Search for periodic sourcesC. Palomba

• Signal duration much larger than typical observation time

• Signal often (but not always) predictable and

• depending on the kind of source

• depending on a (large) number of (poorly known) parameters

Possibility to reduce the false alarm probability to negligible values

• Different approach in the analysis depending if we know the source parameters (targeted search) or not (blind search)

• Blind searches are computationally bound

• Different kinds of periodic sources can be considered:

• isolated NS

• NS in binary systems

• accreting NS (r-modes excitation)• We will focuse attention on the first type and discuss what changes in the other cases

Signal characterization - 1

• Doppler frequency modulation, due to the detector motion and to the source motion

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• Glitches

• Spin-down or spin-up

• Intrinsic frequency modulation, due to a companion, an accretion disk or a wobble

• Amplitude modulation, due to the detector radiation pattern and possibly to intrinsic effects (e.g. wobble)

Signal at the detector

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Data characterization

• Stationarity

• Gaussianity

• Impulsive noise

• Holes in the data

• Correct data timing

Detection of periodic signals

• If we had a monochromatic signal, the most natural strategy would be that of looking for significant peaks in the spectrum of data

• Due to the frequency modulation and spin-down the signal power is spread in a large number of frequency bins• If the signal frequency evolution is known we can correct for the modulation (targeted search)

• Otherwise we need to perform a ‘rough’ search, select some candidates and refine the analysis on them (blind search -> hierarchical methods)

Targeted vs blind searches

• Targeted:

possibility to apply optimal methods

computationally not expensive

upper limits

• Blind:

oriented to detection

computationally expensive

non optimal methods

Targeted search for isolated NS - 1

• Allow to use optimal DA methods

• Nominal sensitivity

• Assume sky position, emission frequency and spin-down are known

• Amplitude, source inclination, polarization angle, initial phase typically are unknown

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Targeted search for isolated NS - 2

• Time-domain methods

• Re-sampling procedure: use a sampling frequency proportional to the varying received frequency

Targeted search for isolated NS - 3

• Heterodyning procedure (Abbott et al., PRL94 181103,

2005): multiply the signal by )(tje

• Allow to take into account complex phase evolution, e.g. using data from radio telescopes, in a straightforward way

• Frequency domain methods

• F-statistics (Jaranowski et al., PRD58 063001, 1998; Abbott et al.

PRD69 082004, 2004) : based on the maximization of the likelihood function

Targeted search for isolated NS - 4

• Can be used as coherent step in a hierarchical procedure

• Analytical signal (Astone et al., PRD65 022001, 2001) : start from a set of short FFTs, compute the analytical signal, correct for the frequency variations, compute the new spectrum

• Can be used as coherent step in a hierarchical procedure

Blind search - 1

• Cannot be performed with optimal methods due to the huge number of points in the source parameter space

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• Assume source position, frequency and spin-down are not known (or only loosely constrained)

Unreachable computing power

Blind search - 2

• Hierarchical methods have been developed which strongly reduce the needed computing power at the price of a small sensitivity loss (Rome, Potsdam…)

• Typically based on alternation between coherent steps and incoherent ones.

• Two kinds of incoherent steps are popular:

• stack-slide (Radon transform)

• Hough transform

• Both methods start from a collection of ‘short’ FFTs: their length is such that a signal would be confined within a frequency bin

Radon transform

t

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1. Compute periodograms from short FFTs

2. Shift (slide) periodograms according to the frequency evolution

3. Sum (stack) the periodograms

Hough transform - 1

• Parameter estimator of patterns in digital images

• Developed in the ’60 by P. Hough to analyze particle tracks in bubble chamber images

• Example: find parameters of a straight line y=mx+q

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y

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q

(xi, yi)

q=yi-mxi

Hough transform - 2

• In our case the HT connects the time-frequency plane to the source parameter space

• On the periodograms select peaks above a threshold

Hough transform - 3

• For each point in the peak-map we have a circle in the sky

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Hough transform - 4

Hough transform - 5

• Slightly less sensitive than Radon (~12% in amplitude)

Hough vs. Radon Sensitivity ratio

Threshold for peak selection

• More robust against non-stationarities and disturbances• Computing power reduced by ~10

Hierarchical method outline

h-reconstructed data

SFDB

peak map

hough transf.

candidates

SFDB

peak map

hough transf.

candidates

coincidences

coherent step

events

• Time domain disturbances

• identification of events through an adaptive threshold (on the CR);

• background estimated from the AR mean of the absolute value and square of the samples;

• events removal.

