**Caltech's Physics 237-2002**

**Gravitational
Waves
**

**PART A: GRAVITATIONAL-WAVE
THEORY AND SOURCES**

** **

*Overview of Gravitational-Wave Science***[Lectures by Kip**]- The nature of gravitational waves [GW's]:
**Week 1, Lecture 1, slides 1 - 18** - The GW spectrum: HF, LF, VLF, ELF bands
- Detection techniques:
- resonant-mass detectors
- interferometers: LIGO and its partners
- LIGO details, noise curves, technology:
**Week 1, Lecture 2****, slides 19 - 37** - LISA
- GW data analysis
- GW sources and science
- Inspiral of compact body into supermassive hole
- Binary black hole mergers
- Neutron-star / black-hole mergers:
**Week 2, Lecture 3 - Part 1, slides 38 - 47** - Neutron-star / neutron-star inspiral
- Spinning neutron stars
- Neutron-star births
- Binaries in our galaxy
- The very early universe
*Introduction to General Relativity***[Lectures by Kip]**- Tidal gravity in Newtonian theory:
**Week 2, Lecture 3 - Part 2** - Motivation: tidal gravity as spacetime curvature
- The Newtonian tidal gravity tensor
- Relative acceleration of freely falling particles
- The mathematics underlying general relativity:
**Week 2, Lecture 4**

- Vectors, tensors, tensor algebra
- Differentiation of tensors, connection coefficients
- Commutators, coordinate and noncoordinate bases:
**Week 3, Lecture 5** - Spacetime curvature: the Riemann and Ricci tensors
- Relativistic tidal gravity; geodesic deviation
- The Einstein field equations
- Motivation via tidal gravity
- "Derivation" of the Einstein equations; number of equations and number of unknowns; contracted Bianchi identitity
*Weak Gravitational Waves [GW's] in Flat Spacetime***[Lectures by Kip]**:**Week 4, Lecture 6**- Wave equation for Riemann tensor
- Transverse-traceless [TT] GW field; + and x polarizations
- A GW's tidal forces (relative motion of freely falling particles)
- Metric perturbations; TT gauge and other gauges
- Proper reference frame of an observer:
**Week 4, Lecture 7** - Physical measurements of GW's in a proper reference frame
- Generation of GW's: The linearized Einstein field equations

- Projecting out the TT GW field
- Slow-motion, weak-stress approximation for GW sources:
**Week 5, Lecture 8** - The quadrupole formula for GW generation
- Derivation in slow-motion, weak-stress approximation
- Validity for slow-motion sources with strong internal gravity and arbitrary stresses
*Propagation of GW's Through Curved Spacetime***[Lectures by Kip]**- Short wavelength approximation; two-lenghscale expansion
- Curved-spacetime wave equation for Riemann tensor
- Solution of wave equation via eikonal approximation (geometric optics) - Foundations
- Geometric optics - Details:
**Week 5, Lecture 9** - gravitons and their propagation; graviton conservation
- rays as graviton world lines; propagation of + and x GW fields
along rays

- + and x polarizations and fields, rays and transport of waves along rays
- gravitational focusing of GW's, e.g. by the sun; diffraction at the focus
- stress-energy tensor for GW's; nonlocalizability of GW energy
- conservation of GW energy and momentum
- conseervation of a graviton's energy and momentum
- Propagation of GW's through homogeneous matter:
**Week 6, Lecture 10** - impact of matter on the waves is always negligible
- propagation through dust, perfect fluid, viscous fluid, elastic medium
- propagation through a cloud of neutron stars
*Generation of GW's by Slow-Motion Sources in Curved Spacetime***[Lectures by Kip]**:**Week 6, Lecture 11**- Strong-field region, weak-field near zone, local wave zone, distant wave zone
- Multipolar expansions of metric perturbation in weak-field near zone and local wave zone
- influence of source's mass and angular momentum
- mass quadrupolar component of GW's; current quadrupolar component
- rates of emission of energy, linear momentum, and angular momentum
- Application to a binary star system with circular orbit
- inspiral rate and timescale
- chirp waveform; chirp mass

*Astrophysical Phenomenology of Binary-Star GW Sources*- GW's from Binary Star Systems:
**Week 7, Lecture 12****[by E. Sterl Phinney]** - GW-driven inspiral of a single binary [review]
- Inspiral evolution of a steady-sstate population of many binaries
- Types of stars: main-sequence stars, white dwarfs [WD], neutron stars [NS], black holes [BH]; their masses and radii
- Binary systems observable by LIGO (and its partners), and by LISA
- Issues relevant to estimating numbers of binary GW sources and
their merger rates

- Cosmology: parameters describing the universe as a whole
- Our Milky Way galaxy: its star-formation history, stellar populations and binary populations
- Use of blue light to extrapolate from rates in Milky Way to rates in the distant universe
- Estimates of numbers of binary GW sources and inspiral/merger
rates: preview of next lecture

- NS/NS rates based on binary-pulsar statistics and blue-light extrapolation
- Population synthesis as foundation for estimates
- Estimates of numbers of binary GW sources [for LISA] and inspiral/merger
rates [for LIGO]:
**Week 8, Lecture 13****[by E. Sterl Phinney]** - Estimates based on observed numbers in our galaxy
- pitfalls
- NS/NS; WD/WD, WD/NS
- Population synthesis
- Foundations for population synthesis:
- stellar structure and evolution
- binary evolution: mass transfers etc.
- Estimates of binary numbers for LISA
- Estimates of NS/NS, NS/BH, and BH/BH numbers for LIGO --
**Week 8, Lecture 14 - Part 1****[by Kip]**

