Formula Used
The line-of-sight comoving distance is computed from the expansion history:
DC(z) = (c / H0) ∫0z dz' / E(z')
where the dimensionless expansion function is:
E(z) = √[ Ωr(1+z)4 + Ωm(1+z)3 + Ωk(1+z)2 + ΩDE(1+z)3(1+w) ]
The transverse comoving distance DM uses curvature via sin/sinh forms, then luminosity distance DL = (1+z)DM and angular diameter distance DA = DM/(1+z).
How to Use This Calculator
- Choose a preset or keep your current values.
- Enter the redshift z for your target object.
- Set H0 and the density parameters for your model.
- Keep curvature on auto, or manually supply Ωk.
- Click Calculate to show results above the form.
- Use the CSV or PDF buttons to export results.
Example Data Table
| Scenario | z | H0 | Ωm | ΩDE | Ωr | Ωk mode | w | What to expect |
|---|---|---|---|---|---|---|---|---|
| Nearby galaxy | 0.05 | 67.4 | 0.315 | 0.685 | 0.00009 | Auto | -1 | Small comoving distance, nearly linear scaling. |
| Moderate redshift | 1.0 | 67.4 | 0.315 | 0.685 | 0.00009 | Auto | -1 | Comoving distance grows faster than z. |
| High redshift | 8.0 | 67.4 | 0.315 | 0.685 | 0.00009 | Auto | -1 | Radiation begins to matter; distances large. |
| Open geometry test | 2.0 | 70.0 | 0.30 | 0.60 | 0.00009 | Manual Ωk=0.10 | -1 | DM exceeds DC due to curvature. |
Comoving Distance Guide
1) What comoving distance represents
Comoving distance tracks separation in coordinates that expand with the universe. If two galaxies follow the Hubble flow, their comoving separation stays nearly constant even while proper distance grows with the scale factor. This helps compare objects across cosmic time using one consistent ruler.
2) Why redshift is the input
Redshift z is directly measured from spectral features. It encodes how much the universe expanded since light was emitted, where 1+z = a0/a. For small z, distance is roughly proportional to z, but at larger z the expansion history makes the relationship nonlinear.
3) What H0 does numerically
H0 sets the overall distance scale through the Hubble distance DH = c/H0. With c = 299,792.458 km/s, H0 = 67.4 km/s/Mpc gives DH ≈ 4,449 Mpc. Larger H0 means smaller distances for the same integral.
4) Matter, radiation, and curvature terms
The calculator uses Ωm(1+z)3 for matter, Ωr(1+z)4 for radiation, and Ωk(1+z)2 for curvature. Radiation is tiny today (often ~9×10−5) but becomes important at high redshift because of the (1+z)4 scaling.
5) Dark energy with w
Dark energy evolves as ΩDE(1+z)3(1+w). The common ΛCDM case uses w = −1, which keeps the dark energy density constant. If you try w = −0.9 or w = −1.1, you can see how late-time expansion changes and how distances shift at moderate redshift.
6) Line-of-sight vs transverse distance
DC is the radial comoving distance from the integral. DM is the transverse comoving distance that includes curvature. For Ωk ≈ 0, DM ≈ DC. With positive curvature (open geometry), DM can exceed DC.
7) Related distances you also get
The output also shows luminosity distance DL = (1+z)DM and angular diameter distance DA = DM/(1+z). These connect to observations: flux scales with 1/DL2, while an object’s apparent size relates to DA.
8) Accuracy and practical tips
Numerical integration uses Simpson’s rule with an even step count. Increasing steps generally improves accuracy, especially for large z. Keep Ω totals near one for a standard Friedmann model, and use the example table as a starting dataset before fine-tuning parameters.
FAQs
1) What units are used in the results?
Distances are displayed in megaparsecs (Mpc), gigaparsecs (Gpc), and billions of light-years (Gly). The conversion uses 1 Mpc ≈ 3.26156 million light-years.
2) Which distance should I report for galaxy surveys?
Many analyses use DC for radial separations and DM for transverse separations. If curvature is near zero, both are nearly the same.
3) Why does changing H0 change all distances?
H0 appears in DH = c/H0, which scales the integral result. Higher H0 reduces distances; lower H0 increases them.
4) When does radiation matter?
Radiation becomes significant at high redshift because it grows as (1+z)4. For z around several to tens, it can noticeably affect E(z) and the integral.
5) What does Ωk auto mode do?
Auto mode computes Ωk = 1 − (Ωm + Ωr + ΩDE). This keeps your model consistent with the density budget unless you intentionally set curvature.
6) What if my Ω totals are not close to one?
The page shows a warning. The calculator still integrates, but the model may be nonstandard, and results can be less interpretable for common cosmology comparisons.
7) How many integration steps should I use?
For z ≤ 3, 4000–8000 steps usually work well. For z ≥ 10 or sensitive comparisons, increase toward 20000–40000. Steps must be even for Simpson’s rule.