Gravitational Wave Astronomy: Data Analysis L · Gravitational Wave Astronomy: Data Analysis L.4....

GWA School, January 2013, A.M. Sintes 1

Alicia M. Sintes Universitat de les Illes Balears

GWA School, Rhodes University, 2013

Gravitational Wave Astronomy: Data Analysis L.4

Data analysis searches for continuous gravitational waves


Content L4

•  Gravitational waves from spinning neutron stars. –  Neutron star overview –  GW emission mechanisms –  Signal model

•  Continuous GW searches –  neutron star populations and searches –  Brief overview of the observations/searches of neutron stars –  Einstein@Home –  Coherent detection methods –  The problem of the wide-parameter search –  Hierarchical strategies –  Semi-coherent methods –  Example: The Hough transform search


Spinning Neutron Stars

(NASA/CXC/SAO)

NASA

Relic of past collapse of a moderately massive star Remnant spin from progenitor, or from having been spun up by accretion Generally magnetized, sometimes very strongly Typical size of 10km, and are about 1.4 solar masses A small fraction of neutron stars are seen as pulsars

If not axisymmetric, will emit gravitational waves Example: ellipsoid with distinct transverse axes


Neutron Stars Sources

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

–  After many 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!


Neutron star overview

•  There are probably ~108-109 neutron stars in the galaxy –  ~105 are radio pulsars (we know of ~2000) –  ~107 are (unseen) dead magnetars –  ~108 are totally unknown

•  We will see GWs from any neutron star that is –  sufficiently lumpy –  sufficiently close –  spinning at a rate that will appear in our band

•  A two-part attack: –  target neutron stars we can see electromagnetically,

reduced parameter space, so good sensitivity detections would usually need ‘new’ physics (quark stars etc…)

–  perform a general search for EM-dark neutron stars, large population our range (reach) is computationally-bound, even with E@H! We need

several different search methods optimised for specific search tasks.


“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


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

��

��

� ��

�

�

�=Ixx�IyyIzz

The strain amplitude h0 refers to a GW from an optimally oriented source with respect to the detector The equatorial ellipticity is highly uncertain, ε~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 ε~10-6. These deformations would result in GW emission at f = ν and f = 2ν. Magnetic mountains

GWA School, January 2013, A.M. Sintes 10 Wobbling 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 precession results in emission at (approximately) the rotation rate ν and twice the rotation rate, i.e. f =ν+νprec and f = 2ν.


•  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) =h0A(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)h# = A# sin"(t)

!

"(t)=#0 +2$f(n )

(n +1)!(T(t) %T(t0))

n+1

n= 0

&

'

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 + =12h0(1+cos2" )

A# =h0 cos"

h0 =4$ 2Gc 4

Izz% fgw2

d

h0 - amplitude of the gravitational wave signal

ι - angle between the pulsar spin axis and line of sight

!

" =Ixx # IyyIzz

- equatorial ellipticity

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


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 populations and 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 / methods? –  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


Demodulation / Matched Filtering for Continuous-Wave Signals

Even a “continuous” gravitational wave is not received as a monochromatic signal

Doppler shift due to Earth’s orbit and rotation Antenna pattern depends on sidereal time of day Intrinsic frequency of source may be changing Source in binary system exhibits additional Doppler shift

Demodulation depends sensitively on source parameters Large-parameter-space searches are limited by computational power

Time

• Fre

quen

cy

Time

Semi-coherent methods use less CPU Stack-slide, Hough transform, etc. Can be used as part of a hierarchical search


10 0 10 1 10 2 10 3 10 -28

10 -27

10 -26

10 -25

10 -24

10 -23

10 -22

GW frequency (Hz)

stra

in

Crab Vela

PSR J0537-6910 (LMC)

PSR J1952+3252 (CTB80)

J0437-4715

Known pulsars Most known (timed) pulsars are out of our band, and their spin-down luminosity is below the 1y LIGO sensitivity

LHO S5 1y targeted search sensitivity


GW observations of neutron stars

19

  Several broad-area searches have placed upper limits on GW fluxes from unknown neutron stars.

  Advanced detectors could plausibly detect Galactic neutron stars in the next five years (but no guarantees).

  To date, LIGO and Virgo have not plausibly detected GW emissions from neutron stars (but analysis of existing data is ongoing).

  For 3 young neutron stars (Crab, Vela, Cassiopeia A), GW observations have placed more stringent limits than EM observations.


