The Hubble Constant

The Hubble constant is the single most consequential dimensional number in cosmology: fix it and you have fixed the age, the size, and the mean density of the observable universe in one stroke. Formally it is nothing exotic — it is the present-day value of the time-dependent Hubble parameter, $H_0\equiv H(t_0)=(\dot a/a)_{t_0}$ , where $a(t)$ is the scale factor of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric normalized to $a(t_0)=1$ . Through the Hubble–Lemaître law $v=H_0 d$ it is the slope of the velocity–distance relation for nearby galaxies; through the Friedmann equations it is the overall normalization of the whole expansion history. Everything downstream — luminosity distances, the critical density, the conversion of a redshift into a look-back time — carries its calibration. Because $H_0$ has dimensions of inverse time, it hands you two headline scales immediately. Its reciprocal is the Hubble time $1/H_0\approx 14\ \mathrm{Gyr}$ , a first guess at the age of the universe; multiply by $c$ and you get the Hubble distance $c/H_0\approx 4.4\ \mathrm{Gpc}$ , a first guess at the horizon. For decades its value was uncertain by a factor of two, which is why cosmologists learned to factor it out as a dimensionless $h$ and quote distances in $h^{-1}\,\mathrm{Mpc}$ and densities in $h^2$ — quarantining the uncertainty. Today the puzzle is no longer the overall scale but a sharp disagreement. The local distance ladder measures $H_0$ directly and lands near $73$ ; the cosmic microwave background, extrapolated forward through $\Lambda$ CDM, predicts near $67$ . That five-sigma gap is the Hubble tension, the thread the rest of this article keeps returning to.

Key takeaways

The Hubble constant $H_0=(\dot a/a)_{t_0}$ is the present value of the Hubble parameter — the fractional expansion rate of the universe today, an inverse timescale, not a speed.
Its mixed units $\mathrm{km\,s^{-1}\,Mpc^{-1}}$ reduce to $H_0\approx 2.2\times10^{-18}\,\mathrm{s^{-1}}$ ; the reciprocal is the Hubble time $t_H=1/H_0\approx 14\,\mathrm{Gyr}$ and $c/H_0\approx 4.4\,\mathrm{Gpc}$ is the Hubble distance.
The reduced constant $h\equiv H_0/(100\,\mathrm{km\,s^{-1}\,Mpc^{-1}})$ absorbs observational uncertainty into $h$ -scaled catalogues: $\rho_{\rm crit}\propto h^2$ , distances $\propto h^{-1}$ .
$H_0$ normalizes the Friedmann equation $H^2=H_0^2[\Omega_r a^{-4}+\Omega_m a^{-3}+\Omega_k a^{-2}+\Omega_\Lambda]$ , so age, density, and every cosmological distance inherit its value.
Local measurements (Cepheid-calibrated Type Ia supernovae) give $H_0\approx 73.0\pm1.0$ ; the Planck CMB fit to $\Lambda$ CDM gives $H_0\approx 67.4\pm0.5$ — a $\approx 5\sigma$ discrepancy, the Hubble tension.
Key idea: $H_0=(\dot a/a)_{t_0}$ is an inverse timescale, not a velocity; its value simultaneously sets the age, size, and critical density of the universe, and the mismatch between the locally measured and CMB-inferred values is a genuine, unresolved cosmological puzzle.

The core insight

$H_0=(\dot a/a)_{t_0}$ is not a velocity but an inverse timescale; its numerical value simultaneously sets the age, size, and critical density of the universe, and the mismatch between locally measured and CMB-inferred values is a genuine, unresolved cosmological puzzle.

