1. Gravitational Wave Detection Overview of Why and How
Dan Burbank and Tony Young
AST5022

2. Introduction Background Physics Sources
Detectors and Detector Implications
Questions
The energy density of gravitational waves is complicated but expressions exist to describe the energy in GW in the short wavelength approximation

3. Gravitational Waves
Speed-of-light wave propagation solution of Einstein’s Field Equations
In general, accelerating mass results in rippled spacetime. GW is the propagation of these ripples.
Weakness of gravitational interaction compared to other forces means GW are hard to detect, small amplitudes.
Carry energy
The energy density of gravitational waves is complicated but expressions exist to describe the energy in GW in the short wavelength approximation

4. Gravity: Intro into GR, Hartle. 2003.
Background Physics
Simplest example is a linearized plane wave
Metric is near flat and can be written as , SR plus small addition (considering 1-D propagation)
Define
where
Define spacetime element
-Metric can be thought of a small perturbation from a flat metric, where the amplitudes h are small perturbations to the flat space metric.
These metric perturbations describe the gravitational wave.
Consider a 1-d propagation in the z direction for defined h and f.
The spacetime element represents a wave of curvature propagating in the positive z direction with the speed of light (c=1).
The size and shape of the propagating ripple in curvature are determined by the function f.
Important to note that the metric displayed by h does not solve Einstein equation exactly, just it solved the equation expanded to first order (meaning linearized) in amplitude.
When amplitudes are very small it is an excellent approximation to the actual solutions (useful for detection since expected amplitudes are thought to be around 10^-21).
Gravity: Intro into GR, Hartle

5. Background Physics
Measuring a change in distance between two test masses in a plane orthogonal to the direction of propagation
Consider a wave traveling in z-direction and two test masses, one at the origin and on the x axis a coordinate distance L* (in the unperturbed flat spacetime).
As the GW passes the length will change as a function of time given by
as h is very small
Simplified this gives the change in L* as
Suppose then
and the fractional change in distance along the x-axis oscillates periodically with half the amplitude of the gravitational wave

6. Background Physics
Given this linearized gravitational wave propagating in the z direction
Leads to no change in separation between two test masses lying along the z-axis
Only x-y separations change as the gravitational wave travels by.
There is stretching of the x and y distances with different phases from the given metric giving the + polarization but a different metric could have been written as
The most general metric is the superposition
Giving the two transverse polarizations + and x (as mentioned in class).
This phase difference gives us the stretching circle we saw in class

7. Background Physics Power radiated in 2 body system.
Through some crazy math done in 1963/1964 you can show that the change in distance between two orbiting bodies is
The distance between the objects changes because the system is emitting gravitational waves that carry energy and angular momentum.
Given the relationship above, you can calculate the lifetime of an orbit of two bodies (assumes circular orbits)
where
As energy is radiated away the orbital characteristics change
In general, as the system radiated energy away, the orbits become more circular and start the shrink (there is an expression for de/dt that I did not put up on here).
-you can also have unbound orbits that become bound through the emission of GW.
GW are emitted with frequency that is twice the orbital frequency
Sun earth system has a lifetime of around 10^23 years
But for more massive objects in closer orbits this will be a lot less
-Gravitational radiation from the motion of two point masses, P. C. Peters, Phys. Rev. ,136, 1224 [1964]
-Gravitational radiation from point masses in a Keplerian orbit, P. C. Peters and J. Mathews, Phys. Rev., 131, 435 [1963]

8. Background Physics
As the orbit decays the frequency of the gravitational waves change and can create noticeable profiles such as
The frequency will increase as the orbital frequency increases and the amplitude will increase
Knowing the exact parameters of this “chirp” frequency can also be used to give a luminosity distance
How chirp frequency changes with time can give you luminosity distance
Amplitude is given by the GW frequency and the GW frequency change with time

9. Primordial GW (GW background)
GW observed in the CMB and are a result of inflation
Current efforts to detect CMB polarization which is the imprint of primordial GWs at the time of CMB production

10. Compact Binary White Dwarfs
Close binary WDs will likely serve as calibration standards
Well modeled
Numerous and relatively nearby
Frequency range ( >10mHz) is within spectral range of Earth-based interferometers
Observations will improve understanding of formation on Type 1a supernovae
The white dwarf binary background is formed as a spaced based interferometer orbits the sun. During different parts of the orbit the detector will be more sensitive in different directions. As white dwarfs are more readily found in the disk, when the detector sensitivity is lined up pointing at the disk it will detect more amplitude from gravitational waves.
This pattern will help scientist distinguish between cbwd and other signals or noise.
White dwarf binary background

