Caltech's Physics 237-2002

Gravitational Waves


Course Outline as Originally Taught

The original order of the lectures was dictated in part by the availability of the guest lecturers.  People studying this course may wish to use the more logical order shown in Part B: Alternative Course Outline

  1. The Physics Underlying Earth-Based GW Interferometers - Week 11, Lecture 19 [by Kip]
    1. Idealized Interferometer: Conceptual design and crude analysis
      1. Encoding GW signal in phase shift of light
      2. Increasing signal strength via bounces in arms
      3. Limit on accuracy of phase measurement
      4. Required laser power; energetic quantum limit
      5. Power recycling
    2. General relativity: Proper reference frame of an accelerated observer
      1. Foundation for analyzing earth-based interferometers
      2. GW acts solely via its tidal force on test masses; negligible coupling to light
      3. TT gauge as an alternative: GW couples solely to light and not at all to test masses
    3. Optics
      1. Gaussian beams; their mathematical description
        1. Gaussian cross section and its evolutionary spreading
        2. Circular phase fronts and their evolution
        3. Eigenfunctions of optical cavity with spherical mirrors
      2. Paraxial Optics - Week 11, Lecture 20 [by Kip]
        1. Paraxial propagator and its use
        2. Application to derive evolution of a Gaussian beam
        3. Eigenmodes of an optical cavity with spherical mirrors
          1. resonances as function of mirror separation; free spectral range
          2. mode matching of Gaussian beam into optical cavity
        4. Mirrors: reflection and transmission coefficients, losses
        5. Properties of optical cavities: finesse, mode cleaning, phase shift as function of mirror separations
    4. Statistical Physics: The theory of random processes
      1. Random process; examples
      2. Fourier transforms, Parcival's theorem
      3. Spectral density; variance
      4. Filtering of random processes; influence on spectral density
      5. Shot noise in light; its spectral density
  2. Overview of Real LIGO Interferometers - Week 12, Lecture 21 [by Alan Weinstein]
    1. Overview of noise sources & how they are controlled
    2. Optics
      1. Fabry-Perot cavity theory; response of reflected light to change of cavity length
      2. Analysis of complicated, linear optical systems; response to mirror motions; Twiddle
      3. Coupling of light into arm cavities: carrier resonates; side bands do not
      4. Properties of cavities: finesse, storage time, pole frequency, gain, visibility, circulating field
      5. Power recycling
      6. Control of arm cavity lengths via Pound-Drever-Hall [PDH] reflection locking 
        1. Phase modulation of input beam 
        2. Demodulation; lock acquisition
      7. Schnupp Asymmetry and Schnupp locking to control the difference in distances from beam splitter to arm-cavity input mirrors (Michelson interferometer)
      8. Hermite-Gaussian modes of arm cavity; their excitation by beam and mirror imperfections and tilts  
      9. Input optics for controlling input beam
        1. Mode cleaner; nested cavities to clean beam
        2. Mode matching telescope
      10. Optimizing the reflectivity of an arm cavity's input test mass [ITM]
    3. Suspensions for mirrors and other optical elements
      1. Pendulum dynamics; filtering seismic noise via pendula
      2. LIGO-I suspension system
      3. Pushing on mirrors with magnetic forces ("actuation")
      4. suspension control system
      5. Summary of the control problem: 4 lengths, ten mirror angles
  3. Thermal Noise in LIGO Interferometers and its Control - Week 12, Lecture 22 [by Phil Willems]
    1. Motivation: Brownian motion of a dust grain buffeted by molecules of an ideal gas
      1. dissipation, mean motion
      2. Fluctuating force as a random process; its correlation function and spectral density
      3. Solving for spectral density of particle position
    2. Fluctuation-dissipation theorem
    3. Damped pendulum: suspension thermal noise derived from fluctuation-dissipation theorem
    4. Dissipation in a LIGO test mass or suspension described via imaginary part of generalized elastic modulus, E(f) = (applied force) / (resulting displacement) = Eo (1+i phi)
      1. Frequency-dependence of loss angle phi: viscous damping, structural damping, damping associated with an internal relaxation process