20

Search for periodic GW signals from known pulsars

116 known pulsars 95% upper limits Abbott et al (LSC & Virgo) ApJ 713, 671 (2010)

  search method makes use of a signal template for each pulsar requires updated ephemeris data to model phase evolution of pulsar signal requires collaboration with radio pulsar astronomers

  Highlights: •  Crab pulsar: h0 < 1.9 x 10−25

GWs <2% of spindown energy •  Vela pulsar: h0 < 2.1 x 10−24

GWs <35% of spindown energy

•  J0537−6910: h0 < 5 x 10−26

At spindown limit •  J1603−7202: h0 < 2.3 x 10−26

Lowest h0 limit •  J2124−3358: ε < 7.0 x 10−8

Lowest ε limit

Jodrell Bank Parkes Telescope Green Bank


21

Linearly polarized

Circularly polarized

All sky surveys for isolated unknown NS

The parameter space for blind searches for weak signals from unknown isolated neutron stars is very large. To probe full parameter space without restricting observation time, need to use semicoherent or incoherent methods. E.g., shift Fourier bins according to Doppler modulation & add power. Different techniques have been designed, each optimized for a different portion of parameter space

All-sky LIGO Search for Periodic Gravitational Waves in the Early S5 Data – PRL 102 (2009) 11110 These are objects which have not been

previously identified at all, and we must search over various possible sky positions, frequencies, and frequency derivatives. They are believed to constitute the overwhelming majority of neutron stars in the Galaxy.


Einstein@home

•  Einstein@Home, a volunteer distributed computing project, where host home or office computers automatically download “workunits” from the servers,carry out analyses when idle, and return results.

•  Distributed using BOINC & run as a screensaver

•  January 1st 2013, Einstein@Home passed the 1 Petaflop computing-power barrier.

•  Since 2009, E@H looks for signals in Arecibo (Parkes) data, using 30% of the search time. Found several new pulsars 22

http://www.einsteinathome.org/


Coherent detection methods

Frequency domain Conceived 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 domain process 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


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 χ2 distribution with 4 degrees of freedom with a non-centrality parameter λ∝(h|h). Fixing ι, ψ and φ0 , 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 Example 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×, ψ, φ0} and Doppler parameters λ


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. ⇒ σk

•  Standard Bayesian parameter fitting problem, using time-domain model for signal -- a function of the unknown source parameters h0 ,ι, ψ and φ0

!

y(t;a) = 14 h0F+(t,")(1+ cos2#)e2i$0 % 1

2 ih0F&(t,")cos#e2i$0


•  We take a Bayesian approach, and determine the joint posterior distribution of the probability of our unknown parameters, using uniform priors on h0 ,cos ι, ψ and φ0 over their accessible values, i.e.

•  The likelihood ∝ exp(-χ2 /2), where

•  To get the posterior PDF for h0, marginalizing with respect to the nuisance parameters cos ι, ψ and φ0 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 / 2d%0d&''' dcos(


Posterior PDFs for CW time domain analyses

Simulated injection at 2.2 x10-21

p

S1 Example shaded area =

95% of total area

p


Crab Pulsar Search for continuous-wave signal Result from first 9 months of S5: Consistent with Gaussian noise

Upper limits on GW strain amplitude h0 Single-template, uniform prior: 3.4×10–25

Single-template, restricted prior: 2.7×10–25 Multi-template, uniform prior: 1.7×10–24 Multi-template, restricted prior: 1.3×10–24

Implies that GW emission accounts for ≤ 4% of total spin-down power

[arXiv:0805.4758 ; ApJL ] C

hand

ra im

age

Mod

el

•  There is observational evidence that identifies the orientation of the pulsar from the geometry of the Crab Pulsar Wind Nebula

•  The values of the inclination angle and polarization angle are well constrained by X-ray observations (Ng and Romani, Ap. J., 2004, 2008)


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


All-Sky 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.

Class f (Hz) τ (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


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

  Type of search → All-sky search - Computational cost increases rapidly with total observation time.

  Type of source → Continuous Sources Very small amplitude ( h0 ∼ 10-26 )

- Long integration time needed to build up enough SNR

- Relative motion of the detector with respect to the source (amplitude and frequency modulated)

- System evolves during the observational period

  Type of Methods →

Coherent Methods

Semi-Coherent Methods

Hierarchical Methods

+ sensitive + computational cost

- sensitive - computational cost

2nd Stage: Follow up the candidates with higher resolution.

1st Stage: Wide-parameter search with low resolution in parameter space.

