Comoving and Proper Coordinates

Every quantitative statement in cosmology begins with a choice of coordinates, and the choice that makes an expanding, homogeneous universe tractable is the split between comoving and proper labels. Because the cosmological principle guarantees a preferred family of observers — the fundamental observers, who see the sky as isotropic and drift with the mean matter — we can hand each of them a spatial label that never changes. As space swells, the coordinate lattice swells with it, and every idealized galaxy stays pinned to its lattice point. All of the expansion is then packed into one function of time, the scale factor $a(t)$ , and the messy time dependence factors cleanly out of the geometry. Proper distance is what a chain of rulers laid end to end at a single instant of cosmic time would read. Since the physical spacing between comoving grid points grows in step with $a(t)$ , the proper distance between two fixed comoving points is $d_p(t)=a(t)\chi$ , where $\chi$ is their frozen comoving separation. Everything measurable — the wavelength of a photon, the separation of two clusters, the physical volume of a survey — inherits the same $a(t)$ scaling, which is exactly why the framework is so powerful. From this one relation the Hubble–Lemaître law drops out as a theorem rather than a postulate: differentiate $d_p=a\chi$ at fixed $\chi$ and you get $v=H(t)\,d_p$ with $H\equiv\dot a/a$ . That recession is coordinate stretching, not flight through space, which is why it may exceed $c$ without offending relativity. This article builds the comoving/proper machinery from the ground up, derives its three headline results, and shows where working cosmologists reach for it every day.

Key takeaways

Comoving coordinates freeze the positions of the fundamental observers (idealized galaxies riding the Hubble flow); in these labels the mean matter distribution is static for all time.
Proper distance is comoving distance times the scale factor, $d_p(t)=a(t)\,\chi$ , so it grows with the universe while the comoving separation $\chi$ stays constant.
Differentiating $d_p=a\chi$ at fixed $\chi$ yields the recession velocity $v=\dot a\,\chi=H(t)\,d_p$ — the Hubble–Lemaître law is forced, not assumed.
The FLRW line element separates a global time factor $a(t)$ from a fixed comoving 3-metric, which is the geometric statement of the comoving/proper split.
Comoving observers keep constant spatial coordinates and their proper time equals cosmic time $t$ ; real galaxies carry small peculiar velocities that redshift away as $v_{\rm pec}\propto a^{-1}$ .
Key idea: factor spacetime into a fixed comoving grid times a time-dependent scale factor — the comoving coordinate $\chi$ is constant for the Hubble flow and proper distance $d_p=a(t)\chi$ carries all the expansion, making $v=H d_p$ automatic.

Where this lives today

The comoving frame is the working coordinate system of practical cosmology, not a pedagogical toy. Cosmological N-body and hydrodynamic simulations — from the Millennium run to modern IllustrisTNG-scale efforts — integrate the equations of motion in comoving coordinates, precisely because the mean density is static there and only the density contrasts evolve; the expansion enters only through $a(t)$ as a time variable and through the $2H\dot x$ Hubble-drag term. Galaxy redshift surveys such as SDSS, DESI, and Euclid map structure in comoving space so that the baryon acoustic oscillation scale — a fixed comoving "standard ruler" of about $150\ \mathrm{Mpc}$ — appears at the same comoving separation at every epoch, turning it into a precision distance probe. Our own motion is read straight off the sky: the cosmic-microwave-background dipole measured by COBE, WMAP, and Planck fixes the Solar System's peculiar velocity at $369.8\ \mathrm{km\,s^{-1}}$ relative to the comoving CMB rest frame, and subtracting that Hubble-flow deviation is a mandatory step in every local measurement of $H_0$ . And the ongoing "Hubble tension" — the roughly $5\sigma$ gap between the early-universe and local-ladder values of $H_0$ — is a debate conducted entirely in the language of scale factor, comoving distance, and proper distance built here.

The physical picture

Paint galaxies as dots on the surface of an inflating balloon. Each dot keeps its latitude–longitude label forever — those are its comoving coordinates — but the physical arc length between any two dots grows as the balloon fills. The inflation factor is the scale factor $a(t)$ , and the physical separation is $a(t)$ times the fixed angular gap. Nothing crawls across the rubber; the dots are carried apart by the substrate itself. That single mental image is the whole subject, and the first figure freezes it at two epochs so the constant labels and the growing distances sit side by side.

