Data analysis for impulsive sources with resonant g.w. detectors Pia Astone ROG collaboration Villa Mondragone International school of Gravitation and Cosmology 7-10 September

M ; T ; Q The Eq of geodetic deviation is the basis for all the experiments to detect g.w. Use of powerful signal analysis tools (adaptive matched filters) to extract the signal from noise (non-stationary) Thermal noise T= 3 K, DL/L = Thermal noise T=300 mK, DL/L =

NI 200 days AU 221 days EX 1036 days NA 831 days AL 852 days ON times of resonant detectors from 1 Jan 1997 up to13 Jun all (almost) parallel-- IGEC data during S1, S2. + +

The expected signal h is a short pulse ( a few ms). The expected value on Earth, if 1% of a solar mass is converted into g.w. in the GC, is of the order of 10 Burst events for a resonant detector: a millisecond pulse, a signal made by a few millisecond cycles, or a signal sweeping in frequency through the detector resonances. E.g.: a stellar gravitational collapse, fall of a body into a BH, the last stable orbits of an inspiraling NS or BH binary, its merging and final ringdown. -18

The signal: detector response u sig (t) [m] to a short (g.w.) pulse Excitation= impulse force applied to the bar (m x oscillator) Energy absorbed by the detector: System transfer function:

V sig (t) =  u sig (t) 0 s s 10.5 here the decay time is   1 = 1/  1 (if  - =   =  1)

The noise Total noise spectrum, at the transducer output, is due to the Brownian noise of the two mechanical oscillators (narrowband) + wide band noise due to electronics: In addition to this noise, due to known effects, we may have some excess noise of non stationary nature, due to unknown noise sources [ V / Hz ] 2 ( the mechanical dissipations correspond to noise force generators with 2-sided spectral density S fx = 4kT e  x m x, S fy = 4kT e  y m y. The spectral densities of the associated displacement are )

SNR= signal to noise ratio SNR is defined as the ratio of the square amplitude of the signal to the noise variance the detector is a linear system the g.w. signal is added to the noise We normally consider the signal (in the detector bandwidth) a delta function, that is we suppose to know the waveform; If we suppose the noise to be Gaussian and stationary,the problem is classical: the detection of a signal of known shape in Gaussian noise THIS IS OPTIMALLY SOLVED USING A MATCHED FILTER

Block diagram of antenna and matched filter F x(j  ) So N(j  ) W ux(j  ) 1 / N(j  ) ( F x W ux / N) * matched filter section St(  ) whitening matche d

The matched filter complex conjugate of the FFT of the antenna response to the pulse noise power spectrum at the antenna output it has the property that the output SNR is maximized it is linear and non-causal ( where and = fo  for impulsive input )

The matched filter for impulsive input (a part from constants) inverse filter smoothing filter (“spectral gain”) Transfer function of the smoothing filter = frequency domain signal at the filter output

The bandwidth after the filter The inverse filter cancels the dynamics of the antenna; the smoothing filter minimizes the contribution of the wideband noise and thus limits the bandwidth. The bandwidth after the filter is much larger than the mechanical bandwidth of the antenna oscillator: in fact the antenna responds in the same way to an excitation due to a g.w. burst and to the brownian noise, and thus the bandwidth is limited only by the wideband noise The ratio fx defines the bandwidth

The SNR after the matched filter SNR here is written in terms of the input force, but it can be shown that it can always be written in terms of the spectral gain. In particular, to reason in terms of h...

The SNR after the matched filter...we define S h (  ) as the “noise spectral density”, [1/Hz], the noise spectrum at the detector input. We realize that G (  ) = 1/S h (  ). If the Fourier transform of the input force is h  g then it is possible to show that: ( h  g  S h (  ) 2

The sensitivity to bursts (1): ( strain sensitivity)

SNR m = SNR o T e T eff The improvement in SNR o obtained by filtering the data (SNR m ) can be expressed in terms of a reduction of the equivalent temperature T e to the effective temperature T eff

The sensitivity to bursts (2): h = Sqrt(T eff ) for 1 ms burst -18

The signal g(t) at the filter output

bandwidth: minus mode only plus mode only

filter input: V sig (t) =  u sig (t) filter output: g(t) [K] 0 seconds 50 Simulated, in the absence of noise decay time:  1 = 1/  1 decay time:   3 = 1/  3

Real data: the arrival of a cosmic ray shower on NAUTILUS Unfiltered data (V 2 ) The signal after filtering (kelvin)

NAUTILUS 1999 Sqrt(T/MQ) cooled at 100 mK Note that the bandwidth depends ONLY on the transducer and amplifier Calibration signal AN EXAMPLE OF STRAIN SENSITIVITY

880 Hz 980 The year 2001 Explorer cooled at 2.6 K Nautilus cooled at 1.3 K Phys Rev Letters (2003) 880 Hz Hz

880 Hz 980 The year 2003 Explorer cooled at 4.2 K Nautilus cooled at 2 K 880 Hz 980 Auriga is also broadband now (since Dec. 2003)

Problems in the detection of g.w. pulses: low SNRs and rarity of the events ignorance of their shape the non stationary noise of the detector the presence of many spurious events

There are different ways of implementing the filter procedures ➔ different ways of taking data (high frequency sampling, aliased sampling, lock-ins); ➔ frequency or time domain procedures; ➔ use of adaptive or non-adaptive procedures; ➔ definition of the threshold and procedure to extract the events; ➔ procedure to extract and use the events features, also to recognize spurious events.

