All-sky search for
gravitational waves from
neutron stars in binary
systems
strategy and algorithms
H.J. Bulten
analysis of PSS from binaries
thesis work of Sipho van der Putten
Sipho van der Putten, R. Ebeling (siesta)
staff involved: JFJ van den Brand, Th.
Bauer, HJB, T.J. Ketel, S. Klous (grid)
theory dept. : G. Koekoek and J.W. van
Holten
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 2
motivation: binary systems
• Virgo/Ligo: better sensitivity at higher
frequency (>10 Hz)
• fixed quadrupole deformation: h f 2
• most high-frequency neutron stars are
in binary systems
– spin-up via gas transfer
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 3
motivation
• Brady et al.
PRD57,2101:
binary
I 2
K 1039 f 2 [ J ]
2
P
dK
I
d I 2
old
dt dt
new?
constant Power
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 4
solitary neutron stars
• solitary neutron star: Doppler shifts from
earth movement
1
fT _ FFT
2TFFT
(v ) rˆ arˆ
f f gw f gw t 5 1011 s 1 f gw T
c c
1
TFFT 1.110 5
, about 1 hour for 1000 Hz
f gw
• Hierarchical search possible, T~ 1h (Rome
group, e.g. Astona, Frasca, Palomba CQG
2005.)
• signal-to-noise ~ Tobs
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 5
solitary neutron stars
• alternative: F-statistics approach
(Ligo, Jaranowski et all PRD58, 063001)
– produce templates that remain in phase
over the template search time
– parameters phase , , f NS ) ; amplitude: h0 , 0 , , ,
(k
– solitary neutron stars: all-sky search
– many templates needed, e.g. Brady et al.
PRD61, 082001
• coherent all-sky search of length of 0.5days
would take 10,000 Tflops (fmax=1000 Hz)
• smaller spin-down, fmax=200 Hz: 5 days
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 6
Binary : Kepler orbitals
• ellipse
T 2 a3 v T 1/ 3
1
v ph vap
a
red shift depends on direction both axes a
• We want to analyze:
– orbital periods from 2 hours – infinite
– masses companion star up to 15 solar masses
– eccentricities up to about 0.7
– frequency shifts up to 0.3%, frequency changes
df/dt up to 10-6 s-2
• 1 mHz shift in 1 second, at f=1000Hz
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 7
frequency shifts
Binary system: T 2.3h, M acc 8M n , 0.6, 56, 0
inproduct major axis: -0.83, minor axis 0.52
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 8
frequency derivative
df a n Binary system: T 2.3h, M acc 8M n , 0.6, 56, 0
inproduct major axis: -0.83, minor axis 0.52
dt c
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 9
frequency shifts
Binary system: T 2.3h, M acc 8M n , 0.6, 56, 0
inproduct major axis: -0.83, minor axis 0.52
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 10
coherence
• phase signal:
s ( k ) t k 1 n (rd rns ) s ( k ) t k
ˆ
(t ) 0 2 f NS f NS (k )!
k 0 (k 1)! c k 0
solitary: rNS may be assumed fixed
phase parameters , , f NS )
(k
amplitude: h0 , 0 , , ,
binary:
rd rns : (t ) depends on extra parameters:
Torbit , orbit , M companion , ( M ns ), major , minor , 0,orbit
• signal should remain in-phase ,e.g. maximally 90
deg. out of phase anywhere during observation time
– frequency within ½ bin - 1/(2Tobs)
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 11
binary neutron stars
• how many extra parameters?
– e.g. orbital period >=2 hours, eccentricity 2hour
• 0< eccentricity < 0.6
• all orientations of semi-major and semi-minor axes
• all starting phases in orbital
• up to 1000 Hz g.w. frequencies
• full parameter scan is not feasible.
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 13
binary neutron stars
• different set of filters: parameterize the
phase as a function of time!
– assume that within Tobs, the frequency can be
described by a second-order function of time
(t ) 0 2 f 0 (t ) t2 t3
2 6
f (t ) f 0 t t2
2
df
t
dt
– third-order effects are assumed to be
negligible.
• scan for presence of signal by calculating the
correlation with the template
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 14
Correlation
• Correlation is given by
Corr ( g , h)
g (t )h( )d G( f ) H * ( f )
FFT G ( f ) H * ( f ) gives array, correlations for lags t 0...N (t )
• presence of signal defined by overlap with filter.
• data is not periodic: make filter equal to zero for
last N/2 samples and shift it maximally N/2
samples to the right
• FFT: interleave, to cover full dataset
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 15
Filter search
data, split in overlapping periods
FFT 1 FFT 2
filter, lag=0 filter, scan to lag = T/2
Filter: zero-padded for half length
check correlations from t=0 to t= ½T (FFT1)
check correlations from t= ½T to t=1T (FFT2)
check correlations from t=1T to t= 1½T (FFT3)
maximum overlap: amplitude and time known
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 16
Filter search
filterfilter
filter
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 17
Example Filters
f (t ) sin , (t ) 2 f 0t
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 18
2 3
filt (t ) sin , (t ) 2 f 0t t t
2 6
d
f (t ) t t 2 , 1.46 104 Hz 2 , 1.39 10 8 Hz 3
dt 2
parameter space
• phase should be given by filter:
– coherent times up to about T=500 seconds:
• for times <500 seconds, fourth-order
corrections due to orbital movements are small
– quadratic change of frequency: can be
parameterized with about 120 parameters
d2 f
106 , fT / 2 62mHz,
dt 2 max
– linear change of frequency:
df
103 , fT / 2 0.25 Hz
dt max
Phase: parameters
• for coherent times up to 500 seconds, the
frequency should be accurate within about
1mHz.