Figure

The comoving grid stretches with the scale factor: galaxies keep fixed comoving labels 0, 1, 2 while the proper distance between neighbouring galaxies doubles from 50 Mpc at a = 0.5 to 100 Mpc at a = 1 today

No dot moves relative to its neighbours through the rubber, yet every dot sees every other dot recede — and the farther the dot, the faster it flees. That apparent recession is nothing but the substrate stretching, exactly the content of $v=H d_p$ . Plotting that relation makes the second, quantitative point: recession velocity is strictly linear in proper distance, so a large enough distance drives $v$ past $c$ . The distance where $v=c$ is the Hubble radius $d_H=c/H_0$ , and the figure marks it for the modern value $H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ .

Figure

Recession velocity is exactly linear in proper distance, v = H₀ dₚ with H₀ = 70 km s⁻¹ Mpc⁻¹. At the Hubble radius d_H = c/H₀ = 4283 Mpc the recession reaches c; beyond it galaxies recede superluminally, which breaks no law because it is grid stretching not local motion

A small ant crawling across the balloon relative to the painted dots has a peculiar velocity — genuine local motion superimposed on the Hubble flow. As the balloon inflates, a fixed crawl speed covers an ever-smaller fraction of the growing surface, which is the geometric reason peculiar momenta damp as $1/a$ . The comoving frame is thus an attractor: left alone, everything settles onto the Hubble flow.

The core equations

The founding definition converts a comoving label into a physical length by multiplying by the scale factor. It is the seed from which every other relation in this article grows, and it is worth reading as a factorization: a frozen geometric part $\chi$ times a single time-dependent number $a(t)$ .

d_p(t)=a(t)\,\chi \qquad (1)

where: $d_p(t)$ — proper (physical) distance at cosmic time $t$ ; $a(t)$ — dimensionless scale factor, normalized $a(t_0)=1$ today; $\chi$ — fixed comoving distance to the object. Reading: at the present epoch $a=1$ , so today's proper distance is numerically equal to the comoving distance. The expansion rate is defined as the logarithmic time derivative of the scale factor. It is a rate, with units of inverse time, and — crucially — it is a function of cosmic time, not a constant; the symbol $H_0$ is reserved for its present-day value.

H(t)\equiv\frac{\dot a(t)}{a(t)} \qquad (2)

where: $H(t)$ — Hubble parameter (expansion rate) at time $t$ ; $\dot a$ — the time derivative $\mathrm{d}a/\mathrm{d}t$ . Reading: $1/H_0\approx 13.97\ \mathrm{Gyr}$ is the Hubble time, the rough age scale set by the current expansion rate. Differentiating equation (1) at fixed comoving position gives the recession velocity. The comoving label does not move, so only $a$ carries time dependence, and the algebra collapses to a proportionality between velocity and distance with slope $H$ .

v=\dot d_p=\dot a\,\chi=H(t)\,d_p \qquad (3)

where: $v$ — recession velocity of a comoving object; $d_p$ — its proper distance. Reading: this is the Hubble–Lemaître law, and it holds for every comoving galaxy with the same $H$ at a given instant — the signature of homogeneity. The geometric container for all of this is the Friedmann–Lemaître–Robertson–Walker line element. It multiplies a fixed comoving spatial 3-metric by $a^2(t)$ while leaving the time part canonical, which is the precise statement that space expands but cosmic time flows uniformly for comoving clocks.

\mathrm{d}s^2=-c^2\mathrm{d}t^2+a^2(t)\left[\frac{\mathrm{d}r^2}{1-kr^2}+r^2\mathrm{d}\Omega^2\right] \qquad (4)

where: $\mathrm{d}s^2$ — spacetime interval; $t$ — cosmic (proper) time of comoving observers; $r,\ \mathrm{d}\Omega^2$ — comoving radial coordinate and solid-angle element; $k$ — spatial curvature constant. Reading: a comoving observer sits at fixed $r,\theta,\phi$ , so along its worldline $\mathrm{d}s^2=-c^2\mathrm{d}t^2$ and its proper time equals $t$ . Integrating the spatial part of (4) at fixed time along a radial ray gives the comoving distance — the curvature-corrected quantity that is truly time-independent for the Hubble flow.

