LISA Aperture Synthesis for
Searching Binary Compact Objects
Aaron Rogan
Washington State University
[email protected]
Collaborator: Sukanta Bose
GWDAW 2003
Space-Based Detectors II Analysis Methods
Supported by NASA: NASA-NAG5-12837
Introduction
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LISA is a network of
three detectors
2 are independent
Total of 6 elementary
data streams
Main Sources of Noise:
• Laser Frequency
Fluctuation
• Relative Craft Motion
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Time delay
interferometry can
eliminate much of the
dominating noise
Introduction Cont’d
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Lasers aborad LISA will
have a frequency
stability of a few parts
in 10-13.
Desired sensitivity
range at least a few
parts in 10-20
Time delay
interferometry uses
time shift operators, Ei.
The time shift operator
described by:
Ei f(t) = f(t-Li/c)
Using these generators
or pseudo-strains LISA
can achieve the desired
levels of sensitivity
Introduction Cont’d
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The pseudo-strains do not
span a vector space
They use data from all six data
streams to cancel noise
Act as a network of 3
independent detectors
The pseudo-strains have
different sensitivities to the
same sky position
An optimal combination of the
pseudo-strains is needed to:
• Maintain the highest signal-tonoise ratio possible over the
entire orbit
• Maintain the highest level of
sensitivity for all sky positions
• Maintain the most efficient
search over all sky positions
The Problem
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To obtain the optimal combination of the
data streams one must consider:
• The pseudo-strains are a function of the orbital
position of LISA
• How to weight each pseudo-strain for a given
orbital position
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The advantages of an optimal combination
are:
• Maintaining the maximum sensitivity to a
wider range of {θ,Φ} values
• Maintaining the maximum sensitivity for all
points on LISA’s orbit
How to Approach the Problem?
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Identify the time domain polarization
amplitudes, h+(t) and hx(t).
Derive the appropriate Fourier domain
polarization amplitudes
Combine the 3 weighted pseudo-strains to
obtain the complete signal, hA(Ω).
Analytically maximize over the following
parameters: {Ψ,ε,δ}
Obtain the optimal statistic, λ|Ψ,Є,δ .
Develop a template bank over remaining
parameters, namely {θ,Ф}.
Determine the computational feasibility of
a search
The Optimal Statistic
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The matched filter is used to obtain the optimal
detection statistic. Before maximization it takes
the following form:
3

A1
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3
A
h ,x
A
( A)
3
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 N   ei Fi
A1 i 1
A
Si , x A

Now maximizing over the source polarization
and inclination angles can be achieved
Signal-To-Noise Ratio
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The SNR for a each
pseudo-strain is plotted
to the right.
The holes indicate
directions that minimize
the SNR
Compare the optimal
SNR to the SNR for a
given pseudo-strain
The optimal statistic
improves the SNR for all
orbital positions
Network Sensitivity
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The optimal statistic also
greatly improves the
sensitivity of LISA
Although a single pseudostrain spans all {θ,Ф} values
• It does not obtain a maximum
sensitivity to all {θ,Ф}
• At any given point in the orbit,
the sensitivity is very limited
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The optimal statistic
advantages are:
• All {θ,Ф} values are maximized
at some point in the orbit
• The likelihood of finding a source
is increased
Developing the Template Bank
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Develop a metric on the parameter space
{Ω,θ,Ф} as outlined by Owen
Project out Ω from the 3-dimensional metric
This new metric will define the overall volume of
your parameter space
Decide on a Minimal Mismatch (MM)
The Minimal Mismatch will fix the number density
of the templates
Determine the grid spacing within this volume
Finally determine the number of templates
I Would Like to Thank The Following Individuals
and Organizations for Their Direct
Contributions to My Research:
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Shawn Seader
Rajesh Kumble Nayak
Washington State University Physics Department
National Aeronautics and Space Administration
If not for the previous work by the following
individuals I would not be here today
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S. Bose
S. Dhurandhar
K. R Nayak
J-Y Vinet
A. Pai
M. Tinto
B. Owen
B. Schutz
T. Price
S. Larson
J.W. Armstrong
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A. Eastabrook
Signal-To-Noise Ratio for the
Optimal Statistic and a single
Pseudo-Strain
Sensitivity of the a single PseudoStrain and the Optimal Statistic for
the Entire Orbit