# Hubble's law **Hubble's law** is the observation that distant galaxies recede from us at a speed proportional to their distance, `v = H_0·d`, with `H_0` in kilometres per second per megaparsec. It is the most direct evidence that the universe is expanding, and is often called the Hubble–Lemaître law, since Georges Lemaître derived it from [[General_relativity|general relativity]] two years before Edwin Hubble published the measurement.[^hubble1929][^lemaitre1927] It does not put us at a centre: in a uniformly stretching space every observer measures the same proportionality. In the microsim below the reader sets `H_0` from 50 to 100 km/s/Mpc and clicks any galaxy to make it the observer. The velocity–distance line is redrawn from that galaxy's point of view and comes out the same — the demonstration that `v = H_0·d` holds for everyone at once. The Hubble time `1/H_0` reads 14.0 billion years at `H_0` = 70 (derived), and the chosen galaxy's [[Balmer_series|Balmer lines]] slide by `lambda_obs = lambda_em·(1 + z)`, so hydrogen-alpha at 6562.8 Å is seen at 7219 Å when `z` = 0.1 (derived): the reader reads the [[Redshift|redshift]] off a [[Spectroscopy|spectrum]] rather than being told it.[^raven-redshift] Hubble's 1929 points sit beside the modern data, their slope seven times too steep.[^hubble1929] On the [[Physics]] flagship this page serves Part IV, Branches and fields, in the section *The expanding universe*, where [[Physical_cosmology|cosmology]] enters as a measurement. Everything downstream — the [[Big_Bang|Big Bang]], the age of the universe, the [[Cosmic_microwave_background|microwave background]] at 2.725 K, [[Big_Bang_nucleosynthesis|nucleosynthesis]], [[Dark_energy|dark energy]] — is calibrated against `H_0`.[^up3-ch11] ## Discovery The law needed three things at once: spectra of the faint spiral nebulae, distances to them, and a theory in which expanding space made sense. The spectra came first, from one observer through the 1910s; the distances a decade later, when the nebulae proved to be galaxies; the theory was already written down and disbelieved. ### Slipher's observations At the Lowell Observatory, Vesto Slipher photographed spiral-nebula spectra with long exposures and measured the displacement of their [[Absorption_spectroscopy|absorption lines]]. In 1913 he reported the Andromeda nebula approaching at about 300 km/s, the largest velocity then known for any object.[^slipher1913] By 1917 he had some twenty-five spirals: a few approach, most recede, several above 1,000 km/s.[^slipher1917] He had the velocities years before anyone had distances to pair with them. ### FLRW equations In 1922 Alexander Friedmann solved the [[Einstein_field_equations|Einstein field equations]] for a universe homogeneous and isotropic but not required to be static, and found it must expand or contract.[^friedmann1922] The geometry uses a scale factor `a(t)` multiplying a fixed comoving grid, so the proper distance between two galaxies is `d(t) = a(t)·x` with `x` constant. Differentiating gives `v = (a'/a)·d` at once: proportionality is a consequence of uniformity in [[Spacetime|spacetime]], not an assumption. ### Lemaître's equation Working independently in Belgium, Georges Lemaître published in 1927 an expanding solution of the same equations and went further by connecting it to data, combining Slipher's velocities with the distances then available to get a linear velocity–distance relation with an estimated constant.[^lemaitre1927] The paper appeared in a little-read Belgian journal, and the 1931 English translation omitted the paragraphs containing that estimate — part of why the result was long credited to Hubble alone. ### Cepheid variable stars outside the Milky Way Distances came from a rung Henrietta Leavitt laid in 1912: among the Cepheid variables of the Small Magellanic Cloud the pulsation period tracks the apparent brightness, and since all lie at nearly one distance the real relation must be between period and luminosity.[^leavitt1912] A Cepheid's period therefore gives its intrinsic brightness, which compared with the brightness seen gives the distance. With the 100-inch telescope at Mount Wilson, Hubble found Cepheids in several spirals in the mid-1920s, far beyond the [[Star|stars]] of our own galaxy.[^hubble1929] ### Combining redshifts with distance measurements Hubble's 1929 paper is three pages long and carries twenty-four galaxies with both a distance and a radial velocity.