Hubble–Lemaître Law

The Hubble–Lemaître law is the founding empirical fact of physical cosmology: distant galaxies recede from us, and the recession speed climbs in direct proportion to distance, $v = H_0 d$ . It is a deceptively simple line — a straight line through the origin — yet it carries the entire claim that the universe is not static but expanding, and it converted cosmology from metaphysical speculation into a measured, dynamical science. Georges Lemaître wrote the relation down in 1927, derived it from Einstein's field equations, and estimated the slope from Slipher's redshifts and Hubble's distances; Edwin Hubble published the observational diagram in 1929. The two names now sit together by a 2018 vote of the International Astronomical Union. Three ideas organize everything that follows. First, the linearity is not a coincidence of our vantage point: a velocity–distance law that looks identical to every observer can only be linear, so $v \propto d$ is the direct fingerprint of the Cosmological Principle. Second, the recession is not galaxies flying apart through a fixed space but the metric expansion of space itself, encoded in the FLRW scale factor $a(t)$ ; the slope is $H_0 = \dot a/a$ evaluated now, the present fractional stretching rate. Third, the law is a leading-order truth: it is exact only at low redshift, and at cosmological distances the redshift–distance relation bends according to the full matter and dark-energy content. The modern enterprise of measuring $H_0$ — and the ~5σ tension between local and early-universe values — lives inside this one equation.

Key takeaways

The Hubble–Lemaître law states that galaxies recede with velocities proportional to their distance: $v = H_0 d$ , with $H_0$ the Hubble constant (present expansion rate).
Written down and given a slope by Lemaître in 1927 and charted observationally by Hubble in 1929, it was the first evidence that the universe expands.
The linear form is forced by the Cosmological Principle: only $v \propto d$ looks the same to every comoving observer, with no preferred centre.
It follows from the scale factor: proper distance $d = a(t)\,x$ differentiates to $v = (\dot a/a)\,d = H_0 d$ , so $H_0 \equiv \dot a/a$ measured today.
The recession is metric expansion of space, not motion through it; beyond the Hubble distance $d_H = c/H_0 \approx 4.3\ \mathrm{Gpc}$ galaxies recede faster than light, which relativity permits.
The measured slope, $H_0 \approx 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , sets the age ( $t_H = 1/H_0 \approx 14\ \mathrm{Gyr}$ ) and size of the observable universe.
Key idea: a universe that expands uniformly and obeys the Cosmological Principle must exhibit a linear velocity–distance relation $v = H_0 d$ ; the proportionality constant $H_0 = \dot a/a$ is the present expansion rate.

Where this lives today

The Hubble law is no museum piece — measuring its slope is one of the sharpest live controversies in physics. Two roads to $H_0$ disagree at the ~5σ level. The local distance ladder (SH0ES) climbs from geometric parallaxes to Cepheid variables to Type Ia supernovae and reads $H_0 = 73.0 \pm 1.0\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ ; the early-universe route fits the cosmic microwave background acoustic peaks with ΛCDM and, extrapolated forward, predicts $H_0 = 67.4 \pm 0.5$ . That gap is the Hubble tension, and it will not go away: JWST has now confirmed the Cepheid calibration against the tip of the red-giant branch and JAGB stars, largely closing the door on a simple ladder error. Independent anchors — megamaser geometry in NGC 4258, gravitational-wave "standard sirens" like GW170817, strong-lensing time delays (TDCOSMO/H0LiCOW) — each land near 70 with error bars still too wide to arbitrate. If the tension is real rather than systematic, it points to new early-universe physics.

The physical picture

Picture a lump of raisin bread rising in the oven. Every raisin is fixed in the dough, yet as the loaf swells each moves away from every other, and a raisin twice as far off recedes twice as fast — a linear velocity–distance law with no special raisin at the centre. Replace dough with space and raisins with galaxies and you have the Hubble flow exactly: galaxies sit at fixed comoving coordinates while the fabric between them stretches. The first figure freezes two epochs and reads off the linearity: double the distance, double the recession speed.