Where this lives today

The Hubble constant is the most actively contested number in physical cosmology, and the contest is a competition of independent methods. On the local side, the SH0ES program rebuilds the distance ladder with ever-tighter control: Gaia parallaxes anchor Milky Way Cepheids, those calibrate Cepheids in supernova-host galaxies, and Type Ia supernovae carry the calibration into the Hubble flow — the 2022 result is $73.04\pm1.04$ . An independent ladder using the tip of the red-giant branch (the CCHP program) sidesteps Cepheids and lands near $69.8$ , between the two camps. The James Webb Space Telescope is now re-imaging these anchors to test whether crowded-field photometry has been biasing the Cepheid rung. On the early-universe side, Planck's CMB fit ( $67.4\pm0.5$ ) is buttressed by independent baryon-acoustic-oscillation plus nucleosynthesis inferences that also prefer $\sim67$ – $68$ , so the low value is not a Planck idiosyncrasy. Meanwhile, geometric routes are maturing: strong-lensing time delays (TDCOSMO/H0LiCOW), water-megamaser distances, and gravitational-wave standard sirens — GW170817 gave a first, crude $H_0\approx70$ , and a large enough catalogue of "dark sirens" could eventually arbitrate the tension with no distance ladder at all. If the gap survives all of these, the theoretical response is to modify the early universe — early dark energy, extra relativistic species raising $N_{\rm eff}$ , or non-standard recombination — since shrinking the sound horizon $r_s$ is the most economical way to reconcile a low-CMB fit with a high local $H_0$ .

The physical picture

Picture a loaf of raisin bread rising in the oven. Every raisin sees every other raisin receding, and the farther one is, the faster it flees — not because any raisin swims through the dough, but because the dough between them expands. The proportionality between recession speed and separation, measured now, is $H_0$ . It is a property of the dough's expansion rate, identical for every observer, and it is not a velocity of anything: it is a rate, an inverse time. Plot the recession velocities of real galaxies against their distances and you recover exactly this — a straight line through the origin whose slope is $H_0$ . The first figure is that plot, the Hubble diagram itself.

Figure

The Hubble diagram: recession velocity rises in direct proportion to distance, v = H₀ d, and the slope of the line through the scattered galaxies is the Hubble constant H₀ ≈ 70 km s⁻¹ Mpc⁻¹ — a galaxy 100 Mpc away recedes at about 7000 km s⁻¹

The same $H_0$ that draws that line also runs the clock. Its reciprocal is roughly how long the universe has been expanding, and — by a near-coincidence in which early deceleration and late acceleration almost cancel — that estimate lands within a few percent of the true age. But the deepest reason the Hubble constant matters right now is that two very different experiments, weighing the same universe, disagree about its value. The distance ladder measures the expansion directly in our cosmic backyard; the microwave background measures a ruler laid down 380 000 years after the Big Bang and predicts what $H_0$ ought to be. The second figure is that confrontation — the current census of $H_0$ measurements, and the gap that will not close.

Figure

The Hubble tension: two independent routes to H₀ disagree far beyond their error bars. The Planck CMB fit to ΛCDM gives 67.4 ± 0.5, the SH0ES Cepheid-supernova ladder gives 73.0 ± 1.0, and the TRGB ladder sits between near 69.8 — a separation of about 5σ

Because $H_0$ is degenerate with distance in almost every measurement, cosmologists routinely quarantine it: quote a galaxy's distance as $h^{-1}\,\mathrm{Mpc}$ , a mass as $h^{-1}M_\odot$ , a density as $h^2\,M_\odot\,\mathrm{Mpc}^{-3}$ , and the reported number is insulated from whichever value of $H_0$ ultimately wins.

The core equations

The definition is deliberately spare: $H_0$ is the logarithmic rate of change of the scale factor, evaluated now. It is a single number extracted from the whole function $H(t)$ at the present instant $t_0$ , where by convention $a(t_0)=1$ .

H_0 \equiv \left(\frac{\dot a}{a}\right)_{t_0} \qquad (1)

where: $H_0$ — Hubble constant, the present expansion rate; $a(t)$ — scale factor of the FLRW metric with $a(t_0)=1$ ; $\dot a$ — its time derivative; $t_0$ — present cosmic time. Read it as a fractional growth rate: a patch of space grows by a fraction $H_0\,\mathrm{d}t$ in a time $\mathrm{d}t$ . Observationally, $H_0$ first appears as the slope of the velocity–distance relation. For nearby galaxies the redshift maps to a recession velocity through $v\approx cz$ , and that velocity grows linearly with proper distance.