[^hubble1929] The distances come from Cepheids where he had them and from the brightest individual stars where he did not; the velocities are almost all Slipher's. Plotted together they scatter widely but rise, and Hubble fitted a line through the origin, obtaining roughly 500 km/s/Mpc from two slightly different solutions. The sample lies within about 2 Mpc and below about 1,100 km/s. That slope is seven times the modern value, and the error is entirely in the distances. His calibration conflated two classes of Cepheid with different period–luminosity relations, and the brightest stars he used in remoter galaxies were often clouds of glowing gas, brighter still. Both mistakes made galaxies seem nearer, and a distance too small with a velocity that is right gives a slope too large.[^freedman2001] The sim plots the 1929 and modern points on one pair of axes: the velocities agree, the distances do not. ### Cosmological constant abandoned Einstein had put the cosmological constant Λ into the field equations in 1917 expressly to permit a static universe, since without it no static solution containing matter exists.[^einstein1917] Once the expansion was established the term had nothing to do, and [[Albert_Einstein|Einstein]] dropped it. It returned seventy years later, when the expansion proved to be accelerating, and Λ is now the leading description of dark energy.[^riess1998][^perlmutter1999] ## Interpretation The commonest misreading is that galaxies fly apart through a fixed space like fragments of an explosion. Relativistically they are nearly at rest in the comoving grid and it is the distances between them that grow, so the redshift is not a [[Doppler_effect|Doppler shift]] but a stretching of every wavelength with the scale factor while the light is in flight — a distinction that matters at large redshift and not at small. ### Redshift velocity and recessional velocity What a spectrograph measures is the dimensionless redshift `z = lambda_obs/lambda_em - 1`; converting it to a velocity is a convention.[^raven-redshift] The redshift velocity `cz` is the non-relativistic reading quoted for nearby galaxies, a restatement of the measurement rather than a claim about motion; above about `z` = 0.1 it approximates no velocity well.[^raven-redshift] The recessional velocity is instead the rate of change of proper distance, `v = H(t)·D`, and it is this that Hubble's law contains. It exceeds the [[Speed_of_light|speed of light]] beyond the Hubble length, contradicting nothing in [[Special_relativity|special relativity]]: no signal overtakes another in any local frame, because the growth is of the space between. ### Expansion velocity vs. peculiar velocity Galaxies also move through the comoving grid, pulled by the [[Gravity|gravity]] of neighbours and the [[Dark_matter|dark matter]] around them, and these peculiar velocities of a few hundred kilometres per second add to the Hubble flow; Andromeda's approach is one.[^slipher1913] They dominate nearby and fade only beyond a few tens of megaparsecs — which governs how a distance ladder is built. ### Time-dependence of Hubble parameter `H` is not a constant of nature but a function of time, `H(t) = a'/a`; `H_0` is its value now. Where matter alone slows the expansion `H` falls as `2/(3·t)`; where a cosmological constant dominates it approaches `sqrt(Lambda·c^2/3)`. Ours has passed from the first regime to the second, so `H` still decreases while the expansion accelerates — compatible because `H` is a ratio, not a speed. ### Idealized Hubble's law The idealisation the sim enacts is a perfectly uniform expansion with no peculiar velocities. Let every galaxy sit at a fixed comoving position, so the separation of galaxies `i` and `j` is `r_ij(t) = a(t)·x_ij`. Differentiating, `r_ij' = (a'/a)·r_ij = H·r_ij` for every pair without exception. The law therefore holds for each galaxy about every other, with the same `H`, and no measurement any of them makes can locate a centre. That is the sim's central control. Clicking a galaxy makes it the observer and recomputes every velocity arrow relative to it: the arrows change completely, the velocity–distance plot does not. Selecting a galaxy also drives the spectrum panel, where the four Balmer lines at 6562.8, 4861.3, 4340.5 and 4101.7 Å are drawn at rest and again at `lambda_em·(1 + z)`, all displaced by one factor.