Amplitude and Phase Fluctuations

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Amplitude and Phase Fluctuations for
Gravitational Waves Propagating through
Inhomogeneous Mass Distribution in the Universe
Ryuichi Takahashi Nagoya Univ. PD RT ApJ
644 80 (2006) RT, Suyama Michikoshi AA 438 L5
(2005)
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Abstract
GW propagates through inhomogeneous mass
distribution in the universe
density perturbation of CDM
incident wave
scattered wave
observer
source
The amplitude and phase of GW are changed by
scattering (or lensing) by density perturbation.
We derive these fluctuations based on a linear
perturbation theory in the wave optics.
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Introduction
? Cosmological Standard Siren
The distance to the binary D is directly
determined by its chirp signal
A amplitude
? Lensing Errors
(Holz Hughes 2005 Kocsis et al. 2006)
The amplitude is changed by the lensing
magnification
(or demagnification)
several in weak lensing
(Bartelmann Schneider 2001)
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Errors on the inferred redshift to a LISA
coalescence event, for SMBH binary
weak lensing error
intrinsic LISA error
redshift error
redshift
Weak lensing errors largely exceed other
sources of uncertainties.
(Kocsis et al. 2006)
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Previous work
(Holz Hughes 2005 Kocsis et al. 2006)
The geometrical optics was assumed.
Only the amplitude is changed by lensing
magnification, but the phase is not changed.
This work
We use the wave optics.
Not only the amplitude but also the phase is
changed.
We discuss lensing effects on both of them.
If the scale of the density perturbation is
smaller than the Fresnel scale
, one should use the wave optics.

(Macquart 2004 RT, Suyama Michikoshi 2005)
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Lensed Waveform
perturbed FRW metric
conformal time scale factor
gravitational potential
wave equation for scalar field
We use the scalar wave, instead of GW, since
both wave equations are same.
(Peters 1974)
Fourier component
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Gravitational Lens Geometry
r comoving distance
two-dimensional vector perpendicular to r
incident wave
scattered wave
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Wave equation
Lensed waveform
incident wave scattered wave
(spherical wave)
Including the effect of U on the first order.
(Born approximation)
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lensed waveform at observer
(Ishimaru 1978)
Lets define K and S as
K amplitude fluctuation
Kgt0 magnification Klt0 demagnification
S phase fluctuation
S phase shift due to the lensing
geometrical optics limit
convergence
time delay
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Amplitude Fluctuation
Fluctuation of the gravitational potential
Poissons eq.
non-linear
density perturbation characterized by CDM power
spectrum
linear
(Eisenstein Hu 1999 Peacock Dodds 1996)
Variance in the amplitude fluctuation
Filter function
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Filter function
k wave number of density perturbation
Fresnel scale
GW does not experience density perturbation of
scale less than Fresnel scale
(Macquart 2004 RT, Suyama Michikoshi 2005)
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Results
rms amplitude fluctuation
geometrical optics
source redshift
decreases for Hz
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Phase Fluctuation in chirp signal
Phase shift
Since time delay does not change the waveform, we
use as the phase fluctuation
?S rms phase difference from ?1 to ?2
In the chirp signal, the wavelength decreases
GW feels the density perturbation of
the smaller scale
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Results
rms phase differences ?S in chirp signal from f1
to f2 Hz
source redshift
in unit of radian
Phase fluctuation is typically several
rad.
Phase can not be measured with an accuracy less
than these values.
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Simple example
Density fluctuation is confined on a thin lens
plane.
observer
source
Phase fluctuation
C constant
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Conclusion
We consider GW propagating through a
inhomogeneous mass distribution in the
Universe, and discuss the lensing effects on
the amplitude and the phase of a waveform.

• rms amplitude fluctuation is 1-10 for
  Hz
• 
  and lt5 for Hz.

2. rms phase fluctuation is rad for
LISA frequency band. The phase can not
measured with an accuracy less than this
value.
3. Measurements of these fluctuations will
provide information about the density
fluctuation of 100pc scale.
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Fresnel scale
Pass difference between pass 1 and 2 is wave
length ?
Pass 1
source
obs.
Pass 2
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In the matched filtering analysis,
the phase can be measured to an accuracy
SN signal-to-noise ratio
Hence if SN gt 10 , the phase fluctuation becomes
important and the phase can be determined with an
accuracy of ?S .
Measurement accuracy for the phase ?