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Bernard F Schutz

International School of Relativistic Astrophysics: John Archibald Wheeler Data Analysis for Gravitational Waves. Bernard F Schutz Albert Einstein Institute – Max Planck Institute for Gravitational Physics, Golm, Germany and School of Physics and Astronomy, Cardiff University, Cardiff, UK

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Bernard F Schutz

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  1. International School of Relativistic Astrophysics: John Archibald WheelerData Analysis for Gravitational Waves Bernard F Schutz Albert Einstein Institute – Max Planck Institute for Gravitational Physics, Golm, Germany and School of Physics and Astronomy, Cardiff University, Cardiff, UK Bernard.Schutz@aei.mpg.de www.aei.mpg.de

  2. Contents • Introduction • Detecting signals with known waveforms • Detecting signals with unknown waveforms • Data analysis in the LSC • Including VIRGO and other detectors • Detecting with high sensitivity: LISA Erice: John Archibald Wheeler School

  3. Introduction Erice: John Archibald Wheeler School

  4. Role of Data Analysis • Aim of detector projects (LIGO, GEO, VIRGO, AURIGA, …) is to identify signals and measure their parameters. • All GW detectors are linearly polarized and are sensitive to the phase of the wave in their waveband. • Measure h(t), time-dependence contains much information • Bar detectors have relatively small bandwidth (Δf/f ~ 0.1 or smaller), interferometers have wide band (10 Hz < f < 2 kHz). • Ground-based GW detectors are noise-dominated even when signals are present. • Mean noise during 1 ms is around h ~ 10-21. • Expected signal amplitudes for key sources, like coalescing binary neutron stars and GW pulsars, is 10-23 or smaller. • Detectors operate in networks, providing both confidence and directional informaton. • Data analysis is a key part of detection. The cleverness and sophistication of detector hardware must be matched in quality by data analysis. Erice: John Archibald Wheeler School

  5. Detector Noise • “Noise” refers to any random process that creates detector output. • Can be intrinsic (e.g. photon shot noise) or external (e.g. ground vibration). • Can be Gaussian (e.g. thermal noise) or non-Gaussian (e.g. laser intensity fluctuations). • Commissioning work includes minimizing non-Gaussian noise. • All data analysis teams include specialists in detector characterization. • Additional disturbances in a data stream include: • Interference: deterministic disturbances, e.g. EM field from power lines. • Confusion: Overlapping signals or strong signals obscuring weaker ones. LISA will have this problem, but not the ground-based detectors. • A detection against noise is a decision based on probability - • There is no such thing as a perfect, absolutely certain measurement. • When signal is weak, statements about confidence of detection must be made with great care. • In the LSC at least one-third of the >400 scientists whose names go on the papers are involved in data analysis and related activities. Erice: John Archibald Wheeler School

  6. Statistical treatment of noise • Data is always sampled at a constant rate (e.g. 16kHz), so we will deal with discrete data sets {xj , j = 0 .. }. • We will assume noise {nj} is • Gaussian, i.e. pdf is the normal distribution • Zero-mean: Enj= 0, where E is expectation value. • Stationary, i.e. characteristics of noise independent of time • Usually work in Fourier domain, because process is stationary and because FFT algorithm brings advantages for data processing. • Variance of noise in Fourier domain is called Spectral Noise Density S: Erice: John Archibald Wheeler School

  7. When we know what signal we are looking for… Erice: John Archibald Wheeler School

  8. Statistical Detection of Known Signals • We suppose we are searching for a signal h = {hj} in a vector space X, and that we understand our (Gaussian stationary) noise. • Let p0(h) be the probability distribution function (pdf) for the pure noise, i.e. when no signal is present. • If we set a detection criterion such that we detect the signal if h is inside a region R of X, then the chance that we will falsely identify a signal when it is not there is called the false-alarm probability and is denoted by PF • Let p1 be the pdf for noise plus the signal h. Then the corresponding detection probability PD is Erice: John Archibald Wheeler School

  9. likelihood prior posterior Frequentist and Bayesian Approaches • For frequentists, the usual wayof identifying signals is the Neyman-Pearson criterion for detection. • We should test the likelihood ratio • We then define the detection region R by placing a threshold k on Λ: • For Bayesians, one begins also with the likelihood ratio of but takes into account explicitly prior knowledge. • One computes the posterior pdf from the likelihood of the signal Bk and prior: While the theoretical dispute between Bayesiansand frequentists has oftenbeen heated, we have foundin the GW data analysis that the practical differencesin the detection of signalsis small. Erice: John Archibald Wheeler School

