1. Gravitational Waves and the R-modes
: The Laser Interferometer Gravitational-wave Observatory
Gravitational Waves and the R-modes
LIGO stands for the Laser Interferometer Gravitational-wave Observatory. I’ll start with a show and tell overview of Gravity Waves and LIGO, then I’ll describe some work I’ve done on one particluar source of Gravity Waves, the R-modes.
Gregory Mendell
LIGO Hanford Observatory

2. Sponsored by the National Science Foundation
Who’s Involved?
Caltech, MIT, and the LIGO Science Collaboration
This is list of who is involved. The project is run by Caltech and MIT. It is funded by NSF. The universities that are heavily involved in writing search code are Caltech, MIT, UWM, UT Brownsville Penn State, and AEI. Most of the code that run during the E7 run was written at UWM, UTB and MIT. Caltech and MIT are involved in all aspects. Many of these groups are involved in detector characterization and instrumentation.
Sponsored by the National Science Foundation

3. The Observatories LIGO Hanford LIGO Livingston
Ironically, it’s LIGO Hanford that is located in the evergreen state!
4 KM arms. Want 1000 km arms for optimal sensitivity, so light is folded 100’s of times in the arms.
LIGO Hanford
LIGO Livingston
Photos:

4. Inside

5. Gravitational Waves
Gravitation = spacetime curvature described by the metric tensor:
Weak Field Limit:
TT Gauge:

6. Gravitational-wave Strain
D. Sigg LIGO-P D

7. How Does LIGO Work? Gravitational-wave Strain:
LIGO is an “ear” on the universe, listening for cosmic spacetime vibrations.
LIGO is a lab looking for GW’s.
Gravitational wave strain = in lab frame seen as tidal strain that stretches in one direction, and squeezes in another direction.
Ligo is unlike other observatories. An X-ray telescope looks for x-rays, but is not meant to learn more about x-rays or the theory behind them. But LIGO is like a lab, doing an experiment to look for GW’s and to learn about General Relativity. It is also an observatory, listening to spacetime vibrations from the cosmos to do astrophysics.
Figures: K. S. Thorne gr-qc/ ; D. Sigg LIGO-P D

8. Astrophysical Sources
LMXBs
Pulsars
Black Holes
Astrophysical Sources
Need compact objects with large amounts of mass and high speed asymmetric motions to produce detectabel gravity waves.
Stochastic Background
Supernovae
Photos:

9. Inspiral (hmax = 10-22 for NS-NS@ 200 Mpc; rate = 3/yr; NS-BH; BH-BH)
“Newtonian” quadrupole formula.
Stochastic (limit GW; cosmic strings; BH from massive pop III stars: h = – 10-21)
Burst (SN at distance of Virgo Cluster: h = – 10-21; rate = 1/yr)
Inspiral (hmax = for 200 Mpc; rate = 3/yr; NS-BH; BH-BH)
Periodic (h = for 10 ms pulsar with maximum ellipticity at 1 Kpc; h = for 2 ms LMXB in equilibrium at 1 Kpc)
GR exist only for quadrupole or higher multipole moments because mass and momentum are conserved. Newtonian quadrupole formula is for systems with all velocities (e.g., escape, orbital, rotational, much less than the speed of light.) Note Q dot dot ~ M X L^2/dt^2 ~ mv^2 ~ asymmetric KE.
The lower bound on SN h is for SN from star with angular momentum of the Crab pulsar. Collapse will have several bounces, with increasing ellipticity. Upper limit on SN for bar mode instability, where collapsing star end up like football, or breaking apart and recollapsing. SN rate is 3-4e-2 for the galaxy. It is .5/yr at 8 Mpc = 3/yr at 15 Mpc.
The stochastic background will be studied by looking at correlations between the sites after filtering within a narrow band for each frequency. Assume that noise is uncorrelated. This will set upper limits on the energy in GW’s from the big bang as a fraction of the closure density. (Many limits exit at low frequencies.) Three could be GW’s from cosmic strings, or resonances
I believe the NS-NS inspiral rate is based on the number of NS binaries detected.
The binary pulsar will insprial in 350 million years. In each case, either the strength of the sources is uncertain by several orders of magnitude, or the event rate (and thus the distance to the nearest source) is uncertain by several orders of magnitude, or the existence of the source is uncertain.)
NS-BH and BH-BH numbers can be much better, but are more uncertain.
1000 known pulsars; should be 10^5 active pulsars, and 10^8 NSs in the galaxy overall. Pulsar birth rate is 1 per 40 years. Should find h < 10^-24 based birth rate to compute closest pulsar to us with age t, using spin down age t = fdot/f, assume fdot due to GWs only. Actual ellipticity is 5 to 8 orders of magnitude small that maximum allowed by crust tensil strength.
Reviews: K. S. Thorne 100 Yrs of Gravitation; P. R. Saulson, Fund. of Interferometric GW Detectors

