The Hidden Universe: Dark Energy, Dark Matter, and Baryons

Dissertation

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

Joseph Eugene McEwen, B.S., M.S.

Graduate Program in Department of Physics

The Ohio State University

2016

Dissertation Committee:

David H. Weinberg, Advisor Christopher M. Hirata Amy Connolly Stuart Raby c Copyright by

Joseph Eugene McEwen

2016 Abstract

The universe is dominated by unknown elements, dark energy and dark matter.

The mysterious nature of the universe is not limited to the dark components. The present day census of ordinary matter (baryons) in the universe is incomplete. This thesis present the work I have done to uncover the nature of dark energy and dark matter as well as understanding the baryon content in the local universe. Chapter

2 presents an analysis of the baryon content in the local universe as traced by X- ray absorption systems. Chapter 3 present the eﬀects of streaming velocities on the baryon acoustic oscillation scale. Chapter 4 is introduces a new galaxy bias model, the environment dependent halo occupation distribution model. Chapter 5 presents a unique algorithm to compute convolution integrals in cosmological perturbation theory.

ii to Billyelou

iii Acknowledgments

Attending a CCAPP seminar in the summer of 2013 I came to an opinion on the skills possessed by astrophysicists. From my perspective, I saw that astrophysicists had mastered a very broad range of physics, from the smallest scales of atomic physics to the largest scales in cosmology. Not only were astrophysicists knowledgable of many scales of physics, they had a keen sense of physical process. They were able to make order of magnitude calculations, reminiscent of the back of the envelope calculations heard of in physics lore. This approach to science was diﬀerent from what I was used to. I was jealous of it. I also found that the division between theory and observation was less pronounced in astrophysics as compared to some of the other physics subﬁelds. It seemed to me that astrophysicists were a sort of a throwback to age when developments in mathematics, experiments, and observations, were less separated. These ideas motivated me to pursue studies related to cosmology.

I would ﬁrst like to thank David Weinberg for giving me the opportunity to explore cosmology, especially given that I had no prior astrophysics experience. David always has my best interest in mind. This I appreciate. David has a keen ability to gauge the value of a project. As much as it may not show, for I am prone to indulge myself with mathematical curiosities, I have actually picked this up. When hearing of a new project for myself or others, I always ask what is the value of the result of this project.

This I believe to be a quality skill to posses.

iv I would not have been able to make as much progress in my PhD without the help of Chris Hirata. Chris oﬀered me an opportunity to work with someone else and expanded my knowledge. Chris’s door is always open, he is always willing to help, and he is one of the most humble people I know. I tremendously beneﬁted from this.

Chris challenged me to understand not just the math, but also the physics. He made me ask myself, why did I get this result? He also helped me organize myself by forcing me to draw ﬂow charts on the board to map out my plan of attack for a problem.

I have also beneﬁtted from many of the postdocs within CCAPP. Eric Huﬀ and

Peter Melchior were always there to help me, especially when I ﬁrst started in cosmology. Adi Zoltov is a great friend and one of the best oﬃcemates one could have.

I always enjoyed coming to the oﬃce, sipping my coﬀee, and gossiping with Adi.

Ashley Ross and Jonathan Blazek are two special postdocs to me. It is a running joke between the three of us that I have oﬃcial oﬃce hours (although I completely ignore them) with Ashley and Jonathan. Ashley and Jonathan are always there to help guide me. They are not just professional colleagues but friends, and probably my two best drinking buddies in Columbus. I have enjoyed the work I have done with

Jonathan. He has taught me how to ask questions and to push things until I fully understand. I also thank John Beacom. John makes sure that all the CCAPP grad students are on track. I thank Stuart Raby for serving on my thesis committee. I took Stuarts particle physics course, and I enjoyed it. I especially enjoyed the end of term projects; they were some of the best learning experiences one can have. I thank

Klaus Honscheid for serving on my thesis committee. I appreciate that Klaus takes his time to understand where I am at on my career path and always oﬀers positive

v and honest advice. I thank Amy Connolly for substituting for Klaus in my thesis defense.

I have also made a number of friends in the physics and astronomy departments.

One cannot get through the stress of graduate school without great friends. I thank

Jessica Brinson for always being behind me. There is no way I could have made it without Jessica. I recognize Kevin George, Matt Warren, Chi Zhang, Chuck Bryant,

Hudson Smith, Shaun Hampton, Satya Nagarajan, and Stan Steers as special friends.

Within CCAPP I have made many close friendships with Shirley Li, Kenny Ng,

Ranjan Laha, Eric Speckard, Xiao Fang, Daniel Martens, Matt Digman, Andres

Salcedo and Ben Webking. Additionally I recognize my long-term friends: Joe Day,

Kimberly Lokovic, Arnulfo Gonzales, Sam Koshy, Zoltan Papp, Aaron Davis, Carlos

Torres, and Jamie Calloway. These people have always been there for me and always will be.

Nothing I have accomplished could be possible without my family. I owe everything to the support my family has given me. My father, Tim McEwen, is my ﬁrst teacher. I vividly remember my father teaching me about the distance to stars, the speed of light, trigonometry, and the pythagorean theorem - all before I even got to the 3rd grade. My science interest is his fault. My mother, Diana Rodriguez-McEwen, challenges me. She challenges me to question my opinions and to acknowledge my biases. To my brothers Jason and Jon, I love you. I hope we can all be together soon.

To my late sister Katy, I miss you, and I wish you were here.

vi Vita

2007 ...... B.S. San Francisco State University

2010 ...... M.S. Long Beach State University

Publications

Research Publications

“ The Photon Underproduction Crisis" Kollmeier J. A., Weinberg D. H., Oppenheimer B. D., Haardt F., Katz N., Davé R., Fardal M., Madau P., Danforth C., Ford A. B., Peeples M. S., McEwen J. The Astrophysical Journal Letters, Volume 789, Issue 2, article id. L32, 5 pp. (2014).

“Streaming Velocities and the Baryon Acoustic Oscillation Scale" Blazek, J. A., McEwen, J. E., Hirata, C. M. Physical Review Letters, Volume 116, Issue 12, id.121303

Fields of Study

Major Field: Physics

vii Table of Contents

Page

Dedication ...... ii

Acknowledgments ...... iii

Vita ...... vi

List of Tables ...... x

List of Figures ...... xi

1. Introduction ...... 1

1.1 Basics of Observational Cosmology ...... 4 1.1.1 The Homogenous Universe ...... 5 1.1.2 Distances ...... 7 1.1.3 Dark Components ...... 9 1.1.4 Correlation Functions ...... 14 1.2 The Larger Picture of this Thesis ...... 15

2. The X-ray Forest...... 29

2.1 Introduction ...... 29 2.2 Observations ...... 31 2.3 Modeling ...... 36 2.3.1 Simulation ...... 36 2.3.2 Spectra ...... 38 2.3.3 Radiation Background ...... 39 2.3.4 Ionization Fractions ...... 41 2.4 Results ...... 44 2.4.1 Incidence of Strong Oxygen X-ray Forest Absorbers . . . . . 44

viii 2.4.2 Correlations of Line Strength ...... 47 2.4.3 Absorber Physical State ...... 56 2.4.4 Absorber Environments ...... 60 2.5 Conclusion ...... 62

3. Streaming velocities and the baryon-acoustic oscillation scale ...... 66

3.1 Introduction ...... 66 3.2 Formalism and bias model ...... 70 3.2.1 Conventions ...... 70 3.2.2 Galaxy biasing model ...... 71 3.3 Eﬀect on 2-point statistics and BAO position ...... 74 3.4 Conclusions ...... 78

4. The eﬀects of assembly bias on cosmological inference from galaxy-galaxy lensing and galaxy clustering...... 81

4.1 Introduction ...... 81 4.2 Halo Occupation Distribution of the HW13 Catalogs ...... 89 4.2.1 Galaxy assembly bias in the HW13 Catalogs ...... 89

4.2.2 HOD Analysis of Mr 19 Galaxies ...... 94 4.2.3 Results for other Galaxy≤ − Samples ...... 100 4.2.4 Summary ...... 103 4.3 Matter Clustering Inference ...... 107 4.4 Conclusion ...... 114

5. FAST-PT ...... 121

5.1 Introduction ...... 121 5.2 Method ...... 126 5.2.1 1-loop Standard Perturbation Theory ...... 127

5.2.2 P22(k) type Convolution Integrals ...... 128 5.2.3 P13(k) type Convolution Integrals ...... 137 5.2.4 Regularization ...... 138 5.3 Performance ...... 141 5.3.1 1-loop Results ...... 141 5.3.2 Renormalization Group Flow ...... 142 5.4 Summary ...... 149

Appendices 151

ix A. Appendix : Streaming velocities and the baryon-acoustic oscillation scale 151

A.1 Details of calculations ...... 151 A.2 Calculating the BAO shift ...... 153 A.3 Eulerian treatment of streaming velocities ...... 154

B. Appendix for FAST-PT ...... 161

B.1 Γ-function identities and evaluations ...... 162 B.2 Mitigation of Edge Eﬀects ...... 163 B.3 RG-ﬂow Integration ...... 164

x List of Tables

Table Page

2.1 X-ray ions analyzed in this paper ...... 33

2.2 Oxygen X-ray detections ...... 36

2.3 Candidate detections from 1 to 8 are found in Fang et al. (2002); Mathur et al. (2003); Fujimoto et al. (2004); Nicastro et al. (2005b); Fang et al. (2010); Zappacosta et al. (2010); Nicastro et al. (2010); Ren et al. (2014), respectively, S/N values are as reported in the observational papers. A indicates that equivalent widths were reported as observed and have† been converted to rest frame values. The work of Buote et al. (2009) is not listed since Fang et al. (2010) conﬁrmed their results...... 36

−3 B.1 Stable RK4 runs for kmin = 10 and kmax = 1 and ∆λ = 0.1...... 165

−3 B.2 Stable STS runs for kmin = 10 and kmax = 10. Results were obtained using STS parameters: µ = 0.1, ∆λCFL = 0.001,Ns = 10...... 166

xi List of Figures

Figure Page

1.1 Left panel shows redshift evolution of Ωm and ΩΛ. Right panel plots the look back time as a function of redshift for a universe with and without a cosmological constant...... 12

1.2 Linear correlation function (right) and linear power spectrum (left). 16

1.3 Abundance of elements 4He, D, 3He, and 7Li predictions from BBN as a function of baryon density and photon to baryon ratio. Bands show the 95% CL range, yellow bands indicate observed light element abundance’s, narrow vertical band sepals CMB measurements of the baryon 4 density. Figure is taken from Olive et al. (2014). Yp = ρ( He)/ρb. . . 18

1.4 BAO feature in matter correlation function (left) and in matter power spectrum (right). Data was simulated using the ﬁtting function of Eisenstein and Hu (1998) with a Planck 2015 cosmology (Planck Col- laboration et al., 2015a). The correlation function plot on the left shows the BAO bump. The power spectrum on the right is divided by the same power spectrum but without the wiggles (NW), highlighting the oscillatory nature of the BAO...... 21

1.5 Schematic illustration of BAO cosmological analysis. First principle

physics combined with values of ρb and ργ (taken from CMB observations) give rs. Knowledge of rs can be used as standard ruler. Line of sight measurements probe H(z), while transverse measurements probe the angular diameter distance DA...... 22

1.6 Left panel plots the stream velocity power spectrum. For k > 1h/Mpc the power spectrum shows little variance, indicating that stream velocities are coherent on these scales. The right panel plots the Jeans scale and stream velocity scale as function of redshift. It is observed that the stream velocity scale is larger compared to the Jeans scale. . 24

xii 1.7 Schematic of gravitational lensing. Image from cfhtlens.org . . . . . 26

2.1 Comparison of the HM12H, HM12, and HM01 ionizing backgrounds at redshift z = 0.1. At lower frequencies the HM12H background takes the form of HM01 but multiplied by 1.5 to match the observed Lyman alpha mean ﬂux decrement. At energies above 3 Rydberg the HM12H background follows HM12, and the HM12 line is therefore obscured in the plot. Vertical lines mark the locations of 1 and 4 Rydberg. . . . . 41

2.2 Ionization fractions for oxygen (top row), carbon (middle row), and neon ions (bottom row) at diﬀerent densities. The left column shows pure collisional ionization, which applies to gas dense enough that photoionization is negligible. Other columns include photoionization with

the HM12H background for gas with an over density ρb/ρ¯b 40 (right) ≈ and ρb/ρ¯b 400 (middle), at z = 0.1. Red curves in the upper right panel show≈ fractions for the HM01 background; in other panels the HM01 and HM12H results would be even closer...... 43

2.3 Cumulative distribution of OVII and OVIII absorbers as a function of rest frame equivalent width threshold. We compare our HM12H results (thick line) to HM01 (thick dashed lines) and the results of Chen et al. 2003 (thin lines), who assumed a constant IGM metallic-

ity Z = 0.1Z . Circled numbers represent observed candidate OVII and OVIII absorption lines. These detections are plotted by equiva-

lent width (horizontal) and quasar redshift (vertical=1/ ln(1 + zQSO)), whose values are listed in Table 2.2. along with literature references. The grey bar marks the approximate detection limit for a 500 ks Chan- dra exposure on a bright X-ray quasar ( 10−11erg s−1 ) (Zappacosta et al., 2010). Our simulation predicts approximately one absorber per ∆z = 1.0 total path length above and equivalent width threshold of 20 kms−1 (OVII) and 10 kms−1 (OVIII)...... 48

2.4 Cumulative distributions of equivalent width for four diﬀerent post- processing analyses applied to the same simulation. The thick solid curve represents our ﬁducial case. Dashed and dot-dashed curves show cases where we increased the intensity of the HM12H background by factors of 10 and 100, respectively, to represent the potential impact of a local illuminating source. The dotted curve shows the result of

setting all the gas particle metallicities to Z = Z , for the HM12H background...... 49

xiii 2.5 Cumulative distributions of equivalent width for CV, CVI, NeVIII, and NeIX. A grey band (left panel) marks the estimated dN/dz and equivalent width range for candidate NeVIII detections (Savage et al., 2005; Narayanan et al., 2011, 2012; Meiring et al., 2013). redThe red line in the left panel shows results obtained by covering each particle with solar metallicity. [taken out, put back in?] ...... 50

2.6 Correlation of OVII and OVIII with UV OVI absorbers. Red special symbols mark the 5 strongest OVII systems, green mark the 5 OVII systems just under 50 km/s, and blue mark the 5 OVII systems just under 20 km/s. Correlations are selected by looking for absorbers on the vertical axis that are within 300 km/s of absorbers that are on the horizontal axis. Reversing this selection process does not change the correlation plot in any signiﬁcant way...... 53

2.7 Correlation of CV and CVI with UV OVI absorbers...... 54

2.8 Correlation of CV or CVI absorbers with OVII or OVIII...... 54

2.9 Correlation of NeVIII absorbers with X-ray OVII or OVIII absorbers. 55

2.10 Correlation plot between HI and OVI, OVII, OVIII. The HI values on the horizontal axis is computed by summing up all HI absorption within 300 km/s of each oxygen absorber...... 55

2.11 OVI, OVII, OVIII, and HI proﬁles for each of the absorbers marked by the special symbols. Left column plots the strongest OVII absorption systems in our simulation, while middle and right columns plot the OVII systems just under 50kms−1 and 20kms−1 ...... 56

xiv 2.12 Absorber metallicity, temperature, and density as a function of equivalent width, for OVI, OVII, and OVIII. The thick line marks the median value of the physical quantity in bins of equivalent width, while the adjacent lines mark the 25th and 75th percentile, 10th and 90th percentile, and the 1st and 99th percentile. Outside of these regions or at high equivalent width we plot individual absorbers as points. Special symbols mark OVII-selected absorbers as in previous plots. Note that an absorber may have diﬀerent physical parameters on diﬀerent panels because temperature and density are weighted by ion density. For example the strongest OVII absorber (red triangle center-middle panel) has a temperature 3 106 K while the associated OVI absorber (red triangle left-middle∼ panel× ) has a temperature 3 104 K...... 59 ∼ × 2.13 The local galaxy environment of the strongest OVII absorber (top row), OVII absorbers with equivalent widths just under 50 km/s (middle row), and OVII absorbers with equivalent widths just under 20 km/s (bottom row). Galaxies are plotted as colored circles. The color bar

on the right marks galaxy mass in units of M . Each central galaxy is encirlced by the halo viral radius determined by the radius for which

ρ = 200 ρcrit. The line of sight is into the page and we have selected a slice that× is within 500 km/s of the absorber redshift. For reference we list the mass of± the largest halos in each of the top ﬁve panels;

log(M) = 13.16, 12.76, 13.5, 13.57, 14.26 M . A star marks the location of an{ additional OVII absorption} system, above an equivalent width threshold of 10 kms−1 ...... 63

2.14 Cumulative distribution of absorber-galaxy nearest neighbor distances (comoving, redshift space) for OVI, OVII, and OVII absorbers (left to right). Top, middle, and bottom rows show results for galaxy mass 10 10 11 thresholds of 2 10 M , 6 10 M and 2 10 M , respectively . In each panel,× solid and dashed× curves show× results for absorbers with EW 10 kms−1 and 50 kms−1 , respectively, and crosses and dot-dashed≥ line show galaxy-galaxy nearest neighbor distributions and Poisson (unclustered) expectation for reference. Filled circles mark the nearest neighbor distance for the 5 strongest OVII and OVIII absorbers...... 64

xv 3.1 Top panel: All contributions to the correlation function from streaming 4 velocities up to (δ ) are shown at z = 1.2, with b1 = b2 = bs = O lin 1, and bv = 0.01. The new advection term (black solid line) is the dominant eﬀect. Bottom panel: The ELG correlation function is shown

for ﬁducial bias values (b1 = 1.5, b2 = 0.25, bs = 0.14) at z = 1.2 with − diﬀerent values of bv. Dashed (dot-dashed) indicates positive (negative) bv. For reference, the thin solid (grey) line shows the linear theory prediction...... 76

3.2 The shift in BAO position due to streaming velocities is shown as a

function of bv/b1. Thick (thin) lines show the shift with (without) the advection term. The solid black lines include the one-loop SPT

correction to the dark matter power spectrum (b2 = bs = 0), while the solid grey line also includes ﬁducial b2 and bs values for the ELG sample. Inset shows detailed behavior for small bv...... 77

4.1 The impact of galaxy assembly bias on the galaxy correlation function, for samples deﬁned by four thresholds in absolute magnitude.

Top panels compare ξgg from the HW13 abundance matching catalog (solid) to that of a scrambled catalog (dashed) in which the eﬀect of galaxy assembly bias is erased by construction. Bottom panels plot p the corresponding galaxy bias factor bg(r) = ξgg(r)/ξmm(r)...... 90

4.2 Galaxy assembly bias in Mr 20 red samples. As in Fig. 4.1, the top panel compares the measured≤ galaxy− correlation function in HW13 to a scrambled version of HW13, and the bottom panel compares results for the galaxy bias factor...... 95

4.3 Distribution of halo environments for halos in four 0.1-dex bins of mass, −1 centered at log Mh/h M = 11.25, 12.25, 13.25, and 14.25 (top left to bottom right). The density contrast δ is measured in a spherical annulus of 1 < r < 5h−1Mpc...... 95

4.4 The Measured HOD in the HW13 catalogs for the Mr 19 sample. ≤ − The grey line shows the global mean occupation function N(Mh) for halos in all environments. Solid and dashed black curvesh showi

N(Mh) for halos in the 20% highest and lowest density environments, h i respectively, as measured by δ1−5. For the global HOD, N(Mh) = 0.5 11.5 −1 h i at Mh = Mmin = 10 h M ...... 100

xvi 4.5 Radial distributions of HW13 Mr 19 satellite galaxies(solid black −1 ≤ − −1 curves), in halos with log Mh/h M = 12 12.05 (left) and log Mh/h M = − 14 14.05 (right). Grey curves show an NFW proﬁle with the mean concentration− expected for this halo mass truncated at the viral radius. 101

4.6 Galaxy-correlation function for the Mr 19 HW13 catalog compared to several HOD realizations. The≤ grey − curve, obscured in the

upper panel, shows ξgg(r) from the HW13 catalog. Dot-dashed and solid black curves show ξgg(r) from catalogs created using the global HOD and environmentally dependent HOD (EDHOD), respectively, measured from the HW13 catalog. The bottom panel shows fractional

deviations from the HW13 ξgg(r). Additional curves show the eﬀect of isotropizing the satellite distributions in the HW13 catalog (heavy dashed) or of combining the environmentally dependent HOD for centrals with the global HOD for satellites (light dashed)...... 101

4.7 Galaxy-matter cross-correlation coeﬃcient (Eqn. 4.4) computed from

the HW13 Mr 19 catalog (grey) or from catalogs created by using the global HOD≤ ( −black solid) or EDHOD (dot-dashed) of this sample. 102

4.8 Mean occupation functions of the HW13 catalogs for Mr 20 (top), ≤ − 21 (middle), 21.5 (bottom) for all halos and for halos in the 20% − − highest or lowest density environment measured by δ1−5, as in Fig. 4.4. Galaxy assembly bias eﬀects are smaller for more luminous samples. . 104

4.9 Fractional deviations of ξgg(r) from global (dashed) and environmentally dependent (solid) HOD catalogs compared to the HW13 catalogs

for the Mr 20, 21, 21.5 samples. Similar to the bottom panel of Fig. 4.6. .≤ . −. . .− . . .− ...... 105

4.10 Cross correlation coeﬃcients rgm(r) from global (dashed) and environmentally dependent (solid) HOD catalogs compared to the HW13

catalogs (grey) for the Mr 20, 21, 21.5 samples. Similar to Fig. 4.7...... ≤ . − . . .− . . .− ...... 105

4.11 Mean occupation for red galaxies with Mr 20 in the HW13 catalog, in the same format as Figs. 4.4 and 4.8.≤ . .− ...... 106

xvii 4.12 Comparison of the galaxy correlation functions (top and middle) and

the galaxy-matter cross correlation coeﬃcient (bottom) for red Mr 20 galaxies computed from the HW13 catalog (grey) and catalogs≤ created− using the global (dashed) or environmentally dependent (dot- dashed) HODs. Compare to Figs. 4.6, 4.7, 4.9...... 106

4.13 HOD and EDHOD fitting results for the Mr 19 sample. Left and ≤ − right panels show ξgg(r) and ξgm(r), respectively. (ED)HOD param- −1 eters are inferred by fitting to ξgg(r) over the range 0.1 30h Mpc and including the total number of galaxies in HW13 as an− additional fitting point. Lower panels show fractional deviations of the best-fit (ED)HOD models from the HW13 correlation functions...... 108

4.14 Galaxy-matter cross correlation coeﬃcient (eq. 4.4). for the Mr 19 HW13 catalog (thick grey) and for the EDHOD model (solid black≤) − and

HOD (dot-dashed) models that best ﬁt the HW13 ξgg(r) as shown in the left panels of Fig. 4.13...... 109

4.15 Galaxy correlation function ﬁtting results for the Mr 20, 21, 21.5 samples in the same format as in Fig. 4.13. . . . .≤ . − . . .− . . .− . . . 110

4.16 Galaxy-matter cross correlation coeﬃcients for the HW13 catalogs and

the EDHOD and HOD model ﬁt to the HW13 ξgg(r), in the same format as Fig. 4.14...... 111

4.17 Accuracy of the matter correlation functions inferred from the ξgg(r) and ξgm(r) measurements of the four HW13 catalogs, using Eqn. 4.5 with rgm(r) computed from the EDHOD (solid) or HOD (dot-dashed) model to ﬁt to ξgg(r). Each panel plots the ratio of the recovered ξmm(r) to the true ξmm(r) measured in the Bolshoi simulation. Grey lines show the eﬀect of using the EDHOD directly measured from the

HW13 catalogs instead of that inferred by ﬁtting ξgg(r)...... 115

4.18 Correlation function and matter correlation recovery for the red Mr ≤ 20 galaxies. Upper panels show the HOD and EDHOD ﬁts to ξgg(r) − and the predicted ξgm(r), in the format of Fig. 4.13. The central panel shows rgm(r) for the HW13 catalog and the two HOD catalogs, as in Fig. 4.14. The bottom panel, analogous to Fig. 4.17, shows ξmm(r) inferred from the measured ξgg(r) and ξgm(r), using rgm(r) shown in the middle panel...... 116

xviii 5.1 Power spectra in the log-periodic universe. Top panel shows the windowed linear power spectrum biased by k−ν (we choose ν = 2), with grey lines indicating the “satellite" power spectra, i.e. the contribution− to the total power spectrum that arises due to the periodic assumption in a Fourier transform. The middle panel plots ∆2(k) = k3P (k)/(2π2), within the periodic universe. This is the quantity that sources the density variance σ2 = R d ln k∆2(k). The bottom panel plots the con- R 2 2 tribution to the displacement variance σξ = d ln k∆ (k)/k ...... 133

5.2 FAST-PT 1-loop power spectrum results versus those computed using a conventional ﬁxed-grid method. The top panel shows FAST-PT results for P22(k) + P13(k) (the dashed line is for negative values). The bottom panel plots the ratio between FAST-PT and the conventional method...... 143

5.3 Estimate of FAST-PT execution time to number of grid points scaling. The left panel plots the average one-loop evaluation time, after initialization of the FAST-PT class. The right panel plots the average time required for initialization of FAST-PT class for 100 runs. For a sample of grid points, the error is computed by taking the standard deviation of 100 runs...... 144

−1 5.4 FAST-PT Renormalization group results for kmax = 5, 50 hMpc . Left panel shows Renormalization group results and SPT{ results} compared to the linear power spectrum (see legend in right panel). Right

panel shows neﬀ = d log P/d log k for Renormalization group, SPT, and linear theory...... 147

5.5 Renormalization group results compared to standard 1-loop calculations and those taken from the Coyote Universe. A plateau at high-k develops due to boundary conditions. Insert shows neﬀ(k) = d log P/d log k...... 148

xix Chapter 1: Introduction

Every science has for its basis a system of principles as ﬁxed and unalterable as those by which the universe is regulated and governed. Man cannot make principles; he can only discover them. -Thomas Paine

For thousands of years cosmologists have sought to answer some of the most captivating questions of all time: how old is the universe, how large is the universe, what is in the universe? If the basic questions, mentioned above, were not captivating enough, the 20th century provided the cosmologist with even more captivating questions to answer. It turns out the universe is expanding. Not like an explosion, but the space between galaxies is increasing itself. Also, the majority of matter in the universe is not the kind we are familiar with. We call this unfamiliar matter “dark matter", because we do not see it directly. From its invisibility, we infer that it must not interact with electromagnetic radiation, such as visible light. We can’t see dark matter, but we know it is there because of its gravitational eﬀects on the matter we do see. Oh, and one more thing, the present day rate of expansion is increasing. The simplest explanation for present day accelerated expansion is another dark component, dark energy!

1 My thesis is unique in that it presents a large amount of material covering a breadth of topics. However, the subject of these topics can all be tied together into a simple single statement about the universe, "where is and what is the stuﬀ we don’t explicitly see?"

My ﬁrst project in cosmology was related to the matter we are more familiar with, the matter composed of protons, neutrons, and electrons, categorized as baryons in the cosmological community (cosmologist include electrons in the baryon category).

The baryon content of the universe can be accounted for by comparing observations of light element abundances to predictions from Big Bang Nucleosynthesis (BBN) and to the spectrum of the cosmic microwave background (CMB). However, BBN and

CMB observations account for the baryon abundance when the universe was much younger. The present day baryon census is incomplete. This scenario is referred to as the “missing baryon" problem. The “missing" terminology is a bit misleading.

We (cosmologists) don’t believe the baryons have actually gone missing. The more accurate description is that, in the process of structure formation, the majority of baryons have been placed in environments that are hard to detect. Chapter 2 presents my work on the theoretical distributions of baryons as traced by X-ray absorption lines within the intergalactic medium (IGM).

Later chapters of this thesis migrate from studies related to the IGM to studies of the large scale structure of the local universe. Large scale structure is a blanket term that refers to the spatial distribution of observables, e.g. galaxies and neutral hydrogen, as well as the distribution of dark matter. Observations of large scale structure give information on the matter content of the universe, the geometry of the universe, and models for cosmological evolution.

2 Chapter 3 assess the impact that the relative velocity between baryons and dark matter in the early universe has on the distribution of galaxies at later times. The relative velocity between baryons and dark matter can trace its origins to the different evolutionary scenarios the two experience, when the universe was younger and dominated by radiation. In short, baryons feel radiation pressure whereas dark matter does not, leading to a velocity difference between the two. The relative velocity effect can influence the formation of galaxies at much later times. In particular the relative velocity effect can influence the baryon acoustic oscillation (BAO) feature in the galaxy distribution. The BAO feature is an uptick in galaxy clustering at a well established scale. The power of the BAO is that it provides for a standard length scale to measure geometries in the universe. Therefore, any physics that can impact the BAO, like the relative velocity effect, needs to be accounted for.

Chapter 4 is concerned with the inference of the dark matter distribution from combined observations of the gravitational lensing of galaxies and the spatial distribution of galaxies (galaxy clustering). Galaxy-lensing plus galaxy clustering analysis requires a galaxy bias model, that is a theoretical model for how galaxies are distributed within the underlying dark matter ﬁeld. A particular galaxy bias model, called Halo Occupation distributions (HOD) models galaxy bias in a statistical fashion. The central tenet of HOD models is that they tie galaxy statistics to host halo mass. It is possible that the real galaxy physics violates this assumption. A possible remedy is to augment the HOD so that galaxy statistics depend on additional halo properties. In Chp. 4 I compare matter clustering inference from galaxy lensing plus galaxy clustering using a standard HOD and a newly formulated extended

3 HOD, to galaxy catalogs known to violate the standard HOD assumption,that galaxy properties depend only on host halo mass.

Chapter 5 introduces a novel method to calculate convolution integrals that appear in the perturbative solution of the cosmological fluid equations. Perturbation theory provides an analytic method to model many observables in cosmology beyond linear theory. In Fourier space the non-linear nature of gravitational evolutions couples modes of different wavelengths, leading to complicated convolution integrals. These integrals can be expensive to solve numerically. The unique aspect of the algorithm presented in Chp. 5 is the speed of the numerical calculation. The exceedingly fast time needed to obtain results improves the utility of perturbation theory. For instance the ability to calculate many perturbative type observables in a sub-sub second time scale is well suited for multi-observational analysis that seek to fit observations to theory, sampling millions of points in parameters space.

This thesis is mostly composed of the projects I have worked on during my PhD.

The later chapters are highly technical and are targeted to the cosmological community. However, to better introduce the topics presented in the later chapters, I provide a brief review of background theory in 1.1 and a synopsis of the larger picture of § my work in 1.2. § 1.1 Basics of Observational Cosmology

The most basic aspects of observational cosmology are reviewed. The focus here is on introducing the mathematical notions relevant to understanding the later chapters.

4 1.1.1 The Homogenous Universe

The starting point for our model of the universe is Einstein’s equation, which

relates the geometric structure of spacetime to the energy content of the universe:

1 Rab gabR gabΛ = 8πTab , (1.1) − 2 − | {z } | {z } mass/energy side geometry side

where Rab and R are the Ricci tensor and scalor, Tab is the stress-energy tensor, gab is the metric ( the dynamical variable in Einstein’s equation), and Λ is the cosmological constant. If no point in space-time is special and the universe looks the same in all directions from all points, then the universe is homogenous and isotropic. The solution to a homogenous and isotropic universe is described by the Friedmann-Lemaître-

Robertson-Walker metric

dr2 ds2 = dt2 + a2(t) + r2dΩ2 , (1.2) − 1 kr2 − where k = 1, 1, 0 corresponding to an open, closed, and ﬂat universe, dΩ2 = dθ2 + − sin2 θdφ2, and a(t) is the scale factor. The scale factor encodes the expansion of

the universe as it relates comoving coordinates to physical coordinates xphysical =

a(t)xcomoving.

Assuming an isotropic and homogenous universe, the stress-energy tensor must

take the form of a perfect ﬂuid Tab = (p + ρ)uaub pgab, where p and ρ are the −

pressure and energy density components of the ﬂuid, and ua is the velocity of the

ﬂuid. For most cases there are two types of ﬂuids, radiation and matter. In the case

of matter the ﬂuid is described by dust and has no pressure. In the case of radiation

the pressure is 1/3 the energy density. The relation between pressure and density is

given by the equation of state parameter w = p/ρ.

5 The solution to the homogenous and isotropic universe are the Friedmans equa-

tions. The ab = 00 component of Eqn. 1.1 gives the ﬁrst Friedmann equation

2 H (a) −3 −4 −2 2 = Ωma + Ωra + Ωka + ΩΛ , (1.3) H0

a˙ where H = a is the Hubble parameter, the subscripts m,r, k,Λ denote matter, radi-

ation, curvature, and cosmological constant (dark energy term) and the related Ωis are the corresponding present day energy densities, scaled by the critical density, the density needed to make the universe ﬂat :

2 ρi 3H0 Ωi = , ρcrit = . (1.4) ρcrit 8πG

Taking the time derivative of a physical coordinate, we derive Hubble’s law

x˙ physical =a ˙x + ax˙ (1.5) = Hxphysical + up ,

where up is the peculiar velocity, i.e. the velocity not associated with Hubble ﬂow.

