1

Time domain targeted pulsar search:

Algorithm and Results

Réjean Dupuis & Graham WoanIGR Glasgow

2

Reasoning behind a time-domain analysis

Searches targeted at known radio pulsars are not computationally expensive, so we can trade some efficiency for clarity and flexibility.

We know the IFO sensitivity is non-stationary and that there are gaps and dropouts. This evolution is handled naturally in a time domain analysis.

The data handling/transport problem can be efficiently reduced by heterodyning (mixing) the raw h(t) channel at near the expected pulsar signal frequency, and band-limiting the result to just a few Hz (gaining a factor of ~1000 in compression).

Pulsars with complex phase evolutions (especially the Crab) can be processed relatively simply.

3

The signal – a reminder

We use the standard model for the detected strain signal from

a non-precessing neutron star:

)sin(cos),()cos()cos1)(,()( 0002

021 ttFhttFhth

offset phase

frequency angular eous)(instantan ssignal'

amplitude strain

angle ninclinatio

angle onpolarisati

patterns beam onpolarisati,

0

0

h

FF

4

Data heterodyning

Data are heterodyned in two stages: A coarse complex heterodyne at a fixed frequency, to reduce

the effective sample rate to 4 Hz and allow for easy data transportation.

A fine complex heterodyne to take account of pulsar slowdown and Doppler shift, reducing its apparent frequency to 0 Hz, and reducing the data rate to (e.g.) 1 sample per 60 s.

Accomplished with LAL routines LALCoarseHeterodyneToPulsar and LALFineHeterodyneToPulsar, which include robust low-pass filtering routines to protect from strong out-of-band signals.

Between these stages, the noise level is estimated from the variance of the data over each 60 s period (assumed constant over this period).

5

Heterodyne output

Once fully heterodyned, the complex times series simply

evolves with the IFO antenna pattern as the pulsar moves

across the sky, and we fit a model to this signal of the form

where a is the vector of the 4 unknown parameters.

00 202

12204

1 cos),()cos1)(,();( ii etFihetFhty a

1 day

GPS seconds

Re[y(t)]

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Model fitting 1

The data from the heterodyne code, {Bk}, are modelled as Gaussian,

with variances estimated from each nominal 60 s stretch. This is fair provided the central limit theorem holds and the data are stationary over the period.

We take a Bayesian approach, and determine the joint posterior distribution of the probability of our unknown parameters, using uniform priors on over their accessible values, i.e.

00 and ,cos , h

)|}({)(}){|( aaa kk BppBp

posterior prior likelihood

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Model fitting 2

The likelihood (that the data are consistent with a given set of

model parameters) is proportional to exp(-2 /2), where

The sum is only over valid data, so dropouts and gaps are

dealt with simply.

Finally we marginalize over the uninteresting parameters to

leave the posterior distribution for the probability of h0:

k k

k tyB2

2 );()(

aa

cosddd}){|( 02/

0

2

eBhp k

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Upper limit definition

The 90% confidence upper limit is set by the value h90

satisfying

90

0 00 d}){|(90.0h

k hBhp

200 10/ h

h90

200 10/}){|( kBhp

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Detection

A detection would appear as a maximum significantly offset

from zero. Note that an upper limit can still be defined(!):

200 10/}){|( kBhp

200 10/ h

h90

Most probable h0

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Note: a naïve calculation would give a 1 upper limit of

The apparent loss on sensitivity of ~12 is due to the precise definition of h0 and the attenuation from the

‘mean’ beam for this (low dec) test source.

Validation 1

Marginalised posterior pdf for h0, resulting from an end-to-end test using

24 h of stationary, fake, Gaussian noise.

signal: h0 = 0, f0=1234 Hz, RA = dec = 0, = 0, = /4

noise: Sh = 9 x 10-17 Hz-1/2 ( = 10-18 at 16384 samples/s)

1 upper limit = 3.1 x 10-22

23106.2/ N

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Validation 2

Marginalized posterior pdf for h0, using 24 h of fake data and Gaussian noise.

signal: h0 = 6 x 10-21 , f0=1234 Hz, RA = dec = 0, = 0, = /4

noise: Sh = 9 x 10-17 Hz-1/2 ( = 10-18 at 16384 samples/s)

}){|( 0 kBhp

0h

(The multiple peaks are an temporary artefact of the fake data generation method.)

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GEO E7 results 1

Within a 4 Hz band around 1283 Hz (PSR J1939+2134), the noise is highly non-stationary,

but the standard deviation weighted data appears Gaussian over 60 s (above). In fact we need to drop to 10 s to resolve stationarity in these data.

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GEO E7 results 2

Assuming just 10 s stationarity, the noise is time-resolved and a

consistent upper limit for PSR J1939+2134 can be determined:

)7|( 0 Ehp

0h

Marginalised posterior probability

90% confidence upper limit: h0 < 4.5 x 10-20

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Prospects for GEO S1

S1 is more sensitive than, but possibly less stationary than, E7:

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Morals and intentions

Method works and can handle some truly horrific conditions.

A good understanding of the noise is vital to the definition of a

reliable upper limit.

Monte Carlo runs will give a final check.

Method still needs to be applied to LIGO data.