\chi=\int_0^r\frac{\mathrm{d}r'}{\sqrt{1-kr'^2}} \qquad (5)

where: $\chi$ — comoving radial distance; $r$ — comoving radial coordinate. Reading: for a flat universe ( $k=0$ ) this reduces to $\chi=r$ , and the proper distance is simply $d_p=a(t)\,r$ . Because a photon's wavelength is a proper length riding between comoving points, it stretches with $a(t)$ , and the ratio of scale factors becomes an observable redshift.

1+z=\frac{a(t_0)}{a(t_e)}=\frac{\lambda_{\rm obs}}{\lambda_{\rm em}} \qquad (6)

where: $z$ — cosmological redshift; $t_e,\ t_0$ — emission and observation times; $\lambda_{\rm em},\ \lambda_{\rm obs}$ — emitted and observed wavelengths. Reading: a redshift is a direct measurement of how much the universe has expanded since the light was emitted. Real galaxies deviate from the pure Hubble flow by a peculiar velocity, and its physical momentum decays as the universe expands — the "Hubble drag" that keeps the comoving frame an attractor.

\vec v_{\rm tot}=\underbrace{H\,\vec d_p}_{\text{Hubble flow}}+\underbrace{\vec v_{\rm pec}}_{\text{local motion}},\qquad p_{\rm pec}\propto a^{-1} \qquad (7)

where: $\vec v_{\rm tot}$ — total physical velocity; $H\vec d_p$ — smooth recession from expansion; $\vec v_{\rm pec}$ — deviation from the comoving frame; $p_{\rm pec}$ — physical peculiar momentum. Reading: only the peculiar part is genuine motion through the local space, and it is bounded by $c$ ; the Hubble part is not.

Derivation AThe Hubble law is forced by comoving coordinates

The goal here is to show that the linear velocity–distance relation is not an empirical accident nor an extra postulate but a strict consequence of writing proper distance as a scale factor times a frozen comoving distance. We start from the founding definition for a fundamental observer, differentiate it with respect to cosmic time while holding the comoving label fixed, and watch the proportionality between recession velocity and distance emerge with the Hubble parameter as its slope — exact, not first order.

Step 1. Write the proper distance to a Hubble-flow galaxy as $d_p(t)=a(t)\,\chi$ , where $\chi$ is its comoving distance and $\dot\chi=0$ . Why: comoving coordinates are constructed precisely so that fundamental observers keep constant labels, which is what freezes $\chi$ and isolates all time dependence into $a(t)$ .

Step 2. Differentiate with respect to cosmic time: $v\equiv \mathrm{d}d_p/\mathrm{d}t=\dot a\,\chi$ , since $\chi$ contributes nothing under the derivative. Why: the product rule has only one surviving term, so the velocity is directly proportional to the same $\chi$ that fixes the distance.

Step 3. Multiply and divide by $a$ : $v=\dfrac{\dot a}{a}\,(a\chi)=\dfrac{\dot a}{a}\,d_p$ , recognizing $a\chi=d_p$ from Step 1. Why: this rewriting trades the unobservable $\chi$ for the observable proper distance, and packages the expansion rate into the ratio $\dot a/a$ .

Step 4. Define $H\equiv\dot a/a$ to obtain $v=H(t)\,d_p$ . Why: the slope is the same for every comoving galaxy at a given instant, so the linear law is an expression of homogeneity, and it is exact for pure Hubble flow rather than a small-velocity approximation. *The proportionality is exact for pure Hubble flow, with $H(t_0)=H_0$ the present-day value.*

Go deeper · Derivation B · 1 minComoving distance from the FLRW line element

Derivation BComoving distance from the FLRW line element

The aim is to extract the comoving distance directly from the geometry, showing that the quantity $\chi$ in $d_p=a\chi$ is the curvature-corrected radial integral of the FLRW metric rather than a loose intuition about a stretching lattice. We fix cosmic time, restrict to a purely radial displacement, take the square root of the spatial part of the line element, and integrate — recovering both the flat-space simplification and the general curved result in one pass.

