Calibrating semi-analytic V T s against reweighted injection V T s - - PowerPoint PPT Presentation

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

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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 “ pmsource

1

, msource

2

, χ1z, χ2zq, other spins typically set to zero

‚ 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 – xV TypΛq ‚ re-weigh injections to get alternate populations, xV TypΛ˚q (Tiwari 2018)

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Technical issues with injection V T’s

‚ Population inference ideally needs V Tpλq – function of intrinsic parameters ‚ Injection code provides xV TypΛq – 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

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Calibrating semi-analytic V T’s to injection V T’s

‚ Solution: assume a parameterized relationship between V Tanalytic and V Tinj, and

• ptimize for the free parameters ζ

V Tinjpλq « V Tanalyticpλq fpλ; ζq

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(RIT) Calibrating V T R&D Call – 2018-10-01 4 / 14

Solving for the calibration prescription

‚ basis functions tgαpλquα with calibration coefficients tζαuα to be solved for fpλ; ζq “ ÿ

α

ζα gαpλq ‚ linear least squares problem:

‚ Compute yk ” xV TyinjpΛkq on a discrete grid of Λk’s ‚ Compute the “design matrix” Hk,α “ ż ppλ | Λkq V Tanalyticpλq gαpλq dλ ‚ Compute the least squares solution to the coefficients ζ “ pHJγHq´1HJγy (γ is the inverse covariance matrix for xV Tyinj estimates)

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Comparison – no calibration

V Tinjpm1, m2q « V Tanpm1, m2q

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Comparison – scalar multiple calibration

V Tinjpm1, m2q « ζ0V Tanpm1, m2q

• D. Wysocki, R. O’Shaughnessy

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Comparison – linear calibration

V Tinjpm1, m2q « pζ0 ` ζ1m1 ` ζ2m2qV Tanpm1, m2q

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Comparison – quadratic calibration

V Tinjpm1, m2q « pζ0 ` ζ1m1 ` ζ2m2 ` ζ3m1m2 ` ζ4m2

1 ` ζ5m2 2qV Tanpm1, m2q

• 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.

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Population inference comparisons – maximum mass

Effect is still there but lesser for maximum mass.

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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 xV Ty 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

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(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

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(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.

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