Mergers of Supermassive Black Hole Binaries in Gas-rich Environments: Models of Event Rates and Electromagnetic Signatures Takamitsu Tanaka Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2011 c 2011 ! Takamitsu Tanaka All rights reserved ABSTRACT Mergers of Supermassive Black Hole Binaries in Gas-rich Environments: Models of Event Rates and Electromagnetic Signatures Takamitsu Tanaka Supermassive black holes permeate the observable Universe,residinginthenuclei of all or nearly all nearby massive galaxies and powering luminous quasars as far as ten billion light years away. These monstrous objects must grow through a combination of gas accretion and mergers of less massive black holes. The direct detection of the mergers by future gravitational-wave detectors will be a momentous scientific achievement, pro- viding tests of general relativity and revealing the cosmic evolution of supermassive black holes. An additional — and arguably equally rewarding — challenge is the concomitant observation of merging supermassive black holes with both gravitational and electromag- netic waves. Such synergistic, “multi-messenger” studies can probe the expansion history of the Universe and shed light on the details of accretion astrophysics. This thesis examines the mergers of supermassive black hole binaries and the ob- servable signatures of these events. First, we consider the formation scenarios for the earliest supermassive black holes. This investigation is motivated by the Sloan Digital Sky Survey observation of a quasar that appears to be powered by a supermassive black hole with a mass of billions of solar masses, already in place one billion years after the Big Bang. Second, we develop semianalytic, time-dependent models for the thermal emission from circumbinary gas disks around merging black holes. Our calculations corroborate the qualitative conclusion of a previous study that for blackholemergersdetectableby aspace-basedgravitational-waveobservatory,agasdisknear the merger remnant may exhibit a dramatic brightening of soft X-rays on timescales of several years. Our results suggest that this “afterglow” may become detectable more quickly after the merger than previously estimated. Third, we investigate whether these afterglow episodes could be observed serendipitously by forthcoming wide-field, high-cadence electromagnetic sur- veys. Fourth, we introduce a new subset of time-dependent solutions for the standard equation describing thin, viscous Keplerian disks. Finally, we apply these solutions to model the electromagnetic emission of accretion disks around supermassive black hole binaries that may be detectable with precision pulsar timing. 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Contents 1Introduction 1 1.1 Supermassive Black Holes . ... 2 1.2 Supermassive Black Hole Binaries . ..... 10 1.2.1 Orbital Evolution of Compact SMBH Binaries . .... 10 1.2.2 Observational Evidence for SMBH Binaries . ..... 16 1.3 Gravitational Waves from Merging SMBH Binaries . ........ 22 1.3.1 Proposed GW-detection Experiments . ... 26 1.4 Multi-messenger Astronomy with Compact SMBH Binaries . ........ 31 1.4.1 Overview of Proposed Electromagnetic Signatures of Merger . 34 1.5 AccretionTheoryOverview . ... 37 1.5.1 Bondi-Hoyle-Lyttleton Accretion . ..... 37 1.5.2 ThinAccretionDisks............................ 38 1.5.3 The Shakura-Sunyaev α Disk....................... 44 1.5.4 Alternative Accretion Flow Models . ... 46 1.5.5 Jets from Accretion Flows . .49 1.6 ThesisSummary................................... 50 1.6.1 Chapter 2: Assembly of the First SMBHs . .. 51 1.6.2 Chapter 3: Afterglows of SMBH Mergers . .. 52 1.6.3 Chapter 4: Afterglows as Birthing Quasars . ..... 53 1.6.4 Chapter 5: Time-dependent Solutions of Accretion Flows . 54 i 1.6.5 Chapter 6: Electromagnetic Counterparts of Pulsar Timing Array Sources.................................... 54 2TheAssemblyofSupermassiveBlackHolesatHighRedshifts 57 2.1 Introduction .................................... .57 2.2 Assumptions and Model Description . ..... 63 2.2.1 TheMergerTree .............................. 64 2.2.2 The Initial Black Hole Population . .... 68 2.2.3 Baryonic and Dark Matter Halo Profiles . ... 69 2.2.4 Mergers of Dark Matter Halos and Black Holes . .... 72 2.2.5 Gravitational Recoil . .75 2.2.6 The Black Hole Accretion Rate . .86 2.2.7 Putting Together the z = 6SMBHMassFunction. 92 2.3 ResultsandDiscussion .. .. .. .. .. .. .. .. ... 94 2.3.1 Building the > 109M SMBHs....................... 94 " 2.3.2 Constraints on the SMBH Mass Function . ..117 2.3.3 Successful Models I: BH Seeds Stop Forming Early . .....120 2.3.4 Successful Models II: Feedback Adjusted to Maintain m-σ Relation . 