10. Figure: D. Sigg LIGO-P980007-00-D
Noise Curves
The initial goal of LIGO is to reach a amplitude strain sensitivity per root hz 3e-23 at 200 Hz.
Shot noise does not increase with f, but detector sensitivity does decrease with f, since shorter period GW’s tend to get smeared out.
Need light storage time less than .5 period naively to prevent this smearing.
Figure: D. Sigg LIGO-P D

11. Signal to Noise Ratio h = signal amplitude
The larger the SNR, the more confident the detection.
The exact definition of SNR is the amplitude of the correlation between the detector output and the template of the signal divided by the time-averaged rms correlation between the detector output and the template of the signal when only noise is present.
The one-sided power spectrum is twice the fourier transform of the autocorrelation of n(t). Let the Fourier transform of n(t) = N(f), with units strain*time. Then n^2(f) = |N(f)|^2/T, where T is the length of the time series. Thus, n^2 has units of strain squared X time = strain squared per Hz. Thus n(f) has units of strain per root Hz.
h = signal amplitude
T = observation time or duration of signal or period of the characteristic frequency of the signal.
n2 = power spectrum of the noise

12. Figure: Brady ITP seminar summer 2000
Sensitivity Curves
Again, these are 4*n(f)/sqrt(10^7). The 4 is due to reduced sensitivity due to amplitude modulation.
Figure: Brady ITP seminar summer 2000

13. Known Possible Periodic Sources
Are neutron stars: the sun compress to size of city. Compact (2GM/Rc2 ~ .2) and ultra dense (1014 g/cm3).
Are composed of (superfluid) neutrons, (superconducting) protons, electrons, + exotic particles (e.g., hyperons) or strange stars composed of an even more exotic up, down, and strange quark soup.
Spin Rapidly (~ .1 Hz to 642 Hz i.e., within the LIGO band.)
LMXBs

14. Neutron (Strange) Star Models
solid crust
Fe, etc. up
down n e- p+
strange n p+
dissipative boundary layer
quark soup
fluid interior
Courtesy Justin Kinney

15. Periodic sources emit GWs due to…
Rotation about nonsymmetry axis
Strain induced asymmetry:
Accretion induced emission
Unstable oscillation modes
Accretion could excite modes, or move spin axis away from symmetry axis.

16. Gravitational-radiation Driven Instability of Rotating Stars
GR tends to drive all rotating stars unstable!
Internal dissipation suppresses the instability in all but very compact stars.

17. Ocean Wave Instability
Wind
Current

18. Perturbations in Rotating Neutron Stars
Neutron star rotates with angular velocity  > 0.
Some type of “wave” perturbation flows in opposite direction with phase velocity /m, as seen in rotating frame of star.
Perturbations create rotating mass and momentum multipoles, which emit GR. 
Rotating neutron star.
/m
Perturbations in rotating frame.
Courtesy Justin Kinney

19. Courtesy Justin Kinney
GR Causes Instability
If  - /m > 0, star “drags” perturbations in opposite direction.
GR caries away positive angular momentum.
This adds negative angular momentum to the perturbations.
This increases their amplitude!
 - /m 
Star drags perturbations in opposite direction. GR drives mode instability.
Courtesy Justin Kinney