v = H_0\, d \qquad (2)

where: $v$ — recession velocity, from redshift $v\approx cz$ at small $z$ ; $d$ — proper distance to the galaxy; $H_0$ — slope of the relation. The law holds cleanly only in the smooth Hubble flow ( $z\gtrsim0.01$ ), where peculiar velocities are a small correction on top of the expansion. Because the historical value was uncertain by a factor of two, the field agreed to carry that uncertainty in a single dimensionless symbol $h$ , and to attach the Hubble time and Hubble distance to it. These two derived scales are what turn an abstract rate into concrete cosmic dimensions.

h \equiv \frac{H_0}{100\ \mathrm{km\,s^{-1}\,Mpc^{-1}}},\qquad t_H=\frac{1}{H_0}\approx 9.78\,h^{-1}\,\mathrm{Gyr},\qquad D_H=\frac{c}{H_0}\approx 2998\,h^{-1}\,\mathrm{Mpc} \qquad (3)

where: $h$ — reduced Hubble constant, $\approx 0.67$ – $0.73$ ; $t_H$ — Hubble time, the order-of-magnitude age; $D_H$ — Hubble distance, the order-of-magnitude horizon; $c$ — speed of light. For $h=0.674$ these are $t_H\approx 14.5\,\mathrm{Gyr}$ and $D_H\approx 4.45\,\mathrm{Gpc}$ . Squaring $H_0$ and combining with Newton's constant produces the density that makes space exactly flat — the natural yardstick against which every cosmic density is measured.

\rho_{\rm crit,0}=\frac{3H_0^2}{8\pi G}\approx 1.88\times10^{-26}\,h^2\ \mathrm{kg\,m^{-3}} \qquad (4)

where: $\rho_{\rm crit,0}$ — present critical density; $G$ — Newton's gravitational constant. Numerically this is about five hydrogen atoms per cubic metre — a sobering measure of how empty the universe is on average. The Hubble constant is also the overall normalization of the expansion history. Writing each energy component as a density parameter $\Omega_{i,0}=\rho_{i,0}/\rho_{\rm crit,0}$ , the Friedmann equation becomes a clean sum of scaling terms multiplied by $H_0^2$ .

H^2(a)=H_0^2\left[\Omega_{r,0}a^{-4}+\Omega_{m,0}a^{-3}+\Omega_{k,0}a^{-2}+\Omega_{\Lambda,0}\right] \qquad (5)

where: $\Omega_{r,0}$ — radiation today ( $\propto a^{-4}$ ); $\Omega_{m,0}$ — matter ( $\propto a^{-3}$ ); $\Omega_{k,0}=1-\sum\Omega_i$ — curvature ( $\propto a^{-2}$ ); $\Omega_{\Lambda,0}$ — dark energy (constant). At $a=1$ the bracket equals unity by construction, so $H(1)=H_0$ . Finally, the two headline determinations that define the present puzzle. They are not a measurement and its error but two independent, internally consistent results that refuse to overlap.

H_0^{\rm local}=73.0\pm1.0,\qquad H_0^{\rm CMB}=67.4\pm0.5\qquad (\mathrm{km\,s^{-1}\,Mpc^{-1}}) \qquad (6)

where: $H_0^{\rm local}$ — SH0ES Cepheid + Type Ia supernova distance-ladder value; $H_0^{\rm CMB}$ — Planck 2018 $\Lambda$ CDM-inferred value. The two differ by $\Delta H_0\approx 5.6$ , which the worked examples below turn into a formal significance.

Derivation AThe critical density set entirely by the Hubble constant

The goal is to show that a spatially flat universe is not flat for free — it demands one specific total density, and that density is fixed completely by $H_0$ with no reference to what the universe is made of. We start from the Friedmann equation with arbitrary curvature, impose flatness to isolate the density that balances the expansion, evaluate it today where $H=H_0$ , and push real numbers through to land on a few hydrogen atoms per cubic metre — the yardstick $\rho_{\rm crit,0}$ against which every $\Omega_i$ is defined.

