gravitational lensing of gravitational waves

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Gravitational lensing of gravitational waves Yousuke Itoh (RESCEU, U. of Tokyo) seminar@ICRR, 2012 June 26 1 Affiliation Tohoku U.(Prof. Futamase)AEI, PotsdamUWM Tohoku U. RESCEU Research Post-Newton LIGO Data Analysis Cosmology KAGRA In collaboration with M. Hattori & T. Futamase (Tohoku Univ.).

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Gravitational lensing of gravitational waves . In collaboration with M. Hattori & T. Futamase (Tohoku Univ.). . Yousuke Itoh ( RESCEU, U. of Tokyo) seminar@ICRR , 2012 June 26. Affiliation : Tohoku U.(Prof. Futamase )  AEI, Potsdam  UWM  Tohoku U.  RESCEU - PowerPoint PPT Presentation

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Page 1: Gravitational lensing  of  gravitational waves

Gravitational lensing of

gravitational waves

Yousuke Itoh   (RESCEU, U. of Tokyo)

seminar@ICRR, 2012 June 26

1

Affiliation : Tohoku U.(Prof. Futamase)AEI, PotsdamUWM Tohoku U. RESCEUResearch : Post-Newton LIGO Data Analysis Cosmology KAGRA

In collaboration with M. Hattori & T. Futamase (Tohoku Univ.).

Page 2: Gravitational lensing  of  gravitational waves

Contents

-1. Announcements.0. Short Introduction1. Young’s (cosmological) double slits experiment2. Can we see interference pattern?3. Gravitational lensing of gravitational waves4. Real world

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• 1st KAGRA Data Analysis School @ RESCEU, 9/3(Mon.)-5(Wed.), 2012

Introductory lectures by the Data analysis subgroup In Japanese (Sorry!) Travel support, Data analysis demonstration, Data analysis practice (octave?), Social gathering (Most important!)    

• 2nd KAGRA Data Analysis School @ NAOJ (Hopefully … )

-1. Announcements

Page 4: Gravitational lensing  of  gravitational waves

Part 0:Short Introduction

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Page 5: Gravitational lensing  of  gravitational waves

Just in case …•Gravitational lens?http://spiff.rit.edu/classes/phys230/lectures/planets/Lens_Nav.swf

HST image of Abell 1689

Chandra (pinkish) & Weak lensing (bluish, pseudo-) images of the bullet clusterhttp://chandra.harvard.edu/photo/2006/1e0657/

Page 6: Gravitational lensing  of  gravitational waves

Introduction•GW suffers from GL effect by galaxies and clusters

•GWs are coherent in many cases and we may be able to see interference pattern.

•Takahashi & Nakamura ApJ 595 1039 (2003)– GL of GWs

– 1e6 ~ 1e9 Mʘ Lens (for the obsolete LISA detector)

– Diffraction effect– Lens mass & source position relative to optical axis.

•Any other astrophysical information from GL of GW phenomena?

Page 7: Gravitational lensing  of  gravitational waves

Part 1:Young’s (cosmological) double slit experiment

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Motion in a spatial interference pattern

http://www.bottomlayer.com/bottom/reality/chap2.html

I (t;x) =¯̄¯̄¯1r1

eiÁ1 + 1r2

eiÁ2¯̄¯̄¯2

' 2 1r2(1 + cos(Á1 ¡ Á2))

Á1(t;x) ¡ Á2(t;x) = 2¼Z tr+td(t;x)tr

f (t0)dt0:= 2¼f ctd(t;x)

I (t;x) » cosµ

2¼f cvµc t + const:

¶+ const:

¢ xv = 2:5years

à 370km=sv

! µ 10"µ

¶ Ã 0:1Hzf

!

Transverse velocity

Page 9: Gravitational lensing  of  gravitational waves

Part 2:Can we see interference pattern?

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Page 10: Gravitational lensing  of  gravitational waves

Probably no in Electro Magnetic Astronomy …

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• Summation of incoherent emitters (bunch of electrons …)

• interfere distractively• Can see interference only when …

𝜃

Page 11: Gravitational lensing  of  gravitational waves

𝜃

Probably no in Electro Magnetic Astronomy …

11

=

𝒍

We can see interference pattern only when the linear dimension of the source (galaxy) is less than ~ 2 cm.Impossible!!

Page 12: Gravitational lensing  of  gravitational waves

Probably yes in Gravitational Wave Astronomy …?

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• Coherent emitter (single source if we are lucky (?) …)

• Interfere constructively.• Can see interference if …

𝜃

Page 13: Gravitational lensing  of  gravitational waves

Probably yes in Gravitational Wave Astronomy …?

13

𝜃

=

=

Page 14: Gravitational lensing  of  gravitational waves

Cosmological Young’s experiment in GWA possible in reality?