Short FFT database - 1

(Astone et al., CQG22, S1197)

Short FFT database - 2

• Construction of the short FFT database

• Maximum length

• Estimation of the average spectrum

• based on AR estimation;

• used for peaks selection and in the Hough transform;

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• 4 SFDB for frequency bands [0,31.25Hz], [31.25Hz,125Hz], [125Hz,500Hz], [500Hz,2kHz]

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Peak map

• Construction based on the ratio R between the spectrum and its AR estimation;

• A threshold is set on R and all the local maxima above it are selected as peaks;

• The threshold is chosen in order to maximize the CR on the Hough map (see next)

• With thr=2.5 we have that ~1/12 of the frequency bins are selected

Hough transform

• For each search frequency takes a Doppler band around it and compute the hough map for all the possible spin-down values

• Computationally heaviest part of the hierarchical analysis• Efficient implementation needed

• Use of computing grids

• Adaptive hough transform for non-stationarities (Palomba et al., CQG22, S1255, 2005)

Hough statistics

• In the case of pure noise the number count in a Hough map follows a binomial distribution

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• In presence of a signal

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• We select candidates putting a threshold on the number count

• The threshold is chosen on the basis of the maximum number of candidates we can manage (e.g. )910

• The choice of the threshold is done maximizing the critical ratio CR:

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• is still nearly optimal and reduce the prob. of peak selection to ~1/12

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Sensitivity - 1

• Loss factor for nominal sensitivity4

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• Nominal sensitivity 8

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• Number of points in the parameter space

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Sensitivity - 2

• The needed computing power is

where is the ‘number of equivalent floating point operations’ needed for pixel increase and the analysis time is assumed to be half of the observation time

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• Computing power of the order of 1Tflops needed for

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• Larger CP available, reduce the spin-down age

Sensitivity - 3

• To compute the ‘effective’ sensitivity loss of the hierarchical method we have to take into account candidate selection

• For the optimal method, the loss due to the selection of 10^9 candidates is (exponential statistics)

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• The Hough number count distribution is binomial

• Using the gaussian approximation, the threshold as a function of the number of selected candidates is

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• The 10^9 sensitivity reduction factor is ~2.2-2.8 depending on the frequency band

• The ‘effective’ loss respect to the optimal method is

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Sensitivity - 4

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Pulsars in binary systems - 1

• Orbital parameters must be taken into account (up to 5)

• Orbital Doppler shift may give more stringent limits to the maximum length of FFTs

(from Dhurandhar & Vecchio, PRD63, 122001)

Pulsars in binary systems - 2

• Optimal methods can be applied only if the system parameters are known or the uncertainties are small

• Otherwise, hierarchical non-optimal methods are needed

Accreting NS (e.g. LMXB)

• Same parameters as in the previous case

• The frequency will change randomly due to fluctuations in the rate of matter accretion

• In LMXB a clustering of frequencies is observed, though the exact rotation frequency is not known

• Possibility to apply coherent methods over short time period (few hours) (Vecchio, GWDAW10)

Summary of results

• Coherent analysis

• Explorer: 0.72Hz, 1 sd, all-sky, h90=1e-22

• LIGO S2: 28 isolated NS, h95>1.7e-24

• LIGO S2: Sco-X1, h95=2e-22

• Incoherent analysis:

• LIGO S2: all-sky, 1 sd, 200-400Hz, h95=4.4e-23

SPARE SLIDES

• Two thresholds enter into the game:

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peaks selection

candidates selection

• False dismissal probability

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