigate the contribution of super-sample covariance to the tradeoﬀ between a wider and a deeper survey. A more detailed model of the survey’s sensitivity as a function of redshift would be necessary to draw useful conclusions from this possible application.

55 mit

2.0e+00 no mit g

1.0e+00 1

v 0.0e+00

-1.0e+00

-2.0e+00

-2.0e+00 -1.0e+00 0.0e+00 1.0e+00 2.0e+00

Figure 3.5: The covariances CeSSC,g, CeSSC,SSC, CeSSC,mit, as defined in eq. (3.8) for the two directions in cosmological parameter space most affected by super-sample covariance. The Gaussian 1 − σ contour is a circle by construction, and the unmitigated super-sample component is diagonal, such that the axes of the ‘no mit’ ellipse align with the axes of the plot. The rotation of the ‘mit’ ellipse due to mitigation shows that the most contaminated direction has changed somewhat after mitigation, which is shown in Table 3.1. Nearly all of the super-sample effect is concentrated to these 2 combinations of parameters. Note that the magnitude of the contamination is not apparent from figure 3.3. Additionally, this plot is not drastically affected by changes to the parametrization, such as fixing wa = 0, while such changes affect figure 3.3 significantly.

56 Strip Circle WFIRST

g, SSC 21.98 20.20 20.26

g, mit 3.92 4.05 4.02

Figure 3.6: Three example geometries tested, with results from a run of SuperSCRAM with each geometry. The black region is the LSST-like survey window, with white cutouts showing three different possible geometries for the WFIRST-like survey with identical angular areas 2098.2 deg2. All parameters other than the angular window are the same as the run summarized in Table 3.1. Without mitigation, the results distinctly favor more compact survey geometries. Mitigation may be more effective for extended geometries, such that the overall constraint with mitigation is slightly better for the "Strip" geometry, although it should be noted that our mitigation strategy is primarily for demonstration purposes and cannot be expected to precisely reflect the actual performance of a realistic mitigation strategy. The more significant result is that, provided WFIRST is completely embedded in an LSST-like survey window, the specific shape of the WFIRST survey window is largely a secondary consideration for the purposes of mitigating super-sample covariance.

3.5 Conclusions

In this paper, we have developed a Fisher matrix formalism and presented the code Su- perSCRAM for evaluating the impact of super-sample covariance on a realistically shaped weak lensing survey geometry. We have further applied our formalism to investigate the possibility of obtaining information from wider, shallower surveys to reduce the impact of super-sample covariance. Overall, we find that mitigation strategies can improve constraining power enough to merit serious consideration for next-generation weak lensing surveys. Further work is needed to investigate other contributions to super-sample covariance beyond the density contrast effect discussed in this paper, such as tidal effects and redshift- space distortions. Including those effects would improve the ability to make detailed conclusions about the relative effectiveness of different survey geometries. Additionally, it would be productive to investigate a wider array of mitigation strategies. In the simplest case, our mean number density observable does not exhaust the information available from a larger wider survey, as a more detailed analysis including observed number density as a function of sky position would likely improve that observable. Additionally, CMB lensing could potentially provide information about densities at much higher redshifts than can be obtained by a shallow galaxy survey, which could significantly improve con-

57 straining power. It may also be possible to use the tangential shear and other weak lensing related observables as further mitigating observables.

Acknowledgments

MCD and CMH were supported by the Simons Foundation award 60052667, NASA award 15-WFIRST15-0008, and the US Department of Energy award DE-SC0019083 in the prepa- ration of this work. We thank Elisabeth Krause for providing scripts to run CosmoLike for calibration of SuperSCRAM, as well as for general discussions. We thank Tim Eiﬂer, Olivier Doré, Xiao Fang, and Benjamin Buckman for useful feedback. The computations in this paper were run on the CCAPP condos of the Ruby and Pitzer Clusters at the Ohio Supercomputer Center. This work is based in part on observations taken by the CANDELS Multi-Cycle Treasury Program with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.

58 Chapter 4 LISA Galactic Binaries in the Roman Microlensing Survey

Matthew C. Digman, Christopher M. Hirata

Abstract

Short-period galactic white dwarf binaries detectable by LISA are the only guaranteed persistent sources for multi-messenger gravitational-wave astronomy. Large-scale surveys in the 2020s present an opportunity to conduct preparatory science campaigns to maximize the science yield from future multi-messenger targets. The Roman microlensing survey will image seven ﬁelds in the galactic bulge approximately 40000 times each. Although the cadence is optimized for detecting exoplanets via microlensing, it is also capable of detecting eclipsing white dwarf binaries. In this paper, we present forecasts for the number of short-period binaries the WFIRST microlensing survey will discover and the implications for the design of electromagnetic surveys.

4.1 Introduction

Short-period galactic white dwarf binaries are the only guaranteed persistent targets for multi-messenger astrophysics with near-future gravitational-wave observatories. Such binaries are expected to be among the most numerous targets for the Laser Interferometer Space Antenna (LISA), a space-based mHz gravitational wave observatory scheduled to launch in 2034. Because most such binaries will evolve minimally over the next 10-20 years, electromagnetic observations performed before LISA launches will help establish baselines for the evolution and behavior of such systems and enhance LISA’s multi-messenger science yield. Multi-messenger observations of white dwarf binaries provide unique opportunities to study the astrophysics, formation, and tidal evolution of such systems. Combining gravitational-wave and electromagnetic data for well-characterized binaries can also be used to test for deviations from general relativity and augment LISA’s calibration. Galactic white 59 dwarf binaries are expected to produce the limiting foreground of gravitational-wave noise at mHz frequencies. Improved population modeling based on electromagnetic studies of these binaries can help mitigate the noise foreground, improve the characterization of all LISA’s gravitational-wave targets, and further augment the multi-messenger science yield. Many current and near-future electromagnetic surveys will provide archival data of ben- eﬁt to multi-messenger science with LISA, including but not limited to Roman, LSST, Euclid, DES, DESI, GAIA, ZTF, TESS. In this work, we focus on the potential of the already-planned microlensing survey aboard the Nancy Grace Roman Space Telescope to detect a population of white dwarf binaries in the galactic bulge that are unlikely to be detected by any other near-future electromagnetic survey.

4.2 The surveys

4.2.1 Roman microlensing survey

The Roman mission, planned for launch in 2025 or 2026, will carry a 2.4 m telescope to the Sun-Earth L2 Lagrange point. Its primary instrument, the wide-field instrument, will have 18 near-infrared detector arrays, each consisting of 4088 × 4088 pixels at a scale of 0.11 arcsec per pixel, a total field of view of 0.28 deg2. Its filter wheel will carry seven imaging filters and two dispersers for slitless spectroscopy. Because the solar arrays are fixed on the side of the observatory, the telescope can point anywhere in the range of 54–126◦ from the sun. The Roman microlensing survey [227] plans to make repeated observations of several fields in the Galactic bulge. The microlensing survey’s primary science objective is to statistically characterize the population of extrasolar planets near or outside the Einstein radius of the lens stars by observing perturbations in the host star’s light curve. The fields will be observed in ∼ 72 day long ‘seasons’ during periods when the Galactic bulge’s orientation is favorable relative to the sun, as seen from L2. In the reference observing scenario, these observations will be carried out mainly in the Wide filter (0.93–2.00 µm), which provides a higher photon count rate and, consequently, signal-to-noise than the standard-width imaging filters. The observatory will cycle through 7 neighboring fields, with an exposure time of 52 s per field, and complete a full cycle in ∼ 900 s.6 We expect that ∼ 6 such seasons will be possible in the 5-year Roman primary mission. More may become possible in an extended mission (there is sufficient propellant for ≥ 10 years, and the infrared focal plane is passively cooled, so its lifetime will not be limited by cryogen). During bulge observations, the largest source of noise in most Roman pixels will be Poisson noise from bulge stars (there are no empty pixels), followed by the zodiacal light and then instrument backgrounds.

6[227] used an exposure time of 47 s; this has been increased to 52; s as of the August 2019 review of the Roman Design Reference Mission.

60 Roman will take advantage of recent improvements in communications infrastructure and provide a data downlink rate of ≥ 11 terabits per day, allowing it to downlink every pixel. This downlink rate makes Roman imaging useful for both blind searches for variable sources and searches where the precise source position is not known in advance. The near- infrared detectors can be read non-destructively, so up to 6 sub-exposures out of each 52 s exposure can be downlinked.

