Cosmology
several distances

A redshift is not yet a distance. The same z is an age, a lookback time, a comoving ruler, a luminosity distance, and an angular-diameter distance. They are different integrals of one expansion history. Slide z from 0 to 20, or type a higher one. The default model is flat Planck 2018, Table 2, column TT,TE,EE+lowE+lensing: H₀ = 67.36, Ω_m = 0.3153, Ω_Λ = 0.6847. A second button sets the SH0ES Hubble constant, 73.04, and keeps those densities, because that paper measures H₀ and not Ω_m. Radiation is left out. The wavelength shift is the redshift desk. The nearby toy d ≈ cz/H₀ is the distance ladder. The Milky Way these numbers are measured from is the galaxy desk.

Educational calculator. Not a fit to the sky and not a catalog distance. Radiation is omitted, the way Hogg’s E(z) omits it. Peculiar velocities are not in the integral. A typed redshift is arithmetic, not a new object. With the Planck central values the no-radiation age is 13.801 Gyr. The paper’s age for that same column is 13.797 ± 0.023 Gyr, and that published age includes radiation. The two sit inside the quoted uncertainty. Angular-diameter distance for these parameters peaks near z = 1.59.

Interactive · Age, lookback, and three distances

Distance versus redshift · log Mpc · 0 to 20 · angular-diameter distance turns over

Comoving
Luminosity
Angular diameter
This z

For the Planck 2018 central values, angular-diameter distance peaks near z = 1.59.

Cosmology · cited central values

Geometry

Objects · cited redshifts only

Redshift z

H0 km/s/Mpc

Ωm Ωm

ΩΛ ΩΛ

The sliders need JavaScript. The anchors are still these. E(z) = √[Ωm(1+z)³ + Ωk(1+z)² + ΩΛ], with Ωk = 1 − Ωm − ΩΛ, and no radiation term. Comoving distance is (c/H0) times the integral of dz/E. Luminosity distance is (1+z) times the transverse comoving distance. Angular-diameter distance divides that transverse distance by (1+z). Lookback time integrates dz/[(1+z)E]. Planck 2018 Table 2, TT,TE,EE+lowE+lensing: H0 = 67.36, Ωm = 0.3153, ΩΛ = 0.6847. The no-radiation age is 13.801 Gyr. The published age, radiation included, is 13.797 ± 0.023 Gyr. At z = 1: lookback 7.954 Gyr, age then 5.847 Gyr, comoving 3402.7 Mpc (11.10 Gly), luminosity 6805.4 Mpc (22.20 Gly), angular diameter 1701.3 Mpc (5.55 Gly), 8.248 kpc per arcsec. Angular-diameter distance peaks at z = 1.588. SH0ES H0 = 73.04 with the same densities gives an age of 12.728 Gyr. Coma uses z = 6853/c. 3C 273 is z = 0.15756751. GN-z11 is z = 10.6034. The CMB button is z* = 1089.92, not the sound horizon.

One redshift, several distances

The ruler today, the faintness, and the angle are not the same length.

Comoving distance is the ruler you would lay down today between here and a galaxy that stayed in the Hubble flow. It only grows. Luminosity distance is the answer to “how far would this have to be, in a static Euclidean space, to look this faint?” Photon energy drops as 1/(1+z), and photons arrive less often by another 1/(1+z), so it is (1+z) times the transverse comoving distance. A standard candle is fainter than the comoving ruler alone would say.

Angular-diameter distance converts an angle on the sky into a proper size at the source, the size the object had when the light left. It is the transverse comoving distance divided by (1+z). The comoving piece grows with z. The division by (1+z) grows too. Past a redshift of about 1.6 in this flat model, the division wins, and the angular-diameter distance turns over. A more distant galaxy can subtend a larger angle than a nearer one of the same proper size. That peak is on the chart. It moves if you change Ωm or H0.

Lookback time is how long the light has been traveling. The age at redshift z is how old the universe was when the light left. Their sum is the age today. A lookback of 8 Gyr is 8 billion light-years of light-travel distance, because a Julian year of light is one light-year. That light-travel distance is not the comoving distance. The comoving distance is longer: space stretched while the photon was en route. cz/H0 matches the comoving distance only while z is small. The distance ladder stops at that toy. This page is the integral.

