Age of the Universe (Hubble Time, Approx.)

The Hubble tension is a persistent disagreement between two families of measurements of H₀. Early-universe probes — chiefly the Planck satellite's map of the cosmic microwave background — give H₀ = 67.4 ± 0.5 km/s/Mpc when interpreted through the standard ΛCDM model. Late-universe probes — Cepheid-calibrated Type Ia supernovae (SH0ES) and the tip of the red giant branch (TRGB) — give H₀ = 73.0 ± 1.0 and 69.8 ± 1.7 respectively. The discrepancy is around 5σ if you trust both error bars. It matters enormously for the age: at H₀ = 67.4, t_H = 14.51 Gyr; at H₀ = 73.0, t_H = 13.39 Gyr. That is a 1.1 Gyr swing in the estimated age from a single disputed parameter.

f is the ratio of the actual ΛCDM age to the Hubble time. It encodes the fact that the universe has not expanded at a constant rate. In a purely matter-dominated (Einstein–de Sitter) universe, expansion follows a ∝ t^2/3, and integrating gives t₀ = (2/3) · t_H — a much younger universe. Add dark energy and the late-time acceleration stretches the age back upward toward t_H. For the Planck 2018 parameters Ω_m = 0.315, Ω_Λ = 0.685, the exact Friedmann integral gives f ≈ 0.951, though the commonly-quoted approximate value is 0.964. The two numbers differ because different authors use slightly different Ω_m; the qualitative point is the same — f is a number close to but slightly less than 1.

It is purely a unit conversion. The Hubble constant has units of inverse time once you convert km/s/Mpc into SI: H₀ = 70 km/s/Mpc = 70 × 1000 m/s / (3.0857 × 10²² m) = 2.269 × 10⁻¹⁸ s⁻¹. The reciprocal gives t_H = 4.408 × 10¹⁷ s. Converting seconds to years (1 yr = 3.15576 × 10⁷ s) gives 1.397 × 10¹⁰ yr = 13.97 Gyr. Assembling all the factors: t_H [Gyr] = 3.0857 × 10¹⁹ / (3.15576 × 10⁷ × 10⁹ × H₀) = 977.79 / H₀. The constant bundles together the megaparsec in kilometres and the year in seconds; nothing physical is hidden in it.

The 13.80 ± 0.02 Gyr figure comes from the Planck 2018 analysis of the cosmic microwave background. By fitting the observed temperature and polarisation power spectra to the six-parameter ΛCDM model, the Planck team simultaneously constrains H₀, Ω_m, Ω_Λ, and the other cosmological parameters, then integrates the Friedmann equation backward in time to obtain the age. Independent checks come from three other directions: the ages of the oldest globular clusters (12.5–13.6 Gyr from main-sequence turnoff and white-dwarf cooling sequences), the uranium/thorium cosmochronometer in metal-poor halo stars (~12.5–14 Gyr), and the cooling sequence of ancient white dwarfs. All four methods agree within their uncertainties, giving a lower bound of about 12.5 Gyr and a best estimate near 13.8 Gyr.

No — that would be a logical contradiction, and it has historically been one of the sharpest checks on cosmological models. In the 1990s, some measurements of H₀ gave values near 80–100 km/s/Mpc, which implied Hubble times of 10–12 Gyr, uncomfortably shorter than the oldest globular clusters at 13–14 Gyr. The discrepancy helped drive the discovery of dark energy: adding a cosmological constant stretches the age back up toward the Hubble time and reconciles the two. Today the situation is comfortable, but the principle remains: the age of the universe must be greater than or equal to the age of anything inside it, and any cosmology that violates that bound is immediately falsified.

The Hubble-time relation t_H = 1/H₀ is model-independent — it depends only on the present expansion rate, not on what the universe is made of or how it evolved. The ΛCDM correction factor f = 0.964, however, is specific to the concordance model with Ω_m ≈ 0.315 and Ω_Λ ≈ 0.685. For a different cosmology — a different Ω_m, a different dark-energy equation of state w, or a modified-gravity model — the exact Friedmann integral would give a different f. The Hubble time would still be exactly 1/H₀; only the correction to the actual age would change. For pure dark energy (Ω_Λ = 1), f = 1; for pure matter (Ω_m = 1), f = 2/3.

Hubble's law states that a galaxy at distance d recedes at speed v = H₀ · d, where H₀ is the Hubble constant. Because the recession speed is proportional to distance, H₀ has units of inverse time — km/s per Mpc, which reduces to s⁻¹. The reciprocal 1/H₀ gives a timescale: the time it would take for a galaxy to reach its current distance from a common origin if the expansion rate had always been H₀. That timescale is the Hubble time. It is not exactly the age of the universe because the expansion rate has changed, but it is the correct order of magnitude and provides the natural reference timescale for cosmological calculations. Cosmologists quote both t_H (from H₀ directly) and t₀ (from the Friedmann integral) because they answer slightly different questions.