4.2.2 LISA

LISA is a constellation of three identical spacecraft orbiting the sun in a cart-wheeling triangle with 2.5 million kilometer arms. Drag-free attitude control systems will hold the gold-platinum test masses at the end of each arm in near-perfect geodesic motion. Laser links between the test masses will allow precise measurement of changes in the relative lengths of the arms. The technique of time-delay interferometry allows the individual phasemeter readouts from the laser links to be interfered in post-processing to construct polarization- sensitive observations of millihertz gravitational waves. LISA’s nominal primary mission is four years, with a possible extension to ten years. Because LISA sources can be observed continuously for years, far longer than is possible with ground-based gravitational-wave detectors, LISA will obtain comparatively good source localizations, opening opportunities to search for faint electromagnetic counterparts for a wide variety of sources. Detecting a source both electromagnetically and gravitationally will break numerous parameter degeneracies, in particular improving measurements of the distance to the source. The primary scientific observable for LISA is the strain in three time-delay interferometry channels as a function of time. While the exact distribution of sources between different classes depends on highly uncertain population modeling, it is expected that LISA will have 10s of thousands of sources observable at any given time, presenting a unique data processing challenge. The bulk of these sources will likely be galactic white dwarf binaries, which will emit gravitational waves at a nearly constant source frequency for the duration of the LISA mission. For high signal-to-noise binaries, high-quality localizations of these sources will be obtained from a combination of Doppler shifting of the observed gravitational-wave frequency as the constellation orbits the sun and the variation in the observed strain amplitudes of the two observable gravitational-wave polarizations due to the rotating antenna gain pattern. Over a dozen already-known galactic white dwarf binaries are guaranteed sources at LISA’s design sensitivity. These are called ‘verification binaries.’ By the time LISA launches, several already-planned time-domain electromagnetic surveys will likely have discovered many more.

61 4.3 Methods

4.3.1 Binary population synthesis

To generate the populations of binaries in the Roman microlensing survey, we use a modified version of COSMIC7 [228], an updated version of BSE [229]. We generate a population of binaries with fixed metallicity Z = 0.018 and a delta-function burst of star formation 8 Gyr ago for stellar ages. Primary masses are drawn from a Kroupa IMF [230] with m1 < 100 m , where m is the mass of the sun. Binaries with m1 < 0.95 m are counted towards the total mass in the microlensing field, but not evolved, because lighter primaries never form compact objects with the physics encoded in COSMIC. We choose secondary masses uniformly from

0 < m2 < m1, although we do not evolve systems with m2 < 0.5 m because they also do not form compact binaries. We select initial eccentricities from a thermal eccentricity distribution, fe = 2e [231]. We set the initial semi-major axis a to be flat in log space from 6 0.5rL < a < 10 r , where rL is the Roche limit of the system, and r is the radius of the sun. We use a modified-Mestel cooling law for white dwarfs, and leave all other parameters to the COSMIC defaults. Higher multiple systems, such as hierarchical triple or quadruple systems, are beyond the scope of this work. The existence of multi-star systems could produce a significant enhancement to the number of LISA-detectable short-period binaries. However, blended starlight from brighter main-sequence companions may make identifying such systems in electromagnetic searches more difficult in some cases. After evolving the systems to the present, we discard all systems which are no longer binaries due to supernovae, mergers, or disruption events. We then discard all systems with 5 final orbital period Pf > 10 s, which are unlikely to be LISA sources, recording only their final stellar mass. Based on the output of a run of TRILEGAL, we expect the total mass of binaries in 8 the Roman microlensing fields to be ∼ 1.5 × 10 m ." We calibrate the number of systems evolved with COSMIC to approximately match the expected mass in the field; drawing binaries from the initial mass distribution until 20, 000, 000 pass our initial mass cuts gives approximately the correct total final mass of binaries. We then assign a new temperature to each binary that survives our output cuts for each version of our simplistic tidal heating model described in Sec. 4.3.2. For the radius of the system, we use the output core radius from COSMIC. Although the temperature and radius chosen this way are not generated from a self-consistent prescription, using the core radius at the lower temperature predicted by COSMIC should give a conservative estimate of the system’s total luminosity. Residual fusion and past accretion episodes are likely sources of additional heat, further increasing electromagnetic detectability. Such mechanisms are

7https://cosmic-popsynth.github.io/

62 beyond the scope of this work. Once the temperature and radius are assigned, we calculate the atmospheric spectra as described in Sec. 4.3.3.

4.3.2 Tidal heating

We adopt the simple tidal heating prescription described in [232]. The contribution to the temperature due to tidal interactions alone, assuming tidal heat is instantaneously re- radiated, is given: 3πκm 1/4 Ttide = , (4.1) 2σPorbPcrittgw where m is the mass of the white dwarf, tgw = 3/2Porb/|P˙orb| is the timescale for the binary to decay due to gravitational waves, κ = I/(mr2) is the coefficient of the moment of inertia, and Pcrit is some characteristic critical period which parametrizes the reduction in tidal heating as the spin period synchronizes to the orbital period. For simplicity, we adopt Pcrit = 3600 s for our default model. We use κ ≈ 0.20306, appropriate for an n = 1.5 polytrope. Once we have the tidal contribution to the temperature, we add the temperature from 4 4 1/4 the cooling age as predicted by COSMIC, Teff = Tcool + Ttide . In practice, Ttide is always larger for the systems of interest here, so including the residual energy from the cooling age of the white dwarf binary makes little difference in Teff. A recent history of accretion could make a more significant contribution to residual heat; the best-fit parameters for the recently detected J1539 [232] suggest it is a detached white dwarf binary. However, it is much hotter than the cooling age or tidal heating alone could explain, suggesting a recent or ongoing accretion episode. Active accretion is beyond the scope of this work, although such systems likely account for a large fraction of the systems that could realistically be detected by the Roman microlensing survey. Additionally, it is possible for low-mass He white dwarfs to maintain an outer hydrogen- burning shell for much longer than a Hubble time, and therefore be much hotter than their cooling age would predict. Especially for younger white dwarfs, flashes of fusion can also heat and inflate the envelope, substantially enhancing the luminosity. Both effects are beyond the scope of this work.

4.3.3 White dwarf atmospheres

For the white dwarf atmospheres, we interpolate a grid of precomputed spectra at various 8 values of TEﬀ and log(g) predicted by [233] for pure hydrogen atmospheres. We use a simple linear interpolation between the four nearest points to the requested TEﬀ and log(g).

8Precomputed spectra were downloaded from http://svo2.cab.inta-csic.es/theory/newov/index. php?model=koester2

63 In the Roman F146 ﬁlter, the spectra are eﬀectively featureless, allowing this interpolation procedure to produce an adequate approximation. Properly accounting for the potential systems with hydrogen depleted atmospheres would have little to no impact on our results. For the handful of systems with parameters outside the precomputed grid, we default to a blackbody spectrum. No systems predicted to be detectable in the Roman microlensing survey have a luminosity primary with parameters outside the precomputed grid. From the atmospheres, we obtain expected brightness for both systems, which are then used to generate light curves, as described in Sec. 4.3.4.

4.3.4 Light curve generation

Light curves for eclipsing white dwarf binaries are generated using ELLC [234]. ELLC models the white dwarfs as triaxial ellipsoids with a Roche potential [235]. We treat neutron stars and black holes as spherical. We adopt a linear limb darkening law with fixed coefficients ldc1 = ldc2 = 0.5. For simplicity, we do not include gravity darkening, Doppler boosting, or heating by companions, although all three effects could be significant. For light curve generation, we also do not include a period derivative, as including it would not affect the detection efficiency. We also set the eccentricity e = 0 for light curve generation. In the absence of multi-star interactions, appreciably nonzero (e > 0.1) eccentricities in LISA’s band are expected to occur only for systems which contain a neutron star or black hole that received a large natal kick from a supernova. Systems with COSMIC-predicted eccentricities e > 0.1 contribute < 0.1% to our predicted number of systems detectable by LISA+Roman. Accounting for eccentricity would make no difference to our reported results. After generating the light curve with ELLC, we apply a dust extinction correction to the light curve as described in Sec. 4.3.5. Finally, we draw the actual observed electron counts for a given exposure, as described in Sec. 4.3.6.

4.3.5 Dust extinction

For the dust extinction as a function of wavelength, we use the RV = 3.1 extinction curve from [236]9.

For the normalization of the extinction correction, we use the AH map [237] used by the

Roman microlensing survey team [227], which uses λH = 1.64 µm as the central wavelength for the H-band ﬁlter [238]10. We linearly interpolate this extinction map when drawing points.

9Downloaded from https://www.astro.princeton.edu/~draine/dust/dustmix.html 10The map is available at https://github.com/mtpenny/wfirst-ml-figures/fields/ GonzalezExtinction.txt. Note the extinctions in the map are AK and must be rescaled using AH = 1.559322AK .

64 To draw random points in the microlensing field, we obtain the chip centers from https: //github.com/mtpenny/wfirst-ml-figures/fields/layout_7f_3.chips. We then assign each realization of a binary to a random chip and select a uniformly random point in (l,b) on the chip to draw its AH from the dust map. The AH drawn this way is uncor- related with the background noise on an individual pixel drawn in Sec. 4.3.6, which may overestimate background noise for deeply extincted fields somewhat. Because we apply the dust extinction correction after generating the light curve, we do not allow the extinction to alter the brightness ratio of the systems. Within the range of white dwarfs of interest for this study, extinction is not a strong enough function of temperature for correctly perturbing the brightness ratio to affect our conclusions.

