Caltech's Physics 237-2002

Gravitational Waves


Course Outline 

  1. Overview of Gravitational-Wave Science [Lectures by Kip]
    1. The nature of gravitational waves [GW's]: Week 1, Lecture 1, slides 1 - 18 
    2. The GW spectrum: HF, LF, VLF, ELF bands
    3. Detection techniques: 
      1. resonant-mass detectors
      2. interferometers: LIGO and its partners
      3. LIGO details, noise curves, technology: Week 1, Lecture 2, slides 19 - 37
      4. LISA
    4. GW data analysis
    5. GW sources and science
      1. Inspiral of compact body into supermassive hole
      2. Binary black hole mergers
      3. Neutron-star / black-hole mergers: Week 2, Lecture 3 - Part 1, slides 38 - 47
      4. Neutron-star / neutron-star inspiral
      5. Spinning neutron stars
      6. Neutron-star births
      7. Binaries in our galaxy
      8. The very early universe
  2. Introduction to General Relativity [Lectures by Kip]
    1. Tidal gravity in Newtonian theory: Week 2, Lecture 3 - Part 2
      1. Motivation: tidal gravity as spacetime curvature
      2. The Newtonian tidal gravity tensor
      3. Relative acceleration of freely falling particles
    2. The mathematics underlying general relativity: Week 2, Lecture 4
      1. Vectors, tensors, tensor algebra
      2. Differentiation of tensors, connection coefficients
      3. Commutators, coordinate and noncoordinate bases: Week 3, Lecture 5
      4. Spacetime curvature: the Riemann and Ricci tensors
      5. Relativistic tidal gravity; geodesic deviation
    3. The Einstein field equations
      1. Motivation via tidal gravity
      2. "Derivation" of the Einstein equations; number of equations and number of unknowns; contracted Bianchi identitity
  3. Weak Gravitational Waves [GW's] in Flat Spacetime [Lectures by Kip]: Week 4, Lecture 6
    1. Wave equation for Riemann tensor
    2. Transverse-traceless [TT] GW field; + and x polarizations 
    3. A GW's tidal forces (relative motion of freely falling particles)
    4. Metric perturbations; TT gauge and other gauges
    5. Proper reference frame of an observer: Week 4, Lecture 7
    6. Physical measurements of GW's in a proper reference frame
    7. Generation of GW's: The linearized Einstein field equations
    8. Projecting out the TT GW field
    9. Slow-motion, weak-stress approximation for GW sources: Week 5, Lecture 8
    10. The quadrupole formula for GW generation
      1. Derivation in slow-motion, weak-stress approximation
      2. Validity  for slow-motion sources with strong internal gravity and arbitrary stresses
  4. Propagation of GW's Through Curved Spacetime [Lectures by Kip]
    1. Short wavelength approximation; two-lenghscale expansion
    2. Curved-spacetime wave equation for Riemann tensor
    3. Solution of wave equation via eikonal approximation (geometric optics) - Foundations
    4. Geometric optics - Details: Week 5, Lecture 9
      1. gravitons and their propagation; graviton conservation
      2. rays as graviton world lines; propagation of + and x GW fields along rays
      3. + and x polarizations and fields, rays and transport of waves along rays
      4. gravitational focusing of GW's, e.g. by the sun; diffraction at the focus
      5. stress-energy tensor for GW's; nonlocalizability of GW energy
      6. conservation of GW energy and momentum
      7. conseervation of a graviton's energy and momentum
    5. Propagation of GW's through homogeneous matter:  Week 6, Lecture 10
      1. impact of matter on the waves is always negligible
      2. propagation through dust, perfect fluid, viscous fluid, elastic medium
      3. propagation through a cloud of neutron stars
  5. Generation of GW's by Slow-Motion Sources in Curved Spacetime [Lectures by Kip]: Week 6, Lecture 11
    1. Strong-field region, weak-field near zone, local wave zone, distant wave zone
    2. Multipolar expansions of metric perturbation in weak-field near zone and local wave zone
      1. influence of source's mass and angular momentum
      2. mass quadrupolar component of GW's; current quadrupolar component
      3. rates of emission of energy, linear momentum, and angular momentum
    3. Application to a binary star system with circular orbit
      1. inspiral rate and timescale 
      2. chirp waveform; chirp mass
  6. Astrophysical Phenomenology of Binary-Star GW Sources 
