Calculate cosmological expansion metrics using the Friedmann-Lemaître-Robertson-Walker (FLRW) metric and ΛCDM model. Convert redshift (\(z\)) into cosmic age, lookback time, comoving distance, luminosity distance, angular diameter distance, scale factor (\(a\)), and Hubble expansion rate \(H(z)\).
Cosmic Epoch Presets:Tap to load
Cosmological Parameters & Redshift Solver
Select cosmological parameters and enter the spectroscopic redshift (\(z\)).
FLRW Quadrature Engine
Cosmological Model Framework
Astronomical Redshift (\(z\))Scale Factor a = 0.5000 (50.0% size)
z =
z = 0.1 (Nearby)z = 1.0 (Intermediate)z = 6.0 (Epoch of Reionization)z = 14.3 (JWST Galaxy)
Hubble Constant (\(H_0\))
km/s/Mpc
Matter Density (\(\Omega_m\))
Dark Energy (\(\Omega_\Lambda\))
Cosmic Lookback Time
7.93 Gyr
Age at Light Emission: 5.86 Gyr • Scale Factor a = 0.500
Relativistic Cosmological Telemetry
🌌 Cosmological Redshift (\(z\))
z = 1.000
⏳ Universe Age at Emission (\(t_{\text{epoch}}\))
5.86 Gyr (5,856 Myr)
📏 Comoving Distance (\(D_C\))
11.08 Gly (3.40 Gpc)
📐 Proper Distance at Emission (\(D_P\))
5.54 Gly (1.70 Gpc)
💡 Luminosity Distance (\(D_L\))
22.16 Gly (6.80 Gpc)
🔭 Angular Diameter Distance (\(D_A\))
5.54 Gly (1.70 Gpc)
🚀 Expansion Rate at Epoch \(H(z)\)
118.9 km/s/Mpc (1.76× H₀)
⚡ Present Recession Velocity (\(v_{\text{rec}}\))
0.77 c (Subluminal)
The Standard ΛCDM Cosmological Model & The Friedmann Metric
The standard model of modern cosmology, known as ΛCDM (Lambda Cold Dark Matter), describes an expanding, homogeneous, and isotropic universe governed by Albert Einstein's field equations of General Relativity. In this framework, the metric expansion of spacetime is characterized by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric:
Friedmann Expansion Rate & Dimensionless Function \(E(z)\):
Comoving Distance (\(D_C\)) factors out cosmic expansion, remaining constant for galaxies locked in the Hubble flow. Proper Distance (\(D_P = D_C / (1+z)\)) represents the actual physical separation measured with a ruler at the exact instant the photons were emitted.
2. Luminosity Distance (\(D_L\))
Calibrates standard candle flux: \(D_L = (1+z) D_C\). Photons lose energy due to redshift (\(1+z\)) and arrive less frequently due to relativistic time dilation (\(1+z\)), dimming distant sources by a factor of \((1+z)^2\).
3. Angular Diameter Distance (\(D_A\))
Relates physical diameter to apparent angular size on the sky: \(D_A = D_C / (1+z)\). Used to calibrate standard rulers like Baryon Acoustic Oscillation (BAO) sound horizons.
4. The Angular Size Turnaround Effect
In expanding spacetime, \(D_A\) peaks at \(z \approx 1.6\) and then decreases at higher redshifts. Consequently, ultra-distant galaxies at \(z = 10\) subtend a larger angular diameter on the sky than intermediate galaxies of the same physical size at \(z = 2\)!
Comprehensive survey of landmark cosmic epochs, lookback times, and relativistic distance scales.
Planck Data
Cosmic Landmark
Redshift \(z\)
Scale Factor \(a\)
Lookback Time
Universe Age
Comoving \(D_C\)
Luminosity \(D_L\)
Present Epoch (Today)
\(0.00\)
\(1.000\)
\(0.00\,\text{Gyr}\)
\(13.79\,\text{Gyr}\)
\(0.0\,\text{Gly}\)
\(0.0\,\text{Gly}\)
Quasar 3C 273 (1st Quasar)
\(0.158\)
\(0.864\)
\(2.00\,\text{Gyr}\)
\(11.79\,\text{Gyr}\)
\(2.14\,\text{Gly}\)
\(2.48\,\text{Gly}\)
Standard Baseline
\(1.00\)
\(0.500\)
\(7.93\,\text{Gyr}\)
\(5.86\,\text{Gyr}\)
\(11.08\,\text{Gly}\)
\(22.16\,\text{Gly}\)
Cosmic Noon (Peak Star Formation)
\(2.00\)
\(0.333\)
\(10.48\,\text{Gyr}\)
\(3.31\,\text{Gyr}\)
\(17.38\,\text{Gly}\)
\(52.14\,\text{Gly}\)
Epoch of Reionization (First Stars)
\(8.00\)
\(0.111\)
\(13.14\,\text{Gyr}\)
\(650\,\text{Myr}\)
\(30.04\,\text{Gly}\)
\(270.36\,\text{Gly}\)
JWST JADES-GS-z14-0 Record Galaxy
\(14.32\)
\(0.065\)
\(13.50\,\text{Gyr}\)
\(290\,\text{Myr}\)
\(34.12\,\text{Gly}\)
\(522.72\,\text{Gly}\)
CMB Surface of Last Scattering
\(1089.9\)
\(0.0009\)
\(13.79\,\text{Gyr}\)
\(380\,\text{kyr}\)
\(45.65\,\text{Gly}\)
\(49.8\,\text{Tly}\)
The Hubble Tension: Early vs. Late Universe Measurement Discrepancy
1. Early Universe CMB (Planck 2018):
Measures temperature anisotropies in the Cosmic Microwave Background formed 380,000 years after the Big Bang. Combined with the standard ΛCDM cosmological model, it predicts a current expansion rate of \(H_0 = 67.4 \pm 0.5\,\text{km/s/Mpc}\).