4.3.6 Roman Exposures

11 For the Roman exposures, we use the Roman F146 filter . From the effective area AF146(ν), we fix the normalization of received light to electrons detected per second using

Z ∞ ref −0.4m(t) Fν N˙ sig(t) = fap10 AF146(ν)dν, (4.2) 0 hν

ref −20 2 where Fν = 3.631 × 10 erg/cm /s/Hz, and m(t) is the light curve in magnitudes as described in Sec. 4.3.4. We consider signal photons hitting a 3 × 3 grid of adjacent pixels,

assuming a constant aperture correction of fap = 0.815. A more optimal aperture correction is beyond the scope of this work. To obtain the expected total count of electrons observed by a single Roman exposure of length dtexp, we compute a dtexp width rolling integral of N˙ sig(t) for a single orbit of the system and interpolate to obtain the expected electron counts at the mid-exposure time speciﬁed by the survey strategy described in Sec. 4.3.7.

The expected number of signal electrons Nsig(t, dtexp) for an exposure is then added to the expected background Nbg(dtexp) for the exposure. For each system, we draw the background count rate N˙ bg from a random 3 × 3 grid of pixels. We then draw the observed electron counts from a Poisson distribution with Nexp(t, dtexp) = Nsig(t, dtexp) + Nbg(dtexp).

We then add a Gaussian random read noise of σcount = 60 electrons (i.e., 9 pixels with

σpixel = 20 electrons/readout/pixel adding incoherently) to the number of electrons observed

Nobs. The process is then repeated for every exposure to obtain the full light curve observed by Roman, which is then processed as described in Sec. 4.3.8.

To account for saturation, we mask all pixels with N˙ bg > 20000 e/s, and reduce our quoted detection eﬃciencies by a factor of the fraction of pixels masked, fmask.

11Eﬀective areas for the F146 ﬁlter were obtained from https://wfirst.gsfc.nasa.gov/science/201907/ WFIRST_WIMWSM_throughput_data_190531.xlsm

65 4.3.7 Survey Strategy

For the Roman microlensing survey, we consider a single 72 day microlensing survey with nvisit = 6912 visits to each ﬁeld, or visiting with a Pvisit = 900 s cadence. For each visit, we take nexp exposures of length texp (e.g., nexp = 1, texp = 52 s).

To reduce aliasing for orbital periods close to small integer fractions of Pvisit = 900 s, we add a Gaussian random variation to the time between visits σvisit = 6 s, which in the real survey could correspond to variation in the time required to acquire the guide stars upon arriving at a ﬁeld.

4.3.8 Chi-squared calculation

For a given run, we ﬁrst phase fold the simulated Roman light curve into nbins = 0 Round [Porb/dt] phase bins with centers linearly spaced in phase on a grid φorb ∈ [0, 0.5], where we fold about the midpoint of the light curve φorb ≡ mod [t/Porb, 1]:  0 φorb φorb ≤ 0.5 φorb = . (4.3) 1 − φorb φorb > 0.5

Folding about the midpoint of the light curve extracts sine-like variation, because we expect the variation from eclipses and ellipsoidal variation to be sine-like. Note that our folding strategy assumes the initial phase of the binary can be eﬀectively ﬁt. Each sample is then placed in the phase bin whose center is closest to the phase at the midpoint of the exposure, and the χ2 is calculated:

nbins ¯ ¯2 X Ni − N χ2 = , (4.4) σ2 i=0 i

where N¯i is the mean number of electrons in the ith phase bin, and N¯ is the mean number 2 samp of electrons over the entire light curve. The variance σi can be estimated in ni exposures out of nexp total exposures with electron the electron count in each sequential exposure given

Nα:

2 2 σlc σi = samp ni " nexp # 1 1 X = N − N¯ . (4.5) nsamp nexp α i α=0

2 Using this estimator of σi corresponds mathematically to testing the null hypothesis that the variation in the light curve is stochastic. For a strongly signal dominated light curve, it will overestimate the variance compared to other possible estimators, such as the

66 samp theoretically expected variance in a bin with ni electrons. However, because using the empirical variance absorbs all experimental sources of variance in the count rate, it is more robust than using a theoretical variance. We then evaluate the signiﬁcance according to a χ2 distribution, with the number of 0 0 samp 0 degrees of freedom k = nbins −1, where nbins is the number of bins with ni > 0. nbins can be less than nbins due to aliasing, although aliasing can be mitigated by avoiding exactly periodic visits as described in Sec. 4.3.7.

4.3.9 Selecting binaries

We count a system as detectable by LISA when the four-year sky and inclination averaged signal-to-noise S/N ≥ 7, as defined by [239]. This condition is approximate but should be sufficient to estimate the population of detectable binaries. We assume the frequency evolution over the four-year primary mission is small enough that it can be ignored when calculating the signal-to-noise. We count a system as detected by Roman when p < 10−10, based on the χ2 discussed in Sec. 4.3.8. The small p cutoff is necessary to avoid false positives due to the large number of pixels searched. To be conservative, we calculate the detectability of a system given only a single microlensing season; including the full microlensing campaign would increase the detection efficiency. In a more theoretically optimal treatment, one could use the LISA sky localizations of high S/N LISA sources to reduce the trials factor for the Roman search, allowing detection with a less conservative p cutoff. For this purpose, we are most interested in sources which can be detected with reasonable high significance before LISA launches, to facilitate using longer time baselines for computing period derivatives and electromagnetic characterization. However, LISA characterization will improve significantly even if only the position of a faint electromagnetic counterpart can be determined. Therefore, Roman archival data can facilitate ongoing improvement in some LISA systems’ localizations even if the Roman microlensing survey fails to detect them at sufficiently high significance.

4.3.10 Trials factor

To compute the trials factor, we assume the search runs over linearly spaced frequency bins −8 −4 of width ∆f = 1/(2tseason) ≈ 8 × 10 Hz from fmin = 10 Hz to fmax = 1/(2texp) ≈ −3 9.6 × 10 Hz, resulting in nf = 118, 387 possible frequency bins. Each frequency bin is

divided into phase bins of width ∆φ = Round [texpf], resulting in a total of nbins = 1, 090, 156 frequency+phase bins. The seven microlensing fields will contain npix = 2, 016, 000, 000 pixel 15 locations to search. Therefore the total number of trials will be ntrials ≈ 2.2 × 10 . Because we report single season detection efficiencies, candidate binaries in the first microlensing

67 season can be veriﬁed in the second season; therefore, in two seasons the expected number 15 2 of false positives is given hnfp,2i ≈ 2.2×10 pcut,1, where pcut,1 is the threshold for identifying −4 a candidate detection in one season. If hnfp,2i = 10 is the maximum tolerable number of −10 false positives, then we require pcut,1 ≈ 2.1 × 10 .

4.3.11 Calculating detection eﬃciency

To calculate the detectability of a system in Roman output by our population synthesis model, each realization of a binary system is given a uniformly random initial phase, a pixel background noise as described in Sec. 4.3.6, and a dust extinction as described in Sec. 4.3.5. All binaries in the Roman microlensing ﬁelds get the same realization of the variation in exposure timing described in Sec. 4.3.7. For each binary we calculate the fdetected(i) at each point on a grid of nincl inclination bins linearly spaced in cos i ∈ [0, 1]. We then integrate to obtain the probability of detection:

Z 1 fdetected = fdetected(i)d cos i. (4.6) 0

For each inclination bin, we do nrun realizations of the systems to compute fdetected(i).

4.4 Known systems

As of this writing, ZTF J153932.16 +502738.8 (J1539) and SDSS J065133.338+284423.37 (J0651) are the two shortest period known detached white-dwarf binaries [232, 240, 241]. Both sources will have LISA S/N & 90 at their observed spectroscopic distances of d ' 2.3 kpc and d ' 1.0 kpc respectively. Because S/N ∝ 1/d, both would still be easily detected as LISA sources (S/N>7) in the galactic bulge at d ' 8 kpc. Both sources are believed to be He-CO binaries. For J0651, the hotter He white dwarf is the primary by luminosity, and the system is detected primarily by the eclipses of the He white dwarf by the CO secondary.