Step 1. In the line element (4) set $\mathrm{d}t=0$ and $\mathrm{d}\Omega=0$ , isolating a radial spatial interval $\mathrm{d}s=a(t)\,\dfrac{\mathrm{d}r}{\sqrt{1-kr^2}}$ . Why: proper distance is measured at a single instant of cosmic time, so the time part is switched off, and along a radial ray only the $\mathrm{d}r$ term survives.

Step 2. Integrate from the origin to the coordinate $r$ of the source: $d_p(t)=a(t)\displaystyle\int_0^r\frac{\mathrm{d}r'}{\sqrt{1-kr'^2}}\equiv a(t)\,\chi$ . Why: the scale factor is constant at fixed $t$ and pulls out of the integral, leaving a purely comoving quantity $\chi$ that depends only on geometry, not on epoch.

Step 3. Evaluate the curvature integral for each sign of $k$ : $\chi=r$ for $k=0$ , $\chi=\arcsin(r)$ for $k=+1$ , and $\chi=\operatorname{arcsinh}(r)$ for $k=-1$ (in units where $|k|=1$ ). Why: the three cases are the closed, flat, and open universes; the flat case collapses to $d_p=a\,r$ , which is the relation used in nearly all practical calculations because observations favour $k\approx0$ . *For $k=0$ the comoving distance is the coordinate $r$ itself and $d_p=a\,r$ .*

Go deeper · Derivation C · 1 minPeculiar momentum redshifts away as one over the scale factor

Derivation CPeculiar momentum redshifts away as one over the scale factor

The purpose is to prove that a galaxy or particle moving relative to the comoving frame is not left alone but continually braked, its physical peculiar momentum decaying as $1/a$ , so that the Hubble flow is a dynamical attractor and photons redshift as a special massless case. We relate physical to comoving velocity, write the nonrelativistic spatial geodesic equation in the FLRW metric, integrate the resulting separable equation, and assemble the momentum.

Step 1. Relate physical peculiar velocity to comoving velocity: $v_{\rm pec}=a\,\dot x$ , where $x$ is the comoving coordinate and $\dot x$ its rate of change. Why: a displacement $\mathrm{d}x$ in comoving coordinates corresponds to a physical displacement $a\,\mathrm{d}x$ , so the physical velocity picks up a factor of the scale factor.

Step 2. Write the nonrelativistic spatial geodesic equation for a free particle in the FLRW metric, $\ddot x+2\dfrac{\dot a}{a}\dot x=0$ . Why: the second term, $2H\dot x$ , is the "Hubble drag" — a friction-like term produced purely by the expanding geometry, with no real force behind it.

Step 3. Treat it as a separable equation for $\dot x$ : $\dfrac{\mathrm{d}\ln\dot x}{\mathrm{d}t}=-2\dfrac{\dot a}{a}$ integrates to $\dot x\propto a^{-2}$ . Why: separating and integrating turns the drag term into a clean power law for the comoving velocity, which is all we need.

Step 4. Form the physical momentum: $p_{\rm pec}=m\,v_{\rm pec}=m\,a\,\dot x\propto a\cdot a^{-2}=a^{-1}$ . Why: bulk streaming motions therefore damp out as the universe grows, and the identical $p\propto1/a$ applied to massless particles gives $1+z=1/a$ , tying the photon redshift to the same drag. *The massless limit reproduces $1+z=a(t_0)/a(t_e)$ of equation (6).*

Limiting cases

The comoving/proper relationship collapses to instructive special cases in the regimes below.

Case	Result	Meaning
Present epoch	$a(t_0)=1\ \Rightarrow\ d_p(t_0)=\chi$	Normalizing the scale factor to unity today makes comoving distance numerically equal to present-day proper distance.
Static universe	$\dot a=0\ \Rightarrow\ H=0,\ v=0$	With no expansion, proper and comoving distances differ only by a constant, and there is no Hubble recession at all.
Flat space	$k=0\ \Rightarrow\ \chi=r,\ d_p=a\,r$	The curvature integral is trivial; the comoving coordinate is the comoving distance, the case observations favour.
Local scales	$d_p\ll c/H\ \Rightarrow\ v\ll c$	Nearby galaxies recede slowly and the naive velocity-as-Doppler reading of small redshifts is safe.
Hubble radius	$d_H=c/H\ \Rightarrow\ v=c$	At $d_H\approx 4283\ \mathrm{Mpc}$ the recession equals $c$ ; beyond it recession is superluminal but harmless.
Peculiar decay	$v_{\rm pec}\propto a^{-1}$	Physical peculiar momentum redshifts away, so comoving observers are the asymptotic attractor of the flow.

