Redshift Distance Calculator

Formula Used

This calculator uses a standard expanding‑universe model with the dimensionless expansion function:

E(z) = \sqrt{ Ω_r(1+z)⁴ + Ω_m(1+z)³ + Ω_k(1+z)² + Ω_Λ(1+z)^3(1+w) }

Curvature is computed as Ω_k = 1 − (Ω_m + Ω_Λ + Ω_r).

The line‑of‑sight comoving distance is found by numerical integration:

D_C(z) = (c/H₀₀^z \frac{dz'}{E(z')}

The transverse comoving distance uses curvature:

• Ωk = 0: D_M = D_C
• Ωk > 0: D_M = (c/H_0CH₀/c)
• Ωk < 0: D_M = (c/H_0CH₀/c)

Finally:

• D_L = (1+z) D_M
• D_A = D_M/(1+z)
• t_L = (1/H₀₀^z \frac{dz'}{(1+z')E(z')}
• μ = 5 log₁₀(D_L/\text{Mpc}) + 25

Numerical mode uses Simpson’s rule. The low‑z checkbox applies a short series approximation intended for very small redshift.

How to Use This Calculator

1. Enter a redshift value z for the source.
2. Choose cosmology values: H0, Ωm, ΩΛ, and optional Ωr, w.
3. Keep numerical mode for general z, and raise steps for precision.
4. Press Compute distance to view results above the form.
5. Use Download CSV or Download PDF to export your results.

Tip: If you want a nearly flat model, keep Ωm + ΩΛ + Ωr ≈ 1. The tool will report the implied curvature Ωk.

1) Redshift and distance context

Cosmological redshift stretches wavelengths by a factor of (1+z). Converting that observable into distance requires an expansion model and an integral along the light path in an FRW universe. This calculator returns standard distance measures used in astronomy and cosmology.

2) Cosmology inputs you control

The Hubble constant H0 sets the overall distance scale. Density parameters shape the expansion history: matter Ωm, dark energy ΩΛ, radiation Ωr, and the equation parameter w. Curvature is derived as Ωk = 1 − (Ωm + ΩΛ + Ωr), enabling flat, open, or closed models.

3) Expansion history through E(z)

Outputs rely on the dimensionless expansion function E(z), with H(z) = H0·E(z). Matter scales as (1+z)^3, radiation as (1+z)^4, curvature as (1+z)^2, and dark energy as (1+z)^{3(1+w)}. Setting w = −1 recovers a constant dark‑energy density.

4) Comoving distances and curvature

The line‑of‑sight comoving distance D_C is computed from (c/H0)∫dz/E(z) using Simpson’s rule. The transverse comoving distance D_M equals D_C for Ωk≈0, but curvature modifies it via sinh (open) or sin (closed), shifting derived distances.

5) Luminosity and angular diameter distances

The luminosity distance D_L = (1+z)D_M controls flux dimming, while the angular diameter distance D_A = D_M/(1+z) links physical size to angle. They satisfy D_L = (1+z)^2 D_A, a convenient validation check.

6) Lookback time, H(z), and modulus

Lookback time integrates ∫dz/((1+z)E(z)) to return t_L in gigayears. The tool also reports H(z) and the distance modulus μ = 5log10(D_L/Mpc)+25, which is standard for supernova and other standard‑candle analyses.

7) Reference dataset using default settings

For a flat model with H0 = 70 km/s/Mpc, Ωm = 0.30, ΩΛ = 0.70, Ωr = 0, and w = −1, the following values show typical scales:

8) Practical workflow for research and teaching

Start with a preset, then adjust H0 and Ωm to match your dataset or paper. Increase integration steps for high redshift or sensitive comparisons, and use the low‑z option only when z is very small. Export CSV for reproducible tables and PDF for reporting.

FAQs

1) What redshift range is practical here?

Use z ≥ 0. For z < 0.1, the low‑z option can be convenient. For z > 5, increase integration steps and keep parameters physical so E(z) remains positive.

2) Why does curvature Ωk change the result?

Curvature modifies the transverse distance D_M. Flat models use D_M=D_C. Open models apply sinh, and closed models apply sin, which shifts D_L and D_A at the same redshift.

3) How do D_C and D_L differ?

D_C tracks comoving separation along the line of sight. D_L is the distance that converts intrinsic luminosity to observed flux, and equals (1+z)D_M.

4) How accurate is the numerical integration?

Simpson’s rule converges quickly for smooth E(z). If you double the step count, results typically stabilize. For high redshift or strong curvature, use several thousand steps and compare repeated runs for sensitivity.

5) What settings represent standard flat ΛCDM?

A common choice is H0≈70 km/s/Mpc, Ωm≈0.30, ΩΛ≈0.70, Ωr=0, and w=−1, which implies Ωk≈0.

6) When should I include radiation Ωr?

Radiation matters most at early times and high redshift. If you are exploring the very early universe, set a small Ωr and keep the step count high. For late‑time applications, Ωr=0 is usually sufficient.

7) What does the distance modulus output assume?

The modulus uses μ = 5log10(D_L/Mpc)+25. That means D_L must be in megaparsecs. You can use μ directly with standard‑candle data to compare inferred and model distances.