Universe Expansion Calculator

Compute recession velocities, Hubble distances, light travel time, and cosmic age based on redshift and H₀. Includes relativistic formulas and ΛCDM context.

Relativistic velocity: v = c·[(1+z)²−1]/[(1+z)²+1]
Hubble's law (low z): v ≈ H₀·D
Cosmological redshift (z ≥ 0). Example: z=1 means wavelength doubled.
z=0.1 (nearby) z=1.0 z=2.5 (quasar) z=1100 (CMB)
Current estimates: Planck 2018 (67.4), SH0ES (73.0), traditional (70).
Planck 2018 H₀=70 SH0ES 73 Traditional
Computing cosmological parameters...

Understanding Universe Expansion

The expansion of the universe is one of the most profound discoveries in modern science. It implies that galaxies are moving away from each other, and that space itself is stretching. Below we explore the key concepts and the current cosmological model.

Hubble's Law (1929): Edwin Hubble found that distant galaxies recede at a speed proportional to their distance: v = H₀·D. The proportionality constant H₀ is the Hubble constant, which sets the scale and age of the universe.

1. Redshift and Scale Factor

Cosmological redshift arises because light waves are stretched by the expansion of space. If the universe expands by a factor (1+z) while light travels, the observed wavelength is multiplied by (1+z). The scale factor a(t) describes how distances change with time; today a₀ = 1, and at redshift z, a = 1/(1+z). For example, at z = 1 the universe was half its present size.

2. Relativistic vs. Classical Velocity

For small redshifts (z ≲ 0.1), the simple formula v = cz works well. But at high redshift, special relativity must be used because velocities approach the speed of light. The correct formula (derived from the relativistic Doppler effect) is:

v = c \frac{(1+z)^2 - 1}{(1+z)^2 + 1}

At z = 1, v ≈ 0.6c; at z → ∞, v → c. Our calculator shows both for comparison.

3. Hubble Constant and Its Tension

The Hubble constant is measured in km/s per megaparsec (Mpc). Current values:

  • Planck 2018 (CMB): 67.4 ± 0.5 km/s/Mpc
  • SH0ES 2022 (supernovae): 73.0 ± 1.0 km/s/Mpc

The 5–6 km/s/Mpc discrepancy, known as the Hubble tension, may indicate new physics beyond the standard ΛCDM model (e.g., early dark energy, modified gravity).

4. Hubble Time and Universe Age

The inverse of H₀ gives a characteristic timescale: t_H = 1/H₀. In a universe that expanded at constant rate, the age would be t_H. For H₀ = 70 km/s/Mpc, t_H ≈ 14.0 billion years. In our ΛCDM universe with matter and dark energy, the actual age is slightly less: about 13.8 billion years (Planck).

5. The ΛCDM Model

The current standard model of cosmology includes:

  • Ordinary matter (baryons) ~5%
  • Dark matter ~27%
  • Dark energy (Λ) ~68%

Dark energy causes the expansion to accelerate, which was discovered in 1998 via supernovae observations. The universe's expansion history is described by the Friedmann equations.

6. Lookback Time and Distance

The light we see from a galaxy at redshift z was emitted when the universe was younger. The lookback time t_L is the difference between the present age and the age at emission. It depends on the cosmological parameters (Ω_m, Ω_Λ). Our calculator's "light travel time" using D/c is a rough estimate; for accurate lookback time, one must integrate the Friedmann equation. For example, at z=1, the true lookback time is about 7.7 Gyr (not 10.4 Gyr as simple D/c would give). We recommend using dedicated cosmology calculators for precise values.

7. Cosmic Microwave Background (z ≈ 1100)

The CMB was emitted when the universe became transparent, at redshift z ≈ 1100. At that time, the scale factor was a = 1/1100, and the universe was just 380,000 years old. This radiation provides our most ancient view of the cosmos.

Key Takeaways: The universe expands, galaxies recede faster the farther they are. Redshift measures the stretching of light. H₀ quantifies the current expansion rate. The simple linear relation v = cz only works nearby; at high z we need relativity. The ΛCDM model, with dark energy, best explains observations.

Further Reading

NASA's WMAP Cosmology 101 • Planck Mission Results • Ned Wright's Cosmology Tutorial

Frequently Asked Questions

For redshifts above ~0.1, the simple v = cz overestimates the true velocity because it ignores special relativity. The relativistic formula accounts for the correct addition of velocities and is valid for any z. At z=1, cz = c (299,792 km/s), while the true velocity is about 0.6c (180,000 km/s).

Measurements of H₀ from the early universe (CMB) give ≈67.4 km/s/Mpc, while late‑universe observations (supernovae, SH0ES) yield ≈73 km/s/Mpc. This discrepancy, known as the "Hubble tension", may indicate new physics beyond the standard ΛCDM model.

The Hubble distance derived here assumes a linear relation and is accurate only for z ≲ 0.5. For precise cosmological distances (comoving, luminosity distance) one needs to integrate the Friedmann equations with matter/dark energy densities. This calculator provides a simple educational approximation.

The scale factor a(t) describes how distances expand with time. It is defined as a = 1/(1+z). At redshift z, the universe was a factor 1/(1+z) smaller than today. For CMB (z=1100), a ≈ 1/1100, meaning the universe was 1100 times smaller.