Step 1. Begin with the Friedmann equation in its general form, $H^2=\tfrac{8\pi G}{3}\rho-\tfrac{kc^2}{a^2}$ , where $k$ encodes spatial curvature. Why: this is the exact dynamical statement relating the expansion rate to the total energy density and the geometry — everything else is a specialization of it.

Step 2. Impose flatness by setting $k=0$ , which collapses the equation to $H^2=\tfrac{8\pi G}{3}\rho$ . Why: a flat universe is precisely the borderline case where the density term alone accounts for the expansion rate, so the required density is uniquely determined by $H$ .

Step 3. Evaluate today, where $H=H_0$ , and solve for the density: $\rho_{\rm crit,0}=\dfrac{3H_0^2}{8\pi G}$ . Why: this defines the critical density as the present total density that makes the geometry exactly Euclidean — the natural normalization for the density parameters.

Step 4. Convert $H_0$ to SI units. For $h=0.7$ , $H_0=0.7\times\dfrac{10^5\,\mathrm{m\,s^{-1}}}{3.086\times10^{22}\,\mathrm{m}}=2.27\times10^{-18}\,\mathrm{s^{-1}}$ . Why: the mixed astronomical units must become inverse seconds before the density comes out in $\mathrm{kg\,m^{-3}}$ .

Step 5. Insert the numbers: $\rho_{\rm crit,0}=\dfrac{3(2.27\times10^{-18})^2}{8\pi(6.674\times10^{-11})}\approx 9.2\times10^{-27}\,\mathrm{kg\,m^{-3}}$ , about five hydrogen atoms per cubic metre. Why: this makes the abstraction tangible — the mean cosmic density is staggeringly low, and it scales as $h^2$ , so it inherits the tension directly. *Recomputing $3H_0^2/8\pi G$ with $H_0=2.27\times10^{-18}\,\mathrm{s^{-1}}$ gives $9.2\times10^{-27}\,\mathrm{kg\,m^{-3}}=1.88\times10^{-26}h^2\,\mathrm{kg\,m^{-3}}$ .*

Go deeper · Derivation B · 1 minThe age of the universe as an integral over the Hubble parameter

Derivation BThe age of the universe as an integral over the Hubble parameter

The goal is to replace the crude estimate "age $\approx 1/H_0$ " with the exact expression, and to see why the crude estimate nonetheless works so well. Because $H=\dot a/a$ is a first-order equation for $a(t)$ , cosmic time can be written as an integral over the scale factor weighted by the inverse expansion rate; the matter and dark-energy content enter only through the dimensionless $E(a)$ . Evaluating the integral for concordance parameters returns a number close to unity — the near-coincidence behind $t_0\approx 1/H_0$ .

Step 1. Rearrange the definition $H=\dot a/a$ into $\mathrm{d}t=\dfrac{\mathrm{d}a}{aH}$ . Why: separating variables converts a statement about rates into an integral over the expansion history, trading time for scale factor.

Step 2. Insert the Friedmann normalization $H=H_0E(a)$ with $E(a)=\sqrt{\Omega_{m,0}a^{-3}+\Omega_{\Lambda,0}}$ , neglecting radiation and assuming flatness. Why: at late times radiation is utterly negligible, so a two-component $E(a)$ captures the age to well under a percent.

Step 3. Integrate from the Big Bang ( $a=0$ ) to today ( $a=1$ ): $t_0=\dfrac{1}{H_0}\displaystyle\int_0^1\dfrac{\mathrm{d}a}{a\,E(a)}$ . Why: the prefactor $1/H_0$ carries all the dimensions, so the integral is a pure dimensionless number that depends only on the density mix.

