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
) by multiplying all masses by
and by replacing the physical distance
by the cosmological luminosity distance
[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
from the
observed signal: the scale-invariance of black hole solutions means that a signal with a redshift of
two and a chirp mass
looks identical to a signal with no redshift and a chirp mass of
. 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:
- Signal-to-noise ratio. The intrinsic measurement uncertainty in the amplitude of the
detector’s response is simply the inverse of the SNR. Since LISA can have an SNR of several
thousand when it observes an SMBH coalescence at high redshift, LISA has great potential for
cosmography.
- Position error. From the detector response one must infer the intrinsic amplitude of the wave,
which means projecting it on the antenna pattern of the detectors. This requires a knowledge
of the source position, and this will be potentially a bigger source of uncertainty because
the sensitivity of LISA depends on the location of the source in its antenna pattern. Recent
work [232, 60] has shown that LISA may be able to achieve position accuracies between one and
ten arcminutes. At, say, three arcminutes error, the amplitude uncertainty will be of order 0.1%.
This error can be reduced to the SNR-limited error if the source can be identified. Although the
coalescence of two SMBHs itself may not have an immediate effect on the visible light from a
galaxy, the host galaxy might be identifiable either because it shows great irregularity (mergers
of black holes follow from mergers of galaxies) or because some years after the merger an X-ray
source turns on (accretion will be disrupted by the tidal forces of the orbiting black holes,
but will start again after they merge) [259]. Other effects that might lead to an identification
include evidence that stars have been expelled from the core of a galaxy, fossil radio jets going
in more than two directions from a common center, and evidence for accretion having stopped
in the recent past.
- Microlensing. If the source is at a redshift larger than one, as we can expect for LISA, then
random microlensing can produce a magnification or demagnification on the order of a few
percent [197
, 132
]. The measured intrinsic amplitude then does not match the amplitude that
the signal would have in an ideal smooth cosmology.
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 [132
]. Nearby coalescences and IMRIs should provide an accurate determination of the Hubble
Constant [205, 252]. Perhaps the most interesting measurement will be to characterize the evolution of the
dark energy, which is usually characterized by inserting a parameter
in the equation of state
of dark energy,
. If
, then the dark energy is equivalent to a cosmological
constant [109] and the energy density will be the same at all epochs. If
, the dark energy is an
evolving field whose energy density diminishes in time. According to [132], gravitational wave
measurements have the potential to measure
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
even
better.