Comoving Distance

What "distance" means in an expanding universe

Ask a Newtonian how far a galaxy is and the question has a single answer: lay a tape measure between you. Ask a cosmologist, and the answer fractures into half a dozen distances that all agree only in the limit of small z. The universe is expanding, so the separation between two galaxies depends on when you measure it; the photon that links them traverses a metric that is itself stretching while it travels; the apparent angular size of an object depends on the geometry of the past light cone in a way that diverges from the brightness scaling.

Cosmologists therefore work with several operationally defined distances, each tailored to a particular measurement. The comoving distance D_C is the one most often used as a reference, because by construction it is the cleanest: it factors out the cosmic stretch entirely, leaving a number that an unaccelerated, unpeculiar-motion galaxy holds constant for all time.

The definition

In an FLRW universe with cosmic scale factor a(t) (so that a(t₀) ≡ 1 today), a radial light ray satisfies c dt = a(t) dχ, where χ is the comoving radial coordinate. Integrating from emission to reception and changing variables from t to redshift z via 1 + z = 1/a:

D_C(z) = c ∫₀^z dz'/H(z')

where H(z) is the Hubble parameter at redshift z. In flat ΛCDM:

H(z) = H₀ √[ Ω_r (1+z)⁴ + Ω_m (1+z)³ + Ω_k (1+z)² + ΩΛ ]

For curved universes the line-of-sight integral above still gives the "comoving line-of-sight distance" D_C; the transverse comoving distance D_M differs by a sin or sinh of (√|Ω_k| H₀ D_C / c) / (√|Ω_k| H₀ / c) — they coincide when Ω_k = 0. Throughout this article we assume flatness; modern CMB+BAO data are consistent with Ω_k = 0 to better than a percent.

Why comoving separation is constant

Consider two galaxies that ride the Hubble flow — no peculiar motion relative to the rest frame of the CMB. Their proper separation today is some number D_p(t₀). At a later epoch when a(t) > 1, their proper separation is D_p(t) = a(t) D_p(t₀). At an earlier epoch when a(t) < 1, the proper separation was smaller by the same factor. The proper distance is time-varying — it tracks the stretch of space itself.

The comoving distance is the proper distance divided by a(t). By construction it removes the time dependence, leaving a number that two Hubble-flow galaxies maintain forever. It is the "label" attached to each location in the universe: not a true coordinate (the FLRW metric is still defined in terms of t and χ), but a tag that does not change while space is stretching.

Numerical values in standard cosmology

Plugging the Planck 2018 best-fit (H₀ = 67.4 km/s/Mpc, Ω_m = 0.315, ΩΛ = 0.685) into the integral, you get the following ladder:

Redshift z	D_C (Mpc)	D_A = D_C/(1+z) (Mpc)	D_L = D_C(1+z) (Mpc)	Lookback (Gyr)
0.1	419	381	461	1.3
0.5	1934	1290	2901	5.1
1.0	3317	1659	6634	7.9
2.0	5253	1751	15760	10.5
3.0	6491	1623	25964	11.5
6.0	8590	1227	60130	12.7
10	9700	882	106700	13.2
1100 (CMB)	13900	12.6	1.53 × 10⁷	13.8
∞ (particle horizon)	14140	0	∞	13.8

Several features deserve attention. D_C grows monotonically and asymptotes to ~14 Gpc — the particle horizon. The angular-diameter distance D_A grows, peaks near z ≈ 1.6, then shrinks: objects beyond that redshift look bigger on the sky than nearer-but-not-too-near galaxies, because the universe was smaller when the light was emitted. The luminosity distance D_L diverges as z → ∞ thanks to the (1+z) factor. And the lookback time at z = 1 is already half the age of the universe.

The family of cosmological distances

From D_C you can derive every other distance by an exact algebraic relation (in a flat universe):

D_p(t)   = a(t) D_C              proper distance (depends on epoch)
D_M      = D_C                    transverse comoving (flat case)
D_A      = D_C / (1+z)            angular-diameter distance
D_L      = D_C (1+z)              luminosity distance
D_L      = D_A (1+z)²             Etherington reciprocity

The Etherington (1933) reciprocity relation is a powerful consistency check: it is a purely geometrical theorem in any metric theory of gravity in which photons travel on null geodesics and photon number is conserved. Observing all three — D_L, D_A, z — independently for the same object tests both photon conservation and the absence of unaccounted cosmic absorption. So far the relation holds at the few-percent level over the redshift range it can be tested.

