All-sky search for gravitational waves from neutron stars in binary systems

All-sky search for gravitational waves from neutron stars in binary

systems

strategy and algorithmsH.J. Bulten

H.J. Bulten - LSC-Virgo PSS June 9, 2008 2

analysis of PSS from binaries

thesis work of Sipho van der PuttenSipho 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


motivation: binary systems

• Virgo/Ligo: better sensitivity at higher frequency (>10 Hz)

• fixed quadrupole deformation:• most high-frequency neutron stars

are in binary systems– spin-up via gas transfer

• maybe other sources with extra variance in frequency (e.g. systems with very high spin-down)

2h f


motivation

• Brady et al.PRD57,2101:

old

binary

new?

239 2

2

10 [ ]2

IK f J

dK d IP I

dt dt

constant Power


solitary neutron stars

• solitary neutron star: Doppler shifts from earth movement

• Hierarchical search possible, T~ 1h (Rome group, e.g. Astona, Frasca, Palomba CQG 2005.)

• signal-to-noise ~

_

11 1

5

1

2

ˆ ˆ( )5 10

11.1 10 , about 1 hour for 1000 Hz

T FFTFFT

gw gw gw

FFTgw

fT

v r a rf f f t s f T

c c

Tf

obsT


solitary neutron stars

• alternative: F-statistics approach (Ligo, e.g. Jaranowski et all PRD58, 063001)

– produce templates that remain in phase over the template search time

– parameters – 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

( )0 0phase , , ; amplitude: , , , ,k

NSf h


Binary : Kepler orbitals• ellipse

• We want to analyze:– frequency shifts up to 0.3%, frequency

changes df/dt up to 10-6 s-2

– this includes :• orbital periods from 2 hours – infinite• masses companion star up to 15 solar masses• eccentricities up to about 0.7• 1 mHz shift in 1 second, at f=1000Hz

a

a

2 3 1/3

1

red shift depends on direction both axes

ph ap

T a v T

v v


frequency shiftsBinary system: 2.3 , 8 , 0.6, 56 , 0

inproduct major axis: -0.83, minor axis 0.52acc nT h M M


frequency derivativeBinary system: 2.3 , 8 , 0.6, 56 , 0

inproduct major axis: -0.83, minor axis 0.52acc nT h M M df a n

dt c


frequency shiftsBinary system: 2.3 , 8 , 0.6, 56 , 0

inproduct major axis: -0.83, minor axis 0.52acc nT h M M


coherence

• phase signal:

• signal should remain in-phase ,e.g. maximally about 60 deg. out of phase anywhere during observation time – frequency within ½ bin - 1/(2Tobs)

1( ) ( )

00 0

( )

0 0

ˆ ( )( ) 2

( 1)! ( )!

solitary: may be assumed fixed

phase parameters , ,

amplitude: , , , ,

binary:

: ( ) depends

k ks sk kd ns

NS NSk k

NS

kNS

d ns

n r rt tt f f

k c k

r

f

h

r r t

0,

on extra parameters:

, , , ( ), , ,orbit orbit companion ns major minor orbitT M M


binary neutron stars

• how many extra parameters?– e.g. orbital period >=2 hours, eccentricity

<=0.6, mass companion <=15 solar masses, frequency <=1000 Hz

– coherent: phase: distance to neutron star within 75 km w.r.t. template anywhere during the coherence time.

– all power coherent within 1 FFT-bin: Tmax = 30s– FFT length 1 hour: signal spreads over 4000

bins.– Tobs = 1 hour:

• detectable difference in orbital period: ~70 ms• a factor of 100,000 in parameter space to scan all

orbital periods between 2 and 4 hours in a blind search


binary neutron stars

• additional parameters:– even with Tobs = 1 hour, at least 100 billion

times as many templates are required to keep the phase of the filter coherent for all possibilities within the boundaries:

• T_orbit => 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


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

– third-order effects are assumed to be negligible.