Step 4. Evaluate for $\Omega_{m,0}=0.31,\ \Omega_{\Lambda,0}=0.69$ : the integral is $\approx 0.955$ , so $t_0\approx 0.955\,H_0^{-1}$ . With $H_0=67.4$ this is $t_0\approx 13.8\,\mathrm{Gyr}$ . Why: the factor $0.955$ being so near one is the whole reason the naive Hubble time is a good age estimate — early deceleration and late acceleration nearly cancel in the integral. *Numerically, $\int_0^1 \mathrm{d}a/(a\sqrt{0.31a^{-3}+0.69})=0.955$ , giving $t_0=13.8\,\mathrm{Gyr}$ for $H_0=67.4$ .*

Go deeper · Derivation C · 1 minSuperluminal recession without breaking special relativity

Derivation CSuperluminal recession without breaking special relativity

The goal is to confront the alarming-looking consequence of $v=H_0 d$ : for large enough $d$ the recession velocity exceeds $c$ , yet nothing in relativity is violated. The resolution is that recession velocity is the time derivative of a proper distance built from an expanding metric, not a local velocity measured by anyone passing anything. We derive the Hubble law itself from the FLRW proper distance, locate the threshold at the Hubble distance, and then state precisely what special relativity does and does not forbid.

Step 1. Write the proper distance to a galaxy at fixed comoving coordinate $\chi$ as $d_p(t)=a(t)\,\chi$ . Why: in FLRW the comoving coordinate of a freely-moving galaxy is constant; all of the separation growth is carried by the scale factor.

Step 2. Differentiate: $\dot d_p=\dot a\,\chi=\dfrac{\dot a}{a}(a\chi)=H\,d_p$ , which is the Hubble law as an exact statement about proper distance. Why: this shows the linear velocity–distance relation is not an approximation but a rigorous identity in FLRW, valid at any epoch with $H=H(t)$ .

Step 3. Set the threshold where $v=c$ : $H_0 d_p=c\Rightarrow d_p=c/H_0=D_H$ , the Hubble distance. Beyond it, $v>c$ . Why: this identifies the Hubble distance as exactly the surface of unit recession-to-light-speed ratio, not as any physical barrier.

Step 4. Reconcile with relativity. Special relativity forbids only local superluminal motion — the relative speed of two objects passing through the same event. Cosmological recession is the stretching of the space between widely separated galaxies, so distant galaxies can and do recede faster than light, and can still be seen if their light was emitted early enough to swim upstream into our past light cone. Why: the speed limit constrains tangent vectors at a single point, and two galaxies at cosmological separation share no common local frame in which to compare velocities.

Limiting cases

The single number $H_0$ is the present value of a function $H(t)$ whose behaviour — and whose relation to the age — depends entirely on what dominates the energy budget.

Case	Result	Meaning
Matter-dominated (Einstein–de Sitter)	$H(t)=\tfrac{2}{3t},\ \ t_0=\tfrac{2}{3}H_0^{-1}$	Age is only two-thirds the Hubble time; $H$ falls as $a^{-3/2}$ under pure gravitational deceleration.
$\Lambda$ -dominated (de Sitter)	$H(t)\to H_\infty=\sqrt{\Lambda/3}=\text{const}$	Expansion becomes exponential, $a\propto e^{H_\infty t}$ ; $H$ stops decreasing and freezes at a constant floor.
Empty (Milne) universe	$H(t)=\tfrac{1}{t},\ \ t_0=H_0^{-1}$	Age equals the Hubble time exactly; no gravity, so coasting expansion — the clean reference limit.
Concordance $\Lambda$ CDM	$t_0\approx 0.955\,H_0^{-1}\approx 13.8\,\mathrm{Gyr}$	Early deceleration and late acceleration nearly cancel, so the true age sits close to the Hubble time.

Common trapthe misconception that H₀ is a speed or a fixed constant of nature

$H_0$ is neither a velocity nor a constant in time. It is the present value of $H(t)$ , an inverse timescale that in $\Lambda$ CDM has been falling throughout cosmic history and is asymptoting toward the constant de Sitter value $\sqrt{\Lambda/3}$ — it is a "constant" only in the sense of being uniform across space at a given epoch, not fixed through time. Two further traps ride along with it. First, the recession "velocity" $v=H_0 d$ can exceed $c$ for large $d$ without violating relativity, because it is the rate of change of a proper distance in an expanding metric, not a local speed. Second, the reduced constant $h$ is a bookkeeping device, not a separate physical quantity: writing a density as $h^2\,M_\odot\,\mathrm{Mpc}^{-3}$ does not remove the uncertainty, it merely names where it lives so that results can be compared cleanly once $H_0$ is pinned down.

