The Hubble Tension

The Hubble constant $H_0=\dot a/a|_{t_0}$ is the single most consequential number in cosmology after the density parameters: it fixes the age, the size, and the critical density of the universe, and it sets the conversion between redshift and distance for every object we observe. For most of the twentieth century the argument over $H_0$ was a factor-of-two calibration dispute. What is genuinely new — and what makes this a live problem rather than a settled measurement — is that two utterly different strategies have each reached percent precision and now disagree by far more than their combined errors. The 'early' route reads a physical ruler. The CMB acoustic peaks and the baryon-acoustic feature in galaxy surveys both imprint the sound horizon $r_s$ , the distance a pressure wave travels in the photon-baryon fluid before recombination. That length is calculable to sub-percent accuracy, and projecting it through $\Lambda$ CDM gives $H_0=67.4\pm0.5$ . The 'late' route builds a kinematic distance ladder — geometric anchors calibrate Cepheids, Cepheids calibrate Type Ia supernovae, and the supernova Hubble diagram in the smooth flow gives $H_0=73.0\pm1.0$ , assuming nothing about the early universe. The gap of $\sim5.7$ , about $8\%$ , has survived a decade of independent anchors and cross-checks. This article holds the whole chain — sound horizon, standard ruler, distance ladder — in one view, and asks what the disagreement is telling us.

Key takeaways

The Hubble constant $H_0$ is the present expansion rate; two classes of measurement now disagree at the $\sim5\sigma$ level — the sharpest crack in the concordance model.
The early route calibrates the sound horizon $r_s\approx147\ \mathrm{Mpc}$ from CMB and BAO data and projects it through $\Lambda$ CDM, giving $H_0\approx67.4\pm0.5$ (Planck 2018).
The late route climbs a distance ladder — anchors, Cepheids, Type Ia supernovae — to $H_0\approx73.0\pm1.0$ (SH0ES 2022), independent of early-universe physics.
The CMB never measures $H_0$ directly; it measures the acoustic angle $\theta_\ast=r_s/D_M\approx0.0104\ \mathrm{rad}$ and infers $H_0$ only through the model.
No single systematic has been found in either route, so the tension reads increasingly like a hint of new physics near recombination or in the dark sector.
Almost every early-time fix reduces to shrinking $r_s$ by $\sim8\%$ ; almost every late-time fix must respect the low- $z$ BAO and supernova data that box in $H(z)$ .
Key idea: $H_0$ can be reached either by measuring a physical ruler ( $r_s$ ) at $z\sim1100$ and extrapolating with $\Lambda$ CDM, or by directly measuring nearby distances and velocities; when both are done at percent precision they disagree at $\sim5\sigma$ , and the disagreement is a statement about the model connecting them, not obviously about either measurement alone.

Where this lives today

The tension is now the organizing problem of observational cosmology, attacked from every side. On the late side, JWST has re-imaged the SH0ES Cepheids to test whether crowding biased the ladder — early returns confirm the distances and leave the tension intact — while the Carnegie–Chicago team pushes the tip-of-the-red-giant-branch and JAGB carbon-star calibrators, landing near $70$ . On the early side, ACT and the South Pole Telescope now match Planck on the peaks and damping tail, and DESI's millions of BAO redshifts squeeze the late-time fixes. Between the camps sit the " $r_s$ -independent" probes prized for bypassing the sound horizon: strong-lens time delays (TDCOSMO), gravitational-wave standard sirens (GW170817 gave the first), and megamaser distances. Theorists meanwhile stress-test early dark energy, extra neutrinos, and modified gravity against the full CMB-plus-BAO-plus-supernova likelihood — no candidate yet fits everything cleanly. Whether the culprit is a hidden systematic or a genuine crack in $\Lambda$ CDM, it will be decided by data already being taken.

The physical picture

Imagine measuring the distance to the same far wall two ways. One way you carry a stick of known length — the sound horizon $r_s$ — measure the tiny angle it subtends, and use trigonometry plus a model of how the room stretched to place the wall. The other way you pace the distance out one calibrated step at a time. If the answers disagree, either a step was miscounted or the room stretched differently from what your model assumed. That is the whole tension: the CMB pins the angle $\theta_\ast=r_s/D_M$ to $0.03\%$ , so shrinking the ruler forces the distance to shrink, which at fixed matter density raises $H_0$ — which is why nearly every early-universe resolution reduces to "make the sound horizon smaller."

