How to compute Hubble's law and cosmological redshift: discovered by Edwin Hubble (USA Mt Wilson 1929) demonstrating that distant galaxies recede with velocity proportional to distance, evidence of Universe expansion. Hubble's formula: v = H_0 * d, where v recession velocity (km/s), H_0 Hubble constant (67.4 km/s/Mpc Planck 2020 vs 73 SH0ES local), d distance (Mpc). Redshift formula: z = (lambda_obs - lambda_em) / lambda_em, for small z: v ≈ c * z. Relativistic formula: v = c * ((1+z)² - 1) / ((1+z)² + 1). Essential tool for cosmology (Planck 2020 ESA, JWST 2021 NASA-ESA-CSA), dark energy discovery (Riess-Perlmutter-Schmidt Nobel 2011), Big Bang theory, stellar/galactic distances, Hubble tension debate 67 vs 73 km/s/Mpc. Hubble time 1/H0 ~14.4 Gyr approximate Universe age. Lookback time = c*z/H_0 for small z.

Hubble's law and redshift formulas

Hubble's Law (1929): v = H_0 * d, recession velocity proportional to distance. H_0 = 67.4 km/s/Mpc (Planck 2020 CMB), 73 km/s/Mpc (SH0ES Riess 2016 supernovae local). 1 Mpc = 3.086e19 km = 3.262 Mly = 3.086e22 m. Example Andromeda M31 d=0.778 Mpc: v=70*0.778=54.5 km/s (actually approaching -110 km/s, local exception).

Redshift z: z = (lambda_observed - lambda_emitted) / lambda_emitted. For small z (z<0.1): v ≈ c*z. Example H-alpha 656.28 nm Cepheid at z=0.02: lambda_obs = 656.28*(1+0.02) = 669.4 nm. 13.1 nm difference measurable HARPS ESO spectrograph. Relativistic redshift: 1+z = sqrt((1+v/c)/(1-v/c)) corrected for v ~ c.

Hubble time/age: t_Hubble = 1/H_0 = 1/(70 km/s/Mpc * 1.022e-12 1/yr) = 13.97 Gyr (with H_0=70). Approximate Universe age, deceleration/acceleration modify. Current age Universe ~13.8 Gyr (Planck 2020 cosmology Lambda-CDM). Lookback time for small z: t_LB = z/H_0 ~ d/c.

Hubble tension: 5-sigma difference between H_0 measurements. CMB-derived (Planck 2020): 67.4±0.5 km/s/Mpc. Local distance-ladder (SH0ES Riess 2022): 73.0±1.4 km/s/Mpc. Implications: new physics beyond standard Lambda-CDM, variable dark energy, sterile neutrinos, modified gravity. Open debate cosmology 2020-2025.

General redshift-distance relation: for accurate cosmology requires Friedmann integral d_L = (1+z)*c*integral(dz'/E(z')), E(z)=sqrt(Omega_M*(1+z)^3 + Omega_Lambda). NED-IPAC Caltech cosmology calculator online. For z1: comoving distance, luminosity distance, angular diameter distance distinct.

History and context

Edwin Hubble (USA 1889-1953) Mt Wilson Observatory California 100-inch telescope in 1929 published "A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae" PNAS 15:168 demonstrating velocity-distance proportionality through Cepheids (pulsating variables) as distance indicators. Actually Vesto Slipher Lowell Observatory Flagstaff Arizona 1912 measured first redshifts spiral galaxies. Henrietta Swan Leavitt (USA Harvard 1908) discovered period-luminosity relation Cepheids. Georges Lemaitre (Belgium 1894-1966) "Big Bang" theory proposed 1927 before Hubble (recognized 2018 Hubble-Lemaitre law). Albert Einstein (Germany-USA 1879-1955) initially accepted static Universe "biggest blunder" cosmological constant 1917, after Hubble accepted expansion 1931. Walter Baade (Germany-USA 1893-1960) Mt Wilson 1944 corrected stars Population I/II doubling Cepheid distances. COBE Cosmic Background Explorer NASA 1989 Mather-Smoot Nobel 2006 perfect CMB spectrum. WMAP NASA 2001-2010 first high-resolution maps. Planck satellite ESA 2009-2013 5' resolution precision H_0=67.4±0.5. SH0ES Hubble Space Telescope Riess team Adam Riess Johns Hopkins University 2016+ Type Ia supernovae local distance ladder H_0=73.0. JWST James Webb Space Telescope NASA-ESA-CSA 2021-active: discoveries galaxies z>10 first 600 million years post Big Bang, JADES JWST Advanced Deep Extragalactic Survey GS-z14-0 z=14.32 record 2024.

