Gravitational Lensing of Gravitational Waves
applied
on Gravitational Waves detectors
Monica ARNAUD VARVELLA
Salerno University (Italy)
Prof. G. Scarpetta
ERGA/LERMA (Paris VI)
Prof. Ph. Tourrenc
M.C.Angonin-Willaime
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Gravitational Waves and Experimental Gravity
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OUTLINE
• Description of Gravitational Lensing
• Electromagnetic and Gravitational Radiation
• A first look at the gravitational lensing of GWs
• Diffraction limit on sources
• Estimation of the number of GW signals
• Future analysis
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What is Gravitational Lensing?
gravitational light deflection
due to the presence
on the optical path of a
gravitational field
SOURCES:
quasars, distant galaxies, stars
Source
Lens
LENSES:
galaxies, stars, galaxy clusters
OBSERVERS:
telescopes (Earth-based and satellites)
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Observer
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Electromagnetic and Gravitational Radiation
Electromagnetic Radiation
• oscillations of EM field through space-time
• incoherent superposition of emission from particles
• dipolar emission
Gravitational Radiation
• oscillations of the "geometry" of space-time
• coherent motions of mass-energy
• quadrupolar emission
same effects of deviation
from geometric optics
• absorption
• scattering
• dispersion by matter
• gravitational focusing
• diffraction
Ref: K.S.Thorne, « Gravitational Radiation» (1983)
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Brief history
• Gravitational deflection of gravitational waves
§ Y. Wang, A. Stebbins, E.L. Turner (1996)
§ G. Bimonte, S. Capozziello et al (1997)
§ A. Ruffa (1999)
§ F. De Paolis, G. Ingrosso et al (2000)
§ T. Wickramasinghe, M. Benacquista (2001)
§ A. F. Zakharov (2002)
§ F. De Paolis, G. Ingrosso et al (2002)
• Gravitational deflection by gravitational waves
§ F. Wintenberg (1967)
§ D. M. Zipoy, B.Bertotti (1968)
§ B. Bertotti, D. Trevese(1971)
§ A. Labeyrie (1993)
§ V. Faraoni (1996) Topic already studied since more 30 years!
§ T. Damour, G. Esposito-Farèse (1998)
§ B. Allen (1999)
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Gravitational Lensing of Gravitational Waves - I
SOURCES: GW candidates
Source
(e.g.: supernovae, compact binary,
black holes.
Accretion, coalescence, binary systems, etc.)
LENSES: objects between sources
and observers
(e.g.: galaxies, black holes)
Lens
OBSERVERS:
Observer
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GW
detectors
resonant-mass and
interferometers (e.g: VIRGO, LIGO)
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Gravitational Lensing of Gravitational Waves - II
3 different cases:
• weak gravitational lensing
* sources in Virgo cluster
* lenses: galaxies in the same direction
or
* lens: the total Virgo cluster
* sources in the same direction
• strong gravitational lensing
* lenses: black holes in Galactic Center
* sources in the same direction
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Which approach for the analysis?
Geometric optics approximation
or
diffraction analysis ?
diffraction limit estimation
diffraction parameter
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cut-off frequency
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GW large frequency domain
Frequency Range
GW ‘Probe’
• Extremely Low Freq. 10−18 ÷ 10−15 Hz
• Very Low Frequency 10−9 ÷ 10−7 Hz
• Low Frequency
10−4 ÷ 1 Hz
• High Frequency
1 ÷ 104 Hz
CMB polarization
Pulsar timing
LISA experiment
Earth-based
detectors
interferometers (VIRGO, LIGO…) and
resonant-mass detectors (Explorer, Nautilus...)
See e.g. C. Cutler - K.S. Thorne, gr-qc/0204090
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Diffraction Limit (I)
4Gπ
y =
c2
M
lens
diffraction parameter
ν GW
c
R
= 2π
lens
Schw artzchild
λ
wave effects in
Gravitational Lensing
y
∞
~1
geometric optics
diffraction
Ref: S. Deguchi -W.D. Watson, Ph.Rev.D, Vol.34, n°6,(1986)
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Diffraction parameter
Diffraction parameter versus GW frequency
SA
I
L
rg
i
V
M = 109 M¤
Galaxy mass
geometric
optics
o
1
diffraction
M = 106 M¤
Massive black hole
Frequency (Hz)
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Diffraction Limit (II)
Cut-off frequency
ω cut-off ∝ (10-1 π Mlens) -1
ω > ω cut-off
ω < ω cut-off
amplification
diffraction
Sun M= M¤
Þ
ω cut-off ~ 104 Hz
Ref: R.J.Bontz, M.P.Haugan, Ap&SS, Vol.78 (1981)
Black hole M = 106 M¤ Þ ω cut-off ~10-1 Hz
Galaxy
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M = 109 M¤ Þ ω cut-off ~ 10-4 Hz
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Work Hypothesis
Earth-based GW
detectors
No diffraction effect on the sources
y >> 1
diffraction parameter
ω > ω cut-off cut-off frequency
Gravitational Lensing
of Gravitational Waves
with geometric optics
approximation
If a GW amplification due to GL effect is possible
does the number of GW signals increase ?
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Number of Signals Estimation (I)
magnification factor A
Þ
2 contributions
(A+ amplification, A-demagnification)
behaviour of
magnification factor
for GWs
Þ
hlensing = hsource A
searching for signals
without lensing
isotropical
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with lensing
directional (depends on β)
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Number of Signals Estimation (II)
• total number of GW signals
2 contributions
magnification factor
N0 Þ N = N0 + ∆ N = 2 N0 + ∆ N+ + ∆ N• variation in the number of GW signals
∆ N / N0 = 1 + ∆ N+/ N0 + ∆ N- / N0
where N0 is the unlensed number of GWs signals
N is the lensed number of GWs signals
∆ N is the variation due to the lensing
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Number of Signals Estimation (III)
2 simple lens models
(1) Schwarzschild lens model
• point mass
• always two images
(2) Singular Isothermal Sphere
• spherical symmetry
• two images only for β < θ E
∆N / N0 ∆N / N0
LENS
(1)
(2)
-7
N10
Virgo cluster
10-5
EG
LIG
-10
Galaxy @ 1 Mpc
10
10-5
IBL
E !!
Black Hole Sag A* 10-11
10-6 !
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Time Delay Method
Analysis of permanent sources (e.g.: pulsars)
with the time delay
between images due to the lensing effect
detectivity condition of (at least) two images
hlensing > hsource A- > hthreshold
un
kn lens model
Estimation for the (1) Schwarzschild
ow
n s Sphere
(2) Singular Isothermal
ou
rce
s !!
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Earthbased
detectors
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Summary and Conclusion
• geometric optics analysis
Þ possible for Earth-based detectors
• increase in the number of GW signals
Þ NEGLIGIBLE !! Þ more deflectors
• lens models
Þ too much « idealistic » !!
• time delay between images of permanent sources
Þ unknown sources (unknown β)
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Future analysis Þ Why ?
the number of signals is not modified
BUT
even ONE source strongly amplified should be very important
• estimation of the amplitude of the effect
for more « realistical » models
analytical elliptical lens models
e.g.: Blandford-Kochanek potential
• searching for real candidates Þ galaxy catalogs
• time delay method
• angular separation
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