Thomas C. K. Ng (The Chinese University of Hong Kong), Maximiliano Isi (Flatiron Institute), Kaze W. K. Wong (Flatiron Institute), Will M. Farr (Flatiron Institute & Stony Brook University)

Abstract

1. Objective: exploring the possibility of amplitude birefringence in gravitational waves (GW)
2. Method: performing parameter estimation (PE) on 71 events from GWTC-2.1 and 3
3. Result: The most precise constraint on amplitude birefringence to date with an order-of-magnitude improvement over previous studies

Paper Summary

GW Amplitude Birefringence:

• Predicted effect in parity-odd theories like Chern-Simons gravity (general relativity (GR) predicts there isn’t)
• Effect: one polarization is enhanced, another is suppressed during propagation

Illustration:

the upper panel of the figure above shows the absolute amplitude of the left and right polarization in the frequency domain. The dotted lines show the original waveforms with GR. With the effect of birefringence, one polarization is enhanced while the other is suppressed, with higher frequency components of the waveforms being modified more. The lower panel shows the same effect in the time domain.

The observed waveform a comoving distance \(d_C\) away from the source can be written as

\[h_{L/R}^{\mathrm{BR}}(f) = h_{L/R}^{\mathrm{GR}}(f) \times \exp\left(\pm\kappa\, d_C \frac{f}{100\,\mathrm{Hz}}\right)\,,\]

where the emitted waveform \(h_{L/R}^{\mathrm{GR}}\) is modified by an exponential birefringent factor to yield the observed waveform \(h_{L/R}^{\mathrm{BR}}\). The overall magnitude of this effect for a given frequency \(f\) is set by an attenuation coefficient, \(\kappa\), which encodes the intrinsic strength of the birefringence. Note that \(\kappa=0\) represents no amplitude birefringence, and the observed waveform is the same as the one predicted by GR.

Method

Aim: constrain the strength of GW amplitude birefringence, which is quantified by \(\kappa\)

• Modify Bilby to include the effect of amplitude birefringence
• Perform PE on 71 BBHs with false alarm rate < 1/yr
• Two sets of collective analyses:
  1. Restricted: assume birefringence is a property of spacetime
     • give a global common value of \(\kappa\) for all events
  2. Generic: assume \(\kappa\) inferred from each event is a random variable drawn from a Gaussian distribution
     • investigate possible diagnostics and explore extra physics

Result

• Individual posteriors colored by magnitude of deviation \(\mu/\sigma\)
• Restricted result (blue region):
  • \(\kappa=-0.019^{+0.038}_{-0.029} \, \mathrm{Gpc^{-1}}\) (90% credible interval)

Hierarchical analysis (Generic Result)

• Lack of birefringence is supported within 90% credibility
• \(\sigma\) posterior peaks visibly away from the origin
  • Indicating some preference for a nonzero variance
• Further observations are needed to determine whether there is true evidence for a nonvanishing variance in this population
• GW200129 is a potentially-contaminated event

Notable Events

GW170818

• Most displaced posterior from \(\kappa=0\)
• GR analysis gives \(\cos\iota=-1\) (\(\iota\): inclination angle)
  • Observed signal can be considered to be purely left-handed

• With birefringence, the same signal can be reconstructed with different \(\iota\), as long as \(d_L\) can also be enhanced to give the right amount of birefringence and overall signal power

• Difficult to determine why the GW170818 data gives this posterior structure
  • May be related to the support it has for precession
  • Birefringence reduces the support for precession (see top-right panel)
• If only a short portion of the signal is observed, adjusting \(\chi_p\) and \(\kappa\) may yield similar results as long as \(d_L\) and \(\iota\) can be tuned accordingly
  • With prior that favors larger distances, we get higher probability density with \(\kappa\neq0\).

GW190521

• Bimodal posterior distribution on \(\kappa\)

• Least support for \(\kappa=0\)

• Two peaks of the \(\kappa\) distribution correspond to two solutions that arise from respective modes in the inclination
• Without certainty about the ratio between two polarization
  • Not possible to distinguish between +ve / -ve \(\kappa\)
• Birefringence changes the posterior on \(\chi_\mathrm{eff}\)
• Situation for GW190521 is similar to that of GW170818
  • Degeneracies are likely increased by the heavier mass of the system, which reduces the observed number of cycles

\(\kappa\) correlated with \(q\) & spins

• Examine the effect of birefringence on other parameters with two events as examples
  • Both events give bimodal \(\kappa\) posteriors, but do not rule out \(\kappa=0\)
• GW170104
  • increase the preference for asymmetric masses
  • decrease the inferred level of precession
• GW191105
  • increase the preference for symmetric masses
• Coupling between \(q\) and \(\kappa\) is likely limited by the fact that changes in \(q\) alone cannot produce significant amplitude modulations

GW200129

• Second-largest deviation from \(\kappa=0\)

• May have been affected by a glitch in the Virgo instrument

• Additional PE runs were performed with only two detectors at a time
  • Results suggest that the Virgo data is responsible for \(\kappa<0\)
• Excluding it from the collective analyses did not significantly impact the results
• Further work is needed to understand the effect of the glitch

Conclusion

• Provided a significant improvement in the constraint on frequency-dependent amplitude birefringence compared to previous studies

• Identified new correlations between attenuation and binary parameters
  • particularly the relevance of spins and their partial degeneracy with \(\kappa\), as well as expected correlations with \(\iota\) and \(d_L\)
• more events are needed to understand the spread in the \(\kappa\) distribution obtained from the hierarchical analysis

More about the author

Thomas Ng is a M.Phil. student at The Chinese University of Hong Kong. More about Thomas: Thomas’s Personal Website