Redshift, Lookback Time & Distance Explorer
Compute FLRW scale factor, lookback time, and comoving/luminosity/angular distances for any redshift z.
In an expanding Friedmann-Lemaître-Robertson-Walker (FLRW) universe, a single redshift z maps to three distinct physical distances depending on the observable: comoving distance d_C(z) (present-day proper distance), luminosity distance d_L(z) = (1 + z) d_C (flux dilution of standard candles), and angular diameter distance d_A(z) = d_C / (1 + z) (apparent angular size of standard rulers).
Interactive Physics Calculator
Governing Equation & Worked Benchmark
Flat ΛCDM Distance & Lookback Integrals
d_C(z) = (c / H₀) ∫₀^z dz' / E(z') ; t_L(z) = (1 / H₀) ∫₀^z dz' / [ (1 + z') E(z') ]Here E(z) = √[ Ω_r(1+z)⁴ + Ω_m(1+z)³ + Ω_Λ ], d_L = (1+z) d_C, and d_A = d_C / (1+z). Notice that d_A(z) peaks near z ≈ 1.6 and decreases at higher redshifts.
| Benchmark Step | Physical Details & Result |
|---|---|
| Scenario | A high-redshift galaxy observed by JWST at z = 6.0 under Planck 2018 parameters (H₀ = 67.4 km/s/Mpc, Ω_m = 0.315). |
| Calculation | Scale factor a = 1 / (1 + 6) = 0.1429 (universe was 7× smaller). Numerical integration gives comoving distance d_C ≈ 8,370 Mpc (27.3 Gly), luminosity distance d_L = 7 × d_C ≈ 58,600 Mpc, angular diameter distance d_A = d_C / 7 ≈ 1,196 Mpc, and lookback time t_L ≈ 12.87 Gyr (cosmic age t(z) ≈ 0.92 Gyr). |
| Physical Takeaway | Although light from z = 6 traveled for 12.87 billion years, metric expansion places the galaxy 27.3 billion light-years away today. |
Physical Mechanism & Interpretation
At low redshifts (z ≪ 0.1), comoving, luminosity, and angular diameter distances all converge to the linear Hubble-Lemaître approximation d ≈ c z / H₀. Above z ≈ 0.1, relativistic expansion breaks this degeneracy: luminosity distance grows rapidly by an extra factor of (1 + z) from photon energy redshift and arrival-rate time dilation.
Conversely, angular diameter distance d_A(z) reaches a maximum of ~1,750 Mpc near z ≈ 1.6 and decreases at higher redshifts, meaning a 10-kiloparsec galaxy at z = 10 actually subtends a LARGER angle on the sky than an identical 10-kiloparsec galaxy at z = 2.
Model Assumptions & Systematic Uncertainties
Changing H₀ between 67.4 km/s/Mpc (Planck) and 73.0 km/s/Mpc (SH0ES) rescales all distances and times inversely by ~8% (d ∝ H₀⁻¹, t ∝ H₀⁻¹).
Related Concepts & Methods
Scientific Sources & Provenance
- Planck 2018 Cosmological Parameters (PR3) (PR3 (2020 A&A 641, A6)) — Planck Collaboration, Aghanim, N., et al. 2020, A&A, 641, A6. Archive / Reference ↗
- SH0ES Local Distance Ladder Hubble Constant (2022 Baseline Cepheid + SN Ia) — Riess, A. G., et al. 2022, ApJL, 934, L7 (A Comprehensive Measurement of the Local Value of the Hubble Constant). Archive / Reference ↗
- JWST / MAST Science Archive & Mission Documentation (Cycle 1–4 Public Releases (2022–2026)) — Rigby, J., et al. 2023, PASP, 135, 048001 (The Science Performance of JWST as Characterized in Commissioning). Archive / Reference ↗