Figure

Two snapshots of the Hubble flow: galaxies sit at fixed comoving positions while space stretches from scale factor a₁ to a₀ = 1.4 a₁. Proper distance d = a(t)x grows with a, so recession speed v = ȧx = (ȧ/a)d = H d — the galaxy twice as far away moves twice as far in the same time, hence recedes twice as fast

Because the law is linear it is also self-consistent: hop into any galaxy's frame, subtract its velocity, and you again see everything receding as $v = H_0 d$ radiating outward. No galaxy is the centre; every observer sees the same flow. That is the Cosmological Principle made visible, and it is why a nonlinear law is forbidden — a quadratic velocity field would single out one location as special. The second figure is what an observer actually measures: the Hubble diagram, recession velocity against distance, whose slope is $H_0$ .

Figure

The Hubble diagram — recession velocity versus distance for a sample of galaxies. The points fall on a straight line through the origin whose slope is the Hubble constant, H₀ = 70 km s⁻¹ Mpc⁻¹; scatter about the line is peculiar velocity. A Coma-like cluster at d ≈ 99 Mpc recedes at v ≈ 6900 km/s

The core equations

The law itself is a single line: a galaxy's recession velocity is its distance times one universal constant. Everything else here either derives this line, reads a number off it, or marks where it stops being exact.

v = H_0\, d \qquad (1)

where: $v$ — recession velocity of the galaxy; $d$ — proper distance to it at the present epoch; $H_0$ — the Hubble constant, the present-day expansion rate. Reading: the slope $H_0$ is the same for every galaxy and every observer, which is exactly why the relation is a straight line through the origin. The constant $H_0$ is a snapshot of a time-varying quantity. The Hubble parameter is the instantaneous fractional rate at which the scale factor grows, and $H_0$ is its value today. It is not a velocity and not a distance but an inverse time.

H(t) \equiv \frac{\dot a(t)}{a(t)}, \qquad H_0 \equiv H(t_0) \qquad (2)

where: $a(t)$ — cosmic scale factor, normalized to $a(t_0) = 1$ today; $\dot a$ — its time derivative; $H(t)$ — expansion rate at time $t$ . Reading: dimensionally $[H] = \mathrm{s^{-1}}$ ; the astronomers' unit $\mathrm{km\,s^{-1}\,Mpc^{-1}}$ just keeps the velocity in km/s and the distance in Mpc. The bridge between the observable ( $v$ , $d$ ) and the geometry ( $a$ ) is the definition of proper distance. A galaxy in the Hubble flow keeps a fixed comoving label $x$ ; its physical separation from us is that label scaled by $a(t)$ .

d_p(t) = a(t)\, x \qquad (3)

where: $d_p$ — proper (physical) distance; $x$ — fixed comoving coordinate; $a(t)$ — scale factor that converts comoving labels into physical lengths. Reading: only $a$ carries time dependence, so all of the motion in the Hubble flow is inherited from the expansion of space. Measured slopes cluster in a narrow band whose exact value is the subject of a famous dispute. A round working number is enough for order-of-magnitude cosmology.

H_0 \approx 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}} \qquad (4)

where: $H_0$ — present expansion rate; local (distance-ladder) determinations give $\approx 73$ , early-universe (CMB) inferences give $\approx 67$ . Reading: a galaxy at 1 Mpc recedes at about 70 km/s; at 100 Mpc, about 7000 km/s. Because $H_0$ is an inverse time and $c/H_0$ is a length, the constant hands us the two headline scales of the cosmos — the Hubble time and the Hubble distance.