Each density component evolves diﬀerently in regards to the scale factor. Matter

density scales inversely proportional to the cube of the scale factor because volume

expansion requires three factors of the scale factor. Radiation picks up an additional

inverse power of the scale factor due to redshifting. Curvature is proportional to

inverse powers of length squared and thus scales inversely proportional to the scale

factor squared. Dark energy from a cosmological constant is constant, and does not

change as the universe expands. The ab = ij components of Eqn. 1.1 combined with

the ﬁrst Friedmann equation gives the second Friedmann equation

a¨ 4πG = (ρ p) . (1.6) a − 3 −

The curvature component Ωk does not arrive from any density component on the right hand side of Eqn. 1.1, but has its origin in the metric itself. In this regard the

6 curvature density is form of ﬁctitious energy density given by

3k ρk = − . (1.7) 8πGa2

Curvature and actual densities are related by

X Ωk = 1 Ωi . (1.8) − i Redshift and scale factor are related by

1 1 + z = . (1.9) a

Conservation of energy is a diﬃcult concept in general relativity, but conservation of stress-energy is well deﬁned. This notion is realized in the 4-divergence of the

ab stress-energy tensor, aT = 0. From conservation of stress-energy, the evolution of ∇ each species of ρi with regards to scale factor can be found

−3(1+w) ρi(a) = ρi(a = 1)a , (1.10) where w is the parameter that sets the equation of state for each density component, i.e. it relates pressure to density p = wρ. For matter w = 0, due to the fact that thermal motions of non-relativistic matter are much less than the speed of light. For radiation w = 1/3. And, for a cosmological constant w = 1. − 1.1.2 Distances

There are several notions of distance in an expanding universe. Distances are some of the most important mathematical objects in cosmology, as knowledge of a distance can help constrain each of the Ωi. The metric in Eqn. 1.2 can be expressed as

2 2 2 2 2 2 ds = dt + a (t) dχ + Sk(χ) dΩ ] , (1.11) − 7 where  sinh[χ] for an open universe  Sk(χ) = χ = r for a ﬂat universe . (1.12) sin[χ] for a closed universe The distance a photon traveling only along the χ direction is known as the comoving

distance. The comoving distance is found by noting that, for a photon, ds2 = 0 (units

of c restored below):

ds2 = c2dt2 + a2(t)dχ2 = 0 (1.13) − Z t dt0 Z z dz0 c Z z dz0 χ = c = 0 = 0 , (1.14) 0 a(t) 0 H(z ) H0 0 E(z ) where

p 3 4 2 E(z) = Ωmz + Ωrz + Ωkz + ΩΛ . (1.15)

Knowledge of an object’s luminosity can be used to obtain information on distance,

as is the case for type Ia Supernova. The luminosity distance is deﬁned as

2 Ls dL = , (1.16) 4πFo

where Ls is the absolute luminosity of the source and Fo is the observed flux. Lu- minosity is defined as the flux times the surface area of a sphere. In an expanding universe, the flux is modified for two reasons: 1) the photons redshift, 2) photons emitted at a time δt apart are measured to be (1 + z)δt apart. Hence the luminosity

can be written as

L = (1 + z)2F surface area . (1.17) ×

2 The surface area is 4πSk(χ) and combing the previous two equations the luminosity distance is found to be

dL = (1 + z)Sk(χ) . (1.18)

8 Knowledge of a transverse distance can be used to gain information on distance, as is the case for BAO observations. The angular diameter distance is deﬁned as

δx d = , (1.19) A δθ where δx is the actual size of an object and δθ is angle subtended. From the FRLW metric δx = a(z)Sk(χ)δθ, hence

S (χ) 1 d = k = d . (1.20) A 1 + z (1 + z)2 L

1.1.3 Dark Components

Dark Matter

Fritz Zwicky’s research was the ﬁrst to give evidence for dark matter (Zwicky, 1933).

Studying the Coma cluster, Zwicky inferred a mass of the cluster via application of the virial theorem. Additionally he measured the amount of light coming from the cluster. The mass to light ratio of the cluster was found to be two orders of magnitude less than that of a single star. Zwicky reasoned that the cluster must contain a form of matter that could not be accounted for by star light. Vera Rubin further advanced the idea of dark matter by studying the rotation curves of galaxies. Galactic rotation curves show a ﬂatting at large distances from the center of the galaxy. This requires the density proﬁle to scale as r−2, in contrast to what should be scaling as r−1/2 if mass were to track light. Additionally evidence for dark matter is found form observations of gravitational lensing and the merging of galaxies. Gravitational lensing is a direct probe of the matter content around galaxies. The mass measured from gravitational lensing exceeds the amount of visible matter, hence, most of the matter must be dark. In the case of galaxy cluster merges, the distribution of mass,

9 measured through lensing, is observed to follow a ballistic trajectory, evidence for

collision-less dark matter. However, the baryonic component (measured through X-

ray emission) is decelerated due to interactions. The existence of old galaxies at

redshifts 10 also support the concept of an unaccounted for form of matter. A large ∼ dark matter component moves the epoch of matter-radiation equality to earlier times.

Since matter perturbations within the horizon only grow in the matter dominated

era, an earlier matter-radiation epoch allows for the growth of structure at earlier

times. Perhaps, the greatest evidence for dark matter, is that a ΛCDM universe ﬁts

observations over many redshifts.

There are a few properties of dark matter that exclude standard model parti-

cles as candidates for dark matter. One, a dark matter particle has to be stable, at

least having a lifetime greater than the age of the universe. Two, dark matter needs

to be “cold", that is that it needs to have peculiar velocities that are much lower

than the velocity dispersion of individual galaxies, so that structure can from. These

properties combined with the characteristic “no electromagnetic interaction aspect"

exclude standard model particles. The leading non-baryonic candidates for dark mat-

ter include: primordial black holes, axions, sterile neutrinos, and weakly interacting

massive particles (see Dark Matter section of Olive et al. (2014) for a full review).

Dark Energy

The simplest explanation for Dark Energy is a cosmological constant Λ. The cosmo-

logical constant appears as an additional term in Einstein’s equation Rab 1/2gabR = −

8πTab + gabΛ (where I have moved the Λ term to the right hand side of the equation).

The Λ term can be viewed as an eﬀective form of stress energy associated with a

constant energy source Tab, = ρΛgab), where ρΛ = PΛ = Λ = Λ/(8πG). From the −

10 perspective of GR developed from an action, the cosmological constant appears as an

addition to the Einstein-Hilbert action

1 Z S = d4x√ g [R 2Λ] . (1.21) 16π − −

The eﬀects of a cosmological constant can be analyzed in terms of the deceleration parameter q, which is deﬁned as follows 1 a(δt) = a(0) +a ˙(0)δt + a¨(0)δt2 + ... 2 a(δt) 1 2 = 1 + H(0)δt q0δt + ... (1.22) a(0) − 2 aa¨ q = . − a˙ 2 Therefore, a negative q corresponds to an accelerated expansion. For a universe

dominated by matter and cosmological constant, the deceleration parameter is

1 q = [Ωm ΩΛ] . (1.23) 2 −

Thus, when ΩΛ > Ωm the expansion rate will increase. The left panel of Fig. 1.1 plots

the evolution for Ωm and ΩΛ and shows one of the biggest puzzles in cosmology, that

the energy density associated with a cosmological constant becomes comparable to

that associated with matter at relatively recent times (which begs the questions, do

we live in special epoch?).

The ﬁrst arguments for dark energy are rooted in the age of the universe. It was

long known that the age of the universe as calculated in a matter dominated universe

could be smaller than the age of the oldest stars (the age of stars in globular clusters is

larger than 11 Gyr). A dark energy component could extend the age of the universe.

The age of the universe (or look-back time) is found by

1 Z ∞ dz t0 = . (1.24) H0 0 E(z)(1 + z)

11 1.0 1.4 1e10

ΩΛ(z)

Ωm(z) 1.2 0.8

1.0

0.6 0.8

0.6 0.4

0.4

0.2 look back time [years] 0.2 with dark energy with out dark energy 0.0 0.0 0 0 1 10 100 1000 0 1 10 100 redshift redshift

Figure 1.1: Left panel shows redshift evolution of Ωm and ΩΛ. Right panel plots the look back time as a function of redshift for a universe with and without a cosmological constant.

The right panel of Fig. 1.1 shows the look-back time (the time it takes a photon to travel from a redshift z to z = 0) of the universe with and without a dark energy

component in the form of a cosmological constant.

The breakthrough in dark energy came in 1998 with the observations of type Ia

Supernova (SN) 1, Riess et al. (1998). Samples of low redshift (z <1) type Ia SN

at at the end of the 90s were of good enough quality to calibrate the absolute mag-

nitude with observations of the their light curves. The light curves, are correlated

1Type Ia Supernova are Supernova whose spectrum does not contain a special line of hydrogen and does contain a spectral line of silicon, and whose explosion occurs when the mass of a white dwarf in a binary system exceeds the Chandrasekhar limit, by absorbing gas from the neighboring star

12 with luminosity, the more luminous the SN the broader the light curve, while the dimmer SN have light curves with a steeper decay. This correlation led to a stan- dardization of the light curve by a single parameter. Type Ia SN appear to have a nearly uniform luminosity with absolute magnitude M = 19.5, thus they act like a − “standard candle" to obtain measurements of the luminosity distance. Observations of the apparent magnitude combined with knowledge of the absolute magnitude can be used to obtain the luminosity distance via the distance modulus:

m M = 5 log [dL] + 25 , (1.25) − 10 where m is the apparent magnitude, M is the absolute magnitude, and dL is the luminosity distance.

Further evidence for dark energy comes from distance measures associated with

“standard rulers". The imprint that acoustic oscillations in the CMB leave in the galaxy correlation function provide such a standard ruler. This topic is discussed in the Baryon Acoustic Oscillation of 1.2. § Much of the focus on the nature of dark energy is concerned with the equation of state. Is dark energy as simple as the cosmological constant, a constant energy density? Or, is dark energy more complex, possibly an energy density that evolves with time? A path to answer this question is to determine if w = 1 or not. Slight − deviations from w = 1 have important consequences for the evolution of the uni- − verse. If w is slightly greater than 1, for example w = .95, the dark energy density − will decrease as the universe expands. If w < 1 the accelerated expansion the uni- − verse will suﬀer the “Big Rip" Caldwell et al. (2003). In the Big Rip scenario, the accelerated expansion of the universe increase. Therefore the horizon continuously decreases and eventually no interaction of any kind can occur and all structure rips

13 apart. A generic from for an evolving dark energy equation of state is given by

w(a) = w0 + wa[1 + a] . (1.26)

In dark energy analysis one ﬁts observables with the above to determine the value of the dark energy w.

1.1.4 Correlation Functions

Observational cosmology is a statistical science. The main observables in cosmology are ensemble averages of fields. The most ubiquitous of these functions are the two point functions of the matter field; the correlation function in configuration space and the power spectrum in Fourier space.

The starting point to build correlation functions is the the matter ﬁeld

ρ(x, t) δ(x, t) = 1 , (1.27) ρ¯(t) − which is a measure of the matter density at a given point relative to the mean.

It is also useful to work in Fourier space Z δ(k, t) = d3xe−ik·xδ(x, t) . (1.28)

Similar deﬁnitions apply for the galaxy ﬁeld, δg(x, t) = ρg(x, t)/ρ¯g(t) 1. − The correlation function, ξ(r), describes the spatial distribution of objects. In

particular, it accounts for the excess clustering from random of ﬁnding two objects

in two regions dV1 and dV2 separated by a distance r,

dP = n1n2[1 + ξ(r)]dV1dV2 , (1.29)

where n1 and n2 are the number density of objects and dP is the probability of ﬁnding

a pair of objects. If the spatial distribution of δ(x, t) is non-random, then ξ(r) is non-

zero. Since, the universe is homogenous and isotropic, the argument of the correlation

14 function is a scalar 2. In terms of the matter ﬁeld the correlation function is deﬁned as an ensemble average

ξ(r) = δ(x)δ(x + r) . (1.30) h i

This idea can be extended to deﬁne the galaxy auto correlation and galaxy matter cross correlations

ξgg(r) = δg(x)δg(x + r) , h i (1.31) ξgm(r) = δg(x)δm(x + r) . h i Note, that when there is no subscript on ξ(r) or δ(x) it is assumed that the reference is in regards to matter.

The power spectrum and correlation function are Fourier transform pairs

Z d3k ξ(r) = eik·rP (k) . (1.32) (2π)3

The power spectrum is deﬁned as the variance in the matter ﬁeld per k bin −

3 0 0 (2π) δD(k + k )P (k) = δ(k)δ(k ) , (1.33) h i where δD(k) is the Dirac delta function and δ(k) is the matter ﬁeld in Fourier space.

Fig. 1.2 shows the dark matter linear correlation function (left) and power spectrum

(right).

1.2 The Larger Picture of this Thesis

Missing Baryons and the X-ray Forest

Chapter 2 is concerned with the distribution of baryons, as traced by X-ray absorption lines (particularly higher ionization states of oxygen and carbon), in the

2This is not a generic feature. For instance this is not the case for the correlation function in redshift space.

15 102 105 peak depends on Ωm

104 101

3 10 P k ∝

) 2 )

k 10 r ( ( 0 ξ 10 P 3 P k − 101 → ∝

100 10-1

10-1

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

1 r [h − Mpc] k [h/Mpc]

Figure 1.2: Linear correlation function (right) and linear power spectrum (left).

16 low redshift intergalactic medium. Similar to the nomenclature for Lyα absorptions systems, we call these X-ray absorbers the X-ray forest. The work in Chp. 2 is theoretical, making predictions for X-ray absorbers based on data analysis from the output of large cosmological simulation that includes recipes for galaxy formation.

Observationally, these absorption systems are diﬃcult to detect, due the low density and high temperatures they reside in. There are number of reasons to study baryon absorption systems in the IGM. Most notably, detections of the X-ray forest can help constrain the assumptions that go into galaxy formation models and any detection may solve the missing baryon problem. Additionally any detection of the

X-ray forest can help constrain our galaxy formation models. This has important cosmological consequences, as in increased knowledge of galaxy formation can improve on our galaxy bias models needed to infer cosmological parameters.

The amount of baryons in the universe is represented by the cosmological parameter Ωb. We know what Ωb should be based on several phenomena: predictions from

BBN, the amplitude and positions of the oscillation peaks in the CMB, and observations of neutral hydrogen in the intergalactic medium. The rates of interaction in Big

Bang Nucleosynthesis (BBN) are dependent on the photon to baryon ratio η = nb/nγ.

The photon density nγ can be observed through the CMB spectrum. Observations of the abundance of light elements H, He, Li, and Be then constrain nb. The amplitudes and positions of the oscillations in the CMB power spectrum depend on Ωb. So, the baryon census at high redshift is well accounted for from BBN and CMB. Fig. 1.3

23 shows the predictions of Ωbh from BBN, as well as observed abundances of light

2 elements, and includes measurements of Ωbh taken from the CMB.

3h is the present day Hubble constant divided by 100

17 Figure 1.3: Abundance of elements 4He, D, 3He, and 7Li predictions from BBN as a function of baryon density and photon to baryon ratio. Bands show the 95% CL range, yellow bands indicate observed light element abundance’s, narrow vertical band sepals CMB measurements of the baryon density. Figure is taken from Olive et al. 4 (2014). Yp = ρ( He)/ρb.

18 At redshifts 2 4 the Lyαforest provides another observation to constrain Ωb. ∼ − Weinberg et al. (1997) has shown that observations of the abundance of Lyαsystems

(as observed through the mean ﬂux decrement) combined with predictions of the photo ionization rate, can infer a lower bound on the baryon density. The inferred

2 lower bound in this analysis is Ωbh 0.0125 which is within the range predicted ≥ 2 form CMB measurements Ωbh = 0.02230 0.00014 (Planck Collaboration et al., ± 2015a).

The works of Fukugita and Peebles (2004) and Shull et al. (2012) provide a census of the present day baryon budget. 10 % of the baryons are found within galaxies or the hot gas in galaxy cluster. The forest can account for 30% of the gas. Where ∼ are the rest of the baryons? Simulations predict a large fraction of the baryons to be located in the shock heated gas, with temperatures 105 107K, commonly referred − to as the warm hot intergalactic medium (WHIM). Due to the low density and high

temperature, observations of baryons within the WHIM are diﬃcult.

Baryon Acoustic Oscillations and Streaming Velocities

At 150 Mpc the galaxy auto correlation function shows an increase in clustering, ∼ albeit small, but still detectable ( see Fig. 1.4 for a visualization). The physics of

baryon acoustic oscillations is understood as follows. Consider a single dark matter

perturbation surrounded by a shell of dark matter and a shell of baryons. Due to

gravitational attraction, the dark matter will fall toward the perturbation. On the

other hand, because baryons are tightly coupled to photons (electrons to photons via

Thomson scattering and electron to protons via Coulomb interactions) the baryon

shell is driven outwards by radiation pressure. The pressure wave moves at speed

greater than half the speed of light. As the universe expands and cools, photon and

19 baryon interactions stall. The photons decouple from the baryons and are free to stream. Electrons recombine 4 with protons to form neutral hydrogen. The radiation pressure ceases, and the baryon shell comes to a halt, leaving a ring of over dense baryons at a distance (the BAO scale). The BAO scale is set by the sound speed cs, which is dependent on baryon and photon densities (ρb and ργ respectively):

Z tdrag Z ∞ cs rs = cs(1 + z)dt = dz , 0 zdrag H(z) (1.34) −1/2 cs = [3(1 + 3ρb/4ργ)] , where tdrag and zdrag refers to time and redshift at the drag epoch, i.e. the time when electrons are released from Compton drag of the photons (they no longer interact with the photons).

The excess density of baryons at the BAO ring, will gravitationally attract matter leading to an accumulation of dark matter at the BAO scale. The excess matter results in an uptick in structure formation at the BAO scale, which is observable in the present day distribution of galaxies.

The physics of the BAO scale is well understood from ﬁrst principles. The parameters used to determine the BAO scale (ρb and ργ) can be calibrated from measurements of the CMB. Therefor BAO can provide for a “standard ruler" to measure the geometry of the universe. Cosmology with BAO is sensitive to two distance measures. Measurements along the line of sight probe the Hubble parameter through

H(z) = c∆z/(sk(z)), while measurements transverse to the line of sight probe the angular diameter distance through DA(z) = s⊥/(∆θ(1 + z)).

The same physics that describes the acoustic oscillation in the CMB also leds to a relative velocity between baryons and dark matter in the early universe. The relative

4Recombination is a famous misnomer in cosmology. The electrons and protons were never previously combined. Hence, they could not recombine.

20 40 1.08

1.06 30

1.04

20 ) k ( 1.02 W ) N r ( P ξ / 2 ) r k ( n

i 1.00 l

10 P

0.98

0 0.96

10 0.94 100.00 120.00 140.00 160.00 180.00 200.00 0.05 0.10 0.15 0.20 0.25 0.30 r [ Mpc] k [1/Mpc]

Figure 1.4: BAO feature in matter correlation function (left) and in matter power spectrum (right). Data was simulated using the ﬁtting function of Eisenstein and Hu (1998) with a Planck 2015 cosmology (Planck Collaboration et al., 2015a). The correlation function plot on the left shows the BAO bump. The power spectrum on the right is divided by the same power spectrum but without the wiggles (NW), highlighting the oscillatory nature of the BAO.

21 c∆z/H(z) DA(z)θ(1 + z)

Figure 1.5: Schematic illustration of BAO cosmological analysis. First principle physics combined with values of ρb and ργ (taken from CMB observations) give rs. Knowledge of rs can be used as standard ruler. Line of sight measurements probe H(z), while transverse measurements probe the angular diameter distance DA.

22 velocity vbc = vb vc is due to the diﬀerent evolutionary scenarios experienced by − dark matter and baryons. As noted previously, dark matter will ﬂow toward an over

density due to the force of gravity. On the other hand, inward of the BAO ring,

baryons experience radiation pressure, and have their trajectories altered compared

to the dark matter. Therefore there is a relative velocity between baryons and dark

matter inside the BAO ring. Outside the BAO ring baryons and dark matter ﬂow

together, as there is no radiation pressure at these scales. The left panel of Fig. 1.6

plots the stream velocity power spectrum. On large scales, outside the BAO scale, the

power spectrum is zero, reﬂective of the zero stream velocity on these scales. Moving

to the right (to smaller scales), the power spectrum crosses the BAO wiggles. On

scales smaller than 1 h−1 Mpc the power spectrum drops and ﬂattens, indicating ∼ little variance in the stream velocity, i.e. coherent ﬂow.

The stream velocity eﬀect has important consequences regarding early structure

formation,(Tseliakhovich and Hirata, 2010a; Tseliakhovich et al., 2011a). Before de-

coupling, baryons and photons act as one ﬂuid with a sound speed 1/√3c = ∼ 1.732 105 kms−1. After decoupling, the neutral hydrogen sound speed drops rapidly × to 6 kms−1. At this time, the relative velocity between baryons and dark matter ∼ have a root mean square velocity of 33 kms−1, which is greater than the sound ∼ speed, hence the relative velocity is super sonic. By redshift 11, the stream velocity is negligible, having decayed away inversely proportional to the scale factor. However, this supersonic stream velocity at higher redshifts impacts the formation of early structure. Simply, baryons in regions with high stream velocity have kinetic energies that make it diﬃcult for small dark matter halos to gravitational trap them. The scale important for galaxy formation is the Jeans scale. The Jeans scale kJ = aH/cs

23 0.8 1000

kbc 0.7 900 kJ

0.6 800

0.5 700 ] c ) k p ( c M b

0.4 / 600 2 v h [ ∆

k 0.3 500

0.2 400

0.1 300

0.0 200 10-3 10-2 10-1 100 10 20 30 40 50 60 k [h/Mpc] z

Figure 1.6: Left panel plots the stream velocity power spectrum. For k > 1h/Mpc the power spectrum shows little variance, indicating that stream velocities are coherent on these scales. The right panel plots the Jeans scale and stream velocity scale as function of redshift. It is observed that the stream velocity scale is larger compared to the Jeans scale.

sets the scale at which gravity can overcome radiation pressure and dark matter halos can hold their baryons. The right panel of Fig. 1.6 compares the Jeans scale to the

p 2 stream velocity scale, kbc = aH/σbc, σbc = vbc. It is observed that stream velocities can have an impact on structure formation down to redshifts 5. ∼ If galaxies have a strong memory of the their early progenitors, stream velocities can impact the clustering of galaxies. This in turn can inﬂuence the BAO feature, potentially biasing BAO cosmological analysis if unaccounted for. The impact of stream velocities on the BAO scale was ﬁrst assessed using perturbation theory in

24 Yoo et al. (2011a); Yoo and Seljak (2013a). In perturbation theory, the response of the galaxy over density to stream velocities is represented by the bias factor bv, which represents the probability to ﬁnd a galaxy in a region with non-zero vbc. The work of Yoo et al. (2011a); Yoo and Seljak (2013a) targeted the stream velocity eﬀect in

Fourier space. A conﬁguration space treatment was handled by Slepian and Eisenstein

(2015a).

Chapter 3 presents my contribution to investigations of the BAO scale and stream velocities within a perturbative framework. The fundamental diﬀerence between this work and previous investigations was the realization made by our group that, because the stream velocity eﬀect was a phenomena at the epoch of decoupling, not at later redshifts, the perturbative analysis required that the stream velocities contribution be evaluated at the initial Lagrangian coordinate rather than the Eulerian coordinate.

My contribution to the group involved the calculation of the shift of the BAO position as a function of stream velocity bias parameter bv. This work has been published in the paper Blazek et al. (2016). Chp. 3 presents the paper in its entirety.

Galaxy-Galaxy lensing and Galaxy Clustering

Galaxy-galaxy lensing (GGL) referees to the distortion of background galaxy images by foreground galaxies. Observations of GGL are a direct probe of the matter content around the foreground galaxies. Like most observables in cosmology, GGL methods are statistical in nature. In practice one averages the lensing signal over many lensing galaxies, thus GGL is a statistical relation between the distribution of galaxies and their surrounding dark matter, it is a function of the galaxy-matter correlation function ξgm(r). Fig. 1.7 provides a schematic illustration of a lensing signal.

25 Figure 1.7: Schematic of gravitational lensing. Image from cfhtlens.org

A route to obtain the distribution of matter in the universe (ξmm(r)), which is proportional to σ8 and Ωm, is to combine observations of GGL (ξgm(r)) and galaxy clustering (ξgg(r)). Mathematically the relation between ξgg, ξgm, and ξmm is described by the bias function b(r) and the cross correlation coeﬃcient rgm(r), so that ξgg =

2 b ξmm and ξgm = brgmξmm. On large scales rgm approaches 1, while b is a constant

2 value and therefore one can obtain ξmm = ξgm/ξgg. However, the lensing signal is strongest on small scales, so one would like to push GGL + galaxy clustering analysis well into the non-linear regime, where there is potentially a fair amount of information to be gained.

The desire to improve upon observations probing the low redshift matter distribution is highlighted by the current 2σ tension between measurements of σ8 and ∼

Ωm from shear measurement and those taken from the CMB (MacCrann et al., 2015).

This tension could be our ﬁrst indication of physics beyond ΛCDM or could be related to systematics in lensing analysis.

26 To push GGL + galaxy clustering analysis to smaller scales requires a model for galaxy bias.

Halo Occupation distribution (HOD) models are one such model. The HOD method describes the distribution of galaxies within dark matter halos in terms of a probability distribution function that specifies the average number of galaxies (of a specific class) that occupy a host halo, along with rules to for the assignment of galaxy positions, velocities, and concentration. The central tenet of the HOD is the the occupation statistics depend only on halo mass. The problem here is that galaxy formation physics may in reality violate the HOD assumption that galaxy statistics depend solely on host halo mass. One avenue for this to happen is that galaxy statistics inherit what is called halo assembly bias. Halo assembly bias is the dependence of halo clustering at fixed mass on halo properties other than halo mass. Numerical simulation have shown that older halos with masses below M? are more clustered than their younger counterparts.

If galaxy formation inherits halo assembly bias, this may challenge the central

HOD tenet that galaxies statistics are solely dependent on host halo mass. This is potentially troublesome from the cosmological perspective, if assembly bias systematically introduces errors in cosmological parameter inference when one uses an HOD approach.

In Chp. 4 I compare the halo catalogs of Hearin and Watson (2013) to a standard

HOD method as well as an HOD extended to include the local dark matter density of a host halo as an additional HOD parameter. The galaxy catalogs of Hearin and

Watson are known to contain a robust level halo assembly bias. The extension of halo environment in the HOD is based on a reasonable assertion that halo environment and

27 assembly bias are related. The work in Chp. 4 is a ﬁrst investigation into the potential impact assembly bias has on cosmological parameter inference form GGL plus galaxy clustering. Additionally, Chp. 4 introduces an extended HOD. The upshot is that assembly bias appears to have little eﬀect on ξmm inference from GGL and galaxy clustering. When the assembly bias signal is robust our newly developed extended

−1 HOD recovers ξmm to scales h Mpc. ∼

28 Chapter 2: The X-ray Forest.

2.1 Introduction

At high redshift, z 2 4, cosmological simulations predict that most of the ∼ − baryonic matter in the universe resides in the diﬀuse photoionized gas within the intergalactic medium (IGM). Baryon census at these redshifts is easily observed through the Lyman alpha forest, i.e. the forest of neutral hydrogen absorption lines in the spectrum of background quasar radiation (Rauch et al., 2001). Observational estimates of the mass density associated with the Lyman alpha forest agree well with the baryon density inferred from big bang nucleosynthesis and from measurements of the acoustic peaks in the cosmic microwave background radiation (Rauch et al., 1997;

Weinberg et al., 1997).

However, at low redshift, the baryon census is more uncertain and apparently incomplete, with 5-10 % of baryons in the stars and neutral/molecular gas components of galaxies, 30% in the diﬀuse photoionized Lyα forest, and 10 30% in the ∼ ∼ − shock-heated medium of clusters, groups, and galactic halos traced by X-ray emission

Cosmological simulations predict that 30 50% of the low redshift baryons reside − in the shock heated intergalactic medium (IGM) at moderate over densities, δ =

5 7 ρb/ρ¯b 5 200, and temperatures 10 10 K , frequently referred to as the warm ≈ − − 29 hot intergalactic medium (WHIM) (Cen and Ostriker, 1999; Davé et al., 1999, 2001).

This gas is too hot to be easily traced by Lyα absorption, but it can potentially be observed through an “X-ray forest" of highly ionized metal lines, such as OVII and OVIII (Davé et al., 2001; Chen et al., 2003). This paper presents theoretical predictions of the X-ray forest from a large volume cosmological smoothed particle hydrodynamics (SPH) simulation and compares them to results from observational searches for the X-ray forest.

These observational searches have been a major investment with both Chandra and XMM-Newton, and the shock heated IGM will continue to be a major driving force for next generation X-ray telescopes with much larger eﬀective areas and high spectral resolution. We review the current observational status of the X-ray forest in 2. In brief, detecting absorption lines of the expected strength pushes the limits § of current X-ray facilities, which results in candidate detections of low statistical signiﬁcance. Absorption at z = 0 along many sight lines through the galactic halo is clearly established, with typical column densities 2 1016 () . However, it remains ∼ × unclear which if any of the z > 0 candidate detections is real.

This paper extends the previous theoretical study of the X-ray forest by our group

(Chen et al., 2003), hereafter C03, improving it along multiple dimensions. Our SPH simulation of a (100 h−1Mpc)3 comoving volume with 2 5763 particles increases both × the simulation volume and mass resolution by a factor of eight over C03’s 2 1443 × simulation. The larger volume is important for statistical predictions because potentially observable X-ray absorbers are rare. In addition, the simulation incorporates self-consistent metal enrichment via galactic outﬂows and includes metal-line cooling physics, while C03 simulation did not have outﬂows and simply imposed a uniform

30 IGM metallicity of Z = 0.1 Z for analysis. Finally, we take advantage of a decade’s worth of observational developments to assess our numerical results.

In this paper we analyze the incidence of CV, CVI, OVII, OVIII, and NeIX, with the absorption lines listed in Table 2.1. We pay particular attention to OVII (21.6

Å) and OVIII (18.97 Å) ions; they have a high abundance, strong oscillator strength, and peak collisional ionization equilibrium at temperatures of (1 2) 106 K, making − × them the prime candidate to trace the higher temperature WHIM. We also examine the correlation of OVII and OVIII absorption with the UV OVI absorption line, which has been the most widely studied WHIM absorber, and we predict the incidence of

UV NeVIII absorbers. After reviewing X-ray forest observations in 2, we describe § the simulation and analysis procedures in 3. Section 4 presents our main results: § the incidence of X-ray absorption, the physical state of the absorbers, and the galaxy environment around strong OVII systems. We summarize our conclusions in 5. § 2.2 Observations

The challenge of X-ray forest observations is readily understood from simple arguments. For the OVII Kα line, typically the strongest X-ray absorption feature, the rest frame equivalent width in the optically thin limit is related to column density by

2 πe osc WOVII = λOVII fOVII NOVII mec · · · N = 3.985 OVII kms−1 (2.1) × 1014 N = 0.287 OVII mÅ , × 1014 osc where fOVII is the transition oscillator strength for OVII and me and c take their usual values. A column density of 2.5 1015 corresponds to an equivalent width × 31 7 mÅ or 100 kms−1 . For a 500 k sec exposure with Chandra LETGS of a bright ∼ ∼ X-ray quasar ( ﬂux 10−11erg cm−2 s−1 in the 0.5 2keV band), a 3σ detection ∼ − ≥ requires an equivalent width 20 mÅ, (Zappacosta et al., 2010), corresponding to a ≥ 15 minimum column density NOVII 7.5 10 . With longer exposure or observing a ∼ × source in outburst, this limiting column density can be pushed down by a moderate

−5 −3 factor. Scaling to gas with over density δ = ρ/ρ¯ = 100 at z = 0 (nH 2 10 cm ) ≈ × −4 and metallicity Z 0.1 Z (nO/nH 10 ), the path length required is ≈ ≈

N L = OVII (nOfOVII) δ −1 n /n −1 f −1 = 1.2 Mpc O H OVII (2.2) × 100 · 10−4 · 1.0 N OVII (1 + z)−3 . × 7.5 1015 · × Thus, for a readily observable system one must intercept a roughly Mpc of over density 100 gas with physical conditions needed to yield a high OVII fraction. For

OVIII the column density (and hence path length) required for the same equivalent width is a factor of two higher, and fOVIII does not exceed 0.5 even for ideal physical conditions (see Figure 2.2).

Given the rarity of systems that can produce detectable absorption, the candidate

X-ray forest detections are inevitably of moderate signal-to-noise ratio (S/N). In a spectrum with 100 resolution elements and Gaussian noise, one expects an average of

2.5 2σ “absorption" features from statistical ﬂuctuations alone (and equal number ≥ of spurious “emission" features). WHIM prospectors have therefore sought to achieve higher S/N by observing X-ray blazars in outbursts e.g., Nicastro et al. (2005b,b), or else to boost statistical signiﬁcance through association with other features such as

OVI or broad HI absorbers in the UV e.g., Mathur et al. (2003); Nicastro et al. (2013)

32 Table 2.1: X-ray ions analyzed in this paper Ion E (keV) λ (Å) Oscillator Strength Ionization Potential (eV) CV Kα 0.3079 40.27 0.0648 392.09 CVI Kα 0.3675 33.73 0.416 490.00 OVII Kα 0.5740 21.60 0.696 738.33 OVIII Kα 0.6536 18.97 0.416 871.40 Ne IX Kα 0.9224 13.45 0.724 1195.80

or structures in the galaxy distribution, which reduces the “look elsewhere" eﬀect.

Most X-ray forest searches have been “blind", selecting targets without reference

to foreground structure, but a few have pursued a targeted strategy (proposed by

Kravtsov et al. (2002)), choosing sight lines that intersect known galaxy structures.