Worked examples

1 · From H₀ to the age, size, and density of the universe (fully worked)

Take the Planck value $H_0=67.4\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ . Find the Hubble constant in inverse seconds, the Hubble time, the Hubble distance, and the critical density.

Step 1. Convert to SI. One megaparsec is $3.086\times10^{19}\,\mathrm{km}$ , so $H_0=\dfrac{67.4\,\mathrm{km\,s^{-1}}}{3.086\times10^{19}\,\mathrm{km}}=\mathbf{2.18\times10^{-18}\ s^{-1}}$ . The kilometres cancel, leaving a pure inverse time.

Step 2. The Hubble time is the reciprocal: $t_H=1/H_0=\dfrac{1}{2.18\times10^{-18}\,\mathrm{s^{-1}}}=4.58\times10^{17}\,\mathrm{s}$ . Dividing by $3.156\times10^{7}\,\mathrm{s\,yr^{-1}}$ gives $t_H=\mathbf{14.5\ Gyr}$ , and the true concordance age is $0.955\,t_H=\mathbf{13.8\ Gyr}$ .

Step 3. The Hubble distance is $D_H=c/H_0=\dfrac{2.998\times10^{5}\,\mathrm{km\,s^{-1}}}{67.4\,\mathrm{km\,s^{-1}\,Mpc^{-1}}}=\mathbf{4448\ Mpc}\approx\mathbf{4.45\ Gpc}$ . Equivalently $D_H=c\,t_H=1.37\times10^{26}\,\mathrm{m}$ .

Step 4. The critical density is $\rho_{\rm crit,0}=\dfrac{3H_0^2}{8\pi G}=\dfrac{3(2.18\times10^{-18})^2}{8\pi(6.674\times10^{-11})}=\mathbf{8.5\times10^{-27}\ kg\,m^{-3}}$ — about five hydrogen atoms per cubic metre. Every one of these four numbers scales with $H_0$ : the age and size as $H_0^{-1}$ , the density as $H_0^2$ , which is exactly why the tension propagates into all of them.

2 · How significant is the Hubble tension? (fully worked)

The local and CMB determinations are $H_0^{\rm local}=73.0\pm1.0$ and $H_0^{\rm CMB}=67.4\pm0.5$ . Treating the errors as independent Gaussians, compute the discrepancy in units of σ.

Step 1. The difference of the central values is $\Delta H_0=73.0-67.4=\mathbf{5.6\ km\,s^{-1}\,Mpc^{-1}}$ .

Step 2. Independent uncertainties add in quadrature: $\sigma_\Delta=\sqrt{\sigma_{\rm local}^2+\sigma_{\rm CMB}^2}=\sqrt{1.0^2+0.5^2}=\sqrt{1.25}=\mathbf{1.12\ km\,s^{-1}\,Mpc^{-1}}$ .

Step 3. The significance is the ratio $\dfrac{\Delta H_0}{\sigma_\Delta}=\dfrac{5.6}{1.12}=\mathbf{5.0\sigma}$ . A five-sigma separation corresponds to a chance probability below one in a million — the conventional threshold for a discovery, which is why the tension is treated as a genuine anomaly rather than a fluke, and why it demands either an unmodelled systematic or new physics.

3 · The empty-universe cross-check

Verify that the Milne (empty) universe gives an age exactly equal to the Hubble time.

Step 1. With no matter, radiation, or $\Lambda$ , the only surviving term in Friedmann is curvature, $H^2=H_0^2\,\Omega_{k,0}a^{-2}$ with $\Omega_{k,0}=1$ , so $\dot a=H_0$ is constant and $a(t)=H_0 t$ .

Step 2. Then $H=\dot a/a=1/t$ , and setting $a(t_0)=1$ gives $t_0=1/H_0=t_H$ — the age equals the Hubble time exactly. This is the reference against which the concordance factor $0.955$ is measured: the real universe is close to Milne in its age, though its dynamics are nothing like coasting.