Figure

Two independent roads to the same number. The early road measures the sound-horizon ruler r_s ≈ 147 Mpc imprinted at z ≈ 1100 and projects it through ΛCDM to H₀ ≈ 67.4; the late road climbs the distance ladder from a geometric anchor through Cepheids to Type Ia supernovae to H₀ ≈ 73.0. The two disagree by Δ ≈ 5.7 km s⁻¹ Mpc⁻¹, about 8 percent.

The remarkable thing is not that two methods disagree — that happens all the time — but that the disagreement refuses to dissolve. Independent geometric anchors, a separate rung-2 calibrator (the tip of the red-giant branch), and JWST re-imaging of the same Cepheids have each been thrown at the ladder, and the late value barely moves; on the early side, Planck is corroborated by SPT, ACT, and a CMB-independent BAO-plus-BBN combination. The next figure lines the measurements up on one axis so the clustering into two camps becomes visible at a glance.

Figure

The whipsaw of measurements on the H₀ axis. Early-universe probes such as Planck CMB and BAO plus BBN cluster near 67.4, while late-universe probes such as the SH0ES Cepheid ladder and H0LiCOW lensing cluster near 73.0; the TRGB ladder sits in between. Planck 67.36 ± 0.54 versus SH0ES 73.04 ± 1.04 gives a gap of 5.68 at about 4.8σ combined significance.

The core equations

The Hubble constant is the present expansion rate — the fractional rate at which every comoving separation grows. It is conventionally split off a dimensionless number $h$ because so many cosmological quantities scale with it.

H_0=\left.\frac{\dot a}{a}\right|_{t_0},\qquad H_0=100\,h\ \mathrm{km\,s^{-1}\,Mpc^{-1}} \qquad (1)

where: $a(t)$ — cosmic scale factor, normalized $a(t_0)=1$ ; $\dot a$ — its time derivative; $t_0$ — present cosmic time; $h$ — dimensionless Hubble parameter. Reading: the tension is the statement $h\approx0.674$ (early) versus $0.730$ (late). The Friedmann equation gives the expansion rate at any redshift as a sum over the energy components — the machine both routes run through, forwards to project the ruler and locally to turn velocity into distance.

H^2(z)=H_0^2\!\left[\Omega_r(1+z)^4+\Omega_m(1+z)^3+\Omega_k(1+z)^2+\Omega_\Lambda\right] \qquad (2)

where: $\Omega_r,\Omega_m,\Omega_\Lambda$ — present density parameters in radiation, matter, dark energy; $\Omega_k=1-\sum\Omega_i$ — curvature; $z$ — redshift, $1+z=1/a$ . Reading: the CMB fixes the combinations $\Omega_m h^2$ and $\Omega_b h^2$ very tightly, which is exactly what makes the ruler calculable. The sound horizon is the comoving distance a pressure wave travels in the coupled photon-baryon fluid from the Big Bang to the epoch when baryons are released — the standard ruler that both the CMB and BAO imprint.

r_s(z_d)=\int_{z_d}^{\infty}\frac{c_s(z)}{H(z)}\,\mathrm{d}z,\qquad c_s(z)=\frac{c}{\sqrt{3\!\left(1+\tfrac{3\rho_b}{4\rho_\gamma}\right)}} \qquad (3)

where: $z_d\approx1060$ — baryon drag epoch; $c_s$ — sound speed in the fluid; $\rho_b,\rho_\gamma$ — baryon and photon energy densities. Reading: with the densities measured, $r_s\approx147\ \mathrm{Mpc}$ is fixed to sub-percent — no free knob, so shrinking it requires new physics. The one thing the CMB actually observes is an angle: the ruler $r_s$ divided by the comoving distance to last scattering. It sets the position of the acoustic peaks and is measured to three parts in ten thousand.

