alicia m. sintes universitat de les illes balears pisa,1 june 2007 data analysis searches for...

Alicia M. Sintes Universitat de les Illes Balears

Pisa,1 June 2007

Data analysis searches for continuous gravitational waves

VESF-school, Pisa, June 2007, A.M. Sintes 2

Content

• Gravitational waves from spinning neutron stars.– Emission mechanisms– Signal model

• Data analysis for continuous gravitational waves– Bayesian analysis– Frequentist approach

• Brief overview of the searches – Directed pulsar search– Wide-parameter search (all sky)

• Coherent methods• Einstein@Home• Hierarchical strategies• Semi-coherent methods

• Summary of results and perspectives


Rotating neutron stars

(NASA/CXC/SAO)

NASA

• Neutron stars can form from the remnant of stellar collapse

• Typical size of 10km, and are about 1.4 solar masses

• Some of these stars are observed as pulsars

• Gravitational waves from neutron stars could tell us about the equation of state of dense nuclear matter

• Our galaxy might contain ~109 NS, of which ~105 are expected to be active pulsars. Up to know ~1700 pulsars have been identified

• search for observed neutron stars

• all sky search (computing challenge)


Neutron Stars Sources

• Great interest in detecting radiation: physics of such stars is poorly understood.

– After 40 years we still don’t know what makes pulsars pulse.

– Interior properties not understood: equation of state, superfluidity, superconductivity, solid core, source of magnetic field.

– May not even be neutron stars: could be made of strange matter!


“Continuous” gravitational waves from neutron stars

Various physical mechanisms could operate in a NS to produce interesting levels of GW emission

As the signal-strength is generally expected to be weak, long integrations times (several days to years) are required in order for the signal to be

detectable in the noise.

Therefore this GW emission has to last for a long time, which characterizes the class of ‘continuous-wave’ signals.

NS might also be interesting sources of burst-like GW emission, f-mode or p-mode oscillations excited by a glitch

crustal torsion modes

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.


Bumpy Neutron Star

Emission mechanisms for continuous GW from spinning NS in the LIGO-VIRGO frequency band

– Non-axisymmetric distortions

– Unstable oscillation modes in the fluid part of the star

– Free precession

Wobbling Neutron Star

Low Mass X-Ray Binaries

Magnetic mountains R-modes in accreting stars


Bumpy Neutron Star

1) Non-axisymmetric distortions

A non-axisymmetric neutron star at distance a d, rotating with frequency ν around the Izz axis emits monochromatic GWs of frequency f = 2ν with an amplitude

h0 4 2G

c 4

Izz fgw2

d

Ixx Iyy

Izz

•The strain amplitude h0 refers to a GW from an optimally

oriented source with respect to the detector •The equatorial ellipticity is highly uncertain, ~10-7. In the most speculative model can reach up to 10-4.

•Accreating neutron stars in binary systems can also have large crust deformations

•Strong internal magnetic fields could produce deformations of up to ~10-6. These deformations would result in GW emission at f = ν and f = 2ν.

Magnetic mountains

VESF-school, Pisa, June 2007, A.M. Sintes 8Wobbling Neutron Star

2) Non-axisymmetric instabilities

At birth or during accretion, rapidly rotating NS can be subject to various non-axisymmetric instabilities, which would lead to GW emission,

The r-mode instability has been proposed as a source of GWs (with frequency f = 4ν/3) from newborn NS and from rapidly accreting NS.

R-modes in accreting stars

3) Free precessionresults in emission at (approximately) the rotation rate ν and twice the rotation rate, i.e. f =ν+νprec and f = 2ν.






Loudest expected signal

From isolated neutron stars (Blandford’s argument):Although there is great uncertainty in the physics of the GW emission mechanisms and the strength of individual sources, one can argue for a statistical upper limit on the expected strongest GW signals from the galactic population of neutron stars, which is almost independent of individual source physics.

h0~410-24

Is the optimistic upper limit, that there is a 50% chance that the strongest signal has at least that amplitude in the LIGO-VIRGO band, assuming NS are uniformly distributed in the galactic disc and a constant overall galactic bith-rate.