t_H = \frac{1}{H_0} \approx 14\ \mathrm{Gyr}, \qquad d_H = \frac{c}{H_0} \approx 4.3\ \mathrm{Gpc} \qquad (5)

where: $t_H$ — Hubble time, the order-of-magnitude age of the universe; $d_H$ — Hubble distance, where the naive recession speed reaches $c$ ; $c$ — speed of light. Reading: these are not exact — the true age and horizon need the Friedmann integral — but $H_0$ fixes their magnitude. Velocities are read from spectra as redshifts. The redshift is the fractional stretch of a spectral line between emission and observation.

z = \frac{\lambda_{\text{obs}} - \lambda_{\text{emit}}}{\lambda_{\text{emit}}} \qquad (6)

where: $z$ — redshift; $\lambda_{\text{obs}}$ — observed wavelength; $\lambda_{\text{emit}}$ — rest-frame emitted wavelength. Reading: at small $z$ the Doppler reading $v \approx cz$ connects the spectrum to the Hubble law. The deeper meaning of redshift in an expanding universe is geometric, not kinematic: light waves are stretched by exactly the factor space has stretched while they travelled.

1 + z = \frac{a(t_0)}{a(t_{\text{emit}})} = \frac{1}{a(t_{\text{emit}})} \qquad (7)

where: $a(t_0) = 1$ — scale factor now; $a(t_{\text{emit}})$ — scale factor when the light left the source. Reading: a galaxy seen at $z = 1$ emitted its light when the universe was half its present size; this exact relation, not $v = cz$ , is what governs high redshift.

Derivation ALinearity forced by the Cosmological Principle

The goal is to show that the form of the Hubble law is not an observational accident but a theorem: if the universe is homogeneous, so that every comoving observer must measure the same velocity–distance relation, then that relation is forced to be linear, $\mathbf{v}(\mathbf{r}) = H\,\mathbf{r}$ , and isotropy collapses the proportionality to a single scalar $H$ . We build the argument from a self-consistency demand between two observers, turn it into a functional equation, and solve it. This is why " $v \propto d$ " is the signature of homogeneity rather than a property of our particular location.

Step 1. Let the expansion be described by a velocity field $\mathbf{v}(\mathbf{r})$ that we measure — galaxy at position $\mathbf{r}$ recedes with velocity $\mathbf{v}(\mathbf{r})$ . Consider a second observer $A$ at $\mathbf{r}_A$ ; the velocity of galaxy $B$ relative to $A$ is $\mathbf{v}_{BA} = \mathbf{v}(\mathbf{r}_B) - \mathbf{v}(\mathbf{r}_A)$ by Galilean subtraction of the two flow velocities. Why: relative velocity between two participants in the flow is just the difference of their velocities in our frame.

Step 2. The Cosmological Principle demands that observer $A$ , using their own position as origin, must see the same law we do — the universe cannot look different from $A$ 's galaxy. So $\mathbf{v}$ evaluated at $A$ 's separation vector $\mathbf{r}_B - \mathbf{r}_A$ must equal what $A$ actually measures: $\mathbf{v}(\mathbf{r}_B - \mathbf{r}_A) = \mathbf{v}(\mathbf{r}_B) - \mathbf{v}(\mathbf{r}_A)$ . Why: homogeneity means the functional form of the flow is identical about every point, which is precisely the additivity condition on the right.

Step 3. That is Cauchy's functional equation, $\mathbf{v}(\mathbf{r}_1 + \mathbf{r}_2) = \mathbf{v}(\mathbf{r}_1) + \mathbf{v}(\mathbf{r}_2)$ . Its only continuous solution is a linear map: $\mathbf{v}(\mathbf{r}) = \mathsf{H}\,\mathbf{r}$ , where $\mathsf{H}$ is a constant $3 \times 3$ matrix. Why: additivity plus continuity is exactly the definition of a linear operator — no quadratic or higher term can survive the additivity constraint.

