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The search for continuous gravitational waves: analyses from LIGO’s second science run Michael Landry LIGO Hanford Observatory on behalf of the LIGO Scientific Collaboration http://www.ligo.org April APS Meeting (APR04) May 1-4, 2004 Denver, CO. Photo credit: NASA/CXC/SAO. Talk overview. - PowerPoint PPT Presentation
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LIGO-G040207-01-Z
The search for continuous gravitational waves: analyses from LIGO’s second science run Michael LandryLIGO Hanford Observatoryon behalf of the LIGO Scientific Collaborationhttp://www.ligo.org
April APS Meeting (APR04)May 1-4, 2004Denver, CO
Photo credit: NASA/CXC/SAO
Landry - April APS, 4 May 2004 2LIGO-G040207-01-Z
Talk overview
• Introduction to continuous wave (CW) sources• CW search group analysis efforts• Review of first science run (S1) results, and a look at
expectations of the S2 run • Time-domain analysis method• Injection of fake pulsars• Results
Landry - April APS, 4 May 2004 3LIGO-G040207-01-Z
CW sources
• Nearly-monochromatic continuous sources of gravitational waves include neutron stars with:
» spin precession at ~frot
» excited oscillatory modes such as the r-mode at 4/3 * frot
» non-axisymmetric distortion of crystalline structure, at 2frot
• Limit our search to gravitational waves from a triaxial neutron star emitted at twice its rotational frequency (for the analysis presented here, only)
• Signal would be frequency modulated by relative motion of detector and source, plus amplitude modulated by the motion of the antenna pattern of the detector
Landry - April APS, 4 May 2004 4LIGO-G040207-01-Z
Source model
• F+ and Fx : strain antenna patterns of the detector to plus and cross polarization, bounded between -1 and 1
• Here, signal parameters are:» h0 – amplitude of the gravitational wave signal
» – polarization angle of signal» – inclination angle of source with respect to line of sight
» 0 – initial phase of pulsar; (t=0), and (t)= t0
2
0 0
1 cost F t; h cos ( ) F t; h cos sin ( )
2h t t
so that the expected demodulated signal is then:
00 cosh;tF2
cos1h;tF4
1;ty 0k
20kk
ii eie a
The expected signal has the form:
Heterodyne, i.e. multiply by: ( )i te
Here, a = a(h0, , , 0), a vector of the signal parameters.
PRD 58 063001 (1998)
Landry - April APS, 4 May 2004 5LIGO-G040207-01-Z
CW search group efforts
• S2 Coherent searches» Time-domain method (optimal for parameter estimation)
– Target known pulsars with frequencies (2frot) in detector band
» Frequency-domain F-statistic* method (optimal for blind detection)– All-sky, broadband search, subset of S2 dataset– Targeted searches (e.g. galactic core)– LMXB (e.g. ScoX-1) search
• S2 Incoherent searches» Hough transform method
» Powerflux method
» Stackslide method
• Future: Implement hierarchical analysis that layers coherent and incoherent methods
• Einstein@home initiative for 2005 World Year of Physics*not the F-statistic associated with statistical literature (ratio of two variances), nor the F-test of the null hypothesis (See PRD 58 063001 (1998))
Landry - April APS, 4 May 2004 6LIGO-G040207-01-Z
First science run: S1
• S1 run: 17 days (Aug 23-Sep 9 02)• Coincident run of four detectors, LIGO (L1, H1, H2), and GEO600• Two independent analysis methods (frequency-domain and time-
domain) employed• Set 95% upper limit values on continuous gravitational waves
from single pulsar PSR J1939+2134, using LIGO and GEO IFO’s: best limit from Livingston IFO:
220 (1.4 0.1) 10h
• Accepted for publication in Phys Rev D 69, 082004 (2004), preprint available, gr-qc/0308050
Landry - April APS, 4 May 2004 7LIGO-G040207-01-Z
S2 expectations
• Coloured spectra: average amplitude detectable in time T (1% false alarm, 10% false dismissal rates):
0 11.4 ( ) /hh S f T• Solid black lines: LIGO and
GEO science requirement, for T=1 year
• Circles: upper limits on gravitational waves from known EM pulsars, obtained from measured spindown (if spindown is entirely attributable to GW emission)
• Only known, isolated targets shown here
LIGO
GEO
Landry - April APS, 4 May 2004 8LIGO-G040207-01-Z
Time-domain analysis method
• Perform time-domain complex heterodyne (demodulation) of the interferometer gravitational wave channel
• Low-pass filter these data
• The data is downsampled via averaging, yielding one value (“Bk”) of the complex time series, every 60 seconds
• Determine the posterior probability distribution (pdf) of the parameters, given these data (Bk) and the model (yk)
• Marginalize over nuisance parameters (cos0) to leave the posterior distribution for the probability of h0 given the data, Bk
• We define the 95% upper limit by a value h95 satisfying:
cosddψdφdata all |apd95.0 00 0
%950
0
h
hh
h95
1
0
strain
Such an upper limit can be defined even when signal is present
Landry - April APS, 4 May 2004 9LIGO-G040207-01-Z
2
k k 2k k 2
k
B y t ;ap B a, exp - exp -χ / 2
2 k
Bk’s are processed data noise
Bayesian analysis
A Bayesian approach is used to determine the posterior distribution of the probability of the unknown parameters via the Likelihood (assuming gaussian noise within our narrow band):
aBpap B|ap kk
posterior prior likelihood
model
The posterior pdf is
Landry - April APS, 4 May 2004 10LIGO-G040207-01-Z
Marginalizing over noise
As we estimate the noise level from the Bk no independent information is lost by treating it as another nuisance parameter over which to marginalize, i.e.
