Periodicity in
gravitational waves
Graham Woan, University of Glasgow
Statistical Challenges in Modern Astronomy
Penn State, June 2006
SCMA IV
June 2006
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Talk overview
• Basics of gravitational wave generation and detection
• Periodic sources from the ground (LIGO/GEO/Virgo)
— Signal types
— Current survey, detection and analysis methods
• Periodic sources from space (LISA)
— Case study: the binary white dwarf confusion problem
— LISA: a Bayesian überchallenge
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Effect of a gravitational wave
• Modulation of the proper distance between free test particles
• Consider simple detector - two free masses whose separation, L, is
monitored
L
• A gravitational wave of amplitude h will produce a strain
between the masses
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L
h

L
2
3
Interferometric detection
Construct a Michelson
interferometer to detect
these strain signals. A strain
results in a change in light
level at the photodetector.
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Current/future Interferometric detectors
GEO600 (British-German)
Hannover, Germany (600m)
LIGO (USA) (4km)
Hanford, WA and Livingston, LA
TAMA (Japan)
Mitaka (300m)
AIGO (Australia), Wallingup
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VIRGO (French-Italian)
Cascina, Italy (3km)
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Detector sensitivity
seismic
thermal
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Response to sky direction
+ polarization
× polarization
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total
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High-frequency (>10 Hz) periodic sources
wobbling neutron stars
low mass X-ray binaries
bumpy neutron stars
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Low frequency ~periodic sources (LISA)
Massive binary black holes
Compact binary systems
Compact objects orbiting massive
black holes (EMRIs)
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The gravitational wave signal
• Take the signal as quasi-sinusoidal GWs from a triaxial, non-precessing
neutron star, modulated by doppler motions and the antenna pattern of
the GW detector:
antenna pattern modulation
(Dupuis, 2006)
Signal model
Signal phase
Barycentric corrections
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The gravitational wave signal
• Model parameters are
—
—
—
—
—
2 axial orientation angles (,)
1 signal amplitude (h0)
2+ spin parameters (0, f0, df0/dt, …)
2 sky location (RA, dec)
4+ further parameters for binary sources
• Many parameters to search/marginalise/maximise over, and
some large dimensions (e.g., f)
• However the signal is believed to be coherent on timescales
of months to years, and ~phase-locked to radio pulses
(should they be available).
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Current LSC periodic wave searches
• We use several methods based on both Bayesian and
frequentist principles:
Coherent searches:
- Time-domain:
- Targeted
- Markov Chain Monte Carlo
Searches over
narrow parameter
space (Bayesian)
- Frequency-domain:
- Isolated
- Binary, Sco X-1
Searches over wide
parameter space
Incoherent searches:
- Hough transform
Excess power
searches
- Stack-Slide
Ultimately, would like to
combine these two in a
hierarchical scheme
(frequentist, with some
Bayesian leanings)
- Powerflux
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Statistical approaches
• Targeted searches (t-domain Bayesian):
— Heterodyne at the expected signal frequency, accounting for
spindown and doppler variations, then determine a marginal pdf for
the strain amplitude (marginalise over all other model parameters,
and the noise floor). Use Markov Chain Monte Carlo when
numerically marginalising over more than four parameters.
• Coherent wide area searches (f-domain, frequentist)
— Use a detection statistic (‘F-statistic’) which is the log likelihood
pre-maximised over (functions of) ,  and . Incoherent
combination of data from different detectors giving a frequentist UL
based on the loudest coincident event.
• Incoherent wide area searches (f-domain, frequentist)
— Stack and slide short power spectra then combine
• after thresholding (Hough method)
• after normalising (Stack-Slide method)
• after weighting for the antenna pattern and noise floor (Powerflux method)
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Coherent vs incoherent
• Clearly, nothing beats coherent searches if computing time is
unconstrained,
• but the trade-off is less obvious for a fixed computing time:
— Incoherent methods relax phase coherence constraints and reduce the
size of the parameter space to the point where it can be searched
exhaustively down to some level
— The search is still big, so only big (high snr) signals are statistically
significant
— A sinusoid that appears in a complex spectrum with a signal-to-noise
ratio  has a signal-to-noise ratio of
   2 (1  2 2 ) -1/ 2
in the corresponding power spectrum, so…
— Incoherent methods are nearly as good as coherent methods once the
signal is big!