Spindown limit for known pulsarsIf the GW emission is powered by the rotational energy of the NS, then one obtains: ≤ sd, h0≤ hsd




• The ‘periodic’ 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

The signal from a NS

h(t) h0 A(t)e(t)

h2(t)

Sh(fgw)0

T

dt• (Signal-to-noise)2 ~


The expected signal at the detector

A gravitational wave signal we detect from a NS will be: – Frequency modulated by relative motion of

detector and source

– Amplitude modulated by the motion of the non-uniform antenna sensitivity pattern of the detector

R


Signal received from an isolated NS

h(t) F(t;) h (t) F(t;) h(t)

F+ and F are the strain antenna patterns. They depend on the orientation of the detector and source and on the polarization of the waves.

h A cos(t)

hAsin(t)

(t)0 2f(n )

(n 1)!(T(t) T(t0))n1

n0

the phase of the received signal depends on the initial phase, the frequency evolution 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, t the time at the detector.


Signal model: isolated non-precessing NS

A 1

2h0(1cos2 )

A h0 cos

h0 4 2G

c 4

Izz fgw2

d

h0 - amplitude of the gravitational wave signal

- angle between the pulsar spin axis and line of sight

Ixx Iyy

Izz- equatorial ellipticity

In the case of an isolated tri-axial neutron star emitting at twice its rotational frequency


Signals in noise. The meaning of probability

The strain x(t) measured by a detector is mainly dominated by noise n(t), such that even in the presence of a ignal h(t) we have

x(t)=n(t)+h(t)

Data analysis methods are not just simple recipes. We want tools capable of—dealing with very faint sources—handling very large data sets—diagnosing systematic errors—avoiding unnecessary assumptions—estimating parameters and testing models

There are different ways of proceeding depending on the paradigm of statistics used.

There are three popular interpretations of the word:Probability as a measure of our degree of belief in a statement

Probability as a measure of limiting relative frequency of outcome of a set of identical experiments

Probability as the fraction of favourable (equally likely) possibilities

We will call these the Bayesian, Frequentist and Combinatorial interpretations.


Algebra of (Bayesian) probability

• If there are two statements X and Y, then joint probability

where the vertical line denotes the conditional statement “X given Y is true” – The Product Rule

• …because p(X,Y)=p(Y,X), we get

which is called Bayes’ Theorem

p(X |Y,I) p(X | I)p(Y | X,I)

p(Y | I)Thomas Bayes

(1702 – 1761 AD)

p(X,Y ) p(Y)p(X |Y)

p(model | data,I) p(data | model,I)p(model | I)

p(data | I)

Likelihood Prior

Evidence

Posterior

We can usually calculate all these terms

Bayes’ theorem is the appropriate rule for updating our degree of belief when we have new data


Algebra of (Bayesian) probability

• We can also deduce the marginal probabilities. If X and Y are propositions that can take on values drawn from and then

this gives use the probability of X when we don’t care about Y. In these circumstances, Y is known as a nuisance parameter.

• All these relationships can be smoothly extended from discrete probabilities to probability densities, e.g.

where “p(y)dy” is the probability that y lies in the range y to y+dy.

