Density Contrast–Peculiar Velocity Relation in the Newtonian Gauge

In general relativistic framework of large scale structure formation theory in the universe, we investigate relation between density contrast and peculiar velocity in the Newtonian gauge. According to the gauge–invariant property of the energy–momentum tensor in the this gauge, the velocity perturbation behaves as the Newtonian peculiar velocity. In this framework, relation between peculiar velocity and density contrast with respect to the Newtonian Peebles formula has an extra correction term which is ignorable for the small scales structures. The relativistic correction of peculiar velocity for the structures with the extension of hundred mega parsec is about few percent which is smaller than the accuracy of the recent peculiar velocity measurements. We also study CMB anisotropies due to the Doppler effect in the Newtonian gauge comparing with using the Newtonian gravity.