For J1539, the extremely hot (Teff ' 48, 900K) CO white dwarf primary is detected due to both strong reflection/reprocessing of light by the otherwise unseen secondary, and total eclipses of the primary. Even after artificially adding our tidal heating prescription in post-processing, the physics encoded by COSMIC cannot produce a short-period He-CO binary with a super-heated CO white dwarf primary such as J1539. It is impossible to know for certain how representative J0651 and J1539 are of the general binary white dwarf population. Because nature did indeed produce both systems, they represent useful benchmarks for the detectability with Roman. For J1539, reflection presents an additional complication because the spectrum for a

Teﬀ ' 48, 900K white dwarf is very blue. The measured CHIMERA g’-band reﬂective

68 +0.159 heating coefficient is heat2 = 3.851−0.147 [232]. Because heat2 > 1, most of the light is reprocessed UV light, rather than true reflected light, and the reprocessing is wavelength- dependent. Modeling the physics of such reprocessing is beyond the scope of this work. Therefore, to give a sense of the range of plausible Roman F146 light curves for J1539, we present two scenarios: one with the best fit g’-band reflection coefficient, heat2 = 3.851, and

one with no reﬂection, heat2 = 0. Additionally, we sample a grid of possible inclinations for both sources, evenly spaced on cos i ∈ [0, 1]. For each inclination, we then generate 100,000 random realizations of the light curves as described in Section 4.3. The other binary parameters are set to the appropriate values from [241] and [232].

4.5 Results

In Figs. 4.3, 4.2, and 4.3, we present the results from our ﬁducial run, summarized in Table 4.1. The most detectable single system generated by any run of our models is a

Porb = 224.4 s He-He binary with a 41, 000 K primary and a 39, 000 K secondary generated

in our model with artiﬁcially enhanced tidal heating, which would have fdetected ' 57% if it were in one of the Roman microlensing ﬁelds. No individual system with an orbital

period Porb > 370 s has fdetected > 25%, but collectively the large number of systems with individually low chances of having an inclination favorable enough to be detectable are the dominant contributions to the expectation values quoted in Table 4.1, rather than the

handful of systems with individually high fdetected.

Type Tide Model Roman LISA Roman+LISA Total No Tides 3.07 ± 0.08 50.3 ± 1.6 0.43 ± 0.04 Total Basic Tides 4.33 ± 0.12 50.3 ± 1.6 1.54 ± 0.10 Total Enhanced Tides 6.23 ± 0.18 50.3 ± 1.6 2.89 ± 0.16 He-He No Tides 2.99 ± 0.08 26.2 ± 1.1 0.42 ± 0.04 He-He Basic Tides 3.91 ± 0.11 26.2 ± 1.1 1.20 ± 0.09 He-He Enhanced Tides 5.50 ± 0.16 26.2 ± 1.1 2.24 ± 0.14 He-CO No Tides 0.05 ± 0.01 19.9 ± 1.0 0.01 ± 0.00 He-CO Basic Tides 0.38 ± 0.05 19.9 ± 1.0 0.33 ± 0.05 He-CO Enhanced Tides 0.70 ± 0.08 19.9 ± 1.0 0.64 ± 0.08

Table 4.1: Expectation values for number of binaries detected with Porb < 3000 s computed based on 20 Monte Carlo realizations of the 7 microlensing ﬁelds12. Quoted 1 − σ errors are purely statistical uncertainties on the expectation values and are far smaller than the inherent systematic uncertainties in the formation, evolution, and modeling of these systems.

69 Figure 4.1: Roman detection probability versus period for our fiducial run with temperatures from cosmic, with LISA S/N shown in color. The higher mass of the He-CO binaries gives them higher LISA S/N at fixed period, while the electromagnetic luminosity of the CO white dwarfs is generally smaller than He white dwarfs, resulting in the apparent inverse correlation between LISA S/N and Roman detection efficiency at fixed period. The small population of binaries separated from the overall trend in the top middle is a population of systems with very young (age< 5 Myr), hot secondaries.

70 Figure 4.2: Roman detection probability versus period for our ﬁducial run with tides enhanced to match the temperatures of the J0651 system, with LISA S/N shown in color. For a given period, approximately 95% of He-He binaries fall above the dashed red line. In this model, all He-He binaries with Pf . 680 s have a > 0.3% chance of being detected in a single season of the nominal Roman microlensing survey. For He-CO binaries, all systems with Pf . 475 s have a > 0.3% chance of being detectable by Roman.

71 Figure 4.3: Roman detection probability versus period for our ﬁducial run with tides enhanced to match the temperatures of the J0651 system, with LISA S/N shown in color. For a given period, approximately 95% of He-He binaries fall above the dashed red line. In this model, all He-He binaries with Pf . 950 s have a > 0.3% chance of being detected in a single season of the nominal Roman microlensing survey. For He-CO binaries, all systems with Pf . 475 s have a > 0.3% chance of being detectable by Roman.

72 4.6 Discussion

In this work, we have only considered statistical false positives due to Poisson fluctuations in background light and detector readout noise. We have endeavored to set an aggressive single-season p value requirement of p < 2.1×10−10 to guarantee that virtually no statistical false positives will remain as candidate detached white dwarf binaries after two microlensing seasons. This requirement is likely very conservative and represents candidate binaries that should be quickly and straightforwardly identified and likely represent the best candidates for follow up with other instruments. A more optimized Bayesian analysis of a simulated dataset for the entire Roman microlensing survey would likely return a higher expected number of true positives while maintaining a low rate of statistical false positives, although such candidates may be more difficult to validate with other instruments. Validation with other instruments is an essential consideration because the bulk of high signal-to-noise false positives in the actual Roman microlensing survey will likely be astrophysical. There are several other types of astrophysical signals that could masquerade as binaries with a comparable period. Rotating stars with spots could exhibit light curve variations on similar periods, as could asteroseismology or pulsations. While such systems may be interesting in their own right, they are contaminants for the purposes of this study. Forecasting the number and period distribution of such systems identifiable with the Roman microlensing survey is beyond the scope of this paper. In most cases, LISA itself will be able to validate the identification of a periodic source as a binary, especially for periods . 900 s where virtually all binaries of white dwarf mass objects will be detectable by LISA at 8 kpc.Other electromagnetic instruments, especially those with higher resolution or larger collecting areas, such as next-generation extremely large telescopes, could follow up candidate binaries to make a more definitive classification. Follow-ups can also provide additional valuable scientific information of value to multi-messenger studies, such as colors and temperatures. Sources that can be identified by both multiple electromagnetic instruments and LISA are appealing multi-messenger science targets.

4.7 Conclusion

In this work, we have presented several forecasts of the number of short-period detached white dwarf binaries detectable by Nancy Grace Roman Space Telescope’s planned microlensing survey based on populations of such binaries simulated using COSMIC. We ﬁnd

12Expected total number of systems from 20 realizations of the microlensing survey is approximated P i using hni ' fdetected/20, with 1-sigma statistical error on the mean approximated using σn ' q i P i 2 i (fdetected) /20 73 that it is probable that at least a handful of detached white dwarf binaries will be detected by the planned survey, and that most such binaries will also be detectable by LISA. The expected number of binaries identified is a strong function of the assumed temperatures of the binary components, which relies on physics such as tides, residual fusion, and accretion history. This physics is not included in the population synthesis code we have used. Additionally, we have excluded binaries undergoing mass transfer from our analysis, which will be significantly heated and, therefore, potentially represent a larger fraction of the total detectable systems. Mass transferring and detached systems are both separately interesting. Ideally, both types of systems can be detected so their evolutionary properties can be compared. In addition to uncertainties in the temperatures, uncertainties in population synthesis as a whole are very large, and will not be substantially reduced until future generations of surveys begin and place observational constraints on formation channels. Indeed several observed short-period binaries are not well-predicted by existing temperature models. Con- sequently, it is not possible to make a precise forecast of the number of detached white dwarf binary systems that will be detectable by Roman. We have attempted to capture some of this uncertainty by presenting three artificial models of tidal heating, which produce results representing a range of point estimates within the realm of plausibility. The planned Roman microlensing survey is not optimized to detect short-period white dwarf binaries. To detect short-period periodic sources, it would be more efficient to conduct a continuous series of exposures in each field, rather than spending a significant fraction of the total survey time slewing and settling. The data gaps associated with the survey cadence also introduce aliasing effects that cause the survey to lose efficiency at certain periods. Aliasing effects can also be mitigated by taking a continuous series of exposures on a single field. Such considerations could motivate using a small fraction of the microlensing survey time allocation to conduct such a series of exposures, which might also be valuable for other scientific purposes within the survey. A survey campaign specifically optimized for identifying multi-messenger science targets in the galactic bulge could also be proposed and conducted in a possible Roman extended mission after the primary mission has been completed. Observations conducted in the first few planned Roman microlensing seasons will be useful for constructing a better-calibrated forecast of the number of systems an optimized survey campaign would be capable of finding. Multi-messenger science is an exciting and rapidly expanding field. Planned survey instruments will be capable of identifying valuable multi-messenger targets. Early identification of some of those targets with Roman will be a scientifically valuable tool to predict and optimize the multi-messenger science yield achievable in concert with LISA.

