Hubble's Law

Hubble's law

v = H₀ · d

Symbol	Meaning	Units
v	Recession velocity	km/s
d	Distance to galaxy	Mpc (megaparsec)
H₀	Hubble constant	km/s/Mpc

1 parsec = 3.086 × 10¹⁶ m ≈ 3.26 light-years. 1 Mpc = 10⁶ pc = 3.26 million light-years.

Measurement methods

Method	Distance ladder rung	H₀ (km/s/Mpc)
Cepheid variables → SN Ia (SH0ES)	Calibrated steps	73.0 ± 1.0
CMB (Planck)	Cosmological model + CMB peaks	67.4 ± 0.5
BAO (galaxy clustering)	Sound horizon as standard ruler	~67-68
Gravitational waves (standard sirens)	Strain + EM counterpart	~70 (preliminary)
Lensing time delays (H0LiCOW)	Time delay lensing	73-74

The 4-9% discrepancy between CMB-based and local-distance-ladder methods is "Hubble tension."

JavaScript — Hubble's law

// H_0 in usable units
const H_0_kmsMpc = 70;  // km/s/Mpc
const Mpc_to_m = 3.086e22;
const H_0_per_s = H_0_kmsMpc * 1000 / Mpc_to_m;  // s⁻¹
console.log(`H_0 = ${H_0_per_s.toExponential(2)} per second`);

// Hubble time
const t_H = 1 / H_0_per_s;
console.log(`Hubble time: ${(t_H / (365.25*86400) / 1e9).toFixed(1)} Gyr`);

// Recession velocity from Hubble's law
function recessionVelocity(distance_Mpc) {
  return H_0_kmsMpc * distance_Mpc;
}

console.log(`Galaxy at 100 Mpc: v = ${recessionVelocity(100)} km/s`);
console.log(`Galaxy at 1000 Mpc: v = ${recessionVelocity(1000)} km/s`);
console.log(`Galaxy at 4000 Mpc: v = ${recessionVelocity(4000)} km/s`);
// 280,000 km/s — close to c

// Distance from redshift (small z)
function distanceFromRedshift(z) {
  // Small-z: v ≈ z·c, so d = v/H_0
  const c_kmps = 299792;
  const v = z * c_kmps;
  return v / H_0_kmsMpc;  // Mpc
}

console.log(`z=0.01: ${distanceFromRedshift(0.01).toFixed(1)} Mpc`);
console.log(`z=0.1: ${distanceFromRedshift(0.1).toFixed(0)} Mpc`);
// At z=1, full GR needed (cosmological model)

// Critical density (boundary between expanding forever and recollapsing)
function criticalDensity(H_0_per_s) {
  const G = 6.674e-11;
  return 3 * H_0_per_s * H_0_per_s / (8 * Math.PI * G);
}

console.log(`Critical density: ${criticalDensity(H_0_per_s).toExponential(2)} kg/m³`);
// ~9.5 × 10⁻²⁷ kg/m³ — about 6 H atoms per m³

// Density parameter Ω
function omega(actualDensity, H_0_per_s) {
  return actualDensity / criticalDensity(H_0_per_s);
}

// Hubble tension: compare two values
function hubbleTension(H_planck, H_local) {
  const sigma = Math.sqrt(0.5 * 0.5 + 1.0 * 1.0);  // combined uncertainty
  return (H_local - H_planck) / sigma;  // sigmas
}

console.log(`Tension: ${hubbleTension(67.4, 73.0).toFixed(1)} σ`);  // ~5 σ

Where Hubble's law matters

• Cosmology. Foundation of expanding universe, Big Bang theory.
• Distance estimation. Redshift → distance for galaxies (with caveats for high z).
• Age of universe. Hubble time gives rough estimate; full calculation requires cosmological model.
• Dark energy detection. Acceleration discovered via SNe Ia at high z (1998 Nobel).
• Standard sirens. Gravitational waves provide independent H₀ measurement.
• Hubble tension. Major puzzle in modern cosmology; potential new physics signal.
• JWST observations. Extending observations to highest redshift galaxies.

Common mistakes

• Treating recession velocity as motion through space. It's space itself expanding; galaxies are approximately at rest locally.
• Believing Hubble's law applies to local objects. Within galaxy clusters, gravity dominates; Hubble's law applies on supercluster scales.
• Assuming v = H₀·d holds for very distant objects. At high z, full general relativistic cosmology needed; simple linear law breaks down.
• Confusing Hubble time with universe age. Hubble time is 1/H₀ — universe age depends on full expansion history. Currently ~13.8 Gyr, close to but different from Hubble time.
• Treating Hubble constant as truly constant. "Constant" in space (uniform expansion), not in time. H_0 is current value; H(t) varies with cosmic time.
• Confusing Hubble tension with measurement error. Tension is statistically significant (5+ σ); not just imprecision but apparently real disagreement.