  10. 2s 20s 200s Matched Filtering • All approaches show that the appropriate linear detection statistic is the matched filter. Its value should exceed some threshold. • In its simplest form one computes the inner product between the data {xj} and the expected signal {hj}. • To understand this, consider an expected sine-wave signal. Then we have • Matched filtering is a generalization of the Fourier transform. Like the FT it improves if the signal duration is longer. Signal to noise ratio ~ Ncycles1/2 • Example: a 200 Hz signal buried in noise with 10 times the amplitude, searched for in data sets of 2s, 20s, and 200s. Erice: John Archibald Wheeler School

  11. Allowing for unknown time-of-arrival • In practice we do the search in the Fourier domain. By the Parseval theorem • This is desirable because we want to search for any arrival time τ, ie for an arbitrarily shifted signal h(t-τ). Its Fourier transform is • The detection statistic in the Fourier domain is then (if τ corresponds to time- sample index j) • This is just the Fourier transform and can be performed by the FFT algorithm. We can therefore search over all arrival times with ~ NlogN operations rather than N2. • This also allows us to weight the filter when the noise is stationary but colored. The square of the optimum signal-to-noise ratio (SNR) is Erice: John Archibald Wheeler School

  12. Signals belong to parametrized families • In practice our signal models h depend on parameters, such as the masses of stars, the eccentricity of a binary, the spin rate of the neutron star, the location of the signal on the sky. The likelihood depends on parameters θn • The frequentist solution is to find the parameter values that maximize Λ: Maximum Likelihood Erice: John Archibald Wheeler School

  13. Chirp from 2 x 1.4 MNS’s from 300Hz to merger. Deterministic Signals: Coalescing Binaries • When a NS-NS, NS-BH, BH-BH system coalesces the signal is in principle known (Will lectures). The inspiral is described by the post-Newtonian waveform. Understanding the merger signal is the domain of numerical relativity. The subsequent ringdown of the final black hole is also understood numerically. • Searches are being done for masses from around 0.5 Mto 20 M. Spins will soon be included. Parameters are chosen with large overlap between nearby templates, so that false dismissal is minimized. • Three or more detectors permit triangulation to find source location. Family dependson m1, m2, s1, s2,eccentricity (zerofor stellar systems),phase at (say)f = 100 Hz. Erice: John Archibald Wheeler School

  14. detector Deterministic signals: GW pulsars • Neutron stars will radiate if they are asymmetric about spin axis. Could be due to crust irregularity, solid core irregularity, free precession, … • Signal in rest frame of pulsar modelled as a slowly changing frequency. Take f, df/dt, d2f/dt2 as parameters in a Taylor expansion. • But long data stretches are needed. For an amplitude of 10-26, we need (10-21/10-26)2 ~ 1010 cycles to reach SNR ~ 1. For a 1 ms pulsar this is 4 months of data. • Modulation by detector motion serious, angular resolution aroundλgw/ 1 AU ~ 1 arcsecond (similar to radio pulsar resolution). • Blind searches require searching ~1013 square arcseconds on the sky! • This is the most compute-intensive of all the searches, and its sensitivity is directly limited by computer power. Einstein@Home is dedicated to this. Erice: John Archibald Wheeler School

  15. When we don’t know what signal we are looking for… Erice: John Archibald Wheeler School

  16. Signal σ =2 Inst 1 σ =10 + × × + Inst 2 σ =10 Avg = 3.34 Avg = -0.46 Stochastic signals • Some signals are known to be totally random. Possible sources: • Big Bang, inflation (Grishchuk talk), phase transitions in early universe • Astrophysical sources, such as binaries, distant supernovae, … • If this random excitation is stronger than detector noise, and if detector noise is understood or can be independently measured, then a stochastic background can be identified (bolometric detection). • If two detectors with independent instrumental noise are available, their outputs can be cross-correlated to look for a common noise. Erice: John Archibald Wheeler School

  17. Well correlatedresponses Poorly correlatedresponses Working with separated detectors • Cross-correlation alone works only when detectors have perfectly correlated signals, which means they have to be in the same place. • Real detectors are separated and have different orientations. • Correlation better for waves arriving from some directions than from others. • Correlation improves for longer wavelengths • Analysis is done in Fourier domain to allow: • Weighting for colored noise • Weighting for frequency response of correlation – optimum filter Erice: John Archibald Wheeler School

