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

Gravitational lensing

Einstein cross

Lensing =>

Light bending

Gravitational lensing

• Strong lensing Determine the mass of astrophysical objects Determine the Hubble constant Determine the structure of galaxies• Weak lensingDetermine the large scale structure of the Universe• Micro lensingInvestigate unknown planets

……

Optical lensing:

High frequency, short wave length geometric optics geodesic

Gravitational wave lensing:

Low frequency, long wave length

wave dynamics

Optical lensing:

High frequency, short wave length geometric optics geodesic

Gravitational wave lensing:

Low frequency, long wave length

wave dynamics

Lensing strengthGW source

Lens source

Earth

Adapted from PRL 80, 1138 (1998)

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 fo

r optics approximation

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]

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

Point lensing model:

SIS (singular isothermal sphere lensing model):

Point lensing model:

SIS (singular isothermal sphere lensing model):

Red lines: numerical wave form for equal mass binary black hole

• 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

: lensed wave form

: wave form without lensing

Fitting factor

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

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.

GW lensing affects the GW source parameters extraction mildly:

Measurement error changes a little; parameters extraction bias is very small

Can not determine the lens parameters at all !!!

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

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.