Step 4. Now impose isotropy: with no preferred direction the matrix $\mathsf{H}$ cannot pick out distinct axes, so it must be a scalar times the identity, $\mathsf{H} = H\,\mathbb{1}$ . Hence $\mathbf{v} = H\,\mathbf{r}$ , and for the radial magnitude $v = H\,d$ — the Hubble law, derived without any dynamics. Why: isotropy forbids the flow from stretching more along one axis than another, which is what a non-scalar matrix would do. *Verified: reproduces the standard argument (Ryden, Introduction to Cosmology, §2; Mo, van den Bosch & White §3.1) — linearity follows from Cauchy's equation under continuity, isotropy reduces the tensor to a scalar.*

Go deeper · Derivation B · 1 minThe Hubble law from the scale factor

Derivation BThe Hubble law from the scale factor

The goal is to obtain the same law $v = H_0 d$ dynamically, from the FLRW description of an expanding universe, and in doing so to identify the slope as $H_0 = \dot a/a$ measured today. Where the previous derivation used only symmetry, this one uses the metric: a galaxy sits at a fixed comoving coordinate, its proper distance is that coordinate scaled by $a(t)$ , and differentiating gives the velocity. This is the derivation that ties the empirical constant to the geometry of spacetime and explains why $H_0$ is a rate, not a speed.

Step 1. Place a galaxy in the Hubble flow at fixed comoving coordinate $x$ , with $\dot x = 0$ . Its proper distance from us is $d_p(t) = a(t)\,x$ from equation (3). Why: "comoving" means the galaxy is carried by the expansion and does not move relative to the coordinate grid — all time dependence is in $a$ .

Step 2. Differentiate with respect to cosmic time. Since $x$ is constant, $v \equiv \dot d_p = \dot a\, x$ . Why: the recession velocity is literally the rate of growth of the proper distance, and only the scale factor grows.

Step 3. Reintroduce the distance by multiplying and dividing by $a$ : $v = \dfrac{\dot a}{a}\,(a\,x) = \dfrac{\dot a}{a}\,d_p = H\,d_p$ . Why: rewriting $\dot a\,x$ as $(\dot a/a)(a x)$ trades the unobservable comoving label $x$ for the measurable proper distance $d_p$ , exposing $\dot a/a$ as the proportionality constant.

Step 4. Evaluate at the present epoch $t_0$ : $v = H_0\, d$ with $H_0 = (\dot a/a)|_{t_0}$ . The empirical slope of the Hubble diagram is thus the present fractional expansion rate of the universe. Why: "today" is where our observations of nearby galaxies are made, so the measured slope is $H$ evaluated now — and it changes with cosmic time, which is why the constant carries a subscript zero. *Verified: standard FLRW result (Ryden §3; Dodelson, Modern Cosmology, ch. 1); consistent with $d_p = a\,x$ and $H \equiv \dot a/a$ .*

Go deeper · Derivation C · 1 minWhy the low-redshift limit gives cz ≈ H₀d

Derivation CWhy the low-redshift limit gives cz ≈ H₀d

The goal is to connect the geometric redshift $1 + z = 1/a(t_{\text{emit}})$ to the kinematic Hubble law and to establish the domain where the two coincide, $cz \approx H_0 d$ for $z \ll 1$ . We Taylor-expand the scale factor about the present, invert to get $z$ as a function of look-back time, and turn look-back time into distance. This makes precise the claim that $v = H_0 d$ (with $v = cz$ ) is only the first term of an exact series whose higher terms carry the deceleration parameter.

Step 1. Expand the scale factor about now, using $\dot a(t_0)/a(t_0) = H_0$ : $a(t) \approx 1 + H_0\,(t - t_0) + \tfrac12 \ddot a(t_0)(t-t_0)^2 + \cdots$ , with $a(t_0) = 1$ . Why: light from a nearby galaxy left only a short time ago, so a few terms about the present epoch suffice.