    5. Dissipation/fluctuation processes for a LIGO test mass
      1. Gas molecules buffeting test mass
      2. Magnetic forces from actuator (which controls mirror)
      3. Internal processes inside the test mass itself:
        1. Analyzed via sum over normal modes of test-mass oscillation [valid only for homogeneous dissipation]
        2. Analyzed via Levin's Direct Method [valid in general]
        3. Conventional internal dissipation (due to imperfections, ...)
        4. Thermoelastic noise
        5. Fused silica vs sapphire for Advanced LIGO (LIGO-II) test masses
        6. Measurements of dissipation
      4. Dissipation in mirror coatings 
      5. Dissipation in suspension wires
  4. Control Systems and Laser Frequency Stabilization - Week 13, Lecture 23  [by Erik Black]
    1. Introduction
      1. What a control system is
      2. Uses of control systems
      3. Simple control system (input, amplifier K, feedback, and output); its oscillatory instability due to time delay
    2. General, linear control theory
      1. Laplace transforms
      2. Transfer function (Kernel) for a linear system, in time domain and in (Laplace-transform) s-space 
      3. Poles of the transfer function in s-space; their relationship to system's stability
      4. Transfer function for simple control system with s-dependent amplifier, K(s)
        1. Open-loop transfer function K(s); closed-loop transfer function K/(1+K)
        2. Nyquist diagram for analyzing stability
        3. Gain margin, phase margin
        4. Bode plot for analyzing stability; stability diagnosed via phase at unity gain point (phase margin)
        5. Bode's gain-phase relations
    3. Laser frequency stabilization via locking to eigenmode of an optical cavity (Pound-Drever-Hall [PDH] locking): an example of linear control theory
      1. Laser frequency adjusted via PZT attached to mirror of laser cavity
      2. Stable Fabry Perot cavity to which laser frequency is locked
      3. Frequency-modulated laser light reflected off locking cavity, demodulated and fed back to laser
      4. Analysis of stability of this PDH  feedback system
        1. Influence of locking cavity's storage time (time delay)
      5. Spectral density of frequency fluctuations for PDH-stabilized laser; magnitude of stabilization
  5. Interferometer Simulations and Lock Acquisition in LIGO - Week 13, Lecture 24, Part 1  [by Matt Evans]
    1. Simulations of all or part of a LIGO interferometer
      1. What a simulation is
      2. Types of simulations:
        1. Frequency domain: fast, but limited to linear systems
        2. Time domain: slower, but necessary for nonlinearities
      3. Example of a simulation:  Control system for a Fabry Perot cavity:
        1. Laser excites Fabry Perot cavity; returning light tapped off by Faraday isolator, detected to produce electronic signal which drives a magnetic actuator that adjusts a cavity mirror to lock the cavity to the laser.
        2. Simulation of the optics, the electronics, the mirror's mechanics, and the electromechanical transducers
        3. Linear parts of system treated via transfer functions
      4. In complex system such as LIGO: subsystems (e.g. the above) treated as modules
      5. Uses of simulations:
        1. Quantify things that can't be measured experimentally
        2. Selectively turn on and off noise sources
      6. LIGO end-to-end (E2E) simulation system
        1. Used to develop and implement lock-acquisition method for LIGO-I
        2. Being prepared for detailed noise tracking in LIGO-I
    2. Lock acquisition in LIGO-I
      1. What is lock acquisition?
      2. Locking a single Fabry Perot cavity
        1. Pound-Drever-Hall (PDH) error signal ("demod signal")
        2. lock acquisition contrasted with maintaining lock once acquired: nonlinear vs. linear
        3. Acquisition error signal = (demod signal)/(cavity power) - linear over length changes ~ wavelength
        4. Control (actuation) force to lock
      3. Locking a LIGO-I interferometer
        1. Four degrees of freedom must be locked using five error signals from three readout ports
        2. 5 x 4 dimensional sensing matrix (degrees of freedom -> error signals)
          1. Invertible in pieces (largest 2x2 piece, then 3x3, then 4x4) -> lock acquisition in stages
        3. 5 states of interferometer, from totally unlocked through partial locks to totally locked
        4. Examples of evolution through the 5 states: experimental data compared with simulations
  6. Seismic Isolation in Earth-Based Interferometers - Week 13, Lecture 24, Part 2  [by Riccardo De Salvo]
    1. Seismic attenuation requirements
    2. Principals of seismic attenuation
      1. Pendulum or oscillator as an example; its transfer function
      2. Chain of oscillators; net transfer function
    3. The Virgo isolation system as an example
