Useful relations in gravitational wave astronomy

Compact binary coalescence (CBC)

Gravitational wave strain and model parameters

Fourier-transformed gravitational wave strain [4]:

Where M is a chirp mass, DL is a luminosity distance, A is the amplitude, Psi is the phase, and C is a geometric detector parameter. The amplitude and the phase are represented as post-Newtonian expansion series.

To the zero'th order,

Gravitational wave frequency evolution in time

Two neutron stars or black holes in a close binary system emit gravitational radiation and get closer and closer together, until they merge. Gravitational wave frequency evolves as a function of time and chirp mass [1]:

Continuous gravitational wave (CW)

Diffraction limited resolution

Sky resolution of laser interferometers (LIGO, Virgo, Kagra, etc.) depends on a gravitational wave frequency as [2]:

Fourier segment duration and loss of SNR

Selecting long segment durations in gravitational wave data analysis for terrestrial-based detectors leads to a higher frequency-domain resolution. However, at the same time, it may lead to a loss of SNR at high frequencies due to sidereal rotation of the Earth with respect to a gravitational wave source. If we can accept only SNR losses of 10% or less, we should constrain the Fourier segment duration by [3]:

References

  1. LIGO Scientific and VIRGO Collaborations, et al. "The basic physics of the binary black hole merger GW150914." Annalen der Physik 529.1-2 (2017): 1600209.
  2. Goncharov, B., Thrane, E., 2018. All-sky radiometer for narrowband gravitational waves using folded data. Physical Review D, 98(6), p.064018.
  3. Thrane, E., Mandic, V. and Christensen, N., 2015. Detecting very long-lived gravitational-wave transients lasting hours to weeks. Physical Review D, 91(10), p.104021.
  4. Cutler, C., & Flanagan, E. E. (1994). Gravitational waves from merging compact binaries: How accurately can one extract the binary’s parameters from the inspiral waveform?. Physical Review D, 49(6), 2658.

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