Comoving coordinate

A comoving coordinate remains fixed for an observer following the homogeneous cosmological expansion.

A comoving coordinate provides labels for the comoving and proper distances comparison; proper separation additionally contains the cosmological scale factor.

Comoving observer

A comoving observer has constant comoving coordinates in a homogeneous expanding universe. Its physical separation from another comoving observer grows in proportion to the scale factor, giving relative recession governed by the Hubble parameter. Light exchanged between these observers undergoes cosmological redshift.

Comoving volume ( $V = a^{3} V_{c}$ )

A comoving volume has fixed boundaries in comoving coordinates. Its physical volume grows as the cube of the scale factor. In a homogeneous adiabatic gas, cosmological entropy conservation makes

s V

constant.

Redshift shell volume at the present epoch ( $d V_{0} = c D_{M}^{2} (z) d z d Ω/ H (z)$ )

On a present FLRW spatial slice, a shell has volume

R_{0}^{3} S^{2} (χ) d χ d Ω

. The radial light-ray relation gives

d χ / d z = c / (R_{0} H)

, yielding the displayed expression in terms of transverse comoving distance. At emission the proper volume is smaller by

(1 + z)^{3}

. For a conserved comoving population its proper density is greater by the same factor, so either epoch gives the same count.

Comoving number density ( $n_{p} = (1 + z)^{3} n_{c}$ )

Comoving number density is object count per comoving volume. With scale factor normalized to unity today, proper density is

n_{p} = a^{- 3} n_{c}

. A conserved population moving with homogeneous expansion has constant

n_{c}

despite declining proper density. Formation, mergers or changing sample selection can make

n_{c}

evolve.

Einstein-de Sitter cumulative number counts ( $N (< z_{f}) = \frac{8 n _{0} c ^{3}}{3 H _{0}^{3}} [1 - (1 + z_{f})^{- 1/2}]^{3}$ )

For constant comoving number density and complete selection in a spatially flat Einstein-de Sitter universe, integrate the present-epoch shell volume up to the population's formation redshift. The radial distance is

D_{M} = 2 c [1 - (1 + z)^{- 1/2}] / H_{0}

, so the count per unit solid angle is

n_{0} D_{M}^{3} (z_{f}) /3

. It tends to a finite value as

z_{f}

tends to infinity because this model's present comoving particle-horizon radius is

2 c / H_{0}

. Formation, mergers or an incomplete observed sample change this ideal count.

Source counts on an FLRW past light cone ( $d N = 4 π a^{2} r^{2} n d t$ )

For proper number density

n (t)

in an FLRW metric

d s^{2} = d t^{2} - a^{2} [d r^{2} / (1 - k r^{2}) + r^{2} d Ω^{2}]

, a radial shell has proper volume

4 π a^{3} r^{2} ∣ d r ∣/ 1 - k r^{2}

. A radial null geodesic obeys

∣ d r ∣ = 1 - k r^{2} d t / a

. Multiplication gives the displayed shell count on a local radial branch of the past light cone. Spatially uniform number density need not be time independent or have conserved comoving number density.

Peculiar velocity

Peculiar velocity is motion relative to the homogeneous Hubble flow. In comoving coordinates,

u = a H x + v,

where

u

is physical velocity and

v

is peculiar velocity.

Redshifting of peculiar momentum ( $a p = constant, aγ v = constant$ )

A freely moving massive test particle in an FLRW metric has physical momentum

p = mγ v

relative to a comoving observer, where

v

is its peculiar velocity. For a radial geodesic, the geodesic Lagrangian gives

C = a^{2} d χ / d τ

constant, and the local radial momentum is

p = ma d χ / d τ = m C / a

. The proper-time normalization gives

d t / d τ = 1 + C^{2} / (a^{2} c^{2})

, so

v = \frac{C / a}{1 + C ^{2} / ( a ^{2} c ^{2} )} .

Consequently 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 = c p

. These statements concern force-free propagation; a bound system need not follow the cosmological expansion.

Radial peculiar velocity in spherical harmonics ( $U = \frac{a H f}{2 π ^{2}} \sum_{ℓ m} (- i)^{ℓ} \int d^{3} k \frac{δ _{k}}{k} j_{ℓ}^{'} (k r) Y_{ℓ m} (\hat{k}) Y_{ℓ m}^{*} (\overset{r}{^})$ )

With

δ (x) = \int δ_{k} e^{- i k \cdot x} d^{3} k / (2 π)^{3}

and logarithmic growing-mode rate

f = d lo g D_{+} / d lo g a

, the linearized cosmological continuity equation gives

v_{k} = - ia H f δ_{k} k / k^{2}

. Apply the Rayleigh plane-wave expansion to its velocity potential and differentiate radially to obtain the displayed expression. The factor

a H

supplies the velocity dimensions. At the present epoch

U / H_{0}

has the same formula with

a H

removed; the commonly used

f ≃ Ω_{m}^{0.6}

is an approximation, not an exact general growth law.

Peculiar-velocity equation

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 v

- \nablaΦ/ a

from the peculiar gravitational potential.

Peculiar-velocity divergence ( $θ$ )

The peculiar-velocity divergence is

θ = \nabla \cdot v

. For irrotational linear perturbations it captures the scalar part of the peculiar velocity field.

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