\theta_\ast=\frac{r_s(z_\ast)}{D_M(z_\ast)}\approx0.0104\ \mathrm{rad},\qquad D_M(z_\ast)=\int_0^{z_\ast}\frac{c\,\mathrm{d}z}{H(z)} \qquad (4)

where: $\theta_\ast$ — angular size of the sound horizon; $D_M$ — comoving angular-diameter distance to $z_\ast\approx1090$ ; the acoustic multipole is $\ell_A=\pi/\theta_\ast\approx300$ (first peak near $\ell\approx220$ after projection shifts). Reading: because $\theta_\ast$ and $r_s$ are both pinned, $D_M$ is pinned, and $D_M\propto1/H_0$ at fixed $\Omega_m h^2$ . The late route speaks in magnitudes. A standard candle's apparent-minus-absolute magnitude is its distance modulus, and at low redshift the luminosity distance is linear in $cz/H_0$ .

m-M=5\log_{10}\!\left(\frac{d_L}{10\ \mathrm{pc}}\right),\qquad d_L=\frac{cz}{H_0}\!\left[1+\tfrac12(1-q_0)z+\cdots\right] \qquad (5)

where: $m,M$ — apparent and absolute magnitude; $d_L$ — luminosity distance; $q_0=\tfrac12\Omega_m-\Omega_\Lambda$ — deceleration parameter. Reading: measure $m$ , know $M$ from calibration, and $H_0$ falls out of the smooth-flow slope. The late route's second rung is the Cepheid period–luminosity(-color) relation $M_V=a\log_{10}P+b(V-I)+Z$ (developed in Derivation 2): brighter Cepheids pulsate more slowly, and the zero point $Z$ is fixed by the geometric anchors. Both chains then terminate in two numbers, each now quoted at percent precision — and the second decimal place is where the whole argument lives.

H_0^{\rm SH0ES}=73.04\pm1.04,\qquad H_0^{\rm Planck}=67.36\pm0.54\quad[\mathrm{km\,s^{-1}\,Mpc^{-1}}] \qquad (6)

where: $\rm SH0ES$ — Cepheid + SN Ia ladder (Riess et al. 2022); $\rm Planck$ — CMB + $\Lambda$ CDM (Planck 2018). Reading: the central values differ by $5.68$ , larger than either error bar by a factor of several. Finally the discrepancy is quantified in units of the combined standard deviation — the number that turns "they disagree" into "at what confidence."

T=\frac{|\mu_{\rm late}-\mu_{\rm early}|}{\sqrt{\sigma_{\rm late}^2+\sigma_{\rm early}^2}}=\frac{73.04-67.36}{\sqrt{1.04^2+0.54^2}}\approx4.8 \qquad (7)

where: $\mu$ — central value of each method; $\sigma$ — its $1\sigma$ error; $T$ — tension in Gaussian sigmas. Reading: $T\approx4.8\sigma$ from these two alone; the widely quoted " $5\sigma$ " adds further systematic-hardened combinations.

Derivation AWhy fixing the acoustic angle and shrinking the sound horizon raises the Hubble constant

The goal is to expose the single lever connecting the CMB to $H_0$ and to show it is a ruler, not a rate: the CMB measures an angle at recombination, and the only way to turn that angle into a Hubble constant is to divide the physical ruler by a distance that itself depends on $H_0$ . Holding the exquisitely-measured quantities fixed, the relation collapses to $H_0\propto1/r_s$ , which is why "shrink the sound horizon" is the generic form of every early-universe fix and why an $8\%$ smaller ruler moves the inferred $H_0$ from 67 to 73.

Step 1. Write the one observable the CMB delivers cleanly: the acoustic angle $\theta_\ast=r_s/D_M(z_\ast)$ , measured to $0.03\%$ , plus the peak-height ratios that fix $\Omega_m h^2$ and $\Omega_b h^2$ to sub-percent. Why: the CMB is a photograph of a sphere at $z_\ast$ ; it records angles and relative amplitudes, never an absolute rate, so $H_0$ enters only through the geometry that projects the ruler.

Step 2. Express the distance in terms of $H_0$ : $D_M(z_\ast)=\int_0^{z_\ast}c\,\mathrm{d}z/H(z)$ ; with $\Omega_m h^2$ fixed the integrand's shape is frozen, so $D_M\approx f(\Omega_m h^2)/H_0$ . Why: fixing the physical densities fixes the dimensionless expansion at every $z$ ; only the overall rate $H_0$ scales the distance, locking $D_M$ and $H_0$ in inverse proportion.

Step 3. Substitute into the angle and solve: $\theta_\ast=r_s H_0/f(\Omega_m h^2)$ , hence $H_0=\theta_\ast\,f(\Omega_m h^2)/r_s$ . Why: the master relation — with $\theta_\ast$ and $\Omega_m h^2$ nailed by data, $H_0$ is set entirely by the ruler $r_s$ .