Periodogram From the Hz the high sensitivity band of the antenna is extracted (  40 Hz) Power Spectrum of the Nautilus data (5 kHz Acquisition Frequency)

In theory the best spectral estimation is obtained using as much data as possible. But various scenarious of non stationary noise are possible: ● Spurious peaks in the spectra; ● “Short” time disturbances in the unfiltered data; ● “Long” time disturbances in the unfiltered data. The adaptive algorithm is the method to estimate a new spectrum from the data. To construct the matched filter we need the power spectrum: given the presence of non stationarities in the system we do not use the theoretical prediction for the noise, but we estimate the noise from the data

A power spectrum and the corresponding filter tranfer function 900 Hz Hz Power spectrum Filter transfer function

To face with these problems, we have implemented three different method to estimate the spectra and hence to build up the filters. WHOLE CLEAN ADAPTED (or varying memory)

WHOLE CLEAN ADAPT The matched filter : W ux *(j  ) S(  ) OUTPUT CHANNELS Adaptive filters use the actual noise spectrum, estimated from the data, to evaluate the filter transfer function. The periodogram P i is used to estimate the spectrum S i : a new periodogram is evaluated every 105 s. The time constant for the spectrum is 1 hour. THE EVALUATION OF THE SPECTRUN IN NON- STATIONARY NOISE IS THE CRUCIAL STEP OF THE FILTERING PROCEDURE

The DAGA2_HF noise estimators for matched filters on non-stationary noise P. Astone, S. D'Antonio, S. Frasca, M. A. Papa All the procedures use the same recursive equation to evaluate the new spectrum, but they differ for the value of the time constant  and for the criterion to accept a periodogram in the average: Whole:  is fixed to 1 hour and all the periodograms are used Clean:  is fixed to 1 hour but only “good” periodograms are used Adapted:  varies according to the variance of the periodogram and all the periodograms are used

The CLEAN MATCHED filter The problem of “short” time disturbances can be resolved by eliminating the correspondent periodogram in the spectral estimation. The choice of the periodogram is done evaluating the integral of the periodogram S and comparing it with its expected value Sp. The periodogram is eliminated if its integral differs from Sp (evaluated over a long time period) by more than one standard deviation. The spectrum obtained with this procedure does not have high excess noise, respect to expectation. This filter gives good results for short disturbances but is not a good filter when data are noisier for long time period!

The integral of the periodograms is over the threshold for about 40 min and the clean filter does not adjourn itself… 23 2 hours of data 25 Presence of “long” time disturbances in the data

The ADAPTED is able to recover the disturbance due to higher noise around the minus mode (the white noise doesn’t change).The transfer functions are very different: the adapted, well adapted to the actual noise characteristics, has a lower gain around the minus mode, where the disturbances acted, compared to the gain of the clean filter, not well adapted to meet the new situation. Spectrum estimated by the ADAPTED Spectrum estimated by the CLEAN

The CLEAN does not use the periodograms whose integral is over the threshold: the spectral estimation is not degraded. The WHOLE uses all the periodograms the spectra estimation is degraded. The CLEAN filter is better than the WHOLE, when the disturbance is over hours Presence of “short” time disturbances in the data

The calibration signals will be one channel of the ROG acquisition system, DAGA2-HF - The best filter is the one that, properly normalized, gives the lower T eff - Calibration signals, added to the noise of the detector, will be used to compare the filters, to evaluate the experimental efficiencies of detection and all the event parameters. Now, let's show some data

0.5 mK 8 Explorer days median=2.2 mK Explorer days median=2.5 mK 2 mK=3.6 * mK 8

Nautilus days median=2.8 mK Nautilus days median=1.6 mK 0.5 mK 8 2 mK=3.6 *10-19

Data of unprecedented sensitivity and very high duty cycles and overlapping times

The threshold mechanism: how to extract the events The threshold changes with time, to follow the changes of the output noise : ADAPTIVE THRESHOLD  = memory of the autoregressive average (10 minutes) ;

The threshold mechanism: how to extract the events The threshold is set on the CRITICAL RATIO CR (which is easily related to the SNR) CR CR = 6 corresponds to SNR (energy)= 19.5 When the signal goes above the threshold an “event” begins. The event ends when the data go below the threshold for a time > “dead time”, which has been set depending on the apparatus, noise, expected signals.

threshold time amplitude Definition of event Any event is identified by various parameters (energy, CR, duration, nmax...)