  18. Unmodeled burst signals • Some signals are difficult to model (e.g. from gravitational collapse) or unexpected. Harder to improve SNR when you have little information. • Teams do various power-based tests, looking for excess power in a range of spectrum, or excess power in time-series, or excess power in a cluster in a time-frequency or wavelet diagram. • Two or more detectors are required for confidence. • Three detectors allow a redundancy test, called the null stream veto. • Since a GW is determined by two functions of time (two polarizations), if its source direction is known then three detectors have redundancy. • There is then a null stream, a linear combination which has no signal. • If a signal is suspected, the null stream should look like typical noise. If it does not, then signal can be vetoed. • Four detectors can test for non-GR signal model, such as scalar polarization. Erice: John Archibald Wheeler School

  19. Data analysis in the Erice: John Archibald Wheeler School

  20. LIGO Scientific Collaboration • All data from 4 interferometers (3 LIGO + GEO) pooled for analysis. • LSC has done joint analysis with TAMA (Japan) and Allegro (bar detector). • Self-governing organization distinct from LIGO but reporting to LIGO. • Any group may join by signing MOU, periodically updated and reviewed • Group commits to useful work on detector development or data analysis. • Group promises to abide by data release and publication rules. • Members entitled to appear in author list of collaboration-wide papers. • LSC coordinates • Development of Advanced LIGO detector (cooperation with GEO) • Operation of detectors • Data analysis and publication • Current spokesman: Peter Saulson (Syracuse University) • Current data analysis coordinator: Maria Alessandra Papa (AEI Potsdam) • Science runs S1 – S4 led to published upper limits with instruments still in commissioning. • Analysis run by 4 teams: pulsars, inspiral, stochastic, bursts. • LIGO has now reached design sensitivity and S5 has been going for 8 months. Erice: John Archibald Wheeler School

  21. How an analysis team works • Analysis teams consist of typically 15-20 active members, many others who contribute at particular times. • Each team has a theorist and an experimentalist as co-chairs. • LSC members required to spend at least 50% of research time on LSC work, and to spend time doing shift-work at the detectors. • Teams are international, meet weekly by telephone conference, hold about 4 face-to-face meetings per year in association with LSC meetings. • Teams write and document code, document their analysis methods. All software and documentation is in open-source repositories. • Code is reviewed in detail, line-by-line, by a team of LSC members from outside the analysis team. Code reviews typically involve an extra telephone conference each week. • Results papers are written jointly, using cvs repositories, and when ready are also externally reviewed in detail before being presented to LSC. All statistical statements, tests, conclusions must be justified. More telephone conferences. • Paper is finally presented to LSC, open for comment from all members for some weeks before being placed on gr-qc and submitted for publication. Conference presentations are given by LSC-wide telecon before going public. Erice: John Archibald Wheeler School

  22. Binary neutron stars S5 search S2 Horizon Distance 1.5 Mpc • First three months of S5 data is analyzed. • Sensitivity of search given in terms of the horizon distance: distance to 1.4+1.4 M optimally oriented & located binary at SNR=8 Erice: John Archibald Wheeler School

  23. Primordial black holes S2 Observational Result Phys. Rev. D. 72, 082002 ( 2005) • This is a MACHO search: small binary objects. No known astrophysical formation mechanism. • S3 search complete • Under internal review • 0.09 yr of data • 1 Milky-Way like halo for 0.5+0.5 M • S4 search complete • Under internal review • 0.05 yr of data • 3 Milky-Way like halos for 0.5+0.5 M • S5 analysis getting under way Rate < 63 per year per Milky-Way-like halo Rate per MW halo per year Total mass Erice: John Archibald Wheeler School

  24. S2 Observational Result Phys. Rev. D. 73, 062001 (2006) Log| cum. no. of events | Rate < 38 per year per Milky-Way-like galaxy signal-to-noise ratio squared Binary black holes • S3 search complete • Under internal review • 0.09 yr of data • 5 Milky-Way like galaxies for 5+5 M • S4 search complete • Under internal review • 0.05 yr of data • 150 Milky-Way like galaxies for 5+5 M Erice: John Archibald Wheeler School

  25. binary neutron star horizon distance Average over run 1 sigma variation binary black hole horizon distance Image: R. Powell Binary black holes S5 search • 3 months of S5 analyzed • Horizon distance (detector sensitivity) versus mass for BBH Erice: John Archibald Wheeler School

  26. Wobbling neutron star Accreting neutron star “Mountain” on neutron star R-modes Radiation from rotating neutron stars Erice: John Archibald Wheeler School

  27. Pointing at known neutron stars • Targeted search of GWs from known isolated radio pulsars • S1 analysis: upper-limit (95% confidence) on PSR J1939+2134: h0 < 1.4 x 10-22 (e < 2.9 x 10-4) Phys Rev D 69, 082004 (2004) • S2 analysis: 28 pulsars (all the ones above 50 Hz for which search parameters are “exactly” known) Erice: John Archibald Wheeler School

