On the perturbation of the luminosity distance by peculiar motions
  We consider some aspects of the perturbation to the luminosity distance
$d(z)$ that are of relevance for SN1a cosmology and for future peculiar
velocity surveys at non-negligible redshifts.
  1) Previous work has shown that the correction to the lowest order
perturbation $\delta d / d = -\delta v / c z$ has the peculiar characteristic
that it appears to depend on the absolute state of motion of sources, rather
than on their motion relative to that of the observer. The resolution of this
apparent violation of the equivalence principle is that it is necessary to
allow for evolution of the velocities with time, and also, when considering
perturbations on the scale of the observer-source separation, to include the
gravitational redshift effect. We provide an expression for $\delta d / d$ that
provides a physically consistent way to measure peculiar velocities and
determine their impact for SN1a cosmology.
  2) We then calculate the perturbation to the redshift as a function of source
flux density, which has been proposed as an alternative probe of large-scale
motions. We show how the inclusion of surface brightness modulation modifies
the relation between $\delta z(m)$ and the peculiar velocity, and that, while
the noise properties of this method might appear promising, the velocity signal
is swamped by the effect of galaxy clustering for most scales of interest.
  3) We show how, in linear theory, peculiar velocity measurements are biased
downwards by the effect of smaller scale motions or by measurement errors (such
as in photometric redshifts). Our results nicely explain the effects seen in
simulations by Koda et al.\ 2013.
  We critically examine the prospects for extending peculiar velocity studies
to larger scales with near-term future surveys.