[^raven-redshift] ILLUSTRATIVE: the sim's galaxies are drawn uniformly at random with no peculiar velocities, so its relation is exact where a real one scatters by several hundred km/s. ### Ultimate fate and age of the universe Extrapolating backwards gives a finite age and forwards a fate, both depending on the matter and dark-energy densities as well as `H_0`. With the parameters from the microwave background the age is 13.8 billion years, and because dark energy now dominates, the expansion continues without limit towards the [[Heat_death_of_the_universe|heat death]].[^planck2018][^up3-ch11] ### Acceleration of the expansion In 1998 and 1999 two teams found distant Type Ia [[Supernova|supernovae]] fainter, hence farther, than a decelerating universe predicts: the expansion has been speeding up for several billion years.[^riess1998][^perlmutter1999] The deceleration parameter `q_0 = Omega_m/2 - Omega_Lambda` is about −0.53 for the standard `Omega_m` = 0.315 and `Omega_Lambda` = 0.685 (derived).[^planck2018] Perlmutter, Schmidt and Riess shared the 2011 Nobel Prize in Physics for it.[^nobel2011] ## Derivation of the Hubble parameter The Hubble parameter is `H = a'/a`, and the Friedmann equation for a homogeneous isotropic universe gives it from the contents: `(a'/a)^2 = (8·pi·G/3)·rho - k·c^2/a^2 + Lambda·c^2/3`. Dividing by `H_0^2` and writing each contribution as a fraction of the critical density gives the working form `H(a) = H_0·sqrt(Omega_r·a^-4 + Omega_m·a^-3 + Omega_k·a^-2 + Omega_Lambda)` with `a` = 1 today, so the parameters sum to one. The critical density `rho_c = 3·H_0^2/(8·pi·G)` is 9.2×10⁻²⁷ kg/m³ at `H_0` = 70, about five and a half hydrogen atoms per cubic metre (derived).[^up3-ch11][^si-units] Each term falls off at its own rate, making the history of `H` a succession of regimes. ### Matter-dominated universe (with a cosmological constant) Neglecting radiation and curvature leaves `H(a) = H_0·sqrt(Omega_m·a^-3 + Omega_Lambda)`, which integrates to `a(t) = (Omega_m/Omega_Lambda)^(1/3)·sinh^(2/3)(3·sqrt(Omega_Lambda)·H_0·t/2)`. Early on the hyperbolic sine reduces to its argument and `a` grows as `t^(2/3)`; late on it becomes an exponential and `H` settles at `H_0·sqrt(Omega_Lambda)`. ### Matter- and dark energy-dominated universe The crossover follows from setting the two terms equal. With `Omega_m` = 0.315 and `Omega_Lambda` = 0.685 they match at `a = (Omega_m/Omega_Lambda)^(1/3)` = 0.77, that is `z` = 0.30; acceleration starts earlier, when `a'' = 0`, which needs `Omega_m·a^-3 = 2·Omega_Lambda` and gives `a` = 0.61, or `z` = 0.63 (derived).[^planck2018] The supernovae that revealed the acceleration lie on both sides of that transition, which is why they could separate the regimes.[^riess1998] ## Units derived from the Hubble constant Having dimensions of inverse time, `H_0` defines a characteristic time, length and volume with no further input. None is a physical boundary; each is a scale set by the present expansion rate. ### Hubble time The Hubble time is `t_H = 1/H_0`. Converting the mixed units — one megaparsec is 3.0857×10¹⁹ km — gives 4.41×10¹⁷ s, or 14.0 billion years, at `H_0` = 70; the slider's 50 to 100 spans 19.6 down to 9.8 billion years (derived).[^si-units] It is the age the universe would have had it always expanded at the present rate; the true age of 13.8 billion years is close only by near-coincidence, early deceleration and recent acceleration almost cancelling in the integral.[^planck2018] ### Hubble length Multiplying by the speed of light gives the Hubble length `c/H_0`, which is 4,280 Mpc at `H_0` = 70, about 14 billion light-years (derived).[^si-units] It is the distance at which the idealised law's recessional velocity reaches `c` — not the edge of the observable universe, which lies much farther out, because light now arriving from there was emitted when it was far closer. ### Hubble volume The Hubble volume is the sphere of that radius, `(4/3)·pi·(c/H_0)^3`, about 3.3×10¹¹ cubic megaparsecs at `H_0` = 70 (derived). It serves as a unit of cosmic volume and a marker of the region within which the linear law needs no correction. ## Determining the Hubble constant Measuring `H_0` means getting a distance and a redshift for one object, and the redshift is the easy half. Every method is therefore a way of getting distances, in two families: ladders built outward from locally calibrated objects, and inferences from the early universe read through a model. ### Earlier measurements After Hubble's 500 km/s/Mpc the value fell repeatedly as the distance scale was rebuilt, most sharply when the Cepheid period–luminosity relation proved to differ between two stellar populations, roughly doubling all extragalactic distances at a stroke. By the 1970s the field had split into camps advocating about 50 and about 100 km/s/Mpc. The Hubble Space Telescope Key Project, built to end the dispute, reported 72 ± 8 km/s/Mpc in 2001.