Unfortunately, the combination of uncertain corrections for a posteriori statistical significance with systematic uncertainties caused by instrumental effects and contin- uum determination means that no intervening X-ray absorbers have been established with complete confidence, and a reliable estimate of the equivalent width or column density distribution of X-ray absorbers is not currently possible. By contrast, Galac- tic OVII and OVIII absorption at redshift consistent with z = 0 is well established along multiple sight lines, as the high column densities N 1016 produce better S/N ≥ and there is no statistical penalty for considering many redshifts, (see Gupta et al.

(2014) for a recent review).

Table 2.2 lists candidate OVII and OVIII absorption systems, ordered chronologi- cally by the papers that ﬁrst reported them. We brieﬂy discuss each of these systems in turn, as well as a few candidate detections of other X-ray forest lines.

33 The ﬁrst non-local X-ray forest sighting was made by Fang et al. (2002), ob-

serving toward PKS 2155-304 with the Chandra Low Energy Transmission Grating

Spectrometer (LETGS) and the Advanced CCD Imaging Spectrometer (ACIS), ﬁnd- ing an OVIII Lyα absorption lines with a 4.5σ statistical signiﬁcance and equivalent width 220 kms−1 . This candidate detection is possibly associated with HI Lyα ≈ clouds and spiral galaxies found along the line of sight (LOS) toward PKS 2155-304

(Shull et al., 1998). However, this seemingly secure detection was not conﬁrmed in

later observations with the Chandra LETG high-resolution camera (HRC)(Williams

et al., 2005; Fang et al., 2007), highlighting the frustrations in detecting the X-ray

forest.

Observing with Chandra LETGS toward H1821+643 (z = 0.297), a line of sight

(LOS) known to intersect six OVI absorbers, Mathur et al. (2003) detected a number

of X-ray forest candidates. A set of OVII and OVIII lines was detected at the redshift

of one OVI system, an OVII line at the redshift of another OVI absorber, and a NeIX

line at the redshift of a third. The statistical signiﬁcance of the individual lines was

> 2σ, which would be inadequate in a blind search of many pixels, but Mathur et al. ∼ (2003) argued that the association with OVI absorption made a substantial but not

overwhelming case for X-ray forest detection.

Fujimoto et al. (2004) carried out a targeted search for WHIM absorption towards

the Virgo Cluster, with observations of LBQS 1228+1116. Using instruments on

XMM-Newton they report an OVIII absorber at the Virgo redshift with a statistical

signiﬁcance of 2σ. Extending targeted searches to the Coma Cluster (z = 0.0231) ≈ Takei et al. (2007) reported a 2.3σ detection of NeIX absorption at the Coma redshift

34 and a 1.9σ detection of OVIII. We omit the latter line form Table 2.2 because of its

< 2σ signiﬁcance.

Buote et al. (2009) carried out a targeted search towards the Sculptor Wall (z =

0.028 0.032), with both XMM-Newton and Chandra, toward H 2356-309 (z = 0.165). − They found OVII absorption at the Sculptor Wall redshift with 3σ signiﬁcance, a

result conﬁrmed and improved by the analysis of Fang et al. (2010), whose results we

report in Table 2.2.

The line of sight toward H 2356-309 intersects other large scale structures that

are possible WHIM hosts. The Pisces-Cetus Supercluster (z = 0.0620) and the more

distant structure at the Sculptor Wall (z = 0.128) were studied by Zappacosta et al.

(2010) with Chandra LETG and XMM-Newton RGS spectra. At the Pisces-Cetus

structure two phases of WHIM were detected: one through CV and the other through

OVII.

Nicastro et al. (2013) observed 1ES 1553+113 (z > 0.4) with Chandra LETG. ∼ Six lines were detected with signiﬁcance of (3.6 4.1 σ) and ﬁve more regarded as − marginal detections. The X-ray component of these detections is through CV and

CVI. No equivalent widths were given in Nicastro et al. (2013), and we therefore have

not used them for comparison with our simulation’s predictions.

Another targeted search was conducted by Ren et al. (2014), observing toward the

blazer Mkn 501, located in the Hercules Supercluster (z 0.03). An OVII absorber ∼ was detected with an equivalent width of 22.3 5.7 mÅ and a signiﬁcance of 2.5σ. ± Although not an X-ray ion, NeVIII is another absorption system that should trace the higher temperature region of the WHIM. Ne VIII has an ionization potential of

239 eV and is found at temperatures of 5 105 106 K, (see Fig. 2.2). There ∼ × −

35 Table 2.2: Oxygen X-ray detections −1 Id. target Ion EW (mÅ)/(kms ) zabs zQSO S/N Association 1 PKS 2155-304 OVIII 14.0 /221 0.05 0.116 4.5 HI Lyα 2 a H1821 + 643 OVII 8.11/100.4 0.121 0.297 > 2 OVI † 2 b H1821 + 643 OVII 11.1/124 0.245 0.297 ∼> 2 OVI † 2 c H1821 + 643 OVIII 8.82/124 0.121 0.297 ∼> 2 OVI 3 M87 OVIII 4.32/68 † 0.004 0.237 2.8∼ – 4a MRK 421 OVII 2.9/41† 0.011 0.03 3.8 HI Lyα 4b MRK 421 OVII 2.1 /29 0.028 0.03 2.8 MRK 421 4c MRK 421 OVIII 8.15/115 0.011 0.03 2.8 HI Lyα 4d MRK 421 OVIII 6.86/102 0.062 0.03 2.8 HI Lyα 5 H 2356-309 OVII 30.0/416 0.03 0.165 3 Sculptor Wall 6 H 2356-309 OVII 28.5/395 0.03 0.165 4 Sculptor Wall 7 PKS 0558-504 OVIII 9.1/144 0.116 0.1372 2.8 broad HLαβ 8 Mkn 501 OVII 22.3/309 0.03 0.04 2.5 Hercules Supercluster

Table 2.3: Candidate detections from 1 to 8 are found in Fang et al. (2002); Mathur et al. (2003); Fujimoto et al. (2004); Nicastro et al. (2005b); Fang et al. (2010); Zappacosta et al. (2010); Nicastro et al. (2010); Ren et al. (2014), respectively, S/N values are as reported in the observational papers. A indicates that equivalent widths were reported as observed and have been converted† to rest frame values. The work of Buote et al. (2009) is not listed since Fang et al. (2010) conﬁrmed their results.

have been a number of NeVIII absorbers detected on Hubble Space Telescope searches

(Savage et al., 2005; Narayanan et al., 2011, 2012; Meiring et al., 2013). In 4 we also § show our NeVIII abundance predictions.

2.3 Modeling

2.3.1 Simulation

In this work we use a modiﬁed version of the Gadget-2 N-body + smoothed particle hydrodynamic (SPH) code (Springel, 2005; Oppenheimer and Davé, 2008). Our cosmology is a ΛCDM universe (inﬂationary cold dark matter with a cosmological con-

−1 −1 stant) with parameters Ωm = 0.25, ΩΛ = 0.75, Ωb = 0.046, h = H0/(100 km s Mpc ) =

36 0.7, ns = 0.96, and an amplitude of ﬂuctuations σ8 = 0.82. We use a cubic box 100

h−1Mpc in length. The simulation starts with 5763 dark matter and gas particles,

8 7 yielding particle masses of 5.8 10 M and 9.6 10 M . The softening length is × × 1.25 h−1kpc comoving, Plummer equivalent.

Cooling process are implemented using primordial abundances of Katz et al.

(1996), with metal line cooling based on tables from Wiersma et al. (2009). Star

formation is modeled with the subgrid recipe from Springel and Hernquist (2003),

−3 where a gas particle above a density threshold of nH = 0.13 cm is modeled as a

fraction of cold clouds embedded in a warm ionized medium, (McKee and Ostriker,

1977). Star formation follows a Schmidt law (Schmidt, 1959), with a timescale scaled

to match the z = 0 Kennicutt relation (Kennicutt, 1998). We use a Chabrier (2003)

initial mass function. Metal enrichment is Type II supernovae (SNe), Type Ia SNe,

and AGB stars.

Outﬂows are crucial to producing the correct galaxy masses and an enriched IGM

and CGM. Here we use the momentum driven wind “vzw" model described in detail

by Oppenheimer et al. (2004); Oppenheimer and Davé (2008), which was motivated

by the arguments of Murray et al. (2005). In this model particles in a star forming

region have a probability of being ejected that is dependent on the star formation

rate p = ∆t η SFR/mp, where ∆t is the time step, η is the mass loading factor, × × mp is the particle mass. Wind particles are ejected perpendicular to the galactic disk.

To allow a wind particles to escape the ISM, hydrodynamic forces are turned oﬀ until

10 −1 either 1.95 10 kms /vwind years have passed or the particle ﬂows to a density × −3 less than 10 percent of the star formation critical density (i.e., to nH < 0.013 cm ).

−1 In the “vzw" model the mass loading factor is η = σ0/σ where σ0 = 150 km s and σ

37 is the halo velocity dispersion estimated from the galaxy stellar mass. Wind particles are ejected at a velocity vwind = 3σ√fL 1, where fL is the luminosity factor in units − of galactic Eddington luminosity (i.e. the critical luminosity required to expel gas from the galactic potential) with values randomly taken from the interval [1.05, 2].

Our choice to use the “vzw" wind model is based on the model’s ability to accurately match a number of observations. These successes include: IGM enrichment seen through CIV at z 2 4 (Oppenheimer and Davé, 2006, 2008), OVI at z 0 ∼ − ∼ (Oppenheimer and Davé, 2009), and metal lines at z 6 (Oppenheimer and Davé, ∼ 2009), matching the galaxy mass-metallicity relation (Davé et al., 2007; Finlator and

Davé, 2008) and enrichment levels as seen in the intragroup gas at z 0 Davé ∼ et al. (2008), suppresses star formation in agreement with high-redshift luminosity functions (Davé et al., 2006), and reproducing the stellar and HI mass functions of galaxies with L L∗ (Davé et al., 2013). ≤ 2.3.2 Spectra

We use our spectral generation code SPECEXBIN to compute absorption spectra and calculate physical properties of the absorption gas; a detailed description of SPECEXBIN can be found in §2.5 of Oppenheimer and Davé (2006) and §2.3 of

Davé et al. (2010). SPECEXBIN ﬁrst computes the average gas density, temperature, metallicity, and velocity of gas in bins along a line of sight by integrating through the kernels of SPH particles that overlap the sightline. Lookup tables generated by

CLOUDY version 13.0 (Ferland et al., 1998) are then used as inputs to generate ionization fractions for the selected ionic species. These lookup tables are created under the assumption of ionization equilibrium in the presence of a uniform photoionizing

38 background see 3.3 . Real space spectra are converted to redshift space incorporating § SPH peculiar velocity, thermal motions, and Hubble ﬂow.

SPECEXBIN outputs a data file that contains redshift, density, temperature, metallicity, optical depth, and mass weighted densities, temperatures, and metallicities for each selected ion along a LOS. Absorption systems are then identified as contiguous regions in which the optical depth is 0.01. Rest-frame equivalent widths ≥ in velocity units are calculated by summing up (1 e−τ )/(1 + z) (c∆z) for each − × pixel that meets the optical depth threshold. We experimented with a variety of methods to select absorbers and define their boundaries, but our X-ray forest results are insensitive to details of these definitions, motivating our simple robust scheme.

In this work we extracted 2500 lines of sight randomly along one axis of the box for a total sampled redshift path length ∆z 80. The number of OVI, OVII, OVIII, ∼ CV, CVI, NeVIII, NeIX identiﬁed are 7330, 6732, 1547, 4604, 3832, 3390, 240, respectively. But note, however, that these identiﬁcations are for an optical depth threshold of 0.01, and typically it is only much stronger lines that are observable, with current instruments.

2.3.3 Radiation Background

The extragalactic background radiation adopted for this work is a composite of those computed by Haardt and Madau (2001) (HM01) and Haardt and Madau (2012)

(HM12), which we call HM12H. Dot-dashed and dotted curves in Figure 2.1 show

HM01 and HM12 backgrounds respectively. The sharp drop at 1 Ryd is the Lyman ≈ break caused by neutral hydrogen opacity in stellar atmospheres and the IGM. The broad maximum at E 30 keV arises from the X-ray peak of quasar spectral energy ≈

39 distributions. While there are numerous diﬀerences of detail between the HM01 and

HM12 models, the crucial one for our purposes is that HM12 adopts a rapidly evolving escape fraction for radiation from star forming galaxies, with the consequence that galaxies make a negligible contribution to the background at z = 0. The much higher galaxy contribution in HM01 makes the background spectrum higher intensity but softer for E 1 4 Ryd. In the X-ray region, the radiation background is empirically ≈ − constrained by direct observations, and the HM01 and HM12 backgrounds are very similar (see HM12 for detailed discussion).

The HI photoionization rate predicted by HM12 is a factor of ﬁve times below that required to reproduce the opacity of the HI Lyman-α forest at low redshift (Kirkman et al., 2007; Danforth et al., 2014), a mismatch that Kollmeier et al. (2014) have dubbed the “photon underproduction crisis." We refer readers to Kollmeier et al.

(2014) for detailed discussion of this problem and possible resolutions.

For our ﬁducial analyses in this paper, we take the HM12 spectrum at energies

> 3 Ryd, and at lower energies we take the HM01 spectrum multiplied by a factor of

1.5, the value required to match the Lyman-α forest opacity. The resulting composite spectrum, HM12H, is shown by the solid curve in Figure 2.1.

As we show below, our absorption predictions are insensitive to the diﬀerences among the three backgrounds. The X-ray absorption lines arise at energies (see Table

2.1) where the HM01 and HM12 backgrounds are similar, and strong lines require high densities where photoionization is unimportant in any case. The choice of background makes a small diﬀerence to our predictions of UV OVI absorption, and an important diﬀerence to our predictions of HI absorption, (which, however, play a small role in this paper).

40 Figure 2.1: Comparison of the HM12H, HM12, and HM01 ionizing backgrounds at redshift z = 0.1. At lower frequencies the HM12H background takes the form of HM01 but multiplied by 1.5 to match the observed Lyman alpha mean ﬂux decrement. At energies above 3 Rydberg the HM12H background follows HM12, and the HM12 line is therefore obscured in the plot. Vertical lines mark the locations of 1 and 4 Rydberg.

2.3.4 Ionization Fractions

Ion fractions are calculated as a function of temperate and density with the photoionization background as an input using the publicly available code CLOUDY (Fer- land et al., 2013); we use version 13.0. Figure 2.2 shows the HM12H ionization fractions for oxygen, carbon, and neon ions at three diﬀerent densities. The left column is for pure collisional ionization, a limit that applies when the density is high enough that photoionization is negligible. The middle and right columns show ion fractions

−4 −3 −5 −3 including photoionization at densities nH = 10 cm and nH = 10 cm , corresponding to overdensities ρb/ρ¯b at z = 0.1 of approximately 400 and 40, respectively.

The behavior for the three elements (oxygen, carbon, and neon) is similar, with shifts in temperature that reﬂect the diﬀerence in ionization potential (see Table 2.1).

The OVII, CV, and NeIX ions are all helium-like, and in collisional equilibrium each

41 of them has an ionization fraction near unity over a factor 4 range in temperature. ∼ At higher temperatures they give way to the hydrogen-like ions OVIII, CVI, and NeX

(not plotted), but these have a maximum fraction 0.5 rather than unity because ∼ the hydrogen-like state is more fragile; by the time the ion fraction of the helium-like state is negligible, most of the atoms are completely ionized and unable to produce line absorption. At lower temperatures, the lithium-like ions OVI and CIV (not plotted), and NeVIII appear, but these states are more fragile still, and in collisional equilibrium they occupy a narrow temperature range with a peak ion fraction 0.2. ∼ Photoionization allows a given ion to persist at temperatures far below those that could be required to maintain it in collisional equilibrium. For oxygen at nH =

10−5cm−3, a density characteristic of intergalactic ﬁlaments, photoionization with the

HM12H background allows OVII to remain signiﬁcant down to T 104 K. At this ∼ density OVII and OVIII now coexist over a wider temperature range, T 2 105 K ∼ × − 6 2 10 K, and even OVIII and OVI have a small overlap in range, with fion 0.1 × ≈ 5 at T 2 10 K. The largest change is for OVI, which now has fion 0.2 0.3 over ∼ × ≈ − 3 4 −4 −3 the broad temperature range T 10 K 5 10 K. However, at nH = 10 cm , ∼ − × photoionization eﬀects for OVI, OVII, and OVIII are almost negligible.

Behavior for carbon and neon is analogous, again with shifts that reﬂect diﬀerences

−4 −3 in energy. For CV, photoionization remains signiﬁcant at nH = 10 cm because there are more UV photons capable of ionizing CIV to CV compared to those capable

−5 −3 of ionizing OV to OVI. At nH = 10 cm , CV is the dominant ionization state over the whole range T = 103 105 K. Conversely, the eﬀects of photoionization are almost − −5 −3 negligible at the higher energies of NeVIII and NeIX, even for nH = 10 cm .

42 Figure 2.2: Ionization fractions for oxygen (top row), carbon (middle row), and neon ions (bottom row) at diﬀerent densities. The left column shows pure collisional ionization, which applies to gas dense enough that photoionization is negligible. Other columns include photoionization with the HM12H background for gas with an over density ρb/ρ¯b 40 (right) and ρb/ρ¯b 400 (middle), at z = 0.1. Red curves in the upper right panel≈ show fractions for the≈ HM01 background; in other panels the HM01 and HM12H results would be even closer.

Red curves in the upper right panel show the impact of adopting the HM01 background instead of the HM12H background. The OVI fraction drops by 0.1-0.2 dex at low temperatures because of the lower UV intensity, and there is smaller but noticeable drop in OVII fraction at low temperatures. However, these shifts are relatively small compared to the overall trends with temperature and density, and the diﬀerences would be much smaller in all other panels of Figure 2, which explains the insensitivity of our results to the choice of radiation background.

43 2.4 Results

2.4.1 Incidence of Strong Oxygen X-ray Forest Absorbers

Our principal predictions are equivalent width statistics for oxygen ions. Figure

2.3 shows the cumulative distribution of OVII and OVIII ions: the x-axis marks the

absorber equivalent width (kms−1 ) and the y-axis represents the number of absorbers

per unit redshift (more precisely we plot dN/d ln(1+z), which converges to dN/dz at

low redshift). The thick solid and dashed curves show results for our ﬁducial HM12H

background and the HM01 background respectively. The thin solid lines show the

results of C03, digitally extracted from their Figure 7. Recall that the C03 simulation

did not include galactic winds and adopted a constant Z = 0.1 Z metallicity when

computing oxygen absorption. A grey band, at an equivalent width 200 kms−1 , ∼ represents the approximate detection threshold for a 500 ks Chandra exposure for a bright point source of 10−11 ergs s−1. The candidate detections from Table 2.2 are marked in numbered grey circles, with a y-axis value corresponding to 1/ ln(1 + zqso), where zqso is the quasar redshift. This y-axis location gives a very rough indication of the contribution of that individual absorber to the cumulative distribution, i.e. the estimate of dN/d ln(1 + z) if one takes just the single absorber to be real and the eﬀective path length searched to be ∆z zQSO. As discussed in 2, current data do ≈ § no allow construction of an “observed" equivalent width distribution because it is too unclear which candidate detections are real, what path length has been searched for a given sensitivity threshold, and how biased the distribution may be as a result of targeting speciﬁc structures for observations.

At high equivalent width (W 50 kms−1 ) our results are fairly similar to those of ≥ C03, but we predict fewer weak OVII and (especially) OVIII absorbers. As we show

44 in 4.3, the metallicity of our strongest absorbers is similar to the 0.1Z assumed by § C03, but the metallicity declines systematically with density, so weaker lines are much

weaker in our current simulation. Although there are diﬀerences in simulation volume,

mass resolution, and input physics, all improved in our new simulation relative to C03,

this metallicity-density relation appears to explain most of the diﬀerence in results.

We predict about one absorber per unit redshift above an equivalent width of

20 kms−1 for OVII and 8 kms−1 for OVIII. At dN/d ln(1 + z) 0.2, the predicted ≈ equivalent widths are 55 kms−1 and 40 kms−1 , respectively. Nearly all candidate

detections (greyed numbers) are at equivalent widths larger than the largest found

in our ∆z 80 simulated path length. The two absorbers (3) and (4) were found ∼ by Nicastro et al. (2005b,a) in a path length ∆z = 0.03, at equivalent widths where

we predict < 1 absorber per redshift. Just as in C03, our predicted equivalent width range is almost disjoint with the range that is detectable with Chandra or XMM-

Newton.

Either essentially all of the candidate detections are spurious, or our simulation

predictions are seriously oﬀ. Figure 2.4 investigates one possible explanation for

this discrepancy, increased ionization by a local illuminating source. To model this

phenomenon we boosted the background intensity Jν, by factors of 10 and 100, and

analyzed 300 lines of sight for each case. The cumulative distributions for these cases

are plotted in Figure 2.4.

Changing the background intensity shifts the ionization balance among

OVI/OVII/OVIII/OIX for ﬁxed temperature and density. It is easiest to interpret this

ﬁgure by examining changes in the expected equivalent widths at a given abundance.

At dN/d ln(1 + z) 4, for example, one can see that boosting Jν by 10 depletes OVI ≈

45 and increases OVII, roughly doubling the OVII equivalent widths from 10 kms−1 to 20 kms−1 . With a factor of 100 boost, the OVI is photoionized to OVIII, so the

OVII equivalent widths return to their previous levels, and OVIII absorbers with W ≈ −1 10 kms appear. A factor of 100 increase in Jν modestly boosts the equivalent widths of strong OVII absorbers, and it has negligible impact on OVIII. Unfortunately, the rare strong absorbers arise in high density gas where photoionization has little eﬀect even at a boosted level. We conclude that local photoionization cannot resolve the discrepancy between our predictions and the candidate detections.

The dotted curve shows the predicted equivalent width distribution if we assign solar metallicity to each SPH particle, instead of using the metallicity computed in the simulation. With a solar metallicity IGM the simulation predicts 3 OVII ≈ systems per unit redshift above W = 100 kms−1 , roughly the equivalent width limit of most of the candidate detections, and about 0.7 OVIII systems per unit redshift above this threshold. Given the short total path length of all candidate X-ray forest observations to date, we would not expect to have a dozen or so true detections, but a solar metallicity IGM could explain some of the observed systems.

In Figure 2.5 we plot the cumulative distribution of CV, CVI, NeVIII, and NeIX absorbers. Again the choice of background intensity does not have an eﬀect on the absorber abundance. The CV ion is the only other absorber that shows signiﬁcant abundance above one absorber per unit redshift for equivalent widths > 10 kms−1 .

The CV ion probes lower temperatures compared to the other X-ray ions, 104 105 − K. The CVI ion traces somewhat hotter gas but does not extend into the higher temperature regions that OVII and OVIII do, and it tops out at a maximum equivalent width of 20 km/s. NeVIII ion is a high temperature UV WHIM absorber. A grey ∼

46 bar in the left panel of Figure 2.5 represents the estimated dN/dz and equivalent width range of NeVIII candidate detections (Savage et al., 2005; Narayanan et al.,

2011, 2012; Meiring et al., 2013). Again our predictions fall below observational estimates, though we caution that these are at higher redshifts than our simulations. The abundance of candidate detections is within our predictions but they are at equivalent widths that our ﬁducial model does not predict.

The NeIX ion (middle panel) probes the high temperature regime, but we again predict few strong absorbers, with a maximum equivalent width of 40 km/s and ∼ an abundance of one absorber per redshift at an equivalent width threshold less than

10 km/s.

2.4.2 Correlations of Line Strength

One approach to identify X-ray forest absorbers is to use more easily detectable

UV lines as a signpost, particularly OVI e.g., Mathur et al. (2003); Nicastro et al.

(2013). Figure 2.6 shows the predicted correlation between OVI and the oxygen or

carbon X-ray absorbers. For each point, the y-axis value marks the equivalent width

of the nearest OVII or OVIII whose central optical depth is within 300 kms−1 along

the line of sight of the OVI (x-axis) absorber’s central optical depth. Special symbols mark the five strongest OVII absorption systems (red), the five OVII systems just under 50 kms−1 (green), and the five OVII systems just under 20 kms−1 (blue).

The order of the special symbols, triangle, square, pentagon, hexagon, and circle

is from strongest to weakest. Understanding the special symbols deserves a bit of

an explanation. When the special symbol is plotted for an ion other than OVII, it

represents the value for that particular ion that is paired with the OVII ion special

47 Figure 2.3: Cumulative distribution of OVII and OVIII absorbers as a function of rest frame equivalent width threshold. We compare our HM12H results (thick line) to HM01 (thick dashed lines) and the results of Chen et al. 2003 (thin lines), who assumed a constant IGM metallicity Z = 0.1Z . Circled numbers represent observed candidate OVII and OVIII absorption lines. These detections are plotted by equivalent width (horizontal) and quasar redshift (vertical=1/ ln(1 + zQSO)), whose values are listed in Table 2.2. along with literature references. The grey bar marks the approximate detection limit for a 500 ks Chandra exposure on a bright X-ray quasar ( 10−11erg s−1 ) (Zappacosta et al., 2010). Our simulation predicts approximately one absorber per ∆z = 1.0 total path length above and equivalent width threshold of 20 kms−1 (OVII) and 10 kms−1 (OVIII).

48 s

Figure 2.4: Cumulative distributions of equivalent width for four diﬀerent post- processing analyses applied to the same simulation. The thick solid curve represents our ﬁducial case. Dashed and dot-dashed curves show cases where we increased the intensity of the HM12H background by factors of 10 and 100, respectively, to represent the potential impact of a local illuminating source. The dotted curve shows the result of setting all the gas particle metallicities to Z = Z , for the HM12H background.

49 Figure 2.5: Cumulative distributions of equivalent width for CV, CVI, NeVIII, and NeIX. A grey band (left panel) marks the estimated dN/dz and equivalent width range for candidate NeVIII detections (Savage et al., 2005; Narayanan et al., 2011, 2012; Meiring et al., 2013). redThe red line in the left panel shows results obtained by covering each particle with solar metallicity. [taken out, put back in?]

50 symbol deﬁnitions. The pairing is deﬁned to be the closest absorber along the same

line of sight and within 300 kms−1 of the special symbol OVII absorber. These special symbols will appear throughout this paper and are used as a case study for

OVII absorption and to highlight the diﬀerent physical state values between OVII and the other oxygen absorbers.

The “diagonal core" in the bottom panel of Figure 2.6 shows that the equivalent widths of moderate strength (W 1 10kms−1 ) OVII absorbers are correlated ∼ − with equivalent widths of associated OVI absorption, albeit with signiﬁcant scatter.

Many of these absorbers are at densities and temperatures where photoionization dominates (see Figure 2.12, below), and the correlation reﬂects stronger absorption at higher densities, but the ratio of OVII to OVI is highly temperature dependent (see

Figure 2.2). The strongest OVII absorbers arise in dense, collisionally ionized gas, and they have a wide range of OVI equivalent width because overlapping temperature range for signiﬁcant OVII and OVI fractions is narrow at high density (Figure 2.2, upper left). Conversely, strong OVI absorbers have a wide range of OVII equivalent width.

For OVIII, there is almost no overlapping temperature range at any density (Fig- ure 2.2), and as a result there is essentially no correlation between OVIII and OVI equivalent width (Figure 2.6 top panel). In fact, we will ﬁnd (Figure 2.12, below) that associated OVIII and OVI absorbers typically arise in gas at diﬀerent temperatures along the same line of sight.

Figure 2.7 show equivalent results for CV and CVI X-ray absorption. The temperature ranges for CV and OVI overlap substantially, (Figure 2.2), and there is correspondingly strong correlation between CV and OVI absorption, though with

51 signiﬁcant outliers in each direction. The CVI temperature range is closer to that of

OVII, and the CVI-OVI correlation resembles that for OVII.

Unfortunately, these results imply that OVI “ signposting" is of limited utility for the X-ray forest absorbers detectable with current instruments. Strong OVI absorption is not a good predictor of strong OVII, OVIII, or CVI (though it is predictive for

CV). Conversely, the strongest X-ray forest lines have a wide range of OVI strength.

However, many of the strongest X-ray absorbers do have associated OVI lines with

W 3kms−1 10 mÅ, so UV correlations may still be of some use in reducing the ≥ ∼ statistical penalty of searching a full spectrum for moderate S/N detections.

Figure 2.8 plots correlations among the X-ray absorbers themselves, i.e., equivalent widths of CV and CVI versus OVII and OVIII. Strong OVII predicts strong CVI and there is fair correlation with CV, but with more scatter. Turning Figure 2.8 on its side, strong CVI is a good signpost for strong OVII, but there is a larger scatter in

OVIII at strong CVI and in OVII and OVIII at strong CV. Conversely, the strength of CV and CVI absorption varies widely at all values of WOVIII. These results can again be understood in terms of the overlap (or disparity) of temperature ranges in

Figure 2.2.

Figure 2.9 plots the correlation between NeVIII absorption in the far UV and

OVII or OVIII absorption in the X-ray. At high densities, the narrow temperature range of NeVIII overlaps the temperature range of OVII but not OVIII, while at lower density photoionization allows OVIII and NeVIII to coexist (Figure 2.2). Figure 2.9 shows that roughly half of strong OVII absorbers have strong associated NeVIII (e.g.,

−1 −1 1/9 absorbers with WOVII > 20kms have WNeVIII > 6kms ), red [if the cut was W(OVII) > 20 and W(NeVIII) > 3 the ratio would be 16/45] and all NeVIII

52 Figure 2.6: Correlation of OVII and OVIII with UV OVI absorbers. Red special symbols mark the 5 strongest OVII systems, green mark the 5 OVII systems just under 50 km/s, and blue mark the 5 OVII systems just under 20 km/s. Correlations are selected by looking for absorbers on the vertical axis that are within 300 km/s of absorbers that are on the horizontal axis. Reversing this selection process does not change the correlation plot in any signiﬁcant way.

−1 −1 absorbers with WNeVIII > 10 kms have WOVII > 20kms . A signiﬁcant fraction of ∼ −1 strong OVIII absorbers have strong NeVIII (e.g., 5/26 wide WOVII > 20kms have

−1 WNeVIII > 6kms ), while roughly half of strong NeVIII absorbers have strong OVIII.

HI Lyα absorption is another possible signpost for X-ray forest lines, perhaps arising in cooler gas along the same line of sight. Our simple line identiﬁcation scheme is quite diﬀerent from that used in HI Lyα surveys, where absorption is more common and the S/N much higher than in X-ray forest searches. Instead of associating lines, we show in Figure 2.4.2 the total rest-frame equivalent width of HI Lyα absorption in the velocity range 300 kms−1 centered on each OVII or OVIII absorber.

53 Figure 2.7: Correlation of CV and CVI with UV OVI absorbers.

Figure 2.8: Correlation of CV or CVI absorbers with OVII or OVIII.

54 Figure 2.9: Correlation of NeVIII absorbers with X-ray OVII or OVIII absorbers.

Figure 2.10: Correlation plot between HI and OVI, OVII, OVIII. The HI values on the horizontal axis is computed by summing up all HI absorption within 300 km/s of each oxygen absorber.

55 Figure 2.11: OVI, OVII, OVIII, and HI proﬁles for each of the absorbers marked by the special symbols. Left column plots the strongest OVII absorption systems in our simulation, while middle and right columns plot the OVII systems just under 50kms−1 and 20kms−1 .

As a visual illustration of these line correlations, Figure 2.11 shows the OVI,

OVII, OVIII, and HI proﬁles for each of the 15 absorbers marked by special symbols in Figures 2.6-2.4.2.

2.4.3 Absorber Physical State

Figure 2.12 plots the distribution of metallicities, temperature, and over densities for OVI, OVII, and OVIII absorbers as a function of equivalent width. In each panel, the thick solid line marks the median value of the physical quantity in bins of equivalent width, while the adjacent lines show the 25/75, 10/90, and 1/99 percentiles

56 of the distribution. At high equivalent width and extremes of the distribution we plot

individual absorbers as points.

There are two subtleties to computing absorber physical properties from the simu-

lation. First, a given real-space pixel along the line of sight typically has contributions

from multiple SPH particles, which have diﬀerent temperatures, densities, and metal-

liciteis. Second, in redshift space, thermal broadening mix regions that are distinct

in real space (the second eﬀect would be present in a gird code calculation, while

the ﬁrst would not.) To more accurately represent each absorber’s physical state,

we calculate the metallicity, temperature, and density by taking the optical depth

weighted value of the quantity of interest: Q X τi i i Q system = X , (2.3) Q τi i where the sum is over all pixels assigned to the absorber.

One consequence of this optical depth weighting is that the OVI or OVIII absorber

associated with a given OVII absorber may have diﬀerent physical properties from the

OVII absorber itself. This eﬀect is evident when comparing the special symbols across

panels in Figure 2.12, especially the temperatures of the strong OVII absorbers (red

and green symbols) in the OVI and OVIII columns. The OVII-weighted temperate of

these absorbers is T 106 K , where the OVI fraction is negligibly small at any density ≈ (Figure 2.2). The OVI-weighted temperatures are T 104.5 105.5 K, where the OVI ≈ − fractions are high given the absorber over density. The OVI absorption associated with strong OVII thus arises in gas of diﬀerent phase, which unfortunately precludes combining OVI and OVII measurements to infer physical conditions (Mathur et al.,

57 2003). The OVIII-weighted temperatures, on the other hand, are only slightly higher

than the OVII-weighted temperatures.

Metallicities and over densities are much less sensitive to optical depth weighting

in most cases, though some systems (e.g., red triangle, green square) are much higher

in density in OVIII relative to OVII.