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Semi-coherent methods
•The trick is to integrate
coherently for an optimal length
of time, then combine these
results incoherently, leading to
hierarchical schemes:
—Coherent sub-steps
—Incoherent combination of
coherent sub-steps
—Fully coherent follow-up of
candidates
a similar scheme is used in
some radio pulsar searches.
Cutler, Gholami, and Krishnan, Phys. Rev. D 72, 042004 (2005)
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Bandwidth issues
• If we know the frequency and phase evolution of the signal
(from radio observations) then robust, well-calibrated
results can be obtained even in a generally poor noise and
interference environment:
Signal frequency is clean
(Abbott et al., PRD, 2004)
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Targeted pulsar search
• No dispersion, but prolonged observations (~years) with a wide-beam
(quadrupole) transit array
• Unknowns: source amplitude, phase, inclination of rotation axis,
polarisation angle. Signal of the form:
with
F , F
 0 ,0



 h0
a




 0
polarisati on beam patterns
signal frequency, fdot
inclinatio n angle(?)
strain amplitude( ?)
polarisati on angle (?)
phase(?)
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Dupuis
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Targeted pulsar search
• Targeted search is done with a simple Bayesian parameter
estimation:
— Heterodyne the data with the expected phase evolution and and bin
to 1 min samples:
— Marginalise over the unknown noise level, assumed Gaussian and
stationary over 30 min periods (-> Student t):
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Targeted pulsar search
• Define the 95% upper limit inferred by the analysis in terms of a
cumulative posterior, with uniform priors on orientation and strain
amplitude:
with a joint likelihood for the stationary segments of
• Numerical marginalisation to get parameter results
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Joint marginals (simulation)
h0 = signal amplitude
0 = initial rotational phase
 = polarisation angle
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Multi-detector analysis
J1920-5959D
• Within a Bayesian framework,
multidetector (‘network’) analysis is
particularly straightforward. The
posterior for the model m is
H1
H2
L1
joint
p(m | d1d2d3)  p(m)p(d1 | m)p(d2 | m)p(d3 | m)
• But the results can be initially surprising –
the joint upper limit can be worse than
some of the contributing individual upper
limits (though this is rare).
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H1
H2
L1
joint
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Tests with signal injections into hardware
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LISA: coming soon…
a real statistical challenge!
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LISA astronomy – quasimonochromatic sources
EMRI sources
Massive black hole mergers
Precision bothrodesy
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Extreme mass-ratio sources
Quasi-periodic orbits showing a
complex “zoom-whirl” structure
Jonathan Gair
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Drasco & Hughes
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White dwarf binary confusion
•
LISA data is expected to contain many (maybe 50,000) signals from white dwarf
binaries. The data will contain resolvable binaries and binaries that just
contribute to the overall noise (either because they are faint or because their
frequencies are too close together). How do we proceed?
• Bayes can sort these out without having to introduce ad-hoc acceptance and
rejection criteria, and without needing to know the “true noise level”
(whatever that means):
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LISA calibration sources
Phinney
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Things that are not generally true
• “A time series of length T has a frequency resolution of 1/T.”
Frequency resolution also depends on signal-to-noise ratio. We know
the period of the pulsar PSR 1913+16 to 1e-13 Hz, but haven’t been
observing it for 3e5 years.
1
.
In fact frequency resolution is 
T  snr
• “you can subtract sources piece-wise from data.”
Only true if the source signals are orthogonal over the observation
period.
• “frequency confusion sets a fundamental limit for low-frequency LISA.”
This limit is set by parameter confusion, which includes sky location
and other relevant parameters (with a precision dependent on snr).
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LISA source identification
•
Toy (zeroth-order LISA) problem (Umstätter et al, 2005):
You are given a time series of N=1000 data points comprising a number of sinusoids
embedded in white gaussian noise. Determine the number of sinusoids, their amplitudes,
phases and frequencies and the standard deviation of the noise.