*Binary Inspiral: Post-Newtonian Gravitational Waveforms for LIGO and Its Partners --*- Matched-filtering data analysis to detect inspiral waves
- Foundations for post-Newtonian approximations to General Relativity
- Mathematical foundations

- Physical effects at various orders
- Post-Newtonian inspiral waveforms for circular orbits and vanishing
spins --
**Week 8, Lecture 14 - Part 2****[by Alessandra Buonanno]** - Expansion parameter v = (pi M f)^1/3
- Phase evolution governed by energy balance
- Waveform in time domain
- Waveform in frequency domain, via stationary-phase approximation
- Influence of spin-orbit and spin-spin coupling: Orbital and spin precession; waveform modulation
- NS/BH binary
- BH/BH binary
- Innermost stable circular orbit (ISCO) and transition from inspiral to plunge
- The IBBH problem: failure of post-Newtonian waveforms in late inspiral; methods to deal with this:
- Pade resummation
- Effective one-body formalism
- Search templates designed to deal with uncertainties in our knowledge of the waveforms
**Supermassive Black Holes [SMBH's] and their Gravitational Waves [for LISA/ --****Week 9, Lecture 15****[by E. Sterl Phinney]**

- Astrophysical phenomenology of SMBH's in galactic nuclei
- Evidence for their existence
- Measurement of SMBH masses via cusp in stellar velocity dispersion (for masses above 10^6 Msun)
- Correlation of SMBH masses with velocity dispersion in galactic bulges
- Number of SMBH's per unit volume in universe; their distribution of masses (for masses above 10^6 Msun)
- Observed quasar and other electromagnetic emission from SMBH's; quiescence of most SMBH's
- Mergers of galaxies
- Statistics of mergers: observational data; predictions of CDM simulations
- Physics of mergers
- Dynamical friction on SMBH's, SMBH binary formation
- Evolution of SMBH binary
- Interaction with stars; loss cone
- 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
- Gravitational radiation reaction
- SMBH merger rates
- Capture and inspiral of stars by a SMBH
- Loss cone and its repopulation
- Tidal disruption of main-sequence stars
- Capture of compact stars [WD, NS, small BH] into highly elliptical orbits
- Evolution of orbital ellipticity during inspiral
- Event rate estimates for captures
- Gravitational waves from SMBH binary inspiral, as measured by
LISA --
**Week 9, Lecture 16****[by Kip]**

- Frequency evolution, signal-to-noise ratios
- Cosmological influences on waves: gravitational redshift; gravitational lensing
- Observables: redshifted masses, luminosity distance, inclination angle
- GW's from inspiral of a compact star (or BH) into a SMBH
- Frequency evolution, signal to noise ratios
- Loss of signal strength due to non-optimal signal processing - caused by complexity of inspiral orbits and resulting complexity of waveforms
- Implications for event rates
- Implications for specifying the level of LISA's noise floor
**GW's from Big Bang: Amplification of Vacuum Fluctuations by Inflation**- Basic idea: same as parametric amplification of classical waves
- Mathematical details
- Background cosmological metric
- Geometric optics propagation of GW's at "late times'
- Wave equation for GW's at all times
- Frozen and decaying solutions when wavelength is much larger than background radius of curvature
- Matching solutions together: resulting wave amplification
*GW's from Neutron-Star Rotation and Pulsation --***Week 10, Lecture 17****[by Lee Lindblom]**- GW's from a structurally deformed, rotating NS
- Deformations maintained by a solid crust
- Deformations maintained by stress of a strong internal magnetic field
- Deformations due to temperature anisotropy induced by accretion of gas onto NS [low-mass X-ray binaries; LMXB's]
- Magnitudes of deformation (ellipticities) detectable by LIGO-I and LIGO-II
- GW's from pulsations in a rotating NS
- Types of pulsational [bar-mode] instabilities: dynamical; secular
- beta = T/W as diagnostic for instabilities

- Instabilities in uniform-density Newtonian stars [Maclaurin Spheroids]
- Mechanisms for forming rapidly rotating NS's:
- Collapse of degenerate stellar cores
- Accretion-induced collapse of a white dwarf
- Spinup by accretion
- Merger of a low-mass NS/NS binary
- NS's formed by collapse: differential rotation, values of beta,bar-mode instabilities, numerical evolution of unstable stars
- realistic models
- models with extreme differential rotationg: instability at small beta
*Numerical Relativity as a Tool for Computing GW Generation --***Week 10, Lecture 18****[by Marc Scheel]**- Motivation: Sources that require numerical relativity for their analysis
- Binary black hole mergers
- Relevance to LIGO & partners, and to LISA
- Estimated event rates for LIGO-I, LIGO-II and LISA
- Inspiral, merger, and ringdown; estimated wave strengths from each
- Rich physics expected in mergers: strong, nonlinear effects; spin-spin and spin-orbit coupling; angular-momentum hangup
- Importance of simulating mergers as foundation for interpreting observations
- Tidal disruption of NS by a BH companion
- Estimated event rate for LIGO-II
- Information carried by waves: NS structure and equation of state
- Possible connection to gamma ray bursts
- Importance of simulations for interpreting observations
- Some other sources: NS/NS mergers, cosmic string vibrations, brane excitations in early universe
- The necessity to use numerical relativity in simulations of these sources
- Mathematical underpinnings of numerical relativity
- 3+1 decomposition of spacetime into space plus time
- Initial data must satisfy "constraint equations"
- Evolve via "dynamical Einstein equations"
- Gauge freedom
- Analogy with electromagnetic theory
- Mathematical details
- Spacetime slicing; lapse, shift, and 3-metric; extrinsic curvature
- Hamiltonian constraint equation
- Momentum constraint equations
- Dynamical equations
- Choices of lapse and shift
- Current state of the art in numerical relativity; current efforts on BH/BH inspiral & merger

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)