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
Il caso di VIRGO
• 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
with
6 x 501 signals
5
For each ADC, a trend SMS,
the following signals: min, max,
mean, rms, slope max, c2
1Structure FrameH
4
General information:GPS time,
frame length………
•
Gauss fit
Frame Header
•
----------------------------------------------------------------------------------------------------------------------------------------------
• Total
reduce
7620
data compression could
the size of a factor 2
Il caso di VIRGO
REDUCED DATA FRAME
Data Type
Number of Signals
Size (kByte)
ADC raw data
1 signal at 20 kHz (4 Bytes) 80
1 computed ITF output
from 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, c2 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
4 signals at 4 kHz (4 Bytes)
Frame Header
1Structure FrameH
4
Details
h value resampled with or without
whitening and noise removal
General information:GPS time,
frame length………
---------------------------------------------------------------------------------------------------------------------------------------------Total
175
Diagramma di flusso dell’analisi dati
Bursts
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
Supernovae
•
•
•
•
•
Impulsive events, GW emitted only in nonspherical 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:
2
A   xi
– Peak Correlation:
i1,N
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
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)  k 
O( f )H * ( f ) i2 ft
e
df
S( f )
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 + df].
– 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 + df].
– Calculate the probability of having obtained the summed power
from Gaussian noise alone using a c2 distribution with 2  dt  df
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 dt, starting frequencies fs and bandwidths df.
[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
o(t )  n(t )  h(t )
~
o~ ( f )  h * ( f )
w( f ) 
Sn ( f )
C (t )   e it w( f )df
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 (r5NS/NS)
VIRGO SNR for
coalescing BH @100
Mpc
1.8 Mpc
for TAMA
30 Mpc
45 Mpc
for GEO
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, grqc/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
Continuous signals
Signals from rotating neutron stars, stars in binary systems
Rotating Neutron Stars
•
Non-axisymmetric rotating NS emit periodic GW at f=2 fspin
 f    
I
 10 kpc 


 45
  6 
2 
 r  10 g/cm  200 Hz   10 
2
h  3 10
•
•
•
 27
SNR can be increased by integrating the signal for long time (months)
Doppler correction of Earth motion ( Df / 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)
How to solve the problem of the limited
computational power.
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”
Time-frequency peak map
Hough map – single annulus
Hough map – source reconstruction
– Requirements
–
–
–
–
SFFT data base
Time-Frequency data quality
Partial Hough Transform Output
Candidate Source data base
525 Gbyte /year
20 GByte/year
40 Tbyte/year
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
•
Tier 1 Lyon – IN2P3, Bologna - CNAF/INFN
•
Tiers 2: Roma 1 and Napoli;
Virgo- Cascina;
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)]
V. Ferrari, S. Matarrese, R. Scheinder: Mont. Not.R. Astron. Soc. 303, (1999) 247 & 258
String cosmology
Stochastic GW Background Detection
• Cross-correlate the output of two
(independent) detectors with a suitable filter
T / 2  /2
kernel:
C(T)  dt d ' s (t)s (t   ' )Q( ' )


T / 2  / 2
• Requires:
1
2
s1( f )s2 ( f )  0, {f} 
(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”:
fL
L  GW ( f ) 
1
c
Stochastic Background Correlation
• Ideally, the stochastic background correlation
increases with integration time as:
1
SNR 
3H02
2
10
 2 ( f ) 2 Df  2
Tint  6 0 GW 
 f S1,n f S2,n f 
– 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
•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) 
T /2
T /2
 dt  dt '
T / 2

s1(t)s2 (t')Q(t  t') ; si (t)  hi (t)  n i (t)
hi=GW signal in detector i
ni= noise in detector i
T / 2

 sin( fT) 
˜
˜
df
'
d
(
f

f
')
s
*
(
f
)
s
(
f
')Q(
f
')
;
d
(
f
)

T



T
1
2
T
 fT 
0
0
2
3H 0
h˜1 * ( f )h˜ 2 ( f ')  d ( f  f ')
GW  f  f ,1,2
3
<=Template for this problem
20 2 f
1
n˜ i * ( f ) n˜ j ( f ')  d ( f  f ')Sij  f 
2
1
n˜ i * ( f ) n˜ j ( f ') n˜ i * ( f ") n˜ j ( f ''') 
Sii  f S jj  f ' d ( f  f ")d ( f ' f ''') 
4
Sij  f Sij  f " d ( f  f ")d ( f ' f ''') 
C(T) 
 df


(
)
Sii  f S jj  f ' d ( f  f ")d ( f ' f ''')
GW(f) = 1/r0 drGW/d(ln[f])
(f, 1, 2) = geometric overlap
reduction factor
depends on
antenna
orientations
Optimal filtering in the presence of
background correlation


3H02
˜
C(T, 1 ,2 )  T  df  
3  GW  f  f ,1, 2  S12  f  Q( f ) ;
2
0
 20 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:
C(T )  C (T / 2)  C (T / 2)
 3H 2


0
˜
C(T )  T  df 
3  GW  f  f , 1 ,2  Q( f )
2
0
 20 f

2
2
2
2
 C  C  C  2 C ,
2
T
 C2   df S1  f S2  f   S122  f  Q˜ ( f )
20
  f , , 
f 

C max d SNR
1
2
GW,mod
el

SNR 

 0  Q˜ ( f ) 
2
 f 3 S1  f S2  f   S12
˜
C
d Q
 f 








Optimal filter for this problem
Detection Confidence
•
Detection computation:
– Coincidence with other GW detectors
– Coincidence with non-GW detectors (optical, , X, n)
– 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
Pn0 ( i ;Ti )  1 e i Ti  probability of at least one event in time Ti
For iTi  1, Pn0  iTi ; if N detectors are statistically independent
N
12...N  1  iTi1;
i 2
For N comparable detectors with comparable time windows
12...N
 reduction of nonGaussian event noise by coincidence reqmnt.
1
N 1
 1T1 j  and iTi  1
– 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
:
Coincidence windows among detectors
• Rejection of statistically uncorrelated
random events
False alarm events per decade
•
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
GOAL
International Network of Detectors
• 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
•
GEO (UK/Germany)
LIGO (U.S.)
VIRGO (Italy/France)
TAMA (Japan)
•
•
AIGO
(planned)
detection
confidence
locate the
sources
decompose the
polarization of
gravitational
waves
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 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)
(http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html)
6) Le verifiche della Local Position Invariance
7) I 3 test standard della Relatività Generale: problematiche sperimentali e limiti sui PPN
8) L’Esperimento 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 1913 + 16, la PSR J0737-3039A/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 d’interazione 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) L’analisi dati negli esperimenti di Onde Gravitazionali
19) Il rumore termico negli esperimenti gravitazionali