  Large-scale bulk flows are commonplace in the universe. This means that
observers living in typical galaxies, like our Milky Way, do not follow the
mean universal expansion, but have peculiar velocities relative to it. Using
relativistic linear cosmological perturbation theory, we show that bulk
peculiar motions introduce a characteristic length scale. The latter is
analogous to the familiar Jeans length and marks the threshold below which the
linear kinematics, as seen by observers living inside the bulk flow, are
dominated by relative-motion effects. On these scales cosmological measurements
can vary considerably between the rest-frame of the bulk flow and that of the
smooth Hubble expansion, due to relative-motion effects alone. In this work, we
look into and compare the measurements of the deceleration parameter. We find
that the associated "peculiar Jeans length" has typical size of the order of
few hundred Mpc. On smaller scales, the deceleration parameter measured by the
bulk-flow observers can be considerably larger (or lower) than its Hubble-frame
counterpart. This, in turn, depends on whether the peculiar motion is slightly
expanding (or contracting) relative to the background universal expansion.
Then, assuming that expanding and contracting bulk flows are randomly
distributed, nearly half of the observers in the universe may be misled to
think that their cosmos is over-decelerated. The rest of them, on the other
hand, may come to believe that the universe is under-decelerated, or even
accelerated in some cases.
