Download Der Gravitationswellendetektor GEO600

Data analysis for continuous
gravitational wave signals
Alicia M. Sintes
Universitat de les Illes Balears, Spain
Villa Mondragone International School
of Gravitation and Cosmology
Frascati, September 9th 2004
Photo credit: NASA/CXC/SAO
Talk overview
 Gravitational waves from pulsars
 Sensitivity to pulsars
 Target searches
 Review of the LSC methods & results
Frequency domain method
Time domain method
Hardware injection of fake pulsars
Pulsars in binary systems
 The problem of blind surveys
 Incoherent searches
The Radon & Hough transform
 Summary and future outlook
Frascati, September 9th 2004, A.M. Sintes
Gravitational waves from pulsars
– Pulsars (spinning neutron stars) are known to exist!
– Emit gravitational waves if they are non-axisymmetric:
Low Mass X-Ray Binaries
Wobbling Neutron Star
Bumpy Neutron Star
Frascati, September 9th 2004, A.M. Sintes
GWs from pulsars: The signal
• The GW signal from a neutron star:
h(t)  h0 Α(t)e
Φ(t)
• Nearly-monochromatic continuous signal
– spin precession at ~frot
– excited oscillatory modes such as the r-mode at 4/3* frot
– non-axisymmetric distortion of crystalline structure, at 2frot
T
•
(Signal-to-noise)2 ~
2
h (t)
0 Sh(f gw )dt
Frascati, September 9th 2004, A.M. Sintes
Signal received from a pulsar
• A gravitational wave signal we detect from
a pulsar will be:
– Frequency modulated by relative motion of
detector and source
– Amplitude modulated by the motion of the
antenna pattern of the detector
R
Frascati, September 9th 2004, A.M. Sintes
Signal received from an isolated pulsar
The detected strain has the form:
h(t )  F (t ; ) h  (t )  F (t ; ) h (t )
F(t,ψ )

F(t,ψ ) 
strain antenna patterns of the detector to plus and cross polarization,
bounded between -1 and 1. They depend on the orientation of the
detector and source and on the polarization of the waves.
h   A cos (t )
the two independent wave polarizations
h  A sin (t )

 (t )  0  2 
n 0
f ( n)
(n  1)!
(T (t )  T (t0 )) n1
the phase of the received signal depends on the
initial phase and on the frequency evolution of the
signal. This depends on the spin-down parameters
and on the Doppler modulation, thus on the
frequency of the signal and on the instantaneous
relative velocity between source and detector
T(t) is the time of arrival of a signal at the solar system barycenter (SSB).
Accurate timing routines (~2s) are needed to convert between GPS and SSB time.
Maximum phase mismatch ~10-2 radians for a few months
Frascati, September 9th 2004, A.M. Sintes
Signal model:
isolated non-precessing neutron star
Limit our search to gravitational waves
from an isolated tri-axial neutron star
emitted at twice its rotational frequency
(for the examples presented here, only)
1
A  h0 (1  cos 2  )
2
A  h0 cos
4 G I zz f gw
h0  4
c
d
2
2
h0 - amplitude of the gravitational
wave signal
 - angle between the pulsar spin
axis and line of sight
Ixx  Iyy
- equatorial ellipticity

Izz
Frascati, September 9th 2004, A.M. Sintes
Sensitivity to Pulsars
Crab pulsar limit
(4 month observation)
Hypothetical population of
young, fast pulsars
(4 months @ 10 kpc)
Crustal strain limit
(4 months @ 10 kpc)
Sco X-1 to X-ray flux
(1 day)
PSR J1939+2134
(4 month observation)
Frascati, September 9th 2004, A.M. Sintes
Target known pulsars
• For target searches only one search template (or a reduced
parameter space) is required
e.g. search for known radio pulsars with frequencies (2frot) in detector
band . There are 38 known isolated radio pulsars fGW > 50 Hz
– Known parameters:
position, frequency and spin-down (or approximately)
– Unknown parameters:
amplitude, orientation, polarization and phase h0 , cos, and 0
• Timing information provided from radio observations
In the event of a glitch we would need to add an extra parameter for
jump in GW phase
• Coherent search methods can be used
i.e. those that take into account amplitude and phase information
Frascati, September 9th 2004, A.M. Sintes
Crab pulsar
Young pulsar (~60 Hz) with large spin-down and timing
noise. Frequency residuals will cause large deviations in
phase if not taken into account.
E.g., use Jodrell Bank monthly crab ephemeris to recalculate
spin down parameters. Phase correction can be applied
using interpolation between monthly ephemeris.
Time
interval
2
(months)
4
6
8
10
12
RMS phase RMS frequency
deviations deviations (Hz)
1.3
1.9E-6
(rad)
157.1
2115.7
12286
46229.3
134652.8
9.7E-5
8.4E-4
3.6E-3
1.0E-2
2.5E-2
Frascati, September 9th 2004, A.M. Sintes
Review of the LSC
methods and results
• S1 run: Aug 23 - Sep 9 2002 (~400 hours).
LIGO, plus GEO and TAMA
“Setting upper limits on the strength of periodic gravitational waves
using the first science data from the GEO600 and LIGO detectors”
Phys. Rev. D69, 082004 (2004), gr-qc/0308050,
Duty cycle
GEO
LLO-4K
LHO-4K
LHO-2K
3x Coinc.
98%
42%
58%
73%
24%
Frascati, September 9th 2004, A.M. Sintes
Directed Search in S1
NO DETECTION EXPECTED
at present sensitivities
Crab Pulsar
Detectable amplitudes with a 1%
false alarm rate and 10% false
dismissal rate
Upper limits on from spin-down
measurements of known radio
pulsars
Predicted signal for rotating
neutron star with equatorial
ellipticity  = d I/I : 10-3 , 10-4 , 10-5
@ 8.5 kpc.
PSR J1939+2134
1283.86 Hz
P’ = 1.0511 10-19 s/s
D = 3.6 kpc
Frascati, September 9th 2004, A.M. Sintes
Two search methods
Frequency domain
Time domain
Conceived as a module in a hierarchical search
process signal to remove frequency variations
due to Earth’s motion around Sun and
spindown
• Best suited for large
parameter space searches
(when signal characteristics are uncertain)
• Straightforward implementation
of standard matched filtering
technique (maximum likelihood
detection method):
Cross-correlation of the signal with the
template and inverse weights with the
noise
• Frequentist approach used to
cast upper limits.
• Best suited to target known
objects, even if phase
evolution is complicated
• Efficiently handless missing
data
• Upper limits interpretation:
Bayesian approach
Frascati, September 9th 2004, A.M. Sintes
Frequency domain search
The input data are a set of SFTs of the time domain data,
with a time baseline such that the instantaneous frequency of a putative signal does
not move by more than half a frequency bin.
The original data being calibrated, high-pass filtered & windowed.
Data are studied in a narrow frequency band. Sh(f) is estimated from each SFT.
Dirichlet Kernel used to combine data from different SFTs
(efficiently implements matched filtering, or other alternative methods).
Detection statistic used is described in Jaranoski, Krolak, Schutz, Phys. Rev.
D58(1998)063001
The F detection statistic provides the maximum value of the likelihood ratio
with respect to the unknown parameters, h0 , cos,  and 0 , given
the data and the template parameters that are known
f0min < f0 < f0max
x(t)
HIGH-PASS FILTER
FT
WINDOW
SFTs
ComputeFStatistic
F(f0)
Frascati, September 9th 2004, A.M. Sintes
Frequency domain method
• The outcome of a target search is a number F* that represents the
optimal detection statistic for this search.
• 2F* is a random variable: For Gaussian stationary noise, follows a
c2 distribution with 4 degrees of freedom with a non-centrality
parameter l(h|h). Fixing ,  and 0 , for every h0, we can obtain a
pdf curve: p(2F|h0)
• The frequentist approach says the data will contain a signal with
amplitude  h0 , with confidence C, if in repeated experiments, some
fraction of trials C would yield a value of the detection statistics  F*
C (h0 )  

2F*
p(2F | h0 )d(2F )
• Use signal injection Monte Carlos to measure Probability
Distribution Function (PDF) of F
Frascati, September 9th 2004, A.M. Sintes
Measured PDFs for the F statistic
with fake injected worst-case signals at nearby frequencies
h0 = 1.9E-21
Note:
hundreds
of
thousands
of injections
were
needed to
get such
nice clean
statistics!
h0 = 2.7E-22
95%
95%
2F* = 1.5: Chance probability 83%
2F*
2F*
2F* = 3.6: Chance probability 46%
h0 = 5.4E-22
95%
h0 = 4.0E-22
95%
2F*
2F* = 6.0: chance probability 20%
2F*
2F* = 3.4: chance probability 49%
Frascati, September 9th 2004, A.M. Sintes
Time domain target search
• Method developed to handle known complex phase evolution.
Computationally cheap.
• Time-domain data are successively heterodyned to reduce the
sample rate and take account of pulsar slowdown and Doppler
shift,
– Coarse stage (fixed frequency) 16384  4 samples/sec
– Fine stage (Doppler & spin-down correction)  1 samples/min  Bk
– Low-pass filter these data in each step. The data is down-sampled via
averaging, yielding one value Bk of the complex time series, every 60
seconds
• Noise level is estimated from the variance of the data over each
minute to account for non-stationarity.  k
• Standard Bayesian parameter fitting problem, using timedomain model for signal -- a function of the unknown source
parameters h0 ,,  and 0
y (t; a)  41 h0F (t, )(1 cos2  )e2i0  21 ih0F (t, ) cose2i0
Frascati, September 9th 2004, A.M. Sintes
Time domain: Bayesian approach
• We take a Bayesian approach, and determine the joint posterior
distribution of the probability of our unknown parameters, using
uniform priors on h0 ,cos ,  and 0 over their accessible values,
i.e.
p(a | {Bk })  p(a)  p({Bk } | a)
posterior
• The likelihood 
prior
exp(-c2 /2),
likelihood
2
where c (a)  
Bk  y (t ; a )
k
2
k
• To get the posterior PDF for h0, marginalizing with respect to the
nuisance parameters cos ,  and 0 given the data Bk
p(h0 | {Bk })   e
c 2 / 2
d0d d cos
Frascati, September 9th 2004, A.M. Sintes
Upper limit definition & detection
The 95% upper credible limit is set by the value h95 satisfying
0.95  
h95
0
p(h0 | {Bk })dh0
Such an upper limit can be defined even when signal is present
A detection would appear as a maximum significantly offset from
zero
h95
h0
p(h0|{Bk})
p(h0|{Bk})
Most probable h0
h95
h0
Frascati, September 9th 2004, A.M. Sintes
Posterior PDFs for
CW time domain analyses
Simulated injection
at 2.2 x10-21
p
shaded area =
95% of total area
p
Frascati, September 9th 2004, A.M. Sintes
S1 Results
No evidence of CW emission from PSR J1939+2134.
Summary of 95% upper limits for ho:
IFO
Frequentist FDS
Bayesian TDS
GEO
(1.90.1) x 10-21
(2.20.1) x 10-21
LLO
(2.70.3) x 10-22
(1.40.1) x 10-22
LHO-2K
(5.40.6) x 10-22
(3.30.3) x 10-22
LHO-4K
(4.00.5) x 10-22
(2.40.2) x 10-22
Previous results for this pulsar:
ho < 10-20 (Glasgow, Hough et al., 1983),
ho < 1.5 x 10-17 (Caltech, Hereld, 1983).
Frascati, September 9th 2004, A.M. Sintes
S2 Upper Limits
Feb 14 – Apr 14, 2003
95% upper limits
• Performed joint coherent
analysis for 28 pulsars using data
from all IFOs
• Most stringent UL is for pulsar
J1910-5959D (~221 Hz) where
95% confident that h0 < 1.7x10-24
• 95% upper limit for Crab pulsar
(~ 60 Hz) is h0 < 4.7 x 10-23
• 95% upper limit for J1939+2134
(~ 1284 Hz) is h0 < 1.3 x 10-23
Frascati, September 9th 2004, A.M. Sintes
Equatorial Ellipticity
• Results on h0 can be interpreted as upper limit on equatorial
ellipticity
• Ellipticity scales with the difference in radii along x and y axes

I xx  I yy
I zz
c4
r h0

 2 
2
4 G f gw I zz
• Distance r to pulsar is known, Izz
is assumed to be typical, 1045 g
cm2
Frascati, September 9th 2004, A.M. Sintes
S2 hardware injections
• Performed end-to-end validation of analysis pipeline by
injecting simultaneous fake continuous-wave signals into
interferometers
• Two simulated pulsars were injected in the LIGO
interferometers for a period of ~ 12 hours during S2
• All the parameters of the injected signals were
successfully inferred from the data
Frascati, September 9th 2004, A.M. Sintes
Preliminary results for P1
Parameters of P1:
P1: Constant Intrinsic Frequency
Sky position: 0.3766960246 latitude (radians)
5.1471621319 longitude (radians)
Signal parameters are defined at SSB GPS time
733967667.026112310 which corresponds to a
wavefront passing:
LHO at GPS time 733967713.000000000
LLO at GPS time 733967713.007730720
In the SSB the signal is defined by
f = 1279.123456789012 Hz
fdot = 0
phi = 0
psi = 0
iota = /2
h0 = 2.0 x 10-21
Frascati, September 9th 2004, A.M. Sintes
Preliminary results for P2
Parameters for P2:
P2: Spinning Down
Sky position: 1.23456789012345 latitude (radians)
2.345678901234567890 longitude (radians)
Signal parameters are defined at SSB GPS time:
SSB 733967751.522490380, which corresponds to a
wavefront passing:
LHO at GPS time 733967713.000000000
LLO at GPS time 733967713.001640320
In the SSB at that moment the signal is defined by
f=1288.901234567890123
fdot = -10-8 [phase=2 pi (f dt+1/2 fdot dt^2+...)]
phi = 0
psi = 0
iota = /2
h0 = 2.0 x 10-21
Frascati, September 9th 2004, A.M. Sintes
GW pulsars in binary systems
• Physical scenarios:
• Accretion induced temperature
asymmetry (Bildsten, 1998;
Ushomirsky, Cutler, Bildsten,
2000; Wagoner, 1984)
• R-modes (Andersson et al,
1999; Wagoner, 2002)
• LMXB frequencies are clustered
(could be detected by advanced
LIGO).
• We need to take into account the
additional Doppler effect produced
by the source motion:
– 3 parameters for circular orbit
– 5 parameters for eccentric orbit
– possible relativistic corrections
– Frequency unknown a priori
Sco X-1
Signal strengths for
20 days of integration
Frascati, September 9th 2004, A.M. Sintes
L1
SFTs
Coincidence Pipeline
Compute F statistic over
bank of filters and
frequency range
Store results above “threshold”
H1
SFTs
Compute F statistic over
bank of filters and
frequency range
Store results above “threshold”
Consistent events
in parameter space?
no
yes
Upper-limit
or
Follow up
FL1 > Fchi2
FH1 > Fchi2
FL1 or FH1
< Fchi2
Pass
chi2
cut ?
Pass
chi2
cut?
Candidates
FL1 < Fchi2
FH1 < Fchi2
no
Rejected
events
Frascati, September 9th 2004, A.M. Sintes
All-Sky and targeted surveys
for unknown pulsars
It is necessary to search for every signal template distinguishable in parameter
space. Number of parameter points required for a coherent T=107s search
[Brady et al., Phys.Rev.D57 (1998)2101]:
Class
f (Hz)
t (Yrs)
Ns
Directed
All-sky
Slow-old
<200
>103
1
3.7x106
1.1x1010
Fast-old
<103
>103
1
1.2x108
1.3x1016
Slow-young
<200
>40
3
8.5x1012
1.7x1018
Fast-young
<103
>40
3
1.4x1015
8x1021
Number of templates grows dramatically with the integration time. To
search this many parameter space coherently, with the optimum sensitivity
that can be achieved by matched filtering, is computationally prohibitive.
=>It is necessary to explore alternative search strategies
Frascati, September 9th 2004, A.M. Sintes
Alternative search strategies
• The idea is to perform a search over the total
observation time using an incoherent (suboptimal) method:
We propose to search for evidence of a signal whose
frequency is changing over time in precisely the
pattern expected for some one of the parameter sets
• The methods used are
– Radon transform
– Hough transform
– Power-flux method
Phase information is lost between data segments
Frascati, September 9th 2004, A.M. Sintes
The Radon transform
• Break up data into N segments
• Take the Fourier transform of each segment and track the Doppler
shift by adding power in the frequency domain (Stack and Slide)
Tobs
T 
Time
Frequency
Tobs
N
Frascati, September 9th 2004, A.M. Sintes
The Hough transform
• Robust pattern detection technique developed at CERN to look for patterns
in bubble chamber pictures. Patented by IBM and used to detect patterns in
digital images
• Look for patterns in the time-frequency plane
• Expected pattern depends on: {a,d, f0, fn}
n
f
f
t
a
d
t
Frascati, September 9th 2004, A.M. Sintes
Hierarchical Hough transform strategy
Pre-processing
raw data
Divide the data set in N chunks
Template placing
Coherent search (a,d,fi)
Construct set of short
FT (tSFT)
in a frequency band
Incoherent search
Peak selection
in t-f plane
Set upper-limit
Hough transform
(a, d, f0, fi)
Candidates
selection
Candidates
selection
Frascati, September 9th 2004, A.M. Sintes
Incoherent Hough search Pipeline
Pre-processing
raw data
Divide the data set in N chunks
Construct set of short FT
(tSFT<1800s)
Incoherent search
Peak selection
in t-f plane
Hough transform
(a, d, f0, fi)
Candidates
selection
Set upper-limit
Frascati, September 9th 2004, A.M. Sintes
Peak selection in the t-f plane
• Input data: Short Fourier Transforms (SFT) of time series
•
~(f)  h~(f)
For every SFT, select frequency bins i such ~
x (f)  n
|~
x (f i ) |2
ri  ~
| n (f i ) |2
exceeds some threshold rth
 time-frequency plane of zeros and ones
• p(r|h, Sn) follows a c2 distribution with 2 degrees of freedom:
~
4 | h (f i ) |2
li 
S n(f i )TSFT
ri  1 
li
 r 2  1  li
2
• The false alarm and detection probabilities for a threshold rth are
a ( r th | S n ) 

r
p (r | 0, S n ) dr  e  rth ,
th
 ( r th | h, S n ) 

r p(r | h, S
n
) dr
th
Frascati, September 9th 2004, A.M. Sintes
Hough statistics
• After performing the HT using N SFTs, the probability that the pixel {a,d,
f0, fi} has a number count n is given by a binomial distribution:
N n
N n
P ( n | p, N )  
n
 p (1  p )
 
n  Np,
 n 2  Np(1  p )
a
p

signal absent
signal present
• The Hough false alarm and false dismissal probabilities for a threshold nth
N n
N n


a
(
1

a
)

 
n  nth  n 
 Candidates selection
nth 1
N
 H ( nth , , N )     n (1   ) N  n
n 0  n 
a H ( nth , a , N ) 
N
• For a given aH, the solution for nth is
nth  Na  2 Na (1  a ) erfc 1 (2a H )
• Optimal threshold for peak selection : rth ≈1.6 and a ≈0.20
Frascati, September 9th 2004, A.M. Sintes
The time-frequency pattern
v(t )  n
f (t )  f 0 (t )  f 0 (t )
c
• SFT data

f (t )  F0 (t )   (t ) (n  Nˆ )
• Demodulated data
n fk
N̂ Fk
f k
F0 (t )  f 0  
TNˆ (t )
k!
k


k
f k  f k  Fk

x (t )  Nˆ
TNˆ (t )  t 
  Time at the SSB for a given sky position
c

k
k 1


 v(t ) 
 x (t )
Fk
Fk
 (t )   F0 (t )  
TNˆ (t ) 
  
