Lanalisi dei dati negli esperimenti per la rivelazione di Onde Gravitazionali

Lanalisi 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 dinformazione 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 Gbyte/day (100 kbyte/s - 1Mbyte/s)

Il controllo globale dellinterferometro 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 TypeNumber of SignalsSize (kByte)Details ADC raw data501 signals monitor, 127 GC and align.,108 susp., 135 det. Bench SMS raw data556 signals2465 from towers and slow mon.,67 from detec.bench SMS trend data6 x 501 signals5For each ADC, a trend SMS, with the following signals: min, max, mean, rms, slope max, Gauss fit Frame Header1Structure FrameH4General information:GPS time, frame length……… Total 7620data compression could reduce the size of a factor 2 Il caso di VIRGO

REDUCED DATA FRAME Data TypeNumber of SignalsSize (kByte)Details ADC raw data1 signal at 20 kHz (4 Bytes)801 computed ITF output from det. bench, at 20 KHz SMS raw data556 signals (18 Sms)2465 from towers and slow mon.,67 from detec.bench SMS trend data6 x 501 signals5For each ADC, a trend SMS, with the following signals: min, max, mean, rms, slopemax, Gauss fit Data Quality 4 structures4fr AC data Moni., GC+Alig., Susp. and det. bench Calibrated Data h(t) 4 signals at 4 kHz (4 Bytes)80h value resampled with or without 4 signals at 4 kHz (4 Bytes) whitening and noise removal Frame Header1Structure FrameH4General information:GPS time, frame length……… Total 175 Il caso di VIRGO

Diagramma di flusso dellanalisi dati

The expected signal h is a short pulse ( a few ms). The expected value on Earth, if 1% of M o is converted into g.w. in the GC, is of the order of 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~ kpc/r axisymmetric collapse –h~ Mpc/r non-axisymmetric collapse Rate: a tens/year in the VIRGO cluster [Zwerger-Müller, ~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 nellintervallo 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 delluscita: – 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 ]

Il filtro lineare ottimo Luscita 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 dingresso o della sua trasformata di Fourier H(f) 2) Le proprietà spettrali del rumore S(f) Luscita del filtro c(t) è

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 t s, a time duration dt (containing N data samples), and a frequency band [f s ; f s + 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 [f s ; f s + 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 t s, durations t, starting frequencies f s 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/

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: h n (m 1,m 2,…) –Correlate templates and ITF output ITF output = noise+signal Correlation of ITF out and template Filtered variable Computational cost to perform transform (FFT): 5 N log2 [N] 1000 s of Sample/s data: 2x109 floating point operations (FLOP) To keep up with data: 2 MFLOP/s (MFLOPS) Perform 20,000 at same time: 40 GFLOPS -> clusters of CPUs 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 (r 5 NS/NS ) VIRGO SNR for coalescing Mpc 1.8 Mpc for TAMA 30 Mpc for GEO 45 Mpc for LIGO 50 Mpc for VIRGO M (M ) SNR

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/ ] 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 f spin SNR can be increased by integrating the signal for long time (months) Doppler correction of Earth motion ( f / f ~ ) must be taken into account 10 9 NS in the galaxy, ~800 known; the blind search requires high computing power and smart algorithms ased on a hierarchical search strategy (Hough transform)

The search divided in steps and in the first step we dont 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 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, h GW. (iii) Baseline separation must be suitably short:

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 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-ph »B. Allen and J. Romano, Phys. Rev. D59, (1999), gr-qc »M. Maggiore, Trieste, June 2000: Gravitational Waves: A Challenge to Theoretical Astrophysics, gr-qc »L.S. Finn and A. Lazzarini, Phys. Rev. D, 15 (2001)

Optimal filtering in the presence of background correlation h i =GW signal in detector i n i = noise in detector i GW (f) = 1/ 0 d GW /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 max 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 T LIGO =0.02s: rate reduction is –For = 1/min, N=4 and T 12 =T 23 = T LIGO =0.02s and T 34 =T max =0.086s: rate reduction is 1.6 x

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 False alarm events per decade LLO 4 km LHO 4 km + 2 km All LIGO Interferometers LIGO + VIRGO LIGO + VIRGO + 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

Tesine - tempo desposizione max. 25 minuti - date possibili Dicembre Gennaio

Tesine Proposte 1) La bilancia di Torsione (Dicke-Braginsky-Adelberger) 2) Test della legge Quadratica Inversa: Rassegne su una o più tecniche 3) Esperimenti di Quinta Forza 4) Misure della costante della Gravitazione Universale 5) Le verifiche della Local Lorentz Invariance (e, più in generale, della Relatività Speciale) ( 6) Le verifiche della Local Position Invariance 7) I 3 test standard della Relatività Generale: problematiche sperimentali e limiti sui PPN 8) LEsperimento di Pound-Rebka: tecnica e significati 9) Il Global Position System (GPS): principio di funzionamento e correzioni relativistiche 10) Il formalismo post-Newtoniano e gli attuali limiti sperimentali per i vari parametri 11) Le Pulsar binarie come laboratorio della Gravitazione: la Pulsar , la PSR J A/B 12) Il Gravitomagnetismo e la sua misura 13) Rassegna sui segnali di onde gravitazionali attesi in relazione alle sensibilità degli apparati 14) Il principio di rivelazione delle Onde Gravitazionali: studio degli effetti dinterazione in sistemi di riferimento differenti 15) I rivelatori risonanti ed i risultati di Weber 16) I Rivelatori interferometrici di OG (anche un problema specifico) 17) Limiti quantistici nei rivelatori di onde gravitazionali e sistemi Quantum Non Demolition 18) Lanalisi dati negli esperimenti di Onde Gravitazionali 19) Il rumore termico negli esperimenti gravitazionali

Sommaria dimostrazione A. Papoulis : Probabilità,variabili aleatorie e proccessi stocastici - ed. Boringhieri Supponiamo a) di conoscere x(t) sia a tempi passati che futuri b) La stazionarietà del rumore Poichè