    1. GW's from Binary Star Systems: Week 7, Lecture 12   [by E. Sterl Phinney]
      1. GW-driven inspiral of a single binary [review]
      2. Inspiral evolution of a steady-sstate population of many binaries
      3. Types of stars: main-sequence stars, white dwarfs [WD], neutron stars [NS], black holes [BH]; their masses and radii
      4. Binary systems observable by LIGO (and its partners), and by LISA
    2. Issues relevant to estimating numbers of binary GW sources and their merger rates
      1. Cosmology: parameters describing the universe as a whole
      2. Our Milky Way galaxy: its star-formation history, stellar populations and binary populations 
      3. Use of blue light to extrapolate from rates in Milky Way to rates in the distant universe
    3. Estimates of numbers of binary GW sources and inspiral/merger rates: preview of next lecture
      1. NS/NS rates based on binary-pulsar statistics and blue-light extrapolation
      2. Population synthesis as foundation for estimates
    4. Estimates of numbers of binary GW sources [for LISA] and inspiral/merger rates [for LIGO]: Week 8, Lecture 13  [by E. Sterl Phinney]
      1. Estimates based on observed numbers in our galaxy
        1. pitfalls
        2. NS/NS; WD/WD, WD/NS
      2. Population synthesis
        1. Foundations for population synthesis: 
          1. stellar structure and evolution
          2. binary evolution: mass transfers etc.
        2. Estimates of binary numbers for LISA
      3. Estimates of NS/NS, NS/BH, and BH/BH numbers for LIGO -- Week 8, Lecture 14 - Part 1  [by Kip]
  7. Binary Inspiral: Post-Newtonian Gravitational Waveforms for LIGO and Its Partners -- 
    1. Matched-filtering data analysis to detect inspiral waves
    2. Foundations for post-Newtonian approximations to General Relativity 
      1. Mathematical foundations
      2. Physical effects at various orders
    3. Post-Newtonian inspiral waveforms for circular orbits and vanishing spins --  Week 8, Lecture 14 - Part 2  [by Alessandra Buonanno]
    4. Expansion parameter v = (pi M f)^1/3
    5. Phase evolution governed by energy balance
    6. Waveform in time domain
    7. Waveform in frequency domain, via stationary-phase approximation
    8. Influence of spin-orbit and spin-spin coupling: Orbital and spin precession; waveform modulation
      1. NS/BH binary
      2. BH/BH binary
    9. Innermost stable circular orbit (ISCO) and transition from inspiral to plunge
    10. The IBBH problem: failure of post-Newtonian waveforms in late inspiral; methods to deal with this:
      1. Pade resummation
      2. Effective one-body formalism
      3. Search templates designed to deal with uncertainties in our knowledge of the waveforms
  8. Supermassive Black Holes [SMBH's] and their Gravitational Waves [for LISA/ --  Week 9, Lecture 15  [by E. Sterl Phinney]
    1. Astrophysical phenomenology of SMBH's in galactic nuclei
      1. Evidence for their existence
      2. Measurement of SMBH masses via cusp in stellar velocity dispersion (for masses above 10^6 Msun)
      3. Correlation of SMBH masses with velocity dispersion in galactic bulges
      4. Number of SMBH's per unit volume in universe; their distribution of masses (for masses above 10^6 Msun)
      5. Observed quasar and other electromagnetic emission from SMBH's; quiescence of most SMBH's
    2. Mergers of galaxies
      1. Statistics of mergers: observational data; predictions of CDM simulations
      2. Physics of mergers
      3. Dynamical friction on SMBH's, SMBH binary formation
    3. Evolution of SMBH binary
      1. Interaction with stars; loss cone
      2. Hangup and ways to overcome it: repopulation of loss cone; effect of binary motion in galaxy core; effect of ellipticity of galactic potential; interaction with gas
      3. Gravitational radiation reaction
      4. SMBH merger rates
    4. Capture and inspiral of  stars by a SMBH
      1. Loss cone and its repopulation
      2. Tidal disruption of main-sequence stars
      3. Capture of compact stars [WD, NS, small BH] into highly elliptical orbits
      4. Evolution of orbital ellipticity during inspiral
      5. Event rate estimates for captures
    5. Gravitational waves from SMBH binary inspiral, as measured by LISA -- Week 9, Lecture 16  [by Kip]
      1. Frequency evolution, signal-to-noise ratios