X {x1, x2,..., xn},

Y {y1, y2,..., ym},

p(x i) p(x i) p(y j | x i)j1..m

p(x i)p(y j | x i)j1..m

p(x i,y j )j1..m

=1

p(x) p(x, y)dy


p(model | data,I) p(data | model,I)p(model | I)

Likelihood PriorPosterior

What we know nowInfluence of our

observationsWhat we

knew before

In the Bayesian approach, we can test our model

Bayesian parameter estimation

• As our data improve (e.g. our sample increases), the posterior pdf narrows and becomes less sensitive to our choice of prior.• The posterior conveys our (evolving) degree of belief in different values of , in the light of our data• If we want to express our result as a single number we could perhaps adopt the mean, median, or mode• We can use the variance of the posterior pdf to assign an uncertainty for • It is very straightforward to define confidence intervals


p(

| d

ata

, I )

95% of area under pdf

12

We are 95% sure thatlies between and

Note: the confidence interval is not unique, but we can define the shortest C.I.

1 2

Bayesian confidence intervals

Bayesian parameter estimation


• Recall that in frequentist (orthodox) statistics, probability is limiting relative frequency of outcome, so :

only random variables can have frequentist probabilities

as only these show variation with repeated measurement. So we can’t talk about the probability of a model parameter, or of a hypothesis. E.g., a measurement of a mass is a random variable, but the mass itself is not.

• So no orthodox probabilistic statement can be interpreted as directly referring to the parameter in question! For example, orthodox confidence intervals do not indicate the range in which we are confident the parameter value lies. That’s what Bayesian intervals do.

Frequentist framework


The method goes like this:– To test a hypothesis H1 consider another hypothesis, called the null hypothesis, H0, the

truth of which would deny H1. Then argue against H0…

– Use the data you have gathered to compute a test statistic obs which has a calculable pdf if H0 is true. This can be calculated analytically or by Monte Carlo methods.

– Look where your observed value of the statistic lies in the pdf, and reject H0 based on how far in the wings of the distribution you have fallen.

Frequentist hypothesis testing – significance tests

p(|H0) Reject H0 if your result lies in here


• H0 is rejected at the x% level if x% of the probability lies to the right of the observed value of the statistic (or is ‘worse’ in some other sense):

and makes no reference to how improbable the value is under any alternative hypothesis (not even H1!).

• We will chose a critical region of *, so that if obs>* we reject H0 and therefore accept H1.

Frequentist hypothesis testing – significance tests

p(|H0)

obs

X% of the area

p(|H0)

p(|H1)


– A Type I error occurs when we reject the null hypothesis when it is true (false alarm)– A Type II error occurs when we accept the null hypothesis when it is false (false dismissal)

both of which we should strive to minimise. According to the Neyman-Pearson lemma, this optimal test is the so-called likelihood ratio:

• For Gaussian noise, one finds

which is the well-known expression for the matched-filtering amplitude. If some of the parameters of the signal h(t;A,) are unknown, one has to find the maximum of lnΛ as a function of the unknown parameters









Frequentist hypothesis testing

Type II error Type I error




Calibrated output: LIGO noise history

Integration times

S1 - L1 5.7 days, H1 8.7

days, H2 8.9 days

S2 - L1 14.3 days, H1 37.9

days, H2 28.8 days

S3 - L1 13.4 days, H1 45.5

days, H2 42.1 days

S4 - L1 17.1 days, H1 19.4

days, H2 22.5 days

S5 (so far...) >1 year


Calibrated output: GEO noise history


Continuous wave searches

• Signal parameters: position (may be known), inclination angle, [orbital parameters in case of a NS in a binary system], polarization, amplitude, frequency (may be known), frequency derivative(s) (may be known), initial phase.

• Most sensitive method: coherently correlate the data with the expected signal (template) and inverse weights with the noise. If the signal were monochromatic this would be equivalent to a FT.

– Templates: we assume various sets of unknown parameters and correlate the data against these different wave-forms.

– Good news: we do not have to search explicitly over polarization, inclination, initial phase and amplitude.

• Because of the antenna pattern, we are sensitive to all the sky. Our data stream has signals from all over the sky all at once. However: low signal-to-noise is expected. Hence confusion from many sources overlapping on each other is not a concern.

• Input data to our analyses:– A calibrated data stream which with a better than 10% accuracy, is a measure of the GW excitation of the detector.