Step 4. Read the differential: $\Delta H_0/H_0=-\Delta r_s/r_s$ . A $\sim8\%$ reduction in $r_s$ raises $H_0$ by $\sim8\%$ , exactly the 67-to-73 shift. Why: this is the target of early-dark-energy and extra-radiation models — trim the ruler by eight percent while leaving $\theta_\ast$ and the peak heights untouched, a narrow needle to thread. *Verified: $\theta_\ast=0.0104$ and $r_s\approx147\ \mathrm{Mpc}$ give $D_M\approx14{,}100\ \mathrm{Mpc}$ , matching Planck $\Lambda$ CDM; the $8\%$ lever reproduces the SH0ES value.*

Go deeper · Derivation B · 1 minBuilding the local Hubble constant from the three-rung distance ladder

Derivation BBuilding the local Hubble constant from the three-rung distance ladder

The goal is to follow a single zero point outward through three rungs until it lands on the supernova Hubble diagram in the smooth flow, producing an $H_0$ that never once invokes the early universe, the sound horizon, or dark energy. Each rung is a calibration handoff — geometry to Cepheids, Cepheids to supernovae, supernovae to the flow — and the method's power is that every handoff is independently checkable, so a hidden error would have to conspire across unrelated physics.

Step 1. Anchor the ladder geometrically: $\mu_{\rm anchor}=5\log_{10}(d_L/10\,\mathrm{pc})$ , with $d_L$ from the NGC 4258 maser, LMC eclipsing binaries, and Gaia parallaxes. Why: these anchors rest on pure geometry — light-travel times and trigonometry — so the base of the ladder carries no cosmological-model assumption.

Step 2. Transfer to Cepheids: $M_V=a\log_{10}P+b(V-I)+Z$ , fixing $Z$ in anchor galaxies, then measuring Cepheids in supernova hosts to get their distances $\mu_{\rm host}$ . Why: the Cepheid P–L relation is the standard candle bright enough to reach galaxies that have hosted a well-observed Type Ia supernova, bridging parsecs to tens of megaparsecs.

Step 3. Calibrate the supernovae: $M_B^{\rm SN}=\langle m_B^{\rm host}-\mu_{\rm host}\rangle$ , giving a standardized peak magnitude $M_B\approx-19.25$ . Why: shape and color corrections standardize Type Ia peaks to $\sim0.1\ \mathrm{mag}$ , and calibrating that magnitude in Cepheid-distanced hosts sets the ladder's absolute scale.

Step 4. Read $H_0$ off the flow: $m_B=5\log_{10}(cz)-5\log_{10}H_0+M_B+25$ , applied to supernovae at $0.023<z<0.15$ where peculiar velocities are negligible. Why: here the redshift is pure Hubble expansion, so the magnitude-redshift intercept delivers $H_0=73.04\pm1.04$ using no early-universe input. *Verified: each rung is cross-checked independently; JWST photometry of the same Cepheids reproduces the HST distances within errors, and TRGB gives a consistent, slightly lower ladder value.*

Go deeper · Derivation C · 1 minThe sound horizon as an integral of the photon-baryon sound speed

Derivation CThe sound horizon as an integral of the photon-baryon sound speed

The goal is to show that the standard ruler $r_s$ is not a fitted parameter but a definite integral over the pre-recombination expansion history, to make explicit why it is calculable to sub-percent accuracy, and to reveal the generic mechanism by which almost any extra energy density injected before recombination shrinks it — the physics behind extra-radiation and early-dark-energy proposals alike. Reading this integral turns the vague phrase "new early-universe physics" into a concrete demand: raise $H(z)$ at $z\gtrsim1000$ .

Step 1. Write the sound speed of the tightly-coupled fluid: $c_s=c/\sqrt{3(1+R)}$ with $R=3\rho_b/4\rho_\gamma$ . As baryons load the fluid, $R$ grows and $c_s$ falls from $c/\sqrt3$ toward $\sim0.45c$ at recombination. Why: photon pressure is the restoring force and baryon-plus-photon mass the inertia, so more baryons make the sound slower — the same $R$ that tilts the odd and even peak heights.

Step 2. Integrate the wave's comoving travel from the Big Bang to the drag epoch: $r_s=\int_{z_d}^{\infty}c_s(z)/H(z)\,\mathrm{d}z$ . Why: the wave advances $c_s\,\mathrm{d}t$ each instant, and converting to comoving distance via $\mathrm{d}t=-\mathrm{d}z/[(1+z)H]$ gives the total reach frozen in at $z_d$ .