2. Late Universe Distance Ladder (SH0ES / HST / JWST):
Uses trigonometric parallax, Cepheid variable stars, and Type Ia supernovae in nearby galaxies to measure local cosmic expansion directly, yielding \(H_0 = 73.0 \pm 1.0\,\text{km/s/Mpc}\). This persistent \(>5\sigma\) statistical discrepancy is known as the Hubble Tension.
Special Relativity prevents matter from moving through space faster than \(c\). In General Relativity, space itself expands. Objects beyond the Hubble Radius (\(R_H = c/H_0 \approx 14.4\,\text{Gly}\)) recede superluminally without violating relativity: \(v_{\text{rec}} = H_0 D_C > c\) for all \(z \gtrsim 1.4\).
2. Particle Horizon vs. Future Event Horizon:
The Particle Horizon (\(46.5\,\text{Gly}\)) marks the boundary of the observable universe today. In contrast, the Cosmic Event Horizon (\(\approx 17.5\,\text{Gly}\)) represents the maximum distance from which a light signal sent today will ever reach us in the infinite future.
Frequently Asked Questions (FAQ)
Authoritative answers to common questions about cosmological expansion, redshift, lookback times, Hubble's Law, and cosmic distances.
Hubble's Law (v = H0 * d) states that galaxies recede from us at speeds proportional to their distance. Rather than galaxies flying through static space, the fabric of spacetime itself is continuously expanding between gravitationally unbound cosmic structures. The expansion rate is quantified by the Hubble Constant H0 (approximately 67.4-73.0 km/s/Mpc).
Cosmological redshift measures the stretching of light wavelengths as photons travel across expanding spacetime: 1 + z = lambda_observed / lambda_emitted. The cosmological scale factor a(t) = 1 / (1 + z) represents the relative physical size of the universe when the light was emitted compared to today (a0 = 1). For example, light observed at redshift z = 1 was emitted when the universe was exactly half (50%) its current linear size.
Comoving Distance (D_C) is the distance between two points measured with coordinates that expand alongside the cosmic Hubble flow, remaining constant over cosmic time. Proper Distance (D_P = D_C / (1+z)) is the physical separation at the time of light emission. Luminosity Distance (D_L = (1+z) * D_C) accounts for the cosmological dimming of light flux due to redshift and photon arrival delays, while Angular Diameter Distance (D_A = D_C / (1+z)) determines the apparent angular size of distant objects on the sky.
Lookback time is calculated by integrating the Friedmann expansion rate: t_L(z) = (1/H0) * integral[0 to z] dz' / ((1+z') * E(z')), where E(z) = sqrt(Omega_r*(1+z)^4 + Omega_m*(1+z)^3 + Omega_k*(1+z)^2 + Omega_Lambda). Subtracting lookback time from the current age of the universe (t0 ≈ 13.787 Gyr for Planck 2018) gives the cosmic age of the universe at the moment the light was emitted.
Yes. Special relativity prevents objects from moving through local space faster than the speed of light c. However, in General Relativity, the metric expansion of spacetime itself is not subject to this speed limit. Galaxies located beyond the Hubble radius (R_H = c/H0 ≈ 4.4 Gpc ≈ 14.4 Gly, corresponding to redshift z > 1.4) recede from us at apparent superluminal velocities without violating any laws of physics.
The Hubble Tension is a statistically significant discrepancy between early-universe Cosmic Microwave Background measurements (H0 = 67.4 ± 0.5 km/s/Mpc from ESA Planck) and late-universe local distance ladder observations (H0 = 73.0 ± 1.0 km/s/Mpc from HST/JWST and Cepheid/Type Ia supernovae). This >5sigma tension suggests potential new physics beyond the standard flat Lambda-CDM cosmological model.
The Lambda-CDM (Lambda Cold Dark Matter) model is the standard cosmological framework. It posits that the universe is spatially flat (Omega_k ≈ 0) and composed of approximately 68.5% Dark Energy (Lambda, accelerating cosmic expansion), 26.5% Cold Dark Matter (non-baryonic matter binding galaxies), 4.9% Ordinary Baryonic Matter (atoms, stars, gas), and 0.01% Radiation (CMB photons and relativistic neutrinos).
Although the universe is approximately 13.8 billion years old, the observable universe has a comoving radius of approximately 46.5 billion light-years (14.26 Gigaparsecs), giving a total diameter of 93 billion light-years. This is because space has been expanding continuously while the ancient light traveled toward Earth over the past 13.8 billion years.
The expansion rate H(z) was significantly higher in the early universe and decreases over time according to the Friedmann equation: H(z) = H0 * sqrt(Omega_m*(1+z)^3 + Omega_Lambda). For example, at redshift z = 1, the expansion rate was approximately 1.75 times faster than today (H(z=1) ≈ 118 km/s/Mpc). In the distant future, as dark energy dominates completely, H(z) approaches a constant value H_infinity = H0 * sqrt(Omega_Lambda) ≈ 56 km/s/Mpc (de Sitter exponential expansion).
The calculator performs numerical Gauss-Legendre quadrature integration of the Friedmann-Lemaître-Robertson-Walker (FLRW) metric equations. It converts any input redshift (z), scale factor (a), lookback time (t_L), or comoving distance (D_C) into full cosmological observables across both Planck 2018 CMB and SH0ES distance ladder parameter sets.
