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.