– phase description of data:
• about 10 phases 0
• about 1 million values of f0
• about 500 values of alpha=df/dt
• about 120 values of beta.
– however: scan with FFT template:
• in time direction: 0 can be determined
• templates can be re-used
• 600,000 templates reduce to about 5000
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 21
shifting in time
• shifting a filter in time by a lag tau
gives a filter with parameters:
1 1
f 0 (t ) t t 2 f 0 (t ) (t ) 2
2 2
1
f 0 f 0 2
2
• you do not have to apply filters with
with
0
T
2
0
f 0 f f 0 f , f determined by ,
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 22
shifting in frequency
• frequency changes are smaller than 1 Hz
within the set of filters
• produce filters in a small frequency band, a
complete set for 1 fixed value of f(t=0).
f max
– reduction of a factor of N F 106
f
• Fourier-transform them
• heterodyne data, or alternatively: compare
the filter in frequency domain with the
appropriate frequency band of the FFT of
the data
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 23
Scan
• Step in frequency: if the filter has small
frequency dependence, you have to step 1
frequency bin. So a filter with a constant
frequency is applied (Fmax/binwidth) times (e.g. 1
million times for an FFT of 1000 second)
• if the filter has large linear or quadratic
dependence, you can step with a stepsize
f max( f filter ) min( f filter )
• total scans needed to analyze 0 - 1000 Hz, 1000
seconds
– about 10,000 filters suffice.
– about 300 million correlations in total (300 million FFTs)
– a few days of CPU-time on a single CPU, current desktop
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 24
Hits
• a hit: overlap is larger than pre-defined threshold
– PSD from FFT from complete set (needs to be
optimized) sets noise threshold
– normalize data in frequency domain to have
mean amplitude of in each bin 2 Nsamp N FFT
ni : FFT from data, normalized to a PSD of 1
N filter / 2
N2
| ni | 2 N samp N FFTbins , i 0
fi fi*
filter
8
,
N filter / 2
N filter
i 0
ni off fi average 0, RMS =
*
2
N Samp N FFTbins
threshold: 4 sigma (on amplitude)
maximum bin in FFT(ni off fi * )
signal overlap estimator:
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 25
Procedure tests
• we tested with white noise, 4096 samples
per second, 1024 seconds FFT:
– filters can pick signal with 20 times smaller
amplitude (time domain) out of the noise (Total
power signal is 800 times smaller than that of
noise)
– overlap filter-signal is 1.0 if signal is equal to
filter+noise: amplitude is reproduced correctly.
– frequency is reproduced correctly (filter gives
only hits in the right frequency band)
– average overlap between filters is about 0.43
(at same frequency)
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 26
First tests
• spectrum : Gaussian-distributed noise with
mean zero and amplitude 1023 N
samp
– one-sided PSD of 10 23
/ Hz
• signals: 10 binary neutron stars:
– frequency between 200 and 250 Hz
– random angles, deformations, etc
– maximum amplitude < 10-23, total power of 10
signals is 0.2 percent of the power in the noise
• FFT lenght 1024 seconds, 2048
samples/sec.
• 30 FFT sets (about 5 hours)
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 27
Overlap of filters, only noise
maximum correlation
for all filters applied
between 0 and 1000 Hz
(81.5 million FFT products,
4096 lags per filter)
N FFTbins
N filter
4* f
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 28
Overlap of filters with signal
maximum correlation with signal
for all filters applied
between 0 and 1000 Hz
(81.5 million FFT products)
N FFTbins
N filter
4* f
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 29
signal-to-noise
N FFTbins
N filter
4* f
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 30
Power spectral density
PSD signal+noise
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 31
PSD, signal only
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 32
PSD, signal only
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 33
Search results
• 30 FFTs, about 5h of data
• analyzed between 100 and 500 Hz
– 2405 different filters
• about 1.3 billion filter multiplications,
28731 hits (10 pulsars+noise)
• pulsars only: 14972 hits
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 34
Search results, all hits
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 35
Search results
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 36
Alternative: cut on power
Cut: 4 sigma on power
FFT –number H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 37
Alternative: cut on power
Cut: 4 sigma on power
7649 hits between 450 and 460 Hz
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 38
highest PSD in data
FFT –number H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 39
PSD: signal only
signal highest PSD
still data spread out over about 30
bins
FFT –number 9 Jun 2008
H.J. Bulten - LSC-Virgo PSS 40
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 41
Summary
• we propose an all-sky search for
gravitational waves from neutron stars in
binary systems
• a complete set of filters (complete to third
order in frequency) is used to
parameterize the signal.
• the correlation of the filters with the data
yield
– time of overlap – with better resolution than
FFT-time
– amplitude and frequency of signal
– first and second derivative of the frequency as
function of time
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 42
Summary
• after first step, amplitude and frequency
of the signal can be parameterized as a
function of time.
– candidates can be followed from 1 FFT to the
next
• Filters can be produced in a small
frequency band
– compared to different frequency bands in the
data
– stepsize in frequency determined by frequency
dependence of filter
• amount of CPU time is manageable
H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 43