Nearby galaxy: z = 0.1
Input: z = 0.1. Result: age ≈ 12.459 Gyr, lookback ≈ 1.345 Gyr, comoving distance ≈ 432.6 Mpc. The low-redshift velocity shortcut is only an approximation; the integral supplies the model distance.
Universe Age, Lookback Time and Distance Calculator
Calculate from
Choose redshift for a direct calculation, or another quantity to solve numerically for z.
Range: 0 to 20,000. Redshift is dimensionless.
Try an example
Presets assume ΛCDM and set ΩΛ for spatial flatness after radiation. The local-H₀ option is a sensitivity example, not a complete survey fit.
Advanced cosmology settings
20–150 km/s/Mpc.
Matter density, 0–2.
Vacuum density, −1 to 2.
Radiation is calculated as ΩR = 4.165 × 10−5/h²; curvature is ΩK = 1 − ΩM − ΩΛ − ΩR.
Keyboard shortcut: Ctrl/Cmd + Enter.
Your answer
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How age and distance change with redshift
Both axes use log(1 + value) scaling so early cosmic time and very large distances remain visible together. Times are in Gyr; distances are in Gpc. Use the checkboxes to hide series. Focus a highlighted point for its exact value.
Worked redshift examples
These reproducible examples use this page’s default Planck 2018 approximation.
Distant quasar: z = 3
Input: z = 3. Result: age ≈ 2.148 Gyr, lookback ≈ 11.656 Gyr, comoving distance ≈ 6.513 Gpc. Its present-day comoving distance exceeds 11.656 billion light-years because space expanded while the light traveled.
Cosmic microwave background: z = 1100
Input: z = 1100. Result: age ≈ 0.367 Myr, lookback ≈ 13.804 Gyr, comoving distance ≈ 13.908 Gpc. This is a background-expansion estimate, not a recombination-physics simulation.
Redshift-to-age reference
| Redshift | Universe age then | Lookback time | Comoving distance today |
|---|---|---|---|
| 0 | 13.804 Gyr | 0 | 0 |
| 0.1 | 12.459 Gyr | 1.345 Gyr | 0.433 Gpc |
| 1 | 5.863 Gyr | 7.941 Gyr | 3.398 Gpc |
| 3 | 2.148 Gyr | 11.656 Gyr | 6.513 Gpc |
| 6 | 0.931 Gyr | 12.873 Gyr | 8.438 Gpc |
| 1100 | 0.367 Myr | 13.804 Gyr | 13.908 Gpc |
Formulas and numerical method
The calculator assumes an FLRW universe with matter, a cosmological constant, radiation, and curvature. These are model-dependent numerical integrals—not distance divided by the speed of light.
E(z) = H(z)/H₀ = √[ΩR(1+z)⁴ + ΩM(1+z)³ + ΩK(1+z)² + ΩΛ]
Lookback time: tL = H₀−1 ∫0z dz′ / [(1+z′)E(z′)]
Age at redshift: t(z) = H₀−1 ∫z∞ dz′ / [(1+z′)E(z′)] = H₀−1 ∫0a da′ / [a′E(a′)]
Line-of-sight comoving distance: DC = (c/H₀) ∫0z dz′ / E(z′)
Curvature: DM = (c/H₀)/√|ΩK| × Sk(√|ΩK|DCH₀/c), where Sk is sinh for ΩK > 0, its argument for ΩK = 0, and sin for ΩK < 0.
Derived measures: DA = DM/(1+z); DL = (1+z)DM; a = 1/(1+z); μ = 5 log₁₀(DL/Mpc) + 25.
Symbols: H₀ is today’s Hubble constant; H(z) is the expansion rate at z; c is light speed; ΩR, ΩM, ΩK, and ΩΛ are present-day radiation, matter, curvature, and vacuum density parameters; a is scale factor; DC is radial comoving distance; DM is transverse comoving distance; DA is angular-diameter distance; DL is luminosity distance.
Conversions: 1 Mpc = 3.26156 million light-years; 1 Gpc = 1,000 Mpc; c × 1 Gyr = 1 billion light-years. The inverse angular scale follows from 1 radian = 206,264.806 arcseconds.
Accuracy, validation, and limits
Integrals use adaptive Simpson quadrature with a relative target near 10−8 and guarded recursion, replacing the former fixed 1,000-step sum. Reverse modes use bracketed bisection. The supported range is 0 ≤ z ≤ 20,000. Values normally settle well beyond the displayed precision.