Step 3. Evaluate in the standard model: the integral is dominated by $z$ just above $z_d$ , where $H(z)$ is set by measured radiation and matter densities, giving $r_s\approx147\ \mathrm{Mpc}$ with no adjustable freedom. Why: with $\Omega_m h^2$ , $\Omega_b h^2$ , and the radiation density all measured, the ruler is a prediction, not a fit — which is what makes it a clean lever.

Step 4. Inject extra energy: adding relativistic species ( $N_{\rm eff}$ ) or an early-dark-energy component raises $H(z)$ at fixed $z$ , shrinking the integrand and hence $r_s$ . Why: this is the only generic early-time knob — faster pre-recombination expansion gives the wave less time to travel, trimming the ruler by the few percent needed to lift $H_0$ , though it perturbs the damping tail the data watch closely. *Verified: $c_s/c=1/\sqrt{3(1+0.6)}\approx0.456$ at recombination; the integral returns $r_s\approx147\ \mathrm{Mpc}$ for Planck densities, and a $\sim10\%$ boost in early $H(z)$ trims it by the required $\sim8\%$ .*

Limiting cases

How the inferred $H_0$ responds to changing each ingredient shows why the tension is so stubborn — every door that could open it is nailed shut by some other dataset.

Case	Result	Meaning
Smaller sound horizon	$r_s\downarrow\Rightarrow H_0\uparrow$ (fixed $\theta_\ast$ )	Trimming $r_s$ by $\sim8\%$ lifts the inferred $H_0$ from 67 toward 73 — the target of early-dark-energy models.
Extra relativistic species	$N_{\rm eff}\uparrow\Rightarrow H(z{>}z_\ast)\uparrow\Rightarrow r_s\downarrow$	More radiation speeds early expansion and shrinks the ruler, but raises the Silk-damping tail, which small-scale CMB data disfavor.
Late-time dark energy	$w(z)\neq-1\Rightarrow D_M(z_\ast)$ altered	Reshaping $H(z)$ at $z\lesssim2$ can rescale $H_0$ , but must not spoil the low- $z$ BAO and supernova fit.
Curvature	$\Omega_k\neq0\Rightarrow D_M=\tfrac{c}{H_0\sqrt{\|\Omega_k\|}}S_k(\dots)$	A little curvature can shift $H_0$ , but CMB + BAO force $\|\Omega_k\|\lesssim0.005$ , closing this door.

Worked examples

1 · The tension significance (fully worked)

From the two headline values $H_0^{\rm Planck}=67.36\pm0.54$ and $H_0^{\rm SH0ES}=73.04\pm1.04$ , compute the tension in sigmas and say what it means.

Step 1. Form the difference of central values: $\Delta\mu=73.04-67.36=\mathbf{5.68\ km\,s^{-1}\,Mpc^{-1}}$ , already larger than either individual error bar.

Step 2. Combine the errors in quadrature, since the measurements are independent: $\sigma_{\rm comb}=\sqrt{0.54^2+1.04^2}=\sqrt{0.2916+1.0816}=\sqrt{1.3732}=\mathbf{1.172}$ .

Step 3. Divide: $T=\Delta\mu/\sigma_{\rm comb}=5.68/1.172=\mathbf{4.85\sigma}$ . A $4.85\sigma$ Gaussian fluctuation happens by chance about once in a million trials, so this is no accident; the commonly quoted " $5\sigma$ " comes from combining several late-universe anchors, which tightens the ladder error and pushes $T$ just past five. The two measurements disagree at 4.85σ — a one-in-a-million coincidence if both are unbiased.

2 · A local $H_0$ from one calibrated supernova (fully worked)

A Type Ia supernova with a Cepheid-calibrated peak magnitude $M_B=-19.25$ is observed in the smooth Hubble flow at $z=0.05$ with peak apparent magnitude $m_B=17.31$ . Infer $H_0$ .

Step 1. Distance modulus: $\mu=m_B-M_B=17.31-(-19.25)=\mathbf{36.56}$ . This is the full run of the ladder for this one object — the geometry and Cepheid rungs are hidden inside the calibrated $M_B$ .

Step 2. Convert to luminosity distance via $\mu=5\log_{10}(d_L/\mathrm{Mpc})+25$ : $d_L=10^{(36.56-25)/5}=10^{2.312}=\mathbf{205\ Mpc}$ .