74 Chapter 5 Conclusion

In this work, I have presented theoretical and computational work exploring several new windows into humanity’s understanding of the universe. The fundamental nature of dark matter and dark energy, explored in Chapters2 and3, represent two of the largest gaps in our knowledge of the physical nature of the universe. Multi-messenger studies using gravitational waves, explored in4, present a powerful new window to explore and advance our understanding in a variety of areas, including astrophysics, cosmology, dark matter, and physics beyond the standard model. Much of this work relies on constructing forecasts for future experimentation. As detectors are better understood and analysis pipelines are developed, the techniques explored here can be expanded and applied to improve the results of those pipelines, and motivate decisions regarding survey design. Motivating multi-messenger studies of white dwarfs will require further exploration of the constraints obtainable with various possible combinations of multi-messenger observations and how they can feed into improved understanding of the underlying physics. It will also require improved modeling of the instrument and data analysis pipelines for LISA to produce more accurate forecasts of the resulting constraints. Because LISA sources are always present in the data streams, obtaining the highest quality catalogue possible will require a simultaneous global fit of all sources present in the entire multi-year dataset. Constructing a global fitting procedure that is as unbiased and efficient as possible is a challenging technical problem. My plans for the next year include directly working on applying Bayesian inference techniques to the development of such pipelines. Opening new windows of understanding onto the universe requires combining both new and old techniques and technologies. Having a solid theoretical understanding of the space where discoveries are likely or possible can motivate accurate and informed decisions about which experiments and experimental designs are likely to produce results that advance our understanding of the universe.

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100 Appendix A Not as Big as a Barn: Upper Bounds on Dark Matter-Nucleus Cross Sections

This appendix contains results relevant to Chapter2.

A.1 Lippmann-Schwinger Equation

k2 It is useful to write E = 2µ , U(r) ≡ 2µV (r) and rearrange Eq. (2.1)

2 2 ∇r + k ψ(r) = U(r)ψ(r). (A.1)

Recognizing Eq. (A.1) as an inhomegenous Helmholtz equation, we can write the general solution in integral form [130, 242]:

Z 0 0 0 3 0 ψ(r) = ψ0(r) + G0 r, r U(r )ψ(r )d r , (A.2)

0 where G0 (r, r ) is the Green’s function for an outgoing wave in the Helmholtz equation:

2 2 0 0 ∇r + k G0 r, r = δ(r − r ) (A.3)

2 2 and ∇r + k ψ0(r) = 0 is a homogeneous solution. The Green’s function is given [242]:

1 0 G r, r0 = − eik|r−r |. (A.4) 0 4π|r − r0|

−3/2 ik ·r Plugging in an incident plane wave for the homogeneous solution, ψ0(r) = (2π) e i ,

where ki ≡ kˆz, we arrive at the Lippmann-Schwinger equation,

Z eik|r−r0| ψ(r) = (2π)−3/2eiki·r − U(r0)ψ(r0) d3r0. (A.5) 4π|r − r0|

101 Now, the goal here is to discover what measurable effect the potential has on the scattered wave. Physically, any measurement we make of the scattered wave must occur long after particle has finished interacting with the potential. Therefore, we may safely assume |r| 0 0 r→∞ 0 −1 |r |, such that |r − r | −−−→ r − ˆr · r + O(r ), such that, defining kf ≡ kˆr, Eq. (A.5) becomes

ikr r→∞ e Z 0 ψ(r) −−−→ψ (r) − U(r0)ψ(r0)e−ikf ·r d3r0 0 4πr eikr ≡ψ (r) + (2π)−3/2 f (k , k ) . (A.6) 0 r i f Physically, this equation represents an incoming plane wave and a radially outgoing spherical wave with scattering amplitude f (ki, kf ) = f (k, θ).

A.2 Born Approximation

Now we want to calculate an approximation to the scattering amplitude for a given potential. If we assume the potential is a perturbation to the incident wavefunction, we can attempt to solve Eq. (A.6) by iteration:

ikr r→∞ e Z 0 ψ(r) −−−→ψ (r) − U(r0)ψ(r0)e−ikf ·r d3r0 0 4πr ikr e Z 0 =ψ (r) − U(r0) ψ (r0) − ... e−ikf ·r d3r0 0 4πr 0 eikr =ψ (r) + (2π)−3/2 (f (1) (k , k ) + ...), (A.7) 0 r i f where we have assumed the correction is small, such that higher-order corrections can be ignored. Then we can read oﬀ our approximation to f (ki, kf ) from Eq. (A.5):

1 Z 0 f (1) (k , k ) = − U(r0)ei(ki−kf )·r d3r0. (A.8) i f 4π

(1) f (ki, kf ) is the ﬁrst Born approximation to f (ki, kf ).

Inspecting Eq. (A.8), we recognize that the ﬁrst Born approximation of f (ki, kf ) is

nothing more than the Fourier transform of the potential. Deﬁning q ≡ ki − kf such that θ 0 0 q = |q| = 2k sin 2 , and assuming the potential to be spherically symmetric U(r ) = U(r ), we can perform the angular integration to obtain: Z ∞ (1) 1 0 0 0 0 f (ki, kf ) = f (q) = − U(r ) sin(qr )r dr . (A.9) q 0 Eq. (A.9) is a useful starting point for analysis. However, before we begin using the result, we should clarify when the approximation breaks down. It can be shown robustly [123] that

102 a sufficient condition for the Born series to converge for all k is that the magnitude of the potential would not be strong enough to support a bound state if it were purely attractive [131], which is to say Z ∞ r|U(r)|dr < 1. (A.10) 0 A useful heuristic condition for the validity of the first Born approximation at a given k can be obtained by simply assuming the first order correction term in Eq. (A.5) must be small in the scattering region, such that ψ(r) ≈ ψ0(r) near r = 0 [128, 129]. Therefore, we require:

ik|r−r0| 1 Z 0 e U(r0)eiki·r d3r0 1. (A.11) 0 4π |r − r |

0 0 0 Taking r = 0, replacing ki · r = kr cos θ , and performing the angular integration, we have

Z ∞ 1 0 2ikr0 0 U(r ) e − 1 dr 1. (A.12) 2k 0 Once we have the scattering amplitude, we can calculate the total cross section as in Eq. (2.4).

A.3 Partial wave analysis

The general scattering amplitude for a spherically symmetric potential in Eq. (A.6) can be

written as an arbitrary expansion in Legendre polynomials Pl (cos θ) [123, 128, 130]:

∞ 1 X f(k, θ) = (2l + 1)eiδl sin (δ ) P (cos θ) . (A.13) k l l l=0

Given the phase shifts δl(k), the total elastic scattering cross section can be readily evaluated:

∞ 4π X σtot = (2l + 1) sin2 (δ ). (A.14) k2 l l=0 A spherically symmetric wave function can be written as a linear combination of Bessel

functions of the ﬁrst and second kind, jl(kr) and nl(kr). The scattered part of the wave function for r > R, where R is some arbitrarily large cutoﬀ radius for the potential, can then also be expanded in terms of Legendre polynomials:

∞ X ψscattered(r, θ) = Al(r)Pl(cos θ), (A.15) l=0 where

iδl Al(r) = e [cos (δl) jl(kr) − sin (δl) nl(kr)] (A.16)

is the radial wave function for the lth partial wave. We can obtain δl(k) by enforcing 103 continuity of the logarithmic derivative of the wave function,

r dAl βl = , (A.17) Al dr at r = R. We can obtain the wave function for r < R by directly integrating the one- dimensional Schrödinger equation,

d2u l(l + 1) l + k2 − 2µV (r) − u (r) = 0, (A.18) dr2 r2 l

where we have deﬁned ul(r) ≡ rAl(r). Note that because we are matching the phase shift with the logarithmic derivative of the wave function, the overall normalization of Al(r) is irrelevant for our purposes and can be chosen arbitrarily. We can then obtain Al(R) for any arbitrary potential V (r) by analytically or numerically evaluating Eq. (A.18) from r = 0 to r = R.

Once we have Al(R), we can obtain δl using

0 kRAljl(kR) − (dAl/d ln r)|r=Rjl(kR) tan(δl) = 0 . (A.19) kRAlnl(kR) − (dAl/d ln r)|r=Rnl(kR)

We have avoided canceling Al(r) in order to preserve signs, to ensure we obtain the correct quadrant for δl(k).

The boundary condition at r = 0 should properly be ul(0) = 0, but for numerical solutions taking the boundary to be at r = 0 is inconvenient because of the 1/r2 centrifugal term. Instead we can take advantage of the arbitrary normalization of Al(r) and ﬁx the boundary conditions ul(rmin) = 1, dul/dr(rmin) = (l + 1)/rmin, where rmin is some small minimum radius. 2l+1 As k → 0, it can be shown [123] that generically δl(k) ∝ k , except in special cases 2l−1 where it is possible to achieve δl(k) ∝ k for a speciﬁc value of l. Inspecting Eq. (A.14), we can see the contributions from l = 0 and l = 1 are the only values of l which can be nonvanishing as k → 0. The l = 0 cross section is called the s-wave cross section.