  28. B0021-72C B0021-72D B0021-72F B0021-72G B0021-72L B0021-72M B0021-72N B0531+21 (Crab) B1516+02A B1820-30A B1821-24 B1937+21 (S1) B1951+32 B0030+0451 J0711-6830 J1024-0719 J1629-6902 J1721-2457 J1730-2304 J1744-1134 J1748-2446C J1910-5959B J1910-5959C J1910-5959D J1910-5959E J1913+1011 J2124-3358 J2322+2057 28 pulsars targeted for S2 There are 38 known isolated radio pulsars with fGW > 50 Hz, including PSR J1939+2134 (used in S1) and the Crab pulsar • Timing information for 28 pulsars: • Radio observations collected over S2/S3 for 18 of these (Michael Kramer) ATNF catalogue used for 10 others • The remaining 10 pulsars have not been included in the analysis because of outdated spin parameters (would require more that one template) Erice: John Archibald Wheeler School

  29. S2 results Best upper-limits: J1910 – 5959D: h0 = 1.7 x 10-24 J2124 – 3358: e = 4.5 x 10-6 Crab: a factor ~ 30 from spin-down limit Dots: UL on h0 Squares: UL one Red and magenta refer to PSRs with no info on fdot Erice: John Archibald Wheeler School

  30. S5 results on pulsars • 32 known isolated, 44 in binaries, 30 in globular clusters Lowest ellipticity upper limit: PSR J2124-3358 (fgw = 405.6Hz, r = 0.25kpc) ellipticity = 4.0x10-7 Erice: John Archibald Wheeler School

  31. Einstein@Home • Matched-filtering for continuous GWs • All-sky, all-frequency search • computationally limited • Aiming at detection, not upper limits • Public outreach distributed computing • Results: S3 showed no evidence of pulsar, S5 ongoing. To participate, sign up at http://www.physics2005.org Erice: John Archibald Wheeler School

  32. S5 incoherent searches preliminary . Erice: John Archibald Wheeler School

  33. LIGO Results on 0h1002 Erice: John Archibald Wheeler School

  34. Catastrophic events involving solar-mass compact objects can produce transient “bursts” of gravitational radiation in the LIGO frequency band: core-collapse supernovae merging, perturbed, or accreting black holes gamma-ray burst engines cosmic string cusps others Precise nature of gravitational-wave burst (GWB) signals typically unknown or poorly modeled. Can’t base such a broad search on having precise waveforms. Search for generic GWBs of duration ~1ms-1s, frequency ~100-4000Hz. Bursts Analysis: Targets possible supernova waveforms T. Zwerger & E. Muller, Astron. Astrophys. 320 209 (1997) Erice: John Archibald Wheeler School

  35. The future: including VIRGO, further expanding the network Erice: John Archibald Wheeler School

  36. VIRGO-LSC joint analysis • An MOU is under negotiation among VIRGO, the LSC, and GEO. It will provide for • Joint data analysis teams working on all data. • Joint publications. • Coordination of observing runs. • Joint data analysis begins when VIRGO finishes commissioning, we hope by end 2006. • As detectors are upgraded (minor and major upgrades) data analysis will continue for a decade or more. • Japanese building cooled detector, can expect that this will join network when it is ready (several years yet). • Other ground-based plans in Australia may lead to a further detector. Erice: John Archibald Wheeler School

  37. Networks of detectors • Networks of detectors can be highly heterogeneous. For optimum analysis one should regard them as a single detector system. • Present methods for analysis for short-duration signals (inspirals, bursts) apply thresholds to detector or filter outputs separately. Candidate events must be in coincidence. • High-amplitude events dominated by non-Gaussian (instrumental) noise • Threshold/coincidence method eliminates most of these. • Method not optimum. • Optimum method is “coherent detection”, essentially adding data together before selecting by threshold. • Simple to see that this is optimum for two co-located detectors. • With separated detectors, do this for each possible source location. • Data must be weighted for detector sensitivity and orientation, shifted appropriately in time. • Still a subject of research, not yet implemented in LSC. Erice: John Archibald Wheeler School

  38. Data Analysiswith high sensitivity:LISA Erice: John Archibald Wheeler School

  39. LISA Project • Scientific leadership is provided by the LIST (LISA International Science Team), co-chaired by Tom Prince (JPL) and Karsten Danzmann (AEI Hannover). • Data analysis has not been as thoroughly studied as the hardware, but efforts underway to ensure that the analysis system is ready on time. • Data analysis planning is overseen by LIST Working Group 1B (co-chaired by Neil Cornish and Bernard Schutz), but most activity is coordinated by ESA and NASA in their separate communities. Structures not yet mature. • LISA Pathfinder coming soon, will give us experience of how to handle “housekeeping” data, but will not produce GW data. Erice: John Archibald Wheeler School

