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Data analysis for continuous gravitational wave signals

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### 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

Wobbling Neutron Star

Bumpy Neutron Star

Gravitational waves from pulsars- Pulsars (spinning neutron stars) are known to exist!
- Emit gravitational waves if they are non-axisymmetric:

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

(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

Signal received from an isolated pulsar

The detected strain has the form:

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.

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

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

- equatorial ellipticity

(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)

Sensitivity to PulsarsTarget 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

RMS frequency

deviations (Hz)

Time interval

(months)

Crab pulsarYoung pulsar (~60 Hz) with large spin-down and timing noise. Frequency residuals will cause large deviations in phase if not taken intoaccount.

E.g., use Jodrell Bank monthly crab ephemeris to recalculate spin down parameters. Phase correction can be applied using interpolation between monthly ephemeris.

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,

NO DETECTION EXPECTED

at present sensitivities

Detectable amplitudes with a 1% false alarm rate and 10% false dismissal rate

Upper limits on from spin-down measurements of known radio pulsars

Crab Pulsar

Predicted signal for rotating neutron star with equatorial ellipticity e = 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

Frequency domain

Conceived 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 domain

process 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

Two search methodsHIGH-PASS FILTER

WINDOW

ComputeFStatistic

f0min < f0 < f0max

x(t)

F(f0)

SFTs

Frequency domain searchThe 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 inJaranoski, 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

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 c2distribution 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)
- Thefrequentistapproach 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

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

h0 = 1.9E-21

h0 = 2.7E-22

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

95%

95%

2F*

2F*

2F* = 1.5: Chance probability 83%

2F* = 3.6: Chance probability 46%

h0 = 5.4E-22

h0 = 4.0E-22

95%

95%

2F*

2F*

2F* = 6.0: chance probability 20%

2F* = 3.4: chance probability 49%

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 Bkof 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

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 0over 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 0given the data Bk

p(h0|{Bk})

p(h0|{Bk})

h95

h95

h0

h0

Upper limit definition & detectionThe 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

Posterior PDFs for CW time domain analyses

Simulated injection

at 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

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

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

- 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 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 = p/2

h0 = 2.0 x 10-21

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 = p/2

h0 = 2.0 x 10-21

Signal strengths for

20 days of integration

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

or

Follow up

L1

SFTs

H1

SFTs

Coincidence PipelineCompute 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”

Consistent events

in parameter space?

no

yes

FL1 > Fchi2

FH1 > Fchi2

FL1 or FH1

< Fchi2

FL1 < Fchi2

FH1 < Fchi2

Pass

chi2

cut ?

Pass

chi2

cut?

no

Candidates

Rejected

events

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

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 method

Phase information is lost between data segments

Time

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

f

d

a

t

t

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}

in a frequency band

raw data

Incoherent search

Peak selection

in t-f plane

Hough transform(a, d, f0, fi)

Hierarchical Hough transform strategyPre-processing

Divide the data set in N chunks

Template placing

Construct set of short FT (tSFT)

Candidates

selection

Set upper-limit

Candidates

selection

raw data

Divide the data set in N chunks

Incoherent search

Peak selection

in t-f plane

Hough transform(a, d, f0, fi)

Incoherent Hough search PipelineConstruct set of short FT (tSFT<1800s)

Candidates

selection

Set upper-limit

Peak selection in the t-f plane

- Input data: Short Fourier Transforms (SFT) of time series
- For every SFT, select frequency bins i such exceeds some threshold rth

time-frequency plane of zeros and ones

- p(r|h, Sn) follows a 2 distribution with 2 degrees of freedom:
- The false alarm and detection probabilities for a thresholdrthare

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:
- The Hough false alarm and false dismissal probabilities for a threshold nth

Candidates selection

- For a given aH, the solution for nth is
- Optimal threshold for peak selection : rth ≈1.6 and a ≈0.20

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

90%

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)): lS

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

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)): lS

- 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

Compute p(n|h0) via Monte Carlo signal injection, using lS, and 0[0,2], [-/4,/4], cos [-1,1].

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

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

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