What is luminosity distance?

Luminosity distance D_L is defined so that the inverse-square law holds: flux F = L / (4π D_L²). Because cosmic expansion redshifts photons and stretches arrival intervals, D_L = (1+z) × D_C, where D_C is the comoving distance. D_L is always larger than D_C for z > 0.

How is this different from the redshift-to-distance calculator?

That tool reports comoving distance, luminosity distance, and lookback time as three parallel outputs. This calculator focuses on luminosity distance and extends it with distance modulus computation and magnitude conversion — useful for standard-candle work.

What cosmological parameters are used?

Defaults are Planck 2018: H₀ = 67.4 km/s/Mpc, Ω_M = 0.315, Ω_Λ = 0.685 (flat universe). You can override these in the Advanced Settings panel.

What is the distance modulus?

The distance modulus μ = m − M connects apparent magnitude m and absolute magnitude M of a source. It equals 5 log₁₀(D_L / 10 pc). For D_L in Mpc, μ = 5 log₁₀(D_L) + 25. A difference of 5 in μ corresponds to a factor of 10 in distance.

Is this accurate for very high redshifts (z > 10)?

The ΛCDM integration used here is accurate for any redshift where the standard cosmological model applies, including z ~ 1000 (CMB). The integration includes radiation density (Ω_r) for precision at z > 100.

Why does D_L grow faster than expected at high z?

At low z, D_L ≈ cz/H₀. At higher z, the (1+z) factor amplifies the comoving distance, and the deceleration/acceleration history of the universe shifts the comoving integral. This is precisely the effect used in supernova cosmology to discover dark energy.

Luminosity Distance Calculator

Name: Luminosity Distance Calculator
Availability: InStock
Author: Nham Vu

Enter a cosmological redshift to get the luminosity distance, distance modulus, and light travel time using standard ΛCDM cosmology.

Redshift Input

Redshift (z)

Valid range: 0 to 1100

Quick presets

Magnitude Conversion (optional)

Fill in one field; the other is computed from the distance modulus.

Absolute magnitude (M)

Type Ia SN: −19.3

Apparent magnitude (m)

Leave blank to compute

Advanced: Cosmological Parameters

H₀ — Hubble constant (km/s/Mpc)

Ω_M — Matter density

Ω_Λ — Dark energy density

Results

Enter a redshift above to compute luminosity distance.

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Summary

Enter a cosmological redshift to get the luminosity distance, distance modulus, and light travel time using standard ΛCDM cosmology.

How it works

Enter the redshift (z) of the object. Typical values: nearby galaxies z < 0.1, distant quasars z > 2.
The calculator numerically integrates the comoving distance D_C = (c/H₀) ∫₀ᶻ dz′/E(z′) using Simpson's rule with 1,000 steps.
Luminosity distance is derived as D_L = (1+z) × D_C.
Distance modulus is computed as μ = 5 log₁₀(D_L / 10 pc) = 5 log₁₀(D_L in Mpc) + 25.
If you supply an absolute magnitude M, apparent magnitude m = M + μ is shown. If you supply m, then M = m − μ.
Adjust H₀, Ω_M, and Ω_Λ in Advanced Settings to explore non-standard cosmologies.

Use cases

Compute the luminosity distance to a supernova type Ia at a given redshift.
Convert a redshift to a distance modulus for standard-candle analysis.
Derive the apparent magnitude of a known absolute-magnitude source at cosmological distances.
Explore how luminosity distance scales differ from comoving distance at high redshifts.
Verify cosmological calculations from papers or coursework.
Compare distances under different cosmological parameter sets.

Frequently Asked Questions

Last updated: 2026-07-22 · Reviewed by Nham Vu