USA contributions: NASA Goddard Space Flight Center Greenbelt MD WMAP COBE Planck collaboration, Caltech IPAC Pasadena Spitzer Hubble JWST. Industry global astrophysical instruments 5+ billion USD/year: Northrop Grumman Falls Church VA James Webb prime contractor, Lockheed Martin Bethesda MD Hubble integration. American Astronomical Society AAS publications. NSF National Science Foundation funding. SDSS Sloan Digital Sky Survey Apache Point Observatory New Mexico 2000-2025 catalog galaxies, DESI Dark Energy Spectroscopic Instrument Kitt Peak Arizona 2021-2026 mapping accelerated Universe expansion. Adam Riess Johns Hopkins shared Nobel Physics 2011 with Saul Perlmutter Berkeley LBL and Brian Schmidt Australian National University discovery dark energy 1998. American Astronomical Society AAS publications.

Redshift-distances astronomical objects table

• Milky Way galaxy (center): z=0, d=8 kpc=26000 ly
• Andromeda M31 Local Group: z=-0.001 (approaching), d=0.778 Mpc=2.5 Mly
• Galaxy M81 Bode: z=-0.000113, d=3.7 Mpc=12 Mly
• M101 Pinwheel: z=0.000804, d=6.4 Mpc=21 Mly
• Virgo Cluster (M87): z=0.0036, d=16.5 Mpc=53 Mly
• Coma Cluster: z=0.023, d=99 Mpc=320 Mly
• Quasar 3C 273 (Hubble galaxy): z=0.158, d=678 Mpc=2.2 Gly
• Hubble Deep Field 1995 galaxies: z=0.5-3, d=1-3 Gly
• Quasar SDSS J1148+5251: z=6.4, d~12.7 Gly
• Galaxy GN-z11 (Hubble 2016): z=11.1, d~13.4 Gly
• Galaxy JADES-GS-z14-0 (JWST 2024): z=14.32, d~13.5 Gly
• CMB Cosmic Microwave Background: z=1090, decoupling redshift
• Big Bang singularity: z infinite, d=14.4 Gly horizon
• H_0 Planck 2020: 67.4±0.5 km/s/Mpc
• H_0 SH0ES Riess 2022: 73.0±1.4 km/s/Mpc
• Universe age Lambda-CDM: 13.8 Gyr

Worked numerical examples

Example 1 - Andromeda M31 Local Group. Distance d=0.778 Mpc=2.5 Mly. Hubble v=H_0*d=70*0.778=54.5 km/s (predicted recession velocity). Actually M31 APPROACHES Milky Way with v=-110 km/s (Doppler blueshift), exception due to Local Group binding gravity. Future collision predicted 4.5 billion years "Milkomeda" merger. Local gravity dominates Hubble flow for d10 Mpc beyond Local Group.

Example 2 - Virgo Cluster M87. Distance d=16.5 Mpc=53 Mly. Hubble v=70*16.5=1155 km/s recession. HARPS ESO spectroscopic measurement confirms z=0.0036 corresponding to 1080 km/s (slightly under due to Local Velocity Anomaly Local Group). M87 black hole 6.5 billion M_sun first EHT photo 2019 (Event Horizon Telescope) Katie Bouman MIT algorithm. Virgo Cluster contains 1500+ galaxies, dominant M87 giant elliptical.

Example 3 - American primordial galaxy high z. z=6.5 (quasar SDSS J1030+0524 1999). Hubble velocity using relativistic formula: v=c*((1+z)²-1)/((1+z)²+1)=c*(56.25-1)/(56.25+1)=c*0.966=2.9e5 km/s = 0.966c. Distance d_LB = c*z/H_0 = 3e5*6.5/70 = 27857 Mpc = 90.9 Gly (linear approximation, not accurate at high z). Cosmology calculator: comoving distance 8.3 Gpc, lookback time 12.7 Gly. JWST recent discoveries: GN-z11 z=11.1 (2016 HST), GS-z14-0 z=14.32 (2024 JWST JADES record). NASA Goddard Greenbelt MD JWST mission control.

Example 4 - American H-alpha redshift Cepheid measurement. H-alpha line emitted 656.28 nm (hydrogen ground state n=3->n=2). Observed on distant galaxy 700.00 nm. Calculation: z = (700-656.28)/656.28 = 43.72/656.28 = 0.0666. Velocity (linear): v=c*z=3e5*0.0666=20000 km/s. Distance: d=v/H_0=20000/70=286 Mpc=932 Mly. Distant galaxy with active quasar or starburst. Typical measurement HARPS ESO Cerro La Silla, GTC OSIRIS Canary Islands, Keck HIRES Mauna Kea Hawaii. American Astronomical Society AAS publications standards.