• scan for presence of signal by calculating the correlation with the template

2 30 0

20

( ) 2 ( )2 6

( )2

t f t t t

f t f t t

dft

dt


Correlation

• Correlation is given by

• 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

*

*

( , ) ( ) ( ) ( ) ( )

( ) ( ) gives array, correlations for lags t 0... ( )

Corr g h g t h d G f H f

FFT G f H f N t


Filter search

FFT 1 FFT 2

filter, lag=0

Filter: zero-padded for half lengthcheck 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

data, split in overlapping periods

filter, scan to lag = T/2


Filter search

filter

filter

filter


Example Filters

0( ) sin , ( ) 2f t t f t

2 30

2 4 2 8 3

( ) sin , ( ) 22 6

( ) , 1.46 10 , 1.39 102

filt t t f t t t

df t t t Hz Hz

dt

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

– linear change of frequency:

3/ 2

max

10 , 0.25T

dff Hz

dt

26

/ 22

max

10 , 62 ,T

d ff mHz

dt


Phase: parameters

• for coherent times up to 500 seconds, the frequency should be accurate within about 1mHz.– phase description of data:

• about 10 phases • 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: can be determined• templates can be re-used• 600,000 templates reduce to about 5000

0

0


shifting in time

• shifting a filter in time by a lag tau gives a filter with parameters:

• you do not have to apply filters with with

2 20 0

20 0

1 1( ) ( ) ( )

2 21

2

f t t t f t t

f f

0 0

0 0

2, determined by ,

T

f f f f f


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).– reduction of a factor of

• Fourier-transform them• heterodyne data, or alternatively:

compare the filter in frequency domain with the appropriate frequency band of the Fourier-transformed data

6max 10F

fN

f


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

• total scans needed to analyze 0 - 1000 Hz, 1000 seconds– about 10,000 filters suffice.– about 300 million correlations in total (300 million FFT products)– about a day of CPU-time on a single CPU, current desktop

max( ) min( )filter filterf f f


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 bin2/ 2

*

0

/ 2*

0

: from data, normalized to a PSD of 1

| | 2 , ,8

average 0, RMS =2

threshold: 4 sigma (on amplitude)

signal overlap

filter

filter

i

Nfilter

i samp FFTbins i ii

Nfilter

i off i Samp FFTbinsi

n FFT

Nn N N f f

Nn f N N

*maximum bin in FFT( ) estimator: i off in f

2 samp FFTN N


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)


First tests

• spectrum : Gaussian-distributed noise with mean zero and amplitude – one-sided PSD of

• 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 length 1024 seconds, 2048 samples/sec.

• 30 FFT sets (about 5 hours)

2310 sampN

2310 / Hz


Overlap of filters, only noise

maximum correlationfor all filters applied between 0 and 1000 Hz(81.5 million FFT products,4096 lags per filter)

4*FFTbins

filter

NN

f


Overlap of filters with signal

maximum correlation with signalfor all filters applied between 0 and 1000 Hz(81.5 million FFT products)(most are <0.001)

4*FFTbins

filter

NN

f


signal-to-noise

4*FFTbins

filter

NN

f

cut, 4


Power spectral density

PSD signal+noise

time (interleaved FFT sets)


PSD, signal only



PSD: 30 FFTs added


PSD of 30 FFTs added


PSD, signal only



Search results

• 30 FFTs, about 5h of data• analyzed between 100 and 500 Hz

– 2405 different filters– in total about 30 h CPU-time on my desktop

PC (dual core ~ 3GHz)

• Applied threshold: 4 sigma• about 1.3 billion filter multiplications,

28731 hits in simulated dataset (10 pulsars+noise)

• analysis on files with signal (10 pulsars) only: 14972 hits


Search results, all hits


correlations with signal-only


Search results, signal+noise


Search results, signal only


Comparison: cut on power

Cut: 4 sigma on power

FFT –number


Alternative: cut on power

Cut: 4 sigma on power

7649 hits between 450 and 460 Hz


highest PSD in data

FFT –number


PSD: signal only

signal highest PSDdata still spread out over about

30 bins here


Summary

• we propose to search for gravitational waves by applying filters that describe the phase of the signal ad hoc (large parameter reduction), up to cubic time dependence– method should be good for short FFT base– less parameters than F-statistics approach– longer FFT base possible than Hough approach

• hits 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


Summary

• first step: for high thresholds these filters are very effective to remove noise– for a time base of 1000 seconds, a 4-sigma

threshold may be set. Full power is recovered, even though it may be spread out over 50 bins or more.

• after first step, amplitude and frequency of the signal can be parameterized as a function of time.– time/frequency dependence is known, filter

is known.– candidates can be followed up from 1 FFT to

the next : a hit predicts possible follow-ups


Concluding remarks

• Second step hierarchical approach needs to be developed yet– how to go to a longer time base

• parameterize frequency as a function of time?• use amplitude?• use hits in consecutive FFTs as

confirmation/rejection?

– how to go to the astrophysical source in case of GW detection

• although we would be glad with the detection anyways...

• Next actions– documenting– further developing framework, run on grid

Documents

All-sky search for gravitational waves from neutron stars in binary systems