• Pulsars in globular clusters

– But diffraction obscures the interference pattern when

• NS/NS binaries

– But freq. derivative & SNR complicates the issue. 14

&

Page 15: Gravitational lensing  of  gravitational waves

No ( ノω ・、 ) in space, but …Yes ( d( o )b ) ♪ ⌒ ⌒ ♪ in a computer

•Time delay•Reach the observer separately in time•Extract the interference pattern in a computer

•SNR•Integrate to get SNR

•Interference disappears•Filtered outputZ

I (t;x) cos(2¼f c¡ T t)dt »Z

cosµ

2¼f cvµc t:

¶cos(2¼f c¡ T t)dt

» sincµ

¼f cµvµ

c ¡ ¡ T¶

Tobs¶

ZI (t;x)dt » const: +

Zcos

µ2¼f cvµ

c t:¶

dt» v independetnt:

•Chirp • Intrinsic frequency (time-)derivative

Page 16: Gravitational lensing  of  gravitational waves

Part 3:Gravitational lensing of gravitational waves

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Page 17: Gravitational lensing  of  gravitational waves

Lensed GW from inspiraling binaryNo relative motion geometrical optics approximation (Takahashi & Nakamura 2003)

M cz;j ! °j M cz; tc;j k ! °ktc;j ;Dopper shift in mass and time due to relative motion

°j =³1 + ~N j ¢~v

¡ = °1°2

= 1 + ( ~N1 ¡ ~N2) ¢~̄+ O(¯ 2):

' 1 + 1:8 £ 10¡ 7µ µ

10"¶ Ã v

370km=s

!;

Гis dopper shift difference between the two images

Page 18: Gravitational lensing  of  gravitational waves

If you like equations (copy from my paper).

Amplitude of the detector beam pattern function

Doppler phase due to the DECIGO orbital motion.

Phase of the detector beam pattern function.

GL amplificationDetector index

Half integer: saddle = ½ etc

Page 19: Gravitational lensing  of  gravitational waves

Beam pattern function (II)

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Shift due to the Doppler effect by the Detector motion

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Beam pattern function (I)

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Page 21: Gravitational lensing  of  gravitational waves

Do you still like equations?GL amplitudeImage index time variable

GL amplitude

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We would like to determine for the following parameters (in case of no relative motion.)

1. Chirp mass2. Phase of coalescence3. Time of coalescence4. Luminosity distance to the source5. Source direction6. Binary orbital plane inclination7. Lens image relative parity8. Time delay9. Relative magnification

Page 23: Gravitational lensing  of  gravitational waves

Geometrical Optics Approx. (GOA)

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Lens

Template

Page 24: Gravitational lensing  of  gravitational waves

Geometrical Optics Approx. (GOA)

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Lens

Template

We can separately detect the two images.

Page 25: Gravitational lensing  of  gravitational waves

To extract interference in GOA (no noise).

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Even without detector noise

In stead,

Page 26: Gravitational lensing  of  gravitational waves

Differential SNR

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Transverse velocity

• When relative transverse velocity is not zero,– Doppler shift to frequency, unobservable a priori.– Doppler shift to the Masses and time variablesUnobservable a priori. – Time variation in the source directionUnobservable for astronomical sources.– But relative Doppler factor is observable

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For those who like equations (if any)…

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Dopper factor

Page 29: Gravitational lensing  of  gravitational waves

SNR as a function of template mismatch in Г

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Differential SNR w or w/o mismatch in Г

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Page 31: Gravitational lensing  of  gravitational waves

Example parameters (Q0957+561)

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Correlation between and time delay

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Along the “ridge”

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Page 34: Gravitational lensing  of  gravitational waves

Part 4:Real world

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Noises

1. Parameters estimation noises2. Detector noises

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Effect of nuisance parameters estimation errors (td = 1.14 years)

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Page 37: Gravitational lensing  of  gravitational waves

Effect of nuisance parameters estimation errors to Г(td = 1.14 years)

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Page 38: Gravitational lensing  of  gravitational waves

Effect of nuisance parameters estimation errors to td (td = 1.14 years)

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Effect of nuisance parameters estimation errors (td = 5.1 years)

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Page 40: Gravitational lensing  of  gravitational waves

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Effect of nuisance parameters estimation errors to Г

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Effect of nuisance parameters estimation errors to td

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Page 42: Gravitational lensing  of  gravitational waves

10 different nuisance parameters set, 1000 noise realizations each (td = 1.4 years)

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Page 43: Gravitational lensing  of  gravitational waves

10 different nuisance parameters, 1000 noise realizations each (td = 5.1 years)

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Page 44: Gravitational lensing  of  gravitational waves

If you think mass/tc should give Г

44It is mathematically true but at lesser accuracy.

Page 45: Gravitational lensing  of  gravitational waves

Summary (as of 10:10, June 26, 2012)

• It is possible that gravitational wave can be gravitationally lensed in principle.

• I could not come up with any natural source that shows observable spatial interference pattern.

• On a computer, we can extract information contained in the interference term.

• For this purpose, I propose a new statistic ζ.• With ζ, we can measure the transverse velocity of

the sources which is hard to detect in a conventional EM GL in the DECIGO/BBO era.

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