•
We could think of this as comparing hypotheses Hm that there are m sinusoids in the data,
with m ranging from 0 to mmax. Equivalently, we could consider this a parameter fitting
problem, with m an unknown parameter within the global model.
signal
parameterised by
giving data
and a likelihood
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Reversible Jump MCMC
• Trans-dimensional moves (changing m) cannot be performed in conventional
MCMC. We need to make jumps from to dimensions
• Reversibility is guaranteed if the acceptance probability for an upward
transition is
where
is the Jacobian determinant of the transformation of
the old parameters [and proposal random vector r drawn from q(r)] to the new
set of parameters, i.e.
.
• We use two sorts of trans-dimensional moves:
— ‘split and merge’ involving adjacent signals
— ‘birth and death’ involving single signals
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Trans-dimensional split-and-merge transitions
• A split transition takes the parameter subvector
from ak
and splits it into two components of similar frequency but about half the
amplitude:
A
A
f
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Trans-dimensional split-and-merge transitions
• A merge transition takes two parameter subvectors and merges them to their
mean:
A
A
f
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Delayed rejection
•
•
Sampling and convergence can be improved (beyond Metropolis Hastings) if a second
proposal is made following, and based on, an initial rejected proposal. The initial
proposal is only rejected if this second proposal is also rejected.
Acceptance probability of the second stage has to be chosen to preserve reversibility
(detailed balance):
acceptance probability for 1st stage:
and for the 2nd stage:
•
Delayed Rejection Reversible Jump Markov Chain Monte Carlo method
‘DRRJMCMC’ Green & Mira (2001) Biometrika 88 1035-1053.
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Initial values
• A good initial choice of parameters greatly decreases the length of the
‘burn-in’ period to reach convergence (equilibrium). For simplicity we
use a thresholded FFT:
• The threshold is set low, as it is easier to destroy bad signals that to
create good ones.
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Simulations
•
•
•
•
•
1000 time samples with Gaussian noise
100 embedded sinusoids of form A cos 2ft  B sin 2ft
As and Bs chosen randomly in [-1 … 1]
fs chosen randomly in [0 ... 0.5]
Noise   1
p( m2 )
Priors
• Am,Bm uniform over [-5…5]
• fm uniform over [0 ... 0.5]
2
•  m has a standard vague inversegamma prior IG(  m2 ;0.001,0.001)
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 m2
 m2
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Results (spectral density)
energy
energy
density
energy
density
frequency
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Results (spectral density)
energy
energy
density
energy
density
frequency
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Joint energy/frequency posterior
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Marginal pdfs for m and 
p(m | { D }, I)  
{ A } m ,{ f } m , m
p(m, { A} m , {B } m , { f } m , | { D }, I) d{ A} m d{B } m d{ f } m d m
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Label-switching
•
•
As set up, the posterior is invariant under signal renumbering – we
have not specified what we mean by ‘signal 1’.
Break the symmetry by ordering in frequency:
1.
2.
3.
4.
5.
6.
Fix m at the most probable number of signals, containing n MCMC steps.
Order the nm MCMC parameter triples (A,B,f) in frequency.
Perform a rough density estimate to divide the samples into m blocks.
Perform an iterative minimum variance cluster analysis on these blocks.
Merge clusters to get exactly m signals.
Tag the parameter triples in each cluster.
f
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Strong, close signals
A
A
B
f
f
1/T
B
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Signal mixing
•
Two signals (red and green) approaching in frequency:
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The full LISA challenge
First the good news:
• only ~1 sample per second, so only ~108-9 data points (fits
on a
)
•
•
•
•
•
Now the bad:
Near-isotropic telescope antenna pattern
10s of thousands of parameterisable quasi-periodic sources
Surely some unexpected source types
Some chirping sources, sweeping through the band
Strongly coloured noise, confusion-dominated at some
frequencies
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The full LISA challenge
• Six Doppler observables,
measuring the beat between the
local laser and received laser
signal in both directions on each
arm
Six Doppler observables, si
• Strong (laser) noise contributions
that must be numerically
“cancelled” to do any astronomy.
You can think of this as a PCA
problem.
Data covariance matrix
(Romano & Woan)
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The full LISA challenge
To dig into the confusion noise and avoid the problems of
source subtraction global Bayesian modelling seems to be
the only game in town, so we will need:
• Quick & dirty methods to get an approximate model of the
sky, prior to global modelling
• Fast likelihood calculation methods
• Well-developed variable-dimension mcmc-like algorithms to
perform the global modelling
• A few years…
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