L’analisi dei dati negli esperimenti per la rivelazione di Onde Gravitazionali

L’analisi dei dati ed i problemi computazionali

• I dati originali prodotti da un interferometro includono i segnali dei

- canali di controllo, - sensori delle sospensioni - sensori ambientali- Duplicazioni d’informazione per ridondanza

864 Gbyte/day 315 Tbytes/year Assumendo un efficiente compressione e on-line processing, i

dati utili possono essere compressi di un fattore compreso tra 10 e 100

Frequenza di produzione dei dati selezionati 9 - 90 Gbyte/day (100 kbyte/s - 1Mbyte/s)

Il controllo globale dell’interferometro e la curva di sensibilità

Calibrazione

• On line data processing– Interferometer control– Data acquisition

• Local Readout

• Frame formatting

• Frame storage on-line (disks-->> data distribution)

• Frame storage off-line (Tapes-->>Raw data archive)

– On line Data analysis• h(t) reconstruction

• Data quality

• On-line filters, triggers and selection candidates

Il caso di VIRGO

• Frames stored on data Distribution– Debug Frames: data acquired and sent to some on line monitoring tool and/or stored on tape for

diagnostic purpose. Usually, the debugging channels are not sent to the Main Frame Builder

– Raw Data Frames: set of data from the multiple signals from the interferometer, the monitor

channels and vetoes. They contain all the ADC channels including trend data and any log message. – Reduced Data Frames: these are frames stored on the data distribution resident on disks.

They contain the reconstructed information ( h(t) ), the data quality, the Slow Monitoring Stations (including the trend data), the summary results of the data selection algorithms.

– Selected Frames: they contain just the original raw data selected in time by the algorithms for

the search of transient events

– Trend Frames: collection of Slow monitoring Stations and data trend formatted to optimize a fast

access for long stretch of data

Il caso di VIRGO

– RAW DATA FRAME• Data Type Number of Signals Size (kByte) Details

• ADC raw data 501 signals 7608 131 monitor, 127 GC and

align.,108 susp., 135 det. Bench

• SMS raw data 556 signals 2 465 from towers and slow mon.,67

from detec.bench

• SMS trend data 6 x 501 signals 5 For each ADC, a trend SMS, with the following signals: min, max, mean, rms, slope max, Gauss fit

• Frame Header 1Structure FrameH 4 General information:GPS time, frame length………

• ----------------------------------------------------------------------------------------------------------------------------------------------

• Total 7620 data compression could reduce the size of a factor 2

Il caso di VIRGO

REDUCED DATA FRAMEData Type Number of Signals Size (kByte) Details

ADC raw data 1 signal at 20 kHz (4 Bytes) 80 1 computed ITF outputfrom det. bench, at 20 KHz

SMS raw data 556 signals (18 Sms) 2 465 from towers and slow mon.,67 from detec.bench

SMS trend data 6 x 501 signals 5 For each ADC, a trend SMS, with the following signals: min, max, mean, rms, slopemax, Gauss fit

Data Quality 4 structures 4 fr AC data Moni., GC+Alig., Susp. and det. bench

Calibrated Data h(t) 4 signals at 4 kHz (4 Bytes) 80 h value resampled with or without 4 signals at 4 kHz (4 Bytes) whitening and noise removal

Frame Header 1Structure FrameH 4 General information:GPS time, frame length………

----------------------------------------------------------------------------------------------------------------------------------------------Total 175

Il caso di VIRGO

Diagramma di flusso dell’analisi dati

The expected signal h is a short pulse ( a few ms).

The expected value on Earth, if 1% of Mo is converted into g.w. in the GC,

is of the order of 10-18

Bursts

Supernovae

• Impulsive events, GW emitted only in non-spherical collapse

• Big uncertainties in the simulation codes: waveform “unpredictable” (templates unreliable)

• Coincidence detection necessary

• Amplitude:

– h~10-21 30 kpc/r axisymmetric collapse

– h~10-21 10 Mpc/r

non-axisymmetric collapse

• Rate: a tens/year in the VIRGO cluster

[Zwerger-Müller, www.mpa-garching.mpg.de/

~ewald/GRAV/grav.html

Algoritmi per la rivelazione di burst• Nessuna conoscenza a priori della forma del segnale • Nel caso degli interferometri che hanno una banda di rivelazione estesa

(da pochi Hz a pochi kHz), non è possibile schematizzare il segnale come una semplice δ-function.