    4. The need for seismic attenuation in all six degrees of freedom:
      1. All feed into horizontal noise that interferometer measures
      2. How to achieve such attenuation
    5. Vertical attenuation: the most serious problem
      1. A solution: cantilever blades, radially compressed
        1. Their transfer function
        2. Example in Virgo
    6. Creep in stressed elements of isolation system
      1. Mechanism of creep
      2. Reduction of creep with time after stress was applied
      3. How to control creep: special materials; freezing dislocations; glassy materials in final attentuation stages
    7. Mechanical resonances in isolation system
      1. Must damp them because of  interferometer's limited dynamic range
      2. Damping techniques: inertial, viscous; active, passive
  7. Large Experimental Science, with LIGO as an Example - Week 14, Lecture 25  [by Barry Barish]
    1. Introduction and Overview: Small science contrasted with large science, in the US:  
      1. Single-investigator mode of small science: 
        1. nature of labs, experiments, financial support
        2. peer review; no direct accountability for research accomplishments
        3. flexibility; effectiveness in promoting new ideas
      2. Large projects  
        1. great differences between those funded by NSF, NASA, DoE, and private sources
        2. Non-private: contrast of projects embedded in large national labs, vs. LIGO;  accountability; peer review for science, management, resources, etc; performance metrics -- threats to extend them to small science
        3. Strategic planning
    2. How to create an effective research environment in a large science project; how to maintain flexibility, with experiment driven by science and ideas.  Different approaches:
      1. NASA: science teams separated from Project (a little less so for LISA than for traditional NASA missions); open data
      2. DOE: umbrella grants to enable scientists to function more nearly as in small science; internal guidelines & reviews within collaborations
      3. Private, e.g. Keck telescopes: CARA Board; less peer review than in non-private sector
      4. LIGO (NSF): in operations mode will evolve into standard peer review structure; see below
    3. Long-range (decades-long) strategic planning:
      1. NASA: top down, from high-level planning committees
      2. Astronomy: decadal review by NAS planel; prioritization of projects; problem of cross-disciplinary projects
      3. Particle physics: combination of National Lab planning + "Road Map"; see below
      4. LIGO: Its orgins were very different from above -- grew out of small science; entrepreneurial
    4. Strategic (long-range) planning in high-energy physics: HEPAP Road Map for next 20 years [recent panel cochaired by Barish]
      1. Issues addressed
      2. Roadmap concept: identify all possible routes toward field's science goals; build decision points at branches
        1. identify possible projects and their science and timelines
        2. not all can be done; identify decision points; decisions to be made by scientists (not bureaucrats or politicians)
        3. Develop funding scenarios for various sets of downstream decisions
    5. LIGO organization and construction: 
      1. Construction phase (1994-2000): vertical organization: tasks, budgets, deliverables, schedules; integration.  Guided by scientists at all levels of organization.
      2. Evolution to an operating research environment (2000 -  ): flat organization - separate groups by task; advanced LIGO (LIGO-II) as a task; LSC broadens participation from Caltech&MIT to many institutions; open data within LSC but not to external world.
      3. LIGO Construction: schedule, milestones - planned and actual dates; costs, commitments, funding vs time;  contingency and its evolution; staffing vs time
    6. LIGO status
      1. Hanford & Livingston sites
      2. GW Coincidences between 3 interferometers at 2 sites
      3. Broad-brush schedule: interferometer construction, commissioning, sensitivity studies & debugging, LIGO-I data run, LIGO-II installation
      4. LIGO beam tube: structure, cover, vacuum achieved; outgasing, bakeout vacuum
      5. LIGO noise sources and their noise curves, from modeling
      6. LIGO test masses; lasers; locking; laser stabilization
      7. Sensitivity in January 2002
      8. Schedule and plans for next several years
  8. Quantum Optical Noise in Advanced LIGO Interferometers - Week 14, Lecture 26  [by Alessandra Buonano and Yanbei Chen]
    1. Introduction: review of interferometers and their sensitivities; references on quantum optical noise; the experimental challenge: prevent quantum properties of detector and light (the "probe") from affecting the GW information we seek
    2. Quantum optical noise in conventional interferometers (LIGO-I, TAMA, VIRGO)
      1. vacuum fluctuations from dark port produce shot noise and radiation pressure fluctuations