Example 5 - American CMB Cosmic Microwave Background. z=1090 (matter-radiation decoupling epoch, 380000 yr after Big Bang). Original CMB emission wavelength: lambda_em = lambda_obs / (1+z) = 1.06 mm / 1091 = 0.97 um. CMB current temperature 2.725 K (Mather Nobel 2006), at z=1090: T_emission = 2.725*(1+1090) = 2973 K (orange visible red!). Lookback time: 13.8 Gyr almost Universe age. Cosmology Lambda-CDM Planck 2020: Omega_M=0.315, Omega_Lambda=0.685, Omega_b=0.049. NASA Goddard COBE 1989 Mather-Smoot Nobel 2006, WMAP 2001-2010 first detailed CMB maps.

Limits and considerations

1. Linear Hubble v=H_0*d valid only for small z (z<0.1, d<400 Mpc): at high z requires Friedmann integral.
2. Hubble tension: H_0 67.4 (Planck CMB) vs 73 (SH0ES local). 5-sigma discrepancy. Possible new physics.
3. Local gravity binding dominates Hubble flow for d<5 Mpc (Local Group, M31 approaching).
4. Cosmology Lambda-CDM: Omega_M=0.3, Omega_Lambda=0.7, accelerated expansion Universe (dark energy 68%).
5. Distance ladder Cepheids-supernovae-Hubble: cumulative systematic uncertainties each step.
6. Comoving vs proper vs luminosity distance: different definitions at z>1.
7. Cosmological redshift z != Doppler velocity z: spacetime expansion vs particle motion.
8. Local Group peculiar velocity (Local Velocity Anomaly) -627 km/s toward Centaurus.
9. K-correction: blueshifted/redshifted stellar spectrum modifies bolometric magnitude.
10. Cosmological time dilation: Type Ia lightcurve stretched (1+z) factor.

Common errors

1. Confusing z (redshift) with v (velocity): for z<0.1 relativistic required.
2. Demanding v>c possible for z>1: cosmological redshift NOT Doppler velocity.
3. Calculating d=v/H_0 for z>0.5: requires Friedmann cosmology integral.
4. Confusing Hubble time 1/H_0 ~14 Gyr with Universe age 13.8 Gyr: not identical.
5. Neglecting Local Group binding gravity: M31 approaching despite Hubble Law.
6. Using H_0=70 without specifying error 67.4 (Planck) vs 73 (SH0ES).
7. Confusing lookback time with comoving distance: at z=6 differences factor 2-3.
8. Calculating CMB distances with linear: redshift z=1090 cosmology integral required.
9. Neglecting K-correction spectrum: bolometric magnitude vs optical filter different.
10. Demanding Hubble tension resolved: open debate 2020-2025, possible new physics.

Numbered applications

1. Cosmology space telescopes: Hubble HST 1990-active, JWST 2021 NASA-ESA-CSA, Planck satellite ESA 2009-2013.
2. Type Ia supernovae distance ladder: Riess-Perlmutter-Schmidt Nobel Physics 2011 dark energy discovery.
3. Primordial cosmology: COBE 1989 NASA Mather-Smoot Nobel 2006, WMAP 2001-2010, Planck.
4. Astronomical spectrographs: HARPS ESO Cerro La Silla Chile, Keck HIRES Mauna Kea, GTC OSIRIS La Palma.
5. Future telescopes: ELT Extremely Large Telescope ESO 39 m 2027 Cerro Armazones, Roman Space Telescope NASA 2027.
6. Cosmological surveys: SDSS Sloan Digital Sky Survey 2000-active New Mexico, DESI Dark Energy Kitt Peak Arizona 2021.
7. USA NASA contributions: NASA Goddard Greenbelt MD mission control, JPL Pasadena CA.
8. Dark energy research: Lambda-CDM debate Phantom modified gravity, dark energy evolution w(z).
9. Hubble tension: open problem 2020-2025 Planck 67.4 vs SH0ES 73 km/s/Mpc, possible new physics.
10. Educational: cosmology courses MIT Caltech Princeton Stanford UT Austin, NSF research funding.

For support

For technical questions or reports: info@calcoi.com. Educational tool for preliminary cosmology. For professional research consult Hubble EP 1929 PNAS 15:168, Lemaitre G 1927 Annales SSB, Friedmann A 1922, Weinberg 'Cosmology' 2008 Oxford, Peacock 'Cosmological Physics' 1999 Cambridge, NED-IPAC Caltech cosmology calculator. Planck Collaboration 2020 A&A 641:A6. SH0ES Collaboration Riess 2022 ApJ. JWST JADES Survey 2024. NASA Goddard Space Flight Center.

Updated July 2026