LISA Response Functions: TheMiddlemen of Gravitational Wave
AstronomyLouis J. Rubbo
Center for Gravitational Wave Physics
at the Pennsylvania State University
Rubbo S&S Spring 2005 1
Talk Outline
The LISA Observatory
Full response model
The LISA Simulator
ApproximationsLow FrequencyRigid AdiabaticExtended LowFrequency
Production
Analysis
Propagation
Detection
Rubbo S&S Spring 2005 2
LaserInterferometer SpaceAntenna
NASA/ESA Mission
Launch date ∼2013
ConfigurationEquilateral formationTrails the Earth by 20◦
〈L〉 = 5 × 106 km
fgw ∈ (10−5, 1) Hz
CharacteristicsNot pointableOmnidirectionalOutputs a set ofindependent timeseries
Rubbo S&S Spring 2005 3
LISA’s Orbital Motion
Orbital and cartwheel period is one year (movie)
Rubbo S&S Spring 2005 4
LISA’s Orbital Motion
Orbital and cartwheel period is one year (movie)Doppler modulations enter as sidebands separatedby the modulation frequency,
fm = 1/year ≈ 3 × 10−8 Hz
Doppler shift, δf ≈ (v/c)f
Guiding center
v/c ≈ 0.994 × 10−4
fgc = 0.3 mHz
Rolling cartwheel motionv/c ≈ 0.332 × 10−5
fr = 16 mHz
Rubbo S&S Spring 2005 4
Target Sources for LISA
Supermassive binaryblack hole mergers
Extreme mass ratioencounters
Single encountersHighly eccentric orbitsInspirals
10-21
10-20
10-19
10-18
10-17
0.0001 0.001 0.01 0.1 1
hf [
per
√H
z]
f [Hz]
EMRi
Binary BkgndResolved
Binaries
MB
H M
erg
ers
LISA th
reshold
sensitivity
Galactic binariesMostly compact objectsToo many of them!
Rubbo S&S Spring 2005 5
Spaceborne Detector Response
Monitor the proper distance between two spacecraft
`ij(t) =
∫ j
i
√
gµνdxµdxν
Metric
ds2 = −(1 + 2φ)dt2 + (1 − 2φ)(dx2 + dy2 + dz2) + hijdxidxj
Proper distance between spacecraft
`ij(t) = ‖~xj(tj) − ~xi(t)‖ +1
2
(
r̂ij(t) ⊗ r̂ij(t))
:
∫ j
i
h(
ξ(ρ))
dρ
ξ(ρ) = t(ρ) − k̂ · ~x(ρ)
Cornish & Rubbo, PRD 67, 022001 (2003)Rubbo S&S Spring 2005 6
Photon Propagation Direction
i
j
r̂ij(ti)
r̂ij(ti) =xj(tj) − xi(ti)
`ij(ti)
`ij(ti) = ‖xj(tj) − xi(ti)‖
= ‖xj(t + `ij(ti)) − xi(ti)‖
Rubbo S&S Spring 2005 7
Spaceborne Detector Response
Measured phase differences
Φij(tj) = Cji(ti) − Cij(tj)
+2πν0
(
nsij(tj) − na
ij(tj) + naji(ti) + δ`ij(ti)
)
Laser phase noise: C(t)
Shot noise: ns(t)
Acceleration noise: na(t)
The six phases differences are combined virtually toform the various signals
Rubbo S&S Spring 2005 8
LISA Signals
Michelson signal
M1(t) = Φ12(t − `21) + Φ21(t) − Φ13(t − `31) − Φ31(t)
#3 #2
#1
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LISA Signals
Michelson signal
M1(t) = Φ12(t − `21) + Φ21(t) − Φ13(t − `31) − Φ31(t)
Time Delay Interferometry
X(t) = Φ12(t − `21) + Φ21(t) − Φ13(t − `31) − Φ31(t)
−Φ12(t − `31 − `13 − `21) − Φ21(t − `31 − `13)
+Φ13(t − `21 − `12 − `31) + Φ31(t − `21 − `12)
Cornish & Hellings, CQG 20, 4851 (2003)
Rubbo S&S Spring 2005 9
The LISA Simulator
The LISASimulator Series
GravitationalWaveforms
Time
CapabilitiesValid for an arbitrary gravitational wave at anyfrequency in the LISA bandOutputs a multitude of signalsIncludes all modulationsIncludes a model of the detector noise
AvailabilityOpen source software (written in C)www.physics.montana.edu/LISA/
Rubbo S&S Spring 2005 10
The LISA Simulator
Version 1 (Spring 2003)Michelson signal from a single vertexNoise is produced in the frequency domain
Rubbo S&S Spring 2005 11
The LISA Simulator