11. Massive Black Hole Binaries
GW from higher redshift massive black hole binaries, MBHB, are likely to provide some understanding of the early evolution of galaxies
Massive Population 3 stars are thought to have seeded the formation of galactic massive black holes
Space-based eLISA will likely be able to observe inspiral, merger, and ring-down phases
Supermassive black hold growth

12. Extreme mass ratio in-spirals, EMRI
The Sag A* black hole provides a relatively nearby potential GW generation associated with stellar masses spiraling into the massive black hole
Space-based eLISA should make it possible to study this type of events in other galaxies

13. C. D. Ott, Class. Quantum Grav. 26 (2009) 06300
Other Sources
Supernova
Pulsars
In Core Collapse Supernova , the order of magnitude of energy taken away by gravitational waves is 1.8x10^46 ergs, total supernova release energy is on the order of 10^51 ergs.
GW signature can tell us the potential explosion mechanism such as neutrino, acoustic, or mhd mechanism all of which effect the GW quadrapole signature differently (move mass differently)
C. D. Ott, Class. Quantum Grav. 26 (2009) 06300

14. How do we detect gravitational waves?
Gravitational waves cannot be detected at a single point.
By Equivalence Principle, gravity can be transformed away at a single point by an appropriate coordinate system*
GW fluctuations, DL, in baseline distance, L, between test mass pairs, have all these observable properties
DL /L  GW propagation
DL /L<10-21
10-4< f < 104
When DLx , DLy 
Laser Interferometer
DLy
DLx
*“Relativity, Gravitation and Cosmology”, TP Cheng, 2nd Ed, p337 ff

15. Current Gravitational Wave Detection Initiatives
Earth based
LIGO, aLIGO
4km baseline interferometers near Hanford, WA and Livingston, LA
aLIGO is LIGO with upgraded capability (fully operational in 2015)
VIRGO
3km baseline interferometer near Pisa, Italy
GEO600
0.6km baseline interferometer near Sarstadt (by Hanover), Germany
KAGRA
2 sets of 3km baseline interferometers underground in Kamioka mine, Japan
Cryogenic cooling of detector components
Space based
eLISA
ESA rescoped LISA after NASA dropped out in 2011 due to lack of funding
eLISA technology demonstration satellite to be launched in 2015
10e6 km baseline interferometer in solar orbit near Earth

16. Signal and Noise for GW Detectors - Conventions
Strain is the dimensionless amplitude of a GW, DL /L, or “h”
Signal , s(t) = n(t) + h(t), where n(t) is noise, also dimensionless
Time average of noise2, 𝑛(𝑡) 2 = lim 𝑇→∞ 𝑇 −𝑇 𝑇 𝑛 𝑡 𝑛 ∗ 𝑡 𝑑𝑡
Power Spectral Density, PSD = Sn(f), has units 1/f and relates to noise2 by 𝑛(𝑡) 2 = 0 ∞ 𝑑𝑓 𝑆 𝑛 (f).
When lower integration limit = 0 this is called “one sided PSD”
Root PSD = 𝑃𝑆𝐷 has units 1/𝑓 , and is the most commonly graphed
Characteristic Strain hcis dimensionless
This is useful in SNR calculation. hc(f) 2=4f2|h(f)|2, hn(f)2 =fSn(f) and
SNR = −∞ ∞ 𝑑 log 𝑓 [ hc hn ]2
Gravitational wave sensitivity curves, CJ Moore, RH Cole, CPL Berry, arXiv: v1 [gr-qc] 4 Aug 2014

17. Noise Sources affecting GW Detectors
Quantum noise
Shot noise scales with sqrt(laser power), whereas signal scales linearly
Power level within the cavity is enhanced by setting the cavity length precisely (via a phase locked loop) to a multiple of the illumination wavelength, leading to a high power density resonance, improving S/N ratio.
Shot noise is also reduced by using a “squeezed light” source
Seismic gravitational gradients
Going underground or into space helps
Thermal noise in test masses and suspensions
Cryogenic cooling helps