Sampling rate 16kHz (LIGO-GEO, 20kHz VIRGO), but since the high sensitivity range is 40-1500 Hz we can downsample at 3 kHz.


Four neutron star populationsand searches

• Known pulsars• Position & frequency evolution known (including derivatives, timing noise, glitches, orbit)

• Unknown neutron stars• Nothing known, search over position, frequency & its derivatives

• Accreting neutron stars in low-mass x-ray binaries• Position known, sometimes orbit & frequency

• Known, isolated, non-pulsing neutron stars• Position known, search over frequency & derivatives

• What searches?– Targeted searches for signals from known pulsars

– Blind searches of previously unknown objects

– Coherent methods (require accurate prediction of the phase evolution of the signal)

– Semi-coherent methods (require prediction of the frequency evolution of the signal)

What drives the choice? The computational expense of the search


Coherent detection methods

Frequency domainConceived as a module in a hierarchical search

• Matched filtering techniques. Aimed at computing a detection statistic.These methods have been implemented in the frequency domain (although this is not necessary) and are very computationally efficient.

• Best suited for large parameter space searches(when signal characteristics are uncertain)

• Frequentist approach used to cast upper limits.

Time domainprocess signal to remove frequency variations due to

Earth’s motion around Sun and spindown

• Standard Bayesian analysis, as fast numerically but provides natural parameter estimation

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

• Efficiently handless missing data

• Upper limits interpretation: Bayesian approach

There are essentially two types of coherent searches that are performed


Summary of directed pulsar searches

• Within the LIGO sensitive band (gw > 50 Hz) there are currently 163 known pulsars

• We have rotational and positional parameter information for 97 of these

• S1 (LIGO and GEO: separate analyses)– Upper limit set for GWs from J1939+2134 (h0<1.4 x 10-22)– Phys. Rev. D 69, 082004 (2004)

• S2 science run (LIGO: 3 interferometers coherently, TDS)– End-to-end validation with 2 hardware injections– Upper limits set for GWs from 28 known isolated pulsars– Phys. Rev. Lett. 94, 181103 (2005)

• S3 & S4 science runs (LIGO and GEO: up to 4 interferometers coherently, TDS) – Additional hardware injections in both GEO and LIGO– Add known binary pulsars to targeted search – Full results with total of 93 (33 isolated, 60 binary) pulsars

• S5 science run (ongoing, TDS) – 32 known isolated, 41 in binaries, 29 in globular clusters


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

C(h0) p(2F | h0)d(2F)2F*


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

S1

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


F-statistics

We can express h(t) in terms of amplitude A {A+, A, , 0} and Doppler parameters














F-Statistics

• Analytically maximize the likelihood over A












Time domain target search

• 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

y(t;a) 14 h0F(t,)(1 cos2)e2i0 1

2 ih0F(t,)cose2i0


• 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

Time domain: Bayesian approach

p(a |{Bk}) p(a)p({Bk} | a)

posterior prior likelihood

2(a) Bk y(t;a)

k

2

k

p(h0 |{Bk}) e 2 / 2d0d dcos


Posterior PDFs for CW time domain analyses

Simulated injectionat 2.2 x10-21

p

shaded area = 95% of total area

p


Progression of targeted pulsars upper limits

Crab pulsar

• Amplitudes of < 10-25 and ellipticities <10-6 for many objects

• Our most stringent ellipticities (10-7) are starting to reach into the range of neutron star structures for some neutron-proton-electron models

0.237h0

10 24

r

1kpc

1Hz2

2

1038kgm2

Izz

APS Meeting, Jacksonville, April 2007


Preliminary S5 results

Used parameters provided by Pulsar Group, Jodrell Bank Joint 95% upper limits from first ~13 months

of S5 using H1, H2 and L1 (97 pulsars)

prelim

inar

yLowest h0 upper limit:

PSR J1435-6100 (gw = 214.0 Hz, r = 3.3 kpc) h0_min = 4.2x10-26

Lowest ellipticity upper limit:PSR J2124-3358 (gw = 405.6Hz, r = 0.25 kpc) = 9.6x10-8

Due to pulsar glitches the Crab pulsar result uses data up to the glitch on 23 Aug 2006, and the PSRJ0537-6910 result uses only three months of data between two glitches on 5th May and 4th Aug 2006

APS Meeting, Jacksonville, April 2007


preliminary• Black curve represents one full year of data for all three interferometers running at design sensitivity• Blue stars represent pulsars for which we are reasonably confident of having phase coherence with the signal model• Green stars represent pulsars for which there is uncertainty about phase coherence


Blind searches and coherent detection methods

• Coherent methods are the most sensitive methods (amplitude SNR increases with sqrt of observation time) but they are the most computationally expensive,

why?– Our templates are constructed based on different values of the signal

parameters (e.g. position, frequency and spindown)

– The parameter resolution increases with longer observations

– Sensitivity also increases with longer observations

– As one increases the sensitivity of the search, one also increases the number of templates one needs to use.


Number of templates

[Brady et al., Phys.Rev.D57 (1998)2101]

The number of templates grows dramatically with the coherent integration time baseline and the computational requirements become prohibitive


S2 run: Coherent search for unknown isolated sources and Sco-X1

• Entire sky search

• Fully coherent matched filtering

• 160 to 728.8 Hz

• df/dt < 4 x 10-10 Hz/s

• 10 hours of S2 data; computationally intensive

• 95% confidence upper limit on the GW strain amplitude range from 6.6x10-23 to 1.0x10-21 across the frequency band

• Scorpius X-1

• Fully coherent matched filtering

• 464 to 484 Hz, 604 to 624 Hz

• df/dt < 1 x 10-9 Hz/s

• 6 hours of S2 data

• 95% confidence upper limit on the GW strain amplitude range from 1.7x10-22 to 1.3x10-21 across the two 20 Hz wide frequency bands

• See gr-qc/0605028


Computational Engine

Searchs offline at:• Medusa, Nemo clusters

(UWM)• Merlin, Morgane cluster

(AEI)• Tsunami (B’ham)• Others


Einstein@home

http://einstein.phys.uwm.edu/• Like SETI@home, but for

LIGO/GEO data• American Physical Society

(APS) publicized as part of World Year of Physics (WYP) 2005 activities

• Use infrastructure/help from SETI@home developers for the distributed computing parts (BOINC)

• Goal: pulsar searches using ~1 million clients. Support for Windows, Mac OSX, Linux clients

• From our own clusters we can get ~ thousands of CPUs. From Einstein@home hope to get order(s) of magnitude more at low cost

• Currently : ~140,000 active users corresponding to about 80Tflops


• Public distributed computing project to look for isolated pulsars in LIGO/GEO data ~ 80 TFlops 24/7• Initially making use of coherent F-statistic method

S3 - no spindown • Using segment lengths of 10 hours• No evidence of strong pulsar signals • Outliers are consistent with instrumental artifacts or bad bands. None of the low significance remaining candidates showed up in follow-up on S4 data.

S4 - one spindown parameter, up to f/fdot ~ 10,000 yr• Using segment lengths of 30 hours• Analysis took ~ 6 months• Post-processing complete. Currently under review.

S5 (R1)- similar to S4• Faster more efficient application• Currently in post-processing stage

Coherent searches onEinstein@home

VESF-school, Pisa, June 2007, A.M. Sintes 45 http://www.boincsynergy.com/stats/

User/Credit History


Current performance

Einstein@Home is currently getting 84 Tflops

http

://w

ww

.boi

ncst

ats.

com

/


All-Sky surveys for unknown gravity-wave emitting 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.

