Messages from 95% ago
A galaxy at z = 6 sent its light when the universe was only ~5% of its current age—your lookback time is about 12.8 Gyr in ΛCDM.
Fossil photons
COSMOTOOLS (Modern) — DL, DC, DA, Lookback, Age
Inputs & Actions
Tip: Press Ctrl/Cmd + Enter to calculate. URL updates so you can bookmark/share your inputs.
Results
Graph
Tip: Drag to pan, wheel/trackpad to zoom. Click to snap the cursor to the nearest z.
Notes
This modern UI preserves Cappi’s original distance & time definitions:
- q₀ = ΩM/2 − ΩΛ, Ωk = 1 − ΩM − ΩΛ
- DC (radial comoving) from ∫0→z dz′ / E(z′), E(z)=√(ΩM(1+z)³ + Ωk(1+z)² + ΩΛ)
- DM uses sin/sinh curvature mapping, DL=(1+z)DM, DA=DM/(1+z)
- Lookback tL from ∫ dz′ /[(1+z′)E(z′)] with the same H₀→Gyr factor as the classic code: Tnorm=9.77810945/h
- “Angle for 1 Mpc” and “Length for 1°” match the original: angle(arcmin)= (60/π) (1+z)² / DL, length(Mpc)= (π/180) DL /(1+z)²
About this Calculator & References
This is a modern UI re-implementation of the classic COSMOTOOLS cosmology calculator by Alberto Cappi (INAF – Osservatorio Astronomico di Bologna). We preserve the same physical definitions and distance relations while updating the design, accessibility, and numerical integration.
Disclaimer. This site is not affiliated with NASA, IPAC, or the NASA/IPAC Extragalactic Database (NED). We simply implement standard FLRW cosmology equations that are widely published in the literature and educational websites.
What equations does this use?
We follow the same definitions as the original COSMOTOOLS page: the expansion function
E(z)=√(ΩM(1+z)3 + Ωk(1+z)2 + ΩΛ),
the line-of-sight comoving distance from ∫dz/E(z), curvature mapping via
sin/sinh, and the usual relations
DL=(1+z)DM and DA=DM/(1+z).
Lookback time uses the classic factor Tnorm=9.77810945/h (with h = H0/100) as in the original script.
Original page
- A. Cappi — COSMOTOOLS (Cosmological calculator)
References (from the original COSMOTOOLS page)
- D. W. Hogg (1999). Distance measures in cosmology.
- A. Cappi (2001). Testing cosmological models with negative pressure.
- H. de Ruiter (2005). Cosmological formulas.
Related resources
- NASA/IPAC Extragalactic Database (NED) — cosmology utilities listing: https://ned.ipac.caltech.edu/ (mentioned on the original page; we are not affiliated)
Suggested citation
If this tool is useful in your work, please cite the original COSMOTOOLS page by Alberto Cappi and the Hogg (1999) note above, and optionally acknowledge this modern UI implementation (Starlight Tools).
5 Fun Facts about Cosmological Distances
Angular size flip
Angular-diameter distance peaks near z ≈ 1.6, so galaxies beyond that redshift actually look larger in arcseconds despite being farther away.
Geometry twist
Curvature sniff test
Just sum ΩM + ΩΛ: if it differs from 1 by even 0.01, Ωk betrays an open or closed universe on the tool’s output cards.
Quick verdict
Scale factor shorthand
Redshift instantly gives a cosmic zoom: the scale factor is a = 1/(1+z), so z = 3 means every proper distance has been stretched by a factor of 4.
Expansion math
Hubble tension stakes
Switching from Planck’s H₀ ≈ 67.7 to SH0ES’ 73 raises luminosity distances by ~8% at z = 1—small numerically, huge for supernova fits.
Parameter drama