8.3 Cosmography: gravitational wave measurements of cosmological parameters

Since inspiral signals are standard candles [332], as described in Section 6, observations of massive black hole coalescences at cosmological distances by space-based detectors can facilitate an accurate determination of the distance to the source. Our earlier expressions for the chirp waveform can be generalized to the cosmological case (a source at redshift z) by multiplying all masses by 1 + z and by replacing the physical distance D by the cosmological luminosity distance DL [227]. If the wave amplitude, frequency, and chirp rate of the binary can be measured, then its luminosity distance can be inferred. It is not, however, possible to infer the redshift z from the observed signal: the scale-invariance of black hole solutions means that a signal with a redshift of two and a chirp mass ℳ looks identical to a signal with no redshift and a chirp mass of ℳ ∕3. To use these distance measures for cosmography, one has to obtain redshifts of the host galaxies.

Before considering how this might be done, we should ask about the accuracy with which the distance can be measured. The relative error in the distance is dominated by the relative error in the measurement of the intrinsic amplitude of the gravitational wave, because the masses will normally be much more accurately measured (by fitting the evolving phase of the signal) than the amplitude. Several factors contribute to the amplitude uncertainty:

The relatively small error boxes within which the LISA coalescences can be localized are promising for identifications, especially if the X-ray indicators mentioned above pick out the host in the error box. These factors and their impact on cosmography measurements have been examined in detail by Holz and Hughes [197], who coined the term “standard siren” for the chirp sources whose distance can be determined by gravitational wave measurements. The potential for cosmographic measurements by advanced ground-based detectors have been considered in a further paper by the same authors and collaborators [132Jump To The Next Citation Point]. Nearby coalescences and IMRIs should provide an accurate determination of the Hubble Constant [205252]. Perhaps the most interesting measurement will be to characterize the evolution of the dark energy, which is usually characterized by inserting a parameter w in the equation of state of dark energy, p = wρ. If w = − 1, then the dark energy is equivalent to a cosmological constant [109] and the energy density will be the same at all epochs. If w > − 1, the dark energy is an evolving field whose energy density diminishes in time. According to [132], gravitational wave measurements have the potential to measure w to an accuracy better than 10% (for advanced ground-based detectors) and around 4% (for LISA). The accuracy with which parameters can be measured improves greatly when one includes in the computation of the covariance matrix the harmonics of the binary inspiral signal that is normally neglected [374]. Arun et al. [50] have shown that the source location in the sky can be greatly improved when the signal harmonics (up to fifth harmonic) are included, which further helps in measuring the parameter w even better.

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