higher order waveforms in the search for gravitational waves from compact binary coalescences david...

Higher order waveforms in the search for gravitational wavesfrom compact binary coalescences

David McKechan

In collaboration with:C. Robinson, B. Sathyaprakash, C. Van Den Broeck

April 2010

Please set your Russian Professor to silent…

Talk outline

• CBC waveforms – higher order?

• Motivations for the use of higher order waveforms

• Brief overview of matched filtering

• Development of a 0.5PN filter

• Results

• Conclusions / future work

CBC waveforms – Just in case you don’t know already…

Compact Binary Coalescence:

A binary system of compact objects (Neutron star / black hole).

The system loses energy via gravitational wave emission.

Consequently the orbital separation decreases and the orbital velocity and frequency increase.

Eventually the two objects merge and form a black hole.

Hulse-Taylor pulsar (PSR B1913+16) and other systems provide conclusive* evidence

of this process occurring in nature.

*Some people prefer the term ‘compelling’.

CBC waveforms – Brief overview of linearised theory

• Einstein’s Linearised Weak-Field Equations:

• h is a perturbation of the flat spacetime metric.

• In vacua we find a solution where h is a superposition of plane waves.

• The gravitational waves travel at the speed of light and are transverse. There are two polarisations (+ and x).

• To understand the nature of gravitational waves we need the general solution.

• Considering low velocities and the solution far away from the source, the solution may be written as a Taylor-expansion of the energy-momentum tensor, Tij , and expressed as moments of its stress components.

• The leading order term in the expansion may be related to the second derivative of the mass quadrupole moment - no monopole or dipole gravitational radiation.

CBC waveforms – A binary system

• Considering the single-body centre-of-mass representation of two point masses in a circular orbit, it is easy to calculate the mass quadrupole moment and its second time derivative.

• One finds that the frequency of the gravitational waves is twice that of the orbital frequency and that the amplitude is proportional to the square of the velocities of the source.

Why? Quadrupole moment looks a bit like this:

Superposition of plane waves with +/- 2w

Qualitative answer: Symmetry of system.

• The total energy loss of a binary system due to the emission of gravitational waves can be calculated by considering the flux of the gravitational waves and integrating over the sphere. Using Kepler’s laws one can then understand the evolution of the system.

CBC waveforms – A binary system - Evolution

CBC waveforms – FBLO First-beyond-leading-order

• What if we also include the higher order terms?

• The FBLO term has some interesting features:• The amplitude it proportional to the cube of the velocity. It is therefore an order of

magnitude below the leading order term (low velocity).

• The frequency of the emitted waves is at one and three times the orbital frequency- the FBLO term introduces a first and third harmonic.

Superposition of waves travelling with +/- w and 3w

• For a system observed with zero inclination angle, there is no radiation at this order – qualitative: motion in direction of the observer cause amplitude modulations!

• We can expect other harmonics and amplitude corrections from even higher order terms.

CBC waveforms – PN formalism

• Linearised theory gives us an idea of the nature of gravitational waves and we have also seen how higher order terms introduce different harmonics.

• However:• We have not considered back reaction – the effect of the energy emission upon the

orbit dynamics…

PN FORMALISM

CBC waveforms – PN formalism

• The PN formalism is an iterative and perturbative approach.

• It results in a series expansion for the amplitude, energy, frequency, luminosity etc. in terms of (v/c).

• The leading order term (v/c)n is the 0PN term, (v/c)n+1 is then the 0.5PN term etc.

• Typically the energy/luminosity is calculated to high PN order and one can then calculate the phase evolution using energy balance equations for the system.

• In data analysis the phase evolution is more important than the amplitude evolution. Therefore, restricted waveforms (RWFs) are usually used, with the amplitude at leading order.

CBC waveforms – Higher order PN waveforms

• The PN formalism at higher order amplitudes leads to similar characteristics as we saw in linearised theory:

if i=pi/4 , then waveform is linearly polarised (+ only)


• For a zero inclination angle we have only the dominant harmonic and its amplitude corrections (motion in direction of observer):


• Equal mass systems have only even harmonics (symmetry of system):

Expect bigger difference between RWF and FWF for larger mass ratios.

CBC waveforms – Higher order PN waveforms - Differences

• Waveform as seen by initial LIGO. Red is RWF, Blue is the difference with the FWF at 2PN.

CBC waveforms – Summary

• Use PN formalism to describe evolution of gravitational waves emitted by CBCs

• Amplitude and phase

• RWF keeps only the leading order term in the amplitude

• FWF contains amplitude corrections.