18. Actual Implementation – complex source and detector
Ground-based interferometers are similar to the Japanese KAGRA design
Laser frequency is selected to resonate in X and Y arm Fabry-Perot etalons
Power recycling used on input and signal recycling used on output
3km arms in vacuum, test masses cooled to 20K
Signal detection uses QND technique
Laser is modulated to create RF sidebands for cavity length control. AS_RF takes that signal to control system.
Interferometer design of the KAGRA gravitational wave detector, Y Aso, et al, arxiv.org , v1

19. Quantum limits on interferometer detection
For a test mass position measurement that generates “back action” results the “standard quantum limit” of uncertainty
SQL for an interferometer is
“Quantum Non Demolition” or QND techniques can evade SQL
M Mass of each identical test mass
W GW angular frequency
ℏPlanck’s constant/2p
L Length of the interferometer’s arms
DL Time evolving difference in arm lengths
h(t)=DL/L Dimensionless gravitational wave signal
𝑆𝑄𝐿 ℎ [ 𝐻𝑧 −0.5 ] = 8ℏ 𝑀 𝐿 2 Ω 2
Quantum noise in second generation, signal-recycled laser interferometric gravitational-wave detectors, A Buonanno and Y Chen, PHY REV D, 64,

20. Beating back seismic noise in Earth-based detectors
Seismic waves passing a GW detector induce density and local gravity fluctuations that mimic the differential GW signal
VIRGO isolation stack
LIGO
Seismic gravity-gradient noise in interferometric gravitational-wave detectors, S Hughes, K Thorne, PHYS REV D, 58,
Detector components are mechanically isolated by “stacks”

21. Example: KAGRA Noise Analysis
Interferometer design of the KAGRA gravitational wave detector, Y Aso, et al, arxiv.org , v1

22. Expected Source Signal and Noise

23. Use of “Null Stream” to Minimize False Positives
Co-locating and co-aligning two identical GW detectors facilitates synthesis of a null data stream*
Example
Detector 1 outputs signal S1 = N1 + h(t)
Detector 2 outputs signal S2= N2 + h(t)
N1 and N2 are uncorrelated noise deviations around the signal h(t), have standard deviations s1 and s2
S1-S2 = h(t)-h(t) + (s1 2 + s22 )0.5 = root sum squared of the detector noise st dev
If a candidate “signal” appears in the null stream, if can be immediately rejected
Seismic gravity gradient noise would not be eliminated, as both detectors would see this externally-sourced noise as a real signal
Null streams can be created for non co-located detectors, canceling signal emanating from a specific direction in the sky… basis for locating sources
*Near optimal solution to the inverse problem for gravitational-wave bursts”, Y Gursel, M Tinto, Phys Rev D 40, 3884

24. eLISA Space Based GW Detector
Laser Interferometer in Space Antenna, LISA, provides unique capabilities
Immune to seismic noise
Long baseline provides Hz GW spectrum sensitivity needed for observing massive black hole mergers
Multiple identical or similar detectors to improve detection confidence
LISA: a mission to detect and observe gravitational waves, O Jennrich, in Gravitational Wave and Particle Astrophysics, Proc SPIE v5500

25. References
Relativity, Gravitation and Cosmology, TP Cheng, 2010, p337 ff
Quantum noise in second generation, signal-recycled laser interferometric gravitational-wave detectors, A Buonanno and Y Chen, PHY REV D, 64,
Seismic gravity-gradient noise in interferometric gravitational-wave detectors, S Hughes, K Thorne, PHYS REV D, 58,
Near optimal solution to the inverse problem for gravitational-wave bursts”, Y Gursel, M Tinto, Phys Rev D 40, 3884
LISA: a mission to detect and observe gravitational waves, O Jennrich, in Gravitational Wave and Particle Astrophysics, Proc SPIE v5500
Interferometer design of the KAGRA gravitational wave detector, Y Aso, et al, arxiv.org , v1
Gravitational wave sensitivity curves, CJ Moore, RH Cole, CPL Berry, arXiv: v1 [gr-qc] 4 Aug 2014
Gravitational radiation from the motion of two point masses. P. C. Peters, Phys. Rev. ,136, 1224 [1964]
Gravitational radiation from point masses in a Keplerian orbit, P. C. Peters and J. Mathews, Phys. Rev., 131, 435 [1963]
Gravity: Introduction to Einstein’s General Relativity, Hartle
C. D. Ott, Classical and Quantum Grav. 26 (2009) 06300
A model for Gravitational Wave Emission from Neutrino-Driven Core-Collapse Supernova. Murphy, J. W., Ott, C. D., & Burrows, A. 2009, arXiv: