data analysis for continuous gravitational wave signals

Data analysis for continuous gravitational wave signals

Alicia M. SintesUniversitat de les Illes Balears, Spain

Villa Mondragone International School

of Gravitation and Cosmology

Frascati, September 9th 2004

Photo credit: NASA/CXC/SAO

Frascati, September 9th 2004, A.M. Sintes

Talk overview

Gravitational waves from pulsars Sensitivity to pulsars Target searches

Review of the LSC methods & resultsFrequency 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


Gravitational waves from pulsars

– Pulsars (spinning neutron stars) are known to exist!

– Emit gravitational waves if they are non-axisymmetric:

Wobbling Neutron Star

Bumpy Neutron Star

Low Mass X-Ray Binaries


GWs from pulsars: The signal

• The GW signal from a neutron star:

• 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

Φ(t)eΑ(t)hh(t) 0

dt)(fS

(t)hT

gwh0

2

• (Signal-to-noise)2 ~


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


Signal received from an isolated pulsar

The detected strain has the form:

)();()();()( thtFthtFth

)

)

(t,ψF

(t,ψF 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.

)(sin

)(cos

tAh

tAh

0

10

)(0 ))()((

)!1(2)(

n

nn tTtTn

ft

the two independent wave polarizations

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 (~2s) are needed to convert between GPS and SSB time. Maximum phase mismatch ~10-2 radians for a few months


Signal model: isolated non-precessing neutron star

d

fI

c

Gh

hA

hA

gwzz2

4

2

0

0

20

4

cos

)cos1(2

1

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)

h0 - amplitude of the gravitational wave signal

- angle between the pulsar spin axis and line of sight

zz

yyxx

I

II - equatorial ellipticity


Hypothetical population of young, fast pulsars

(4 months @ 10 kpc)

Sensitivity to Pulsars

PSR J1939+2134(4 month observation)

Crab pulsar limit(4 month observation)

Crustal strain limit(4 months @ 10 kpc)

Sco X-1 to X-ray flux(1 day)


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

• 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

00 and ,cos , h


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.

RMS frequency deviations (Hz)

RMS phase deviations (rad)

Time interval(months)

2 1.3 1.9E-64 157.1 9.7E-56 2115.7 8.4E-48 12286 3.6E-3

10 46229.3 1.0E-212 134652.8 2.5E-2


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,

GEO LLO-4K LHO-4K LHO-2K 3x Coinc.

Duty cycle 98% 42% 58% 73% 24%


Directed Search in S1

NO DETECTION EXPECTED

at present sensitivities

PSR J1939+2134

1283.86 Hz

P’ = 1.0511 10-19 s/s

D = 3.6 kpc

Predicted signal for rotating neutron star with equatorial ellipticity =I/I : 10-3 , 10-4 , 10-5 @ 8.5 kpc.

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


Two search methods

Frequency domainConceived as a module in a hierarchical search

• 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.

Time domainprocess signal to remove frequency variations

due to Earth’s motion around Sun and spindown

• Best suited to target known objects, even if phase evolution is complicated

• Efficiently handless missing data

• Upper limits interpretation: Bayesian approach


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, , given the data and the template parameters that are known

00 and ,cos , h

x(t) F(f0) HIGH-PASS FILTER

WINDOWFT ComputeFStatistic

f0min < f0 < f0max

SFTs


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 2 distribution with 4 degrees of freedom with a non-centrality parameter (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*

• Use signal injection Monte Carlos to measure Probability Distribution Function (PDF) of F

*2 00 )2d()|2()(

FFhFphC


Measured PDFs for the F statistic with fake injected worst-case signals at nearby frequencies

2F* = 1.5: Chance probability 83%

2F*2F*

2F*2F*

2F* = 3.6: Chance probability 46%

2F* = 6.0: chance probability 20% 2F* = 3.4: chance probability 49%

h0 = 1.9E-21

95%95%

95%95%

h0 = 2.7E-22

h0 = 5.4E-22 h0 = 4.0E-22

Note: hundreds of thousands of injections were needed to get such nice clean statistics!


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 time-domain model for signal -- a function of the unknown source parameters h0 ,, and 0

00 202

12204

1 cos),()cos1)(,();( ii etFihetFhty a


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.

• The likelihood exp(-2 /2), where

• To get the posterior PDF for h0, marginalizing with respect to the nuisance parameters cos , and 0 given the data Bk

)|}({)(}){|( aaa kk BppBp

posterior prior likelihood

k k

k tyB2

2 );()(

a

a

cosddd}){|( 02/

0

2

eBhp k


Upper limit definition & detection

The 95% upper credible limit is set by the value h95 satisfying

Such an upper limit can be defined even when signal is present

A detection would appear as a maximum significantly offset from zero

95

0 00 d}){|(95.0h

k hBhp

h95

h95

Most probable h0

p(h 0|

{Bk}

)

p(h 0|

{Bk}

)

h0h0


Posterior PDFs for CW time domain analyses

Simulated injectionat 2.2 x10-21

p

shaded area = 95% of total area

p


S1 Results

No evidence of CW emission from PSR J1939+2134.

Summary of 95% upper limits for ho:

IFO Frequentist FDS Bayesian TDS

GEO (1.90.1) x 10-21 (2.20.1) x 10-21

LLO (2.70.3) x 10-22 (1.40.1) x 10-22

LHO-2K (5.40.6) x 10-22 (3.30.3) x 10-22

LHO-4K (4.00.5) x 10-22 (2.40.2) 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).


S2 Upper LimitsFeb 14 – Apr 14, 2003

• 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

95% upper limits


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

xx yy

zz

I I

I

40

2 24 gw zz

c r h

G f I

• Distance r to pulsar is known, Izz is assumed to be typical, 1045 g cm2


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


Preliminary results for P1

Parameters of P1:

P1: Constant Intrinsic FrequencySky position: 0.3766960246 latitude (radians)

5.1471621319 longitude (radians)Signal parameters are defined at SSB GPS time733967667.026112310 which corresponds to a wavefront passing:LHO at GPS time 733967713.000000000LLO at GPS time 733967713.007730720In the SSB the signal is defined byf = 1279.123456789012 Hzfdot = 0phi = 0psi = 0iota = /2h0 = 2.0 x 10-21


Preliminary results for P2

Parameters for P2:

P2: Spinning DownSky position: 1.23456789012345 latitude (radians)

2.345678901234567890 longitude (radians)Signal parameters are defined at SSB GPS time:SSB 733967751.522490380, which corresponds to awavefront passing:LHO at GPS time 733967713.000000000LLO at GPS time 733967713.001640320In the SSB at that moment the signal is defined byf=1288.901234567890123fdot = -10-8 [phase=2 pi (f dt+1/2 fdot dt^2+...)]phi = 0psi = 0iota = /2h0 = 2.0 x 10-21


GW pulsars in binary systems

Signal strengths for 20 days of integration

Sco X-1

• 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


Candidates

Rejected events

Pass chi2 cut ?

FL1 > Fchi2FH1 > Fchi2

FL1 < Fchi2FH1 < Fchi2

Pass chi2cut?

Consistent eventsin parameter space?

Compute F statistic over bank of filters and frequency range

Store results above “threshold”

Compute F statistic over bank of filters and frequency range

Store results above “threshold”

FL1 or FH1 < Fchi2

Coincidence Pipeline

yes

no

no

Upper-limitor

Follow up

H1 SFTs

L1 SFTs


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]:

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

Class f (Hz) (Yrs) NsDirected 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


Alternative search strategies

• The idea is to perform a search over the total observation time using an incoherent (sub-optimal) 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 methodPhase information is lost between data segments


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)

N

TT obs

obsT

Frequency

Tim

e


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: {,, f0, fn}

n

t

f

t

f


Hierarchical Hough transform strategy

Set upper-limit

Pre-processing

Divide the data set in N chunks

raw data

Construct set of short FT (tSFT)

Coherent search (,fi)

in a frequency band

Template placing

Incoherent search

Hough transform(f0, fi)

Peak selection in t-f plane

Candidates

selection

Candidates

selection


Incoherent Hough search Pipeline

Set upper-limit

Pre-processing

Divide the data set in N chunksraw data

Construct set of short FT (tSFT<1800s)

Incoherent search

Hough transform(f0, fi)

Peak selection in t-f plane

Candidates

selection


Peak selection in the t-f plane

• Input data: Short Fourier Transforms (SFT) of time series

• For every SFT, select frequency bins i suchexceeds some threshold th

time-frequency plane of zeros and ones

• p(|h, Sn) follows a 2 distribution with 2 degrees of freedom:

• The false alarm and detection probabilities for a threshold thare

2

2

|~|

|~|

)(fn

)(fx

i

ii

ii

iSFTin

ii )T(fS

)(fh 12

1|

~|4 2

2

d),|(),|(

,d),0|()|(

nnth

nnth

ShpSh

eSpS

th

th

th

(f)h(f)n(f)x~~~


Hough statistics

• After performing the HT using N SFTs, the probability that the pixel {,, f0, fi} has a number count n is given by a binomial distribution:

• The Hough false alarm and false dismissal probabilities for a threshold nth

Candidates selection

• For a given H, the solution for nth is

• Optimal threshold for peak selection : th ≈1.6 and ≈0.20

)1(,

)1(),|(

2 pNpNpn

ppn

NNpnP

n

nNn

presentsignal

absentsignal

p

1

0

)1(),,(

)1(),,(

th

th

n

n

nNnthH

N

nn

nNnthH

n

NNn

n

NNn

)2(erfc)1(2 1Hth NNn


The time-frequency pattern

• SFT data

• Demodulated data

c

ttftftf

nv

)()()()( 00

Time at the SSB for a given sky position

k

kN

k tTk

fftF

)(

!)( ˆ00

c

NtxttT

N

ˆ)()(ˆ

kkk Fff N̂

n

kF

kf

c

txtT

k

F

c

ttT

k

FtFt

k

kN

kk

kN

k )()(

)!1(

)()(

!)()(

1

ˆˆ0

v

)ˆ()()()( 0 NttFtf n


Validation code-Signal only case- (f0=500 Hz)

• bla


Validation code-Signal only case- (f0=500 Hz)


Noise only case


Results on simulated data

Statistics of the Hough maps

f0~300Hz

tobs=60days

N=1440 SFTs

tSFT=1800s =0.2231

<n> = N=321.3074

n=(N(1-))=15.7992


Number count probability distribution

1440 SFTs

1991 maps

58x58 pixels


Frequentist Analysis

• Perform the Hough transform for a set of points in parameter space ={,,f0,fi} S , given the data:

HT: S N n(

• Determine the maximum number count n*n* = max (n( S

• Determine the probability distribution p(n|h0) for a range of h0

30% 90%


Frequentist Analysis

• Perform the Hough transform for a set of points in parameter space ={,,f0,fi} S , given the data:

HT: S N n(

• Determine the maximum number count n*n* = max (n( 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

N

nn

%

n|hp*

95

00.95

•Compute p(n|h0) via Monte Carlo signal injection, using S , and 0

[0,2], [-/4,/4], cos [-1,1].


Set of upper limit. Frequentist approach.

95 %

N

nn

%

n|hp*

95

00.95

n*=395 =0.295

h095%


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

obs

n

T

SNh 4/1

0 5.8

obs

n

T

Sh 4.110

obs

n

T

SNh 4/1

0 6.7


Computational Engine

Searchs offline at:

• Medusa cluster (UWM)

• Merlin cluster (AEI)


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

data analysis for continuous gravitational wave signals

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