Worked examples

1 · Redshift to distance for a nearby galaxy (confidence builder)

A galaxy is measured at redshift $z=0.10$ . Estimate its recession velocity and its present proper distance, taking $H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ .

Step 1. For small $z$ the redshift is approximately the recession velocity in units of $c$ : $v\approx cz=(3.0\times10^5\ \mathrm{km\,s^{-1}})(0.10)=\mathbf{3.0\times10^4\ km\,s^{-1}}$ . This linearization is the low- $z$ tail of equation (6); it is good to a few percent at $z=0.1$ .

Step 2. Invert the Hubble law $v=H_0 d_p$ for the proper distance: $d_p=v/H_0=(3.0\times10^4)/70=\mathbf{4.3\times10^2\ Mpc}\approx 428\ \mathrm{Mpc}$ . Converting, $428\ \mathrm{Mpc}\times3.26\ \mathrm{Mly/Mpc}\approx 1.4\ \mathrm{Gly}$ .

Step 3. Sanity check: $428\ \mathrm{Mpc}$ is about a tenth of the Hubble radius $d_H=4283\ \mathrm{Mpc}$ , consistent with $v\approx 0.1c$ . The galaxy recedes at 3.0×10⁴ km/s and lies about 428 Mpc away today.

2 · The Hubble radius and superluminal recession (fully worked)

Find the Hubble radius for $H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , and the recession velocity of a galaxy at present proper distance $d_p=6000\ \mathrm{Mpc}$ . Is it superluminal, and is that a problem?

Step 1. The Hubble radius is where $v=c$ : $d_H=c/H_0=(2.998\times10^5\ \mathrm{km\,s^{-1}})/(70\ \mathrm{km\,s^{-1}\,Mpc^{-1}})=\mathbf{4283\ Mpc}\approx 14.0\ \mathrm{Gly}$ . This is the distance at which the pure Hubble recession reaches the speed of light.

Step 2. Apply $v=H_0 d_p$ at $6000\ \mathrm{Mpc}$ : $v=(70)(6000)=4.2\times10^5\ \mathrm{km\,s^{-1}}$ , i.e. $v/c=4.2\times10^5/2.998\times10^5=\mathbf{1.40\,c}$ . The galaxy recedes at $1.4$ times the speed of light.

Step 3. This violates nothing. The $1.4c$ is a rate of change of proper distance driven by grid stretching, not a local velocity; the galaxy is at rest in its own comoving frame and no photon passes anything faster than $c$ locally. We can even still receive its light if the Hubble sphere later overtakes it in comoving terms. d_H = 4283 Mpc; the galaxy at 6000 Mpc recedes at 1.40 c, superluminally and legally.

3 · Scale factor, wavelength stretch, and Hubble drag (fully worked)

Light from a quasar reaches us at redshift $z=7$ . (a) By what linear factor was the universe smaller at emission? (b) A Lyman-α photon leaves at rest wavelength $121.6\ \mathrm{nm}$ — what wavelength do we observe? (c) If the Sun's peculiar velocity today is $370\ \mathrm{km\,s^{-1}}$ , what would an undisturbed peculiar velocity have been at $z=1$ ?

Step 1. From equation (6), $a(t_e)=1/(1+z)=1/8=\mathbf{0.125}$ , so the universe was $1+z=\mathbf{8}$ times smaller in linear scale (and $8^3=512$ times denser) when the light left. Every proper length, including photon wavelengths, was smaller by this same factor.

Step 2. The observed wavelength stretches by $1+z$ : $\lambda_{\rm obs}=(1+z)\lambda_{\rm em}=8\times121.6\ \mathrm{nm}=\mathbf{973\ nm}$ . A far-ultraviolet line arrives in the near infrared — which is why high- $z$ quasars are hunted with infrared instruments.

