Redshift Distance Calculator (ΛCDM)

Sample Distances vs. Redshift

Flat ΛCDM (Planck 2018 defaults). D_A peaks near z ≈ 1.6.

Understanding Redshift & Cosmological Distances

This redshift distance calculator converts a measured redshift z into several standard cosmological distances used by astronomers and cosmologists. In a flat ΛCDM model (Lambda Cold Dark Matter), we specify the Hubble constant H₀ (km/s/Mpc) and the matter density parameter Ω_M. Flatness implies Ω_Λ = 1 − Ω_M, where Ω_Λ accounts for dark energy (Λ). With these, the expansion history enters through E(z) = √(ΩM(1+z)3 + ΩΛ).

What each distance means

• Comoving distance (D_C) – the “map distance” in today’s Universe, integrating expansion effects from now back to redshift z. It is computed as DC = DH ∫0z dz/E(z), where DH = c/H0 is the Hubble distance.
• Angular diameter distance (D_A) – links a source’s physical size to its observed angular size via θ = size / DA. In ΛCDM, DA = DC / (1+z). Notably, at high redshift, D_A can decrease with increasing z, so very distant objects can appear larger than somewhat nearer ones.
• Luminosity distance (D_L) – relates a source’s intrinsic luminosity to its observed flux: F = L / (4π DL2). In ΛCDM, DL = DC(1+z).

Choosing cosmological parameters

The default preset uses Planck 2018 values (e.g., H₀ ≈ 67.4, Ω_M ≈ 0.315), which fit the cosmic microwave background under ΛCDM. Many local-distance ladder analyses (e.g., SH0ES) report a higher H₀ (≈ 73 km/s/Mpc). Because D_H = c/H₀, a higher H₀ yields smaller distances at the same redshift. For transparent reporting, cite the H₀ and Ω_M you use.

Practical notes for observers

• Low redshift (z ≪ 1): Hubble’s law v ≈ H0 D is a good first approximation, but peculiar velocities can be a significant fraction of the signal.
• Photometry & SEDs: If you derive luminosities from fluxes, consider K-corrections and bandpass effects; D_L alone isn’t the whole story.
• Angles to sizes: Convert arcseconds to radians before applying size = θ · DA. One arcsecond is approximately 4.848 × 10−6 radians.
• Units: This tool reports Gpc for readability; internally the integration is performed in Mpc via D_H.

Method overview

The calculator performs a fast numerical integration of 1/E(z) from 0 to your chosen redshift, using the speed of light c = 299,792.458 km/s. All computation runs entirely in your browser for privacy and responsiveness. Whether you are estimating a galaxy’s physical size from imaging, converting observed fluxes to luminosities, or teaching the difference between D_L, D_A, and D_C, this redshift calculator provides a clear, ΛCDM-consistent foundation for distance determinations.

FAQs

Which cosmology does this use?

Flat ΛCDM with adjustable H₀ and Ω_M (so Ω_Λ=1−Ω_M).

Which distances are reported?

Comoving (D_C), angular diameter (D_A), and luminosity (D_L), plus the Hubble distance D_H.

Are computations private?

Yes—everything runs locally in your browser; nothing is uploaded.

What units are used?

Outputs are shown in Gpc (with Mpc basis internally for integration).

How it works

This calculator assumes a spatially flat ΛCDM cosmology. It computes the comoving distance via numerical integration of 1/E(z). Angular diameter and luminosity distances follow from standard relations. All computation is client-side for privacy.

References

• Planck Collaboration (2018), arXiv:1807.06209
• NED Cosmology Calculator, NASA/IPAC Extragalactic Database
• Hogg, D.W. (1999), “Distance measures in cosmology,” arXiv:astro-ph/9905116