OVII absorption with W > 50 kms−1 arises at temperature and overdensity where

collisional ionization dominates over photoionization. These physical conditions are

generally favorable to strong OVIII absorption as well, though at high overdensity

and T < 1.5 106 K the OVIII fraction drops precipitously. At lower equivalent × widths, W < 20 kms−1 , the median OVIII absorber arises in photoionized gas with

T 104 K, though a few percent of absorbers arise in dense, collisionally ionized ∼ gas. At all equivalent widths, the median OVI absorbers are moderate overdensity

(ρ/ρ¯ 10 100), cool (T 104 K), and photo ionized, with a tail of denser, hotter ∼ − ∼ systems. Higher equivalent width correlates with higher density for OVI, OVII, and

OVIII, though the range of ρ/ρ¯ at a given W is always larger.

The median metallicites of strong OVII and OVIII absorbers are Z 0.1 Z , ∼ which helps explain the convergence of our equivalent width distribution to that of

CO3 (who assumed Z = 0.1 Z throughout the IGM) at high equivalent width (Figure

2.3). In our current simulation the average IGM metallicity decreases at lower gas

densities, but the absorption is biased toward higher metallicity gas, and the median

−1 metallicity is 0.1 Z even for WOVII 10 kms . The strongest OVII/OVIII ∼ ∼

absorber has Z 0.5 Z , but such high metallicites are rare outside of galaxies, and ∼

only 1% of absorbers have comparably high Z/Z at lower equivalent width. ∼

58 Figure 2.12: Absorber metallicity, temperature, and density as a function of equivalent width, for OVI, OVII, and OVIII. The thick line marks the median value of the physical quantity in bins of equivalent width, while the adjacent lines mark the 25th and 75th percentile, 10th and 90th percentile, and the 1st and 99th percentile. Outside of these regions or at high equivalent width we plot individual absorbers as points. Special symbols mark OVII-selected absorbers as in previous plots. Note that an absorber may have diﬀerent physical parameters on diﬀerent panels because temperature and density are weighted by ion density. For example the strongest OVII absorber (red triangle center-middle panel) has a temperature 3 106 K while the associated OVI absorber (red triangle left-middle panel) has a temperature∼ × 3 104 K. ∼ ×

59 2.4.4 Absorber Environments

Recent searches for the X-ray forest absorption have found candidate detections

associated with large scale structure (Buote et al., 2009; Fang et al., 2010; Ren et al.,

2014), and highly ionized metals are expected in low mass galaxy groups or in the

circum-galactic medium (CGM) of shock-heated gas halos. Characterizing the galaxy

environment of absorbers is thus important both for practical purposes (where should

one search for absorbers) and for understanding the physics of the X-ray forest.

In Figure 2.13 we plot the local galaxy environment of the 15 special symbol

absorbers highlighted in our earlier plots. Each panel is a 6 h−1 6 h−1 Mpc slice × showing galaxies within 500 kms−1 of the absorber redshift. Around each central ± galaxy (the most massive galaxy in its parent halo), we plot the halo virial radius calculated as the radius for which the mean enclosed density is 200 ρcrit. × A survey of Figure 2.13 shows that even strong OVII absorbers (top two rows) occupy a range of galactic environments. Most of them lie (at least in projection) within the virial radius of a galaxy halo, but the two in the second column (red and green squares) do not. Two of these absorbers (right column, red and green circles)

13 reside in halos with M > 6 10 M , and several others occupy lower mass halos × in dense galaxy concentrations. However, the strongest absorber (red triangle, top

13 left panel) occupies a moderate mass halo (M = 4 10 M ) in a moderate galaxy × over density. The weaker, W 20 kms−1 (bottom row, note that at this threshold ≈ we predict approximately one absorber per unit redshift) still reside within or near halo virial boundaries, but they appear to occupy somewhat lower density galactic environments on average. In addition to the systems shown in Figure 2.13, we have examined environments of 15 additional absorbers with W 20 50 kms−1 to ≈ − 60 conﬁrm that the trends, and variation, seen in Figure 2.13 are representative of the

larger absorber population.

To quantify these environmental trends, Figure 2.14 plots the cumulative distri-

bution of 3-dimensional redshift-space distances between absorbers and the nearest

galaxy in the simulation. Top, middle, and bottom rows show results for galaxy pop-

10 10 11 ulations above stellar mass thresholds of 2 10 M , 6 10 M , and 2 10 M , × × × while the three columns show results for OVI, OVII, and OVIII absorbers (left to right).

In each panel we show nearest neighbor distributions for absorbers above equivalent width thresholds of 10 kms−1 and 50 kms−1 and, for comparison the nearest neighbor distributions for galaxies above the stellar mass threshold (crosses) or for randomly distributed points (dot-dashed curve, computed analytically). In the middle and right columns, ﬁlled cirlces mark the nearest neighbor distances for the ﬁve strongest OVII and OVIII absorbers, respectively.

Beginning with the OVII panels, we see that the strongest absorbers are highly

11 correlated with massive (Mgal 2 10 M ) galaxies. All ﬁve of the strongest ≥ × OVII absorbers lie within 2 h−1 Mpc of such a galaxy, and 50% of absorbers with

W 60 kms−1 lie within 1 h−1 Mpc of a galaxy(check). Comparing the dashed lines ≥ or the ﬁlled circles down the middle column shows that the nearest galaxy to a strong

11 OVII absorber is often but not always above Mgal = 2 10 M . The weaker OVII × absorbers, with W 10 kms−1 , are much less concentrated around galaxies, showing ≥ a nearest-neighbor distribution similar to that of galaxies themselves (but still much narrower than that of unclustered random points).

61 Results for OVIII absorbers are generally similar, but in this case the nearest-

neighbor galaxies are more often lower mass, and there is no longer much diﬀerence

in the clustering of W 50 kms−1 and W 10 kms−1 absorbers. For OVI, the ≥ ≥ clustering of both W 10 kms−1 and W 50 kms−1 is similar to that of typical ≥ ≥ galaxies.

Our simulation predicts that strong OVII is more likely found within 1 2 h−1 − Mpc of galaxy

2.5 Conclusion

In this work we have used a large volume hydro-simulation to make predictions

for the X-ray forest in the local universe ( z = 0.1). Our goal is to probe the warm

hot intergalactic medium, thought to contain a substational fraction of the “missing

baryons", through X-ray absorption features in the spectrum of distant quasars. The

bulk of the “missing baryon" fraction in the WHIM is believed to be shock heated

by gravitational collapse to temperatures > 105 K, most easily traced by X-ray ab-

sorption of heavy ions. This work is an evolution of previous work of a decade ago,

Chen et al. (2003). In addition to simulation improvements we are able to compare

our results to ten years of observational studies of the X-ray forest with Chandra and

XMM-Newton. In particular we have examined the predicted absorption of six ions,

OVII, OVIII, CV, CVI, NeVIII, and NeIX in the intergalactic medium, paying special

attention to the most abundant X-ray forest ion OVII. Additionally we have studied

the physical state and galaxy environment of OVI, OVII, and OVIII.

Our results are as follows are.

62 Figure 2.13: The local galaxy environment of the strongest OVII absorber (top row), OVII absorbers with equivalent widths just under 50 km/s (middle row), and OVII absorbers with equivalent widths just under 20 km/s (bottom row). Galax- ies are plotted as colored circles. The color bar on the right marks galaxy mass in units of M . Each central galaxy is encirlced by the halo viral radius determined by the radius for which ρ = 200 ρcrit. The line of sight is into the page and we have selected a slice that is within× 500 km/s of the absorber redshift. For reference we list the mass of the largest± halos in each of the top ﬁve panels; log(M) = 13.16, 12.76, 13.5, 13.57, 14.26 M . A star marks the location of an additional OVII{ absorption system, above an} equivalent width threshold of 10 kms−1 .

63 Figure 2.14: Cumulative distribution of absorber-galaxy nearest neighbor distances (comoving, redshift space) for OVI, OVII, and OVII absorbers (left to right). Top, 10 middle, and bottom rows show results for galaxy mass thresholds of 2 10 M , 6 10 11 × × 10 M and 2 10 M , respectively . In each panel, solid and dashed curves show results for absorbers× with EW 10 kms−1 and 50 kms−1 , respectively, and crosses and dot-dashed line show galaxy-galaxy≥ nearest neighbor distributions and Poisson (unclustered) expectation for reference. Filled circles mark the nearest neighbor distance for the 5 strongest OVII and OVIII absorbers.

64 Our main results for OVII and OVIII abundance is captured in Figure 2.3, • which shows results for the abundance of absorbers per unit redshift above an

equivalent width threshold. We ﬁnd that at equivalent widths greater than 20

and 10 kms−1 one should ﬁnd one OVII and OVIII absorber per redshift.

Compared to observations our model does not predict absorber abundance or • equivalent width strengths that match candidate X-ray forest observations (Fig-

ure 2.3, Table 2.2). In order to get equivalent width strengths that are in agree-

ment with observations we require the IGM to have solar metallicity (Figure

2.4). Given the general low signiﬁcance of the candidate X-ray observations and

in combination with our theoretical results the subject of the missing baryons

in the high temperature WHIM is still murky. Perhaps all observations are false

or our understanding of the chemical make up of the IGM is severely oﬀ or that

simulations are missing a key physics ingredient.

We can make no guarantee that X-ray forest ions are correlated, in the sense • that a detection of one guarantees the detection of another. Coexistence of ions

requires preferable temperature and density conditions, which do not occur

often enough to produce substantial ion correlation. However, simultaneous

detections of X-ray forest ions substantially constrains the temperature and

density environment.

OVII is the most likely observable X-ray forest tracer. Strong OVII absorption • (equivalent width > 50 kms−1 ) is more likely found within 1-2 h−1 Mpc of

galaxy, in densities that only allow collisional ionization, and at temperatures

106 K. ∼

65 Chapter 3: Streaming velocities and the baryon-acoustic oscillation scale

3.1 Introduction

Baryon-acoustic oscillations (BAOs) have emerged as one of the major probes of

the expansion history of the Universe. In the early Universe, the ionized baryons

were kinematically coupled to the cosmic microwave background (CMB) of photons

via Thomson scattering. This baryon-photon ﬂuid supported sound waves, sourced by

primordial perturbations, that could travel a comoving distance rd prior to decoupling.

This distance is precisely constrained by CMB observations to be rd = 147.3 0.3 ± Mpc Planck Collaboration et al. (2015b). After decoupling, the baryons became eﬀectively pressureless at large scales, and the perturbations in the baryons and dark matter grew together in a single combined growing mode. Thus at low redshift, all tracers of the matter density, either in dark matter or baryons, are predicted to show a feature in their correlation function at a position rd – or equivalently oscillatory features in their power spectrum P (k), with spacing ∆k = 2π/rd. This feature acts as a standard ruler, enabling galaxy redshift surveys to measure the distance- redshift relation D(z), and (using the radial direction) the expansion rate H(z). The

BAO scale is of interest because its distinctive shape and large scale make it less

66 dependent on nonlinear evolution and galaxy formation physics than the broad-band power spectrum Albrecht et al. (2006); Seo et al. (2008); Padmanabhan and White

(2009); Seo et al. (2010); Mehta et al. (2011). However, the small amplitude of the feature makes it detectable only in very large surveys Seo and Eisenstein (2007b).

The early detections of the BAO in the clustering of low-redshift galaxies Cole et al. (2005); Eisenstein et al. (2005) have given way to a string of results of ever- increasing precision Percival et al. (2007, 2010); Blake et al. (2011a,b); Padmanabhan et al. (2012); Xu et al. (2012); Kazin et al. (2013); Anderson et al. (2014); Kazin et al. (2014); Cuesta et al. (2015). High-redshift measurements have become possible by using quasar spectra to trace large-scale structure in the autocorrelation function of the Lyman-α forest and in its cross-correlation with quasars Busca et al. (2013);

Kirkby et al. (2013); Slosar et al. (2013); Font-Ribera et al. (2014); Delubac et al.

(2015). Taken together, these measurements have become one of the most important constraints on dark energy models Aubourg et al. (2014a). These successes have motivated a suite of future spectroscopic surveys to measure BAOs more precisely, including the Prime Focus Spectrograph Takada et al. (2014) and the Dark Energy

Spectroscopic Instrument (DESI) Levi et al. (2013) in the optical, and Euclid Laureijs et al. (2011b) and the Wide-Field Infrared Survey Telescope (WFIRST) Spergel et al.

(2015) in the infrared. They have also spawned novel concepts for measuring BAOs such as radio intensity mapping Wyithe et al. (2008); Chang et al. (2008) as planned for e.g. the Canadian Hydrogen Intensity Mapping Experiment (CHIME) Bandura et al. (2014).

The same acoustic oscillations that give rise to the BAO also leave the baryons with an r.m.s. velocity of 33 km/s relative to the dark matter at decoupling, coherent

67 over scales of many comoving Mpc. This velocity is cosmologically small at late times, since it decays 1/a. However, it was realized in 2010 that the sound speed in ∝ neutral hydrogen at the decoupling epoch is only 6 km/s, so this “streaming velocity” is supersonic Tseliakhovich and Hirata (2010b) and hence is more important than standard Jeans-like filtering in determining the scale on which baryons can fall into dark matter potential wells. The filtering mass of the cold IGM (before re-heating by astrophysical sources) is increased typically by a factor of 8 relative to what it ∼ would be without the streaming velocities Tseliakhovich et al. (2011b). Moreover, the streaming velocity has order-unity spatial variations and a power spectrum showing prominent acoustic peaks Tseliakhovich and Hirata (2010b). The effects of streaming velocities on gas accretion and cooling have been a subject of intense analytical and numerical investigation Stacy et al. (2011); Maio et al. (2011); Greif et al. (2011);

McQuinn and O’Leary (2012); Fialkov et al. (2012); Naoz et al. (2013); Asaba et al.

(2015).

It was soon realized that these ingredients implied that small, high-redshift galaxies whose abundance was modulated by the streaming velocity would show an unusual

BAO signature Dalal et al. (2010), and in some models the BAO signature in the pre- reionization 21 cm signal could be strongly enhanced relative to the strength of the

BAOs in the matter clustering alone Visbal et al. (2012); McQuinn and O’Leary

(2012); Fialkov (2014). Moreover, if low-redshift galaxies have any memory of the streaming velocity, then low-redshift BAO measurements could be biased Dalal et al.

(2010); Yoo et al. (2011b). While the direct eﬀect of streaming velocities is on the

6 small scale structure (< few 10 M ), feedback processes associated with reionization ∼ × or metal enrichment could inﬂuence the subsequent evolution of more massive galaxies

68 in a way that is diﬃcult to predict from ﬁrst principles Dalal et al. (2010); Yoo et al.

(2011b). In the absence of a first-principles theory, this effect can be parameterized in terms of the “streaming velocity bias” bv, which is the excess probability to find a galaxy in a region with r.m.s. streaming velocity versus a region with zero streaming velocity. Studies based on perturbation theory have found that the BAO ruler shrinks

(stretches) for bv > 0 (bv < 0) Yoo et al. (2011b); Yoo and Seljak (2013b); Slepian and Eisenstein (2015b).

In this paper, we compute the effect of streaming velocities on the BAO feature including all leading-order terms; we find that the largest term was missing from previous work. The galaxy density, a scalar, cannot depend on the direction of the streaming velocity, but only on its magnitude (or square). In linear perturbation theory with Gaussian initial conditions, the density (an odd moment) cannot correlate with the velocity-squared (an even moment), one must go to the next order to obtain a nonzero result. Previous investigations included three such effects: (i) the nonlinear evolution of the matter density field; (ii) nonlinear galaxy bias; and (iii) the autocorrelation of the streaming velocity field. We show that to consistent order in perturbation theory, two additional terms appear: (iv) the dependence of the galaxy abundance on the local tidal field Baldauf et al. (2012); and (v) an “advection term,” since galaxy properties depend on the past streaming velocity at their Lagrangian position. We find that for plausible bias parameters, the tidal effect is small, but the advection term greatly enhances the shift in BAO position and impacts the shape and amplitude of the BAO feature. Because knowing the correct BAO scale is required to relate observed galaxy clustering to underlying cosmological physics, understanding

69 the impact of streaming velocities is critical if we are to obtain unbiased results from the future generation of high-precision measurements.

3.2 Formalism and bias model

We ﬁrst construct a model for the distribution of galaxies. We describe our notations, choice of cosmology, and normalization conventions in §3.2.1, before proceeding in §3.2.2 to building the model for the distribution of galaxies. The supplemental ma- terials include three appendix sections with more details on the calculations in this paper.

3.2.1 Conventions

We work in real space in this paper, leaving the redshift-space treatment to future work. The ﬁducial cosmology is the base 6-parameter Planck + “everything” model

2 2 Planck Collaboration et al. (2015b): ﬂat ΛCDM with Ωbh = 0.02230; Ωmh =

−1 −1 −9 −1 0.14170; H0 = 67.74 km s Mpc ; As = 2.142 10 (at k = 0.05 Mpc ); × ns = 0.9667; and τ = 0.066.

The streaming velocity ﬁeld vbc vb vc can be computed on large scales in ≡ − linear perturbation theory, and scales 1/a once the baryons have decoupled and ∝ are eﬀectively pressureless. Following the notation of Ref. Slepian and Eisenstein

(2015b), we deﬁne the normalized streaming velocity ﬁeld to be

vbc(x, a) vbc(x, a) vs(x) = = , (3.1) 2 0 1/2 σvbc v (x , a) 0 h bc ix where the average in the denominator is taken over all positions x0. By dividing out

σvbc, we obtain a normalized streaming velocity that is independent of redshift and is of order unity. Note that some authors Yoo et al. (2011b); Yoo and Seljak (2013b)

70 have deﬁned an alternative variable ur, equivalent to √3 vs here, which has an r.m.s.

value of 1 per axis (see Eq. 3.4).

At linear order, the relative velocity ﬁeld in Fourier space can be written as

ˆ vs(k) = iTv(k, z)kδlin(k, z), (3.2) −

where Tv (Tv,b Tv,c)/Tm is the ratio of transfer functions that map initial curvature ∝ − fluctuations into late-time matter and velocity fluctuations. Note that with this definition, even though vs(k) is redshift-independent, Tv decays as 1/D(z), where ∝ D(z) is the growth factor. The appropriate transfer function can be obtained from a

Boltzmann code – we used both CAMB Lewis et al. (2000b) and CLASS Blas et al.

(2011), obtaining consistent results – and the normalization of Eq. (3.1) at any desired

redshift z can be obtained by enforcing the integral:

Z kmax 2 k Pm,lin(k, z) 2 2 Tv(k, z) dk = 1, (3.3) 0 2π | |

where Pm,lin(k, z) is the linear matter power spectrum. The choice of kmax is set by the

minimum scale relevant for the formation of the relevant tracer, e.g. its Lagrangian

radius. In practice, we ﬁnd that σvbc is nearly insensitive to the choice of kmax unless

ﬂuctuations below the pre-reionization baryonic Jeans scale are included. These ﬂuc-

−1 tuations are not relevant for galaxy formation, and we thus choose kmax = 10h Mpc .

2 We will also need the tidal ﬁeld magnitude s = sijsij. Here the traceless-

−2 1 symmetric dimensionless tidal tensor sij is given by sij(x) = ( i j δij)δ(x). ∇ ∇ ∇ − 3 3.2.2 Galaxy biasing model

We now write a model for the overdensity of a given tracer of large-scale structure,

δg. This tracer population could be e.g. galaxies, Lyman-α absorption, or the unre-

solved H i 21 cm emissivity. While the detailed physics of the formation and evolution

71 of these tracer populations remains an outstanding problem, nonlinear galaxy biasing McDonald and Roy (2009b) provides a useful framework to study the streaming velocity effect. This theory is based on the idea that galaxy formation is local, with the only long-range physics being gravity. Under these assumptions, the galaxy overdensity measured on scales large compared to the range of galaxy formation physics should depend only on the density and tidal fields, the local streaming velocity, and their past history (since galaxy formation, while local in space, is obviously not local in time). At small scales, additional terms can appear involving derivatives of the density or tidal field, but as we are interested in large scales we do not include these. Note that any terms involving past history should be based on the history at

fixed Lagrangian position, since small-scale structure, metal enrichment, and similar properties are advected by large-scale velocity fields rather than remaining in a fixed

Eulerian cell.

Since galaxy overdensity is a scalar, its dependence on vs must be at least quadratic.

The leading corrections to the linear galaxy 2-point function are from terms of (δ4 ) O lin 3 in the linear density ﬁeld δlin, since terms of (δ ) vanish for Gaussian initial con- O lin ditions. It follows that the galaxy biasing model we require should go up to (δ3 ). O lin Since our primary interest is the contributions coming from streaming velocities, we

3 neglect (δ ) contributions to the density ﬁeld that do not involve vs (e.g. Saito O lin et al. (2014b)). The tracer density is then given by:

b2 2 2 bs 2 2 δg(x) = b1δ(x) + [δ (x) σ ] + [s (x) s ] + 2 − 2 − h i ··· 2 2 + bv[v (q) 1] + b1vδ(x)[v (q) 1] s − s −

+ bsvsij(x)vs,i(q)vs,j(q) + , (3.4) ···

72 where δ denotes the (nonlinear) dark matter density ﬁeld and σ2 is the variance in density ﬂuctuations. The Lagrangian position (i.e. comoving position of the particles just after the Big Bang) is denoted q to distinguish it from Eulerian position x. In

this formulation, it is the linear vs that is evaluated at q, while the advection of

the density and tidal ﬁelds is already included through the perturbative expansion of

the density ﬁeld. Although Eq. (3.4) expresses the astrophysical motivation for the

advection contribution, this term can be derived from a purely Eulerian perspective,

as shown in Appendix C. Indeed, as we demonstrate, the advection term is required

to preserve Galilean invariance.

The deﬁnitions of the bias coeﬃcients are not standardized: while our bv is equiva-

lent to that of Slepian and Eisenstein (2015b), br in Yoo and Seljak (2013b) is related

1 to both via br = 3 bv. Note also that there are multiple combinatoric conventions for

b2.

The mapping between x and q can be expanded to order δlin, since we are only

3 concerned with contributions to δg up to (δ ). Lagrangian and Eulerian positions O lin are related by x(q, η) = q + Ψ(q, η). The Lagrangian displacement is given to linear

−2 order by Ψ = δlin(x, η) (this is the Zel’dovich approximation, combined with −∇∇ the fact that at leading order we do not need to distinguish x and q in the argument

of a perturbation ﬁeld). Any ﬁeld ϕ then maps according to

−2 ϕ(q) = ϕ(x Ψ) = ϕ(x) + ϕ(x) δlin(x, η) + . (3.5) − ∇ · ∇∇ ···

At the required order,

2 2 −2 2 v (q) = v (x) + [ i δlin(x)][ iv (x)]. (3.6) s s ∇ ∇ ∇ s

73 3.3 Eﬀect on 2-point statistics and BAO position

0 We are interested in the tracer auto-correlation function ξgg(r) = δg(x)δg(x ) , h i where r = x x0. In this work, we consider terms that contribute at up to one-loop, − i.e. (δ4 ): O lin

2 2 2 1 2 2 2 ξgg(r) = b δ δ + b1b2 δ δ + b1bs δ s + b δ δ 1h | i h | i h | i 4 2h | i 1 2 2 2 1 2 2 + b s s + b2bs δ s 4 sh | i 2 h | i 2 −2 2 + 2b1bv δ vs + δ i δ ivs h | i h |∇ ∇ ∇ i 2 2 2 2 2 2 2 + b2bv δ v + bsbv s v + b v v , (3.7) h | si h | si vh s| si where we use the shorthand A B A(x)B(x0) . h | i ≡ h i

We note that there is no term proportional to b1b1v or b1bsv, since by parity a scalar or tensor must have zero average correlation with a vector at the same position, and hence all contractions vanish when Wick’s theorem is applied. In the following, we denote the streaming velocity correlations in Eq. (3.7) as ξδv2 , ξadv, ξδ2v2 , ξs2v2 , and

ξv2v2 , respectively. See Appendix A for the details of how all relevant correlations are calculated.

We use Wick’s theorem to simplify the advection term:

−2 2 ξadv(r) = δ i δ ivs h |∇ ∇ ∇ i −2 0 0 0 = 2 δlin(x)[ i δlin(x )][vs,j(x ) ivs,j(x )] h ∇ ∇ ∇ i 2 0 = Ls δlin(x)∇ vs(x ) , (3.8) 3 h · i where

−2 Ls = [∇ δlin(x)] vs(x) , (3.9) h ∇ · i 74 and we note that only one of the three contractions in the second line of Eq. (3.8) is

0 nonzero (the others vanish since by isotropy the symmetric tensor ivs,j(x ) has zero ∇ −2 0 0 correlation with the vectors i δlin(x ) or vs,j(x ) at the same point). ∇ ∇ To illustrate the impact of streaming velocities and this new advection term, we

show results for a ﬁducial sample of emission line galaxies (ELGs) at z = 1.2, such

as that relevant for DESI, Euclid, and WFIRST. Unless otherwise noted, we assume

2 b1 = 1.5, b2 = 0.25, and bs = (1 b1) = 0.14. However, the impact of streaming 7 − −

velocities depends primarily on the ratio bv/b1 for the tracer in question, and thus

our results qualitatively hold for other samples.

Due to nonlinear evolution, the BAO in the dark matter correlation δ δ is shifted h | i from its linear position. To model this, we include the one-loop standard perturbation theory (SPT) contributions to the matter power spectrum Bernardeau et al. (2002b).

As can be seen in Fig. 3.2, these terms lead to a 0.2% shift in the BAO at z = ∼ 1.2. We note that SPT does not provide the ideal model for the evolved BAO – we leave a more detailed treatment of this eﬀect for future work. The inclusion of these nonlinear terms alters the impact of streaming velocities when ﬁtting the

BAO position – nonlinear broadening makes the BAO feature more sensitive to the shift from streaming velocities. Note that Ref. Yoo and Seljak (2013b) modeled the nonlinear matter power spectrum using Haloﬁt Smith et al. (2003), which does not include nonlinear evolution of the BAO (see their Fig. 3).

Streaming velocity contributions to the correlation function (including all prefac- tors) are plotted in the top panel of Fig. 3.1. For reasonable bias values, ξδv2 had been considered the primary streaming velocity term. The new advection eﬀect is larger by a factor of 5. The bottom panel of Fig. 3.1 shows the ELG correlation function ∼

75 0.2

0.0

0.2

0.4 ⇠ 2 v ⇠adv 0.6 ⇠2v2

2 2 ] 0.8 ⇠s v 2 ⇠v2v2 [Mpc 40 gg ⇠ 2 r

10 linear

bv =0 0 b = 0.01b v ± 1 b = 0.05b v ± 1 80 100 120 140 160 180 200 r [Mpc]

Figure 3.1: Top panel: All contributions to the correlation function from streaming 4 velocities up to (δlin) are shown at z = 1.2, with b1 = b2 = bs = 1, and bv = 0.01. The newO advection term (black solid line) is the dominant effect. Bottom panel: The ELG correlation function is shown for fiducial bias values (b1 = 1.5, b2 = 0.25, bs = 0.14) at z = 1.2 with different values of bv. Dashed (dot-dashed) indicates − positive (negative) bv. For reference, the thin solid (grey) line shows the linear theory prediction.

with diﬀerent values of bv – the impact on both the shape and position of the BAO feature is apparent.

To quantify the shift of the BAO peak due to relative velocity effects, we employ a method similar to Seo et al. (2008); Yoo and Seljak (2013b), fitting the shifted power spectrum to a template with flexible broadband power – see Appendix B for more details. Figure 3.2 shows the BAO shift as a function of bv/b1, both including and ignoring contributions from nonlinear galaxy bias and BAO evolution. For positive

76 3.0 t 0.5 n 2.5

e 0.4 c

r 2.0 0.3

e 0.2 p

1.5 0.1 n

i 0.0

) 1.0 0.1 1 0.2 -0.004 -0.002 0.000 0.002 0.004 − 0.5 α (

f 0.0 i

h SPT + advect.

s 0.5 SPT

O Linear + advect.

A 1.0 Linear

B ELG + advect. 1.5 -0.04 -0.02 0.00 0.02 0.04 bv /b1

Figure 3.2: The shift in BAO position due to streaming velocities is shown as a function of bv/b1. Thick (thin) lines show the shift with (without) the advection term. The solid black lines include the one-loop SPT correction to the dark matter power spectrum (b2 = bs = 0), while the solid grey line also includes ﬁducial b2 and bs values for the ELG sample. Inset shows detailed behavior for small bv.

bv/b1 streaming velocities damp the BAO feature and shift it to smaller scales. For negative bv/b1, streaming velocities enhance and quickly dominate the BAO feature as bv increases, leading to a saturation in the eﬀective shift. Note that we diﬀer from | | Ref. Yoo and Seljak (2013b) by an overall factor of 2 in the numerical evaluation of

(δ4 ) terms and ﬁnd a correspondingly smaller shift in the BAO position from the O lin non-advection terms they consider.

77 3.4 Conclusions

We have examined the impact of streaming velocities on the tracer correlation

function, considering all contributions at (δ4 ) and including two terms not con- O lin sidered in previous work. While we ﬁnd the correlation of the tidal ﬁeld and the

streaming velocity to be small, the contribution from advection is signiﬁcant, domi-

nating the total eﬀect of streaming velocities on the BAO feature. The importance

of advection is due to the rapid change in streaming velocity correlations at the BAO

scale. For a simple illustration, consider a single δ-function overdensity that has

evolved to decoupling (z 1020). Dark matter at all separations infalls towards the ≈ overdensity. Within the acoustic scale, baryons are roughly in hydrostatic equilib-

rium. Just inside the acoustic scale, baryons move outward due to radiation pressure,

while just outside this scale, baryons match the dark matter infall (e.g. Figure 2

of Ref. Slepian and Eisenstein (2015b)). Thus, the streaming velocity, vbc, rapidly

changes at the acoustic scale, and advection can move tracers separated by roughly

this scale between regions of diﬀerent vbc. Indeed, this eﬀect is nearly maximal, since

the ﬁrst-order displacement is almost entirely anti-correlated with the relative veloc-

ity direction (correlation coeﬃcient of 0.9). The qualitative behavior expected ∼ − from this simpliﬁed picture can be seen in Figure 3.1: at the BAO scale, advection

has carried in tracers that formed at slightly larger scales, where vbc is much smaller.

Thus, for positive (negative) bv/b1 the observed correlation function is suppressed (en-

hanced). The overall eﬀect is to shift the observed BAO feature to smaller (larger)

scales and to suppress (enhance) its amplitude.

The eﬀect of advection boosts the impact of bv, dramatically increasing the range of parameter space over which streaming velocities are relevant to large-scale structure

78 surveys. Conversely, advection makes bv signiﬁcantly easier to detect, providing a potential window into the astrophysics of streaming velocities and tracer formation.

For instance, DESI will obtain an overall BAO-scale measurement of order 0.2%

(1σ), corresponding to the shift induced by streaming velocities at bv/b1 0.004 Levi ∼ et al. (2013). The ultimate impact of streaming velocities will depend on the as-yet- unknown value and sign of bv, as well as other possible bias terms related to differences in the baryon and CDM fluids (see Appendix C). Their direct effect is to suppress

5 2/3 the infall of baryons into halos by a fractional amount (10 M /Mhalo) (e.g. ∼ Tseliakhovich et al. (2011b)). This scaling results from the fact that the suppression is proportional to v2, and is (1) when the streaming-enhanced ﬁltering scale is the halo s O −5 mass, which suggests a contribution to bv of order a few 10 at galaxy mass scales. ×

We view this as a “soft” lower bound on bv in the sense that e.g. reionization physics may be much more important and thus dominate bv, but there is no reason for a precise cancellation that would give a total bv 0. On the other hand, very large values ≈

( bv /b1 > 0.1) would disrupt the qualitative agreement with current observations. | | ∼ The eﬀect of streaming velocities may also be relevant for other luminous tracers of large-scale structure, notably the Lyman-α forest and (possibly) unresolved 21 cm emission; these tracers are sensitive to a range of mass scales down to the post-

9 reionization Jeans scale (few 10 M ), and their bv may be correspondingly larger. ×

We leave a more detailed consideration of astrophysical eﬀects that impact bv for future work. We also defer consideration of reconstruction (which may have signiﬁcant impact on displacements) Eisenstein et al. (2007); Padmanabhan et al. (2009) and redshift-space distortions in the context of streaming velocities.

79 While we have primarily considered the impact of streaming velocities on the

position of the BAO feature, it is clear from Fig. 3.1 that the BAO shape is also significantly altered. Although it is not typically not considered in cosmological analyses, these results suggest that the shape may help to separate the effect of streaming velocities from geometric effects. We will consider implications of changes to the BAO shape in future work.

80 Chapter 4: The eﬀects of assembly bias on cosmological inference from galaxy-galaxy lensing and galaxy clustering.

4.1 Introduction

A central challenge of contemporary cosmology is to determine whether accelerat- ing cosmic expansion is caused by an exotic “dark energy" component acting within

General Relativity (GR) or whether it instead reﬂects a breakdown of GR on cosmological scales. One general route to distinguishing dark energy from modiﬁed gravity is to test whether the growth of structure (measured through redshift space distortions, gravitational lensing, or galaxy clustering) is consistent with GR predictions given constraints on the expansion history from supernovae, baryon acoustic oscillations (BAO), and other methods (see reviews by Frieman et al. 2008; Weinberg et al.

2013). Intriguingly, many (but not all) recent estimates of low redshift matter clustering are lower than predicted from cosmic microwave background (CMB) anisotropies evolved under a ΛCDM framework (see, Mortonson et al. 2014; Aubourg et al. 2014b;

Planck Collaboration 2015, ΛCDM = inflationary cold dark matter universe with a cosmological constant). If this discrepancy is confirmed, it could be the first clear indication that ΛCDM is an incomplete description of cosmology, and it would hint in

81 the direction of modiﬁed gravity explanations A promising route to measuring matter clustering is to combine galaxy clustering with galaxy-mass correlations inferred from galaxy-galaxy lensing (e.g. Mandelbaum et al. 2013; More et al. 2015). This approach necessarily requires a model for galaxy bias, i.e. for the relation between galaxy and dark-matter distributions. In this paper we examine how the theoretical uncertainties associated with modeling galaxy bias inﬂuence the matter clustering inferred from combinations of galaxy and galaxy matter clustering.