104 Appendix B Forecasting Super-Sample Covariance in Future Weak Lensing Surveys with SuperSCRAM

This appendix contains results relevant to Chapter3.

B.1 Basis Decomposition

In SuperSCRAM, we use a form of a spherical Fourier-Bessel basis, similar to the one described in ref. [243]. We use a spherical Bessel function for our radial basis and the real spherical harmonics for our angular basis, so that the elements of our basis can be written:

R ψα(r, θ, φ) = jlα (kαr)Ylαmα (θ, φ) , (B.1)

R where Ylm(θ, φ) is the real spherical harmonic, kα is a solution of jlα (kαRmax) = 0, and α is an index running over all possible sets (kα, lα, mα). To evaluate the covariance matrix with

a ﬁnite number of modes, we choose a cutoﬀ kmax and take all combinations of (kα, lα, mα)

which have kα < kmax. This cutoﬀ procedure has the eﬀect of gradually decreasing the

number of modes included for a given lα as lα gets larger, and should converge faster than

an arbitrary lmax cutoﬀ. The real spherical harmonics can be deﬁned in terms of Legendre polynomials: √  2 sin |m|φ if m < 0 r s  2(l + 1) (l − |m|)! |m|  Y R = (−1)|m|P (cos θ) 1 if m = 0 . (B.2) lm 4π (l + |m|)! l √  2 cos |m|φ if m > 0

105 We can then write a density ﬂuctuation mode in this basis as Z α 1 R 3 δ (kα) = δ(r)jlα (kαr)Ylαmα (ˆr)d r Nα 3 r 1 Z d k Z max Z ˆ = δ(k) drr2j (k r) d2ˆr eikrk·ˆrY R (ˆr) 3 lα α lαmα Nα (2π) 0 4πilα Z d3k Z rmax = δ(k)Y R (kˆ) drr2j (k r)j (kr) , (B.3) 3 lαmα lα α lα Nα (2π) 0

where Nα is a normalization factor. In the second equality we use the Fourier transform relation δ(r) = (2π)−3 R d3k eik·rδ(k), and in the third equality we use the identity Z 2 R ik·r lα R ˆ d ˆr Ylαmα (ˆr)e = 4πi jlα (kr)Ylαmα (k) . (B.4) S2

The normalization factor Nα is deﬁned Z 3 R R Nα = d r Ylαmα (ˆr)Ylαmα (ˆr)jα(kαr)jα(kαr) Z R R 2 = Iα(kα, rmax) Ylαmα (ˆr)Ylαmα (ˆr) d ˆr Ω

= Iα(kα, rmax), (B.5)

where R Y R (ˆr)Y R (ˆr) d2ˆr = 1. The integral I (k , r ) is deﬁned as Ω lαmα lαmα α α max

Z rmax 2 Iα(k, rmax) = dr r jlα (kαr)jlα (kr) 0 π r 1 Z rmax = dr rJlα+1/2(kαr)Jlα+1/2(kr) 2 kαk 0 h i k J (kr )J 0 (k r ) − kJ (k r )J 0 (kr ) π r α lα+1/2 max lα+1/2 α max lα+1/2 α max lα+1/2 max √max = 2 2 2 kαk k − kα π r kαJ (krmax)J (kαrmax) − kJ (kαrmax)J (krmax) √max lα+1/2 lα−1/2 lα+1/2 lα−1/2 = 2 2 2 kαk k − kα π r kαJ (krmax)J (kαrmax) √max lα+1/2 lα−1/2 = 2 2 , 2 kαk k − kα (B.6)

106 where in the last step we have use Jlα+1/2(kαrmax) = 0, from the deﬁning property of kα. In the special case k = kα, we can simplify Nα = Iα(k, rmax) using Bessel function identities:

π r kαJ (krmax)J (kαrmax) √max lα+1/2 lα−1/2 Nα = Iα(kα, rmax) = lim 2 2 k→kα 2 kαk k − kα

πrmax Jlα+1/2(rmax)Jlα−1/2(kαrmax) = lim 2 2 k→kα 2 k − kα 0 πr rmaxJ (krmax)Jlα−1/2(kαrmax) = lim max lα+1/2 k→kα 2 2k h l +1/2 i 2 α J (kr ) − J (k r ) J (k r ) πr krmax lα+1/2 max lα+3/2 α max lα−1/2 α max = lim max k→kα 2 2k 2 πrmax = − Jlα+3/2(kαrmax)Jlα−1/2(kαrmax). (B.7) 4kα

−1 αβ αβ With these deﬁnitions, we can calculate the covariance matrix CSSC = FSSC in our basis: (4π)2 Z d3k d3k Cαβ = hδαδβi = 1 2 hδ(k )δ∗(k )iY R (kˆ )Y R (kˆ ) SSC 3 3 1 2 lαmα 1 lβ mβ 2 NαNβ (2π) (2π) Z rmax Z rmax 2 2 × dr r jlα (kαr)jlα (k1r) dr r jlβ (kβr)jlβ (k2r) 0 0 (B.8) (4π)2 Z d3k = P δδ(k)I (k, r )I (k, r )Y R (kˆ )Y R (−kˆ ) 3 α max β max lαmα 1 lβ mβ 1 NαNβ (2π) Z 2 2 δδ = k dkP (k)δlα,lβ δmα,mβ Iα(k, rmax)Iβ(k, rmax) , πNαNβ

δδ 3 3 0 δδ where P (k) is the matter power spectrum, and we used (2π) δD(k + k )P (k) = hδ(k)δ∗(k0)i.

B.1.1 Response of Observables to Density Fluctuations

To calculate the response of an observable OI to a density ﬂuctuation mode δα, we can use the chain rule:

∂OI Z zmax ∂OI ∂δ¯ = dz (z) (z), (B.9) α ¯ α ∂δ 0 ∂δ ∂δ provided [∂OI /∂δ¯](z) can be calculated. In SuperSCRAM, the integral in eq. (B.9) is accomplished by calculating the integrand on a grid of z values {zi} and using the trapezoidal

107 α rule. To calculate [∂δ/∂δ¯ ](zi), we expand the mean density ﬂuctuation δ¯(zi), in our basis:

X 3 Z ri+1 1 ZZ δ¯(z ) = dr r2j (k r)δ (k ) √ dΩ Y R (ˆr) , (B.10) i r3 − r3 lα α α α 2 πa lαmα α i+1 i ri 00 Ω | {z } alαmα

where r is the comoving distance in the range ri ≤ r < ri+1, and alαmα are the real spherical harmonic coeﬃcients of a given survey window function Ω. We can use eq. (B.10) to write the derivative:

∂δ¯ 3 Z ri+1 1 2 √ α (zi) = 3 3 dr r jlα (kαr) alαmα . (B.11) ∂δ ri+1 − ri ri 2 πa00

B.2 Projected Power Spectra

0 The angular correlation function wAB(ˆn · nˆ ) of the line of sight projections of two ﬁelds A and B can be expanded in terms of its angular power spectrum CAB (`) [244]:

X 2` + 1 wAB(ˆn · nˆ0) ≡ A(ˆn)B(ˆn0) = CAB(`)P (ˆn · nˆ0), (B.12) 4π ` `

0 where nˆ and nˆ are unit vectors in the direction of observation and P` are Legendre polynomials. For a given ﬁeld A, there is a weight function qA which relates the ﬁeld to its line of sight projection A˜(ˆn), such that [244]: Z A˜(nˆ) = dr qA(r)A(rnˆ), (B.13)

where r is the comoving coordinate. In terms of qA and qB, the angular cross-power spectrum can be written: D E Z Z 2k2dk CAB(`) ≡ A˜ B˜∗ = dr dr q (r )q (r ) j (kr )j (kr )P AB(k), (B.14) lm lm 1 2 A 1 B 2 π ` 1 ` 2

1 where j`(kr) are spherical Bessel functions, and k = (` + 2 )/r. In practice, this integral is inconvenient to compute numerically due to the rapid oscillations of the spherical Bessel 1 AB functions for kr & ` + 2 . Therefore, most authors write C using the Limber approxima- 1 −1 tion, which can be taken by expanding this expression to lowest order in (` + 2 ) 1,

Z rmax A B AB ∼ q (r)q (r) δδ ` + 1/2 C (`) = dr 2 P k = . (B.15) 0 r r SuperSCRAM currently implements eq. (B.15) in the ShearPower class. Note that for 1 1 ` 2 many authors take ` + 2 ≈ `, although SuperSCRAM does not, both because the correction does not aﬀect the code’s execution time, and for consistency with other

108 implementations such as the one in CosmoSIS [245]. The next order correction is suppressed 1 −2 by a factor of O((l + 2 ) ), which should be negligible for next generation weak lensing surveys.

For the weight functions qA(r) for shear-shear lensing, implemented as QShear, we use [193, 194] 3H2Ω r q (r) = 0 m g (r) (B.16) γi 2c2 a i where Z ri+1 0 0 0 0 gi(r) = pi(r )(r − r)/r dr (B.17) ri 0 where pi(r ) is the probability density function for source galaxies in tomographic bin i.