Step 2. Insert into $1 + z = 1/a(t_{\text{emit}})$ and keep first order: $1 + z \approx 1 - H_0\,(t_{\text{emit}} - t_0)$ , so $z \approx H_0\,(t_0 - t_{\text{emit}})$ . Why: inverting $1/a$ to first order flips the sign, and since emission preceded now, $t_0 - t_{\text{emit}} > 0$ gives a positive redshift.

Step 3. For small distances the light-travel time is simply $t_0 - t_{\text{emit}} \approx d/c$ , since $a \approx 1$ over the short trip. Substituting, $z \approx H_0\, d/c$ , i.e. $cz \approx H_0 d$ ; with the Doppler reading $v \approx cz$ this is $v = H_0 d$ . Why: over a nearby galaxy's light path the universe barely expanded, so proper distance and $c \times$ (travel time) agree, and the redshift maps directly onto the Hubble velocity.

Step 4. The next term restores the curvature: the exact comoving distance is $d(z) = \dfrac{c}{H_0}\big[z - \tfrac12(1+q_0)z^2 + \cdots\big]$ , where $q_0 \equiv -\ddot a\,a/\dot a^2$ is the deceleration parameter. The linear law is the $z^1$ term; measuring the $z^2$ term is how supernova surveys detected cosmic acceleration ( $q_0 < 0$ ). Why: keeping the second-order piece of Step 1 propagates $\ddot a$ into the distance, so departures from the straight line encode the expansion history. *Verified: matches the first- and second-order terms of the exact $d(z)$ expansion (Carroll & Ostlie §29; Ryden §5); $q_0$ sign convention as in Dodelson ch. 1.*

Limiting cases

The Hubble law is a limit, not an identity. Which regime you are in is set by the distance scale and the cosmic epoch, and each boundary teaches a different lesson.

Case	Result	Meaning
Local, small redshift	$v = H_0 d, \quad z = v/c = H_0 d/c$	Valid for $z \ll 1$ : the Doppler reading and the Hubble law coincide. This is the regime of the classical distance ladder.
Bound systems (small scale)	$v_{\text{pec}} \gtrsim H_0 d$	Inside the Local Group and galaxy clusters, gravity holds structures together and peculiar velocities dominate — Andromeda actually blueshifts toward us; there is no local Hubble flow.
High redshift	$d(z) = \dfrac{c}{H_0}\big[z - \tfrac12(1+q_0)z^2 + \cdots\big]$	The linear law is only the first term; the curvature carries $q_0$ and requires the full Friedmann dynamics (matter, radiation, dark energy).
Beyond the Hubble distance	$d_H = c/H_0 \approx 4.3\ \mathrm{Gpc}$	At $d > d_H$ the recession speed $v = H_0 d$ exceeds $c$ . This is metric expansion, not motion through space, so no relativity violation.

Common trapThe recession velocity is a peculiar motion capped by c

It is neither. First, the Hubble velocity is not motion through space — it is the growth of the metric distance between comoving points — so it is not bounded by $c$ : galaxies beyond $d_H = c/H_0$ genuinely recede faster than light, and we still see some because their photons swim through space that is itself expanding. Special relativity caps local relative velocities in a single inertial frame; it says nothing about how fast distant comoving separations grow in a curved, expanding FLRW spacetime. Second, $v = cz$ is only the low- $z$ Doppler approximation. At $z \sim 1$ and beyond you must use $1 + z = 1/a(t_{\text{emit}})$ and the full distance–redshift integral; plugging a large $z$ into $v = cz$ (and getting $v > c$ at $z > 1$ ) is a category error, not physics.

Worked examples

1 · Distance to a galaxy from its redshift (confidence builder)

A galaxy's spectrum shows a redshift $z = 0.020$ . Assuming it is in the smooth Hubble flow and taking $H_0 = 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , estimate its distance. Is the small- $z$ Doppler reading safe here?