• Tutti i filtri proposti sono inesorabilmente sub-ottimali, nel senso che sono più o meno lontani dalle prestazioni del Filtro Lineare Ottimo ( vedi trapsaprenza successiva).

• Alcuni filtri applicati sui dati “sbiancati”:– Valore di conteggio nell’intervallo di tempo elementare: il filtro

seleziona select tutti gli intervalli elemntari che hanno un segnale al di sopra di una data soglia ( il piu’ semplice )

– Filtro a Norma: il massimo della funzione di autocorrelazione dell’uscita:

– Peak Correlation:P(N,k) = Σ x(i+k) f([i-N/2]t)

dove f(t) =exp (-t2 /2 τ2) definita in [-3 τ,3τ ] €

A = x i2

i=1,N

∑

Il filtro lineare ottimo

L’uscita del rivelatore è o(t) , sommma del segnale h(t)e e del rumore n(t):

o(t)=h(t)+n(t) Il filtro adattato è una tecnica lineare di “ patter-matching “ che

permette di esaltare il rapporto Segnale su Rumore (SNR). Per applicare questa tecnica noi dobbiamo porre in ingresso le

seguenti informazioni: 1) Ipotesi sulla forma del segnale d’ingresso o della sua

trasformata di Fourier H(f) 2) Le proprietà spettrali del rumore S(f)

L’uscita del filtro c(t) è

€

c(t) = kO( f )H*( f )

S( f )∫ e−i2πftdf

Filtro Adattato • Nelle telecomunicazioni un filtro adattato o filtro ottimo (è ottenuto correlando

un segnale conosciuto con un segnale incognito per rivelare la presenza di un marcatore all'interno del segnale incognito.

• Ciò è equivalente ad effettuare l'operazione di convoluzione tra il segnale incognito ed una versione tempo-invertita del segnale noto.

• Il filtro adattato è il filtro lineare ottimo per la massimizzazione del Rapporto segnale/rumore (SNR) in presenza di rumore stocastico additivo.

• I filtri adattati sono comunemente usati in ambito radar, in cui un segnale conosciuto viene trasmesso, ed il segnale riflesso è esaminato per la ricerca di elementi comuni con il segnale trasmesso. Altre applicazioni del filtro adattato si ritrovano nell'elaborazione digitale delle immagini, ad esempio per incrementare il rapporto SNR in fotografie a raggi X.

Derivazione del Filtro adattato

Caso Generale

Burst Searches: an example of algorithm:Excess Power Statistic (W. Anderson et al.)

• The algorithm [1]:– Pick a start time ts, a time duration dt (containing N data samples),

and a frequency band [fs; fs + f].– Fast Fourier transform (FFT) the block of (time domain) detector

data for the chosen duration and start time.– Sum the power in the frequency band [fs; fs + f].– Calculate the probability of having obtained the summed power from

Gaussian noise alone using a 2 distribution with 2 t f degrees of freedom.

– If the probability is significantly small for noise alone, record a detection.

– Repeat the process for all desired choices of start times ts, durations t, starting frequencies fs and bandwidths f.

[1] A power filter for the detection of burst sources of gravitational radiation in interferometric detectors. Authors: Warren G. Anderson, Patrick R. Brady, Jolien D. E. Creighton, Eanna E. Flanagan. gr-qc/0001044

Coalescing Binaries• Compact stars (NS/NS, NS/BH, BH/BH)• Inspiral signal accurately predictable

– Newtonian dynamics– Post-Newtonian corrections (3PN, (v/c)11/2) [L.Blanchet et al., CQG 13,

1996]

• Detection rate (predicted event rate VIRGO sensitivity)

– <1/yr for NS/NS– 2-3/yr for BH/BH

chirp

•The classic search method is based on the Matched filter.–Signal hypothesis (a chirp dependent on some physical parameters). FFT+ zero padding.–Spectral Noise estimation from the data. FFT–Correlation between them in the frequency domain (scalar product)–Anti FFT

The optimum linear filter• Signal known, optimal filtering possible

– Produce a set of templates with different mass and orbit parameters: hn(m1,m2,…)