Step 3. Peculiar momentum decays as $p_{\rm pec}\propto a^{-1}$ . Between $z=1$ ( $a=0.5$ ) and today ( $a=1$ ) the scale factor doubled, so the momentum today is half its earlier value; running it backward, the undisturbed peculiar velocity at $z=1$ was $v_{\rm pec}(z{=}1)=370/0.5=\mathbf{740\ km\,s^{-1}}$ . a₍e₎ = 0.125 (8× smaller); 121.6 nm is observed at 973 nm; the peculiar velocity was ~740 km/s at z = 1.

Physics of the portrait

The comoving/proper split is inseparable from the man who first read an expanding metric as a physical prediction rather than a mathematical curiosity, and who understood the fundamental observers as a real, drifting cosmic frame.

Origins · 1 minGeorges Lemaître and how this was discovered

Georges Lemaître

1894–1966 · Belgian priest and cosmologist, the expanding universe

Georges Lemaître trained as a civil engineer, served as an artillery officer in the First World War, was ordained a Catholic priest in 1923, and then studied under Eddington at Cambridge and at Harvard and MIT — an unusually broad path into cosmology. In 1927, two years before Hubble's observational paper, he published a solution of Einstein's equations describing an expanding universe and derived from it the linear relation between a galaxy's recession velocity and its distance, complete with an estimate of the proportionality constant now called the Hubble parameter. The work appeared in a little-read Belgian journal and went largely unnoticed until Eddington had it translated in 1931 — a translation that, it later emerged, quietly omitted the paragraph containing Lemaître's numerical estimate. Lemaître's picture is exactly the one this article formalizes: a family of fundamental observers carried apart by the geometry, each seeing the others recede, with proper distances that scale in time while the observers keep fixed positions in the expanding frame. He went further in 1931, arguing that running the expansion backward implied a "primeval atom" — the conceptual seed of the Big Bang — and he lived to hear of the 1965 discovery of the cosmic microwave background that vindicated the idea, dying the following year. Einstein, who had first told him "your calculations are correct, but your physics is abominable," came to endorse the expanding-universe model he had helped make rigorous.

History

1922

— Friedmann finds the expanding and contracting solutions of Einstein's field equations

1927

— Lemaître derives the velocity–distance law from an expanding metric of fundamental observers

1929

— Hubble publishes the measured linear redshift–distance relation for nearby galaxies

1935

— Robertson establishes the isotropic, homogeneous line element in its general form

1937

— Walker completes the uniqueness proof, fixing the FLRW metric

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., Ch. 3–5 (comoving coordinates and proper distance) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 29 · Dodelson, Modern Cosmology, 2nd ed., Ch. 2 (the FLRW metric and geodesics) · Mo, van den Bosch & White, Galaxy Formation and Evolution, §3.1 · Binney & Tremaine, Galactic Dynamics, 2nd ed., cosmology appendix. All relations are standard FLRW results. The three key checks are: (1) $\dot d_p=\dot a\,\chi=Hd_p$ follows from differentiating $d_p=a\chi$ at fixed $\chi$ ; (2) the comoving-distance integral from the FLRW spatial metric gives $\chi=r$ for $k=0$ ; (3) $p_{\rm pec}\propto a^{-1}$ follows from the spatial geodesic equation with Hubble drag, and its massless limit reproduces $1+z=a(t_0)/a(t_e)$ . Anchor numbers: $H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}\Rightarrow d_H=c/H_0=4283\ \mathrm{Mpc}$ and $1/H_0=13.97\ \mathrm{Gyr}$ ; the CMB-dipole solar velocity $369.8\ \mathrm{km\,s^{-1}}$ is the Planck value; $z=7\Rightarrow a_e=0.125$ , and Lyman-α $121.6\ \mathrm{nm}\to973\ \mathrm{nm}$ . The normalization $a(t_0)=1$ is a convention.

Prerequisites. The Cosmological Principle · Homogeneity and Isotropy in Cosmology · The Expansion of the Universe and the Scale Factor. Leads to. The FLRW Metric · The Hubble–Lemaître Law · Cosmological Redshift · The Hubble Constant.

Ask the tutor: Ask me to walk a redshift all the way to a proper distance for your favourite galaxy, or to show exactly why a galaxy receding at $1.4c$ still sends us its light.