Galaxy-galaxy lensing (GGL) is a direct probe of the total matter content around a galaxy and provides a statistical relationship between galaxy and matter distributions.

Speciﬁcally GGL produces a tangential shear distortion of background galaxy images around foreground galaxies or clusters (see Bartelmann and Schneider 2001 for a detailed review and 2 of Mandelbaum et al. 2013 for details that are relevant to § this paper). With adequate photometric redshifts of background and foreground objects, the mean tangential shear can be converted to the projected excess surface mass density ∆Σ(R), where R is the 2-dimensional radial distance transverse to the line of sight. The excess surface mass density can be related to an integral of the galaxy-matter cross correlation function (Sheldon et al. 2004) Z R Z ∞ h 2 0 0 0 ∆Σ(R) =ρ ¯ 2 r ξgm(r , z)dzdr R 0 −∞ Z ∞ (4.1) 0 i ξgm(r , z)dz , − −∞ 3 with ρ¯ = Ωmρcrit,0(1 + z) . With suﬃciently good measurements, ∆Σ(R) in Eqn. 4.1 can be inverted to yield the product of the mass density and galaxy-mass correlation function, Ωmξgm(r). Here we will assume that this inversion can be carried out and also that the projected galaxy correlation function can be inverted to yield the 3- dimensional, real-space correlation function ξgg(r) of the same galaxies for which

82 Ωmξgm(r) is measured by GGL. In practice, cosmological analyses may proceed by

forward modeling to predict projected quantities rather than inversion to 3-d (e.g.

Mandelbaum et al. 2013; More et al. 2015; Zu and Mandelbaum 2015). For our

present purpose of understanding the complications (potentially) caused by complex

galaxy bias, it is most straightforward to focus on the 3-d quantities themselves.

The correlation functions ξgg(r) and ξgm(r) are related to the matter auto-correlation

function ξmm(r) by

2 ξgg(r) = bg(r)ξmm(r) , (4.2)

ξgm(r) = bg(r)rgm(r)ξmm(r) , (4.3)

where

ξgm rgm = p (4.4) ξmmξgg is the galaxy-matter cross-correlation coeﬃcient. Equations 4.2, 4.3, and 4.4 are

general and may be taken as deﬁnitions of the scale dependent bias factor bg(r) and

cross correlation coeﬃcient rgm(r). We note that the quantity rgm(r) in real space is

not constrained to be less than or equal to one in magnitude, unlike the shot-noise

corrected counterpart in Fourier space (Guzik and Seljak 2001).

Using Eqns. 4.2, 4.3, 4.4 one can combine observations of ξgg(r) and Ωmξgm(r) to

determine 2 2 [Ωmξgm(r)] 1 Ωmξmm(r) = 2 . (4.5) ξgg(r) · [rgm(r)] 1/2 Thus, given a theoretical model for rgm(r), one can infer the product Ωmξmm, with an overall amplitude proportional to Ωmσ8(z), where σ8(z) is the rms matter ﬂuctuation

−1 −1 −1 amplitude in 8h Mpc spheres at redshift z and h = H0/100 kms Mpc . To a

83 ﬁrst approximation, it is the z = 0 value of Ωm that is constrained, though for high- redshift lens and source samples the nature of the constrained parameters becomes more complex and depends on what auxiliary observational constraints are being imposed.

Under fairly general conditions, one expects rgm to approach unity on large scales, where ξgg(r) 1 (see Baldauf et al. 2010). However, because ∆Σ(R) is an integrated ≤ quantity, it is aﬀected by small scale clustering even at large projected separation R and is therefore potentially susceptible to uncertainties in non-linear galaxy bias. To mitigate this problem, Baldauf et al. 2010, constructed a ﬁltered GGL estimator that eliminates small scale contributions. This approach was put in practice by Mandel- baum et al. (2013), who applied the SDSS GGL at R > 2 and 4h−1Mpc to derive

0.57 constraints on σ8 and Ωm. They found σ8(Ωm/0.25) = 0.80 0.05, about 2σ below ± the Planck+ΛCDM prediction.

While the method in Baldauf et al. (2010) and Mandelbaum et al. (2013) is already competitive with other probes of low-z structure, one could do better by incorporating smaller scales, and thus increasing the signal-to-noise ratio of the GGL measurement.

This requires a description of the relation between galaxies and mass that extends to non-linear scales.

Halo occupation distribution (HOD) modeling oﬀers one approach to tie galaxies and dark-matter distributions down to non-linear scales (Jing et al. 1998; Peacock and Smith 2000; Seljak 2000; Ma and Fry 2000; Scoccimarro et al. 2001; Berlind and

Weinberg 2002). The HOD speciﬁes P (N Mh), the conditional probability that a halo | of mass Mh hosts N galaxies of a speciﬁed class, as well as the spatial and velocity distribution of galaxies within host halos. Yoo et al. (2006) showed that if one chooses

84 HOD parameters to match galaxy clustering measurements, then the predicted GGL

signal depends on the adopted cosmological model, increasing with σ8 and Ωm in

both the large scale linear regime and on smaller scales. Several variants of the HOD

modeling approach to GGL have been described in the literature (Leauthaud et al.

2011; Yoo and Seljak 2012; Cacciato et al. 2013). More et al. (2015) measured GGL

by SDSS-III BOSS galaxies (Dawson et al. 2013) using imaging from the CFHTlens

survey (Heymans et al. 2012), and applying the methods of van den Bosch et al.

+0.044 +0.019 (2013) they obtained σ8 = 0.785−0.044 for Ωm = 0.310−0.020 at the 68% conﬁdence interval. Recently, Zu and Mandelbaum 2015 have applied a modiﬁed HOD method to the SDSS main galaxy sample (Strauss et al., 2002), obtaining an excellent joint

ﬁt to clustering and GGL for a cosmological model with σ8 = 0.77 and Ωm = 0.27.

The philosophy of deriving cosmological constraints from such modeling is to treat HOD quantities as “nuisance parameters" that allow one to marginalize over uncertainties associated with galaxy formation physics (Zheng and Weinberg 2007).

Standard HOD modeling assumes P (N Mh) is uncorrelated with the halo’s large scale | environment (in this paper we take environment to be the surrounding dark matter density) at ﬁxed halo mass. If P (N Mh) does depend on large scale environment, this | will change the predicted galaxy clustering and galaxy-mass correlation for given set of HOD parameters. The risk is then that modeling with an environment-independent

HOD may leave systematic bias in the cosmological inferences and/or underestimate the derived cosmological parameter uncertainties associated with galaxy formation physics.

The simplest formulation of excursion set theory (Bond et al., 1991) predicts that halo environment is correlated with halo mass but uncorrelated with formation history

85 at ﬁxed mass (White, 1999), motivating the idea of an environment independent HOD.

However, N-body simulations show that the clustering of halos of ﬁxed mass varies

systematically with formation time or concentration (Sheth and Tormen 2004; Gao

et al. 2005; Wechsler et al. 2006; Harker et al. 2006; Jing et al. 2007). The dependence

of halo clustering on formation time or concentration is strongest for old halos well

below M∗. For halos above M∗ there are indications that the situation is reversed

(Wechsler et al. 2006). In general the dependence of halo clustering on halo properties other than mass is termed assembly bias (Gottlöber et al. 2001; Sheth and Tormen

2004; Gao et al. 2005; Avila-Reese et al. 2005; Harker et al. 2006; Wechsler et al. 2006;

Wang et al. 2007; Croton et al. 2007; Maulbetsch et al. 2007; Bett et al. 2007; Wetzel et al. 2007; Angulo et al. 2008; Dalal et al. 2008; Fakhouri and Ma 2009; Faltenbacher and White 2010). The physical origin of assembly bias remains unclear, though a number of explanation have been proposed (Dalal et al., 2008; Lacerna and Padilla,

2011, 2012). Some level of assembly bias may arise from correlated eﬀects of long wavelength modes on halo formation times, breaking the uncorrelated random walk assumption that underlies the minimal excursion set model (?). Assembly bias can also arise in the non-linear regime from tidal truncation of low mass halo growth in the environment of high mass halos. Observationally, assembly bias has yet to be unanimously conﬁrmed, however, a few groups have made progress (Lacerna et al.,

2014; Miyatake et al., 2016; Lin et al., 2016)

If galaxy properties are tightly coupled to halo formation history, then a galaxy population can inherit assembly bias from its parent halos. Such galaxy assembly bias implies that P (N Mh) depends on halo environment (or halo clustering) at ﬁxed | mass. Limited work has been carried out measuring the galactic assembly bias signal

86 in hydrodynamic simulations. In some simulations, the HOD has shows little to no dependence on halo environment (Berlind et al. 2003; Mehta 2014), which suggests that stochasticity in the galaxy formation physics in these simulations erases signa- tures of halo assembly bias. However, recent work of Chaves-Montero et al. (2015) has shown a galactic assembly bias signal in the EAGLE simulation (Schaye et al.,

2015), boosting the galaxy-correlation function by 25% on scales greater than ∼ 1h−1Mpc. Although the analysis themselves are different, the differing conclusions ∼ of Mehta (2014) and (Chaves-Montero et al., 2015) in simulations of volume suggest that the presence of galaxy assembly bias in hydrodynamic simulations depends on the adopted physical description of star formation and feedback. Semi-analytic models predict a significant assembly bias effect in galaxy clustering for some galaxy populations (Croton et al. 2007), particularly red galaxies of low stellar mass.

Abundance matching (AM) is an alternative route to populating dark-matter halos with galaxies (e.g. Kravtsov et al. 2004; Vale and Ostriker 2004; Tasitsiomi et al. 2004;

Conroy and Wechsler 2009; Guo et al. 2010; Simha et al. 2009; Neistein et al. 2011;

Watson et al. 2012; Rodríguez-Puebla et al. 2012; Kravtsov 2013; Chaves-Montero et al. 2015). Simple versions of abundance matching monotonically tie galaxy luminosity or stellar mass to some proxy for the halo or subhalo gravitational potential well, such as halo mass or maximum circular velocity. For subhalos, AM recipes typically use Mh or Vmax at time of accretion, with the expectation that tidal stripping will aﬀect subhalo mass but not its stellar content (Conroy et al., 2006; Reddick et al.,

2013). With a subhalo mass at accretion recipe, AM is fairly successful at reproducing the galaxy content of halos in hydrodynamic cosmological simulations (Simha et al. 2009, 2012; Chaves-Montero et al. 2015). Abundance matching can easily be

87 extended to incorporate scatter between between halo mass and galaxy properties

and has been shown to be remarkably successful at reproducing observed evolution

of galaxy clustering and other aspects of galaxy evolution.

Recently Hearin & Watson (2013; hereafter HW13) have extended the AM idea

to galaxy color. Their age matching technique monotonically maps a measure of halo

formation time to galaxy color at ﬁxed stellar mass. Applied to the Bolshoi ΛCDM

N-body simulation (Klypin et al., 2011), this prescription produces good agreement

with observed luminosity and color dependent clustering and GGL observations of

SDSS galaxies, despite having essentially no free parameters (Hearin et al., 2014).

(However, Zu and Mandelbaum 2016 show that age-matching at ﬁxed stellar mass

over predicts the GGL signal of the most luminous blue galaxies.)

By comparing clustering in the HW13 galaxy catalogs to “scrambled" catalogs,

that eliminate correlations with halo formation history, Zentner et al. (2014) show

that the HW13 catalogs exhibit signiﬁcant galaxy assembly bias. For stellar mass

threshold samples, this assembly bias arises because HW13 assign stellar mass based

on Vmax, and at ﬁxed Mh the halos that form earlier tend to have higher concentrations

and higher Vmax. For color selected samples, the direct mapping between formation time and color imprints a stronger assembly bias signature.

The HW13 catalogs adopt a physically plausible and empirically successful description of galaxy formation physics, so even if they are not correct in all details, we would like cosmological inference methods based on HOD models to be insensitive to galaxy assembly bias at this level.

In this paper we examine the degree to which galaxy assembly bias can aﬀect matter clustering inference results from GGL + galaxy clustering analysis. We begin

88 by examining the HOD and its environmental dependence in the HW13 catalogs, confirming findings of Zentner et al. (2014) but recasting them in a more HOD-specific form. We then turn to the implications of GGL modeling, focusing our attention on the cross-correlation coefficient rgm(r), which is the quantity needed to recover Ωmξmm.

We show that HOD models fit to the galaxy correlation function of the HW13 catalogs yield accurate predictions (at the 2 5% level of precision allowed by the Bolshoi − simulation volume) for rgm(r), even though they are incomplete descriptions of the bias in these galaxy populations. We concentrate mainly on galaxy samples defined by luminosity thresholds. However, we also consider a sample of red galaxies above a luminosity threshold, in part to examine a case with near-maximal assembly bias effects, and in part because red galaxy samples allow accurate photometric redshifts, which make them more attractive for observational GGL studies. Our bottom line, illustrated in Fig.4.17, is that HOD modeling of galaxy clustering and GGL allows

2 −1 for accurate recovery of Ωmξmm(r) on scales r > 1h Mpc, even in the presence of ∼ galaxy assembly bias as predicted by HW13.

4.2 Halo Occupation Distribution of the HW13 Catalogs

4.2.1 Galaxy assembly bias in the HW13 Catalogs

The abundance and age-matching catalogs of HW13 are built from the Bolshoi

N-body simulation (Klypin et al., 2011), which uses the Adaptive Reﬁnement Tree

(ART) code (Kravtsov et al. 1997; Gottloeber and Klypin 2008 ) to solve for the evolution of 20483 particles in a 250h−1Mpc periodic box. The mass of each particle

8 −1 −1 is mp 1.9 10 h M . The force resolution is 1 h kpc. The cosmological ≈ × ≈ parameters are: Ωm=0.27, ΩΛ = 0.73, Ωb = 0.042, ns = 0.95, σ8 = 0.82, and

89 103 104 104 104 HW13 HW13 scrambled 103 103 103 102

2 2 2

) 10 10 10

r 1 ( 10 g

g 1 1 1 ξ 10 10 10

100 100 100 100

Mr -19.0 Mr -20.0 Mr -21.0 Mr -21.5 10-1 10-1 10-1 10-1 1.20 1.40 2.2 4.0 1.35 1.15 2.0 3.5 1.30 1.8 3.0 1.25

) 1.10 r

( 1.20 1.6 2.5 g

b 1.05 1.15 1.4 2.0 1.10 1.00 1.2 1.5 1.05 0.95 1.00 1.0 1.0 0.1 1.0 10.0 0.1 1.0 10.0 0.1 1.0 10.0 0.1 1.0 10.0 1 1 1 1 r [h− Mpc] r [h− Mpc] r [h− Mpc] r [h− Mpc]

Figure 4.1: The impact of galaxy assembly bias on the galaxy correlation function, for samples deﬁned by four thresholds in absolute magnitude. Top panels compare ξgg from the HW13 abundance matching catalog (solid) to that of a scrambled catalog (dashed) in which the eﬀect of galaxy assembly bias is erased by construction. Bottom p panels plot the corresponding galaxy bias factor bg(r) = ξgg(r)/ξmm(r).

90 −1 −1 H0 = 70 km s Mpc . Bolshoi catalogs and snapshots are part of the Multidark

Database and are available at http:www.multidark.org. Halos are identiﬁed by the

(sub)halo ﬁnder ROCKSTAR (Behroozi et al., 2013), which uses adaptive hierarchical

reﬁnement of friends-of-friends groups in six phase-space dimensions and one time

dimension. Halos are deﬁned within spherical regions such that the average density

inside the sphere is ∆vir 360 times the mean matter density of the simulation. ≈ To create galaxy catalogs HW13 follow a two step process. First galaxies of a particular luminosity are assigned to (sub)halos based on an abundance matching scheme.

Their abundance matching algorithm requires that the cumulative abundance of SDSS galaxies brighter than luminosity L is equal to the cumulative abundance of halos and subhalos with circular velocities larger than Vmax, ng(> L) = nh(> Vmax). Speciﬁcally

HW13 uses the peak circular velocity Vpeak (Reddick et al., 2013), which is the largest

Vmax that the halo or subhalo obtains throughout its assembly history. The second

step, age-matching, assigns colors by imposing a monotonic relation between galaxy

colour and halo age at ﬁxed luminosity, matching to the observed colour distribution

in SDSS. The redshift deﬁning halo age is set to the maximum of (1) the highest

12 −1 redshift at which the halo mass exceeds 10 h M , (2) the redshift at which the

halo becomes a subhalo, (3) the redshift at which the halo’s growth transitions from

fast to slow accretion, as determined by the ﬁtting function of Wechsler et al. (2002).

Criterion (3) determines the age for most halos and subhalos. These catalogs are

publicly available at http://logrus.uchicago.edu/ aphearin/. ∼ When abundance matching is based on halo mass e.g. Conroy et al. (2006), then

the resulting population of central galaxies, has no assembly bias by construction.

However, at ﬁxed halo mass, halos that form earlier are more concentrated and thus

91 have higher Vmax, so even luminosity thresholded samples exhibit galaxy assembly bias in the HW13 catalogs (Zentner et al. 2014). We consider four samples deﬁned by absolute magnitude thresholds Mr 5 log h 19, 20, 21, 21.5 (hereafter we − ≤ − − − − omit the 5 log h for brevity). The -20 and -21 samples bracket the characteristic galaxy

luminosity L∗, with -21 thresholds yielding the overall best clustering measurements

in the SDSS main galaxy sample (Zehavi et al. 2011). The -19 threshold corresponds

to fairly low luminosity galaxies with high space density, while -21.5 corresponds

to rare, high luminosity galaxies. We also consider a sample of red galaxies with

Mr 20 and g r 0.8 0.3(Mr + 20.0). Because color is tied monotonically to ≤ − − ≥ − halo formation time in HW13, this selection yields a near-maximal degree of galaxy assembly bias. In addition to testing our methods under extreme conditions, this sample is observationally relevant because red galaxies allow relatively accurate photometric redshifts, making them attractive for galaxy-galaxy lensing measurements in large imaging surveys such as the Dark Energy Survey (Rozo et al., 2015). The number of galaxies in the Bolshoi simulation volume is 244766, 96595, 17250, 3954 for the Mr 19, 20, 21, and 21.5 samples, respectively, and 56591 for the red ≤ − − − −

Mr 20 samples. Of these, a fraction fcen = 0.75, 0.77, 0.81,.85, 0.77 are central ≤ − galaxies of their host halos, and a fraction fsat = 1 fcen are satellite galaxies located − in subhalos.

Fig. 4.1 compares galaxy correlation functions measured from the luminosity- threshold HW13 catalogs to those from scrambled versions of the same catalogs. p Lower panels show galaxy bias deﬁned by bg(r) = ξgg(r)/ξmm(r). Scrambled cat-

alogs are constructed by binning central and satellite systems in host halo mass,

randomly reassigning centrals to other halos within the mass bin, then randomly

92 reassigning satellite systems to these centrals. By construction scrambling removes

any galaxy assembly bias present in the original HW13 catalog, i.e., any correlation

between the galaxy content of a halo and any halo property other than mass. For

full details of the scrambling process see Zentner et al. (2014), who present a similar

clustering analysis.

For the Mr 19 sample, bg(r) is about 10% higher in the HW13 catalog relative ≤ − −1 to the scrambled catalogs at r > 3 h Mpc. Both catalogs show a drop in bg(r) as

r decreases from 3h−1Mpc to 0.5h−1Mpc, then a rise on still smaller scales. The two

correlation functions converge at r < 0.5 h−1Mpc.

The Mr 20 sample shows similar behavior, but the diﬀerences between scram- ≤ − bled and unscrambled correlation functions are somewhat smaller. For luminous,

Mr 21 galaxies, the diﬀerence in the large scale bias is only 3%, and the two ≤ − ∼ correlation functions are essentially converged at r < 1 h−1Mpc. For the most lumi- ∼ nous sample, Mr 21.5, any diﬀerences are smaller still, and consistent with noise ≤ −

in ξgg(r).

The trends in Fig. 4.1 make sense in light of the previous studies of halo assembly

bias, which show that the dependence of clustering on formation time is strongest

for low mass halos and declines as the halo mass approaches the characteristic mass

M∗ of the halo mass function (Gao et al., 2005; Harker et al., 2006; Wechsler et al.,

2006). The minimum halo mass for Mr 19 galaxies is low, and halos in denser ≤ − environments are more likely to host HW13 galaxies because they have earlier for-

mation times and higher circular velocities at ﬁxed mass. This preferential formation

in dense environments accounts for the higher large scale bias factor of the HW13

catalog relative to the scrambled catalog. As the luminosity threshold and minimum

93 host halo mass increase, the bias factor grows but the impact of assembly bias di-

13 −1 minishes. For Mr 21.5 the minimum halo mass is Mh 10 h M (see Fig. ≤ − ≈ 4.8 below), and any residual impact of assembly bias on the galaxy population is no longer discernible.

In the lower luminosity samples, bg(r) becomes scale-dependent (at the 10 20% − level) at the transition between the 2-halo regime of ξgg(r), where galaxy pairs come from separate halos, and the 1-halo regime dominated by galaxy pairs within a single halo. Any halo massive enough to contain two galaxies is far above the minimum mass threshold for a central galaxy, so any assembly bias eﬀects in the 1-halo regime will arise from the satellite galaxy population. The convergence of correlation functions at small r suggests that assembly bias eﬀects in the HW13 catalog are driven by the central galaxy population rather than satellites, a point we demonstrate explicitly in below.

Figure 4.2 shows galaxy correlation functions and galaxy bias results for red Mr ≤ 20 galaxies (our maximal galaxy assembly bias sample), again comparing HW13 to − scrambled catalogs. The large diﬀerence in bias factors in the bottom panel of Fig.

4.2 is indicative of the strong galaxy assembly bias for low luminosity red galaxies in HW13. As with the luminosity threshold case, ξgg(r) for colour selected HW13 and scrambled catalogs converges on small scales, indicating the assembly bias is also primarily due to central galaxy populations.

4.2.2 HOD Analysis of Mr 19 Galaxies ≤ −

The HOD speciﬁes the probability P (N Mh) that a halo of mass Mh contains | N galaxies of a speciﬁed class, together with auxiliary prescriptions that specify the

94 104 HW13 103 HW13 scrambled 2

) 10 r ( g

g 1 ξ 10 Red 100 Mr -20.0 10-1 10-1 100 101 1.8

1.7

1.6 ) r

( 1.5 g

b 1.4

1.3

1.2 0.1 1.0 10.0 1 r [h− Mpc]

Figure 4.2: Galaxy assembly bias in Mr 20 red samples. As in Fig. 4.1, the top panel compares the measured galaxy correlation≤ − function in HW13 to a scrambled version of HW13, and the bottom panel compares results for the galaxy bias factor.

4500 700 4000 600 log(M ) 11.25 log(M ) 12.25 3500 h ∼ h ∼ 3000 500 2500 400 2000 300 1500 200 1000 500 100 0 0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 log(Mh ) 14.25 120 10 ∼ log(M ) 13.25 100 8 h ∼ 80 6 60 4 40 20 2 0 0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0

log(δ1 5) log(δ1 5) − −

Figure 4.3: Distribution of halo environments for halos in four 0.1-dex bins of mass, −1 centered at log Mh/h M = 11.25, 12.25, 13.25, and 14.25 (top left to bottom right). The density contrast δ is measured in a spherical annulus of 1 < r < 5h−1Mpc.

95 spatial and velocity distributions of galaxies within halos (Benson et al., 2000; Berlind

and Weinberg, 2002). Following Guzik and Seljak (2001) and Kravtsov et al. (2004),

we separate P (N Mh) into contributions from central and satellite galaxies, |

P (N Mh) = P (Ncen Mh) + P (Nsat Mh) . (4.6) | | |

We adopt the parameterization of Zheng et al. (2005), which provides a good ﬁt to the

theoretical predictions of hydrodynamic simulations and semi-analytic models and a

good ﬁt to observed galaxy correlation functions (Zehavi et al. 2005, 2011; Coupon

et al. 2012, 2015).

The mean occupation of central galaxies is described by

1h log Mh log Mmin i Ncen = 1 + erf − . (4.7) h i 2 σlog Mh

The parameter Mmin sets the scale where Ncen = 1/2 while σlog M controls the h i h sharpness of the transition from Ncen = 0 to Ncen = 1. Physically σlog M repre- h i h i h sents the scatter between halo mass and central galaxy luminosity. A large scatter corresponds to a soft transition and a small scatter to a sharp transition.

Satellite occupancy is determined by

α Mh Mcut Nsat = exp . (4.8) h i M1 − Mh

The HOD parameter M1 is approximately the mass scale at which halos have an

average of one satellite. At larger halo masses the satellite occupancy increases as a

power-law with slope α. Mcut controls the scale at which the power law is truncated at low mass.

The total mean occupancy is the sum of central and satellite mean occupancies

N = Ncen + Nsat . (4.9) h i h i h i 96 A host halo is assigned a central galaxy by Bernoulli sampling with Eqn. 4.7 serving as the probability for success. If a host halo is determined to contain a central, the number of satellites assigned to the central is done by Poisson sampling with Eqn.

4.8 serving as the average.

We want to examine the dependence of the HOD on the large scale environment of halos at ﬁxed Mh. We deﬁne halo environment by the dark matter density contrast

−1 δ1−5 measured in a spherical annulus of 1 < r < 5h Mpc. Figure 4.3 shows the distribution of δ1−5 in four narrow bins of log Mh. As expected, the higher mass halos tend to reside in higher density regions, giving rise to the well known mass dependence of halo bias. To remove the trend from our analysis, we rank the halos by δ1−5 in narrow (0.2-dex) mass bins, so we can compare the HOD of halos in, e.g., the 20% highest or lowest density environment relative to other halos of nearly equal mass. We have experimented with diﬀerent deﬁnitions of environment and found that our overall results are insensitive to, e.g., changing the radii of the spherical annulus, including the central 1h−1Mpc, or incorporating distance to nearest large halo as an environmental measure.

Fig. 4.4 illustrates the HOD dependence on host halo environment in the HW13 catalog. The solid grey curve shows N(Mh) for the global HOD computed by h i counting galaxies in 0.2 dex bins of Mh with out reference to environment. Solid

(dashed) curves show N(Mh) computed for the 20 % of halos with the highest h i (lowest) density environments in each mass bin. The shape of the measured HOD curves in Fig. 4.4 is similar to the functional form predictions of Eqns. 4.7 and

4.8: a sharp rise in N from zero to one associated with central galaxies, and a h i shallow plateau between N = 1 2, followed by a steepening to a power law. h i −

97 The environmental dependence is visually evident, primarily for low mass host halos

where N < 1. In the language of HOD parameters, the HOD for halos in higher h i

density environments (top 20 %) has a lower Mmin and a larger σlog Mh than the global

HOD, and the reverse is true in low density environments. Halos with Mh 1 2 ∼ − × 11 −1 10 h M are therefore more likely to host a central galaxy with Mr 19 if they ≤ − reside in a high density environment, giving rise to the higher bias factor seen in Fig.

4.1, relative to the scrambled catalog which has the environment independent, global

HOD by construction. The satellite N(Mh) shows little dependence on environment, h i though the highest mass halos are only present in the dense environments.

How much of the assembly bias in the HW13 catalog is explained by this depen-

−1 dence of N(Mh) on the 5h Mpc environment? To answer this question, we con- h i struct catalogs with an environment-dependent HOD (EDHOD) and compare their

clustering to that of the HW13 galaxies. As a ﬁrst step, we measure N(Mh) in 30 h i

bins of environment δ1−5 (and the same 0.2-dex mass bins). This bin-wise EDHOD

automatically incorporates environmental dependence for both central and satellite

galaxies. In the terminology introduced by Hearin et al. (2015) one can regard our

EDHOD as a “decorated HOD" with δ1−5 as the additional control variable.

After choosing the number of central and satellite galaxies in each halo by drawing

from P (N Mh, δ1−5), we must determine the positions of galaxies within the halos. | A standard approach is to place the central galaxy at the halo center-of-mass and

distribute satellite galaxies with a Navarro et al. (1997) type proﬁle (hereafter NFW)

so that n(r) ρNFW(r). However, we find that the satellite galaxy distribution in the ∝ HW13 catalogs differs substantially from an NFW profile, a consequence of satellites being placed within subhalos in the AM scheme (Nagai and Kravtsov 2005; Zentner

98 et al. 2005). Figure 4.5 illustrates this diﬀerence, comparing the HW13 satellite

profiles in two narrow bins of halo mass to an NFW profile. The radial profile of HW13

satellites is much ﬂatter than an NFW proﬁle, and it extends beyond the viral radius

because Rockstar halos are aspherical while rv is deﬁned with a spherical overdensity.

We therefore use the measured HW13 radial profiles rather than an NFW form to create our EDHOD catalogs. We found that clustering on scales r 2h−1Mpc would ≤ be substantially different if we imposed an NFW profile for the satellite distribution.

In Fig. 4.6 we compare galaxy auto-correlation results taken from our measured

(ED)HOD catalogs and from HW13 for Mr 19 samples. The top panel plots ≤ −

ξgg(r) while the bottom panel shows the fractional diﬀerence in ξgg(r) compared to

HW13. As seen in the lower panel, randomizing the host-centric satellite angles in the

HW13 catalog, thus removing the eﬀects of halo ellipticity and substructure, depresses

−1 ξgg(r) by up to 10% below 1 h Mpc but has negligible eﬀect at larger separations.

The standard HOD model underpredicts the HW13 ξgg(r) even at large separations,

an indication of the impact of galaxy assembly bias, as already seen in Fig. 4.1. The

EDHOD model, on the other hand, matches HW13 almost perfectly at r > 7 h−1Mpc.

−1 However, the EDHOD ξgg(r) rises 5% above HW13 at r 5 h Mpc and falls 10% ≈ below at r 1 2 h−1Mpc, before converging to the isotropized satellite case at still ≈ − smaller scales. The dashed curve shows the eﬀect of imposing an EDHOD for central

galaxies but using the global HOD for satellites. These results are nearly identical

to those of the full EDHOD model, demonstrating that for this sample it is central

galaxy environment dependence that matters. We conclude that incorporating the

environmental dependence of central galaxy occupations reproduces the large scale

bias of the HW13 catalog but leaves a 5-10% residual in ξgg(r) on non-linear scales.

99 103 global HOD 102 bottom 20 % top 20 % ® ) 101 h M ( 0 N 10 M 19.0 r − -1 10 11.5 1 Mmin =10 h− M ¯ 1011 1012 1013 1014 1015 1 M [h− M ] ¯

Figure 4.4: The Measured HOD in the HW13 catalogs for the Mr 19 sample. ≤ − The grey line shows the global mean occupation function N(Mh) for halos in all h i environments. Solid and dashed black curves show N(Mh) for halos in the 20% h i highest and lowest density environments, respectively, as measured by δ1−5. For the 11.5 −1 global HOD, N(Mh) = 0.5 at Mh = Mmin = 10 h M . h i

Figure 4.7 plots the Mr 19 cross correlation coeﬃcient rgm(r) for the HW13, ≤ − HOD, and EDHOD catalogs. The HW13 curve remains close to unity (within 0.5%) at r > 1 h−1Mpc, with a drop and rise inside 0.4 h−1Mpc. The EDHOD prediction is strikingly similar, matching HW13 to 1% or better at r > 0.4 h−1Mpc and showing similar form at smaller scales. Even the global HOD prediction is similar, deviating

−1 by 1.5% at r > 0.4 h Mpc, despite the much larger deviation in ξgg(r) seen in

Fig. 4.4. These results are our ﬁrst indication that using a standard HOD versus an environment-dependent HOD has little impact on matter clustering inferences.

4.2.3 Results for other Galaxy Samples

Figure 4.8 plots the measured HOD for HW13’s Mr 20 (top), Mr 21 ≤ − ≤ −

(middle), and Mr 21.5 (bottom) galaxy samples, comparing the global HOD to ≤ − that of halos in the top 20th and bottom 20th percentile in environmental density,

100 104 104 log(M ) 12.025 log(M ) 14.025 h ∼ h ∼

103 103 ) r ( n 102 102

HW13 1 1 10 ρ(r)-NFW 10 0.1 1.0 0.1 1.0

r/rv r/rv

Figure 4.5: Radial distributions of HW13 Mr 19 satellite galaxies(solid black −1 ≤ − −1 curves), in halos with log Mh/h M = 12 12.05 (left) and log Mh/h M = 14 14.05 (right). Grey curves show an NFW− proﬁle with the mean concentration expected− for this halo mass truncated at the viral radius.

103 HW13 HW13 + isotropic sat. dist. 102 EDHOD

) HOD r 1 (

g 10 EDHOD-cen. + HOD-sat. g ξ 100 Mr 19.0 -1 − 10 0.3 0.2 g

g 0.1 ξ

n 0.0 l 0.1 ∆ 0.2 0.3 0.1 1.0 10.0 1 r [h− Mpc]

Figure 4.6: Galaxy-correlation function for the Mr 19 HW13 catalog compared to ≤ − several HOD realizations. The grey curve, obscured in the upper panel, shows ξgg(r) from the HW13 catalog. Dot-dashed and solid black curves show ξgg(r) from catalogs created using the global HOD and environmentally dependent HOD (EDHOD), respectively, measured from the HW13 catalog. The bottom panel shows fractional deviations from the HW13 ξgg(r). Additional curves show the eﬀect of isotropizing the satellite distributions in the HW13 catalog (heavy dashed) or of combining the environmentally dependent HOD for centrals with the global HOD for satellites (light dashed).