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


Coherent wide-parameter searches

• The second effect of the large number of templates Np is to reduce the sensitivity compared to a targeted search with the same observation time and false-alarm probability: increasing the number of templates increases the number of expected false-alarm candidates at fixed detection threshold. Therefore the detection-threshold needs to be raised to maintain the same false-alarm rate, thereby decreasing the sensitivity.

• Note that increasing the number of equal-sensitivity detectors N improves the SNR in the same way as increasing the integration time Tobs. However, increasing the number of detectors N does — contrary to the observation time Tobs — not increase the required number of templates Np, which makes this the computationally cheapest way to improve the SNR of coherent wide-parameter searches.


Hierarchical strategies

Set upper-limit

Pre-processing

Divide the data set in N chunks

raw data GEO/LIGO

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


Hierarchical method for ‘blind’ searches

h-reconstructed data

Data quality

SFDB

Average spectrum estimation

peak map

Hough transf.

candidates

peak map

Hough transf.

candidatescoincidences

coherent step

events

Data quality

SFDB

Average spectrum estimation

The procedure involves two or more data sets belonging to a single or more detectors


Incoherent power-sum methods

• The idea is to perform a search over the total observation time using an incoherent (sub-optimal) method:

• Three methods have been developed to search for cumulative excess power from a hypothetical periodic gravitational wave signal by examining successive spectral estimates:

– Stack-slide (Radon transform)– Hough transform– Power-flux method

They are all based on breaking up the data into segments, FFT each, producing Short (30 min) Fourier Transforms (SFTs) from h(t), as a coherent step (although other coherent integrations can be used if one increasing the length of the segments), and then track the frequency drifts due to Doppler modulations and df/dt as the incoherent step.

cohT

Fre

quen

cy

Time


Differences among the incoherent methods

What is exactly summed?

• StackSlide – Normalized power (power divided by estimated noise) Averaging gives expectation of 1.0 in absence of signal

• Hough – Weighted binary counts (0/1 = normalized power below/above SNR), with weighting based on antenna pattern and detector noise

• PowerFlux – Average strain power with weighting based on antenna pattern and detector noise Signal estimator is direct excess strain noise(circular polarization and 4 linear polarization projections)


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 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

i | ˜ x (f i) |2

| ˜ n (f i) |2

i 4 | ˜ h (f i) |2

Sn(f i)TSFT

i 1i

2

2 1 i

(th | Sn ) p( th

| 0,Sn )d e th ,

(th | h,Sn ) p( th

| h,Sn )d

˜ x (f) ˜ n (f) ˜ h (f)


Standard 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

P(n | p,N)N

n

pn (1 p)N n

n Np, n2 Np(1 p)

p signal absent

signal present

H (nth,,N)N

n

n (1 )N n

nn th

N

H (nth,,N)N

n

n (1 )N n

n0

n th 1

nth N 2N(1 ) erfc 1(2H )


Noise only case


Frequentist upper limit

• 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

0.95 p n|h0

95% nn*

N

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


0.1% 30.5%

87.0% 1

Number count distributionfor signal injections

p(n|h0) ideally binomial for a target search, but:– Amplitude modulation of

the signal for different SFTs

– Different sensitivity for different sky locations

– Random mismatch between signal & templates

‘smear’ out the binomial distributions

L1: 200-201 Hz, n* =202

1000 injections


Hough S2: UL Summary Feb.14-Apr.14,2003

Detector L1 H1 H2

Frequency (Hz) 200-201 259-260 258-259

h095% 4.43x10-23 4.88x10-23 8.32x10-23

• S2 analysis covered 200-400Hz, over the whole sky, and 11 values of the first spindown (Δf = 5.55×10 – 4 Hz, Δf1 = –1.1×10– 10 Hz s– 1)

• Templates: Number of sky point templates scales like (frequency)2

– 1.5×105 sky locations @ 300 Hz

– 1.9×109 @ 200-201 Hz

– 7.5×109 @ 399-400 Hz

• Three IFOs analyzed separately

• No signal detected

• Upper limits obtained for each 1 Hz band by signal injections: Population-based frequentist limits on h0 averaging over sky location and pulsar orientation


Improvements for S4

• We use a weighted Hough to give more weight to SFTs having greater SNR. Weights are proportional to the beam pattern functions and inversely proportional to the SFT noise floor.