      2. Two-photon formalism for analyzing these noises
      3. Application of this formalism to one arm cavity of the interferometer: shot noise; radiation-pressure noise
      4. Input-output relations for the full interferometer [input is vacuum fluctuation at dark port and GW force on mirrors; output is GW signal plus noise]
      5. Spectral density of quantum optical noise (shot and radiation pressure noise) deduced from input-output relations
    3. Free-mass standard quantum limit [SQL] (for conventional interferometers)
      1. Deduced from variation of quantum optical noise with laser power
      2. Key issue: absence of shot/radiation-pressure correlations; correlations could invalidate the limit
      3. Similarity to Heisenberg microscope
    4. Ways to beat the SQL
      1. In conventional interferometer:  measure a different quadrature of output light, one which posseses shot/radiation-pressure correlations
      2. Change the test-mass dynamics: via a signal-recycling mirror (LIGO-II) or "optical-bar" configuration
    5. Quantum optical noise in signal-recycled interferometers (LIGO-II)
      1. Shot/radiation-pressure correlations
      2. "Optical-spring" test-mass dynamics
      3. Optical-mechanical instability; control system to overcome it
      4. Effects of optical losses
    6. Other noise sources and total noise in LIGO-II; the severity of thermoelastic noise
      1. Lowering thermoelastic noise by flattening the light beams
    7. Beyond LIGO-II: How to improve the sensitivity further without radical changes of interferometer's optical topology:
      1. At low frequencies: reduce thermal noise via cryogenic cooling of test masses; reduce radiation pressure noise via larger test masses, lower optical power; seismic noise and seismic gravity-gradient noise
      2. At high frequencies: reduce coating and substrate absorption so arm-cavity light power can be increased; narrow-band the noise curve
    8. Beyond LIGO-II: New optical topologies
      1. Speed-meter interferometers
      2. Intracavity readout designs
  9. Doppler Tracking of Spacecraft for GW Detection in the LF Band - Week 15, Lecture 27, Part 1 [by John Armstrong (JPL)]
    1. The doppler-tracking method of GW detection
      1. Jargon and references
      2. 3-pulse response of doppler signal to a gravitational wave
      3. Principal noise sources and their control
        1. Multiple pulse charcateristics of noise from various sources
        2. Clock jitter [instabilities of frequency standard]
        3. Plasma scintillation [fluctuations in dispersion of doppler signal in interplanetary plasma]
        4. Tropospheric scintillation (due to fluctuations in water vapor  causing index of refraction to fluctuate); water vapor radiometers to remove scintillation from data
        5. Mechanical vibrations in the tracking radio telescope
    2. Doppler-tracking observations to date: about 160 hours total from 1980 through 1997 on 8 spacecraft, including one three-spacecraft experiment
    3. Data analysis for various types of signals
      1. Some details for bursts, chirps, sinusoids
      2. Some other techniques tried that might be useful in other GW experiments: wavelets, Karhunen-Loeve expansion, bispectral analysis, multi-taper spectral analysis
    4. Cassini: the current-generation observatory
      1. Launch, orbit, observation windows (when spacecraft is downwind from the sun)
      2. The GW experiment on Casini 
        1. KA-band translator for 2-way coherent signal
        2. First observations - Nov 26 2001 - 4 January 2002; quick-look data; removal of plasma scintillation via multi-frequency data; other noises
        3. Expected sensitivities as function of location on sky and GW frequency
        4. Net sensivity (rms noise) for GW bursts, stochastic background, periodic GW's 
    5. Beyond Cassini: 
      1. Main obstacles to improvement 
      2. Conclusion: perhaps 10-fold improvement, but at very high cost.
  10. Pulsar Timing for GW Detection in the VLF Band - Week 15, Lecture 27, Part 2  [by Kip]
    1. Introduction: comparison of wave bands and detection sensitivities; 
      1. Energy density ~ (h f)^2 so at lower frequencies f, expect signals to be stronger
      2. Current sensivity of pulsar timing in VLF band) compared with those of doppler tracking and LISA (LF band) and earth-based interferometers (HF band) 
    2. The basic principles of pulsar-timing searches for GW's
      1. The signal: pulse arrival times -- actual compared to predicted if no GW's
      2. The influence of GW's on pulse arrival times
      3. GW sensitivity as function of "residuals" (noise) in pulse arrival times
    3. Best past sensitivities
    4. Most promising source:  Stochastic background from superposition of waves from many supermassive black hole binaries, with masses ~ 10^9 Msun.  