Version 1 (Spring 2003)Michelson signal from a single vertexNoise is produced in the frequency domain
Version 2 (Summer 2003)Michelson from each vertex and TDI signals {X, Y, Z}Noise is produced in the time domainReturns time and frequency domain results
Rubbo S&S Spring 2005 11
The LISA Simulator
Version 1 (Spring 2003)Michelson signal from a single vertexNoise is produced in the frequency domain
Version 2 (Summer 2003)Michelson from each vertex and TDI signals {X, Y, Z}Noise is produced in the time domainReturns time and frequency domain results
Rubbo S&S Spring 2005 11
The LISA Simulator
Version 1 (Spring 2003)Michelson signal from a single vertexNoise is produced in the frequency domain
Version 2 (Summer 2003)Michelson from each vertex and TDI signals {X, Y, Z}Noise is produced in the time domainReturns time and frequency domain results
Version 3 (Spring 2005)User friendlyOnly time domain resultsMore TDI signals
Rubbo S&S Spring 2005 11
Intrinsic Detector Noise
Michelson noise realization
Standard sensitivity curve (green) is from the OnlineCurve Generator by Shane Larson
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-15
-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
log
(hf)
Hz-1
/2
log ( f ) Hz
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AM Canum Venaticorum
-20.9
-20.6
-20.3
-20
-19.7
-19.4
-2.7115 -2.7114 -2.7113 -2.7112 -2.7111
log
(hf)
Hz-1
/2
log ( f ) Hz
Interacting white dwarf binary
r ≈ 100 pc
fgw = 1.94 mHz
Monochromatic in its rest frameRubbo S&S Spring 2005 13
Supermassive BH Merger
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-15
-5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5
log(
h f) H
z-1/2
log( f ) Hz
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-16
-15
-5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5
log(
h f) H
z-1/2
log( f ) Hz
M1 = M2 = 106M�
z = 1 (DL = 6.5 Gpc)
tc = 1.00075 years
Simulation used 2PN waveforms from
Blanchet, Iyer, Will, & Wiseman, CQG 13, 575 (1996)
Rubbo S&S Spring 2005 14
The Need for Approximations
The full response is...analytically difficult to handletime consuming to evaluateoverkill in detail
Rubbo S&S Spring 2005 15
The Need for Approximations
The full response is...analytically difficult to handletime consuming to evaluateoverkill in detail
Response approximations are helpful because...analytically simplernumerically fastthey can give physical insight into what the detectoris actually doing
Rubbo S&S Spring 2005 15
Difficulties with Spaceborne Detectors
Orbital motion of the detectorBreathing mode in the triangular formation, L → L(t)
Second order in the orbital eccentricity (ε ≈ 0.01)
Rubbo S&S Spring 2005 16
Difficulties with Spaceborne Detectors
Orbital motion of the detectorBreathing mode in the triangular formation, L → L(t)
Second order in the orbital eccentricity (ε ≈ 0.01)
Point aheadSpacecraft are movingSpeed of light is finite
Rubbo S&S Spring 2005 16
Difficulties with Spaceborne Detectors
Orbital motion of the detectorBreathing mode in the triangular formation, L → L(t)
Second order in the orbital eccentricity (ε ≈ 0.01)
Point aheadSpacecraft are movingSpeed of light is finite
Finite arm sizeAbove the transfer frequency gravitational waves “fitinside” the arms
f∗ ≡c
2πL≈ 9.54 mHz
Transfer functions account for the finite size of thearms
