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A freely falling particle's physical momentum is measured in the local orthonormal frame of a comoving observer. With proper peculiar velocity $\mathbf{v}=ad\mathbf{x}/dt$ and Lorentz factor $\gamma =(1-v^2{)}^{-1/2}$, it is $m\gamma \mathbf{v}$. It differs by one factor of the scale factor from the conserved comoving momentum. For a nonrelativistic particle, its peculiar velocity therefore decays as $a^{-1}$.

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.

The scale factor $a(t)>0$ measures how spatial lengths in an FLRW metric change with cosmic time $t$. Choose $a$ to be a dimensionless quantity with $a(t_0)=1$ at a reference epoch, usually today. For two comoving observers, their fixed radial comoving distance $\chi$ and their proper distance in cosmology $d(t)$ satisfyThe proper distance in cosmology is measured along a spatial slice at one cosmic time; it is not the distance that a travelling photon covers between emission and reception. Only ratios of the scale factor have a normalization-independent meaning: replacing $a$ by $Ca$ and $\chi$ by $\chi /C$ leaves $d$ unchanged.

For example, take $\chi =100\mathrm{Mpc}$, where $\mathrm{Mpc}$ means a million parsecs. At $a=0.5,1,1.5$, the same pair of comoving observers has proper distance in cosmology $50,100,150\mathrm{Mpc}$ respectively. Every such separation changes by the same factor, although more distant pairs gain more distance. The illustration treats the marked objects as ideal comoving observers; internal motions in galaxies need not follow this background scaling.

The beauty of the scale factor is that one function describes the changing spatial ruler for every comoving observer. Spatial homogeneity gives no preferred centre of cosmic expansion: in a flat spatial patch, relative positions obey $\mathbf{r}-{\mathbf{r}}_O=a(\mathbf{x}-{\mathbf{x}}_O)$, so recentering on any comoving observer preserves Hubble's law. The evolution of the scale factor is determined by the Friedmann equations, while its ratio between emission and reception determines cosmological redshift.

In synchronous coordinates, fixed spatial-coordinate worldlines are timelike geodesics, and $t$ is their proper time. The metric has unit lapse function and zero shift vector in time units consistent with the displayed factor $c^2$. Indeed $g_{tt}=c^2$ and $g_{ti}=0$ give ${\Gamma }^{\mu }{}_{tt}=0$. Locally, an irrotational freely falling congruence permits this construction before crossings of the coordinate worldlines. Comoving observers in a Friedmann-Lemaître-Robertson-Walker metric provide the standard cosmological example.

The equation of state $p=-\varrho$ makes the cosmological continuity equation imply constant energy density ${\varrho }_0$. For ${\varrho }_0>0$, put $H_{\Lambda }^2=8\pi G{\varrho }_0/3$. Solving both the Friedmann equation and Friedmann acceleration equation givesUp to shifts of cosmic time and time reversal, the positive scale factors are $A{e}^{H_{\Lambda }t}$ for $k=0$, $H_{\Lambda }^{-1}\cosh (H_{\Lambda }t)$ for $k=1$, and $H_{\Lambda }^{-1}\sinh (H_{\Lambda }t)$ on $t>0$ for $k=-1$. They are the flat, closed and open slicings of de Sitter spacetime. The open zero is a coordinate boundary with finite Ricci scalar $12H_{\Lambda }^2$ and finite Kretschmann scalar $24H_{\Lambda }^4$.

For the expanding flat branch, the cosmological event horizon has comoving radial distance ${\int }_t^{\infty }ds/a(s)=1/[H_{\Lambda }a(t)]$ and proper distance in cosmology $H_{\Lambda }^{-1}$. An event on the horizon reaches the central comoving observer only at infinite reception time; events beyond it never do.