Calibrating semi-analytic V T ’s against reweighted injection V T ’s Daniel Wysocki and Richard O’Shaughnessy Rochester Institute of Technology LIGO R&D Call Monday, October 1, 2018 LIGO-T1800427-v1 D. Wysocki, R. O’Shaughnessy (RIT) Calibrating V T R&D Call – 2018-10-01 1 / 14
A tale of two V T ’s ‚ semi-analytic V T ’s ‚ single-detector model using a reference PSD, operating for some time T , based on a detection threshold ‚ gives us V T on a grid of intrinsic parameters, λ ‚ e.g., λ “ p m source , m source , χ 1 z , χ 2 z q , other spins typically set to zero 1 2 ‚ injection V T ’s ‚ perform an injection campaign, and count the number of detections by your pipeline (e.g., pyCBC, GstLAL, cWB) to get an average V T for the injected population, Λ – x V T yp Λ q ‚ re-weigh injections to get alternate populations, x V T yp Λ ˚ q (Tiwari 2018) D. Wysocki, R. O’Shaughnessy (RIT) Calibrating V T R&D Call – 2018-10-01 2 / 14
Technical issues with injection V T ’s ‚ Population inference ideally needs V T p λ q – function of intrinsic parameters ‚ Injection code provides x V T yp Λ q – function of population hyperparameters ‚ Injection code limited to broad populations – lots of injections needed to cover all possible narrow populations, due to Monte Carlo error ‚ Reweighting code is rather slow, problematic for already slow population inference methods D. Wysocki, R. O’Shaughnessy (RIT) Calibrating V T R&D Call – 2018-10-01 3 / 14
Calibrating semi-analytic V T ’s to injection V T ’s ‚ Solution: assume a parameterized relationship between V T analytic and V T inj , and optimize for the free parameters ζ V T inj p λ q « V T analytic p λ q f p λ ; ζ q D. Wysocki, R. O’Shaughnessy (RIT) Calibrating V T R&D Call – 2018-10-01 4 / 14
Solving for the calibration prescription ‚ basis functions t g α p λ qu α with calibration coefficients t ζ α u α to be solved for ÿ f p λ ; ζ q “ ζ α g α p λ q α ‚ linear least squares problem: ‚ Compute y k ” x V T y inj p Λ k q on a discrete grid of Λ k ’s ‚ Compute the “design matrix” ż H k,α “ p p λ | Λ k q V T analytic p λ q g α p λ q d λ ‚ Compute the least squares solution to the coefficients ζ “ p H J γ H q ´ 1 H J γ y ( γ is the inverse covariance matrix for x V T y inj estimates) D. Wysocki, R. O’Shaughnessy (RIT) Calibrating V T R&D Call – 2018-10-01 5 / 14
Comparison – no calibration V T inj p m 1 , m 2 q « V T an p m 1 , m 2 q D. Wysocki, R. O’Shaughnessy (RIT) Calibrating V T R&D Call – 2018-10-01 6 / 14
Comparison – scalar multiple calibration V T inj p m 1 , m 2 q « ζ 0 V T an p m 1 , m 2 q D. Wysocki, R. O’Shaughnessy (RIT) Calibrating V T R&D Call – 2018-10-01 7 / 14
Comparison – linear calibration V T inj p m 1 , m 2 q « p ζ 0 ` ζ 1 m 1 ` ζ 2 m 2 q V T an p m 1 , m 2 q D. Wysocki, R. O’Shaughnessy (RIT) Calibrating V T R&D Call – 2018-10-01 8 / 14
Comparison – quadratic calibration V T inj p m 1 , m 2 q « p ζ 0 ` ζ 1 m 1 ` ζ 2 m 2 ` ζ 3 m 1 m 2 ` ζ 4 m 2 1 ` ζ 5 m 2 2 q V T an p m 1 , m 2 q D. Wysocki, R. O’Shaughnessy (RIT) Calibrating V T R&D Call – 2018-10-01 9 / 14
Population inference comparisons The scale of the change in results may be concerning for some parameters, most importantly the rate. D. Wysocki, R. O’Shaughnessy (RIT) Calibrating V T R&D Call – 2018-10-01 10 / 14
Population inference comparisons – maximum mass Effect is still there but lesser for maximum mass. D. Wysocki, R. O’Shaughnessy (RIT) Calibrating V T R&D Call – 2018-10-01 11 / 14
Conclusions ‚ Semi-analytic V T ’s are still a necessary evil ‚ Can calibrate against injection V T ’s to improve accuracy significantly ‚ Can be applied to any x V T y table ‚ (e.g., GstLAL; including spin dependence; redshift dependence; etc) ‚ allows cross-checking V T ’s between pipelines, and captures systematic effects across parameter space ‚ Disagreement reduced from biased high 39%–93% to symmetric 5% ‚ Easy fix – should definitely utilize this in O2 Populations Paper, runs ongoing now ‚ Note: we can easily choose any basis functions we want, these are just the first we tried D. Wysocki, R. O’Shaughnessy (RIT) Calibrating V T R&D Call – 2018-10-01 12 / 14
Generalizing method ‚ Method generalizes beyond just mass calibrations ‚ Can do spin and redshift calibrations as well ‚ Beyond calibration: idea of basis functions for V T can be used for efficient population estimation, including redshift distributions ‚ Ongoing project, upcoming paper by Wysocki & O’Shaughnessy D. Wysocki, R. O’Shaughnessy (RIT) Calibrating V T R&D Call – 2018-10-01 13 / 14
References I Vaibhav Tiwari. Estimation of the sensitive volume for gravitational-wave source populations using weighted Monte Carlo integration. Classical and Quantum Gravity , 35:145009, 145009, July 2018. doi : 10.1088/1361-6382/aac89d . D. Wysocki, R. O’Shaughnessy (RIT) Calibrating V T R&D Call – 2018-10-01 14 / 14
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