(Matched filtering)

(Stack Slide, Power Flux, Hough)

Reduce the number of candidates to be followed up → Improve the sensitivity keeping the computational cost.


Semi-coherent 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

Freq

uenc

y

Time


Differences among the semi-coherent 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)


StackSlide/Hough/PowerFlux differences for S4

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


The Hough Transform

•  Robust pattern detection technique.

cntvtftftf!!!

=")()(ˆ)(ˆ)( ...)(ˆ)(ˆ 00 +−+= ttfftf )(ˆ tf )(ˆ tf

Detector RF SSB

•  For isolated NS the expected pattern depends on the parameters: { }...,,,, ff δα

•  We use the Hough Transform to find the pattern produced by the Doppler modulation (due to the relative motion of the detector with respect to the source) and spin-down of a GW signal in the time – frequency plane of our data:


The Hough Transform Procedure:

NTT obs=!

obsT

1

2

Select just those that are over a certain threshold ρth. 3

ρth

ρ

Frequency t

f

t

f

Break up data (x(t) vs t) into segments

Take the FT of each segment and calculate the corresponding normalized power in each case (ρk )


The Hough Transform

To improve the sensitivity of the Hough search, the number count can be incremented not just by a

factor +1, but rather by a weight ωi that depends on the response function of the detector and the noise floor estimate → greater contribution at the more sensitive sky locations and from SFTs which have low noise.

The thresholds n0 and ρth are chosen based on the Neyman – Pearson criterion of minimizing the false dismissal (i.e. maximize the detection probability) for a given value of false alarm.

Procedure:

t

f

t

f

α δ

n

n0


Hough Transform Statistics

)1(2 qqN !="

Without Weights With Weights

After performing the Hough Transform N SFTs, the probability that the pixel

has a number count n is given by

( ) nNn ppnN

np −−

= 1)(

{ }ff !,,, 0!"

( )

++=

=

2

21 kk

th Oe

eq

th

th

λλρ

η ρ

ρSignal absent

Signal present 2k!k! )(

)(~42

kncoh

kk fST

fh=!SNR for a single

SFT

!=

=N

iiinn

1" !

=

=N

ii N

1"

( )( )

2

2

222

1 !

"!

nn

enp#

#

=

( )!=

"=N

iiii

1

22 1 ##$%

The probability for any pixel on the time - frequency plane of being selected is:

)1(2 !!" #= N

qNn =Signal present Signal absent

Signal present Signal absent

!=

=N

iinn

1Number Count

Mean

Variance )1(1

22 qqN

ii !="

=

#$

!=

+=N

iii

thqqNn12

"#$

Mean

Variance

qNn = !Nn =

Number Count

=p


Frequentist upper limit

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

HT: S → N λ → n(λ)

•  Determine the maximum number count n* n* = max (n(λ)): λ ∈ 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|h095%( )

n= n*

N

"

Compute p(n|h0) via Monte Carlo signal injection, using λ ∈ S , and φ0 ∈[0,2π], ψ ∈[-π/4,π/4], cos ι∈[-1,1].


0.1% 30.5%

87.0% 1

Number count distribution for signal injections

p(n|h0) ideally binomial for a target search (no weights), 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


The S4 Hough search

•  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 upper limits set in 0.25 Hz bands

•  Multi-IFO and single IFOs have been analyzed

Best UL for L1: 5.9×10-24

for H1: 5.0×10-24

for Multi H1-H2-L1: 4.3×10-24


H1 (Hanford 4-km) and Multi-IFO S4 All-sky pulsar search upper limits

Phys. Rev.D77,022001(2008)

PowerFlux: Comparing linear to circular polarization limits Linear amplitude = 0.5 × h0

worst-pulsar Circular amplitude = h0best-pulsar

Typical: h0worst-pulsar ~ (3-4) × h0

best-pulsar

Best multi-H1-H2-L1 UL: 4.28×10-24


S4 Semicoherent Search Astrophysical Reach


Outlook

•  Periodic gravitational wave searches by the VIRGO and LSC are making significant progress

•  No detection yet but upper limits are becoming astrophysically interesting –  gravitational wave observations are providing new information

unobtainable by previous EM observations •  Advanced LIGO should beat spin-down limits for many pulsars •  Opportunity for fruitful interaction between GW searches and

astrophysics

Date post:	21-Aug-2020
Category:	Documents
Upload:	others
View:	3 times
Download:	0 times

Gravitational Wave Astronomy: Data Analysis L · Gravitational Wave Astronomy: Data Analysis L.4....

Documents