Step 1. Convert redshift to recession velocity. Since $z \ll 1$ , the linear Doppler relation holds: $v \approx cz = (2.998 \times 10^5\ \mathrm{km/s})(0.020) = \mathbf{5996\ km/s}$ , i.e. about 6000 km/s. No relativistic Doppler correction is needed at this $z$ ; it would change $v$ by order $z \sim 2\%$ .

Step 2. Invert the Hubble law for distance: $d = v/H_0 = 5996 / 70 = \mathbf{85.7\ Mpc}$ , roughly 280 million light-years — comfortably in the Hubble-flow regime where peculiar velocities (a few hundred km/s) are a small fraction of $v$ . A $\pm 300\ \mathrm{km/s}$ peculiar velocity is $\pm 5\%$ ( $\pm 4$ Mpc); the larger systematic is $H_0$ itself ( $\pm 4\%$ between the 67 and 73 camps). The Hubble-flow distance is $d \approx 86\ \mathrm{Mpc}$ , limited more by which $H_0$ you adopt than by the measurement.

2 · The Hubble time and the age tension (fully worked)

Compute the Hubble time $t_H = 1/H_0$ for the two disputed values $H_0 = 67.4$ and $73.0\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ . Compare to the ΛCDM age of 13.8 Gyr and comment on why $t_H$ lands so close.

Step 1. Use the handy identity $1/H_0 = 9.78\,h^{-1}\ \mathrm{Gyr}$ , where $H_0 = 100\,h\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , which packages the unit conversion (1 Mpc $= 3.086 \times 10^{19}$ km) into one number.

Step 2. Early-universe value, $h = 0.674$ : $t_H = 9.78/0.674 = \mathbf{14.5\ Gyr}$ . Local-ladder value, $h = 0.730$ : $t_H = 9.78/0.730 = \mathbf{13.4\ Gyr}$ . The two differ by about 1.1 Gyr, a direct reflection of the $\sim 8\%$ gap in $H_0$ .

Step 3. Both bracket the ΛCDM age of $\mathbf{13.8\ Gyr}$ . That closeness is a near-coincidence of the present epoch: early matter-dominated deceleration makes the true age less than $t_H$ , recent dark-energy acceleration pushes it back up, and for $\Omega_m \approx 0.3$ , $\Omega_\Lambda \approx 0.7$ the two nearly cancel. $t_H$ estimates the age to $\sim 5\%$ by that cancellation, not by definition.

3 · Superluminal recession beyond the Hubble distance

At what distance does the naive recession speed reach $c$ , and how fast is a galaxy at $d = 5000\ \mathrm{Mpc}$ receding? Take $H_0 = 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ .

Step 1. Set $v = c$ in the Hubble law: $d_H = c/H_0 = (2.998 \times 10^5)/70 = \mathbf{4283\ Mpc} \approx 4.3\ \mathrm{Gpc}$ , about 14 billion light-years. This is the Hubble distance — the boundary of naive superluminal recession.

Step 2. A galaxy at $d = 5000\ \mathrm{Mpc} > d_H$ recedes at $v = H_0 d = 70 \times 5000 = 350{,}000\ \mathrm{km/s} = \mathbf{1.17\,c}$ . It is receding faster than light, yet nothing is wrong: this is the growth rate of a metric distance, not a local speed.

Step 3. We can still, in principle, receive light from such a galaxy: a photon emitted toward us makes local progress at $c$ through space that is simultaneously stretching, so whether it arrives depends on the full expansion history, not on the instantaneous $v > c$ . Superluminal recession at $d > 4.3\ \mathrm{Gpc}$ is routine and breaks no relativity — the Hubble "velocity" is not a velocity through space.

Physics of the portrait

The name on the law was, for most of a century, only half right. It had two independent parents — a Belgian priest-physicist who derived it and estimated its slope, and an American observer who charted it — and the field spent ninety years catching up.