Physics of the portrait

The Hubble constant carries one man's name across a century of cosmology, but the story it tells is of a measurement that has never stopped being contested — from a first slope wrong by a factor of seven to a modern disagreement in the second significant figure.

Origins · 1 minEdwin Hubble and how this was discovered

Edwin Hubble

1889–1953 · observational cosmology and the extragalactic distance scale

Edwin Hubble was a Missouri-born lawyer-turned-astronomer who, working at the 100-inch Hooker telescope on Mount Wilson, did more than anyone to turn cosmology into an observational science. In 1924 he resolved individual Cepheid variables in the "Andromeda nebula" and used their period–luminosity relation — Henrietta Leavitt's discovery — to prove it lay far outside the Milky Way, settling the Great Debate and establishing that the universe is populated by galaxies. Then in 1929, combining his own distances with Vesto Slipher's radial-velocity measurements, he published the linear relation $v=H_0 d$ that now bears his name (shared, by a 2018 IAU resolution, with Georges Lemaître, who had derived and estimated it two years earlier). Hubble's original slope was about $500\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ — too large by a factor of about seven, because his Cepheid calibration was off, which is why his law implied an absurdly young universe. The subsequent history of the constant is a history of correcting that error: Baade's recognition of two Cepheid populations halved it in the 1950s, Sandage and Tammann versus de Vaucouleurs fought the "factor-of-two" war through the 1970s and 80s, and the Hubble Space Telescope Key Project finally converged near $72$ in 2001. Hubble himself never accepted that the redshifts implied genuine expansion, preferring to leave their interpretation open — a caution that reads, in hindsight, like the same skepticism now trained on the tension between $67$ and $73$ .

History

1912

— Slipher measures the first spiral-nebula radial velocities, most of them redshifted

1922

— Friedmann derives the expanding-universe solutions of general relativity

1927

— Lemaître publishes the velocity–distance relation and estimates its slope

1929

— Hubble reports the linear law v = H₀ d from Cepheid distances

2001

— the HST Key Project measures H₀ = 72 ± 8, ending the factor-of-two era

2020

— Planck (67.4) versus SH0ES (73) hardens the discrepancy into the Hubble tension

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., ch. 2 and 5 (the Hubble constant, expansion history, and age) · Dodelson & Schmidt, Modern Cosmology, 2nd ed., ch. 1–2 (FLRW and the Friedmann equations) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 27–29 (the distance scale and cosmology) · Mo, van den Bosch & White, Galaxy Formation and Evolution, ch. 3 (cosmological framework and $h$ -scaling) · Binney & Tremaine, Galactic Dynamics, 2nd ed. (distance-scale conventions). The key numerical results are: (1) $t_H=1/H_0=9.78\,h^{-1}\,\mathrm{Gyr}$ ; (2) $D_H=c/H_0=2998\,h^{-1}\,\mathrm{Mpc}$ ; (3) $\rho_{\rm crit,0}=3H_0^2/8\pi G=1.88\times10^{-26}h^2\,\mathrm{kg\,m^{-3}}$ ; (4) the age integral $\int_0^1\mathrm{d}a/(a\,E(a))=0.955$ for $\Omega_m=0.31,\ \Omega_\Lambda=0.69$ ; (5) the tension significance $(73.0-67.4)/\sqrt{1.0^2+0.5^2}=5.0\sigma$ . The measured values are $H_0^{\rm CMB}=67.4\pm0.5$ (Planck 2018 Results VI, A&A 641 A6), $H_0^{\rm local}=73.0\pm1.0$ (Riess et al., SH0ES, ApJL 934 L7, 2022), and the TRGB value near $69.8$ (Freedman et al., CCHP).

Prerequisites. The Expansion of the Universe and the Scale Factor · The Hubble–Lemaître Law · The Friedmann Equations · Cosmological Redshift. Leads to. Critical Density and the Density Parameters · The FLRW Metric · Spatial Curvature of the Universe.

Ask the tutor: Ask me to walk you through converting $H_0$ into a cosmic age from scratch, or to lay out exactly which new-physics fixes could dissolve the Hubble tension and which are already ruled out.

Is the Hubble "constant" actually constant?

