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

2. GWDAW9 2 Parameters of waveform: Optimize ALL SKY searches UNKNOWN PULSARS for UNKNOWN PULSARS Due to detectors’ motion wrt Solar System Barycenter; Due to detectors’ motion wrt Solar System Barycenter; => Amplitude and Phase modulation => Amplitude and Phase modulation Initial work was done by Brady and Creighton, who considered a two stage Hierarchical Search P. Brady and T. Creighton, PRD 61, 082001 (2000) Demodulation performed by F-Statistic

3. GWDAW9 3  Problem: Full coherent searches for unknown pulsars by using the present computational resources is not feasible Full coherent searches for unknown pulsars by using the present computational resources is not feasible Searching for young fast pulsars over the whole sky and including two spin-down parameters just for 10 days data, requires a 10 17 Computer Flops. Why Hierarchical Search Need an inexpensive sub-optimal techniques to Need an inexpensive sub-optimal techniques to discard uninteresting regions in parameter space discard uninteresting regions in parameter space Optimal method: A full coherent search Optimal method: A full coherent search  Example:

4. GWDAW9 4 The Stack-Slide method  Computing power spectrum for each segment  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. sufficient to confine a putative signal to ~ 1 frequency bin in each stack. (One example of semi-coherent method)

5. GWDAW9 5 Frequency Time  Shift the individual power spectra relative to each other (Slide)  Add corrected power spectra in the frequency domain

6. GWDAW9 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. GWDAW9 7 Two different methods of incorporating new data I st stage II nd stage III rd stage 1. Re-use already-analyzed data

8. GWDAW9 8 2. Ignore previously analyzed data until final coherent follow up (Fresh Mode) time I st stageII nd stageIII rd 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. GWDAW9 9 The search parameters to optimize N i : Number of stacks T i : Time-baseline of each stack  i : Mismatch in signal power Variables for each incoherent stage Variables for final coherent stage N coh = 1 T obs : Total observation time  coh : Mismatch in signal power  coh : Mismatch in signal power Given : Amount of data, T obs Weakest signal strength we wish to detect, h 0 Set false dismissal in each stage (= few %) Number of incoherent stages : n We numerically solve for: Optimal values of the search parameters By minimizing Computational Cost subject to constraints

10. GWDAW9 10 Results

11. GWDAW9 11 Example: All sky search for young fast pulsars (Taking Fresh data in each stage) Minimum spin-down age: 40 years Minimum spin-down age: 40 years Maximum frequency searched over: 1000 Hz Maximum frequency searched over: 1000 Hz False dismissal rates: 1 st stage = 10%, subsequent stages = 1% False dismissal rates: 1 st stage = 10%, subsequent stages = 1% Weakest detectable signal has 1-year SNR = 39.72 Weakest detectable signal has 1-year SNR = 39.72 Total data available: one year Total data available: one year What is the Computational Cost to find such a pulsar? What is the best hierarchical strategy?

12. GWDAW9 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) 1.320e+15 (O) 2. 4.245e+13 (F) 4.422e+13 (O) 4.422e+13 (O) 3. 9.764e+12 (F) 1.057e+13 (O) 1.057e+13 (O)

13. GWDAW9 13 Stage (days) N T obs (days) 1 2.630 0.7669 6 15.775 1 2.630 0.7669 6 15.775 2 3.013 0.0738 7 21.091 2 3.013 0.0738 7 21.091 3 32.838 0.8124 10 328.384 3 32.838 0.8124 10 328.384 Computational Cost: (Number of Operations) 1st 2nd3rd Coherent (stages) 1st 2nd3rd Coherent (stages) 1.828e+20 4.030e+19 1.384e+18 2.679e+16 Minimum Computational Power required: 9.764e+12 Flops Optimal search Parameters Based on taking Fresh data in each stage

14. GWDAW9 14 Signal strength required for different spin-down ages when we fix the Computational Cost to be 10 13 Flops.

15. GWDAW9 15 Computational Cost vs minimum spin-down age for fixed signal strength

16. GWDAW9 16 Conclusions 3-stage hierarchical searches significantly better than 1 or 2 stages, 3-stage hierarchical searches significantly better than 1 or 2 stages, but no point going beyond 3 stages. but no point going beyond 3 stages. Have solved for the optimum search strategy, and found Have solved for the optimum search strategy, and found the minimum computational cost for given sensitivity the minimum computational cost for given sensitivity Have not considered cost of Monte Carlo simulations or memory Have not considered cost of Monte Carlo simulations or memory issues issues

17. GWDAW9 17

18. GWDAW9 18 Computational Cost vs DeltaT1 for fixed values of other parameters’ points

19. GWDAW9 19 Computational Cost vs N1 for fixed values of other parameters’ points

20. GWDAW9 20 Computational Cost vs Mu1 for fixed values of other parameters’ points

21. GWDAW9 21 Computational Cost vs Mu2 for fixed values of other parameters’ points

22. GWDAW9 22 Computational Cost vs Mu3 for fixed values of other parameters’ points

23. GWDAW9 23 Differences between BC work and the current work They just considered 2 stages hierarchical search 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 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 They did not consider any final coherent stage In current work we used the F-Statistic, but they ignored the polarization. In current work we used the F-Statistic, but they ignored the polarization. They just considered re-use old data for each stage They just considered re-use old data for each stage