B.3 Analytic Polygon Geometry

Using Stokes’s theorem, for a spherical polygon survey window, which has N sides which are great circle arcs, we can write

N ZZ X 1 Z a = dΩ Y R (ˆr) = ∇Y R (ˆr) · ˆz ds, (B.18) lm lm l(l + 1) lm n n=1 ∂Ωn Ω | {z } ≡∆alm,n where ∂Ωn denotes integration over the boundary of the nth arc and zˆn is a unit vector orthogonal to the two unit vectors whose tips touch the ends of the nth arc, such that if ˆpn is the unit vector at the start of the nth arc, ˆzn ≡ ˆpn+1 × ˆpn/|ˆpn+1 × ˆpn|, leaving

ˆyn ≡ ˆzn × ˆpn. The integral is most simple to evaluate if the great circle is along the equator π at θ = 2 , so for each side we rotate to a coordinate system where the side is along the 0 equator, calculate ∆alm,n and rotate back to the global coordinate system using

m0=l X n 0 ∆alm,n = Dlmm0 ∆alm0,n, (B.19) m0=−l

n where Dlmm0 is a spherical harmonic rotation matrix element.

109 π The integrand restricted to the equator θ = 2 is given. Recalling eq. (B.2), we have

R R ∂Ylm(θ, φ) ∇Ylm(ˆr) · ˆzn = θ=π/2 ∂z θ=π/2 R ∂Y (ˆr) = − sin θ lm (B.20) ∂θ θ=π/2 √  2 cos(|m|φ) m > 0 s  (2l + 1) (l − |m|)! |m|  = −(l + |m|)(−1)|m| P (0) 1 m = 0 4π (l + |m|) l−1 √  2 sin(|m|φ) m < 0 where in the last step we have have used the recurrence relation

dP m(x) (1 − x2) l = lxP m(x) − (l + m)P m (x). (B.21) dx l l−1 Then, because the side has been rotated along the equator, we need only integrate φ

from 0 to the side length βn and ﬁnd

Z β 0 1 R ∆alm,n = ∇Ylm(ˆr) dφ (B.22) l(l + 1) 0  1  sin(βn|m|) m > 0 m r  |m| (−1) 2l + 1p R π  = (l − |m|)(l + |m|)Y(l−1)m , 0 βn m = 0 l(l + 1) 2l − 1 2   1  |m| (1 − cos(βn|m|)) m < 0

m R π where we have substituted P(l−1)(0) for Y(l−1)m( 2 , 0) using eq. (B.2). The side length is given by βn ≡ atan2 [|ˆpn+1 × ˆpn|, ˆpn+1 · ˆpn].

Now, to ﬁnd the spherical harmonic rotation matrices, we obtain the Euler angles αn,

ψn, and γn for the z − x − z Euler rotation necessary to rotate an arc from the global frame into a frame where the side lies along the equator in the x−y plane. as depicted in ﬁgure B.1. −1 We deﬁne ψn ≡ − cos (ˆznz), γn ≡ atan2 [−ˆznx, ˆzny] , αn ≡ −atan2 [−ˆynz, ˆxnz]. Here are, by construction, the Euler angles for a z − x − z Euler rotation which transforms from a local coordinate system where the nth arc lies in the x − y plane to the global coordinate system, so that

l n X z x z 0 0 Dlmm = Dlmm1 (γn)Dlm1m2 (ψn)Dlm2,m (αn), (B.23) m1,m2=−l

110 Figure B.1: Applying Euler rotations to rotate an arc of angle β, highlighted in red, to a coordinate system where the side is along the equator, highlighted in yellow. The contribu- 0 tion ∆alm,n is simple to calculate along the equator. Then the contribution to the geometry 0 on the sky ∆alm,n may be obtained by applying the rotations to the matrix ∆alm,n.

where the z-rotation matrices are given such that:  0 0 cos(|m|α)al|m| + sin(|m|α)al−|m| m > 0 l  X z 0 0 D 0 (ω)a 0 = . (B.24) lmm lm al|m| m = 0 0  m =−l  0 0 − sin(|m|α)al|m| + cos(|m|α)al−|m| m < 0

x x We use an angle-doubling algorithm to compute the matrix Dl . Writing Dl as a (2l + 1) × x (2l + 1) matrix, Dl (ψ) = El(ψ) + 1, and El(ψ) is computed by recursively applying the ψ ψ 2 angle doubling formula El = 2El( 2 ) + [El( 2 )] to the inﬁnitesimal rotation matrix El. This −1 ˜ matrix is constructed as El() = Ml ElMl, where the inﬁnitesimal complex-basis rotation around x is

˜ Elmm0 () = −i[Lx]mm0 −i h p p i = δ 0 (l + 1 − m)(l + m) + δ 0 (l − m)(l + 1 + m) , (B.25) 2 m,m +1 m,m −1

111 and Ml transforms to the real spherical harmonic basis:   √1 if m > 0 and m0 = m  2  −i √1 if m > 0 and m0 = −m  2  (−1)m  √ if m < 0 and m0 = −m 2 Mlmm0 = (B.26) (−1)m i √ if m < 0 and m0 = m  2  1 if m = 0 and m0 = 0   0 otherwise.

In SuperSCRAM, the number of doublings can be speciﬁed by the user; for our

WFIRST+LSST demo, 31 doublings are suﬃcient to recover alm up to l = 100 to within a maximum error of . 0.1%, and approximately 82 doublings will converge all elements of the rotation matrices up to l = 200 to within the limits of IEEE 745 64 bit precision. The repeated multiplications of (2l+1)×(2l+1) matrices are relatively time consuming for large l. An advantage of the analytic solution over a pixel based geometries is that the results are less vulnerable to pathological input survey geometries, such as a geometry with many narrow stripes (perhaps chip gaps) masked out, aligned so that none of the stripes contained any pixel centroids, which could produce poorer quality results in a pixel based geometry. Additionally the analytic solution can in principle be calculated for arbitrarily large l, while the pixelated solution is limited by the resolution of the pixelation scheme. The various manipulations required to calculate eq. (B.19) are computed by the alm_utils module in SuperSCRAM. The PolygonGeo class provides a polygonal geometry with great circle arc sides which uses these analytic calculations, and the PolygonPixelGeo class provides the pixelated equivalent.

B.4 Galaxy Number Density

To simulate a sample redshift distribution for a future weak lensing survey, we take all galaxies in the CANDELS [246, 247] GOODS-S catalogue with i band magnitude below a selected cutoff (i<24 is used as a default). We then calculate a simple smoothed number density as a function of redshift using a Gaussian smoothing kernel with user specified width σ and reflecting boundary conditions at z = 0:

2 2 dN X 1 − (z−zi) − (z+zi) (z) = √ e 2σ2 + e 2σ2 , (B.27) dzdΩ i 2πσΩ where Ω is the area of the CANDELS survey. Then we can calculate n(z):

1 dN dz n(z) = (z) (z). (B.28) r(z)2 dzdΩ dr

112 Now, we want to use n(z) to calculate b(z), the bias. Because we are simply demonstrating a possible mitigation strategy and not attempting to model it in detail, we adopt a simple approximation in which we use this n(z) to extract a Mmin for the halo mass function and use it to calculate the bias, neglecting any subtleties from the mass-luminosity relationship. We use the Sheth-Tormen halo mass function [248–250],

Z ∞ dn Z ∞ ρ¯ d ln σ−1 n(Mmin, z) = dM = fST(σ)dM, (B.29) Mmin dM Mmin M dM

2 −3 where ρ¯ is the background matter density in units of h M Mpc , and σ(M, z) is given by

2 Z ∞ 2 G (z) 2 2 σ (M, z) = 2 k dkPlin(k)W (k, M), (B.30) 2π G0 0 where W (k, M) = W (k, R) = 3j1(kr)/(kr) is the Fourier transform of a spherical top hat 1 1/3 window function, and R = h [3M/(4πρ¯)] , Plin(k) is the linear power spectrum and G(z) is the linear growth factor. The Sheth-Tormen fST(σ) itself is

r 2 p 2 2a σ −a δc σ2 fST(σ) = A 1 + 2 e , (B.31) π aδc where A = 0.3222, a = 0.707, and p = 0.3 are empirically ﬁt parameters, and we take

δc = 1.686 to be the critical overdensity for spherical collapse to avoid considering any cosmology dependence. From here numerically solve n(Mmin, z) = n(z) to obtain Mmin. Then to calculate the linear bias, we use

2 aδc 2 − 1 2p b(σ) = 1 + σ + , (B.32) 2 p δc aδc δc 1 + σ2

2 2 where σ = σ (Mmin, z). SuperSCRAM could be extended to use improved ﬁtting functions such as in refs. [251] or [252], although the Sheth-Tormen functions are suﬃcient for the demonstration in this paper. In SuperSCRAM, eqs. (B.32) and (B.28) are calculated by the NZCandel class.

