Example data table
| Redshift z | H0 (km/s/Mpc) | Ωm | ΩDE | Lookback (approx, Gyr) |
|---|---|---|---|---|
| 1.0 | 67.4 | 0.315 | 0.685 | ~7.8 |
| 3.0 | 67.4 | 0.315 | 0.685 | ~11.6 |
| 1100 | 67.4 | 0.315 | 0.685 | ~13.8 |
Values are indicative and depend on chosen parameters.
Formula used
The lookback time is the difference between the universe age today and the universe age at a chosen redshift.
This calculator evaluates the integral numerically using Simpson’s rule on ln(a) for stability at very small scale factors.
How to use this calculator
- Enter a redshift value (z) for the observed object.
- Choose your Hubble rate and density parameters.
- Enable auto-closure if you want ΩDE computed automatically.
- Select an accuracy level for faster or finer integration.
- Press Calculate to see results above the form.
- Use the download buttons to export your output.
Article
1) What lookback time tells you
Lookback time is how long light traveled before reaching you. It links an observed redshift to a past cosmic epoch. When z increases, the scale factor a = 1/(1+z) decreases, so you are viewing an earlier universe state.
2) Redshift, scale factor, and expansion rate
Redshift measures how much expansion stretched wavelengths. The expansion history is captured by E(a)=H(a)/H0. This calculator combines radiation, matter, curvature, and dark energy to compute E(a) and then integrates 1/E(a) over ln(a).
3) Density parameters and the closure check
Ωm, Ωr, Ωk, and ΩDE describe today’s fractional energy budget. Their sum is near 1 for many models. If you enable auto-closure, ΩDE becomes 1 − Ωm − Ωr − Ωk. The result panel reports (sum − 1) so you can verify consistency.
4) Why radiation matters at high redshift
At early times, radiation scales as a⁻⁴ and quickly dominates over matter (a⁻³). For z above a few thousand, Ωr strongly affects the integral and the age at z. Including Ωr improves results for recombination-era comparisons and very early epochs.
5) Curvature effects you can explore
Curvature contributes as Ωk/a². Small Ωk values can still shift ages at moderate redshift because a⁻² grows faster than matter but slower than radiation. Try Ωk slightly positive or negative to see how open or closed geometry changes lookback time.
6) Changing w alters late-time history
The parameter w controls dark energy evolution through a^{-3(1+w)}. With w = −1, dark energy stays constant. If w is higher than −1, dark energy fades faster in the past, typically increasing the age at a given z for the same H0 and densities.
7) Numerical method and accuracy choices
The integral for cosmic time is evaluated with Simpson’s rule using x = ln(a). This keeps steps well-behaved near tiny a. Higher accuracy increases the number of integration panels and lowers a_min, improving early-time resolution at the cost of runtime.
8) Interpreting results in practical work
Use lookback time to connect observations to cosmic history: star formation epochs, galaxy assembly, and event rates. Compare the “age at z” to astrophysical timescales. Export CSV for reports, or PDF for sharing a reproducible snapshot of inputs and outputs.
Tip: Compare multiple z values using the same cosmology to map a timeline.
FAQs
1) Is lookback time the same as distance?
No. Lookback time is a time interval. Distance depends on the cosmology and the chosen distance definition. Two objects with the same z share lookback time, but their measured distances can be expressed in several different ways.
2) Why can lookback time approach the universe age?
At very high redshift, you observe light from near the beginning. The “age at z” becomes small, so lookback time approaches the age today. It never exceeds the age today within a physically consistent model.
3) What does the closure delta mean?
Closure delta is (Ωm + Ωr + Ωk + ΩDE) − 1. Values near zero indicate a consistent density budget. If auto-closure is enabled, delta mainly reflects rounding and input limits rather than a real mismatch.
4) Should I keep Ωr nonzero?
For z below a few hundred, Ωr has a small effect. For very large z, it matters more. If you want early-universe ages, include Ωr. For quick late-time estimates, default Ωr is usually sufficient.
5) Can I use w values other than −1?
Yes. This calculator supports constant-w dark energy. It is a simplified model, but it helps you test sensitivity. Keep w within a reasonable range and compare against w = −1 to see how late-time expansion changes.
6) Why does higher H0 reduce ages?
H0 sets the overall time scale through 1/H0. A larger H0 implies a faster expansion rate today, which generally reduces both the age today and the age at a given redshift, decreasing times across the board.
7) What accuracy setting should I choose?
Use Standard for fast estimates, High for careful comparisons, and Very high for extreme redshift tests. If the page feels slow, reduce accuracy. If results change noticeably with accuracy, keep the higher option for stability.