      1. Estimated wave strength: Omega ~ 10^-11, nearly independent of frequency
      2. Prospects for reaching this level: good, if moderate resources are put into the effort.
      3. Problem of very few bits of information in VLF band.
  11. LIGO Data Analysis - Week 15, Lecture 28  [by Albert Lazzarini]
    1. The context: LIGO-I noise curve and anticipated signal strengths
    2. LIGO data attributes
      1. Data channels: GW signal (32 kB/sec) plus many auxiliary  channels (~1 MB/sec) that monitor the state and environment of interferometer
      2. Data format: common to all interferometer projects
      3. Uses of auxiliary-channel data: reduce noise in GW channel; monitor instrument behavior
      4. The data from January 2002 observations: noise spectra; expected improvements in near future
    3. Some signal processing theory and methods
      1. Theory of random processes: brief summary [see also Week 11, Lecture 20]
      2. Fast Fourier transforms; 90% of LIGO cpu computational time is here; their computational cost; capabilities of arrays of Pentium processors
      3. Pre-processing data to remove ugly instrumental effects
      4. Time-frequency methods: general theory; time-frequency spectrograms; time-frequency characteristics of various types of GW's (stochastic, periodic, ringdown, bursts, chirps)
      5. stacking Fourier transforms vs fully coherent transform
      6. Optimal filters in general; brief overviews of applications to inspiral of compact binaries; stochastic background waves (one detector output serves as filter for other); spinning neutron stars; GW bursts
    4. Optimal filtering for parametrizable waveforms
      1. General theory; derivation of the optimal filter
      2. Wave detection contrasted with parameter extraction
      3. Binary inspiral: matched filtering with a family of templates
        1. intrinsic vs extrinsic parameters
        2. 2-parameter template family when spins are negected
        3. data analysis flow
        4. tests in last January's LIGO-I data 
        5. setting event rate limits with 1994 LIGO prototype data
    5. Stochastic background searches
      1. General method: cross correlation of outputs of two detectors; buildup of signal to noise with integration time
      2. Optimal filter when searching for background with known spectrum using detectors whose noise is correlated; effect of correlations on measured upper limits
    6. Hypothesis testing: maximum likelihood; Baysean statistics; false alarm probability compared with detection probability
    7. Searching for (transient) bursts of GW's
      1. General theory of search strategies
      2. Excess power statistic (especially useful when have limited knowledge of waveforms, e.g. today for BH/BH mergers)
    8. Analysis of data from a network of detectors
      1. LIGO network; international network
      2. Coincidence analysis: rejection of uncorrelated random events
      3. Event localization on the sky
      4. Joint data analysis: validation of detections
  12. The Long-Term Future of LIGO: Facility Limits, and Techniques for Improving on LIGO-II
    1. Facilities Limits (limits on sensitivity due to the LIGO environment, vacuum system, ...) -  Week 16, Lecture 29, Part 1  [by Kip]
      1. Overview
      2. Noise due to scattering of light in the LIGO beam tube
        1. Noise mechanism
        2. Baffles to reduce the noise
        3. Random teeth on the baffles: reduce the noise and destroy coherent superposition of noise via different scattering routes
        4. Net scattering noise: from backscatter off baffles' surfaces and diffration off baffles' teeth
      3. Noise due to fluctuating dispersion of light beam in vacuum system's residual gas
        1. Noise mechanism
        2. Magnitude of noise as function of vacuum pressure
      4. Seismic gravitational noise (due to fluctuating gravitational pulls of density inhomogeneities caused by ambient seismic waves)
        1. Noise mechanism
        2. Modeling of the seismic waves and their noise
        3. Magnitude of noise and uncertainties
      5. Human gravitational noise (mostly due to jerkiness of human walking)
        1. Noise mechanism
        2. Magnitude of noise as function of distance from humans to test masses
      6. Comparison of facilities limits with LIGO-II sensitivities
    2. Techniques for Improving on LIGO-II - Week 16, Lecture 29, Part 2  [by Ronald W.P. Drever]
      1. Beating the Standard Quantum Limit (shot noise & radiation pressure noise):  See last part of Week 14, Lecture 26 by Chen
      2. Reducing seismic noise: "straightforward" but not easy
      3. Reducing suspension thermal noise: Replace fibers by ribbons (planned for LIGO-II)
      4. Reducing internal thermal noise (the toughest problem): Cryogenically cool the test masses
        1. Japanese plans for LCGT (Large-scale Cryogenic Gravitational-wave Telescope); Japanese R&D
        2. Problem of heating the test mass by laser beam; bleading off the heat
        3. How cooling helps: reduced rms thermal motion; higher mechanical Q so reduced thermal fluctuations
      5. Reduce mirror heating in presence of high optical power (so power can be higher): Use diffractive optics so light beam does not pass through the mirror  and beam-splitter substrates
        1. Example of mode cleaning cavity with diffractive optics
        2. Example of diffractive beam splitter
        3. Examples of fully diffractive interferometers
