DC · radial comoving distance
Present-day line-of-sight map separation, with cosmic expansion factored out. Useful for redshift maps and radial separations.
Modern COSMOTOOLS · advanced adjustable FLRW model
Advanced Redshift and Cosmology Calculator
Model limit: includes matter (ΩM), spatial curvature (Ωk), and a cosmological constant (ΩΛ, w = −1), but omits radiation, evolving dark energy, and neutrino mass evolution. Redshift is limited to 0 ≤ z ≤ 20; near the upper end, treat results as a matter + curvature + Λ approximation, not precision early-universe or CMB cosmology.
Inputs
Observed wavelength shift; supported range 0–20.
Present-day expansion rate.
Present matter density relative to critical.
Cosmological-constant density, with w = −1.
Press Ctrl/Cmd + Enter to calculate. Display rounding preserves full internal precision.
Batch mode: up to 20 redshifts
Paste a comma-separated list or one redshift per line. The current cosmology and units apply to every row.
Results
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Distance and time across redshift
Distance mode plots luminosity distance DL, radial comoving distance DC, and angular-diameter distance DA. A keyboard cursor reports values at a selected redshift.
Chart cursor loading.
Keyboard: focus the chart, then use ←/→ to move, +/− to zoom, and Home to reset. Pointer users can tap or click to position the cursor. Use the buttons on touch screens.
Accessible chart data table and CSV
Eleven evenly spaced samples from the current chart view provide a text alternative to the canvas.
What each result means
DM · transverse comoving distance
Comoving separation across the sky after applying spatial curvature. It connects angle to transverse comoving size.
DA · angular-diameter distance
Use this to turn apparent angular size into physical size at emission: size = angle × DA.
DL · luminosity distance
Use this with observed flux and intrinsic luminosity: flux = luminosity/(4πDL²).
Lookback time
Time elapsed between emission and today. The light-travel distance c × lookback time is not the same as comoving distance.
Age at redshift
Age of the universe when the observed light was emitted. Current age = age at emission + lookback time.
Distance modulus
Difference between apparent and absolute magnitude, μ = 5 log10(DL/Mpc) + 25.
Angular scale
Physical kiloparsecs represented by one arcsecond (and its reciprocal), useful for image and instrument measurements.
Worked cosmology examples
Expected values use H₀ = 67.4 km/s/Mpc, ΩM = 0.315, ΩΛ = 0.685, no radiation, and displayed rounding.
1. Distance to a z = 1 galaxy
Inputs: Planck 2018 baseline, z = 1.
Expected: DL = 6802.526 Mpc; DC = DM = 3401.263 Mpc; DA = 1700.631 Mpc.
Interpretation: use DL for flux and DA for apparent size.
2. Convert 1 arcsecond at z = 1
Inputs: same model and redshift.
Expected: 1 arcsec = 8.245 physical kpc (0.12128 arcsec/kpc).
Interpretation: a 0.5 arcsec feature spans about 4.12 kpc.
3. Age when the light was emitted
Inputs: same model and redshift.
Expected: current age 13.796 Gyr, lookback time 7.951 Gyr, age at emission 5.846 Gyr.
Check: 5.846 + 7.951 ≈ 13.796 Gyr.
4. Planck versus higher H₀
Inputs: z = 1 and ΩM = 0.315; compare H₀ = 67.4 with 73.04 while holding densities fixed.
Expected DL: 6802.526 versus 6277.249 Mpc.
Interpretation: higher H₀ lowers distance at fixed density parameters, here by about 7.7%.
Method, equations, and assumptions
This calculator evaluates an FLRW expansion history containing non-relativistic matter, curvature, and a cosmological constant. Ωk is derived as 1 − ΩM − ΩΛ; H₀ is in km s−1 Mpc−1, distances are calculated internally in Mpc, and times in Gyr.
E(z) = H(z)/H₀ = [ΩM(1+z)³ + Ωk(1+z)² + ΩΛ]1/2 DC = (c/H₀) ∫0z dz′/E(z′) DM = (c/H₀)/√|Ωk| × {sinh(√Ωk H₀DC/c), Ωk>0; H₀DC/c, Ωk=0; sin(√|Ωk| H₀DC/c), Ωk<0} DL = (1+z)DM, DA = DM/(1+z) t(a) = H₀−1 ∫0a √a′ da′ / √(ΩM + Ωka′ + ΩΛa′³), a=1/(1+z)- Time relationship: t(z) = t₀ − tL. The scale-factor integral begins at a = 0, so the current age reaches the Big Bang rather than stopping at an arbitrary maximum redshift.