Nautilus 2001: the bandwidth was of the order of 1 Hz or smaller Time dispersion : it is a function of the bandwidth (after filter) and SNR s s 0.1 Nautilus 2003: the bandwidth was of the order of 10 Hz g(t)

Example of time dispersion Dt [s] as a function of the (amplitude) SNR Nautilus 2001: the bandwidth was of the order of 1 Hz or smaller Nautilus 2003: the bandwidth was of the order of 10 Hz 0 SNR Dt [s]

Coincidences: the pulse correlation Coincidences among events of different detectors are the basic point of the search for bursts, in a network of detectors. Coincidences are done within a given “window”, which can be fixed or may vary, according to the event time uncertainties. To compare coincidences with the expected background we normally use the pulse correlation (the Poisson statistics is not good in case of non-stationary rates, which is our case). The pulse correlation is a non-parametric procedure, introduced by J. Weber. The background rate is evaluated by adding, N times, a bias time  to the events of one detector. The mean value  c of C(  is then compared to C(0)

Coincidences: the pulse correlation Because of non-stationarities, the shape of C(  ) may not be uniform, thus the evaluation of the chance probability of getting C(0) deserves some cares. We can use: the Poisson probability with parameter  c; the histogram of C(  ), evaluating how many times C(  ) > C(0) But, both methods have to be used with care. A deep study of the features in the time delay histogram is needed

seconds Coincidences: the time delay histogram

Allegro-Explorer : Jun-Dec 1991 (180 days) Phys. Rev D 59, 1999 Explorer-Nautilus-Niobe : Dec 1994-Oct 1996 (Explorer- Nautilus: 57 days; Explorer-Niobe: 56 days) Astrop. Phys. 10, 1999 IGEC : Phys. Rev. Letters, 85, 2000 Explorer-Nautilus 1998 : CQG, 18, 2001 Explorer-Nautilus 2001 : CQG, 19, 5449 (2002) IGEC : PRD 68, (2003) Coincidence analyses done among resonant detectors: Coincidences with Astro-Particle detectors

IGEC search for burst g.w. in the years PRD 68, (2003) new upper limit on the rate of g.w. bursts

Net observation times ( data-New IGEC protocol) 1 detector: 1322 days 1.2 detectors: 713 days 2.3 detectors: 178 days 3.4 detectors: 29 days 4.5 detectors: 0 days The total span of the time of the analysis is 4 years=1460 days and: – the time coverage is 90%, over 4 years

Upper limit on the rate r of g.w., with the IGEC detectors 95% CL 90% CL r/yr

Burst signals for a bar detector: we use to model them as 'delta' signals ➔ G. w. from the core collapse: Muller catalog 1. G. w. from neutron stars at different evolutionary stages (Ferrari, Miniutti, Pons astro-ph/ and CQG 20, S841 presented at GWDAW2002 in Kioto by V. Ferrari) : hot joung stars: damped sinusoids with f(t) and  (t) cooled stars : damped sinusoids with f and , for the QNMs (  'moderate' ;  'small' -->the spectrum becomes 'flat') 2. G. w. from the Ringdown of BHs: damped sinusoids (   s M/M 0 f ~ 12kHz/(M/M 0 ))

Burst signals for a bar detector: we use to model them as 'delta' signals ➔ Is this reasonable, given the actual bandwidth ? ➔ Which sources are suitable to do coincidences within a network of bars and interferometers ? Approaches: ➔ Analytical ➔ Simulations, adding fake signals to the noise of the detectors..in progress..

Use of Energy filters and Antenna pattern ● The sensitivity of each detector varies with time ● The sensitivities of the various detectors are different ● The same signal generates events with energies different for each detector (due to the noise effect, related to SNR) selection algorithm based on the event energies CQG, 18 (2001) Practical problems of coincidence analysis:

A&A,398 (2003)

Event SNR Differential probability SNR=signal to noise ratio of the signal (here: 10, 20, 50) The x-axis gives the signal to noise ratio of the event The y-axis gives the differential probability for the SNR of the event The basics of the use of energy filters: s ignals (from the source) and events (what we observe after filtering)

SNR of the threshold Probability of detection SNR=signal to noise ratio of the signal (here: 10, 20, 50) The x-axis gives the signal to noise ratio of the threshold The y-axis gives the probability of detection for a given SNR-t The basics of the use of energy filters: probability of detection as a function of the threshold

=0.57 GD GC Explorer and Nautilus: coincidences in the year 2001 Rog:CQG 19, 5449 (2002) P.A., G. D' Agostini,S. D' Antonio CQG 20 (2003) sidereal time, in hours

Review critically how our beliefs are modified by the actual observation -> Bayesian analysis P. Astone,G. D'Agostini,S. D'Antonio CQG 20 (2003) Ingredients of the inference are: -->the data; -->the knowledge of the detectors; -->hypotheses on the underlying physics; -->the physical quantity with respect to which we are uncertain is the g.w. rate on Earth, r, and the model responsible for g.w. emission; -->we are rather sure about b, but not about the number which will actually be observed; -->what is certain is the number n c of coincidences;

On times for the coincident observation with the Explorer and Nautilus detectors 2001:126 days 2003:160 days

Nautilus 2004: 180 days with T eff < 3 mK

Explorer 2004: 60 days with T eff < 3 mK

Web sites, of resonant detectors: Allegro gravity.phys.lsu.edu Auriga Explorer, Nautilus Niobe