      6. Magnetic levitation to reduce suspension noise; recent experiments in Drever's lab
      7. Alternative optical topologies
        1. Herriott delay line
        2. GEO600 topology
        3. Sagnac topology (being developed at Stanford)
  13. Resonant-Mass ("Bar") GW Detectors for the HF Band - Week 16, Lecture 30 [by William O. Hamilton (LSU)]
    1. Historical remarks; Joseph Weber's pioneering contributions; others' contributions
    2. Basic elements of a resonant-mass detector, and how it works
      1. Vacuum chamber and cryostat
      2. Seismic isolation system
      3. Bar -- fundamental end-to-end mode excited by GW
      4. Small mechanical oscillator attached to end of bar to amplify bar's mechanical motion
      5. Mechanical-electrical transducers to convert oscillator's motion into electrical signal
        1. general discussion of transducers
        2. parametric transducer: basic principle; analogy with a child pumping a swing
        3. the superconducting inductive transducer used in the LSU resonant-mass detector "Allegro"; squid amplifier and its noise
        4. back-action noise on the bar's normal mode
      6. Thermal noise in bar
    3. The full mechanical-electrical system for the LSU detector Allegro
      1. Equations of motion for system with noise sources
      2. Measured noise at transducer output; two noise peaks due to the two coupled mechanical resonances 
      3. Calibrating the detector by applying mechanical noise to the bar using a capacitive electomechanical transducer
      4. Resulting GW noise curve
      5. Comparison with predictions of equations of motion: good agreement; dominant noises at resonances -- squid current noise and transducer's brownian (thermal) noise
    4. Experience with Allegro
    5. Prospects to search for a stochastic background using Allegro and the Livingston LIGO interferometer
    6. TIGA and Spherical Bars: Looking toward the future
      1. Isotropic sensitivity
      2. Disentangling motions of five quadrupole modes using six transducers
    7. IGEC: The international network of bar detectors - Week 16, CaJAGWR Seminar [by William O. Hamilton (LSU)]
      1. Data collection record since 1997
      2. Network's upper limits on Fourier transform of GW field, h(f) at resonant frequency, during 1998, as a function of time
      3. Upper limits on GW bursts during 1997 - 2000
    8. Some results from the LSU detector Allegro
      1. Noise as a function of time, and noise curve
      2. Search for periodic waves (e.g. pulsars)
    9. Prospects for future improvements:
      1. Cool to lower temperatures - Auriga performance
      2. Improve SQUID amplifiers - Trento/Lignaro work
      3. Improved transducer with tighter coupling to resonant mass: broadening the frequency band of high sensitivity (in process this summer at LSU in collaboration with U. Maryland)
    10. Identifying a GW burst amidst noise: an audio analogy
    11. Spherical detectors: current status and plans -- in Italy, Netherlands and Brazil; projected sensitivity compared with Advanced LIGO (LIGO-II)
  14. LISA (Laser Interferometer Space Antenna) for GW Detection in LF Band: Conceptual Design -  Week 17, Lecture 31  [by William Folkner (JPL)]
    1. The context: Noise curves and GW sources for LISA and for LIGO; white-dwarf  / white-dwarf background noise for LISA.
    2. History of ideas for a LISA type GW detector: 1978 - 1998; motivations for changes of conceptual design as time passed
    3. Noise estimates for current LISA design
      1. The noise curve, in detail
      2. Shot noise and what determines it
      3. Influence of arm length
    4. Spacecraft formation and orbits; influence of time-varying arm lengths:
      1. Time-varying separation between spacecraft; time-varying doppler shift
      2. Local frequency standard to deal with varying doppler shifts; noise in frequency standard
      3. Pointing changes to deal with spacecraft motions; pointing noise
      4. An alternative spacecraft formation that has been explored: triangle orbiting earth rather than sun; comparison with LISA's design
      5. Variation of antenna pattern with time modulates source amplitudes; gives information about directions to sources
      6. Cancelling laser phase noise by combining signals from arms, with time delays based on estimated arm lengths
      7. How errors in arm-length knowledge degrade this phase-noise cancellation
    5. Overview of spacecraft and launch
      1. An individual spacecraft: science module [lasers, telescopes, proof masses]; thermal shields; radio antennas; propulsion module
      2. Launch vehicle; launch configuration
    6. Payload [science module] on each spacecraft
      1. Telescopes and their pointing
      2. Drag-free system; its proof masses; accelerations
      3. Optical system; optical bench; telescope detail
    7. Thermal and laser noise
      1. Thermal stability: solar luminosity fluctuations; thermal stabilization; expected thermal fluctuations and their affects
      2. Laser frequency noise; factors that influence it
    8. Disturbance-Reduction System [DRS] (Drag-free system)
      1. Proof masses and sensors for their motions
        1. Heritage from previous missions: TRIAD, GP-B, GRACE, CHAMP, ...
        2. Proof-mass shape: sphere vs cube; choice of cube
        3. Capacitive sensor configuration
      2. Acceleration noise of proof masss; various contributions: spacecraft gravity, patch fields on proof masses and capacitive sensors, magnetic forces, gas-pressure, thermal photon pressure, ...