Rubbo S&S Spring 2005 16
Low Frequency Approximation
Work to linear order in the orbital eccentricity(Rigid Detector)
Ignore relative motion of the spacecraft
Ignore transfer functions
Cutler, PRD 57, 7089 (1998)Cornish & Rubbo, PRD 67, 022001 (2003)
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
10-5 10-4 10-3 10-2 10-1 100
r(f
)
f Hz
0.96
0.98
1
10-3 10-2
Rubbo S&S Spring 2005 17
Low Frequency Approximation
Noiseless Michelson signal (i.e. monochromatic source)
M(t) = F+(t)A+ cos(
2πft + ΦD(t))
+ F×(t)A× sin(
2πft + ΦD(t))
= A(t) cos(
2πft + ΦD(t) + ΦP (t))
Amplitude, Frequency, and Phase Modulations
A(t) =√
(A+F+(t))2 + (A×F×(t))2
ΦD(t) = 2πfR sin(θ) cos(2πfmt − φ)
ΦP (t) = − tan−1(A×F×(t)/A+F+(t))
Rubbo S&S Spring 2005 18
Low Frequency Approximation
Noiseless Michelson signal (i.e. monochromatic source)
M(t) = F+(t)A+ cos(
2πft + ΦD(t))
+ F×(t)A× sin(
2πft + ΦD(t))
= A(t) cos(
2πft + ΦD(t) + ΦP (t))
Amplitude, Frequency, and Phase Modulations
A(t) =√
(A+F+(t))2 + (A×F×(t))2
ΦD(t) = 2πfR sin(θ) cos(2πfmt − φ)
ΦP (t) = − tan−1(A×F×(t)/A+F+(t))
Mono AM FM PM Full
Rubbo S&S Spring 2005 18
Rigid Adiabatic Approximation
Work to linear order in the orbital eccentricity(Rigid Detector)
Ignore relative motion of the spacecraft
Include transfer functions
Rubbo, Cornish, & Poujade, PRD 69, 082003 (2004)
0.8
0.85
0.9
0.95
1
10-5 10-4 10-3 10-2 10-1 100
r(f
)
f Hz
0.96
0.98
1
0.1 1
Rubbo S&S Spring 2005 19
The Need For Speed
The galaxy has a lot of ∼monochromatic binaries
Rubbo S&S Spring 2005 20
The Need For Speed
The galaxy has a lot of ∼monochromatic binaries
Time DomainTo prevent aliasing takes a lot of data pointsTime domain ⇒ slow
Rubbo S&S Spring 2005 20
The Need For Speed
The galaxy has a lot of ∼monochromatic binaries
Time DomainTo prevent aliasing takes a lot of data pointsTime domain ⇒ slow
Frequency DomainModulations occur over a few frequenciesFrequency domain ⇒ fastAnalytical Fourier transform of the Low FrequencyApproximation was done by
Cornish & Larson, PRD 67, 103001 (2003)Extended Low Frequency Approximation
Timpano, Rubbo, & Cornish, Hopefully Soon
Rubbo S&S Spring 2005 20
Extended Low Frequency Approximation
Work to linear order in the orbital eccentricity(Rigid Detector)
Ignore relative motion of the spacecraft
Expand transfer functions to second order in (f/f∗)
Include linear chirping
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
1e-05 0.0001 0.001 0.01 0.1
r(f
)
f Hz
0.96
0.98
1
0.001 0.01
Rubbo S&S Spring 2005 21
Galactic Background
Speed means we can build gravitational wavebackgrounds in a reasonable amount of time
N ≈ 4 × 107 galactic binaries
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-4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1
log
(hf)
Hz-1
/2
log ( f ) Hz
Rubbo S&S Spring 2005 22
Punch Lines
A complete forward model of the LISA observatory, validfor arbitrary gravitational waves, has been worked out
The LISA SimulatorSoftware package for simulating the response to anarbitrary gravitational wave
Response ApproximationsApproximations allow quite simulations and insightinto the detectorLow Frequency ApproximationRigid Adiabatic ApproximationExtended Low Frequency Approximation
Rubbo S&S Spring 2005 23
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