The model includes ΩR = 4.165 × 10−5/h² and curvature. It assumes w = −1 and does not separately evolve massive neutrinos, changing dark energy, inhomogeneity, peculiar velocity, recombination, or measurement uncertainty. The default closes ΩΛ after its massless-radiation term, whereas Astropy Planck18 includes a 0.06 eV neutrino; that physical-model difference is larger than the integration error.
Reference comparison
Checked 18 July 2026 against Astropy Planck18 (H₀ = 67.66, ΩM = 0.30966) and the equations/implementation described by Wright. This tool’s age and comoving distance differ from Astropy by at most about 0.21% across these checks, chiefly because of neutrino and closure treatment.
| z | Age: this tool | Age: Astropy | DC: this tool | DC: Astropy |
|---|---|---|---|---|
| 0 | 13.804 Gyr | 13.787 Gyr | 0 Mpc | 0 Mpc |
| 0.1 | 12.459 Gyr | 12.442 Gyr | 432.6 Mpc | 432.6 Mpc |
| 1 | 5.863 Gyr | 5.851 Gyr | 3398.4 Mpc | 3395.6 Mpc |
| 3 | 2.148 Gyr | 2.144 Gyr | 6512.6 Mpc | 6504.0 Mpc |
| 6 | 0.931 Gyr | 0.929 Gyr | 8437.8 Mpc | 8425.1 Mpc |
| 1100 | 0.367 Myr | 0.367 Myr | 13908.1 Mpc | 13886.3 Mpc |
Which cosmological distance is which?
- Lookback time is elapsed travel time. Its equivalent light-travel distance is c × lookback time.
- Comoving distance today factors out expansion and describes the source’s present coordinate separation.
- Proper distance at emission is radial comoving distance multiplied by the scale factor then.
- Transverse comoving distance includes spatial curvature and controls transverse geometry.
- Angular-diameter distance converts apparent angle to physical size at emission.
- Luminosity distance converts observed flux to intrinsic luminosity and includes redshift dimming.
Frequently asked questions
What does lookback time mean?
Lookback time is the elapsed time between the emission of the light we observe and today. Multiplying it by c gives the light-travel distance, not the object's present-day separation.
Why can comoving distance exceed 13.8 billion light-years?
The universe expanded while the light traveled. Lookback time records the travel duration, while comoving distance describes the object's present-day separation in the expanding coordinate system, so it can be much larger.
Which distance should I use for brightness or angular size?
Use luminosity distance for flux and intrinsic-luminosity calculations. Use angular-diameter distance, or the reported kpc per arcsecond scale, to convert apparent angular size to physical size at emission.
Is cosmological redshift a velocity?
Not generally. At very small redshift, cz approximates recession velocity, but at cosmological redshift the wavelength stretch mainly traces expansion and must be interpreted with a cosmological model.
Why do results vary between cosmology calculators?
Results depend on H0, matter, dark energy, curvature, radiation, neutrino treatment, and numerical method. Check that calculators use the same parameters and physical assumptions before comparing them.
What happens at redshift z = 0?
Lookback time and all source distances are zero, scale factor is 1, and age at emission equals the present age of the universe. Distance modulus and angular scale are undefined at zero distance.
Which cosmology preset should I choose?
Planck 2018 is the best general default for standard flat Lambda-CDM comparisons. Choose WMAP9 to reproduce older literature, the illustrative local-H0 option to explore sensitivity, or Custom when a source specifies parameters.
How accurate are high-redshift results?
Adaptive integration is numerically stable through z = 20000, but the physical model is simplified. Radiation is included, while massive neutrinos, evolving dark energy, and early-universe perturbation physics are not; at high z, model choice dominates rounding error.
Review and references
Author and internal scientific-method reviewer: Starlight Robotics, publisher and maintainer of browser-based scientific calculation tools. Reviewed and updated . Numerical integration and reference values were checked during this review; no external individual reviewer is claimed.
The FLRW distance implementation follows the equations presented by Hogg and Wright; the adaptive integration, reverse solver, interface, and reference tests were independently implemented for this page.
- D. W. Hogg, Distance measures in cosmology (2000).
- Planck Collaboration, Planck 2018 results VI: Cosmological parameters, A&A 641, A6 (2020).
- E. L. Wright, A Cosmology Calculator for the World Wide Web, PASP 118, 1711 (2006).
- Astropy Planck18 realization documentation.
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