Only in space, not in time. At any single cosmic epoch $H$ is the same for every observer and in every direction — that spatial uniformity is what "constant" refers to. But $H(t)$ changes as the universe evolves: in $\Lambda$ CDM it was enormous in the early radiation era, has been decreasing ever since, and is now levelling off toward the constant de Sitter value $\sqrt{\Lambda/3}$ as dark energy takes over. The subscript in $H_0$ is essential — it means "evaluated today". Quoting $H_0$ without it invites the very confusion that leads people to treat the expansion rate as a fixed number like the speed of light.

Why quote H₀ in km s⁻¹ Mpc⁻¹ instead of just inverse seconds?

Physically $H_0$ is an inverse time — $2.2\times10^{-18}\,\mathrm{s^{-1}}$ for the CMB value — but that form hides the observation it came from. The mixed units make the Hubble–Lemaître law read off directly: a galaxy one megaparsec away recedes at $H_0$ kilometres per second, so $67\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ literally means " $67$ km/s of extra recession for every megaparsec of distance". Astronomers measure velocities in km/s (from Doppler shifts) and distances in Mpc, so the compound unit keeps the raw observables visible. You convert to inverse seconds only when you need the Hubble time or the critical density.

What is the difference between H₀ and h, and why bother with h?

$h$ is just $H_0$ divided by $100\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ , so a value near $0.7$ . Its purpose is historical damage control: when $H_0$ was uncertain by a factor of two, any result that depended on it — a galaxy's luminosity, a cluster's mass, the critical density — would have inherited that uncertainty and been useless to compare across papers. By writing distances as $h^{-1}\,\mathrm{Mpc}$ , masses as $h^{-1}M_\odot$ , and densities as $h^2\,M_\odot\,\mathrm{Mpc}^{-3}$ , the $h$ -dependence is displayed explicitly and factored out, so two measurements can be compared cleanly and re-scaled the moment $H_0$ is pinned down. It quarantines the uncertainty; it does not remove it.

How can two "precise" measurements of H₀ disagree by 5σ — isn't one just wrong?

Possibly, but neither camp can find the error, and that is what makes the tension interesting. Both determinations are internally consistent and reproduced by independent groups: the low value comes not only from Planck but from ground-based CMB and BAO-plus-nucleosynthesis data; the high value comes not only from Cepheid-calibrated supernovae but partly from lensing time delays and masers. A $5\sigma$ gap means that if both are unbiased, the odds of such a split by chance are below one in a million. So either an unrecognized systematic lurks in one method — the leading suspects being the Cepheid photometry or the CMB modelling — or the extrapolation from $z=1100$ to today passes through new physics that $\Lambda$ CDM omits.

Does a bigger H₀ mean a younger universe?

Yes, and directly. The age scales roughly as $1/H_0$ : a faster present expansion rate means the universe reached its current size in less time, so $t_0$ shrinks. Quantitatively, the concordance age is $t_0\approx0.955/H_0$ , so the local value $H_0=73$ implies an age near $13.0\,\mathrm{Gyr}$ while the CMB value $67.4$ gives $13.8\,\mathrm{Gyr}$ . This is a mild secondary front in the tension: the higher local $H_0$ pushes the age down toward — and in some combinations below — the ages of the oldest known globular clusters and metal-poor stars, so cosmic chronology offers a modest independent check on which value is right.

Could resolving the tension require abandoning the Big Bang or ΛCDM entirely?

Almost certainly not the Big Bang — the framework of an expanding, cooling universe is supported by far too much independent evidence (the CMB blackbody, light-element abundances, structure growth) to be at risk from a $\sim9\%$ disagreement in one parameter. What is at stake is the six-parameter $\Lambda$ CDM model. The most economical fixes leave the Big Bang untouched and shrink the sound horizon $r_s$ — early dark energy that briefly boosts $H$ before recombination, or extra relativistic species raising $N_{\rm eff}$ — letting a CMB fit accommodate a higher local $H_0$ . Late-time dark-energy modifications tend to spoil other measurements. So the likely outcome is an extension of $\Lambda$ CDM, not its overthrow.