– Correlate templates and ITF output

∫ −=

⋅=

+=

dffwetC

fS

fhfofw

thtnto

ti

n

)()(

)(

)(~

)(~)(

)()()(

*

ω

ITF output = noise+signal

Correlation of ITF out and template

Filtered variable

Computational cost to perform transform (FFT): 5 N log2 [N]

ﾐ 1000 s of 16384 Sample/s data: 2x109 floating point operations (FLOP)

ﾐ To keep up with data: 2 MFLOP/s (MFLOPS)

ﾐ Perform 20,000 at same time: 40 GFLOPS -> clusters of CPUs

ﾐ 90+% of CPU time involved in f<->t transformations

E.Cuoco

How Far Can They See?

• The detector sensitivity can be measured by the distance to an ideally oriented NS/NS coalescing binary system that would produce a SNR=5 (r5

NS/NS)

• VIRGO SNR for coalescing BH @100 Mpc

1.8 Mpc for TAMA

30 Mpc for GEO

45 Mpc for LIGO

50 Mpc for VIRGO

M (M) 10 20 30 50 100

SNR 1.5 3 3.5 5 4

•Standard candles: distance and redshift of the source can be found out of the waveform of a NS/NS [Schutz, Nature, 1986]

•Test for GR: accurate measurements of inspiral waveform can test gravity in the strong field regime [Damour, Esposito-Farese, gr-qc/9803031]

•Nuclear physics: before coalescence waveform sensitive to the equation of state [Cutler et al., PRL, 70, 1993]

• The search of continuous signals• The search method is based on a hierarchical method.

– Filling a Short FFT data base ( the data are dived in different band of frequencies)

– Construction of Time Frequency maps

– Hough Transform

– Candidate Selection

– Coherent search in the selected frequency ranges (Zooming, Doppler correction , FFT…..)

– New iteration

• The search is applied on the reduced data transferred in Rome and stored in the SFFT data base

Signals from rotating neutron stars, stars in binary systems

Continuous signals

Rotating Neutron Stars

• Non-axisymmetric rotating NS emit periodic GW at f=2 fspin

• SNR can be increased by integrating the signal for long time (months)

• Doppler correction of Earth motion ( f / f ~ 10-4 ) must be taken into account

• 109 NS in the galaxy, ~800 known; the blind search requires high computing power and smart algorithms ased on a hierarchical search strategy (Hough transform)

⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛⋅≈ −−

6

2

24527

10Hz 200g/cm 10

kpc 10103

εfI

rh

The search divided in steps and in the first step we don’t use the phase information.

An example of incoherent step followed by a coherent one (Complex FFT)

• create Data Base of short FFT and then derive the

periodograms.•Create a time-frequency map of the peaks above a threshold• For each spin-down parameters point and each

frequency value, create a sky map (“Hough map”); to

create a H map, sum an annulus of “1” for each peak;

an histogram is then created, that must have a

prominent peak at the “source”

How to solve the problem of the limited computational power.

Time-frequency peak map

Hough map – single annulus

Hough map – source reconstruction

– Requirements – SFFT data base 525 Gbyte /year– Time-Frequency data quality 20 GByte/year– Partial Hough Transform Output 40 Tbyte/year – Candidate Source data base 20 GByte/year

• Scheduled tests– Search limited to the frequency interval 10 Hz - 1.25 kHz. The computation

will cover few weeks of data taken by the CITF. – Method applied to the data of resonant antennas: data taken in a small

bandwidth of 2 Hz around the two peaks in the 900 Hz region

– 8000 SI95 and 10 Tbyte of storage disk.

• Computational power and storage hierarchically distributed in the Virgo INFN sections and laboratories of the collaboration and structured in the classes of computer centers

• Tier 0 Virgo- Cascina;

• Tier 1 Lyon – IN2P3, Bologna - CNAF/INFN

• Tiers 2: Roma 1 and Napoli;

Use of GRID technology

Stochastic Background

Cosmological origin: it is the result of processes that happened immediately after the Big-Bang. If measured, it will allow to discriminate various cosmological models

Astrophysical origin: it is the result of more recent event (redshift z order of 2-5). It is due to unresolved processes of gravitational collapses. It will provide information on star formation rates, supernova rates, black holes......