Literacy · the words on the readout

E(z): Hogg’s equation 14: E(z) = √[Ω_m(1+z)³ + Ω_k(1+z)² + Ω_Λ]. H(z) = H₀ E(z). Ω_k = 1 − Ω_m − Ω_Λ. There is no radiation term. A photon’s flight and a clock both come from dividing by E(z). Flat means Ω_k = 0, so the transverse distance equals the line-of-sight comoving distance.
Age: The age today integrates dz/[(1+z)E(z)] from 0 to infinity, times 1/H₀. The age at z stops the integral at that redshift and runs it to infinity. Planck’s published 13.797 ± 0.023 Gyr includes radiation. This page does not, and it prints 13.801 Gyr for the table’s central values.
Lookback: Age today minus age at z. Hogg’s equation 30. It is the light-travel time. In billion light-years of light-travel distance it is the same number. It is not a comoving distance.
Comoving: D_C = (c/H₀) ∫ dz/E(z). Hogg’s equation 15. The line-of-sight ruler today. At low z it sits near cz/H₀. c = 299,792.458 km/s. For H₀ = 67.36, c/H₀ = 4450.6 Mpc.
Luminosity: D_L = (1+z) D_M. Hogg’s equation 21. D_M is the transverse comoving distance, equation 16. In a flat universe D_M = D_C. This is the distance a magnitude uses when the source is cosmological. The distance ladder does not switch over to it.
Angular diameter: D_A = D_M/(1+z). Hogg’s equation 18. Proper size per radian at the time of emission. One arcsecond is π/(180×3600) radians, so the scale in kpc per arcsecond is D_A in megaparsecs, times 1000, times that angle. The nearby version of the same angle, without redshift, is the angular size desk. Hogg notes that D_A turns over instead of growing forever. For these Planck parameters the turnover is at z = 1.588.
Flat or curved: Flat locks Ω_Λ = 1 − Ω_m. Curved lets both densities move, and Ω_k is whatever is left. Positive Ω_k uses sinh. Negative Ω_k uses sin. If that sine goes negative, the ray has passed the antipode of a closed universe and the distances stop being a simple ruler.
Gly: Billion light-years, using 3.2616 light-years per parsec, the same four-decimal rounding as the magnitude and distance-ladder desks. The gigayear is different: 10⁹ Julian years of 365.25 × 86400 seconds, with the IAU parsec, so that H₀ becomes a time. Light-travel Gly equals lookback Gyr. Comoving Gly does not.