- Numerics: adaptive Simpson quadrature with relative target 10−9 for distance/time and 2 × 10−8 for the nested volume integral, recursion depth up to 20, c = 299,792.458 km/s, and Hubble time 9.777922217/h Gyr (Mpc converted with the IAU parsec and a 365.25-day Julian year). All expansion-function samples must be finite and real.
- Curvature: sinh applies to open geometry, the identity to flat geometry, and sin to closed geometry. Comoving volume is integrated from dV/dz = 4π(c/H₀)DM²/E(z).
- Low redshift: D ≈ cz/H₀ when z ≪ 1. At cosmological redshift, z is not simply a velocity; use the full expansion model.
- Excluded physics: Ωr = 0, no separate photon/neutrino components, and no w ≠ −1 or evolving dark energy. The enforced z ≤ 20 range prevents CMB-era use but does not make radiation negligible for every precision application.
Numerical verification
Reference values below were generated with Astropy FlatLambdaCDM using H₀ = 67.4 km/s/Mpc, ΩM = 0.315, ΩΛ = 0.685, and TCMB = 0 so the reference model matches this calculator’s no-radiation assumption. Acceptance is relative difference below 0.01% for the shown quantities.
| z | Astropy DC (Mpc) | Astropy DL (Mpc) | Lookback (Gyr) | Age at z (Gyr) | Expected difference |
|---|---|---|---|---|---|
| 0.1 | 434.110995 | 477.522094 | 1.350010 | 12.446225 | <0.01% |
| 1 | 3401.262917 | 6802.525835 | 7.950613 | 5.845622 | <0.01% |
| 3 | 6506.062225 | 26024.248901 | 11.654228 | 2.142007 | <0.01% |
| 6 | 8424.083829 | 58968.586804 | 12.866765 | 0.929470 | <0.01% |
Reference implementation: Astropy cosmology documentation. Values independently reproduced for this release; Astropy is not affiliated with Starlight Tools.
Tool version: 2.0 · Last reviewed: 18 July 2026 · Calculation author/reviewer: Starlight Robotics · Method: client-side double-precision JavaScript and adaptive quadrature.
Cosmology calculator FAQs
Which cosmological model does this calculator use?
A matter + curvature + cosmological-constant FLRW model. Dark energy is fixed at w = −1. Radiation, massive-neutrino evolution, and other dark-energy equations of state are excluded.
What is the difference between comoving and light-travel distance?
Comoving distance describes present-day map separation after factoring out expansion. Light-travel distance is c multiplied by lookback time. Expansion means the two are generally unequal.
Why do luminosity and angular-diameter distances differ?
Expansion changes observed flux and apparent size differently: DL = (1+z)DM, while DA = DM/(1+z). Choose DL for luminosity/flux and DA for physical/apparent size.
What does Ωk mean?
Ωk = 1 − ΩM − ΩΛ. Positive is open, zero is flat, and negative is closed spatial geometry. The flat lock keeps Ωk at zero.
Can redshift be converted directly to velocity?
Only at very low redshift is v ≈ cz a useful Hubble-law approximation. Cosmological redshift depends on the expansion history and is not simply a special-relativistic recession velocity.
Which preset should I choose?
Planck 2018 is a useful standard flat-ΛCDM baseline. Choose the exact cosmology required by your paper, catalog, or course. The SH0ES option is a higher-H₀ comparison, not a complete independent density fit.
What redshift range is supported?
0 ≤ z ≤ 20 is enforced. Radiation becomes increasingly important toward high redshift, so do not use this no-radiation tool for precision early-universe or CMB work.
How accurate are the calculations?
The integrator targets relative precision near 10−9. Typical standard-model results agree with the matching Astropy configuration well inside 0.01%; display rounding does not change internal precision.
Are calculations private?
All calculations run in your browser and are not uploaded by this tool. Calculating updates the URL with single-mode parameters, so shared URLs and browser history can contain those values.
Modern COSMOTOOLS history and references
This interface is inspired by Alberto Cappi’s classic COSMOTOOLS. It is not a claim of identical physics: the historic listing describes Λ = 0 models, while this implementation supports an explicit ΩΛ term and modern FLRW distance relations.
References: D. W. Hogg, Distance measures in cosmology; A. Cappi, Testing cosmological models with negative pressure. This site is not affiliated with NASA, IPAC, NED, Astropy, or Alberto Cappi.