LISA Response Functions: The Middlemen of Gravitational ...

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LISA Response Functions: The Middlemen of Gravitational Wave Astronomy Louis J. Rubbo [email protected] Center for Gravitational Wave Physics at the Pennsylvania State University Rubbo S&S Spring 2005 1

Transcript of LISA Response Functions: The Middlemen of Gravitational ...

Page 1: LISA Response Functions: The Middlemen of Gravitational ...

LISA Response Functions: TheMiddlemen of Gravitational Wave

AstronomyLouis J. Rubbo

[email protected]

Center for Gravitational Wave Physics

at the Pennsylvania State University

Rubbo S&S Spring 2005 1

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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))

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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-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

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Supermassive BH Merger

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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

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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

-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

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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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