B.5 Growth Factor

Simply modifying the calculation of the linear growth factor accounts for most of the correction to the matter power spectrum from dark energy with a variable equation of state. In general relativity, the linear growth factor G(a) evolves according to

3 Ω H2 0 = G00(a)a2H2(a) + G0 a¨ + 2aH2(a) − m 0 G, (B.33) 2 a3

113 where the primes denote derivatives with respect to a and the dots are derivatives with respect to cosmic time, and H(a) is the Hubble rate [170]. In the presence of an arbitrary set of perfect ﬂuids with densities {Ωi(a)} and equations of state {wi(a)}, the Hubble rate is

2 H(a) X 3 R 1(1+w (a0)) 1 da0 = Ω e a i a0 (B.34) H2 i 0 i

G(a) and we can use D(a) ≡ a to arrive at a diﬀerential equation for D(a):

1 X 7 3 1 X 3 D00(a) = − D0(a) − w (a) Ω (a) − D(a) (1 − w (a)) Ω (a) (B.35) a 2 2 i i a2 2 i i i i these equations take simple forms for constant w solutions, such as matter, with wm = 0, 1 1 radiation with wr = 3 , and curvature wk = − 3 , and a cosmological constant wΛ = −1. 3 R 1(1+w(a0)) For dark energy, there are closed form solutions to the integral e a for all dark energy parametrizations of w(a) considered in this paper. SuperSCRAM can be extended 3 R 1(1+w(a0)) to other parametrizations by simply providing functions returning w(a) and e a . The CosmoPie class can then evaluate D(a) using SciPy’s odeint to solve eq. (B.35).

B.6 Code Tests

We have performed a number of tests to verify the various modules of SuperSCRAM perform as expected. Most of the code is covered by unit tests using pytest. The FisherMatrix class and its associated algebra_utils module have unit tests for all their functions in fisher_tests.py and algebra_tests.py, which run every test with a suite of input matrices, including randomly generated matrices. For FisherMatrix, every test is also run for FisherMatrix objects with every possible internal state, to verify the external behavior of the functions is independent of internal state, as described in subsection 3.3.6. The PolygonGeo, PolygonPixelGeo, and RectGeo classes have unit tests in polygon_geo_tests.py which verify that their results are both consistent between classes for the same geometries, and that their calculated alm correctly describe the window function for the input survey geometry. Note that if results for l > 85 are needed, SuperSCRAM requires the arbitrary precision mpmath package to avoid numerical overﬂows at double precision. δδ The module’s predictions for ∂P are compared to the results of [253] power_response ∂δ¯ in power_response_test.py, and qualitatively agree at the level expected given that they have convolved their power spectrum with a window function and we have not. γγ The response of a shear shear lensing observable Cl to a density perturbation as dis-

114 cussed in subsection B.1.1 and Appendix B.8 and calculated by ShearPower should have ∂Cγγ functional dependences of the form l (z ) ∝ ∆zs Cγγ, where ∆z is the width of a reso- ∂δ¯ s zi l s lution z slice integrated over in (B.9) and zi is the average z of the tomographic bin. The ∂Cγγ functional dependences of l (z ) are checked by . ∂δ¯ s power_derivative_tests.py Our Python implementation of Halofit used by the MatterPowerSpectrum class described in subsection 3.3.5 is very similar to the Fortran implementation of the Takahashi Halofit prescription available in CAMB [214, 217], with some modifications to facilitate our pre- δδ scriptions for arbitrary w(z), and computing ∂P . The agreement with the CAMB output ∂δ¯ is tested in power_comparison_tests.py. Our implementation agrees with CAMB’s imple- −1 mentation of the Takahashi Halofit prescription to within 0.2% for k . 10 Mpc . Most of the residual difference is because we blend the transition between the linear and nonlinear δδ power spectrum to avoid sharp spikes in ∂P in eq. (B.9), while CAMB’s implementation ∂δ¯ uses a sharp transition at k = 0.005h Mpc−1. Our prescription for an arbitrary w(a) prescription implemented in the WMatcher class, discussed in Appendix B.7 is tested in w_matcher_tests.py. We test that our results for our w0wa model with w(a) = w0 + (1 − a)wa match the results of [220], and that our 36 bin jdem model gives results consistent with the w0wa prescription for similar w(a). We also test that both w0wa and jdem models correctly recover the known w0 for constant w(a) models. 2 R d3k The variance in a window with window function W (k) is given σ = (2π)3 P (k)W (k) 2 ∂δ¯ αβ ∂δ¯ and can be written in our formalism σ = ∂δα C ∂δβ . The script variance_demo.py com- pares results from SuperSCRAM using PolygonGeo to a code which integrates the linear matter power spectrum directly for a rectangular window function. For an approximately cubic 500Mpc window, SuperSCRAM converges to within about 2% of the σ2 predicted by SuperSCRAM which directly integrates the matter power spectrum. The variance_demo.py module also enables convergence testing our basis decomposition. In general the decomposition converges faster when the volume of the geometry is increased, or when zmax is decreased. The script super_survey_tests.py does a full run of the sequence necessary to run SuperSCRAM, including various consistency checks on the calculated eigenvalues. In addition to the testing modules described here, we have verified by conducting multiple runs that the results converged to O(0.01%) with respect to all the various parameters, except for the number of basis elements.

B.7 Modifying the Matter Power Spectrum

Motivated by the procedure in ref. [220], for a given w(z), the w_matcher module calculates an eﬀective equation of state for dark energy W(z) which represents, at a given z, the

115 constant weﬀ which reproduces the same comoving distance to last scattering,

Z zlss dz0 Z zlss dz0 0 = 0 (B.36) z E(z , w = W(z)) z E(z , w = w(z)) where E(z) = H(z)/H0 and zlss is the redshift of last scattering. Currently, the WMatcher class precomputes the left-hand side of eq. (B.36) on a grid of possible w and z values and interpolates to match the right-hand side. Then, the amplitude of the z = 0 linear matter 2 2 power spectrum must be rescaled using Plin(k, z = 0) → G (z)Plin(k, z = 0) where G (z) is obtained from

G(z, w = W(z))2 G(z, w = w(z))2 G2(z) = . (B.37) G(0, w = W(z)) G(0, w = w(z))

Note that this condition is equivalent to eq. (2.3) in ref. [220], but this form is clearer in parametrizations where σ8 is not a ﬁxed parameter. The w_matcher module matches this condition by precomputing a grid of possible G(z, w) values and interpolating for a given W(z). Once W(z) is evaluated, the model is treated as having that constant w value in all respects for calculations involving that z value; for example, we must obtain a new linear power spectrum, because the small k transfer function depends on w.

B.8 Separate Universe Response

The response of the matter power spectrum to a long-wavelength density fluctuation can be approximated by treating the overdense region as a ’separate universe’, which can be used to find the response of an observable to density fluctuations. The response of the linear matter power spectrum to a fluctuation δ¯(t) is [187, 253–255]

d ln P δδ(k, a) 68 1 d ln k3P δδ(k, a) lin = − lin . (B.38) dδ¯(t) 21 3 d ln k 26 The one-loop correction simply picks up an additional factor of 21 ,

d ln P δδ (k, a) 68 1 d ln k3P δδ (k, a) 26 P δδ (k, a) − P δδ(k, a) 1-loop = − 1-loop + 1-loop lin , (B.39) ¯ δδ dδ(t) 21 3 d ln k 21 P1-loop(k, a) and for the Haloﬁt power spectrum, we follow the prescription in ref. [253] by absorbing the 13 ¯ 13 ¯ factor of Ge(˜a) = 1 + 21 δ(t) G(a) into σ˜8 = 1 + 21 δ(t0) σ8, so that

d ln P δδ (k, a) 13 d ln P δδ (k, a) 1 d ln k3P δδ (k, a) Halofit = Halofit + 2 − Halofit . (B.40) dδ¯(t) 21 d ln σ8 3 d ln k

Because at present we only consider w(z) models as perturbations around w =constant models, we can ignore any correction to these terms due to a variable w(z). The response

116 of the shear-shear lensing power spectrum can be approximated by plugging the response of the separate universe matter power spectrum into eq. (B.15):

∂CAB(`) Z rmax qA(r)qB(r) ∂P δδ ` + 1/2 ∼ dr k = . (B.41) ¯ = 2 ¯ ∂δ 0 r ∂δ r ∂CAB (`) The response of the observable to ﬂuctuations in our basis, ∂δα can then be calculated using the chain rule as described in subsection B.1.1, obtaining:

∂CAB(`) Z rmax qA(r)qB(r) ∂P δδ ` + 1/2 ∂δ¯ ∼ dr k = (r). (B.42) α = 2 ¯ α ∂δ 0 r ∂δ r ∂δ

117