Why D_A decreases at high z — the rest of the story

A galaxy of proper diameter ℓ at redshift z subtends an angle θ = ℓ / D_A(z) on the sky. Since D_A peaks near z ≈ 1.6 and then declines, a fixed-physical-size object looks largest at intermediate z, and grows again toward the CMB. This counterintuitive geometry is direct evidence of the expansion: the light reaching us from z = 1100 was emitted when the universe was 1100 times smaller, and the same physical patch was therefore angularly enormous compared to the same patch today. CMB anisotropies of physical size ~150 Mpc (the sound horizon at recombination) appear at angular scale ~1°, fixed by exactly this D_A behaviour.

BAO and supernovae — how D_C is actually measured

The geometry of the universe enters observational cosmology through two main standardisations: standard candles and standard rulers. Both pin down D_C, but at orthogonal directions of the line of sight.

Standard candles (supernovae). Type Ia supernovae are standardisable: after correcting for light-curve shape and colour (the Phillips-Tripp-Riess relation), their peak luminosity has a residual scatter of ~0.12 mag — about 6 % in flux. Observing flux at known redshift gives the luminosity distance D_L = √(L / 4πF). Converting via D_L = D_C(1+z) yields D_C(z). The Hubble diagram of distant SNe Ia is, after the (1+z) factor, a direct measurement of D_C — and was how the 1998 acceleration was discovered.

Standard rulers (BAO). At recombination, the universe imprinted a characteristic length scale on the matter distribution: the sound horizon at the drag epoch, r_drag ≈ 147 Mpc comoving. This scale is visible today as a faint peak in the galaxy two-point correlation function at the same comoving separation. Observing the BAO peak transverse to the line of sight measures D_A(z) = r_drag / θ_BAO; observing it along the line of sight measures c/H(z) = r_drag / Δz_BAO. Combining the two breaks the degeneracy and pins down both H(z) and D_C(z) at the redshift of the galaxy sample.

The two methods complement: supernovae trace the integrated D_C, BAO traces both D_C and its derivative dD_C/dz = c/H. Cross-checks between them constrain the dark-energy equation of state.

Where D_C sits in the distance ladder

The whole edifice of cosmological distances is built outward in a "ladder," each rung calibrated by the rungs below it:

1. Parallax (Gaia) — direct trigonometric distance to nearby stars to ~10 kpc.
2. Cepheids and TRGB — period-luminosity and tip-of-the-red-giant-branch standardisations calibrated by parallax. Reach 10–100 Mpc.
3. Type Ia supernovae — peak luminosity calibrated by Cepheid-hosting galaxies. Reach z ≳ 2.
4. BAO + CMB — the comoving sound horizon r_drag pins the absolute scale; angular and radial BAO peaks measure D_C(z) and H(z) at multiple redshifts.

D_C is the natural reporting variable at the upper rungs: it integrates all the local expansion history below z, it is what every survey ultimately quotes, and it is the quantity that connects directly to the CMB sound-horizon angle θ_* — the single most precisely measured number in cosmology.

Particle horizon vs cosmic event horizon

Two related-sounding quantities that confuse first-time students:

Horizon	Definition	Comoving value (flat ΛCDM)
Particle horizon	D_C(z = ∞) — most distant point whose light has reached us by now	~14.2 Gpc
Hubble distance	c / H₀	~4.45 Gpc
Cosmic event horizon	Most distant point from which light emitted now will ever reach us	~5 Gpc (~16 Gly proper today)

In a matter-only Einstein-de Sitter universe, the cosmic event horizon is infinite — there is no fundamental limit to the future reach of an emitted photon. In a universe with a cosmological constant Λ > 0, the late-time acceleration eventually moves objects outside our reach: a photon emitted now from a galaxy currently 16 Gly away (proper) will be redshifted to nothing before it crosses the gap. New light from galaxies inside our particle horizon, but outside our event horizon, will keep arriving for a finite (proper) time and then dim away. Λ-dominated universes have an event horizon; Λ-free ones do not.

Comoving volume and the survey window

Every galaxy survey reports counts per unit comoving volume, not proper volume, because the latter changes with time. The differential comoving volume is

dV_C/dz = 4π D_C² × c / H(z)
        (all-sky, flat universe)

This peaks around z ~ 2.5 in flat ΛCDM: at lower z the D_C² geometric factor is small, at higher z the c/H factor falls. The peak is why deep redshift surveys (DESI, Euclid) cluster their targets near z ~ 1–2: that is where each unit of redshift contributes the most volume to the survey. The integrated comoving volume out to the CMB is ~3700 Gpc³.

Hubble tension as a tension in D_C

The "Hubble tension" is usually quoted as a discrepancy in H₀ — 67.4 km/s/Mpc from Planck CMB versus 73 km/s/Mpc from Cepheid-SN ladders. Geometrically, this is a tension in D_C. To first order, D_C scales as 1/H₀ when other parameters are fixed; a 9 % disagreement in H₀ propagates to a 9 % disagreement in the absolute distance scale at every redshift.