24. GWDAW9 24 Full details of the search parameters For Fresh data ############ INPUT ############ ############ INPUT ############ Fresh mode Fresh mode All sky search All sky search Minimum spindown age: 40.00 years Minimum spindown age: 40.00 years Maximum frequency searched over: 1000.00 Hz Maximum frequency searched over: 1000.00 Hz False dismissal rates: 0.100 0.010 0.010 False dismissal rates: 0.100 0.010 0.010 Smallest signal that can cross thresholds: 5.00e-05 Smallest signal that can cross thresholds: 5.00e-05 1.00-Year Signal to Noise Ratio (SNR): 39.72 1.00-Year Signal to Noise Ratio (SNR): 39.72 Total data available: 1.00 years Total data available: 1.00 years Length of sfts if sft method used: 1800.00 seconds Length of sfts if sft method used: 1800.00 seconds ############ OUTPUT ########### ############ OUTPUT ########### False alarm rates: 6.047e+13 5.096e+08 1.000e+00 False alarm rates: 6.047e+13 5.096e+08 1.000e+00 False alarm prob.: 2.028e-06 3.167e-08 0.000e+00 False alarm prob.: 2.028e-06 3.167e-08 0.000e+00 Npf: 1.205e+11 3.205e+13 8.101e+20 Npf: 1.205e+11 3.205e+13 8.101e+20 Npc: 1.025e+08 1.100e+10 3.205e+13 Npc: 1.025e+08 1.100e+10 3.205e+13 NpfCoh: 1.597e+24 NpfCoh: 1.597e+24 Thresholds: 2.631e+01 3.548e+01 4.609e+02 Thresholds: 2.631e+01 3.548e+01 4.609e+02 Computational Cost: Computational Cost: Incoherent Part: 1.344e+20 1.121e+18 1.323e+18 Incoherent Part: 1.344e+20 1.121e+18 1.323e+18 Coherent Part: 4.842e+19 3.918e+19 6.131e+16 Coherent Part: 4.842e+19 3.918e+19 6.131e+16 Total: 1.828e+20 4.030e+19 1.384e+18 2.679e+16 Total: 1.828e+20 4.030e+19 1.384e+18 2.679e+16 Stage DeltaT (days) muMax N Tobs (days) SNR Stage DeltaT (days) muMax N Tobs (days) SNR 1 2.865 0.7669 5.507 15.775 8.26 1 2.865 0.7669 5.507 15.775 8.26 2 2.865 0.0738 7.363 21.091 9.55 2 2.865 0.0738 7.363 21.091 9.55 3 34.163 0.8124 9.612 328.384 37.66 3 34.163 0.8124 9.612 328.384 37.66 Minimum computational power required: 9.764e+12 Flops Minimum computational power required: 9.764e+12 Flops

25. GWDAW9 25 Full details of the search parameters For re-use old data ############ INPUT ############ ############ INPUT ############ Re-use old data mode Re-use old data mode All sky search All sky search Minimum spindown age: 40.00 years Minimum spindown age: 40.00 years Maximum frequency searched over: 1000.00 Hz Maximum frequency searched over: 1000.00 Hz False dismissal rates: 0.100 0.010 0.010 False dismissal rates: 0.100 0.010 0.010 Smallest signal that can cross thresholds: 5.00e-05 Smallest signal that can cross thresholds: 5.00e-05 1.00-Year Signal to Noise Ratio (SNR): 39.72 1.00-Year Signal to Noise Ratio (SNR): 39.72 Total data available: 1.00 years Total data available: 1.00 years Length of sfts if sft method used: 1800.00 seconds Length of sfts if sft method used: 1800.00 seconds ############ OUTPUT ########### ############ OUTPUT ########### False alarm rates: 4.473e+13 3.344e+08 1.000e+00 False alarm rates: 4.473e+13 3.344e+08 1.000e+00 False alarm prob.: 1.410e-06 1.310e-14 0.000e+00 False alarm prob.: 1.410e-06 1.310e-14 0.000e+00 Npf: 1.284e+11 1.032e+14 2.278e+21 Npf: 1.284e+11 1.032e+14 2.278e+21 Npc: 1.033e+08 1.217e+10 1.032e+14 Npc: 1.033e+08 1.217e+10 1.032e+14 NpfCoh: 1.597e+24 NpfCoh: 1.597e+24 Thresholds: 2.689e+01 5.565e+01 5.158e+02 Thresholds: 2.689e+01 5.565e+01 5.158e+02 Computational Cost: Computational Cost: Incoherent Part: 1.461e+20 3.579e+18 8.418e+17 Incoherent Part: 1.461e+20 3.579e+18 8.418e+17 Coherent Part: 4.961e+19 4.102e+19 5.094e+16 Coherent Part: 4.961e+19 4.102e+19 5.094e+16 Total: 1.957e+20 4.460e+19 8.927e+17 9.528e+15 Total: 1.957e+20 4.460e+19 8.927e+17 9.528e+15 Stage DeltaT (days) muMax N Tobs (days) SNR Stage DeltaT (days) muMax N Tobs (days) SNR 1 2.861 0.7618 5.605 16.038 8.32 1 2.861 0.7618 5.605 16.038 8.32 2 2.861 0.0700 10.433 29.852 11.36 2 2.861 0.0700 10.433 29.852 11.36 3 38.842 0.8074 9.404 365.250 39.72 3 38.842 0.8074 9.404 365.250 39.72 Minimum computational power required: 1.057e+13 Flops Minimum computational power required: 1.057e+13 Flops