What is the difference between comoving and proper distance in one sentence?

Proper distance is the physical separation you would measure with rulers at one instant of cosmic time, while comoving distance factors out the expansion so that it stays constant for galaxies riding the Hubble flow. The two are tied by a single factor, $d_p(t)=a(t)\,\chi$ , where $a(t)$ is the scale factor normalized to $1$ today. Because of that normalization, comoving distance is numerically equal to today's proper distance, which is why cosmologists quote distances in comoving megaparsecs — the number then does not drift as the universe grows.

Why can recession velocities exceed the speed of light?

Because a recession velocity is not a velocity through space. It is the rate at which a proper distance grows as the comoving grid stretches, $v=H d_p$ , and there is no local frame in which any object passes another faster than light. Beyond the Hubble radius $d_H=c/H\approx4283\ \mathrm{Mpc}$ the recession is formally superluminal, yet nothing physical is transmitted at that rate. Special relativity limits only the peculiar velocity — genuine local motion relative to the comoving frame — and that always remains below $c$ . The superluminal recession of distant galaxies is a coordinate statement, not a causal one.

What exactly is a "fundamental" or "comoving" observer?

A fundamental observer is an idealized observer at rest with respect to the local mean matter distribution, so that the universe looks isotropic to them — no CMB dipole, no net galaxy streaming. Such observers are the ones assigned fixed comoving coordinates, and their clocks read cosmic time $t$ , the same $t$ that appears in the scale factor. Real galaxies are only approximately fundamental: they carry small peculiar velocities, typically a few hundred kilometres per second, on top of the smooth Hubble flow. The Milky Way's departure from a fundamental observer is exactly what the CMB dipole measures, and correcting for it is how we recover the comoving rest frame.

Does the space between atoms in my body expand too?

No, and the reason sharpens what comoving coordinates mean. The Hubble flow describes only systems that are not bound and are at rest in the comoving frame. Atoms, planets, stars, and galaxies are held together by forces — electromagnetic, or gravitational binding far stronger than the cosmic expansion — so they sit at fixed proper size, not fixed comoving size. As the universe expands, a bound object's comoving coordinates actually shrink. Expansion is a statement about the smooth, unbound background; it does not stretch rulers, atoms, or the Local Group, only the comoving lattice on which those bound islands float.

If comoving coordinates are just labels, why bother with them?

Because they make the mathematics of an expanding universe honest and simple. In comoving coordinates the mean matter density is static, structure formation reduces to the growth of density contrasts, and the only time dependence in the background geometry is a single function $a(t)$ . Simulations exploit this directly: evolving dark matter in comoving space removes the overall expansion from the bookkeeping, leaving only the interesting dynamics. Observationally, features fixed in comoving space — most famously the baryon acoustic oscillation scale near $150\ \mathrm{Mpc}$ — become standard rulers whose apparent size at different redshifts measures the expansion history.

How does the peculiar velocity "know" to decay as the universe expands?

There is no force doing it — the decay is pure geometry, encoded in the $2H\dot x$ term of the geodesic equation and nicknamed Hubble drag. Picture a free particle given a kick relative to the comoving grid: as it travels, the comoving points ahead of it are themselves receding, so each unit of comoving ground it covers takes it a diminishing physical step relative to the growing frame. Working this through gives physical momentum $p_{\rm pec}\propto a^{-1}$ . Bulk streaming motions therefore damp out over cosmic time, the comoving Hubble flow acts as an attractor, and the identical law applied to photons produces the cosmological redshift.

Is cosmological redshift a Doppler shift?

Not quite, though the two coincide at low redshift. A Doppler shift comes from relative motion through space; the cosmological redshift comes from the stretching of the photon's wavelength along with the comoving grid, $1+z=a(t_0)/a(t_e)$ . For small $z$ the two pictures agree numerically, $v\approx cz$ , which is why the low- $z$ Hubble law reads like a Doppler law. But at large $z$ the Doppler interpretation breaks down — a $z=7$ photon has not been "moving away at seven times some velocity"; its wavelength has simply been stretched by a factor of eight as the universe grew eightfold in linear scale. The redshift is a scale-factor ratio, cleanest read as expansion, not motion.