• Number count n is not an integer anymore

• Using the weights does not lead to any loss in computational efficiency or robustness. It has also been generalized to the Multi-IFO case

• Mean number count is unchanged due to normalization of weights. Standard deviation always increases:

• Number count threshold for a given false alarm:

n wini

i1

N

wi Ni1

N

wi F

( i) 2 F( i) 2

Sn( i)

thNNn exp )1( wn

)2(erfc)1(2 12

Hth wNn


Contribution of different IFOs

Figure plots the relative noise weights from H1, H2 and L1

All i

IFO i

w

w


Loudest events for every 0.25 Hz Multi interferometer case 50-1000 Hz

Significance defined as s=(nmax-<n>)/

Prelim

inary


The S4 Hough search

• As before, input data is a set of N 1800s SFTs (no demodulations)

• Weights allow us to use SFTs from all three IFOs together:1004 SFTS from H1, 1063 from H2 and 899 from L1

• Search frequency band 50-1000Hz• 1 spin-down parameter. Spindown

range [-2.2,0]×10-9 Hz/s with a resolution of 2.2×10-10 Hz/s

• All sky search• All-sky upper limits set in 0.25 Hz

bands• Multi-IFO and single IFOs have

been analyzed Best ULfor L1: 5.9×10-24

for H1: 5.0×10-24

for Multi H1-H2-L1: 4.3×10-24

Prelim

inary


Improvements due to the weights

Comparison of the All-sky 95% upper limits obtained by Monte-Carlo injections for the multi-IFO case.

The average improvement by using weights in this band is 9.25% for the multi-IFO case, but only ~6% for the single IFO


StackSlide/Hough/PowerFlux differences

StackSlide Hough PowerFlux

Windowing Tukey Tukey Hann

Noise estimation Median-based floor tracking

Median-based floor tracking

Time/frequency decomposition

Line handling Cleaning Cleaning Skyband exclusion

Antenna pattern weighting

No Yes Yes

Noise weighting No Yes Yes

Spindown step size 2 x 10-10 Hz/s 2 x 10-10 Hz/s Freq dependent

Limit at every skypoint

No No Yes

Upper limit type Population-based Population-based Strict frequentist


H1 (Hanford 4-km) and Multi-IFO Results

PRELIMINARY


L1 (Livingston 4-km) and Multi-IFO Results

PRELIMINARY


S5 incoherent searches preliminary PowerFlux results

Preliminary




E@H-Hierarchical search

Basic algorithm

• Started test-run (~3 months)– currently working on improving stability & reliability

• Will be followed by a full 1-year run• Should permit a search that extends hundreds of pc into the Galaxy• This should become the most sensitive blind CW search possible with current

knowledge and technology


Sensitivity estimate of different E@H Runs




Searches for Continuous Waves, present, past and future




Conclusions

• Analysis of LIGO data is in full swing, and results from LIGO searches from science runs 4, 5 are now appearing.– Significant improvements in interferometer sensitivity since S3.– In the process of accumulating 1 year of data (S5).– Known pulsar searches are beginning to place interesting upper

limits in S5

– All sky searches are under way and exploring large area of parameter space

• Veto strategies and data conditioning are becoming more relevant in order to reduce the selection threshold and improve sensitivity

alicia m. sintes universitat de les illes balears pisa,1 june 2007 data analysis searches for...

Documents

spinning neutron stars

accreating neutron stars

neutron stars sources

accreting stars

nonaxisymmetric neutron

continuous gw

pulsars gravitational

sintes universitat