[^freedman2001] ### Precision cosmology and the Hubble tension Two methods now reach better than two per cent and disagree. The microwave-background route uses the angular size of the acoustic peaks imprinted at recombination as a standard ruler and infers the present rate through the [[Lambda-CDM_model|ΛCDM model]]; Planck gives `H_0` = 67.4 ± 0.5 km/s/Mpc.[^planck2018] The distance-ladder route calibrates Type Ia supernovae with Cepheids and measures the expansion directly; SH0ES gives 73.04 ± 1.04 km/s/Mpc.[^riess2022] The gap of 5.6 km/s/Mpc is about five times the combined uncertainty (derived) — too large for accident, and known as the Hubble tension. A decade of work on crowding and metallicity corrections, on anchors and on foregrounds has found no error of that size on either side. ### Possible resolutions of the Hubble tension Either a systematic error has escaped that search, or ΛCDM is incomplete. Because the microwave value is model-dependent, the most discussed new physics acts before recombination and shortens the sound horizon — early dark energy, or extra relativistic species — raising the inferred `H_0` without disturbing the peaks. Late-time changes are harder to square with supernova data. Methods using neither ladder nor model are the natural arbiters: time delays between the images of a [[Gravitational_lens|lensed]] quasar, and [[Gravitational_wave|gravitational-wave]] standard sirens with identified hosts. Their uncertainties still exceed the gap.[^planck2018][^riess2022] ## Measurements of the Hubble constant The history of the constant is the history of the distance scale, and the table marks its turning points. The 1929 value is wrong by a factor of seven, not because the velocities were wrong but because the distances were; the 2001 value carries an eleven per cent uncertainty; the two modern values are under two per cent and do not overlap. | Year | Source | `H_0` (km/s/Mpc) | Distances from | |---|---|---|---| | 1929 | Hubble | ≈ 500 | Cepheids, brightest stars; 24 galaxies[^hubble1929] | | 2001 | HST Key Project | 72 ± 8 | Cepheid-calibrated indicators[^freedman2001] | | 2018 | Planck, within ΛCDM | 67.4 ± 0.5 | microwave-background acoustic peaks[^planck2018] | | 2022 | SH0ES | 73.04 ± 1.04 | Cepheids, Type Ia supernovae[^riess2022] | ## See also - [[Expansion_of_the_universe]] - [[Redshift]] — the Balmer strip the sim's spectrum panel draws - [[Big_Bang]] - [[Cosmic_microwave_background]] - [[Lambda-CDM_model]] - [[Physical_cosmology]] - [[Edwin_Hubble]] - [[Dark_energy]] - [[Supernova]] - [[General_relativity]] ## Notes `H_0` is the present value of a quantity that changes with time; where this page writes `H_0` without qualification it means the value now, and `H(t)` or `H(a)` where the history matters. The mixed unit km/s/Mpc is conventional rather than natural: dividing by the number of kilometres in a megaparsec, 3.0857×10¹⁹, converts it to inverse seconds, the form used in every equation here. All footnotes are collected under References. ## References [^hubble1929]: Hubble, E. (1929). "A relation between distance and radial velocity among extra-galactic nebulae." *Proceedings of the National Academy of Sciences* 15 (3): 168–173. https://doi.org/10.1073/pnas.15.3.168 — the twenty-four galaxies, the linear fit, and the two solutions for the constant near 500 km/s/Mpc. [^up3-ch11]: OpenStax (Sanny, J.; Ling, S. J.). *University Physics Volume 3* (2016), Ch. 11 Particle Physics and Cosmology, pp. 493–540 (page to pin): Hubble's law, the expansion and the Big Bang, the critical density, the microwave background and Olbers' paradox. https://openstax.org/books/university-physics-volume-3 [^raven-redshift]: Raven, W. *Atomic Physics for Everyone: An Introduction to Atomic Physics, Quantum Mechanics, and Precision Spectroscopy* (2025), pp. 88, 90 and 91 (Portal Book 046; sub-manual 04 §2.4): the astronomers' form `z = lambda_obs/lambda_em − 1` and `f_obs = f_em/(1 + z)` [p. 88], the warning that `v = cz` is a non-relativistic mapping [p. 90], and the Balmer rest wavelengths Hα 6562.8 Å, Hβ 4861.3 Å, Hγ 4340.5 Å, Hδ 4101.7 Å [p. 91]. Hα at 7219.1 Å for `z` = 0.1 is derived from these. [^slipher1913]: Slipher, V. M. (1913). "The radial velocity of the