101 1.10 HW13 HOD 1.05 EDHOD ) r ( 1.00 m g r

0.95 M 19.0 r − 0.90 0.1 1.0 10.0 1 r[h− Mpc]

Figure 4.7: Galaxy-matter cross-correlation coeﬃcient (Eqn. 4.4) computed from the HW13 Mr 19 catalog (grey) or from catalogs created by using the global HOD (black solid≤) or − EDHOD (dot-dashed) of this sample.

as in Fig. 4.4. Like the Mr 19 sample, the Mr 20 sample shows an increase ≤ − ≤ − (decrease) in N for low mass host halos residing in higher (lower) density environ- h i ments. For brighter samples, the environmental dependence is weaker, and essentially indiscernible for Mr 21 or Mr 21.5. These results are consistent with the ≤ − ≤ − weakening impact of galaxy assembly bias at higher luminosities seen in Fig. 4.1.

Figure 4.9, analogous to the lower panel of Fig. 4.6, plots the fractional difference in ξgg(r) measured from the HW13 catalog and from catalogs constructed using the global HOD or the environmentally dependent HOD. Results for Mr 20 are ≤ − similar to those for Mr 19: incorporating environmental dependence removes ≤ − the 10% offset in the large scale bias factor found for the global HOD, but there ∼ −1 are still 5-10 % differences in ξgg(r) for r 0.5 5h Mpc. For Mr 21 there ≈ − ≤ −

is only a small bias oﬀset for the global HOD model, and deviations in ξgg(r) for

the EDHOD model are consistent with random ﬂuctuations. For Mr 21.5, all ≤ − three models give consistent results. Fig. 4.10 shows results for the EDHOD and

102 HOD cross-correlation coeﬃcient compared to HW13 for our brighter samples. For

Mr 20 and 21, EDHOD and HOD rgm(r) results track the HW13 results well ≤ − − −1 on scales greater than 1h Mpc. Results for Mr 21.5 are dominated by noise. ∼ ≤ − Color selection has the potential to introduce stronger galaxy assembly bias be-

cause of the direct connection that the HW13 age-matching prescription introduces

between colour and halo formation time. Figure 4.11 shows the HOD environmen-

tal variation for the red Mr 20 galaxies in the HW13 catalog. Comparison to ≤ − Fig. 4.8 shows that environmental dependence is indeed stronger than that of the

full Mr 20 sample; in particular, the increased N(Mh) in dense environments ≤ − h i 12.5 −1 continues up to 10 h M halos. However, there is still no indication of an envi-

ronmental dependence of the satellite HOD.

Figure 4.12 shows the deviations in the galaxy correlation function and galaxy-

mass cross correlation coeﬃcient for our red Mr 20 galaxy sample. As expected, ≤ − diﬀerences between the global HOD model and the HW13 catalog are larger than those

for the full Mr 20 sample, with a 20 % difference in the large scale bias factor. The ≤ − EDHOD model again removes this large scale offset but leaves significant deviations

−1 in the 0.5 5h Mpc range. Crucially, however, the values of rgm(r) computed from − these three catalogs still match, at the 2% level or better for r 1h−1Mpc. ≥ 4.2.4 Summary

As shown previously by Zentner et al. (2014), the HW13 galaxy catalogs exhibit

substantial impact of assembly bias on galaxy clustering, particularly for low lumi-

nosity or colour selected samples. In the case of luminosity threshold samples, galaxy

assembly bias arises because HW13 tie luminosity to halo Vmax, which is correlated

103 103 global HOD 102 bottom 20 % top 20 % ® ) 101 h M ( 0 N 10 M 20.0 r − -1 10 12.0 1 Mmin =10 h− M ¯ 1011 1012 1013 1014 1015 1 M [h− M ] ¯

103 global HOD 102 bottom 20 % top 20 % ® ) 101 h M ( 0 N 10 M 21.0 r − -1 10 12.9 1 Mmin =10 h− M ¯ 1011 1012 1013 1014 1015 1 M [h− M ] ¯

103 global HOD 102 bottom 20 % top 20 % ® ) 101 h M ( 0 N 10 M 21.5 r − -1 10 13.7 1 Mmin =10 h− M ¯ 1011 1012 1013 1014 1015 1 M [h− M ] ¯

Figure 4.8: Mean occupation functions of the HW13 catalogs for Mr 20 (top), 21 (middle), 21.5 (bottom) for all halos and for halos in the 20% highest≤ − or lowest − − density environment measured by δ1−5, as in Fig. 4.4. Galaxy assembly bias eﬀects are smaller for more luminous samples.

104 0.2 0.1 Mr < 20.0 0.0 − 0.1 0.2 )

r 0.2 (

g 0.1 M < 21.0 g r ξ 0.0 − 0.1 n l 0.2 ∆ 0.2 0.1 Mr < 21.5 0.0 − 0.1 0.2 0.1 1.0 10.0 1 r [ h− Mpc]

Figure 4.9: Fractional deviations of ξgg(r) from global (dashed) and environmentally dependent (solid) HOD catalogs compared to the HW13 catalogs for the Mr 20, 21, 21.5 samples. Similar to the bottom panel of Fig. 4.6. ≤ − − −

1.10 1.05 Mr < 20.0 1.00 − 0.95 0.90 1.15

) 1.10

r 1.05 M < 21.0 ( 1.00 r m −

g 0.95

r 0.90 0.85 1.15 1.10 1.05 1.00 0.95 M < 21.5 0.90 r 0.85 − 0.1 1.0 10.0 1 r [ h− Mpc]

Figure 4.10: Cross correlation coeﬃcients rgm(r) from global (dashed) and environmentally dependent (solid) HOD catalogs compared to the HW13 catalogs (grey) for the Mr 20, 21, 21.5 samples. Similar to Fig. 4.7. ≤ − − −

105 103 global HOD 102 bottom 20 % top 20 % ® ) 101 h M ( 0 N 10 RedM 20.0 r − -1 10 12.5 1 Mmin =10 h− M ¯ 1011 1012 1013 1014 1015 1 M [h− M ] ¯

Figure 4.11: Mean occupation for red galaxies with Mr 20 in the HW13 catalog, in the same format as Figs. 4.4 and 4.8. ≤ −

104 103 HW13

) 2 HOD

r 10 ( EDHOD g 1

g 10 ξ 100 Red Mr < =-20.0 10-1

g 0.4 g

ξ 0.2

n 0.0 l 0.2

∆ 0.4 1.10 )

r 1.05 ( 1.00 m

g 0.95 r 0.90 0.1 1.0 10.0 1 r[h− Mpc]

Figure 4.12: Comparison of the galaxy correlation functions (top and middle) and the galaxy-matter cross correlation coeﬃcient (bottom) for red Mr 20 galaxies computed from the HW13 catalog (grey) and catalogs created using≤ the − global (dashed) or environmentally dependent (dot-dashed) HODs. Compare to Figs. 4.6, 4.7, 4.9.

106 with formation time at ﬁxed Mh. For colour selected samples the connection to halo assembly is imposed directly by HW13’s age matching procedure.

In HOD terminology, the assembly bias manifests itself as a increase in N(Mh) h i for central galaxies of halos in denser than average environments, and a correspond-

ing decrease of N(Mh) in low density environments. We ﬁnd no evident eﬀect of h i halo environment on the satellite galaxy occupation. Constructing HOD mock cata-

logs that incorporate the environmental dependence measured in the HW13 catalog

removes the large scale oﬀset in ξgg(r) that arises with an environment-independent

HOD model. However, deviations of ξgg(r) at the 5-20 % level (depending on scale and

galaxy sample) remain between the HW13 catalogs and catalogs constructed from an

EDHOD. Nonetheless, the HW13 catalogs, EDHOD catalogs, and (to a lesser extent)

−1 HOD catalogs yield similar predictions for rgm(r) at scales r > 1 h Mpc, typically

within the statistical ﬂuctuations arising from the ﬁnite size of the simulated cata-

logs. This similarity suggests that the impact of galaxy assembly bias on ξgg(r) and

ξgm(r) will cancel in cosmological analysis, a point we address more directly in the

next section.

4.3 Matter Clustering Inference

In an observational analysis, one does not know the HOD or EDHOD of a galaxy

sample a priori but infers it by ﬁtting the observed galaxy clustering. In a joint GGL

+ clustering analysis, the goal is to simultaneously infer the values of the cosmological

parameters that determine ∆Σ(R). A complete version of such an analysis would

likely involve forward modeling of the projected clustering and GGL observables, with

details that depend on the data sets being analyzed and on the external constraints

107 103 103 HW13 HOD fit 102 102 EDHOD fit ) ) r r

1 ( 1 ( 10 10 g m g g ξ ξ 100 100 Mr < 19.0 10-1 − 10-1 ) )

r 0.15 r

0.2 ( ( 0.10 g m

g 0.1

g 0.05 ξ 0.0 ξ 0.00 0.1 0.05 n n

l 0.10 0.2 l 0.15 ∆ 0.1 1.0 10.0 ∆ 0.1 1.0 10.0 1 1 r [ h− Mpc] r [ h− Mpc]

Figure 4.13: HOD and EDHOD fitting results for the Mr 19 sample. Left and ≤ − right panels show ξgg(r) and ξgm(r), respectively. (ED)HOD parameters are inferred −1 by fitting to ξgg(r) over the range 0.1 30h Mpc and including the total number of galaxies in HW13 as an additional− fitting point. Lower panels show fractional deviations of the best-fit (ED)HOD models from the HW13 correlation functions.

adopted on the cosmological parameters (e.g., from CMB measurements). Here we consider an idealized analysis in which de-projection has been used to translate wp(R) and ∆Σ(R) into the 3-d quantities ξgg(r) and Ωmξmm(r). HOD or EDHOD parameters

2 are inferred by ﬁtting ξgg(r), and equation (4.5) is used to infer Ωmξmm(r) from the GGL measurement. We want to know whether this approach would yield unbiased

2 estimates of rgm(r), and thus of Ωmξmm(r), given the galaxy assembly bias present in the HW13 abundance matching model. Our approach is inherently numerical, as we are using populated N-body halos to calculate ξgg(r), ξgm(r) and rgm(r) on all scales

We construct an EDHOD model by allowing the central galaxy HOD parameters

Mmin and σlog Mh to have a power-law dependence on δ1−5, in equations:

log Mmin = A + γ log(δ1−5) , (4.10)

log σlog Mh = B + β log(δ1−5) . (4.11)

108 1.10 HW13 HOD fit 1.05 EDHOD fit ) r ( 1.00 m g r

0.95 M 19.0 r − 0.90 0.1 1.0 10.0 1 r[h− Mpc]

Figure 4.14: Galaxy-matter cross correlation coeﬃcient (eq. 4.4). for the Mr 19 HW13 catalog (thick grey) and for the EDHOD model (solid black) and HOD≤ ( −dot- dashed) models that best ﬁt the HW13 ξgg(r) as shown in the left panels of Fig. 4.13.

Note that our EDHOD model has seven parameters, A, B, γ, β, M1,Mcut, α , whereas { } the HOD presented in 2 has ﬁve. (ED)HOD parameters are inferred by using § a downhill simplex method to minimize a sum of squares function, P (Di i model − i 2 i 2 ~ DHW13) /(DHW13) . The data vector D is galaxy-correlation function in the range r = 0.1 30 h−1Mpc in 30 equal logarithmic bins, with the total number of galaxies − included as an additional ﬁtting point for each sample selection.

Figure 4.13 shows the results of fitting the Mr 19 galaxy sample. The EDHOD ≤ − model achieves a good overall fit to the HW13 ξgg(r), with fluctuating deviations up to 5% in the region of the 1-halo to 2-halo crossover. The best-fit HOD model has ∼ a large scale offset of 10% in ξgg(r) (5 % in bg); while several HOD parameters can be adjusted to increase the large scale bias, doing so would alter the small scale ξgg(r) in a way that worsens the overall fit. This kind of offset could be a diagnostic for galaxy assembly bias, but we have not explored whether it can be erased by giving more freedom to the assumed radial profile of satellites within halos. The right panel

109 104 104 HW13 3 3 10 HOD fit 10 EDHOD fit

2 ) 2 )

10 r 10 r ( ( g m

g 1 1 10 g 10 ξ ξ

100 100 Mr < 20.0 10-1 − 10-1 ) )

r 0.15 r

0.2 ( ( 0.10 g m

g 0.1

g 0.05 ξ 0.0 ξ 0.00 0.1 0.05 n n

l 0.10 0.2 l 0.15 ∆ 0.1 1.0 10.0 ∆ 0.1 1.0 10.0 1 1 r [ h− Mpc] r [ h− Mpc]

104 104 HW13 3 3 10 HOD fit 10 EDHOD fit

2 ) 2 )

10 r 10 r ( ( g m

g 1 1 10 g 10 ξ ξ

100 100 Mr < 21.0 10-1 − 10-1 ) )

r 0.15 r

0.2 ( ( 0.10 g m

g 0.1

g 0.05 ξ 0.0 ξ 0.00 0.1 0.05 n n

l 0.10 0.2 l 0.15 ∆ 0.1 1.0 10.0 ∆ 0.1 1.0 10.0 1 1 r [ h− Mpc] r [ h− Mpc]

104 104 HW13 3 3 10 HOD fit 10 EDHOD fit

2 ) 2 )

10 r 10 r ( ( g m

g 1 1 10 g 10 ξ ξ

100 100 Mr < 21.5 10-1 − 10-1 ) )

r 0.15 r

0.2 ( ( 0.10 g m

g 0.1

g 0.05 ξ 0.0 ξ 0.00 0.1 0.05 n n

l 0.10 0.2 l 0.15 ∆ 0.1 1.0 10.0 ∆ 0.1 1.0 10.0 1 1 r [ h− Mpc] r [ h− Mpc]

Figure 4.15: Galaxy correlation function ﬁtting results for the Mr 20, 21, 21.5 samples in the same format as in Fig. 4.13. ≤ − − −

110 1.10 HW13 HOD fit 1.05 EDHOD fit ) r ( 1.00 m g r

0.95 M 20.0 r − 0.90 0.1 1.0 10.0 1 r[h− Mpc]

1.10

1.05 ) r ( 1.00 m g r

0.95 M 21.0 r − 0.90 0.1 1.0 10.0 1 r[h− Mpc]

1.10

1.05 ) r ( 1.00 m g r

0.95 M 21.5 r − 0.90 0.1 1.0 10.0 1 r[h− Mpc]

Figure 4.16: Galaxy-matter cross correlation coeﬃcients for the HW13 catalogs and the EDHOD and HOD model ﬁt to the HW13 ξgg(r), in the same format as Fig. 4.14.

111 of Fig. 4.13 shows ξgm(r) predicted by the HOD or EDHOD model that best ﬁts

ξgg(r). Deviations from the HW13 ξgm(r) are similar to those for ξgg(r), but reduced

in magnitude by a factor of two.

Figure 4.14 compares the cross-correlation coeﬃcients for Mr 19 samples ≤ − in the HW13, HOD, and EDHOD catalogs. In each case we use the simulation’s

true ξmm(r) and the ξgg(r) and ξgm(r) computed numerically from the corresponding

−1 catalog, calculating rgm(r) from Eqn. 4.4. At r > 1 h Mpc, the EDHOD and HOD

models reproduce the HW13 result to 1 % or better, despite the 5-10 % deviations

in ξgg(r). For this sample, all three models predict rgm(r) very close to one on these

scales. Within the 1-halo regime, the EDHOD and HOD ﬁts continue to track the

HW13 result, with deviations of a few percent.

Figure 4.15 shows similar results for the Mr 20, 21, and 21.5 samples. ≤ − − − Results in 2 show that the impact of galaxy assembly bias decreases with increasing § luminosity threshold. Consistent with this behavior, the EDHOD model ﬁts the

HW13 ξgg(r) substantially better than the global HOD model, but the diﬀerence

between EDHOD and HOD ﬁts becomes less signiﬁcant for the higher thresholds.

Results for these brighter, sparser samples become progressively noisier. As seen

previously for Mr 19, deviations in ξgg(r) are mirrored in ξgm(r), with a factor ≤ − 2 reduction in amplitude. ∼ Figure 4.16 shows the cross-correlation coeﬃcients for these higher luminosity

samples. For Mr 20 and Mr 21, the EDHOD and HOD models again ≤ − ≤ − −1 track the HW13 results at r > 1 h Mpc, with 1-2 % deviations in rgm(r) that

appear consistent with random ﬂuctuations. In the Mr 21 case, this percent- ≤ −

level agreement holds even when rgm(r) has climbed to 1.05, so it is not simply a

112 consequence of all three models predicting rgm 1.0. For the Mr 21.5 samples, ≈ ≤ − results are too noisy for percent-level tests, but there is no evidence for a systematic diﬀerence between the HW13 cross-correlation and those predicted by the HOD or

EDHOD ﬁts.

Given measurements of ξgg(r) and ξgm(r), and a cross correlation coeﬃcient rgm(r) inferred from an HOD or EDHOD model ﬁt, one can calculate the underlying matter correlation function ξmm(r) via Eqn. 4.5. (Because GGL constrains Ωmξgm, it is

2 Ωmξmm that is constrained, but here we omit the Ωm dependence for simplicity.) Solid black curves in Fig. 4.17 show the principal results of this paper, the accuracy within which the true matter correlation function of the Bolshoi simulation is recovered by applying this procedure to the ξgg(r) and ξgm(r) measurements from the HW13 catalogs, using an EDHOD model to infer rgm(r). For the Mr 19 ≤ − samples, recovery is accurate to better than 2% for r > 1 h−1Mpc. Deviations for the brighter samples are larger, but they appear consistent with statistical ﬂuctuations.

Deviations are larger inside 1 h−1Mpc, but not drastically so. Results for the global

HOD fits are similar to those for the EDHOD fits, even though the global HOD model does not produce good fits to ξgg(r). Grey solid curves in Fig. 4.17 show the results of using the true EDHOD measured directly from the HW13 catalogs (as described in 2). Fitting the EDHOD to ξgg(r) yields better recovery of ξmm(r) than using the § directly measured EDHOD.

A red galaxy sample presents a stringent test of our methodology because of the strong galaxy assembly bias imprinted by the HW13 age-matching procedure, and secondarily because the HOD paramerization that we use is designed for luminosity threshold samples. Figure 4.18 shows results for the red Mr 20 galaxy sample, ≤ −

113 similar to those shown in Figs. 4.13-4.17 for the luminosity threshold samples. For red galaxies, the global HOD model produces a poor fit to ξgg(r), with a 10 % large scale offset and a maximum deviation of 25 %. The EDHOD fit has deviations of

5-10 % in the 1 10 h−1Mpc range. Nonetheless, the EDHOD model matches the − HW13 cross-correlation coeﬃcient to 1 % or better at r > 2 h−1Mpc, and to 2 % or

−1 better at r > 1 h Mpc. Even for the global HOD model, the agreement in rgm is generally better than 3 % at r > 1 h−1Mpc.

The end result, shown in the bottom panel of Fig. 4.18, is that our EDHOD

−1 modeling allows recovery of ξmm(r) with accuracy of 2 % or better at r > 2 h Mpc when applied to the Mr 20 red galaxy population of the HW13 catalog. Even ≤ − though this model does not fully represent the galaxy assembly bias present in the

HW13 catalog, errors in ξgg(r) and ξgm(r) cancel in a way that accurately estimates matter clustering. In contrast to the luminosity-threshold cases shown in Fig. 4.17, the EDHOD modeling signiﬁcantly outperforms the global HOD modeling in this case, so the additional complication appears worthwhile at least in situations where the global HOD model yields a poor ﬁt to ξgg(r).

4.4 Conclusion

The combination of galaxy clustering and GGL is a powerful complement to cosmic shear measurements of matter clustering, applicable to the same data sets but with statistical and systematic errors that are at least partly independent. Fully ex- ploiting the combination requires a model of galaxy correlations and galaxy-matter cross correlations that extends to the non-linear regime. Halo occupation methods are a natural approach to this problem, as advocated by Yoo et al. (2006); Cacciato

114 1.08 1.04 1.06 HOD ﬁt EDHOD ﬁt 1.04 1.02 EDHOD true

, 1.02

mm 1.00 1.00

/ξ 0.98

mm 0.98 ξ 0.96

0.96 Mr 19.0 0.94 Mr 20.0 ≤ − 0.92 ≤ − 1 10 1 10 1.10 1.3

1.2 1.05

true 1.1 ,

mm 1.00 1.0 /ξ 0.9 mm

ξ 0.95 Mr 21.0 0.8 Mr 21.5 0.90 ≤ − 0.7 ≤ − 1.0 10.0 1.0 10.0 1 1 r [h− Mpc] r [h− Mpc]

Figure 4.17: Accuracy of the matter correlation functions inferred from the ξgg(r) and ξgm(r) measurements of the four HW13 catalogs, using Eqn. 4.5 with rgm(r) computed from the EDHOD (solid) or HOD (dot-dashed) model to fit to ξgg(r). Each panel plots the ratio of the recovered ξmm(r) to the true ξmm(r) measured in the Bolshoi simulation. Grey lines show the effect of using the EDHOD directly measured from the HW13 catalogs instead of that inferred by fitting ξgg(r).

115 104 104 HW13 3 3 10 HOD fit 10 EDHOD fit

2 ) 2 )

10 r 10 r ( ( g m

g 1 1 10 g 10 ξ Red ξ 100 100 Mr < 20.0 10-1 − 10-1 ) )

r 0.15 r

0.2 ( ( 0.10 g m

g 0.1

g 0.05 ξ 0.0 ξ 0.00 0.1 0.05 n n

l 0.10 0.2 l 0.15 ∆ 0.1 1.0 10.0 ∆ 0.1 1.0 10.0 1 1 r [ h− Mpc] r [ h− Mpc]

1.10 HW13 HOD fit 1.05 EDHOD fit ) r ( 1.00 m g r

0.95 Red M 20.0 r − 0.90 0.1 1.0 10.0 1 r[h− Mpc]

1.10 HOD ﬁt EDHOD ﬁt 1.05 true ,

mm 1.00 /ξ mm ξ 0.95

Red M 20.0 r ≤ − 0.90 1.0 10.0 1 r[h− Mpc]

Figure 4.18: Correlation function and matter correlation recovery for the red Mr ≤ 20 galaxies. Upper panels show the HOD and EDHOD ﬁts to ξgg(r) and the pre- − dicted ξgm(r), in the format of Fig. 4.13. The central panel shows rgm(r) for the HW13 catalog and the two HOD catalogs, as in Fig. 4.14. The bottom panel, analogous to Fig. 4.17, shows ξmm(r) inferred from the measured ξgg(r) and ξgm(r), using rgm(r) shown in the middle panel.

116 et al. (2009, 2012); Leauthaud et al. (2011); Yoo and Seljak (2012); Cacciato et al.

(2013); Coupon et al. (2015); More et al. (2015). However, modeling that assumes an environment independent HOD could yield a biased result if the galaxy population is significantly affected by assembly bias, which can alter the galaxy content of halos of fixed mass in different large scale environments. In this paper, we have used the abundance matching and age-matching catalogs of Hearin and Watson 2013 (HW13), which exhibit substantial galaxy assembly bias, to show that systematic error in the recovery of matter clustering remain small even down to scales of 1h−1Mpc. ∼ The HW13 catalogs ties its galaxy properties to halo assembly history by using

Vmax (rather than halo mass) as a ranking parameter for luminosity and formation time as a ranking parameter for color. As shown in Zentner et al. (2014) and 2 of this § paper, the HW13 scheme thereby translates halo assembly bias into galaxy assembly bias. This galaxy assembly bias has substantial (5-50 %) impact on galaxy correlations and cross-correlations, with the largest effects for low luminosity or color-defined samples. The HOD of central galaxies in these samples is boosted in dense environments (defined by overdensity in a 1 5 h−1Mpc spherical annulus) and suppressed − in low density environments. We find no significant variation of the satellite galaxy

HOD with environment for any of these samples. By incorporating the environment dependence of the central galaxy HOD, we construct EDHOD catalogs that reproduce the large scale bias of the HW13 catalogs, but these still have 5-10 % deviations of

−1 ξgg(r) at scales of 1 5 h Mpc. Nonetheless, these catalogs predict nearly the same − galaxy-matter cross-correlation coefficient rgm(r) as the corresponding HW13 catalog on scales r > 1 h−1Mpc, typically to < 2% or within the statistical fluctuations of the finite simulation volume. Catalogs that use the global HOD with no environment

117 dependence preform only slightly worse in reproducing rgm(r), even though they have

substantial errors in ξgg(r) and ξgm(r) individually.

Our most important results come from treating the HW13 catalogs as a source

of “observed" ξgg(r) and ξgm(r), then attempting to recover the true matter cor-

relation function ξmm(r) measured directly from the Bolshoi simulation by ﬁtting

(ED)HOD parameters to ξgg(r). We consider luminosity threshold samples Mr ≤

19, 20, 21, and 21.5, and color-selected sample of red galaxies with Mr 20. − − − − ≤ − We construct EDHOD catalogs that adopt power-law trends of the central galaxy

Mmin and σlog Mh parameters with δ1−5 (Eqns. 4.10 and 4.11), determining their pa-

−1 rameters by ﬁtting ξgg(r) in the range 0.1 30 h Mpc. We also ﬁt global HOD models − with no environment dependence and construct corresponding catalogs. We calculate rgm(r) numerically from these catalogs and apply this function to the measured ξgg(r)

and ξgm(r) to infer ξmm(r), via Eqn. 4.5. While this procedure is idealized compared to a true observational analysis, it enables us to use numerical calculations instead of analytic approximations (which are not sufficiently accurate on these scales), and it isolates the key physical issue — the influence (or lack of influence) of galaxy assembly bias on the cross-correlation coefficient.

For dense galaxy samples that yield precise measurements, our procedure recovers

−1 −1 ξmm(r) to 2% or better on scales r > 1 h Mpc (Mr 19), 1.5 h Mpc (Mr 20), ≤ − ≤ − −1 or 2 h Mpc (Mr 20 red). For sparser, high luminosity samples, recovery of ≤ − −1 ξmm(r) appears consistent with statistical uncertainties at r > 1 h Mpc. Results

using HOD or EDHOD are usually similar, but the EDHOD recovery is more accurate

for the red galaxy sample, which has the strongest assembly bias.

118 On linear scales, where galaxy bias is described by a single parameter bg, the de-

tailed physics that produces that bias does not matter for GGL + clustering analysis,

a point exploited by the observational study of Mandelbaum et al. (2013). Our anal-

ysis of the HW13 catalogs, which exhibit fairly complex assembly bias because of the

prescription used to create them, suggest that this insensitivity continues down to

−1 1 h Mpc scales, even though bias becomes scale-dependent and rgm can deviate ∼ from unity in this regime.

There are many interesting avenues for future investigations. One is to apply

HW13-like prescriptions to larger simulations and higher redshifts. This would enable higher precision tests of ξmm(r) recovery, especially for sparse samples of luminous

galaxies like those in the SDSS Luminous Red Galaxy survey (Eisenstein et al. 2001)

or the SDSS-III BOSS sample (Dawson et al. 2013). It would also be informative to

“stress-test" our ﬁndings against models with more extreme galaxy assembly bias or

non-linear rgm values further from unity. We would also like to apply these methods

to catalogs created with semi-analytic galaxy formation models, and to hydrodynamic

simulations with large enough volume to yield good statistics for ξgg(r) and ξgm(r).

In addition to exhibiting diﬀerent (and perhaps weaker, cf. Mehta (2014); Chaves-

Montero et al. (2015)) galaxy assembly bias, hydrodynamic simulations are important

for predicting the impact of gas physics, star formation, and feedback on the small

scale mass distribution (e.g., van Daalen et al. 2014).

For observational applications, signiﬁcant work is needed to develop a forward

modeling framework that directly predicts observables such as wp(R) and ∆Σ(R).

A critical element of such a framework is a numerically calibrated procedure to pre-

dict ξgg(r) and ξgm(r) as a function of cosmological and EDHOD parameters, since

119 existing analytic approximations have errors at the several percent level on scales of interest (e.g., Yoo et al. 2006; Cacciato et al. 2013) and do not allow for HOD environmental variations. Zheng and Guo (2015) describe efficient procedures for exploring the HOD parameter space, which will be useful for this daunting computational task. GGL measurements from the SDSS main galaxy sample already yield tight constraints on the galaxy-halo connection for fixed cosmological parameters (Zu and Mandelbaum 2015), and application of the methods described here could yield competitive new constraints on the amplitude of low redshift matter clustering. On the several year timescale, measurements from the Dark Energy Survey and the Sub- aru Hyper-Suprime Camera could easily tighten these constraints to the one-percent level. The weak lensing surveys of LSST, Euclid, and WFIRST seek a further order- of-magnitude improvement in measurement precision, presenting a stiff challenge for theoretical modeling and a superb opportunity to test our understanding of dark energy and gravity on cosmological scales.

120 Chapter 5: FAST-PT

5.1 Introduction

The large-scale structure of the universe provides numerous probes of the under-

lying cosmological model, including the source of present-day accelerated expansion.

Current and upcoming surveys (Levi et al., 2013; Dawson et al., 2013; Laureijs et al.,

2011a; Dark Energy Survey Collaboration et al., 2016; Spergel et al., 2013) will pro-

vide impressive statistical power to test the ΛCDM (cosmological constant plus cold dark matter) paradigm as well as potential modiﬁcations (see Ref. Weinberg et al.

(2013) for a review). Connecting the predictions of these models to observables from tracers of large-scale structure requires understanding the role of physics on a wide range of scales, including the growth of dark matter structure and the formation of galaxies and other luminous objects. On small scales, numerical simulations are required to solve for the full nonlinear growth (e.g. Springel et al. (2006)). Perturbative techniques provide an analytic approach to describe structure on mildly nonlinear scales and are particularly valuable in that they can be quickly calculated for diﬀer- ent sets of cosmological parameters (without running a new simulation) and provide physical intuition into the relevant processes.

121 A number of cosmological observables can be modeled in the weekly non-linear regime, including the clustering of galaxies and other luminous tracers, as well as weak gravitational lensing and cross-correlations between these probes (e.g. “galaxy-galaxy lensing.”). For instance, perturbation theory has long been used to describe corrections to the scale-dependence of dark matter clustering, including both two-point and higher-order statistics (see, e.g., Bernardeau et al. (2002a)). Perturbative techniques can also eﬀectively predict the nonlinear shift and broadening of the baryon acoustic oscillation (BAO) feature Crocce and Scoccimarro (2008); Sugiyama and Spergel

(2014) – a powerful “standard ruler” for studying the evolution of geometry in the universe – including the potential impact of streaming velocities between baryons and dark matter in the early universe (Yoo et al., 2011a; Yoo and Seljak, 2013a; Slepian and Eisenstein, 2015a; ?). The utility of perturbative techniques is not limited to dark matter. The relationship between dark matter and luminous tracers will generally include a nonlinear “biasing” relationship, resulting in correlations that are naturally described in a perturbative expansion (e.g. McDonald (2006); McDonald and Roy

(2009a); Saito et al. (2014a)). These biasing techniques have been used in large surveys to constrain cosmological parameters Mandelbaum et al. (2013); Kwan et al.

(2016), as well as to constrain neutrino mass Saito et al. (2011); Zhao et al. (2013).

The velocity ﬁeld of dark matter and luminous tracers, which sources “redshift-space distortions” in clustering measurements can also be modeled analytically beyond linear theory (e.g. Scoccimarro (2004); Vlah et al. (2012)). Similarly, correlations of intrinsic galaxy shapes (known collectively as “intrinsic alignments”) can be described perturbatively and nonlinear perturbation theory may be especially important for

“tidal torquing” (quadratic) alignment models (e.g. Hirata and Seljak (2004); Blazek

122 et al. (2015)). Even in cases where N-body simulations are needed to reach the desired accuracy, a fast perturbation theory code is still valuable, since interpolation from grids of N-body simulations can be used to compute the non-perturbative correction to an observable , rather than trying to interpolate the much larger “raw” O value of . O A generic feature of nonlinear perturbation theory is the coupling of modes at

diﬀerent scales through kernels that capture the physics of structure growth. As a

result, these nonlinear corrections typically appear as convolutions over the power

spectrum or related functions of the wavevector. In this paper, we primarily consider

the most ubiquitous of these approaches, standard perturbation theory (SPT) (e.g.

Bernardeau et al. (2002a)). However, integrals with a similar structure are found in

other approaches as well, including Lagrangian perturbation theory (LPT, Sugiyama

(2014)), renormalized perturbation theory (RPT, Crocce and Scoccimarro (2006)),

renormalization group perturbation theory (RGPT, McDonald (2007, 2014)), the ef-

fective ﬁeld theory (EFT, Baumann et al. (2012); Carrasco et al. (2012); Pajer and

Zaldarriaga (2013); Hertzberg (2014)) approach to structure formation, and renormal-

ization and time ﬂow frameworks Audren and Lesgourgues (2011), which can include

scale-dependent propagators for the ﬂuctuation modes (e.g. arising from massive neu-

trinos). Therefore, it is of great utility that the cosmological community have access

to eﬃcient and accurate methods to compute these integrals.

In this paper we provide a new algorithm, FAST-PT, to calculate mode coupling integrals in perturbation theory. As a ﬁrst example of our method we focus on 1-loop order perturbative descriptions of scalar quantities (e.g. density or velocity

123 divergence). In particular, we present examples for standard 1-loop SPT and renor-

malization group results. A generalization to arbitrary-spin quantities (e.g. intrinsic

alignments, a spin-2 tensor ﬁeld) and other directionally dependent power spectra

(e.g. redshift-space distortions and secondary CMB anisotropies) will be presented in

a follow-up paper.

