Optimized Search Strategies for Continuous Gravitational Waves

1. Optimized Search Strategiesfor Continuous Gravitational Waves Iraj Gholami Curt Cutler, Badri Krishnan Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut) Golm, Germany GWDAW9 Friday, 17 December 2004Annecy, France

2. Optimize ALL SKYsearches for UNKNOWNPULSARS Parameters of waveform: Due to detectors’ motion wrt Solar System Barycenter; => Amplitude and Phase modulation Demodulation performed by F-Statistic • Initial work was done by Brady and Creighton, who considered a two stage Hierarchical Search P. Brady and T. Creighton, PRD 61, 082001 (2000)

3. Why Hierarchical Search • Optimal method: A full coherent search • Problem: Full coherent searches for unknown pulsars by using the present computational resources is not feasible • Example: Searching for young fast pulsars over the whole sky and including two spin-down parameters just for 10 days data, requires a 1017 Computer Flops. • Need an inexpensive sub-optimal techniques to discard uninteresting regions in parameter space

4. TheStack-Slide method (One example of semi-coherent method) • Break up data into shorter lengths (Stacks) • Phase correction in each stack using a mesh of correction points sufficient to confine a putative signal to ~ 1 frequency bin in each stack. • Computing power spectrum for each segment

5. Frequency Time • Shift the individual power spectra relative to each other (Slide) • Add corrected power spectra in the frequency domain

6. Hierarchical Search • Perform search in several (n) stages • At each stage consider only candidates that survived previous stage • Do finer search with more data near surviving candidates

7. Two different methods of incorporating new data 1. Re-use already-analyzed data Ist stage IInd stage IIIrd stage

8. 2. Ignore previously analyzed data until final coherent follow up (Fresh Mode) time Ist stage IInd stage IIIrd stage For both above methods, we consider one final coherent stage that search over entire data coherently, but taking just those candidates could pass the last incoherent stage.

9. The search parameters to optimize Number of incoherent stages : n Variables for each incoherent stage Variables for final coherent stage Ni: Number of stacks Ti: Time-baseline of each stack mi: Mismatch in signal power Ncoh = 1 Tobs : Total observation time mcoh : Mismatch in signal power • Given : • Amount of data, Tobs • Weakest signal strength we wish to detect, h0 • Set false dismissal in each stage (= few %) • We numerically solve for: • Optimal values of the search parameters By minimizing Computational Cost subject to constraints

10. Results

11. Example: All sky search for young fast pulsars (Taking Fresh data in each stage) • Minimum spin-down age: 40 years • Maximum frequency searched over: 1000 Hz • False dismissal rates: 1st stage = 10%, subsequent stages = 1% • Weakest detectable signal has 1-year SNR = 39.72 • Total data available: one year What is the Computational Cost to find such a pulsar? What is the best hierarchical strategy?

12. Optimized Computational Cost vs Number of Stages Based on taking Fresh data in each stage 1. 1.320e+15 (F) 1.320e+15 (O) 2. 4.245e+13 (F) 4.422e+13 (O) 3. 9.764e+12 (F) 1.057e+13 (O)

13. Optimal search Parameters Based on taking Fresh data in each stage Stage (days) N Tobs (days) 1 2.630 0.7669 6 15.775 2 3.013 0.0738 7 21.091 3 32.838 0.8124 10 328.384 Computational Cost: (Number of Operations) 1st 2nd 3rd Coherent (stages) 1.828e+20 4.030e+19 1.384e+18 2.679e+16 Minimum Computational Power required: 9.764e+12 Flops

14. Signal strength required for different spin-down ages when we fix the Computational Cost to be 1013 Flops.

15. Computational Cost vs minimum spin-down age for fixed signal strength

16. Conclusions • 3-stage hierarchical searches significantly better than 1 or 2 stages, but no point going beyond 3 stages. • Have solved for the optimum search strategy, and found the minimum computational cost for given sensitivity • Have not considered cost of Monte Carlo simulations or memory issues

18. Computational Cost vs DeltaT1 for fixed values of other parameters’ points

19. Computational Cost vs N1 for fixed values of other parameters’ points

20. Computational Cost vs Mu1 for fixed values of other parameters’ points

21. Computational Cost vs Mu2 for fixed values of other parameters’ points

22. Computational Cost vs Mu3 for fixed values of other parameters’ points

23. Differences between BC work and the current work • They just considered 2 stages hierarchical search • They fixed the confidence level to be 99%, but if you reduce the number of candidates in the last stage to the few one, therefore you can have a confidence level more than 99%. • They did not consider any final coherent stage • In current work we used the F-Statistic, but they ignored the polarization. • They just considered re-use old data for each stage

24. Full details of the search parametersFor Fresh data • ############ INPUT ############ • Fresh mode • All sky search • Minimum spindown age: 40.00 years • Maximum frequency searched over: 1000.00 Hz • False dismissal rates: 0.100 0.010 0.010 • Smallest signal that can cross thresholds: 5.00e-05 • 1.00-Year Signal to Noise Ratio (SNR): 39.72 • Total data available: 1.00 years • Length of sfts if sft method used: 1800.00 seconds • ############ OUTPUT ########### • False alarm rates: 6.047e+13 5.096e+08 1.000e+00 • False alarm prob.: 2.028e-06 3.167e-08 0.000e+00 • Npf: 1.205e+11 3.205e+13 8.101e+20 • Npc: 1.025e+08 1.100e+10 3.205e+13 • NpfCoh: 1.597e+24 • Thresholds: 2.631e+01 3.548e+01 4.609e+02 • Computational Cost: • Incoherent Part: 1.344e+20 1.121e+18 1.323e+18 • Coherent Part: 4.842e+19 3.918e+19 6.131e+16 • Total: 1.828e+20 4.030e+19 1.384e+18 2.679e+16 • Stage DeltaT (days) muMax N Tobs (days) SNR • 1 2.865 0.7669 5.507 15.775 8.26 • 2 2.865 0.0738 7.363 21.091 9.55 • 3 34.163 0.8124 9.612 328.384 37.66 • Minimum computational power required: 9.764e+12 Flops

25. Full details of the search parametersFor re-use old data • ############ INPUT ############ • Re-use old data mode • All sky search • Minimum spindown age: 40.00 years • Maximum frequency searched over: 1000.00 Hz • False dismissal rates: 0.100 0.010 0.010 • Smallest signal that can cross thresholds: 5.00e-05 • 1.00-Year Signal to Noise Ratio (SNR): 39.72 • Total data available: 1.00 years • Length of sfts if sft method used: 1800.00 seconds • ############ OUTPUT ########### • False alarm rates: 4.473e+13 3.344e+08 1.000e+00 • False alarm prob.: 1.410e-06 1.310e-14 0.000e+00 • Npf: 1.284e+11 1.032e+14 2.278e+21 • Npc: 1.033e+08 1.217e+10 1.032e+14 • NpfCoh: 1.597e+24 • Thresholds: 2.689e+01 5.565e+01 5.158e+02 • Computational Cost: • Incoherent Part: 1.461e+20 3.579e+18 8.418e+17 • Coherent Part: 4.961e+19 4.102e+19 5.094e+16 • Total: 1.957e+20 4.460e+19 8.927e+17 9.528e+15 • Stage DeltaT (days) muMax N Tobs (days) SNR • 1 2.861 0.7618 5.605 16.038 8.32 • 2 2.861 0.0700 10.433 29.852 11.36 • 3 38.842 0.8074 9.404 365.250 39.72 • Minimum computational power required: 1.057e+13 Flops