Relic Stochastic Background

• Imprinting of the early expansion of the universe

• Need two correlated ITFs• Standard inflation produces a

background too low• Hope from string models

[Buonanno et al., PRD (1997)]

String cosmology

V. Ferrari, S. Matarrese, R. Scheinder: Mont. Not.R. Astron. Soc. 303, (1999) 247 & 258

Stochastic GW Background Detection

• Cross-correlate the output of two (independent) detectors with a suitable filter kernel:

• Requires:(i) Two detectors must have overlapping frequency

response functions i.e.,(ii) Detectors sensitive to same polarization state (+,

x) of radiation field, hGW.(iii) Baseline separation must be suitably “short”:

C(T) = dt−T / 2

T / 2

∫ dτ−τ /2

τ /2

∫ ' s1(t)s2(t−τ' )Q(τ' )

s1( f)s2( f )≠0, {f}∉∅

L <λGW( f)⇒fLc

<1

• Ideally, the stochastic background correlation increases with integration time as:

– Assumes no additional sources of correlated noise• cannot discriminate with a single measurement

– Mutual orientation dependence of GW background signal may be exploited to discriminate among possible correlated sources

SNR∝3H0

2

10π2 Tintγ2( f0)ΩGW

2 Δff6S1,n f S2,n f

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

12

Stochastic Background Correlation

•References:»P.F. Michelson, Mon. Not. Roy. Astron. Soc. 227, 933 (1987).»N. Christensen, Phys. Rev. D46, 5250 (1992)»E. Flanagan, Phys. Rev. D48, 2389 (1993), astro-ph9305029»B. Allen and J. Romano, Phys. Rev. D59, 102001 (1999), gr-qc9710117»M. Maggiore, Trieste, June 2000: Gravitational Waves: A Challenge to Theoretical Astrophysics, gr-qc-0008027»L.S. Finn and A. Lazzarini, Phys. Rev. D, 15 (2001)

Optimal filtering in the presence of background correlation

€

C(T) = dt−T / 2

T / 2

∫ dt−T / 2

T / 2

∫ ' s1(t)s2(t')Q(t − t') ; si(t) = hi(t) + ni(t)

C(T) = df0

∞

∫ df0

∞

∫ ' δT ( f − f ')˜ s *1 ( f )˜ s 2( f ')Q( f ') ; δT ( f ) ≡ Tsin(πfT)

πfT

⎧ ⎨ ⎩

⎫ ⎬ ⎭

˜ h 1 * ( f ) ˜ h 2( f ') = δ( f − f ')3H0

2

20π 2 f3 ΩGW f( )γ f ,

r Ω 1,

r Ω 2( )

˜ n i * ( f ) ˜ n j ( f ') =1

2δ( f − f ')Sij f( )

˜ n i * ( f ) ˜ n j ( f ') ˜ n i * ( f ") ˜ n j ( f ' ' ') =1

4(Sii f( )S jj f '( )δ( f + f ")δ( f '+ f ' ' ') +

Sij f( )Sij f "( )δ( f + f ")δ( f '+ f ' ' ') +

Sii f( )S jj f '( )δ( f + f ")δ( f '+ f ' ' '))

hi=GW signal in detector ini= noise in detector i

GW(f) = 1/0 dGW/d(ln[f])(f, 1, 2) = geometric overlap

reduction factor depends on antenna orientations

<=Template for this problem

Optimal filtering in the presence of background correlation

C(T,

r Ω 1,

r Ω 2) =T df

0

∞

∫ ±3H0

2

20π2 f 3 ΩGW f( )γ f ,r Ω 1,

r Ω 2( )+S12 f( )

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ ̃ Q ( f) ;

max

C(T) = C+(T /2)−C−(T /2)

C(T) =T df0

∞

∫ 3H0

2

20π 2 f 3 ΩGW f( )γ f ,r Ω 1,

r Ω 2( )

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ ̃ Q ( f)

σC2 = C2 − C 2 =2σC

2+,−

σC2 =

T2

df0

∞

∫ S1 f( )S2 f( )+S122 f( )( ) ˜ Q ( f)[ ]

2

SNR=CσC

⇒ δ SNR[ ]δ ˜ Q [ ]

=0 ⇒ ˜ Q ( f ) =γ f ,

r Ω 1,

r Ω 2( )ΩGW,model f( )

f 3 S1 f( )S2 f( )+S122 f( )( )