Sources · the parameters and the presets

Cited, not invented

Hogg 1999, “Distance measures in cosmology,” arXiv:astro-ph/9905116 — eq. 4, D_H = c/H₀; eq. 14, E(z), with no radiation; eq. 15, line-of-sight comoving distance; eq. 16, transverse comoving distance; eq. 18, angular-diameter distance; eq. 21, luminosity distance; eq. 30, lookback time. He writes that the linear Hubble law holds only at small redshift. The age is the same integral as the lookback, run from z out to infinity.
Planck Collaboration 2020, A&A 641, A6 (arXiv:1807.06209) — Table 2, column TT,TE,EE+lowE+lensing, 68% limits. Central values used here: H₀ = 67.36 ± 0.54 km s⁻¹ Mpc⁻¹, Ω_m = 0.3153 ± 0.0073, Ω_Λ = 0.6847 ± 0.0073, age 13.797 ± 0.023 Gyr, z_* = 1089.92 ± 0.25. The abstract rounds H₀ to 67.4 ± 0.5 and Ω_m to 0.315 ± 0.007. The redshift and distance-ladder desks use that rounded 67.4. This desk uses the table. The published age includes radiation. The integral on this page does not. The same table’s r_* = 144.43 ± 0.26 Mpc is the comoving sound horizon, not the distance to last scattering, and this page does not compute it.
Riess et al. 2022, ApJL 934, L7 (arXiv:2112.04510) — baseline SH0ES H₀ = 73.04 ± 1.04 km s⁻¹ Mpc⁻¹. The button uses the central value. The paper does not publish an Ω_m or Ω_Λ for that button. The densities stay at the Planck 2018 central values.
Colless and Dunn 1996, ApJ 458, 435 (arXiv:astro-ph/9508070) — the main Coma cluster, centered on NGC 4874, mean cz = 6853 km s⁻¹, σ_cz = 1082 km s⁻¹. This desk sets z = cz/c with c = 299,792.458 km/s. The paper prints cz. The velocity distribution is non-Gaussian. The same mean is the Hubble-flow preset on the distance ladder.
SIMBAD — 3C 273 — adopted redshift 0.15756751 ± 0.0005, referenced to Koss et al. 2022, ApJS 261, 2. The same figure is on the redshift desk. The distances here are the integrals at that z.
Bunker et al. 2023, arXiv:2302.07256 — GN-z11 systemic redshift z = 10.6034 ± 0.0013 from emission lines other than Lyman-alpha. The paper’s Lyman-alpha centroid is offset by 555 ± 32 km/s. This button uses 10.6034 and does not add the offset. The same choice as the redshift desk.
NIST CODATA — speed of light in vacuum — c = 299,792,458 m/s exactly, so 299,792.458 km/s.
IAU 2012 Resolution B2 — the astronomical unit is 149,597,870,700 m. The parsec is (648000/π) AU. The gigayear uses that parsec and a Julian year of 365.25 days of 86400 seconds. The Gly column uses 3.2616 light-years per parsec, the rounding on the magnitude desk, and it is not that time conversion.

Classroom cosmology calculator. Radiation left out. Peculiar velocities left out. Planck 2018 Table 2 central values, and SH0ES H0 = 73.04 with those densities held fixed. Object buttons use the cited redshifts only. Not a catalog. Earthrise preview: NASA / Bill Anders (public domain).

Keep going

The shift

Curves	Comoving, luminosity, and angular-diameter distance, from z = 0 to z = 20. The vertical axis is logarithmic megaparsecs, so the angular-diameter turnover stays visible beside the much larger luminosity distance.
Marker	The selected redshift, when it is on the chart. The CMB button is past z = 20, so the marker leaves the chart and the readout keeps the number.
Peak	A dot on the angular-diameter curve. For the Planck button it sits near z = 1.59. It moves when the densities move.

Ages	Today, at the selected z, and the lookback, in Gyr.
Distances	Comoving, luminosity, and angular diameter, in Mpc and Gly.
Arcsecond	Proper kiloparsecs per arcsecond at that angular-diameter distance.
cz/H0	The nearby toy, shown beside the comoving distance so the gap is visible.

Chart	Left and Right nudge the redshift while the canvas is focused.
Buttons	Enter or Space selects. The pressed button is aria-pressed. Targets are at least 44 pixels on a side.
Phone	Below 900 pixels the long section row becomes a single Sections control. Open it for the same links. The site menu is the Menu button in the top bar.
This dialog	The question mark, or ?, opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it.

Cosmology
several distances

The ruler today, the faintness, and the angle are not the same length.

Cited, not invented

Redshift

Distance ladder

Galaxy

Angular size

Cosmology desk

The chart

Controls

Readouts

Keyboard and touch

Redshift	The slider runs from 0 to 20 and gives more room at small z. The number field accepts 0 to 2000, so the CMB redshift fits.
Planck	H0 = 67.36, Ωm = 0.3153, ΩΛ = 0.6847, flat. Table 2 central values.
SH0ES	H0 = 73.04. Ωm and ΩΛ stay at the Planck central values.
Flat	ΩΛ is locked to 1 − Ωm. Curved unlocks ΩΛ. Ωk is whatever is left over.
Objects	Coma, 3C 273, GN-z11, and the CMB last-scattering redshift. Each note names the source.

Cosmologyseveral distances

The ruler today, the faintness, and the angle are not the same length.

Cited, not invented

Redshift

Distance ladder

Galaxy

Angular size

Cosmology
several distances