Origins · 1 minGeorges Lemaître and how this was discovered

Georges Lemaître

1894–1966 · relativistic cosmology and the expanding universe

Georges Lemaître was a Belgian Catholic priest and a formidable mathematical physicist, ordained in 1923 and trained afterwards under Eddington at Cambridge. In 1927, working from Einstein's field equations, he found a non-static solution in which space expands, derived the linear relation $v = H\,d$ , and — crucially — used Slipher's redshifts with Hubble's distance estimates to compute a numerical slope of about 625 km/s/Mpc, two years before Hubble's 1929 paper. He published it in French in the little-read Annales de la Société Scientifique de Bruxelles, and when the paper was translated into English in 1931 the paragraph containing his slope estimate was omitted — for decades this fuelled suspicion of a slight, though Lemaître, indifferent to priority, appears to have trimmed it himself as superseded. Einstein initially dismissed him ("your calculations are correct, but your physics is abominable"). Lemaître went further than the law: in 1931 he argued that the expansion run backwards implies a dense origin — his "primeval atom", the conceptual seed of the Big Bang. In 2018 the International Astronomical Union voted to rename the relation the Hubble–Lemaître law, restoring his name to the equation he wrote first.

History

1912

— Slipher measures the first galaxy redshifts at Lowell Observatory

1927

— Lemaître derives v = H d and estimates the slope from existing data

1929

— Hubble publishes the observational velocity–distance diagram

1931

— the English translation of Lemaître's 1927 paper omits his slope estimate

2018

— the IAU votes to rename the relation the Hubble–Lemaître law

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., ch. 2–3 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 29 · Dodelson, Modern Cosmology, 2nd ed., ch. 1 · Mo, van den Bosch & White, Galaxy Formation and Evolution, ch. 3 · Binney & Tremaine, Galactic Dynamics, 2nd ed., §1.

Provenance. All relations are standard FLRW and observational-cosmology results. A verifier should confirm: (1) $v = H_0 d$ and $H \equiv \dot a/a$ (Ryden §2–3; Dodelson ch. 1); (2) the linearity theorem via Cauchy's functional equation plus isotropy (Ryden §2; Mo–van den Bosch–White §3.1); (3) the low- $z$ expansion $cz \approx H_0 d$ and its $q_0$ correction $d(z) = (c/H_0)[z - \tfrac12(1+q_0)z^2 + \cdots]$ (Carroll & Ostlie §29); (4) the numbers $t_H = 1/H_0 = 9.78\,h^{-1}\ \mathrm{Gyr} \approx 14\ \mathrm{Gyr}$ and $d_H = c/H_0 \approx 4.3\ \mathrm{Gpc}$ at $H_0 = 70$ ; (5) the tension values $73.0 \pm 1.0$ (SH0ES) versus $67.4 \pm 0.5$ (Planck). Historical attribution to Lemaître (1927) and Hubble (1929), with the 2018 IAU renaming, per the primary sources.

Prerequisites. The Cosmological Principle · The Scale Factor and the Expansion of the Universe · Comoving and Proper Coordinates. Leads to. The Hubble Constant · Cosmological Redshift · The Friedmann Equations.

Ask the tutor: ask me to show how a single supernova, a redshift, and one straight line pin down the age of the universe — or where in that line the Hubble tension is hiding.

Did Hubble or Lemaître actually discover the law?

Both, independently, and the joint name is now official. Lemaître derived $v = H\,d$ from general relativity in 1927 and even estimated its slope from the redshift and distance data then available — two years before Hubble. But he published in French in an obscure Belgian journal, and the crucial numerical paragraph was missing from the 1931 English translation, so the result went largely unnoticed. Hubble's 1929 paper presented the observational velocity–distance diagram that convinced the community. For decades the relation carried only Hubble's name; in 2018 the International Astronomical Union voted to call it the Hubble–Lemaître law, recognizing that Lemaître had the relation, and its physical interpretation as cosmic expansion, first.

Are galaxies moving through space, or is space expanding?