)0(,1
| jj
jp
a
j
0
d |},{|}{
aBpBp jkk a
j
0
d ,|}{||}{
jkjk BppBp aaaWhere is constant
for each 30 , i.e. 30mj
kB
We assign Jeffreys prior to sigma, so that
giving a (marginalized) likelihood of
which can be evaluated analytically for gaussian noise.
Landry - April APS, 4 May 2004 11LIGO-G040207-01-Z
Compute likelihoods
Analysis summary
k2
k k
1p B a
B ynn
k
Heterodyne, lowpass,
average, calibrate: Bk
Model: yk
Compute pdf for h0
Compute upper limits
aBpap B|ap kk
Raw Data
uniform priorson h0(>0), cos
Landry - April APS, 4 May 2004 12LIGO-G040207-01-Z
S2 hardware signal 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
• Fake signal is sum of two pulsars, P1 and P2• All the parameters of the injected signals were
successfully inferred from the data
Landry - April APS, 4 May 2004 13LIGO-G040207-01-Z
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
Landry - April APS, 4 May 2004 14LIGO-G040207-01-Z
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
Landry - April APS, 4 May 2004 15LIGO-G040207-01-Z
Pulsar timing
• Analyzed 28 known isolated pulsars with 2frot > 50 Hz.» Timing information has been provided using radio observations collected over
S2/S3 for 18 of the pulsars (Michael Kramer, Jodrell Bank). » Timing information from the Australia Telescope National Facility (ATNF)
catalogue used for 10 pulsars
• An additional 10 isolated pulsars are known with 2frot > 50 Hz but the uncertainty in their spin parameters is such that a search over frequency is warranted
• Crab pulsar heterodyned to take timing noise into account
Landry - April APS, 4 May 2004 16LIGO-G040207-01-Z
Preliminaryresults for PSR B0021-72L
• Posterior probability density for PSR J1910-5959D
• Flat prior for h0 (h0>0), Jeffreys prior for , i.e. p() 1/
L1H1H2joint
Landry - April APS, 4 May 2004 17LIGO-G040207-01-Z
Preliminaryresults for the Crab pulsar
• Posterior probability density for PSR B0531+21
• Crab pulsar heterodyned to take timing noise into account
• Flat prior for h0 (h0>0), Jeffreys prior for , i.e. p() 1/
L1H1H2joint
Landry - April APS, 4 May 2004 18LIGO-G040207-01-Z
Preliminary
upper limits for 28 known pulsars
h0 UL range Pulsar
10-23-10-22 J1939+2134, B1951+32, J1913+1011, B0531+21
10-24-10-23
B0021-72C, B0021-72D, B0021-72F, B0021-72G, B0021-72L, B0021-72M, B0021-72N, J0711-6830, B1820-30A, J1730-2304, J1721-2457, J1629-6902, J1910-5959E, J2124-3358, J1910-5959C, J0030+0451, J1024-0719,
J1910-5959D, J2322+2057, B1516+02A, J1748-2446C, J1910-5959B, J1744-1134, B1821-24
Blue: timing checked by Jodrell Bank
Purple: ATNF catalogue
Landry - April APS, 4 May 2004 19LIGO-G040207-01-Z
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
Landry - April APS, 4 May 2004 20LIGO-G040207-01-Z
Preliminary ellipticitylimits for 28 known pulsars
UL range Pulsar
10-2-10-1 B1951+32, J1913+1011, B0531+21
10-3-10-2 -
10-4-10-3 B1821-24, B0021-72D, J1910-5959D, B1516+02A, J1748-2446C, J1910-5959B
10-5-10-4
J1939+2134, B0021-72C, B0021-72F, B0021-72L, B0021-72G, B0021-72M, B0021-72N, B1820-30A, J0711-6830, J1730-2304,
J1721-2457, J1629-6902, J1910-5959E, J1910-5959C, J2322+2057
10-6-10-5 J1024-0719, J2124-3358, J0030+0451, J1744-1134
Blue: timing checked by Jodrell Bank
Purple: ATNF catalogue
Landry - April APS, 4 May 2004 21LIGO-G040207-01-Z
Summary and future outlook
• S2 analyses» Time-domain analysis of 28 known pulsars complete
» Broadband frequency-domain all-sky search underway
» ScoX-1 LMXB frequency-domain search near completion
» Incoherent searches reaching maturity, preliminary S2 results produced
• S3 run» Time-domain analysis on more pulsars, including binaries
» Improved sensitivity LIGO/GEO run
» Oct 31 03 – Jan 9 04
» Approaching spindown limit for Crab pulsar