Optimized hierarchical continuous-wave searches Badri Krishnan Curt Cutler, Iraj Gholami AEI, Golm G Z LSC meeting, March 2004 ASIS session

Motivation Full coherent searches for unknown pulsars not computationally feasible Full coherent searches for unknown pulsars not computationally feasible Require incoherent, sub-optimal methods Require incoherent, sub-optimal methods Illustrative example: The stack slide search Illustrative example: The stack slide search Example: Searching for young, fast pulsars over the whole sky and including two spin-down parameters for 10 days data requires a Flops computer P.Brady and T.Creighton, PRD 61, (2000)

The Stack-Slide method Basic idea: Take the Fourier transform of each segment and track the Doppler shift by adding power in the frequency domain (Stack and Slide) Basic idea: Take the Fourier transform of each segment and track the Doppler shift by adding power in the frequency domain (Stack and Slide) First step : Break up data into N segments First step : Break up data into N segments Calculate power spectrum or DeFT for each segment Calculate power spectrum or DeFT for each segment

Frequency Time Add power after frequency bins are shifted according to the time-frequency pattern

General hierarchical pipeline Break data into segments Analyze each segment coherently Combine segments incoherently Select candidates Analyze candidates coherently using all available data Acquire more data Detection or upper limit Incoherent step can be either stack-slide or Hough

The search parameters N i : Number of stacks T i : Time-baseline of each stack  i : Mismatch in signal power X i  : Threshold on summed power Variables for each incoherent stageVariables for final coherent stage T obs : Total observation time  coh : Mismatch in signal power Given : Available computer power C 0 Data of a certain time duration T obs Weakest signal strength we wish to detect h 0 Desired confidence level Number of incoherent stages : n We want to know: Optimal values of the search parameters Minimize computational cost subject to constraints

Two different search modes Take fresh data in each stage Take fresh data in each stage Re-use old data Re-use old data time I st stageII nd stage III rd stage I st stage II nd stage III rd stage

   False alarm rate for i th stage Determines number of candidates for next stage to analyze    False dismissal rate for i th stage Determines weakest signal that can be detected Statistics Summed power follows chi-square distribution with 2N d.o.f False alarm and false dismissal rates : Total false alarm rate always determined by final coherent stage Choose thresholds for each stage by fixing false dismissal rate

Template counting Parameter space metric calculated by Brady & Creighton Number of points in coarse grid : Number of points in fine grid :

Computational costs Computational cost of each stage is essentially cost of calculating FFT plus cost of summing the power For each point in parameter space, number of floating point operations for first, intermediate and coherent stages: F (i) = Number of candidates which survive the ith stage

Optimization strategy Basic strategy is to minimize Total computational cost subject to constraint that amount of analyzed data is lesser than available data False alarm rate is not really a constraint because false alarm rate is set by final coherent step Computationally limited searches can only see strong signals and when we do see them, it is usually easy to build up confidence Want to analyze data in (roughly) real time Function to be optimized is S(x) = 1 if 0 < x < 1 and very large otherwise

Preliminary results Example of search criteria : Total observation time : 1 year Signal strength we wish to detect : Allowed false dismissal rate for each stage : 1% Mismatch in coherent stage : 0.10 Don’t reuse old data Max number of spindowns included : 3 All-sky search Largest frequency searched over : f max = 1000 Hz Smallest spindown age :  min = 40 yr Optimization carried out by simulated annealing and amoeba method

Two stage search with coherent follow-up:  Length of each stack : 0.37 days ; 0.25 days  Number of stacks : 199 ; 956  Observation time : 73 days ; 240 days  Power mismatch : 0.51 ; 0.23  Computational requirement : 8.8 x Flops Three stage search with coherent follow-up:  Length of each stack : 0.17 days ; 0.15 days ; 0.48 days  Number of stacks : 511; 1028 ; 261  Observation time : 84 days ; 149 days ; 125 days  Power mismatch : 0.48 ; 0.33 ; 0.02  Computational requirement : 7.8 x Flops Single stage search with coherent follow-up:  Length of each stack : 1.9 days  Number of stacks : 39  Observation time : 74 days  Power mismatch : 0.53  Computational requirement : 6.3 x Flops

Conclusions Optimization scheme for hierarchical stack-slide search presented Tells us what the search pipeline parameters must be Expect similar results for hierarchical Hough search also Does not consider cost of Monte Carlo simulations or memory issues Shows that hierarchical schemes are absolutely essential for large parameter space blind searches