The Effect of Higher The Effect of Higher HarmonicHarmonic

Corrections on Non-Corrections on Non-SpinningSpinning

Massive BH Binaries Massive BH Binaries for LISA.for LISA.

Ed Porter (AEI)Ed Porter (AEI)

&&

Neil Cornish (MSU)Neil Cornish (MSU)

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Two Main Questions:Two Main Questions:

GWDAW 12, December 2007, Boston, USA

1) 1 detector vs. 2: Can we still do GW astronomy? Will the higher harmonics play a role?

2) HCPN vs. RPN:

What are the effects on distance and sky resolution?

What correlations are broken/enhanced by the extra harmonics?

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HCPN Waveforms.HCPN Waveforms.

GWDAW 12, December 2007, Boston, Germany

The strain of the GW at LISA is

where

and the dimensionless velocity is defined in terms of the orbital frequency by

and the phase evolution is kept at 2-PN order.

The importance of the HHCs are the H’s ~ (m1 – m2) / (m1 + m2) at odd orders.

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Monte Carlo.Monte Carlo. Short run, 1000 pointsShort run, 1000 points Different red-shifted chirp massesDifferent red-shifted chirp masses Mass ratios of 1 and 0.1Mass ratios of 1 and 0.1 Distance of 10 GpcDistance of 10 Gpc Time of observation of 1 yearTime of observation of 1 year Time of coalescence of 0.999 yrsTime of coalescence of 0.999 yrs Maximum allowed frequency of 5 mHz.Maximum allowed frequency of 5 mHz. All other parameters randomized.All other parameters randomized. LISA cutoff frequency of 0.01 mHz with red noise.LISA cutoff frequency of 0.01 mHz with red noise. One detector ( X ) and two detector cases ( AE )One detector ( X ) and two detector cases ( AE ) Low frequency approximationLow frequency approximation

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Monte Carlo.Monte Carlo.

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The SMBH is defined by the parameter set :

Use a Fisher Matrix

and a correlation matrix

Fisher is calculated using numerical differencing

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Effect of HHCs on SNREffect of HHCs on SNR

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Some Effects known:

Enhances SNR for high masses

Overestimates SNR for low masses.

NOT A GOOD WAY TO JUDGE THE EFFECT OF HHCs!

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Effect of HHCs on Distance Effect of HHCs on Distance EstimationEstimation

We can not see the highunequal mass binaries.

While the very high equalmass systems can be seen,we can not determine thedistance.

1 vs. 2 detector LISAdependent on mass ratio

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Effect of HHCs on Sky Resolution

At very high equal mass, the HHCs resolve the source as long as all links are working

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What Happens When We Use What Happens When We Use HHCsHHCs

We need to understand exactly what changes when we include HHCs

Investigate the correlation coefficients matrix

Look at the ratio of AE0 / AE2 coefficients

Look at 107 solar mass case where HHCs are expected to play a major role

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What Happens When We Use What Happens When We Use HHCsHHCs

The unequal mass case

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What Happens When We Use What Happens When We Use HHCsHHCs

The equal mass case : major correlation breaking

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What Happens When We Use What Happens When We Use HHCsHHCs

The equal mass case : intermediate correlation breaking

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What Happens When We Use What Happens When We Use HHCsHHCs

The equal mass case : correlation tightening

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ConclusionConclusion SNR is not the best way to compare the effect of HHCsSNR is not the best way to compare the effect of HHCs In some cases, with HHCs 1 detector is as good as 2 In some cases, with HHCs 1 detector is as good as 2

without HHCswithout HHCs HHCs can improve sky resolution by 2~5HHCs can improve sky resolution by 2~5 At very high mass, HHCs enhance detection and sky At very high mass, HHCs enhance detection and sky

resolutionresolution HHCs can improve distance estimation by 2~10HHCs can improve distance estimation by 2~10 Similar results for mass estimationSimilar results for mass estimation Need to run longer Monte CarloNeed to run longer Monte Carlo Confident of our results : same codes used for MLDCsConfident of our results : same codes used for MLDCs and have previously compared Fisher results against and have previously compared Fisher results against

MCMC chains.MCMC chains. HHCs can even give visible improvement for low mass HHCs can even give visible improvement for low mass

systems.systems.

GWDAW 12, December 2007, Boston, USA