1. Gravitational lensing effects on parameters estimation in gravitational wave detection with advanced detectors Zhoujian Cao Institute of Applied Mathematics, AMSS Cowork with Li-Fang Li and Yan Wang 2014-6-30 2nd International Beijing Gravitational Waves Workshop

2. Gravitational lensing Einstein cross Lensing => Light bending

3. Gravitational lensing Strong lensing  Determine the mass of astrophysical objects  Determine the Hubble constant  Determine the structure of galaxies Weak lensing Determine the large scale structure of the Universe Micro lensing Investigate unknown planets ……

4. Optical lensing: High frequency, short wave length  geometric optics  geodesic Gravitational wave lensing: Low frequency, long wave length  wave dynamics

5. Optical lensing: High frequency, short wave length  geometric optics  geodesic Gravitational wave lensing: Low frequency, long wave length  wave dynamics

6. Lensing strength GW source Lens source Earth Adapted from PRL 80, 1138 (1998)

7. Lensing strength Adapted from PRL 80, 1138 (1998) Wave dynamics lensing is stronger than optics lensing The fraction of lensed event for wave dynamics is larger than that for optics approximation

8. Event rate for GW lensing LISA will definitely see GW lensing event during its mission [Takahashi, and Nakamura, ApJ 595, 1039 (2003); Sereno, Sesana, Bleuler, Jetzer, Volonteri and Begelman, PRL 105, 251101 (2010)] For ET, 0.1 – 10 per year considering inspiralling binary NS only [Piorkowska, Biesiada, and Zhu, JCAP 2013, 022]

9. GW lensing with wave dynamics is the complex amplification factor of lensing effect Potential function Character function Time delay function Moylan, McClelland, Scott, Searle and Bicknell, arXiv: 0710.3140

10. Point lensing model: SIS (singular isothermal sphere lensing model):

11. Point lensing model: SIS (singular isothermal sphere lensing model): Red lines: numerical wave form for equal mass binary black hole

12. Signal loss due to wrong wave form template? How does the lensing affect parameters extraction from GW? How much information we can get from GW lensing? Related questions to advanced detectors

14. : lensed wave form : wave form without lensing Fitting factor

18. Signal loss For lensing mass 1000 solar masses, SNR may decrease 20% For lensing mass 10^6 solar masses, SNR may decrease 30% ET events may decrease from 10s per year to less than 10 per year AdvLIGO, lensing signal detection becomes impossible from possible

19. Without lensing: PDF got by MCMC for gravitational wave signal without lensing. The green dotdashed line in these plots mark out the 1-sigma parameter region got through Fisher matrix method. The red solid line marks out such region for MCMC method. The left plots are for Advanced LIGO, and the right ones are for ET. From top row to bottom row the plots correspond to BH-BH, BH- NS and NS-NS respectively.

20. GW lensing affects the GW source parameters extraction mildly: Measurement error changes a little; parameters extraction bias is very small

22. Can not determine the lens parameters at all !!!

23. But the narrow band gives us stringent relation between the lens parameters Multi-messenger astronomy: combining other observations will gives good measurement of lens objects

24. Summary Signal loss due to wrong wave form template? Yes. When lensing mass increases, the FF may decrease to 70% or even more How does the lensing affect parameters extraction from GW? The effect is mild. The GW source parameters measurement error changes a little; the parameters bias is small How much information we can get from GW lensing? With GW detection alone, we can not determine the lens source parameters at all. But a stringent relation between lens source parameters can be got. Multi-messenger astronomy observation is needed to get accurate lens source parameters.