  40. Similarities with ground-based sources • LISA sources have many similarities with ground-based sources. • Inspiralling supermassive black-hole binaries emit signals identical with those at high frequencies, rescaled by the mass. • Binary star systems in the Galaxy emit signals very like those of GW pulsars: • Slowly-changing frequency (orbit decay raises f in this case) • Modulation by motion of LISA (less severe due to longer wavelength) • Differences: higher harmonics present if orbit is eccentric, interactions between stars may lead to other effects. • EMRI signals unlike any ground-based source, but experience with blind GW pulsar searches will help us to handle searching the extremely large parameter space of these signals. • LISA data stream much smaller than ground-based. Easier to do analysis. Erice: John Archibald Wheeler School

  41. LISA as a network of detectors • LISA is really three detectors. • Each vertex produces an interferometric signal. • For long wavelengths (λ > arm-length) these signals are correlated, and a null stream (linear combination with zero GW signal) can be formed. • For short wavelengths these signals have distinct forms. Still possible to form null stream for signals from any particular direction, as for ground-based networks. • LISA has extensive on-board processing to remove frequency noise. Will return three data streams, called TDI signals. (TDI = time-delay interferometry) • As LISA orbits, its antenna pattern rotates. From the induced amplitude and phase modulation it can determine directions to sources. • Additional information at short wavelengths from time-delays between TDI streams. Erice: John Archibald Wheeler School

  42. Wgw =10-10 High-SNR GW Observing • LISA observations will have high SNR, up to 104 in amplitude. • For many sources, LISA will face signal confusion • Binaries in the Galaxy below ~1 mHz blend into a “binary sea” that cannot be resolved: too many sources per frequency bin. • Above 1 mHz, EMRI capture signals are visible out to z ~ 0.5; more distant capture events provide the main background against which detection must take place. Olber’s Paradox avoided only by high-z cutoff in sources. • Resolvable binary systems above 1 mHz must be separated from non-stochastic EMRI interference. • Transient signals, such as from SMBH binary coalescence, must be separated from binary and EMRI background. Erice: John Archibald Wheeler School

  43. Approach to LISA data analysis • Requirement is to resolve overlapping signals. This involves not just detecting them but also measuring all parameters needed to remove them from the data stream. • Main data-analysis approach will be iterative: • Solve for strong sources approximately, subtract them. • Solve for next strongest, subtract, go back to strongest and remove their residuals better. • Binary orbital parameters improve with time, so their signals can be subtracted better after 2nd year. So transient events (black hole mergers) in first year also improve after 2 years. • Data products: source detections and parameters, cleaned-up data streams, full data streams. • Highly integrated analysis system required, but no decisions yet by agencies on how or where this analysis will be done, what proprietary data rights the scientists will have, etc. Erice: John Archibald Wheeler School

  44. Challenges of LISA data analysis • Confusion challenge • Source identification not unique. Must use intelligent principles to identify “best” identifications. How to guarantee that iterative scheme finds the globally “best” solution? What is the right search method? • Network challenge • LISA actually has 3 interferometer signals, optimum combinations depend on source location and polarisation. Modulation complicates this. • Computational challenge • Parameter space for EMRIs is huge. Even with anticipated improvements in computing, a hierarchical search will be needed. Not clear how to do this against a background of weaker EMRIs. • Theory challenge • Some signal templates not yet known well enough, including EMRIs and BH merger waveforms. • Organizational challenge • There is no legacy analysis system: it must be designed in scientific community but be highly integrated. Erice: John Archibald Wheeler School

  45. What is happening now • Hardware • LPF being built, lessons learned applied to LISA design. • Astrium Friedrichshafen is doing the formulation study for ESA • Data analysis • ESA has formed a Data Analysis Study Team to coordinate work of more than 50 institutions in Europe. • JPL has held meetings of US scientists to distribute work. • ESA (ESTEC) and NASA (JPL) will run independent but coordinated efforts developing algorithms in the community. • In Europe, groups must be funded by national agencies (PPARC). In the US, the NASA budget restrictions leave little room for funding data analysis development. • LIST provides overall coordination, issues mock data challenges. • Mock Data Challenges: first will be released at LISA Symposium this month. Periodic releases, increasing in complexity, as stimulus to community and as demonstration of competence. • Conclusion • Many opportunities for key contributions, leadership! Erice: John Archibald Wheeler School

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