Andromeda Nebula." *Lowell Observatory Bulletin* 2 (8): 56–57. [^slipher1917]: Slipher, V. M. (1917). "Nebulae." *Proceedings of the American Philosophical Society* 56: 403–409 — the enlarged sample of spiral nebulae, predominantly receding. [^friedmann1922]: Friedmann, A. (1922). "Über die Krümmung des Raumes." *Zeitschrift für Physik* 10 (1): 377–386. [^lemaitre1927]: Lemaître, G. (1927). "Un univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques." *Annales de la Société Scientifique de Bruxelles* A47: 49–59. [^leavitt1912]: Leavitt, H. S.; Pickering, E. C. (1912). "Periods of 25 Variable Stars in the Small Magellanic Cloud." *Harvard College Observatory Circular* 173: 1–3. [^einstein1917]: Einstein, A. (1917). "Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie." *Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften*: 142–152. [^riess1998]: Riess, A. G.; et al. (1998). "Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant." *The Astronomical Journal* 116 (3): 1009–1038 (DOI to pin). [^perlmutter1999]: Perlmutter, S.; et al. (1999). "Measurements of Ω and Λ from 42 High-Redshift Supernovae." *The Astrophysical Journal* 517 (2): 565–586 (DOI to pin). [^nobel2011]: The Nobel Prize in Physics 2011 (Saul Perlmutter, Brian P. Schmidt, Adam G. Riess), "for the discovery of the accelerating expansion of the Universe through observations of distant supernovae". NobelPrize.org. https://www.nobelprize.org/prizes/physics/2011/summary/ [^freedman2001]: Freedman, W. L.; et al. (2001). "Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant." *The Astrophysical Journal* 553 (1): 47–72 (DOI to pin) — `H_0` = 72 ± 8 km/s/Mpc, and the review of how the extragalactic distance scale was rebuilt after Hubble. [^planck2018]: Planck Collaboration (2020). "Planck 2018 results. VI. Cosmological parameters." *Astronomy & Astrophysics* 641: A6 (DOI to pin) — `H_0` = 67.4 ± 0.5 km/s/Mpc, `Omega_m` = 0.315, `Omega_Lambda` = 0.685, an age of 13.8 Gyr, and a spatially flat geometry, all within ΛCDM. [^riess2022]: Riess, A. G.; et al. (2022). "A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team." *The Astrophysical Journal Letters* 934 (1): L7 (DOI to pin) — `H_0` = 73.04 ± 1.04 km/s/Mpc. [^si-units]: Bureau International des Poids et Mesures. *The International System of Units (SI)*, 9th ed. (2019) — the speed of light is exactly 299 792 458 m/s by definition. The parsec is the distance at which one astronomical unit subtends one arcsecond, 1 pc = 3.0857×10¹⁶ m, so 1 Mpc = 3.0857×10¹⁹ km; the Hubble time, Hubble length, Hubble volume and critical density quoted here are computed from these values (derived). ### Bibliography - OpenStax, *University Physics Volume 3* (2016), Ch. 11 Particle Physics and Cosmology — the open-access account of Hubble's law and the Big Bang at the level of a first course (Portal Book 079). - Raven, W. *Atomic Physics for Everyone* (2025), Ch. 4 Atoms in Motion, pp. 88–91 — the redshift relation and the Balmer rest wavelengths the sim's spectrum panel uses (Portal Book 046). - Hubble, E. (1929), *PNAS* 15: 168–173 — three pages, twenty-four galaxies, and the plot that opened the subject. ## External links - [*University Physics Volume 3*](https://openstax.org/books/university-physics-volume-3) — OpenStax, CC BY; Chapter 11 carries the cosmology background (Portal Book 079) - [Hubble, "A relation between distance and radial velocity among extra-galactic nebulae" (1929)](https://doi.org/10.1073/pnas.15.3.168) — the original paper, open access at PNAS - [The Nobel Prize in Physics 2011](https://www.nobelprize.org/prizes/physics/2011/summary/) — the accelerating expansion, with the lectures - Further sites are listed in the Wikipedia pair's *External links*; none is reproduced here until its URL has been checked. <!-- MATTERSIM:BEGIN g24 — Matter & Energy Cluster microsim (framework build, specs/sims/Hubble's_law.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework), pending deploy:** *Hubble's law* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Hubble's_law.html` is live. <!-- pending: <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Hubble's_law.html" data-title="Hubble's law"></div> --> <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Hubble's_law) : [Wikitube](https://en.wikitube.io/wiki/Hubble's_law) · pinned revision [1373657064](https://en.wikipedia.org/w/index.php?oldid=1373657064) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P70 · sim pending (matter/Hubble's_law).*