Time domain targeted pulsar search: Algorithm and Results

1. Time domain targeted pulsar search:Algorithm and Results Réjean Dupuis & Graham Woan IGR 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:

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. 1 day Re[y(t)] GPS seconds

6. 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. posterior prior likelihood

7. Model fitting 2 • The likelihood (that the data are consistent with a given set of model parameters) is proportional to exp(-2/2), whereThe 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:

8. Upper limit definition • The 90% confidence upper limit is set by the value h90 satisfying h90

9. Detection • A detection would appear as a maximum significantly offset from zero.Note that an upper limit can still be defined(!): Most probable h0 h90

10. 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,  = /4noise: Sh = 9 x 10-17 Hz-1/2 ( = 10-18 at 16384 samples/s) • 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. 1 upper limit = 3.1 x 10-22

11. 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,  = /4noise: Sh = 9 x 10-17 Hz-1/2 ( = 10-18 at 16384 samples/s) • (The multiple peaks are an temporary artefact of the fake data generation method.)

12. 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.

13. 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: 90% confidence upper limit: h0 < 4.5 x 10-20 Marginalised posterior probability

14. Prospects for GEO S1 • S1 is more sensitive than, but possibly less stationary than, E7:

15. 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.