Looking for a Stochastic Gravitational Wave Background

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Looking for a Stochastic Gravitational Wave
Background
Laser Interferometry and Pulsar Timing

• Stefan Ballmerfor the LIGO Scientific
  Collaboration

International Pulsar Timing Array Meeting, August
1-2, 2008 Arecibo, PR
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VIRGO Observatory Tuscany, Italy One
interferometer (3km)
LIGO Livingston Observatory Louisiana, USA One
interferometer (4km)
GEO600 Niedersachsen, Germany One interferometer
(600m)
LIGO Hanford Observatory Washington, USA Two
interferometers (4 km and 2 km arms)
Ground-based Observatories of the LIGO / VIRGO
collaboration

• Adapted from The Blue Marble Land Surface,
  Ocean Color and Sea Ice at visibleearth.nasa.gov

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Interferometer Sensitivity
end test mass
4 km Fabry-Perot cavity
recycling mirror
input test mass
20000 W
300 W
6 W
50/50 beam splitter
GW signal
(Numbers for LIGO 4km)
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Stochastic Analysis Basics

• Signal in detector i
• Stochastic signal

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Cross correlation

• Cross-correlation between 2 detectors
• What is Q? Depends If we look forisotropic
  background

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Cross correlation

• Cross-correlation between 2 detectors
• What is Q? Depends If we look forun-polarized
  source at ?

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What about Pulsar Timing?

• Signal from pulsar i
• Compared to LIGO
• Antenna function FiA has a different shape
  (Estabrook/Wahlquist 1975)
• Additional Pulsar term (noise due to GW)
• No time delay term (GW frequency lower)

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Hellings Downs or Overlap Reduction
FunctionIts all geometry!

• General definition of geometry factor ?
• For pulsars
• Time delay Phasor ? 1 ? ? frequency indep.!
• No signal loss at higher freq. for isotropic
  background
• No additional directional information from time
  delay

Time delay
Antennafunctions
Shape
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Looking for an isotropic background
PTA Hellings Downs
LIGO Overlap Reduction Function
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Geometry factor ? for LIGO

• Poor resolution at DC
• Time delay provides better resolution at AC

LIGO ? at DC
LIGO ? at 100Hz
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Geometry factor ? for 2 pulsars

• Small pulsar separations provide good resolution
• No additional resolution at AC

PTA ? for 15deg separation
PTA ? for 90deg separation
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So what about spatial resolution?

• How well can I resolve a point source?
• i.e. what is my point spread function?
• Answer Calculate the Fisher matrix G?,?
• LIGO Resolution dependent on H(f)
• PTA Resolution independent of frequency

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Point Spread Function for LIGO
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Point Spread Function for PTA(10 random pulsars)

• Depends on number and location of pulsars
• For this talk
• Assumed 10 fictional pulsars, randomly spaced

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Point Spread Function for PTA(10 random pulsars)

• Un-polarized source at RA15h, DECL30deg

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Point Spread Function for PTA(10 random pulsars)

• Un-polarized source at RA18h, DECL 0deg

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Conclusion

• The LIGO stochastic analysis methods is portable
  to a Pulsar Timing Array search
• 1 detector 1 pulsar
• LIGO isotropic Hellings Downs search
• Except
• Use of a cross-correlation kernel Q(f) (optimal
  filtering)
• LIGO directional search also portable
• Only difference No time delay term for PTA

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The

• End

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Extra

• Slides

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Some Definitions (sorry)

• The Signal
• The Measurements
• Detector X-Power as function of time and
  frequency.
• They are all independent, i.e.

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The Expectation Value

• Expectation value of Measurementdepends on time
  due to earth rotation
• This defines the geometry factor

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So farLooking for Point Sources

• For each pixel
• Optimal to find isolated Point Sources
• Computationally inexpensive
• Neighboring pixels correlated (blurred map)
• S4 result Phys. Rev. D 76, 082003 (2007)

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What about that blur?

• Blur described by covariance (Fisher) matrix
• In principle not hard to calculate,
• But G needs to be inverted to calculate noise
• Size ( Pixel)2
• No exploitable symmetry for Pixel basis
• S. Mitra et. al., to be published in PRD
• Fisher matrix not fully inverted

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Spherical Harmonics

• Maximum likelihood estimation gives
• i.e. same as before
• But symmetries imply structure
• Parity ?l odd ? G 0 (exact)
• Rotational symmetry m ? m ? G 0 (approx.)