20. The R-modes
The r-modes corresponds to oscillating flows of material (currents) in the star that arise due to the Coriolis effect.  The r-mode frequency is proportional to the angular velocity,  .
The current pattern travels in the azimuthal direction around the star as exp(it + im)
For the m = 2 r-mode:
Phase velocity in the corotating frame: -1/3 
Phase velocity in the inertial frame: +2/3 

21. Courtesy Lee Lindblom

22. Courtesy Lee Lindblom

23. R-mode Instability Calculations
Gravitation radiation tends to make the r-modes grow on a time scale GR
Internal friction (e.g., viscosity) in the star tends to damp the r-modes on a time scale F
The shorter time scale wins:
GR  F : Unstable!
GR  F : Stable!

24. Key Parameters to Understanding the R-mode Instability
Critical angular velocity for the onset of the instability
Saturation amplitude

25. Viscous Boundary Layers

26. Add Magnetic Field… B

27. Magneto-viscous Boundary Layer With Alfven Waves

29. MVBL Critical Angular Velocity
Mendell 2001, Phys Rev D ; gr-qc/

30. R-mode Movie
1.4 solar mass star is initially rotating at 95% breakup angular velocity. Best SNR is for narrow-banded detector.
See:
Lee Lindblom, Joel E. Tohline and Michele Vallisneri (2001), Phys. Rev. Letters 86, (2001). R-modes in newborn NS are saturated by breaking waves.
Computed using Fortran 90 code linked wtih the MPI library on CACR’s HP Exemplar V2500.
Owen and Lindblom gr-qc/ : r-modes produce 100 s burst with f ~ Hz and optimal SNR ~ for r = 20 Mpc and enhanced LIGO.

31. Basic Detection Strategy
Coherently add the signal
Signal to noise ratio ~ sqrt(T)
Can always win as long as
Sum stays coherent
Understand the noise
Do not exceed computational limits

32. Figure: D. Sigg LIGO-P980007-00-D
Amplitude Modulation
Sensitivity is reduced by amplitude modulation. M is rotations about Euler angles to bring h^TT from the wave frame into the lab frame. Theta and phi are unsual polar coordinates, and Psi is the polarization or angle of rotation of the xy plane of the wave frame.
Figure: D. Sigg LIGO-P D

33. Phase Modulation
The phase is modulated by the intrinsic frequency evolution of the source and by the Doppler effect due to the Earth’s motion
The Doppler effect can be ignored for

34. DeFT Algorithm
AEI: Schutz & Papa gr-qc/ ; Williams and Schutz gr-qc/ ; Berukoff and Papa LAL Documentation

35. Taylor expand the phase

36. Advantages of DeFT Code
Pkb is peaked. Can sum over only 16 k’s
Complexity reduced from O(MN  number of phase models) to O(MNlog2N + M  number of phase models).
Unfortunately, number phase models increased by factor of M/log2MN over FFT of modulated data. FFT is O(MNlog2MN  number of phase models/MN.)
But memory requirements much less than FFT, and easy to divide DeFT code into frequency bands and run on parallel computing cluster.
Need 1010 – 1020 phase models, depending on frequency band & number of spin down parameters, for no more than 30% power loss due to mismatch.
To do FFT’s with sample rate of 4096 Hz, need ~100 MB RAM per hour of data, can only hold 5-10 hours of data in memory at a time.
Can FFT 10^10 data points in 1 sec on 1 TB computer.

37. Basic Confidence Limit
Probability stationary white noise will result in power greater than or equal to Pf:
Threshold needed so that probability of false detection = 1 – .
Brady, Creighton, Cutler, Schutz gr-qc/

38. Maximum Likehood Estimator
Jaranowski, Krolak, Schutz gr-qc/

39. LDAS = LIGO Data Analysis Systems

40. LDAS Hardware 14.5 TB Disk Cache Beowulf Cluster
LHO data rate is 20 GB/ hour (fills up your hard drive every hour.) 100 node beowulf ~ 20 Gflop supercomputer.
Beowulf Cluster

41. LDAS Software

42. Interface to the Scientist