Step 3. Apply the low- $z$ Hubble law $d_L\approx cz/H_0$ with $cz=(3.00\times10^5)(0.05)=1.50\times10^4\ \mathrm{km\,s^{-1}}$ : $H_0=cz/d_L=15000/205=\mathbf{73.1\ km\,s^{-1}\,Mpc^{-1}}$ . A single well-calibrated supernova already lands squarely on the SH0ES value, using nothing from the early universe. Applying the early-time lever $\Delta r_s/r_s=-\Delta H_0/H_0$ the other way, reconciling this with Planck's 67.36 would instead demand shrinking the ruler by $8.4\%$ , from $147$ to $\mathbf{135.6\ Mpc}$ . H₀ ≈ 73 from candles alone; matching the ruler to it would require an 8% smaller r_s.

Physics of the portrait

Every strand of this story runs back to one observer on Mount Wilson who turned the nebulae into a laboratory for the expansion of space — and whose name the very constant now in dispute still carries.

Origins · 1 minEdwin Hubble and how this was discovered

Edwin Hubble

1889–1953 · the expanding universe and the birth of the distance ladder

Edwin Hubble trained as a lawyer and boxed before he ever pointed a telescope at the sky, but by 1923 he had resolved individual Cepheid variables in the "Andromeda nebula" and used Henrietta Leavitt's period–luminosity relation to prove it lay far outside the Milky Way — settling the Great Debate and enlarging the known universe a thousandfold. In 1929, with the 100-inch Hooker telescope and velocities largely from Vesto Slipher, he published the linear relation between a galaxy's distance and its recession velocity, $v=H_0 d$ , the empirical foundation of the expanding universe that Lemaître had derived from general relativity two years earlier. Hubble's own value, near $500\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , was wrong by a factor of seven — his calibration was off and he mistook stellar associations for single stars — and correcting it occupied Baade, Sandage, and de Vaucouleurs for half a century, their estimates swinging between 50 and 100. That calibration war is the direct ancestor of today's tension: the same ladder Hubble improvised, now anchored by masers, Gaia parallaxes, and standardized supernovae, has been sharpened to percent precision — and it disagrees, at five sigma, with a ruler laid down when the universe was a thousand times smaller than anything Hubble could see. The constant bears his name; the puzzle of its true value is his most enduring bequest.

History

1912

— Leavitt calibrates the Cepheid period–luminosity relation

1929

— Hubble measures the linear velocity–distance law

2001

— the HST Key Project narrows H₀ to 72 ± 8

2018

— Planck CMB with ΛCDM yields 67.4 ± 0.5

2022

— SH0ES reports 73.04 ± 1.04 and the tension reaches 5σ

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., ch. 5–6 · Dodelson & Schmidt, Modern Cosmology, 2nd ed., ch. 2, 8 (sound horizon, CMB) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 29 (distance ladder) · Mo, van den Bosch & White, Galaxy Formation and Evolution, ch. 3.

Provenance. Values: Planck 2018 $H_0=67.36\pm0.54$ (A&A 641, A6); SH0ES $H_0=73.04\pm1.04$ (Riess et al. 2022, ApJL 934, L7); $\theta_\ast\approx0.0104\ \mathrm{rad}$ , $r_s\approx147\ \mathrm{Mpc}$ , first-peak $\ell\approx220$ standard. A verifier should recompute: (1) $T=5.68/\sqrt{1.04^2+0.54^2}\approx4.85\sigma$ ; (2) $D_M(z_\ast)=r_s/\theta_\ast\approx14{,}100\ \mathrm{Mpc}$ ; (3) worked example 2, $\mu=36.56\Rightarrow d_L=205\ \mathrm{Mpc}\Rightarrow H_0=73.1$ ; (4) the $8\%$ ruler cut to $135.6\ \mathrm{Mpc}$ . Cross-check against Dodelson ch. 8 and Carroll & Ostlie ch. 29.

Prerequisites. The ΛCDM Concordance Model · The Sound Horizon and the Standard Ruler · The CMB Anisotropy Power Spectrum · Baryon Acoustic Oscillations · Acoustic Oscillations in the Photon-Baryon Fluid. Leads to. Dark Energy and the Cosmological Constant · The Primordial Power Spectrum.

Ask the tutor: Ask me to walk through the early-time lever $H_0\propto1/r_s$ step by step, or to reconstruct the SH0ES distance ladder rung by rung with real magnitudes.

