Peculiar velocity

Peculiar velocity is motion relative to the homogeneous Hubble flow. In comoving coordinates,where $\mathbf{u}$ is physical velocity and $\mathbf{v}$ is peculiar velocity.

A freely moving massive test particle in an FLRW metric has physical momentum $p=m\gamma v$ relative to a comoving observer, where $v$ is its peculiar velocity. For a radial geodesic, the geodesic Lagrangian gives $C=a^{2}d\chi /d\tau$ constant, and the local radial momentum is $p=mad\chi /d\tau =mC/a$. The proper-time normalization gives $dt/d\tau =\sqrt{1+C^2/(a^2{c}^2)}$, soConsequently nonrelativistic peculiar velocity scales as $a^{-1}$, while the exact relativistic law keeps $|v|<c$. The momentum magnitude of a free photon also scales as $a^{-1}$, giving cosmological redshift through $E=cp$. These statements concern force-free propagation; a bound system need not follow the cosmological expansion.

With $\delta (\mathbf{x})=\int {\delta }_k{e}^{-i\mathbf{k}\cdot \mathbf{x}}{d}^{3}k/(2\pi {)}^3$ and logarithmic growing-mode rate $f=d\log D_{+}/d\log a$, the linearized cosmological continuity equation gives ${\mathbf{v}}_k=-iaHf{\delta }_k\mathbf{k}/k^2$. Apply the Rayleigh plane-wave expansion to its velocity potential and differentiate radially to obtain the displayed expression. The factor $aH$ supplies the velocity dimensions. At the present epoch $U/H_0$ has the same formula with $aH$ removed; the commonly used $f\simeq {\Omega }_m^{0.6}$ is an approximation, not an exact general growth law.

For a fluid in an expanding universe, the peculiar-velocity equation is the Euler equations for an inviscid fluid after decomposing physical velocity into Hubble flow and peculiar velocity. It contains the Hubble-drag term $H\mathbf{v}$ and force $-\nabla \Phi /a$ from the peculiar gravitational potential.

The peculiar-velocity divergence is $\theta =\nabla \cdot \mathbf{v}$. For irrotational linear perturbations it captures the scalar part of the peculiar velocity field.