FAST-PT can calculate the SPT power spectrum, to 1-loop order to the same level of accuracy as conventional methods, on a sub-second time scale. In the context of Monte Carlo Markov chain (MCMC) cosmological analyses, which may explore

> 106 points in parameter space, the extremely low recurring cost of our method is

particularly relevant. The FAST-PT recurring cost to calculate the 1-loop power

spectrum at N = 3000 k values is 0.01s. This speed is even more valuable for ∼ multi-probe cosmological analyses. For instance, a gravitational lensing plus galaxy

clustering analysis may require, e.g., the matter and galaxy power spectra in real and

redshift space, nonlinear galaxy biasing contributions, and the intrinsic alignment

power spectra, at each point in cosmological parameter space.

FAST-PT takes a power spectrum, sampled logarithmically, as an input. Special

function identities are then used to rewrite the angular dependence of the mode-

coupling kernels in terms of a summation of Legendre polynomials. The angular

integration for each of these components can be performed analytically, reducing the

numerical evaluation to one-dimension. Because of the uniform (logarithmic) sam-

pling we are able to utilize Fast Fourier Transform (FFT) methods, thus enabling

computation of the mode-coupling integrals in (N log N) operations, where N is O the number of samples in the power spectrum. Our approach is similar in struc-

ture to the evaluation of logarithmically sampled Hankel transforms Talman (1978);

124 Hamilton (2000), which have been used to transform power spectrum into correla-

tion functions (and vice versa). It also draws on the realization that convolution

integrals in spherical symmetry – even convolutions of integrands with spin – can be

expressed using Hankel transforms with the angular integrals performed analytically

(e.g. Slepian and Eisenstein (2015a)). We implement the FAST-PT algorithm in a

publicly-available package. The code is written in Python, making use of numpy and

scipy libraries, and has a self-contained structure that can be easily integrated into

a larger project. We provide a public version of the code along with a user manual

and example implementations at https://github.com/JoeMcEwen/FAST-PT.

Recently, Schmittfull et al. Schmittfull et al. (2016) have presented a related

method for fast perturbation theory integrals, based on the same mathematical prin-

ciples. Our Eq. (5.21) encapsulates the same approach as their Eq. (31), combined

with the logarithmically sampled Hankel transform. However, the numerical approach

is diﬀerent: the decomposition of an arbitrary power spectrum P (k) into power laws

of complex exponent is treated as fundamental (and is kept explicitly in the code);

the near-cancellation of P22 + P13 is handled by explicit regularization; and the P13

integral is solved using a diﬀerent method (based only on scale invariance). Finally,

we present a fast implementation of RGPT.

This paper is organized as follows: in 2, we provide the theory to our method, § motivating our approach by considering the 1-loop SPT power spectrum. In 3, we § provide results for 1-loop corrections to the power spectrum and demonstrate an implementation of the renormalization group approach of McDonald (2007, 2014). In

4, we summarize our results, including a discussion of other potential applications § of FAST-PT, and provides a brief description of the publicly-available code. The

125 appendices provide additional details of our numerical calculations and the structure of the terms under consideration.

5.2 Method

This work presents an algorithm to eﬃciently calculate mode-coupling integrals of the form

Z d3q K(q, k q)P (q)P ( k q ) , (5.1) (2π)3 − | − | where K(q1, q2) is a mode-coupling kernel that can be expanded in Legendre polynomials and P (q) is an input signal logarithmically sampled in q. The motivation for this method is mildly-nonlinear structure formation in the universe, although it can be more generally considered as a technique to evaluate a range of expressions in the form of Eq. (5.1).

For clarity we list our conventions and notations:

fast Fourier transform and inverse fast Fourier transform are denoted as FFT • and IFFT;

Fourier transform pairs have the 2π placed in the denominator of the wavenum- • ber integral, as is standard in cosmology:

Z Z d3k Φ(k) = d3r Φ(x) e−ik·r Φ(r) = Φ(k) eik·r; (5.2) ↔ (2π)3

“log” always refers to natural log and we will use log explicitly when we are • 10 referring to base 10;

represents a convolution (discrete or continous); • ⊗

126 the Legendre polynomials will be denoted l (to avoid confusion with power • P

spectra P ), normal Bessel functions of the ﬁrst kind are denoted Jµ(t), and

spherical Bessel functions of the ﬁrst kind are denoted jl(t), all with standard

normalization conventions Abramowitz and Stegun (1964);

i = √ 1 (never to be confused with an index); • −

“log sampling” means that the argument of the input signal is qn = q0 exp(n∆), • where n = 0, 1, 2, ... and ∆ is the linear spacing between grid points; and

we use the convention that when calculations require discrete evaluations, for • example as in the case of discrete Fourier transforms, we index our vectors,

while when evaluations are performed analytically we omit the index.

In this section we begin by reviewing SPT (§5.2.1); the reader who is already

experienced with SPT may skip directly to 5.2.2. §5.2.2 describes our main result: a §

rearrangement of the mode-coupling integral that allows P22 and related integrals to

be computed in order N log N operations. The P13 integral is simpler than P22, but brute-force computation of P13 is in fact slower than the FAST-PT method for P22, so we describe our fast approach to P13 in §5.2.3. Finally, in §5.2.4 we describe our numerical treatment of the cancellation of infrared divergences in P22 and P13.

5.2.1 1-loop Standard Perturbation Theory

When fluctuations in the density field are small, δ(k) 1, non-linear structure formation in the universe can be modeled by solving the cosmological fluid equations using perturbation theory (see Ref. Bernardeau et al. (2002a) for a comprehensive review of Eulerian perturbation theory). For this paper we only sketch out the most

127 basic elements of perturbation theory, focusing on the integrals we evaluate. The matter ﬁeld written as a perturbative expansion in Fourier space is

δ(k) = δ(1)(k) + δ(2)(k) + δ(3)(k) + ... , (5.3) where the first order contribution δ(1)(k) is the linear matter field and each higher- order term represent non-linear contributions. Non-linear effects manifest themselves as mode-couplings in Fourier space, consequently each δ(n)(k) is a convolution integral

(1) over n copies of the linear ﬁeld δ (q) with a kernel Fn(q1, ..., qn):

Z 3 3 n d q1 d qn X δ(n)(k) = ... δ3 (k q )F (q , ..., q )δ(1)(q ), ..., δ(1)(q ) , (5.4) (2π)3 (2π)3 D j n 1 n 1 n − j=1

3 where δD(k) is the three-dimensional Dirac delta function. The power spectrum P (k) is deﬁned as an ensemble average of the matter ﬁeld δ(k),

δ(k)δ(k0) = (2π)3δ3 (k + k0)P (k) . (5.5) h i D

The ﬁrst non-linear contribution to the power spectrum comes from ensemble averages taken up to ([δ(1)]4): O

δ(k)δ(k0) = δ(1)(k)δ(1)(k0) + δ(2)(k)δ(2)(k0) + 2 δ(1)(k)δ(3)(k0) + ... , (5.6) h i h i h i h i which deﬁnes the one-loop power spectrum

P1-loop(k) = Plin(k) + P22(k) + P13(k) , (5.7)

(2) (2) 0 3 3 0 (1) (3) 0 3 3 where δ (k)δ (k ) = (2π) δ (k + k )P22(k) and 2 δ (k)δ (k ) = (2π) δ (k + h i D h i D 0 k )P13(k).

5.2.2 P22(k) type Convolution Integrals

We ﬁrst focus on P22(k), leaving the evaluation of P13(k) to a later subsection.

P22(k) is a convolution integral that takes two copies of the linear power spectrum

128 Plin(k) as inputs:

Z 3 d q 2 P22(k) = 2 Plin(q)Plin( k q ) F2(q, k q) . (5.8) (2π)3 | − | | − |

The F2 kernel is 5 1 q1 q2 2 2 F2(q1, q2) = + µ12 + + µ12 7 2 q2 q1 7 (5.9) 17 1 q1 q2 4 = 0(µ12) + + 1(µ12) + 2(µ12) , 21P 2 q2 q2 P 21P

where we have deﬁned µ12 = q1 q2/(q1q2) = qˆ1 qˆ2, which is the cosine of the angle · ·

between q1 and q2. Squaring this and substituting into Eq. (5.8), we ﬁnd that the

P22(k) power spectrum expanded in Legendre polynomials is

Z 3 d q1 h1219 671 32 1 2 −2 P22(k) = 2 0(µ12) + 2(µ12) + 4(µ12) + q q 2(µ12) (2π)3 1470P 1029P 1715P 3 1 2 P

62 −1 8 −1 1 2 −2 i + q1q 1(µ12) + q1q 3(µ12) + q q 0(µ12) Plin(q1)Plin(q2) , 35 2 P 35 2 P 6 1 2 P (5.10)

where we have deﬁned q2 = k q1 and used the q1 q2 symmetry to combine − ↔ terms. We note that the last Legendre component in Eq. (5.10) will eventually lead to a divergent expression in the FAST-PT framework. In §5.2.4 we discuss this type of divergence (which can appear in other contexts) and explicitly show the cancellation.

Each Legendre component of Eq. (5.10) is a speciﬁc case of the general integral

Z 3 d q1 α β Jαβl(k) = q q l(µ12)P (q1)P (q2) . (5.11) (2π)3 1 2 P

Note that we have now omitted the subscript “lin” on the power spectrum and carry

on our calculations for a general input power spectrum. For SPT calculations the

input power spectrum should be Plin(k), however there are cases when a general

power spectrum input is required, such as renormalization group equations. Our

method of evaluation draws on several key insights from the literature. The ﬁrst

129 is that the Legendre polynomial can be decomposed using the spherical harmonic

addition theorem, and that in switching between real and Fourier space one may

use the spherical expansion of a plane wave to achieve separation of variables; see

the Appendix of Ref. Slepian and Eisenstein (2015a). The second is the fast Hankel

transform Talman (1978); Hamilton (2000). We also have to address a number of

subtleties to make these ideas useful for the 1-loop SPT integrals.

Our goal in this section is to develop an eﬃcient numerical algorithm to evaluate

integrals of the form Eq. (5.11). Combining the results for the relevant values of

(α, β, l) will then allow us to construct P22(k) or other similar functions. For instance,

in terms of these components, Eq. (5.10) reads h1219 671 32 P22(k) = 2 J0,0,0(k) + J0,0,2(k) + J0,0,4(k) 1470 1029 1715 (5.12) 1 1 62 8 i + J (k) + J (k) + J (k) + J (k) . 6 2,−2,0 3 2,−2,2 35 1,−1,1 35 1,−1,3 To evaluate Eq. (5.11) we ﬁrst Fourier transform to conﬁguration space and then expand the Legendre polynomials in spherical harmonics, using Eq. (B.1): Z d3k J (r) = eik·rJ (k) αβl (2π)3 αβl Z 3 3 d q1 d q2 = ei(q2+q1)·rqαqβP (µ)P (q )P (q ) (2π)3 (2π)3 1 2 l 1 2 l Z 3 3 4π X d q1 d q2 = eiq1·reiq2·rqαqβY (qˆ )Y ∗ (qˆ )P (q )P (q ) . 2l + 1 (2π)3 (2π)3 1 2 lm 1 lm 2 1 2 m=−l (5.13)

R ∞ 2 The q1 and q2 integrals can each be broken into a radial ( 0 dq1 q1) and angular

R 2 ( S2 d qˆ1) part; the angular parts do not depend on the power spectrum and can be evaluated analytically using Eq. (B.4):

l 4π(4π il)2 X Z ∞ Z ∞ J (r) = Y (ˆr)Y ∗ (ˆr) dq q2+αj (q r)P (q ) dq q2+βj (q r)P (q ). αβl (2π)6(2l + 1) lm lm 1 1 l 1 1 2 2 l 2 2 m=−l 0 0 (5.14)

130 Additionally we make use of the orthogonality relation, Eq. (B.2), to eliminate the sum over m:

l Z ∞ Z ∞ ( 1) α+2 β+2 Jαβl(r) = − 4 dq1q1 jl(q1r)P (q1) dq2q2 jl(q2r)P (q2) . (5.15) 4π 0 0

Equation (5.15) can be considered as one component of a correlation function. For instance, the correlation function ξ22(r) [the Fourier counterpart to P22(k)] is built from Eq. (5.15) with the same α, β, l combinations and pre-factors as in Eq. (5.10).

Equation (5.15) is the product of two Hankel transforms (terms in brackets) with the relevant prefactor. We denote the bracketed terms in Eq. (5.15) as Iαl(r) and

Iβl(r). To evaluate Iαl(r), we ﬁrst take the discrete Fourier transformation of the power spectrum (biased by a power of k):

N−1 N/2 X P (kn) −2πimn/N X ν+iηm cm = Wm e P (kn) = cmk , (5.16) kν ↔ n n=0 n m=−N/2 where N is the size of the input power spectrum, ηm = m 2π/(N∆), m = ×

N/2, N/2 + 1, ..., N/2 1,N/2 and ∆ is the linear spacing, i.e. kn = k0 exp(n∆). − − − ∗ For real power spectrum the Fourier coeﬃcients obey cm = c−m. Here Wm is a window function that can be used to smooth the power spectrum.5 Using discrete FFTs allows a signiﬁcant reduction in computation time. However, these methods require that the function being transformed is (log-)periodic. In the case of FAST-PT, this procedure is equivalent to performing calculations in a universe with a power spectrum, biased by a power-law in k, that is log-periodic. This universe has divergent power on large

5 1 1 If no smoothing is desired, we would set Wm = 1 for all m except for W±N/2 = 2 . The 2 ensures that the counting of both m = N/2 in the second sum in Eq. (5.16) is the correct inverse transform. However, in our numerical implementation± we always include a window function that goes smoothly to zero to prevent “ringing” in the interpolated P (k); see Appendix B.2.

131 or small scales, depending on the choice of ν. Figure 5.2.2 shows the resulting power

spectrum, with a window function applied at the periodic boundaries. In order for

perturbation theory to make sense, the large-scale density variance should be ﬁnite,

R k 02 0 0 i.e. 0 k P (k )dk should be ﬁnite, and the small-scale displacement variance should

R ∞ 0 0 ν be ﬁnite, i.e. k P (k ) dk should be convergent (see Fig. 5.2.2). Since P (k)/k is log-periodic, this means that FAST-PT will require biasing with 3 < ν < 1 (this − − paper chooses ν = 2). − In most cases, suﬃciently far from the boundaries, the impact of the periodic nature of the P (k) is negligible. However, while P22(k) and P13(k) are well-behaved

in standard methods with CDM power spectra, they are both inﬁnite in FAST-

PT where the satellite features at extremely large scales (k 0) produce inﬁnite → displacements. This is the same inﬁnity found in power-law spectra and is of no

physical concern: since displacement is not a physical observable, Galilean invariance

guarantees that the divergent parts of P22(k) and P13(k) will cancel as long as the

displacement gradient or strain is ﬁnite. In §5.2.4 we will address the numerical

aspects of this cancellation and show how to perform a well-behaved 1-loop SPT

calculation in the FAST-PT framework.

Continuing our evaluation: Z ∞ α+2 Iαl(r) = dk k jl(kr)P (k) 0 N/2 Z ∞ X ν+2+α+iηm = cm dk k jl(kr) m=−N/2 0 r N/2 ∞ (5.17) π X Z = c r−3−ν−α−iηm dt t3/2+ν+α+iηm J (t) 2 m l+1/2 m=−N/2 0 r N/2 π X = c g r−3−ν−α−iηm 2Qαm , 2 m αm m=−N/2

132 102 ν

k 10-2 / ) -6 k 10 ( P 10-10

106 ) 102 k -2

( 10 -6

2 10 10-10 ∆ 10-14 10-18

2 106 k 102 / -2

) 10 -6 k 10 ( 10-10 2 10-14 -18 ∆ 10 10-9 10-7 10-5 10-3 10-1 101 103 105 107 k

Figure 5.1: Power spectra in the log-periodic universe. Top panel shows the windowed linear power spectrum biased by k−ν (we choose ν = 2), with grey lines indicating the “satellite" power spectra, i.e. the contribution to the− total power spectrum that arises due to the periodic assumption in a Fourier transform. The middle panel plots ∆2(k) = k3P (k)/(2π2), within the periodic universe. This is the quantity that sources the density variance σ2 = R d ln k∆2(k). The bottom panel plots the R 2 2 contribution to the displacement variance σξ = d ln k∆ (k)/k .

133 where in the third equality we have exchanged the Bessel function of the ﬁrst kind for a p spherical Bessel function, jν(z) = π/(2z) Jν+1/2(z) and performed the substitution

t = kr. In the last equality we have evaluated the integral according to Eq. (B.5) and

1 3 deﬁned gαm g(l + ,Qαm) and Qαm + ν + α + iηm. ≡ 2 ≡ 2 Strictly speaking, the convergence criteria for Eq. (5.17) are α < 1 ν and − − α + l > 3 ν. For ν = 2 we thus require (i) α < 1 and (ii) α + l > 1. All terms − − − − with α = 2 violate (i), while the α = 2, l = 0 term also violates (ii). The violations

of condition (i) can be cured if we apply an exponential cutoﬀ in the power spectrum

to force the integral to converge, i.e. in Eq. (5.17) we insert a factor of e−k and take

the limit as 0+; this yields the same result and is equivalent to smoothing out → the “wiggles” in the Bessel functions at high k.6 The violation of condition (ii) comes

from the low k’s and is more problematic: the physical result for I−2,0(r) is divergent,

and this will be cured in §5.2.4. The ﬁnal result for the Jαβl(r) correlation component

is then ( 1)l Jαβl(r) = − Iαl(r)Iβl(r) 4π4 N/2 N/2 (5.18) ( 1)l X X Qαm+Qβn −6−2ν−α−β−iηm−iηn = − cmcngαmgβn 2 r . 8π3 m=−N/2 n=−N/2

6This can be proven by inserting a factor of e−t in Eq. (B.5) and taking the limit as 0+. Following Eq. (6.621.1) of Gradshteyn and Ryzhik (1994), the integral can be expressed→ in µ+κ+1 µ+κ+2 −2 terms of the hypergeometric function 2F1( 2 , 2 ; µ + 1; ). The transformation formula, Eq. (9.132.2) of Gradshteyn and Ryzhik (1994), can then be− used to express a hypergeometric function of large argument −2 in terms of functions of argument approaching 0. Using − → −∞ limz→0 2F1(α, β; γ; z) = 1 and the Γ-function duplication formula suﬃces to prove this generalized version of Eq. (B.5).

134 To obtain the power spectrum, we Fourier transform Eq. (5.18) back to k-space: Z ∞ 2 Jαβl(kq) = dr 4πr j0(kqr)Jαβl(r) 0 N/2 N/2 ( 1)l Z ∞ sin( ) X X Qαm+Qβn −5−2ν−α−β−i(ηm+ηn) kqr = − 2 cmgαmcngβn2 dr r , 2π kq m=−N/2 n=−N/2 0 (5.19)

where in the ﬁrst equality homogeneity converts the 3-dimensional Fourier transform

into a Bessel integral, and then we have used j0(z) = sin(z)/z. The integral over r

can be evaluated using the f-function of Eq. (B.7) via the substitution t = kqr and

leads to

N/2 N/2 ( 1)l X X Qh −p−2+iτh Jαβl(kq) = − cmgαmcngβn2 k fh , (5.20) 2π2 q m=−N/2 n=−N/2

where we have deﬁned fh = f(p + 1 iτh), p = 5 2ν α β, τh = ηm + ηn, − − − − −

and Qh = Qαm + Qβn. Note that τh (and hence fh) and Qh depend only on the sum m + n.

In what follows, we will transform a double summation over m and n into a discrete convolution, indexed by h, such that h = m + n N, N + 1, ..., N 1,N . This ∈ {− − − } leads to:

N/2 N/2 ( 1)l 3+2ν+α+β X X −p−2+iτh iτh Jαβl(kq) = − 2 cmgαmcngβn fhk 2 2π2 q m=−N/2 n=−N/2 ( 1)l 2+2ν+α+β X iτh −p−2+iτh = − 2 [cmgαm cngβn]hfh2 k π2 ⊗ q h (5.21) ( 1)l 2+2ν+α+β −p−2 X iτh = − 2 k Chfh2 exp(iτh log k0) exp(iτhq∆) π2 q h ( 1)l 2+2ν+α+β −p−2 iτh = − 2 k IFFT[Chfh2 ] , π2 q where in the second equality we have replaced n = h m and in the third and fourth − P equality the sum over m is written as a discrete convolution m cmgαmch−mgβ,h−m =

135 [cmgαm cngβn]h = Ch. Also, due to the log sampling of kq the ﬁnal sum over ⊗

P iτh h in Eq. (5.21) is actually an inverse discrete Fourier transform, i.e. h Ahkq =

P 7 h Ah exp(i2πhq/[2N]) , and can thus be evaluated quickly using an FFT. Equa-

tion (5.21) is the main analytical result of this work, it allows one to evaluate P22(k)

type integrals quickly, scaling with N log N.

Since in FAST-PT P (k)/kν is log-periodic, there are discontinuities in the power

N∆ spectrum at kmin = k0 and kmax = k0e . This means that when Fourier-space

methods are applied, the series of Eq. (5.16) will exhibit ringing; the FAST-PT

user has several options for controlling this behavior. The power spectrum can be

windowed in such a way that the edges of the array are smoothly tapered to zero

(of course, this must be done outside the k-range that contributes signiﬁcantly to the

mode-coupling integrals). The location of the onset of the tapering is controlled by the

user. The Fourier coeﬃcients cm can also be ﬁltered so that the highest frequencies

are damped. We use the same window function to ﬁlter the Fourier coeﬃcients

and smooth the edges of the power spectrum – the functional form is presented in

Appendix B.2. In practice, while we always apply a ﬁlter to the cm coeﬃcients, we

choose to directly window the power spectrum only within our renormalization group

routine (see Appendix B.3). We have also written the code in such a way that the

user can easily implement their own window function. One can also “zero pad” the

input power spectrum, adding zeros to both sides of the array. The contributions of

the mode-coupling integrals from the large-scale satellite power spectrum (k < kmin)

7 In the last two lines of Eq. (5.21) a shift, exp(iτh log k0), in the Fourier transform appears. In practice, our code does not compute this shift which also appears in the initial Fourier transform and thus cancels. Additionally, to conform to Python Fourier conventions we drop the positive end point in the ﬁnal FFT.

136 heavily contaminate P22(k) at k < 2kmin (range restricted by the triangle inequality).

We thus recommend zero-padding by a factor of 2. ≥

5.2.3 P13(k) type Convolution Integrals

The P13(k) integral does not share the same form as P22(k), since the wavenumber structure is diﬀerent: it describes a correction to the propagator for Fourier mode k due to interaction with all other modes q. The structure of P13(k) is thus P (k) times an integral over the power in all other modes:

3 Z ∞ k 2 P13(k) = 2 Plin(k) dr r Plin(kr)Z(r) , (5.22) 252(2π) 0 where

12 158 3 r + 1 Z(r) = + 100 42r2 + (7r2 + 2)(r2 1)3 log , (5.23) r4 − r2 − r5 − r 1 | − | and r = q/k. Upon making the substitution r = e−s, Eq. (5.22) becomes 3 Z ∞ k 2 P13(k) = 2 Plin(k) dr r Plin(kr)Z(r) 252(2π) 0 3 Z ∞ k −3s log k−s −s = 2 Plin(k) ds e Plin(e )Z(e ) (5.24) 252(2π) −∞ k3 Z ∞ = 2 Plin(k) ds G(s)F (log k s) , 252(2π) −∞ − where in the ﬁnal line we reveal the integral as a continuous integral with the following

−3s −s s definitions G(s) e Z(e ) and F (s) Plin(e ). In the discrete domain we have ≡ ≡ ds ∆, log kn = log k0 + n∆, and sm = log k0 + m∆, so that the discrete form is → N−1 Z ∞ X ds G(s)F (log k s) ∆ GD(m)FD(n m) , (5.25) −∞ − → m=0 − where in the final line we define the discrete functions GD(m) G(sm) and FD(m) ≡ ≡ F (m∆), so that we have

3 kn P13(kn) = Plin(kn)∆[GD FD][n] . (5.26) 252(2π)2 ⊗ 137 2 Thus P13(k), which at ﬁrst appears to involve order N steps (an integral over N

samples at each of N output values kn) can in fact be computed for all output kn in

N log N steps.

5.2.4 Regularization

As mentioned above, we need to regularize the divergent portion in P22(k) with

P13(k). In standard calculations in a ΛCDM universe, the suppression of power on

large scales [P (k) kn, n > 1] controls this divergence, allowing the numerical ∝ − evaluation of each term separately. The relevant cancellation will then occur upon addition of the terms, as long as suﬃcient numerical precision has been achieved.

However, because the FAST-PT method relies on FFTs, the “true” underlying power

spectrum is log-periodic, leading to non-vanishing power on inﬁnitely large (and small)

scales. These divergences are thus numerically realized and must be analytically re-

moved before evaluation. Physically the divergences are due to the artiﬁcial breaking

of local Galilean invariance when the 1-loop SPT power is split into P22(k) and P13(k):

a long-wavelength (q k) velocity perturbation displaces small-scale structure without aﬀecting its evolution, but since the perturbative expansion terms δ(n) are deﬁned with respect to a stationary background, each term in perturbation theory shows a divergence even when the physically relevant sum does not. This fact is well-known in the context of P22 + P13 Vishniac (1983) and has been generalized to higher orders

Jain and Bertschinger (1996); Scoccimarro and Frieman (1996).

We construct our regularization scheme so that it preserves the 1-loop contribution to power spectrum, i.e.

P22(k) + P13(k) = P22,reg(k) + P13,reg(k) , (5.27)

138 where the subscript “reg” stands for regularization, by subtracting out the contribution

to P13(k) from small q = kr in Eq. (5.22), and adding it to the J2,−2,0(k) contribution in P22(k) to obtain a regularized P22,reg(k). We ﬁrst expand the kernel in Eq. (5.22) in a Laurent series around small r:

928 4512 416 2656 r2Z(r) = 168 + r2 r4 + r6 + r8 + ... . (5.28) − 5 − 35 21 1155

If P13(k) were dominated by contributions from large-scale modes (i.e. r 1), as occurs when there is an infrared divergence, then we could make the replacement

2 r Z(r) 168 and ﬁnd that P13(k) approaches → − 3 Z ∞ Z 3 168 k 1 2 d q Plin(q) P13(k) 2 Plin(k) drPlin(kr) = k Plin(k) 3 2 . (5.29) → −252(2π) 0 −3 (2π) q

We then subtract this oﬀ from the kernel Z(r) so that

168 12 10 2 3 2 2 3 r + 1 Zreg(r) = Z(r) + = + + 100 42r + (7r + 2)(r 1) log . r2 r4 r2 − r5 − r 1 | −(5.30)|

The regularized version of P13(k) is k3 Z ∞ P (k) = P (k) dr r2P (kr)Z (r) 13,reg 252(2π)2 lin lin reg 0 (5.31) 3 Z ∞ k 3s log k+s s = 2 Plin(k) ds e Plin(e )Zreg(e ) , 252(2π) −∞ which can be evaluated numerically in the same manner that was presented in 5.2.3. §

To regularize J2,−2,0(k) we take the power that we subtracted from P13(k)

2 Z 3 k d q Plin(q) ∆P (k) = P13(k) P13,reg(k) = P (k) , (5.32) − − 3 (2π)3 q2

and add it to J2,−2,0(k). To do this, we ﬁrst take the Fourier transform of Eq. (5.32): Z 3q 1 Z 3q Z 3q ( ) d 1 iq1·r d 1 iq1·r 2 d 2 Plin q2 ∆ξ(r) = 3 e ∆P (q1) = 3 e q1Plin(q1) 3 2 (2π) −3 (2π) (2π) q2 Z ∞ Z ∞ 1 4 = 4 dq1 q1Plin(q1)j0(q1r) dq2 Plin(q2) . −12π 0 0 (5.33)

139 1 Since J2,−2,0(r) appears in ξ22(r) with a factor of 3 – see Eq. (5.12) – it follows that

3∆ξ(r) should be added to J2,−2,0(r) if we want to preserve the sum P22(k) + P13(k)

in the regularization process. This leads to a regularized J2,−2,0(r):

J[2,−2,0 reg](r) = J2,−2,0(r) + 3∆ξ(r) Z ∞ Z ∞ 1 4 = 4 dq1 q1Plin(q1)j0(q1r) dq2 Plin(q2)[j0(q2r) 1] . 4π 0 0 − (5.34)

The left bracket of Eq. (5.34) proceeds in the same manner as presented in §5.2.2.

The right bracket in Eq. (5.34) requires some additional work: Z ∞ I−2,0,reg = dq2 Plin(q2)[j0(q2r) 1] 0 − N/2 Z ∞ X ν+iηn sin(q2r) = cn dq2 q2 1 q2r − (5.35) n=−N/2 0 N/2 X −1−ν−iηn reg = cnr gn , n=−N/2

where the integral may be evaluated by substituting z = kr and ﬁnding:

Z ∞ reg reg reg ν+iηn sin z reg πQn reg gn (Qn ) = dz z 1 = Γ(Qn ) sin = f(Qn ) (5.36) 0 z − 2

reg 8 reg reg and Qn = ν + iηn. The last equality uses Eq. (B.7), and ensures that gn (Qn ) can be evaluated using the same numerical machinery used for the Jαβl(k) integrals. The

ﬁnal result for J[2,−2,0,reg](k) is completely analogous to the method in 5.2.2, with §

reg Qh Q−2,n the only exception that gn is replaced by gn and the factor 2 is replaced by 2 in Eq. (5.20). FAST-PT allows the user to specify which case is desired.

8This integral is valid for its range of convergence, 3 < ν < 1. A straightforward way to prove this is to insert a factor of e−z, with small− and positive,− into the integrand; then iz −iz reg expanding sin z = (e + e )/(2i) leads to a sum of three Γ-functions, two with Γ(Qn ) and one reg + with Γ(Qn + 1). Taking the limit of 0 causes the latter to drop out and the remaining two to give Eq. (5.36). →

140 5.3 Performance

We now discuss the results from the FAST-PT algorithm. Unless otherwise noted,

results are based on the input linear power spectrum generated by the Boltzman

solver CAMB (Lewis et al., 2000a), assuming a ﬂat ΛCDM cosmology corresponding

to the recent Planck results Planck Collaboration et al. (2015a). Timing results were

obtained on a MacBook Pro Retina laptop computer, with a 2.5 GHz Intel Core i5

processor and running OS X version 10.10.3. We used Python version 2.7.10, numpy

1.8.2, and scipy 0.15.1.

5.3.1 1-loop Results

To test our method we evaluated the 1-loop SPT correction to the power spectrum,

P22(k)+P13(k). We sample the power spectrum for 3000 k-points from log kmin = 4 10 − to log10 kmax = 2 and we additionally pad our input signal with 500 zeros at both ends of the array. A typical run for a sample of this size takes FAST-PT a total time

0.02 seconds on a laptop. We recommend that FAST-PT users sample the input ∼ power spectrum on a grid larger than desired and then trim the output to the desired range to avoid wrapping eﬀects. We take this approach and present our results on a grid from kmin = 0.003 to kmax = 50. The top panel Fig. 5.2 plots our FAST-

PT results, while the bottom panel plots the ratio of our FAST-PT calculations to a conventional method.9 We observe that our new method can recover the 1-loop

9 The “conventional” method is a ﬁxed-grid 2D integration code. Here P22 was computed by putting k on the z-axis and writing q in cylindrical coordinates. The azimuthal integral is trivial. We sample the integrand logarithmically in the radial direction q⊥, and stretch the vertical direction according to qz/k = 1+sinh(20υ)/[2 sinh(20)], with υ > 1. This samples half of space (so the result must be doubled) and by uniformly sampling in υ, it places− higher resolution near q k, which ≈ is important to correctly sample the contribution to P22 from advection by very long-wavelength modes. The P13 integral was log-sampled in r.

141 power spectrum to less than a percent accuracy. The noise observed in the bottom

panel of Fig. 5.2 is due to noise in the input power spectrum from CAMB; any

integration method must interpolate this noise, and this results in noise in the output

spectrum P22 + P13 which diﬀers depending on the method. At high k, the noise in P22 + P13 is larger than (and of opposite sign to) the noise in Plin(k), which is a phenomenon common to diﬀusion problems and is the correct mathematical solution to SPT, where re-normalization or re-summation techniques are not used (see §5.3.2).

The sharp spike around k = 0.1h/Mpc is due to the zero crossing of the 1-loop power

spectrum, where ratios of corrections suﬀer from a “0/0” ambiguity. We conclude that

diﬀerences between FAST-PT results and those from our conventional method are negligible on the scales of interest.

Fig. 5.3 plots estimated run time versus grid size. A solid black line in the left panel plots the average recurring time (i.e. the time of execution after initialization of the FAST-PT class) for 100 runs. The grey band covers the area enclosed by ± one standard deviation. The right panel plots the average initialization time for 100

runs, i.e. the time to initialize the FAST-PT python-class and evaluate all functions

that only depend on grid size (for example gαn). The total time for one one-loop

evaluation is the addition of the black line in the right and left panels. Run time can

vary across machines, so Fig. 5.3 serves only as an estimate.