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

Choose two orientations of one detector { 1, 1’ }, for which (f, 1, 2) = - (f, 1

’, 2), denote C+, C- values of integrated correlation in these two orientations:

Optimal filter for this problem

Detection Confidence

• Detection computation:

– Coincidence with other GW detectors

– Coincidence with non-GW detectors (optical X, )

– Matched filtering for known signals

– Correlations for stochastic background

• Environmental monitoring on site

– ‘fast’ monitoring of seismic, acoustic, e.m. noise

– ‘slow’ monitoring of tilts, temperature, weather

Coincidence windows among detectors• Rejection of statistically uncorrelated random events

– Coincidence window duration determined by baselines, always less than 2*13000km/(300000km/s) = 0.086s

– For = 1/min ,N=3 and TLIGO =0.02s: rate reduction is 10-7

– For = 1/min, N=4 and T12=T23= TLIGO =0.02s and T34=Tmax=0.086s: rate reduction is 1.6 x 10-10

€

Pn≠0(λ i;Ti) =1− e−λ iTi ⇒ probability of at least one event in time Ti

For λ iTi <<1, Pn≠0 ≈ λ iTi; if N detectors are statistically independent :

λ12...N ≈ λ1 λ iTi1

i= 2

N

∏ ;

For N comparable detectors with comparable time windowsλ12...N

λ1

≡ reduction of nonGaussian event noise by coincidence reqmnt.

≈ λ1T1 j[ ]N−1

and λ iTi <<1

Coincidence windows among detectors

• Rejection of statistically uncorrelated random events

• Two Sites - Three Interferometers– Single Interferometer - limited by non-gaussian noise

~70/hr– Hanford -- 2x coincidence requirement (x1000 reduction)

~1/day– Hanford + Livingston -- 3x coincidence (another x5000 reduction)

<0.1/yr

Background events per hour per interferometer

Fal

se

ala

rm e

ven

ts p

er d

ecad

e

LLO 4 km

LHO 4 km + 2 km

All LIGO Interferometers

LIGO + VIRGO

LIGO + VIR

GO + GEO

GOAL

• A number of projects are bringing detectors on line during the next few years• Operated as a phased array, they will augment the chances for detection by excluding

backgrounds and localizing sources• True coincidences will be within milliseconds of each other • detection

confidence

• locate the sources

• decompose the polarization of gravitational waves

International Network of Detectors

LIGO (U.S.)

GEO (UK/Germany) VIRGO (Italy/France)

TAMA (Japan)

AIGO(planned)

FINIS FINIS

Tesine

- tempo d’esposizione max. 25 minuti

- date possibili 14 -15 Dicembre 15-16 Gennaio

Tesine Proposte

1) La bilancia di Torsione (Dicke-Braginsky-Adelberger)2) Test della legge Quadratica Inversa: Rassegne su una o più tecniche3) Esperimenti di Quinta Forza4) Misure della costante della Gravitazione Universale5) Le verifiche della Local Lorentz Invariance (e, più in generale, della Relatività Speciale)

(http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html)6) Le verifiche della Local Position Invariance7) I 3 test standard della Relatività Generale: problematiche sperimentali e limiti sui PPN8) L’Esperimento di Pound-Rebka: tecnica e significati9) Il Global Position System (GPS): principio di funzionamento e correzioni relativistiche10) Il formalismo post-Newtoniano e gli attuali limiti sperimentali per i vari parametri11) Le Pulsar binarie come laboratorio della Gravitazione: la Pulsar 1913 + 16, la PSR J0737-3039A/B12) Il Gravitomagnetismo e la sua misura13) Rassegna sui segnali di onde gravitazionali attesi in relazione alle sensibilità degli apparati14) Il principio di rivelazione delle Onde Gravitazionali: studio degli effetti d’interazione in sistemi di riferimento differenti15) I rivelatori risonanti ed i risultati di Weber16) I Rivelatori interferometrici di OG (anche un problema specifico)17) Limiti quantistici nei rivelatori di onde gravitazionali e sistemi Quantum Non Demolition18) L’analisi dati negli esperimenti di Onde Gravitazionali19) Il rumore termico negli esperimenti gravitazionali