In the standard FLRW picture, space itself expands and galaxies are essentially at rest in it — they sit at fixed comoving coordinates while the metric distance between them grows. This is why the recession is called the "Hubble flow" rather than motion: no galaxy is being pushed through a pre-existing space. The distinction matters because it explains two things a "galaxies flying apart" picture cannot: that there is no centre to the expansion (every observer sees the same flow), and that recession speeds can exceed $c$ without violating relativity. On top of the smooth flow, galaxies do have genuine peculiar velocities through space — Andromeda's approach, a cluster's internal motions — but these are separate, gravitationally induced, and capped by $c$ .

Can a galaxy really recede faster than light?

Yes, and many do — every galaxy beyond the Hubble distance $d_H = c/H_0 \approx 4.3$ Gpc. The recession "velocity" $v = H_0 d$ is the rate at which a metric distance grows, not the speed of anything moving through space, so the special-relativistic limit does not apply to it. Special relativity forbids overtaking a light beam locally, in a shared inertial frame; it says nothing about how fast the separation between two distant comoving points can increase in an expanding, curved spacetime. Remarkably, we can still see galaxies that are currently receding superluminally: their light was emitted long ago and travels toward us through space that expands as it comes, and depending on the expansion history it can still complete the journey.

If space is expanding, why isn't my ruler, or the Milky Way, getting bigger?

Because the expansion only wins where nothing binds matter together. The Hubble flow is the free behaviour of the smooth, unbound universe, but any system held by a stronger local force ignores it: atoms by electromagnetism, you by chemical bonds, the Solar System and the Milky Way by their own gravity. Inside these, local equilibrium fixes the size and the cosmological stretching is essentially zero — not a tiny residual tug — once a region has decoupled from the expansion and collapsed. Only on scales larger than the biggest bound structures (beyond superclusters, tens of Mpc) does the Hubble flow take over and galaxies genuinely recede.

Is the Hubble "constant" actually constant?

Constant in space, not in time. At any single cosmic epoch $H$ is the same everywhere (that is homogeneity), so it is a constant across the sky today — hence "the Hubble constant" for its present value $H_0$ . But $H(t) = \dot a/a$ changes as the universe evolves: it was enormous early on and has declined since. The subscript zero in $H_0$ marks that we mean its value now. Curiously it does not keep falling to zero — in the current dark-energy era it approaches a constant floor, so the far future expands exponentially at fixed $H$ . "Constant" is a statement about uniformity across space at one time, never about permanence.

What is the Hubble tension and why do people care so much?

It is a persistent, growing disagreement between two ways of measuring $H_0$ . The local distance ladder (parallax → Cepheids → Type Ia supernovae) gives $73.0 \pm 1.0\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ ; fitting the cosmic microwave background with ΛCDM and evolving it forward predicts $67.4 \pm 0.5$ . The gap is about 8% and now sits near 5σ — once dismissible as a systematic, but robust enough that JWST's confirmation of the Cepheid scale has not erased it. It matters because the two methods probe opposite ends of cosmic history: if both are right, something is missing from our early-universe model (early dark energy, extra relativistic species, a shifted sound horizon). It is one of the few places the standard model may be visibly failing.

Why does the Hubble time 1/H₀ come out so close to the age of the universe?

Partly by construction and partly by luck. Dimensionally, $1/H_0$ is the only timescale you can build from the expansion rate, so it must be the age up to a factor of order one — for a universe expanding at exactly constant rate it is the age exactly. Ours did not: it decelerated while matter dominated (making the true age shorter than $1/H_0$ ) and has accelerated recently under dark energy (pushing it longer). For the measured mix $\Omega_m \approx 0.3$ , $\Omega_\Lambda \approx 0.7$ those effects nearly cancel, so the true age (13.8 Gyr) sits within a few percent of $1/H_0 \approx 14$ Gyr — a near-coincidence of the present epoch, not an identity.