BAO measurements at intermediate z, when combined with the CMB sound-horizon scale, favour the low-H₀ branch. The Cepheid-SN local ladder favours the high-H₀ branch. These two routes to D_C(z) differ at the few-percent level over different redshift windows. Resolving the tension means either finding an unaccounted-for systematic in one of the rungs, or accepting that the function D_C(z) deviates from flat ΛCDM in a way that current data cannot fit with a single H₀. The "tension" lives in the geometry — in D_C.

Numerical pitfalls

• Confusing D_C with proper distance. D_C does not change with time for Hubble-flow galaxies; proper distance does. They are equal today by convention (a(t₀) = 1), so casual references to "the galaxy is at 200 Mpc" almost always mean D_C — but only because we measure today.
• Light-travel distance. Press releases sometimes report c × t_lookback, the distance light would travel in a static space over the same time. This is neither D_C nor D_p; it overstates nearby distances and understates high-z ones. Cosmologists almost never use it.
• Confusing D_A with D_C at low z. At z ≪ 1, all four (D_C, D_M, D_A, D_L) coincide to first order. The differences kick in at z ~ 0.1 and grow rapidly thereafter. Quoting D_C ≈ 4000 Mpc for a z = 1 galaxy and then treating it as the "angular size" distance is a common student error.
• D_A non-monotonicity. If you blindly invert θ to recover D_A from a high-z fixed-size object, you find D_A decreasing at z > 1.6. Many software packages output |D_A| or sign-flip it — verify your sign conventions before computing.
• Particle horizon ≠ event horizon. The particle horizon is the maximum D_C from which light has already reached us; the event horizon is the maximum D_C from which light emitted now will ever reach us. They have different formulas and very different values in ΛCDM.
• Curvature. In the curved (Ω_k ≠ 0) case, D_C (line-of-sight) and D_M (transverse) are no longer equal. D_M = (c/H₀√|Ω_k|) sinn(√|Ω_k| H₀ D_C / c) where sinn is sin, identity, or sinh for closed, flat, open. Almost every observational formula uses D_M, not D_C, in the angular-diameter and luminosity definitions. Flat universes hide this distinction.

Worked example: comoving distance to z = 1

For flat ΛCDM with H₀ = 67.4 km/s/Mpc, Ω_m = 0.315, ΩΛ = 0.685:

D_C(1) = c ∫₀^1 dz / [H₀ √(Ω_m (1+z)³ + ΩΛ) ]
       = (c / H₀) × I,
  I    = ∫₀^1 dz / √(0.315 (1+z)³ + 0.685)

Numerically (Simpson's rule with 10 intervals):
  z=0.0  integrand = 1/√(0.315 + 0.685)        = 1.0000
  z=0.1  integrand = 1/√(0.315 × 1.331 + 0.685)= 0.9136
  z=0.5  integrand = 1/√(0.315 × 3.375 + 0.685)= 0.6555
  z=1.0  integrand = 1/√(0.315 × 8.000 + 0.685)= 0.4527

  I ≈ 0.7459

  c/H₀ = 2.998 × 10⁵ km/s / 67.4 km/s/Mpc
       = 4448 Mpc

  D_C(1) ≈ 0.7459 × 4448 Mpc ≈ 3317 Mpc

So a galaxy at z = 1 sits at about 3.3 Gpc comoving — ten billion light-years today. Its D_A is 1.66 Gpc (the universe was half its present size when the light left), and its D_L is 6.63 Gpc (the brightness drops as the square of D_L, not D_C).

Where comoving distance shows up

• Survey catalogues. SDSS, DESI, Euclid, LSST all quote galaxy positions in comoving (z, RA, Dec) and convert to D_C as needed.
• Power spectrum P(k). The matter power spectrum is a function of comoving wavenumber k (units of h/Mpc), conjugate to comoving distance. BAO is a feature at k ≈ 0.06 h/Mpc — the inverse sound horizon.
• Comoving correlation function ξ(r). Galaxy clustering is reported as ξ(r) at comoving separation r; the BAO peak appears at r ≈ 105 h⁻¹ Mpc.
• N-body simulations. Particle positions are stored in comoving coordinates, so the box does not "expand" between snapshots — only a(t) does.
• Standard candles and rulers. SN Ia Hubble diagram (D_L), BAO peak (D_A and c/H), CMB sound-horizon angle θ_* (D_M to z = 1100) — all are comoving-distance probes.
• Gravitational-wave sirens. Binary inspirals are self-calibrating standard sirens that directly measure D_L without a distance ladder. Combined with EM redshift, they give an independent D_C(z) ladder.