TNˆ (t ) 
k k!

 c
 k (k  1)!
 c




Frascati, September 9th 2004, A.M. Sintes
Validation code
-Signal only case- (f0=500 Hz)
• bla
Frascati, September 9th 2004, A.M. Sintes
Validation code
-Signal only case- (f0=500 Hz)
Frascati, September 9th 2004, A.M. Sintes
Noise only case
Frascati, September 9th 2004, A.M. Sintes
Results on simulated data
Statistics of the
Hough maps
f0~300Hz
tobs=60days
N=1440 SFTs
tSFT=1800s
a=0.2231
<n> = Na=321.3074
n=(Na(1-a))=15.7992
Frascati, September 9th 2004, A.M. Sintes
Number count probability distribution
1440 SFTs
1991 maps
58x58 pixels
Frascati, September 9th 2004, A.M. Sintes
Frequentist Analysis
• Perform the Hough transform for a set of points in parameter space
l={a,d,f0,fi} S , given the data:
HT: S  N
l  n(l)
• Determine the maximum number count n*
n* = max (n(l)): l  S
• Determine the probability distribution p(n|h0) for a range of h0
30%
90%
Frascati, September 9th 2004, A.M. Sintes
Frequentist Analysis
• Perform the Hough transform for a set of points in parameter space
l={a,d,f0,fi} S , given the data:
HT: S  N
l  n(l)
• Determine the maximum number count n*
n* = max (n(l)): l  S
• Determine the probability distribution p(n|h0) for a range of h0
• The 95% frequentist upper limit h095% is the value such that for repeated
trials with a signal h0 h095%, we would obtain n  n* more than 95% of
the time
0.95 
 pn|h )
N
95%
0
n  n*
•Compute p(n|h0) via Monte Carlo signal injection, using l  S , and 0
[0,2],  [-/4,/4], cos [-1,1].
Frascati, September 9th 2004, A.M. Sintes
Set of upper limit.
Frequentist approach.
n*=395  =0.295
 h095%
0.95 
95 %
 pn|h )
N
95%
0
n  n*
Frascati, September 9th 2004, A.M. Sintes
Comparison of sensitivity
• For a matched filter search directed at a single
point in parameter space, smallest signal that
can be detected with a false dismissal rate of
10% and false alarm of 1% is
• For a Hough search with N segments, for same
confidence levels and for large N:
• For the Radon transform
• For, say N = 2000, loss in sensitivity is only
about a factor of 5 for a much smaller
computational cost
Sn
h 0 11.4
Tobs
h 0  8.5 N 1/ 4
Sn
Tobs
h 0  7.6 N
Sn
Tobs
1/ 4
Frascati, September 9th 2004, A.M. Sintes
Computational Engine
Searchs offline at:
• Medusa cluster (UWM)
• Merlin cluster (AEI)
Frascati, September 9th 2004, A.M. Sintes
Summary and future outlook
• S2 run (Feb 14, 2003 - Apr 14, 2003)
–
–
–
–
Time-domain analysis of 28 known pulsars
Broadband frequency-domain all-sky search
ScoX-1 LMXB frequency-domain search
Incoherent searches.
• S3 run (Oct 31, 2003 – Jan 9, 2004)
– Time-domain analysis on more pulsars, including binaries
– Improved sensitivity LIGO/GEO run. Approaching spin-down limit
for Crab pulsar
• Future:
– Implement hierarchical analysis that layers coherent and
incoherent methods
– Grid searches
– Einstein@home initiative for 2005 World Year of Physics
Frascati, September 9th 2004, A.M. Sintes