      3. Ground tests with torsion-pendulum facilities
      4. Control system
      5. Thrusters and their performance
    9. LISA test flight
  15. LISA's Lasers and Optics - Week 17, Lecture 32  [by Robert Spero (JPL)]
    1. Introduction: Comparison and contrast of LISA and LIGO
    2. LISA's light beams: 
      1. parameters; spreading (far-field limit), 
      2. why must receive, photodetect and transmit new beam back ("transpond" the light) rather than reflecting off a mirror
    3. Detection of incoming beam: 
      1. shot noise prevents simple photodetection 
      2. reduce shot noise by beating incoming beam against local oscillator light
      3. modulation & demodulation of local oscillator light to reduce noise
      4. possible designs for transponding system: DC lock, frequency offset lock, and offset-cancelled lock (current preference)
    4. Three-spacecraft phase-monitoring system (current baseline design):
      1. 1 master laser, three slave lasers, 4 phase measurements; 3 semi-independent 2-arm interferometers
      2. Time-delay interferometry [TDI] as an attractive alternative
    5. Laser frequency noise and its control
      1. Analysis when GW wavelength is long compared to spacecraft separation [for pedagogical simplicity]; suppression of laser noise by near equality of arm lengths
      2. Problem of influence of round trip time delay on laser frequency control
      3. Laboratory experiments on laser frequency stability
    6. Time-delay interferometry [TDI] as a way to remove laser frequency noise
      1. TDI as a transponder-free scheme: all lasers are free running
      2. Phase-meter for monitoring phase difference between incoming beam and local laser
      3. Combine phase differences with appropriate time delays to cancel laser frequency noise
      4. Uncertainty in (time-varying) arm lengths produces error in cancellation; demonstration that 30 meter accuracy in arm-length knowledge is adequate
      5. Details of how phase meter works
      6. Measurement of arm lengths to 30 meter accuracy
    7. Noise due to fluctuations in pointing of laser beams
  16. LISA's Distrubance Reduction System [DRS] (Drag-Free System) - Week 18, Lecture 33  [by Bonny Schumaker (JPL)]
    1. Review of LISA: concept, orbit, spacecraft, optics, baseline parameters that affect the DRS
    2. Requirements and general approach:
      1. Requirements on proof mass: nongravitational accelerations; centering in housing; alignment with measuring optics
      2. How these requirements arise from the science we want LISA to do, plus practical issues
      3. Acceleration requirement compared to achievements on past space missions and earth-based experiments
      4. LISA's DRS contrasted with accelerometers
    3. The DRS control system (system to control proof-mass and spacecraft degrees of freedom)
      1. Basic design
      2. Mathematical model
      3. Solution of model to get disturbance matrix: How various disturbance sources influence proof-mass acceleration, spacecraft acceleration, and effective acceleration of proof-mass / spacecraft gap
    4. Disturbance sources; their magnitudes; implications for DRS design and control-system parameters 
      1. Spacecraft external disturbances: predominantly fluctuations of solar radiation pressure, and thruster noise
      2. Direct proof-mass disturbances: magnetic forces, cosmic rays, residual gas, laser photons, radiometric force, thermal radiation pressure; noise forces in proof-mass readout & actuation system
        1. Most serious issue, for baseline design: noise in the capacitive readout & actuation system that has been chosen as the baseline design
      3. Proof mass - spacecraft coupling forces: gravity gradients, coulomb image charges, ...
        1. Most serious issues: again associated with capacitive readout & actuation system
      4. Implications and summary
    5. Capacitive readout & actuation systems: Heritage and ground demonstrations to date; importance of tests on the ground as well as in space; torsion-pendulum facility for ground tests
    6. Baseline design of DRS system and alternative options
      1. The baseline design
      2. Thruster configuration and requirements
      3. Spherical proof mass as an alternative to cubes
      4. Optical readout system as an alternative to capacitive
      5. Gravitational actuation of proof mass as an alternative to capactive (electrostatic) actuation
    7. Summary
  17. Time-Delay Interferometry [TDI] for LISA - Week 18, Lecture 34  [by John Armstrong (JPL)]
    1. The context: 
      1. Review of LISA; its main noise sources and their magnitudes
      2. Why conventional Micheson-interferometer method of cancelling laser frequency noise will not work for LISA: large, time-varying difference in arm lengths
    2. Basic idea of TDI
      1. View unequal-arm LISA as symmetric system of 12 one-way links
      2. From 12 data channels with appropriate time delays based on estimates of arm lengths, construct TDI observables which cancel the leading noises while keeping GW signals
    3. Details of TDI
      1. The nature of each data channel: fractional frequency shift of incoming laser light compared to local laser
      2. Noises on each channel: laser phase noise, shot noise, proof-mass acceleration noise, noise in metrology data
      3. Noise-cancelling combinations of time-delayed channel signals
        1. GW-carrying combinations
        2. Sagnac combination 
    4. Computation of LISA sensitivity to periodic waves -- sensitivity averaged over sky and over GW polarizations
      1. Computation is done for each GW-carrying, noise-cancelling combination of data channels, using Monte Carlo sampling of sky directions and polarizations