125 2.4 Conclusions ..................................... 135 3Time-DependentModelsfortheAfterglowsofSupermassiveBlack Hole Merg- ers 141 3.1 Introduction .................................... .141 3.2 Modeling the Binary-Disk System . .....147 3.2.1 The Circumbinary Disk at Decoupling . ...147 3.2.2 A Simple Model for the Viscous Evolution . ...152 3.3 Observable Features of the Time-Dependent Afterglow . ..........162 3.3.1 Bolometric Light Curve . 162 3.3.2 SpectralEvolution ............................. 167 3.3.3 Possible Reprocessing of the X-ray Signature . .......171 ii 3.3.4 Possible Effects of Advection and Super-Eddington Winds . 174 3.4 Conclusion...................................... 184 3.A Properties of the Circumbinary Disk After Decoupling . ..........188 3.B Green’s Function for the Viscous Evolution of the Disk Surface Density . 198 4WitnessingtheBirthofaQuasar 205 4.1 Introduction .................................... .205 4.2 A Simple Model for the Afterglow Population . ......211 4.2.1 Modeling Afterglow Light Curves . ..211 4.2.2 Modeling the Population of Afterglow Sources . ......217 4.3 ResultsandDiscussion .. .. .. .. .. .. .. .. ...219 4.3.1 Basic Parameter Dependencies . ..219 4.3.2 Counts of Birthing Quasars in X-ray and Optical Surveys.......225 4.4 Conclusions ..................................... 235 5ExactTime-dependentSolutionsfortheThinAccretionDiskEquation:Bound- ary Conditions at Finite Radius 239 5.1 Introduction .................................... .239 5.2 Green’s-Function Solutions with Boundary Conditions at R = 0.......242 5.2.1 Zero Torque at R = 0............................244 5.2.2 Zero Mass flow at R = 0..........................247 5.3 Green’s-Function Solutions with Boundary Conditions atFiniteRadii. 251 5.3.1 Zero Torque at Rin .............................251 5.3.2 Zero Mass Flux at Rin ...........................254 5.4 Conclusion...................................... 259 6ElectromagneticCounterpartsofSupermassiveBlackHoleBinaries Resolved by Pulsar Timing Arrays 261 6.1 Introduction .................................... .261 6.2 Plausible Hosts of PTA-resolved Binaries . ........265 iii 6.2.1 ThePTAErrorBox .............................266 6.2.2 InterloperCounts.............................. 267 6.2.3 Expected Counts of Interloping Galaxies . .....274 6.3 Accretion disks Around PTA-source Binaries . ........280 6.3.1 Disk Properties and Binary Decay . ..282 6.3.2 Surface Density Evolution of the Circumbinary Gas . .......290 6.3.3 Thermal Emission of Accreting PTA Sources . ....296 6.4 Conclusions ..................................... 304 6.A Green’s-function Solution for the Thin-disk Equation with Moving Inner Boundaries......................................305 7Conclusion 309 7.1 Gravitational-wave Astronomy . .....310 7.2 The Prospects for EM Observations of SMBH Binaries . ........312 7.3 Future Modeling of EM Signatures of SMBH Binaries . .......314 iv List of Figures 1.1 The M σ relation.................................. 8 − 1.2 Schematic diagram of SMBH binary evolution . ...... 14 1.3 The dual active nuclei in NGC6240 . .... 18 1.4 The three stages of black-hole coalescence . ......... 26 1.5 Model signal-to-noise contours for the proposed LISA detector . 29 2.1 Merger tree outputs of dark matter halo mass functions at high redshifts . 67 2.2 Distributions of gravitational recoil velocities for random spin magnitudes . 79 2.3 Radial motion of a recoiling SMBH in a spherical mass distribution . 84 2.4 Estimates of the maximum mass accreted by a black hole before redshift 6 . 90 2.5 Model SMBH number densities at redshift 6 (Pop III seeds) .......... 97 2.6 Model occupation fractions of SMBHs in DM halos and SMBH-to-halo mass ratios(PopIIIseeds) ................................ 102 2.7 Model event rates for LISA from SMBH mergers above redshift 6 (Pop III seeds).........................................105 2.8 Model SMBH number densities at redshift 6 (massive seeds).........107 2.9 Model occupation fractions of SMBHs in DM halos and SMBH-to-halo mass ratios(massiveseeds). .. .. .. .. .. .. .. .. .. .109 2.10 Model event rates for LISA from SMBH mergers above redshift 6 (massive seeds).........................................110 2.11 Dependence of the SMBH mass function at redshift 6 on model parameters 113 v 2.12 The relative contribution from seed BHs formed at different redshifts to the mass of the most massive SMBHs . 122 2.13 SMBH mass functions and occupation fractions in models with satisfying the m-σ relation ...................................128 2.14 LISA event rates in models with satisfying the m-σ relation . .130 2.15 Accretion rates of SMBHs in models satisfying the m-σ relation . 134 3.1 Surface-density snapshots of circumbinary