5.3.2 Renormalization Group Flow

The renormalization group (RGPT) method of McDonald (2007, 2014) provides a

more accurate model for the power spectrum than SPT (Carlson et al., 2009; Widrow

et al., 2009; Orban and Weinberg, 2011), providing signiﬁcant improvement to both

142 103 3

] 2

h 10 / c p

M 1

[ 10

) k ( 3

1 0

P 10 + ) k

( -1 2 10 2 P

10-2 0.01 0.10 1.00 10.00 1.010

1.005

1.000

0.995

ratio to conventional method 0.990 0.01 0.10 1.00 10.00 k [h/Mpc]

Figure 5.2: FAST-PT 1-loop power spectrum results versus those computed using a conventional ﬁxed-grid method. The top panel shows FAST-PT results for P22(k) + P13(k) (the dashed line is for negative values). The bottom panel plots the ratio between FAST-PT and the conventional method.

143 0.08 average of execution time average time to initialize 0.010 0.07

0.06 0.008

0.05

0.006 0.04

0.03

0.004

time [seconds] 0.02

0.01 0.002

0.00 500 1000 1500 2000 2500 3000 500 1000 1500 2000 2500 3000 number of grid points number of grid points

Figure 5.3: Estimate of FAST-PT execution time to number of grid points scaling. The left panel plots the average one-loop evaluation time, after initialization of the FAST-PT class. The right panel plots the average time required for initialization of FAST-PT class for 100 runs. For a sample of grid points, the error is computed by taking the standard deviation of 100 runs.

144 the structure of the BAO feature and the broadband power at smaller scales (higher k). The RG evolution equation is

dP (k, λ) = G [P (k, λ),P (k, λ)] , (5.37) dλ 2 where G2[P,P ] is the standard 1-loop correction to the power spectrum, i.e. P22(k) +

P13(k) with the caveat that the input power spectrum need not be the linear power spectrum. The parameter λ is a “coupling” strength parameter proportional to the growth factor squared. One can imagine that Eq. (5.37) represents a time-evolution of the power spectrum (in an Einstein-deSitter universe) starting at P (k, λ = 0) =

Plin(k), moving forward in time by a small step, using perturbation theory to up- date the power spectrum, and then using the updated power spectrum as the initial condition for the next step, iterating until one reaches λ = 1.

However, despite the potential advantages of the RG approach, it can be quite numerically intensive. Eq. (5.37) is a stiﬀ equation and becomes unstable when the integration step size is too large, and it requires an evaluation of the 1-loop SPT kernel at every step. Conventional computational methods are thus extremely time consuming. The speed of FAST-PT makes this calculation signiﬁcantly more feasible.

We have compared our RG ﬂow results with those obtained from the Copter code

Carlson et al. (2009); Carlson (2013), a publicly available code written in C++. We have found that for RG ﬂow our code can obtain results in substantially less time.

For instance, on a 200 point grid, from kmin = 0.01 to kmax = 10, our FAST-PT RG

flow results take 5 seconds, while Copter RG flow results take over 5 minutes. ∼ In Appendix B.3 we explain our integration routine, as well as document RG-flow run times for various grid sizes. A FAST-PT user must consider the stiff nature of

145 Eq. (5.37) when choosing a step size for the integration; we recommend that they

consult Appendix B.3.

The left hand panel of Fig. 5.4 shows our renormalization group and SPT results

compared to linear theory. In our analysis we performed two renormalization group

−1 −1 runs: one to kmax = 5 hMpc and another to kmax = 50 hMpc . Our results are consistent with the plots found in McDonald (2007) (note that in our runs we include the BAO feature). The right hand panel of Fig. 5.4 plots the eﬀective power law index as a function of k, neﬀ = d log P/d log k. Here we see two characteristic features

of RG evolution, the damping of the BAO and neﬀ approaching a ﬁxed point value of

1.4. ∼ − Figure 5.5 shows the eﬀect of a boundary condition within our numerical algo-

rithm. We integrate Eq. (5.37) for some kmin and kmax. The kmax boundary does

not allow for power to continuously ﬂow from larger to smaller scales, as would occur

for inﬁnite boundary conditions. As a result power builds up at high-k causing the

plateau observed in Fig. 5.5. The insert in Fig. 5.5 shows nneﬀ. We do see that be-

fore the onset of the plateau neﬀ does approach 1.4 and this designates a region ∼ −

where RG results at ﬁnite kmax reproduce the asymptotic behavior as kmax . To → ∞ qualify the accuracy of the RG method in the weakly non-linear regime, we also plot

results from the FrakenEmu emulator, which is based of the Coyote Universe

simulations Heitmann et al. (2014). In the vicinity of k 0.1, it is observed that RG ∼ methods better follow the fully non-linear results of the Coyote Universe.

Figure 5.5 also shows another interesting feature, the removal of noise in the

RG-framework. As mentioned earlier, linear power spectrum generated by CAMB

contains low-level noise. This noise is most easily visualized through a derivative, for

146 5.0 0.5 linear SPT 1-loop 4.5 0.0 Renormalization Group (kmax =5 h/Mpc) Renormalization Group (kmax =50 h/Mpc)

4.0 0.5

3.5 r

a 1.0 e ) n k i

l 3.0 ( f f P e / 1.5 n P 2.5

2.0 2.0

1.5 2.5

1.0 3.0 0.10 1.00 0.10 1.00 k [h/Mpc] k [h/Mpc]

−1 Figure 5.4: FAST-PT Renormalization group results for kmax = 5, 50 hMpc . Left panel shows Renormalization group results and SPT results compared{ } to the linear power spectrum (see legend in right panel). Right panel shows neﬀ = d log P/d log k for Renormalization group, SPT, and linear theory.

instance neﬀ. One can see that neﬀ for the linear power spectrum in Fig. 5.5 is noisy,

particularly at large k. Under the RG evolution this noise is washed away, as seen in

the RG neﬀ results. This is a result of the fact that noise in Plin(k) results in “negative”

noise features in P13(k). Under the RG ﬂow, this feature causes noise initially present

to be smeared away in the nonlinear regime. This is also what happens in the real

universe, since features in the power spectrum at small ∆k correspond to correlations

at large real-space scales 2π/∆k, which are smeared out by advection; this eﬀect ∼ is responsible for the familiar BAO peak smearing Seo and Eisenstein (2007a).

147 105

104

103

102 0.5 0.0 )

k 1 10 0.5 ( ) P 1.0 k (

0 f 10 f e 1.5 n 2.0 10-1 Renormalization Group 2.5 one-loop SPT 3.0 10-2 0.001 0.010 0.100 1.000 10.000 Coyote Universe k Linear 10-3 0.00 0.01 0.10 1.00 10.00 k [h/Mpc]

Figure 5.5: Renormalization group results compared to standard 1-loop calculations and those taken from the Coyote Universe. A plateau at high-k develops due to boundary conditions. Insert shows neﬀ(k) = d log P/d log k.

148 5.4 Summary

In this paper we have introduced FAST-PT, a simple code that can calculate power spectrum in standard perturbation theory to 1-loop order. FAST-PT oﬀers

the community an eﬃcient method to calculate convolution integrals such as those

appearing in SPT and similar approaches. The code is modular and written in a

high-level language (Python), but due to algorithmic improvements it is extremely

fast. The keys to the algorithm are locality (expressing the conﬁguration-space mode

coupling integral Jαβl in terms of the product of two integral transforms of the power spectrum); scale independence of the physics of gravity (hence simple behavior of power laws); and the FFT (which enables log-spaced data to be converted into a superposition of power laws and vice versa). The recurring cost of the 1-loop SPT calculations is presented in Fig. 5.3, for a linear power spectrum sampled on a 3000 point grid, one can expect to obtain results in 0.01 seconds. The time for RG ∼ results in tabulated in Tables B.1 and B.2. For a linear power spectrum sampled on

500 point grid from kmin = 0.001 to kmax = 10, RG results are obtained in a few seconds.

We have demonstrated FAST-PT in the context of 1-loop SPT and the RG ﬂow.

However, we are working to extend the technique to other problems in cosmological perturbation theory. For instance, when the evolution of ﬂuctuation modes is given by a scale-dependent propagator, the time- and scale-dependence of each mode can no longer be separated Audren and Lesgourgues (2011). Such a scenario arises in the presence of massive neutrinos, where growth of structure is suppressed on small scales due to free-streaming Saito et al. (2008, 2009); Blas et al. (2014). Solving for nonlinear evolution in such a scenario can be done using a time-ﬂow approach

149 Pietroni (2008), requiring many evaluations of mode-coupling integrals. It is similar

to the RG ﬂow described above, but with some extra complications, particularly due

to the scale dependence of the propagators (note that our Jαβl integrals include only

power-law dependences on the magnitudes of q1 and q2), and the fact that the Green’s function solution for the bispectrum (needed to reduce the power spectrum solution to a mode-coupling integral) involves products of power spectra at unequal time. We are investigating the extent to which these complications can be grafted onto FAST-PT.

Another future extension is to mode coupling of vector or tensor ﬁelds, as arise in the calculation of nonlinear intrinsic alignment correlations (e.g. Hirata and Seljak (2004);

Blazek et al. (2015)). While ﬁelds of arbitrary spin introduce additional indices, the basic features of the FAST-PT framework apply to this case. An extension of the code to such ﬁelds will be presented in a future paper.

Our Python code is publicly available at https://github.com/JoeMcEwen/FAST-PT and includes a user manual. We also provide Python scripts to reproduce 1-loop power spectrum, renormalization group results, and animations for renormalization group results.

150 Appendix A: Appendix : Streaming velocities and the baryon-acoustic oscillation scale

A.1 Details of calculations

It is convenient to work calculations in Fourier space. We use the relationship between fields in configuration and Fourier space: f(k) = R d3x e−ik·x f(x). We define the power spectrum in terms of the ensemble average over the Fourier space density fields:

δ(k)δ(k0) = (2π)3δ(3)(k + k0)P (k). (A.1) h i In perturbation theory we write the matter ﬁeld as a series expansion: δ(k) = δ(1)(k)+

δ(2)(k) + , in which terms of order 2 and higher represent non-linear evolution of ··· the matter ﬁeld. The second order contribution to the density contrast is

Z d3k δ(2)(k) = 1 F (k , k )δ(1)(k )δ(1)(k ), (A.2) (2π)3 2 1 2 1 2 where we deﬁne k2 k k1, µ12 k1 k2/(k1k2), and the second order density kernel ≡ − ≡ · is 5 1 k1 k2 2 2 F2(k1, k2) = + µ12 + + µ12. (A.3) 7 2 k2 k1 7 In Fourier space the squared tidal tensor is

Z d3k s2(k) = 1 S (k , k )δ(k )δ(k ), (A.4) (2π)3 2 1 2 1 2 151 2 1 where S2(k1, k2) = µ . At one-loop, the correlations in Eq. (7) can be expressed 12 − 3 as Fourier transforms of the corresponding power spectra:

2 P (k) = L kT (k)P (k), adv 3 s v lin Z 3 d k1 P 2 (k) = 2 F (k , k )P (k )P (k ), δδ (2π)3 2 1 2 lin 1 lin 2 Z 3 d k1 P 2 2 (k) = 2 P (k )P (k ), δ δ (2π)3 lin 1 lin 2 Z 3 d k1 P 2 (k) = 2 F2(k1, k2)µ12Tv(k1)Tv(k2) δv − (2π)3

Plin(k1)Plin(k2), × Z 3 d k1 P 2 2 (k) = 2 µ12Tv(k1)Tv(k2)Plin(k1)Plin(k2), δ v − (2π)3 Z 3 d k1 P 2 2 (k) = 2 µ12S2(k1, k2)Tv(k1)Tv(k2) s v − (2π)3

Plin(k1)Plin(k2), × Z 3 d k1 2 2 2 P 2 2 (k) = 2 µ T (k )T (k )P (k )P (k ), v v (2π)3 12 v 1 v 2 lin 1 lin 2 Z 3 d k1 P 2 2 (k) = 2 S (k , k )P (k )P (k ), δ s (2π)3 2 1 2 lin 1 lin 2 Z 3 d k1 2 P 2 2 (k) = 2 S (k , k )P (k )P (k ), and s s (2π)3 2 1 2 lin 1 lin 2 Z 3 d k1 P 2 (k) = 2 F (k , k )S (k , k ) δs (2π)3 2 1 2 2 1 2

Plin(k1)Plin(k2). (A.5) ×

Here the Fourier-space expression for Ls, deﬁned in Eq. (9), is

Z d3k T (k) L = v P (k). (A.6) s (2π)3 k lin

Note that Ls can be interpreted as the r.m.s. contribution to the comoving displacement that is correlated with the streaming velocity (opposite directions) at the same position. At z = 1.2, Ls = 7.7 Mpc, compared with an r.m.s. displacement of

152 Ψ2 1/2 = 8.5 Mpc. These power spectra must be multiplied by the relevant nu- h i merical pre-factors, including bias values, shown in Eq. (7). The full galaxy power spectrum is then

2 1 2 P (k) = b P (k) + b b P 2 (k) + b P 2 2 (k) g 1 NL 1 2 δδ 4 2 δ δ 1 2 1 + b P 2 2 (k) + b b P 2 (k) + b b P 2 2 (k) 4 s s s 1 s δs 2 2 s δ s h i + 2b1bv Pδv2 (k) + Padv(k)

1 1 2 + b b P 2 2 (k) + b b P 2 2 (k) + b P 2 2 (k), (A.7) 2 2 v δ v 2 s v s v v v v where PNL(k) = P11(k) + P22(k) + P13(k) is the non-linear power spectrum calculated in standard perturbation theory to one-loop order.

A.2 Calculating the BAO shift

Following [3, 47], we use the following template power spectrum:

2 5 X j X j Pmodel(k) = cjk Pevo(k/α) + ajk , (A.8) j=0 j=0 where the coeﬃcients ai and ci are marginalized as nuisance parameters and α represents the shift in the BAO peak (α > 1 corresponds to a shift towards smaller

7 scales). Our fitting template differs slightly from [3, 47] in that we omit the a7k term. We found that including this term provided too much flexibility, leading to persistent residual noise in the fit, although results were otherwise nearly identical.

The nonlinear damping of the BAO is captured by the evolved power spectrum:

2 2 −k Σm/2 Pevo(k) = [Plin(k) Pnw(k)] e + Pnw(k), (A.9) − where Pnw is the no-wiggle (i.e. BAO-removed) power spectrum, computed from the

−1 ﬁtting formula of [58], and Σm = 4.8h Mpc is the ﬁducial damping factor at z = 1.2,

153 although it is treated as a free parameter in our analysis. While this template includes nonlinear damping of the BAO, it does not include the corresponding shift in BAO position ( 0.2% when bv = 0). ∼ The power spectrum in Eq. A.7 (corresponding to the Fourier-space analog of

Eq. 7) is ﬁt to the template by minimizing χ2 assuming the standard covariance for

P (k) (e.g. [59]):

Z kmax 3 2 2 d k [Pg(k) Pmodel(k)] χ = V 3 − 2 , (A.10) kmin (2π) 2[Pg(k) + 1/n¯] where we have chosen the density of galaxies n¯ = 3 10−4h3 Mpc−3, and the integra- × tion is from 0.02 < k < 0.35h Mpc−1.

A.3 Eulerian treatment of streaming velocities

Although the physics of galaxy formation should be aﬀected by the streaming velocity at the Lagrangian positions of tracers, these velocities can be equivalently expressed at the Eulerian positions. Indeed, as long as all relevant terms are included, there is always a mapping between Eulerian and Lagrangian biasing (e.g. [49]). In this appendix, we demonstrate how consistent Eulerian treatment produces the same advection contribution to the streaming velocity bias.

We can write the streaming velocity as the Eulerian expansion:

(1) (2) vbc(x) = v (x) + v (x) + . bc bc ···

This expansion is analogous to the Eulerian treatment of the density ﬁeld, and the higher-order streaming velocity contributions can be derived in similar fashion (e.g.

[60]). We start with the continuity and Euler equations for two ﬂuids: cold dark

154 matter and baryons. We assume curl-free velocity ﬁelds, which can be expressed in

terms of the velocity divergence: θ(k) = ik v(k). Working in an Einstein-de Sitter · universe, the Fourier space equations are:

∂δx(k, a) a (a) + θx(k, a) H ∂a Z 3 d k1 k k1 = 3 · 2 θx(k1, a)δx(k2, a) and (A.11) − (2π) k1 ∂θx(k, a) 3 (a) a + θx(k, a) + (a)δm(k, a) H ∂a 2H Z 3 2 d k1 k k1 k2 = 3 2 · 2 θx(k1, a)θx(k2, a), (A.12) − (2π) 2k1k2

where “x” denotes the relevant ﬂuid (baryons or CDM), (a) = aH(a) is the confor- H mal Hubble parameter, and δm is the total matter density perturbation. Since we are interested in the relative velocity between dark matter and baryons, we express these equations in terms of total and relative densities and velocity divergences:

δm = fcδc + fbδb; θm = fcθc + fbθb

δbc = δb δc; θbc = θb θc, (A.13) − − where fc and fb are the CDM and baryon fractions, respectively, of the total matter, and fc + fb = 1. The evolution of the total and relative quantities is then given by:

∂δm(k, a) a (a) + θm(k, a) (A.14) H ∂a Z 3 d k1 k k1 = 3 · 2 θm(k1, a)δm(k2, a) + fbfcθbc(k1, a)δbc(k2, a) , (A.15) − (2π) k1 ∂θm(k, a) 3 (a) a + θm(k, a) + (a)δm(k, a) (A.16) H ∂a 2H Z 3 2 d k1 k k1 k2 = 3 2 · 2 θm(k1, a)θm(k2, a) + fbfcθbc(k1, a)θbc(k2, a) , (A.17) − (2π) 2k1k2 ∂δbc(k, a) a (a) + θbc(k, a) (A.18) H ∂a 155 Z 3 d k1 k k1 = 3 · 2 [(fc fb)θbc(k1, a)δbc(k2, a) − (2π) k1 − (A.19)

+ [θbc(k1, a)δm(k2, a) + θm(k1, a)δbc(k2, a)] , (A.20) ∂θbc(k, a) (a) a + θbc(k, a) (A.21) H ∂a Z 3 2 d k1 k k1 k2 = 3 2 · 2 (fc fb)θbc(k1, a)θbc(k2, a) + 2θbc(k1, a)θm(k2, a) . (A.22) − (2π) 2k1k2 −

Setting the right hand sides of these equations to zero, we can solve for the linear evolution. The (standard) growing-mode solution for total matter ﬂuctuations is:

δ(1)(k, a) a and θ(1)(k, a) = (a)δ(1)(k, a) a1/2. (A.23) m ∝ m −H m ∝

The relative velocity equation is a ﬁrst-order diﬀerential equation, and its solution evolves as θ(1)(k, a) a−1. The other two modes are the decaying matter density bc ∝ (1) −3/2 (1) mode, δm (k, a) a , and a mode with δ (k, a) constant and zero velocities. ∝ bc We are interested in the nonlinear interaction when the growing density mode and the relative velocity mode are present in the initial conditions, not just the growing density mode.

(2) We want the second-order relative velocity, θbc (k, a). This has contributions from both “total-relative” terms (θbcθm) and “relative-relative” terms (θbcθbc). Since the total and relative perturbations must be of similar magnitude at recombination, but relative velocities decline as a−1 whereas total velocities grow as a1/2, by z 1 ∝ ∝ ∼ the relative-relative coupling terms are suppressed by a factor of 104 relative to the ∼ total-relative terms and can thus be safely ignored.

The equation for the second-order relative velocity is then

(2) Z 3 2 ∂[aθbc (k, a)] d k1 k k1 k2 (1) (1) = 3 2 ·2 θbc (k1, a)δm (k2, a). (A.24) ∂a (2π) k1k2 156 The right-hand side is independent of a, so the solution is

Z 3 2 (2) d k1 k k1 k2 (1) (1) aθbc (k, a) = (a ai) 3 2 ·2 θbc (k1)δm (k2), (A.25) − (2π) k1k2

(2) where ai is the initial scale factor, i.e. where θbc (k, a) = 0, which we take to be the time of recombination. At redshifts relevant for BAO measurements, ai/a 1, and so Z 3 2 (2) d k1 k k1 k2 (1) (1) θbc (k, a) = 3 2 ·2 θbc (k1)δm (k2). (A.26) (2π) k1k2 Alternatively, we could make the following power-law ansatz for the leading time- dependent term at each order (e.g. [61]):

θ(n)(k, a) = (a)an−3/2θ(n)(k), bc H bc (n) n−3/2 (n) δbc (k, a) = a δbc (k). (A.27)

Requiring Eq. (A.21) to hold at each order and dropping the “relative-relative” terms as before yields:

Z 3 2 (n) d k1 k k1 k2 (n 1)θbc (k) = 3 2 ·2 − − (2π) k1k2 n−1 X (l) (n−l) θ (k1)θ (k2). (A.28) × bc m l=1

For n = 2, we recover the solution above.

−1 Because its linear mode decays as a , it is somewhat counterintuitive that vbc has a significant nonlinear contribution. However, because it couples to the total matter perturbation, there are nonlinear corrections which grow in fractional importance as the matter fluctuations grow. The ratio of second- to first-order velocity perturbations

θ(2)(k, a)/θ(1)(k, a) scales as a, i.e. in proportion to the growth factor of the growing bc bc ∝ matter mode.

157 Our next step is to determine the galaxy biasing terms that arise from the second-

order streaming velocity. Using the deﬁnition of the velocity divergence and normal-

izing by the r.m.s. streaming velocity, the second-order streaming velocity is:

Z 3 (2) d k1 k1 k2 (1) (1) vs (k) = k 3 2· 2 k1 vs (k1)δm (k2). (A.29) (2π) k1k2 ·

This expression can be written in conﬁguration space (using index notation):

(2) h −2 (1) −2 (1) i v (x) = i ( j kv )( j δ ) s,i ∇ ∇ ∇ ∇ s,k ∇ ∇ m (1) −2 (1) (1) −2 (1) = [ iv ] j δ + v i j δ , (A.30) ∇ s,j ∇ ∇ m s,j ∇ ∇ ∇ m

(1) −2 (1) (1) where the irrotationality of vs was used to show that j kv = v , and ∇ ∇ ∇ s,k s,j 2 where all quantities are evaluated at Eulerian position x. Finally, we can write vs (x)

up to (δ3 ): O lin

v2(x) =v(1) v(1) + 2v(1) v(2) s s · s s · s (1) 2 (1) (1) −2 (1) =[v ] + 2v [ iv ] j δ s s,i ∇ s,j ∇ ∇ m (1) (1) −2 (1) + 2v v i j δ s,i s,j ∇ ∇ ∇ m (1) 2 (1) (1) −2 (1) =[v ] + 2v [ jv ] j δ s s,i ∇ s,i ∇ ∇ m 1 + 2v(1)v(1) s(1) + δ(1) s,i s,j ij 3 m =[v(1)]2 + ∇ [v(1)]2 [∇ −2δ(1)] s { s }· ∇ m 2 2 + 2v(1)v(1)s(1) + v(1) δ(1). (A.31) s,i s,j ij 3 s m

(1) (1) (The irrotationality of vs was used again to swap the indices in iv .) The second ∇ s,j term is the advection contribution (see Eq. 6), while the ﬁnal two terms can be

absorbed into the deﬁnition of b1v and bsv in Eq. (4), indicating that they would naturally appear even had we not originally included them.

158 In the context of separate baryon and CDM ﬂuids, symmetry allows additional

biasing terms, proportional to δbc or θbc. Most notably, a θbc contribution would

correlate with the linear bias to produce a term with the same scale dependence as

ξadv. Such a bias contribution would thus be partially degenerate with bv, although they each additionally produce unique correlations. Although such bias terms may may be physically motivated, they are distinct eﬀects and can be self-consistently set to zero in our present treatment. The dominant contributions (in terms of redshift

2 scaling) to vbc at each perturbative order will not contain a linear contribution from

δbc (see Eqs. A.28 and A.21), while θbc is the derivative of vbc and is thus suppressed by a factor of k. We leave consideration of these terms for future work.

Finally, we show that the advection contribution is required to maintain Galilean invariance, which was actually broken in previous presentations of streaming velocity bias. Galilean invariance requires that the laws of physics (e.g. galaxy formation) are the same in all inertial reference frames. In particular, if we are interested in galaxy correlations at some scale k, then large-scale perturbations at wave numbers k0 k can only affect the statistics of the galaxy field via the local density or tidal field. The displacement (or velocity, or gravitational field – these are all related in cosmological perturbation theory) can have no effect. Mathematically, since the displacement field in linear perturbation theory in Fourier space is Ψ(k0) = i(k0/k02)δ(k0), effects of the

displacement ﬁeld are characterized by inverse powers of k0. Terms that have these

inverse powers of k0 in the long-wavelength limit must cancel in any physical theory.

To see how Galilean invariance works in the case of streaming velocities, we con-

sider the eﬀect of large-scale power, only present up to some long wavelength kL, on

the observed tracer correlations at k kL. From Eq. (A.5) and the non-symmetrized

159 F2, we see that Pδv2 contains a term:

Z 3 d k1 2 k2 2 Pδv (k) 2 3 µ12 Tv(k1)Tv(k2) ⊃ − (2π) k1

Plin(k1)Plin(k2). (A.32) ×

For k1 k, k2 k, and the dependence on µ12 can be trivially averaged, yielding ≈ the contribution:

Z 3 2 d k1 k 2 Pδv (k) 3 Tv(k1)Tv(k) ⊃ − 3 (2π) k1

Plin(k1)Plin(k). (A.33) ×

−1 The k1 factor is problematic, as it suggests that the displacement ﬁeld from the long-wavelength mode k1 < kL is aﬀecting galaxy power at scale k. This implies that, in order to be physical, Eq. (A.7) must have another term proportional to b1bv that cancels this divergence. The only other candidate is the advection term Padv; plugging Eq. (A.6) into Eq. (A.5), we see that the advection term is

Z 3 2 d k1 Tv(k1) Padv(k) 3 Plin(k1) kTv(k)Plin(k). (A.34) ⊃ 3 (2π) k1

Thus the sum Pδv2 + Padv contains no inverse powers of k1 and obeys Galilean invariance, although each term does not individually. This cancellation is analogous to the one that occurs in the sum P22 + P13 in the one-loop SPT power spectrum.

A subtlety in this argument is that at small k1, Tv(k1) k1, since the relative ∝ velocities of baryons and dark matter are sourced by the gradient in photon pressure in the early Universe (we have checked this scaling numerically from CLASS outputs).

Thus the inverse power of k1 in Eq. (A.33) is not realized in practice. However, the arguments in this appendix should be valid irrespective of the origin of the streaming velocities and hence the functional form of Tv.

160 Appendix B: Appendix for FAST-PT

sectionMathematical Identities In this work we have used a number of common mathematical identities. These identities are easily found in any standard mathematical physics text, (e.g. Abramowitz and Stegun (1964)). However, to make our paper self-contained we list those relevant to our paper. In 5.2.2 we used the following § special function identities: the addition theorem

l 4π X ∗ Pl(qˆ1 qˆ2) = Ylm(qˆ1)Y (qˆ2); (B.1) · 2l + 1 lm m=−l

the special case thereof,

l X 2l + 1 Y (θ, φ)Y ∗ (θ, φ) = ; (B.2) lm lm 4π m=−l

the orthonormality relation

Z 2 ˆ ˆ ∗ ˆ d k Ylm(k)Yl0m0 (k) = δll0 δmm0 ; (B.3) S2

and the expansion/decomposition of a plane wave:

Z l 2 ∗ iq·r l ∗ iq·r X l X ∗ d qˆ Ylm(qˆ)e = 4πi jl(qr)Ylm(ˆr) e = 4π i jl(qr) Ylm(qˆ)Ylm(ˆr) . 2 ↔ S l m=−l (B.4)

161 B.1 Γ-function identities and evaluations

We make extensive use of the following integral (see pg. 486 of Ref. Abramowitz and Stegun (1964)):

Z ∞ κ κ Γ [(µ + κ + 1)/2] κ dt t Jµ(t) = 2 = 2 g(µ, κ) , κ < 1/2 , (κ + µ) > 1 , 0 Γ [(µ κ + 1)/2] < < − − (B.5) where we deﬁne the Γ-function ratio:

Γ [(µ + κ + 1)/2] g(µ, κ) = . (B.6) Γ [(µ κ + 1)/2] − A second useful integral is

Z ∞ πρ √π 1 1 f(ρ) dt tρ−1 sin t = Γ(ρ) sin = 2ρg , ρ (B.7) ≡ 0 2 2 2 − 2

for 1 < ρ < 1. The second equality is Eq. (3.761.4) of Ref. Gradshteyn and Ryzhik − < (1994). The last expression is an evaluation of the integral Eq. (B.5) and the relation

r πt sin t = J (t). (B.8) 2 1/2

We use the second or third expressions to deﬁne f(ρ) via analytic continuation to

values of ρ outside the domain of convergence of the integral.

The numerical evaluation of g(µ, σ) in FAST-PT uses the scipy gamma function

for most values. However, when the argument to the Γ-function has a large com-

plex value numerical overﬂows may occur. Therefore when σ > 200 we use an |= | asymptotic form for our evaluations [Eq. (6.1.40) of Ref. Abramowitz and Stegun

(1964)]:

∞ 1 X B2m log Γ(z) (z 1/2) log z z + log 2π + , (B.9) ≈ − − 2 2m(2m 1)z2m−1 m=1 −

162 for z in arg z < π. Here B2m are the Bernoulli numbers; we find that only → ∞ | | 1 1 the first two terms B2 = and B4 = are necessary at σ > 200 to achieve an 6 − 30 |= | error of < 10−13 in log Γ(z). To avoid overflows, the logarithms are differenced to give log g(µ, κ) and this result is exponentiated.

The function f in Eq. (B.7) may be numerically evaluated using a Γ-function

routine, but we prefer to use our routines for g. This is because f itself is well- behaved near ρ = 0 (the Γ function has a simple pole and the sine function has a single zero), but the Γ-function expression in Eq. (B.7) is ill-behaved. In contrast, our implementation of the function g is well-behaved at ρ = 0.

B.2 Mitigation of Edge Eﬀects

Fourier methods are susceptible to ringing effects due to discontinuities in the input signal. To mitigate ringing we smoothly tapper the array edges of the Fourier coefficient array cm with a window function defined to have continuous first and second derivatives:  x−xmin 1 x−xmin  x −x 2π sin 2π x −x x < xleft  left min − left min W (x) = 1 xleft < x < xright , (B.10)  xmax−x 1 xmax−x  sin 2π x > xright xmax−xright − 2π xmax−xright

where xleft and xright are input parameters that determine the position of the tapering.

For a typical run we dampen the high frequency Fourier modes by applying the window function to cm. In this case the position where the tapering begins is at an

m = 0.75 N/2, where N is the size of the input array. ± × Fourier analysis assumes the input signal to be periodic. This often leads a a wrap-

around eﬀect in our results, i.e. the leakage between low-k to high-k. To alleviate

this eﬀect we allow for zero-padding of the input power spectrum. For an input

163 power spectrum sampled on a k-grid of a few thousand points, we add 500 zeros ∼ to both ends. Wrap-around eﬀects can also be mitigated by using an input power

spectrum sampled over a larger k-range than desired and then trimming on output, we recommend one take this approach in combination with ﬁltering of the Fourier coeﬃcients and zero-padding.

B.3 RG-ﬂow Integration

Numerical integration of Eq. (5.37) will quickly develop instabilities when using a simple integration routine (e.g. Euler integration). These instabilities are highly sensitive to kmax and the linear grid spacing ∆. For kmax 1 we have found that ≤ a fourth order Runge-Kutta (RK4) method will produce stable results. However,

−3 for kmax > 1, stable RK4 results require an integration step ∆λ greater than 10 , decreasing rapidly with increasing kmax. An integration step this small will increase computation time substantially. To decrease computation time we have implemented the super time step (STS) method of Alexiades et al. (1996). Super time step methods are a class of integrators developed to solve parabolic equations, often for diﬀusion problems. They belong to the family of Runge-Kutta-Chebyshev methods and have the advantage that they can decrease the computation time by increasing the stability region. For each integration step ∆λ, the STS method takes Ns inner Euler steps P δλj, where j = 1, 2, ..., Ns, such that ∆λ = j δλj. The δλj are chosen by

π(2j 1) −1 δλj = ∆λCFL (µ 1) cos − + (1 + µ) , (B.11) − 2Ns

where ∆λCFL is the usual Courant-Friedrichs-Lewy stability step, µ is a damping

factor (related to a ratio of eigenvalues) and is between 0 and 1. Equation (5.37) is not

164 ∆ log k grid points run time [seconds] 0.14 50 0.24 0.069 100 0.27 0.013 500 0.30 0.0045 1500 0.45

−3 Table B.1: Stable RK4 runs for kmin = 10 and kmax = 1 and ∆λ = 0.1.

a parabolic partial differential equation; it is an integro-differential equation, which behaves as a diffusion equation under certain limiting circumstances. As such we do not have a rigorous theory for selecting ∆λCFL and µ, and we chose their values by numerical experiment. For kmax = 10 and 2000 grid points we have chosen ∆λCFL =

0.001, µ = 0.1, and Ns = 10.A FAST-PT user has the option to specify ∆λCLF, µ, and Ns. In tables B.1 and B.2 we document RG-ﬂow run times for various grid sizes.

These tables should serve as guidance when choosing the integration routine and/or routine parameters.

We also control stability by filtering the right hand side of Eq. 5.37 at each integration step with the window function presented in Appendix B.2. The tapering of the window function begins at log kmin + 0.2 and log kmax 0.2. Applying this − window function smooths any sharp features introduced by the edge effects, slowing the development of instabilities due to the stiff nature of the differential equation.

165 ∆ log k grid points run time [seconds] 0.018 500 2.77 0.0092 1000 3.55 0.0046 2000 5.10

−3 Table B.2: Stable STS runs for kmin = 10 and kmax = 10. Results were obtained using STS parameters: µ = 0.1, ∆λCFL = 0.001,Ns = 10.

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