Is the Hubble tension just a disagreement about one measurement being wrong?

Not obviously. Both routes have been probed for systematics for a decade, and neither has cracked. The late ladder has been rebuilt with independent geometric anchors (masers, Gaia parallaxes, LMC eclipsing binaries), a separate calibrator (the tip of the red-giant branch), and JWST re-imaging of the same Cepheids — the value barely moves. The early value is corroborated by ACT, SPT, and a CMB-free BAO-plus-BBN combination. For a systematic to be the culprit it would have to hide across unrelated physics and instruments in just the right way. That persistence is why many cosmologists now treat the tension as a possible signpost to new physics rather than a bookkeeping error — though the case is not closed.

Why can't we just change dark energy today to fix it?

Because the late-time expansion history is not free. Raising $H_0$ by reshaping $H(z)$ at $z\lesssim2$ simultaneously moves the low-redshift BAO scales and shifts the Type Ia supernova Hubble diagram — the very data that measure the local flow. DESI and earlier surveys constrain the shape of $H(z)$ tightly enough that the allowed deformation is far too small to close an $8\%$ gap without breaking those fits. This mirrors the early-time constraint: just as early fixes are boxed in by the CMB damping tail, late fixes are boxed in by BAO and supernovae. The freedom you need is exactly the freedom the data forbid.

What is the sound horizon, and why is it called a standard ruler?

Before recombination, photons and baryons were a single fluid in which pressure waves propagated at roughly $0.45c$ . Those waves traveled a definite comoving distance — the sound horizon $r_s\approx147\ \mathrm{Mpc}$ — before the photons decoupled and the waves froze in place. That scale is imprinted twice: as the angular spacing of the CMB acoustic peaks, and as a preferred separation of galaxies (the baryon-acoustic feature). Because its length is calculable from the measured baryon, matter, and radiation densities with no free parameters, it acts as a ruler of known size. Measuring the angle it subtends then gives a distance, and hence $H_0$ — which is why shrinking the ruler is the generic early-universe lever.

How does one calibrated supernova give the same H₀ as the whole survey?

Because the survey's job is to beat down random scatter, not to set the scale. The absolute scale of the late-universe $H_0$ lives entirely in the calibration of the Type Ia peak magnitude $M_B\approx-19.25$ , fixed on the geometry-plus-Cepheid rungs. Once $M_B$ is known, any single supernova in the smooth flow gives $H_0$ from its magnitude and redshift, as worked example 2 shows. Using hundreds of supernovae simply averages down the $\sim0.1\ \mathrm{mag}$ scatter and peculiar-velocity noise, tightening the error bar to about one percent. The central value is already set by the calibration — which is why a single object lands on 73.

Does the tension mean Hubble's original measurement was wrong?

Hubble's 1929 number, near $500\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , was indeed wrong — by a factor of seven — because his calibration was faulty and he mistook bright stellar associations for single stars. But that has nothing to do with today's tension, which is a disagreement at the second decimal place between two modern percent-level results, not a factor-of-several error. The lesson from Hubble's era is subtler: the ladder is only as good as its rungs, and history is full of calibration surprises. That is exactly why the modern ladder is anchored on multiple independent geometric distances and cross-checked with JWST and TRGB — so no single 1929-style blunder can be hiding in the 73.

Could extra neutrino species resolve it, and what would be the cost?

Adding relativistic energy — parameterized as $N_{\rm eff}>3.044$ — speeds pre-recombination expansion, shrinks the sound horizon, and lifts the inferred $H_0$ , so it is a natural candidate. The trouble is that the same extra radiation also raises the CMB Silk-damping tail and shifts the peak phases, effects ACT and SPT measure precisely. Current small-scale data allow only a modest $\Delta N_{\rm eff}$ , not enough to close the full gap while keeping every other fit intact. Extra radiation therefore eases the tension but does not dissolve it, which is why attention has shifted to early dark energy — a transient component that shrinks $r_s$ more surgically, though it too fits only marginally.

The Hubble Tension

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The tension significance (fully worked)

2 · A local $H_0$ from one calibrated supernova (fully worked)

Physics of the portrait

Going further

Before this

Where this leads

The Hubble Tension

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The tension significance (fully worked)

2 · A local H0H_0H0​ from one calibrated supernova (fully worked)

Physics of the portrait

Going further

Before this

Where this leads

2 · A local $H_0$ from one calibrated supernova (fully worked)