Seth Timpano Louis Rubbo Neil Cornish Characterizing the Gravitational Wave Background using LISA.
LISA Response Functions: The Middlemen of Gravitational ...
Transcript of LISA Response Functions: The Middlemen of Gravitational ...
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LISA Response Functions: TheMiddlemen of Gravitational Wave
AstronomyLouis J. Rubbo
Center for Gravitational Wave Physics
at the Pennsylvania State University
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Talk Outline
The LISA Observatory
Full response model
The LISA Simulator
ApproximationsLow FrequencyRigid AdiabaticExtended LowFrequency
Production
Analysis
Propagation
Detection
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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
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LISA’s Orbital Motion
Orbital and cartwheel period is one year (movie)
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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
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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!
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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
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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)‖
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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
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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)
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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/
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The LISA Simulator
Version 1 (Spring 2003)Michelson signal from a single vertexNoise is produced in the frequency domain
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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
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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
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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
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Intrinsic Detector Noise
Michelson noise realization
Standard sensitivity curve (green) is from the OnlineCurve Generator by Shane Larson
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-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
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-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
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Supermassive BH Merger
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-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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-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)
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The Need for Approximations
The full response is...analytically difficult to handletime consuming to evaluateoverkill in detail
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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
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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)
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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
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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
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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
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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))
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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
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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
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The Need For Speed
The galaxy has a lot of ∼monochromatic binaries
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The Need For Speed
The galaxy has a lot of ∼monochromatic binaries
Time DomainTo prevent aliasing takes a lot of data pointsTime domain ⇒ slow
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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
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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
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1e-05 0.0001 0.001 0.01 0.1
r(f
)
f Hz
0.96
0.98
1
0.001 0.01
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Galactic Background
Speed means we can build gravitational wavebackgrounds in a reasonable amount of time
N ≈ 4 × 107 galactic binaries
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log
(hf)
Hz-1
/2
log ( f ) Hz
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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
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