Motivations for the user of higher-order waveforms

• 1. Parameter Estimation

(10,100)Msun, FWF at 2PN in amplitude, Overhead initial LIGO, i=pi/4

Structure


• 1. Parameter Estimation

Van Den Broeck & Sengupta used the covariance matrix toshow that with the 2.5PN FWF:

In Advanced LIGO the error in time-of-coalescence (arrival-time) may reduce by a factor of five compared to the RWF at low masses and by a much larger factor at high masses.

The individual component masses of the binary are expected to be found with errors as low as a few percent in Advanced LIGO, as opposed to being poorly determined by the RWF.


• 2. Mass reach – Detectors are limited by lower cutoff frequency.

(10,100)Msun, FWF at 2PN in amplitude, Overhead initial LIGO, i=pi/4

Power at higher frequencies


• 2. Mass reach – Advanced LIGO Expected SNR

m1/m2=4, FWF at 2PN in amplitude, Overhead initial LIGO, i=pi/4

Brief overview of matched filtering

Matched filter withsignal template

hInput data x Output SNR ρ

We write SNR as

Brief overview of matched filtering

• RWF Template (as seen in the detector)

• Analytically maximise the SNR over φ0

• Maximum is given by the sum of squares of the two quadratures

• Previous gravitational wave searches for CBCs have used this filter (LSC-Virgo S5 etc.)

Sky position, polarisation, inclination.

Development of a 0.5PN filter

• FWF Template at 0.5PN in amplitude

• Cannot easily analytically maximise the SNR over φ0,1,3

• But don’t we just have 2 extra templates?

• Why not just filter each term separately, maximising over the phase as before and compute the sum of squares?

• Yes, provided there is no correlation between the harmonics, i.e., the cross terms should be equal to zero:


• If there was no cross-correlation the following quantity would always equal unity:

m1/m2=4, FWF at 2PN in amplitude, Overhead initial LIGO, i=pi/4


• Let’s start from the beginning…

We have 3 harmonics and 2 quadratures so we effectively want to filter with 6 components. Let’s think of the template as a six component vector.

The cross-correlations between the components means that they are not orthogonal.

To matched filter with this template we require it to be normalised:

To deal with the cross-correlations we would like the components to be orthogonal


• Let’s transform the template components so that they are orthogonal…

Diagonalises A


• If we normalise each component individually, then the template will also be normalised

• Great, so the SNR is given by:

• Is it maximised?

YES! It is fairly easy to show and the result is old (BCV etc.), besides we’re tired of these derivations!

Let’s get to the results….

Development of a 0.5PN filter – Initial Results

• Monte-Carlo Trial – Signal injected into noise at SNR 10.• Signal 2PN in amplitude and phase.• RWF template bank, minimal match = 0.99 20-100Msun• Detection with RWF template and 0.5PN template.

DETECTION GOOD!PARAMETER ESTIMATION BAD!

Development of a 0.5PN filter – Poor parameter estimation

• Compare ambiguity of templates – should be sharply peaked around signal parameters• NSBH signal (1, 10) Msun

SECONDARY PEAK!

Development of a 0.5PN filter – Poor parameter estimation

• Do the wrong harmonics pick up the signal?

Sub-dominant template harmonic may have good match with dominant signal harmonic

Development of a 0.5PN filter – Constrain the SNR

• Use the information that we know about the waveform to constrain the SNRBasic idea – check contributions to the SNR from the three harmonics are as expectedMust work in original basis – the new basis has no physical meaningNumerically Maximise ratio of sub-dominant:dominant harmonic over sky angles

Development of a 0.5PN filter – Constrain the SNR

• How does the constraint work?For a template compute the maximum allowed ratios of the expected values using the fitted functions.Orthonormalise the template, compute the SNR.Inverse transform the SNR to the original basis.Compare the ratios with the maximum allowed.If either the SNR from the first or third harmonic is larger than expected discard the SNR for that template.

BUT does it work?

Results

• Monte-Carlo rerun – SNR 10

Results

• Monte-Carlo extensive parameter estimation studySNR = [8, 16, 64]; Total Mass = [30, 45, 67.5, 80]; eta = [0.05, 0.075, 0.111, 0.167, 0.25]

Summary: Parameter estimation always better or as good with 0.5PN filter Improvements are better at higher masses RWF does better for symmetric systems so less improvement

Results – BONUS!

• SNR time-series when filtering noise should be chi^2 distributed with six degrees of freedom

Tested a time series of 26144 points without the constraint.Then….

Conclusions / Future Work

• Higher-order waveforms can be useful in the search for gravitational waves from CBCs:• Improved Parameter estimation – especially at high mass• Signal-based veto

• Need to develop test the algorithm on real data.

• Implement the numerical maximisation routine as part of the algorithm so the relative expected SNRs are calculated for each template, rather than using a fit.

• Pass my viva!

higher order waveforms in the search for gravitational waves from compact binary coalescences david...

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