Bernard F Schutz

International School of Relativistic Astrophysics: John Archibald Wheeler

Data Analysis for Gravitational Waves

Bernard F SchutzAlbert Einstein Institute – Max Planck Institute

for Gravitational Physics, Golm, Germanyand

School of Physics and Astronomy, Cardiff University, Cardiff, UK

[email protected]

www.aei.mpg.de

Bernard F Schutz Albert Einstein Institute

06 June 2006

Erice: John Archibald Wheeler School

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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


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Introduction


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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.


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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.


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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:


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When we know what signal we are looking for…


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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


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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: aBpap B|ap kk

posterior prior likelihood

While the theoretical dispute between Bayesiansand frequentists has often

been heated, we have foundin the GW data analysis

that the practical differencesin the detection of signals

is small.


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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 ~ Ncycles

1/2

– Example: a 200 Hz signal buried in noise with 10 times the amplitude, searched for in data sets of 2s, 20s, and 200s.

2s 20s

200s


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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


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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


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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.


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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 around λgw/ 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.

orbitR

wave

detector


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When we don’t know what signal we are looking for…


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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.

Inst 1σ =10

Inst 2σ =10

×

Avg = -0.46

Signal σ =2

×

Avg = 3.34

+

+


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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

Well correlatedresponses

Poorly correlatedresponses


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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.


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Data analysis in the


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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.


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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.


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● Sensitivity of search given in terms of the horizon distance: distance to 1.4+1.4 M optimally oriented & located binary at SNR=8

Binary neutron stars S5 search

S2 Horizon Distance

1.5 Mpc

• First three months of S5 data is analyzed.


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Primordial black holesS2 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

Total mass

Rat

e pe

r M

W h

alo

per

year


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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

signal-to-noise ratio squared

Rate < 38 per year per

Milky-Way-like galaxy

S2 Observational Result

Phys. Rev. D. 73, 062001 (2006)

Log|

cum

. no.

of e

vent

s |


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Binary black holes S5 search

binary neutron star

horizon distance

binary black hole

horizon distance

• 3 months of S5 analyzed• Horizon distance (detector

sensitivity) versus mass for BBH

Image: R. Powell

Average over run

1 sigma variation


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Radiation from rotating neutron stars

Wobbling neutron star

R-modes

“Mountain” on neutron star

Accreting neutron star


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• 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)

Pointing at known neutron stars


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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)

B0021-72CB0021-72DB0021-72FB0021-72GB0021-72LB0021-72MB0021-72N

B0531+21 (Crab)B1516+02AB1820-30AB1821-24B1937+21 (S1)B1951+32B0030+0451

J0711-6830J1024-0719J1629-6902J1721-2457J1730-2304J1744-1134J1748-2446C

J1910-5959BJ1910-5959CJ1910-5959DJ1910-5959EJ1913+1011J2124-3358J2322+2057


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S2 results

Dots: UL on h0

Squares: UL on

Red and magenta refer to PSRs with no info on fdot

R

fIh zz

20

4

2

0 c

G16

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


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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


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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


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S5 incoherent searches preliminary

.


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LIGO run H-L H1-H2 Frequency Range Obs. Time

S1 PRD 69(2004)

< 44 (H2-L1)

Cross-correlated instrumental noise

found40-314 Hz

64 hours(08/23/02 – 09/09/02)

S2unpublished

< 0.035 (H1-L1)

Cross-correlated instrumental noise

found 50-300 Hz

387 hours(02/14/03 – 04/14/03)

S3 In PRL

< 8.4e-4(H1-L1)

Trying to account for instrumental

noise in bounding

69-156 Hz218 hrs)

(10/31/03 – 01/09/04)

S4 Starting Analysis

447 hrs (H1-L1)510 hrs (H1-H2)

(02/22/05 – 03/24/05)

LIGO Results on 0h1002


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Bursts Analysis: TargetsCatastrophic 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.

0 10 20 30 40 50–1.5

–1

–0.5

0

0.5

1

1.5

Am

plit

ud

e [

x 10

–20]

Time [msec]

Gravitational waveformsfrom stellar–core collapse

(10kpc from the earth)

A1B1G1A3B3G1A4B1G2

possible supernova waveforms T. Zwerger & E. Muller, Astron. Astrophys. 320 209 (1997)


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The future: including VIRGO,further expanding the network


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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.


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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.


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Data Analysiswith high sensitivity:

LISA


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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.


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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.


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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.


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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.

gw


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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.


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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.


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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!

Documents

Bernard F Schutz