      2. Resulting sensitivity curves for the various GW combinations
      3. Dependence of sensitivity on arm length
      4. How sensitivity curves change if spacecraft triangle shape is changed
    5. Uses of TDI:
      1. On-orbit calibration of instrumental noise
      2. Separation of GW background from instrumental noises
    6. Practical problems due to: 
      1. Frequency offsets of lasers with respect to each other
      2. Spacecraft relative motion
      3. Noise in oscillators used for downconverting photodetector fringe rates, ...
      4. How to deal with these problems
    7. Summary
  18. The Big-Bang Observatory [BBO]:  A Possible Follow-On Mission to LISA - Week 19, Lecture 35, Part 1  [by William M. Folkner (JPL)]
    1. Scientific goal for a post-LISA mission: detect and study waves from inflation and other processes in the very early universe
      1. Sensitivity goal: reach one or two orders of magnitude below predicted GW's from  standard slow-roll inflation
      2. Frequency window where foreground sources can be removed and inflationary waves are strongest: between LIGO and LISA -- f ~ 0.1 Hz => arm lengths 100 times shorter than LISA
      3. Possible noise curve for BBO; digging into the noise by cross correlating outputs of detectors (as is planned for LIGO's stochastic GW searches)
    2. BBO conceptual design
      1. Spacecraft configuration and orbits: 
        1. two LISA-type triangles, in star-of-David configuration; to be cross correlated for stochastic GW search
        2. two other LISA-type triangles, 120 degrees apart in orbit around sun; cross correlate outputs to triangulate on foreground sources and remove them; detect and remove every NS/NS, NS/BH and BH/BH merger in universe,with masses below ~ 10^4 Msun
      2. Measurement system requirements: acceleration noise 1/10 of LISA; optical noise 1/1000 of LISA
      3. Parameters to achieve this: 
        1. laser power: 100 W
        2. telescope diameter: 3 m
        3. laser stability; telescope optics; ...
      4. How noises scale with parameters
      5. Discussion
  19. GW's from Inflation and GW Detection in ELF Band via Anisotropy of CMB Polarization - Week 19, Lecture 35, Part 2  [by Marc Kamionkowski]
    1. The Cosmic Microwave Background [CMB]
      1. Its nature and physical origin
      2. Surface of last scattering; size of causally connected regions
      3. Why so isotropic? only good explanation: inflation
    2. Inflation: basic ideas
      1. Inflaton scalar field and its potential; slow roll; evolution of its vacuum energy density; influence on universal expansion: inflation
      2. Evolution of expansion factor of universe: pre-inflation, inflation, radiation-dominance, matter dominance
      3. Smoothing of universe during inflation; explanation of observed isotropy of CMB
      4. Inflation also predicts universe is spatially flat -- as has now been confirmed observationally
    3. GW production by inflation: 
      1. Explanation as analog of Hawking radiation from a black hole
      2. Derivation as inflation's parametric amplification of vacuum fluctuations [see also Week 9, Lecture 16]
      3. Predicted rms h: proportional to square of energy scale of inflation divided by square of Planck mass => If we can measure h, can infer energy scale of inflation
      4. Predicted spectrum; comparison with LISA and LIGO sensitivities; main hope to detect is by influence on CMB in ELF band
    4. Influence of inflationary GW's on CMB
      1. Anisotropy of temperature:
        1. limit on h and on energy scale of inflation from observed temperature anisotropy; comparison with energy scales for GUT and other possible causes of inflation
        2. Temperature anisotropy is also produced by density fluctuations; cannot cleanly separate influence of density fluctuations from GWs
      2. Anisotropy of polarization:
        1. GW's produce anisotropy in EM radiation at epoch of last scattering
        2. This anisotropy of EM intensity causes scattered radiation to be polarized
        3. Density perturbations also produce polarization
        4. GW-induced polarization is distinguishable from density-induced polarization via polarization pattern: GW pattern has nonvanishing curl
    5. Prospects to detect CMB polarization and its nonvanishing curl, and thereby measure energy scale of inflation
      1. MAP, Planck, and post-Planck CMB missions
      2. post-Planck could reach inflation energy scale 2 x 10^15 GeV (1/15 current limit)
      3. Constraint on sensitivity: density-induced polarization has a tiny but finite curl due to weak gravitational lensing, which mimics GW-induced polarization

Links to this course's other web pages:

    Course home page
    Course description
    Course Outlines
        Part A: Gravitational-Wave Theory and Sources
        Part B: Alternative Course Outline in a more logical order than the above
    Course Materials (videos of lectures, reading, homework, solutions)