      2. Cosmological influences on waves: gravitational redshift; gravitational lensing
      3. Observables: redshifted masses, luminosity distance, inclination angle
    6. GW's from inspiral of a compact star (or BH) into a SMBH
      1. Frequency evolution, signal to noise ratios
      2. Loss of signal strength due to non-optimal signal processing - caused by complexity of inspiral orbits and resulting complexity of waveforms
        1. Implications for event rates
        2. Implications for specifying the level of LISA's noise floor
  9. GW's from Big Bang: Amplification of Vacuum Fluctuations by Inflation
    1. Basic idea: same as parametric amplification of classical waves
    2. Mathematical details
      1. Background cosmological metric
      2. Geometric optics propagation of GW's at "late times'
      3. Wave equation for GW's at all times
      4. Frozen and decaying solutions when wavelength is much larger than background radius of curvature
      5. Matching solutions together: resulting wave amplification
  10. GW's from Neutron-Star Rotation and Pulsation --   Week 10, Lecture 17 [by Lee Lindblom] 
    1. GW's from a structurally deformed, rotating NS
      1. Deformations maintained by a solid crust
      2. Deformations maintained by stress of a strong internal magnetic field
      3. Deformations due to temperature anisotropy induced by accretion of gas onto NS [low-mass X-ray binaries; LMXB's]
      4. Magnitudes of deformation (ellipticities) detectable by LIGO-I and LIGO-II
    2. GW's from pulsations in a rotating NS
      1. Types of pulsational [bar-mode] instabilities: dynamical; secular
      2. beta = T/W as diagnostic for instabilities
      3. Instabilities in uniform-density Newtonian stars [Maclaurin Spheroids]
      4. Mechanisms for forming rapidly rotating NS's: 
        1. Collapse of degenerate stellar cores
        2. Accretion-induced collapse of a white dwarf
        3. Spinup by accretion
        4. Merger of a low-mass NS/NS binary
      5. NS's formed by collapse: differential rotation, values of beta,bar-mode instabilities, numerical evolution of unstable stars
        1. realistic models
        2. models with extreme differential rotationg: instability at small beta
  11. Numerical Relativity as a Tool for Computing GW Generation -- Week 10, Lecture 18  [by Marc Scheel] 
    1. Motivation: Sources that require numerical relativity for their analysis
      1. Binary black hole mergers
        1. Relevance to LIGO & partners, and to LISA
        2. Estimated event rates for LIGO-I, LIGO-II and LISA
        3. Inspiral, merger, and ringdown; estimated wave strengths from each
        4. Rich physics expected in mergers: strong, nonlinear effects; spin-spin and spin-orbit coupling; angular-momentum hangup
        5. Importance of simulating mergers as foundation for interpreting observations
      2. Tidal disruption of NS by a BH companion
        1. Estimated event rate for LIGO-II
        2. Information carried by waves: NS structure and equation of state
        3. Possible connection to gamma ray bursts
        4. Importance of simulations for interpreting observations
      3. Some other sources: NS/NS mergers, cosmic string vibrations, brane excitations in early universe
      4. The necessity to use numerical relativity in simulations of these sources
    2. Mathematical underpinnings of numerical relativity
      1. 3+1 decomposition of spacetime into space plus time
      2. Initial data must satisfy "constraint equations" 
      3. Evolve via "dynamical Einstein equations"
      4. Gauge freedom
      5. Analogy with electromagnetic theory
    3. Mathematical details
      1. Spacetime slicing; lapse, shift, and 3-metric; extrinsic curvature
      2. Hamiltonian constraint equation
      3. Momentum constraint equations
      4. Dynamical equations
      5. Choices of lapse and shift
    4. Current state of the art in numerical relativity; current efforts on BH/BH inspiral & merger
Links to this course's other web pages:

    Course home page
    Course description
    Outlines of Part B:
        Part B: Gravitational-Wave Detection: original outline
        Part B: Gravitational-Wave Detection: alternative outline, with the order of the lectures made more logical
    Course Materials (videos of lectures, reading, homework, solutions)