The Large - Scale Velocity

1 Bulk Flows as a Cosmological Probe Hubble's Law, now spectacularly connrmed by the work of 27], 35], and 39], tells us that the distances of galaxies are proportional to their observed recession velocities, at least at low redshifts: cz = H 0 r : (1) However, this is not exactly correct. Galaxies have peculiar velocities above and beyond the Hubble ow indicated by Eq. (1). We denote the peculiar velocity v(r) at every point in space; the observed redshift in the rest frame of the Local Group is then: cz = H 0 r + ^ r (v(r) ? v(0)) ; (2) where the peculiar velocity of the Local Group itself is v(0), and ^ r is the unit vector to the galaxy in question. In practice, we will measure distances in units of km s ?1 , which means that H 0 1, and the uncertainties in the value of H 0 discussed by Freedman and Tammann in this volume are not an issue. Thus measurements of redshifts cz, and of redshift-independent distances via standard candles, yield estimates of the radial component of the velocity eld. What does the resulting velocity eld tell us? On scales large enough that the rms density uctuations are small, the equations of gravitational instability can be linearized, yielding a direct proportionality between the divergence of the velocity eld and the density eld at late times 33], 34]: r v(r) = ? 0:6 (r) : (3) This equation is easily translated to Fourier space: ik ~ v(k) = ? 0:6 ~ (k) ; (4) which means that if we deene a velocity power spectrum P v (k) ~ v 2 (k) in analogy with the usual density power spectrum P (k), we nd that P v (k) = 1:2 k ?2 P (k) : (5) There are several immediate conclusions that we can draw from this. Peculiar velocities are tightly coupled to the matter density eld (r). Therefore, peculiar velocities are a probe of the matter power spectrum; any bias of the distribution of galaxies relative to that of matter is not an issue. Moreover, Eq. (5) shows that it is in principle easier to probe large spatial scales with peculiar velocities than with the density eld, because of the two extra powers of k weighting for the velocity power spectrum. Eq. (3) shows that a comparison of the velocity eld with the …