Hubble Parameter H(z) & Expansion History
Despite being called the 'Hubble constant' at its present-day value H₀, the Hubble parameter H(t) = (da/dt)/a changes continuously with cosmic time. It was much larger in the early universe, decreased as gravity decelerated expansion during the radiation- and matter-dominated eras, and is now asymptotically approaching a constant value (~56 km/s/Mpc) as dark energy dominates.
Physical Mechanism & Dynamics
From the Friedmann equation, H(z) scales with the dominant energy component: H(z) ∝ (1 + z)² during radiation domination (z > 3,400), H(z) ∝ (1 + z)^(3/2) during matter domination (3,400 > z > 0.3), and H(z) → H₀ √Ω_Λ during cosmological-constant domination. Note that while H(t) decreases toward a constant positive floor, the physical recession velocity of any individual galaxy v = da/dt · χ switched from decelerating (d²a/dt² < 0) to accelerating (d²a/dt² > 0) at transition redshift z_acc ≈ 0.64.
Astronomers measure H(z) directly at high redshifts using radial Baryon Acoustic Oscillations (which measure Δz across the known sound horizon r_d) and Lyman-alpha forest cross-correlations in quasar spectra.
Quantitatively, this mechanism is governed by Dimensionless Expansion Rate and Deceleration Parameter (E(z) ≡ H(z) / H₀ = √[ Ω_m (1 + z)³ + Ω_Λ ] ; q(z) = Ω_m(z)/2 − Ω_Λ(z)). Today q₀ = Ω_m/2 − Ω_Λ ≈ 0.315/2 − 0.685 = −0.53 (negative q means accelerating scale factor d²a/dt² > 0).
Dimensionless Expansion Rate and Deceleration Parameter
E(z) ≡ H(z) / H₀ = √[ Ω_m (1 + z)³ + Ω_Λ ] ; q(z) = Ω_m(z)/2 − Ω_Λ(z)Today q₀ = Ω_m/2 − Ω_Λ ≈ 0.315/2 − 0.685 = −0.53 (negative q means accelerating scale factor d²a/dt² > 0).
Quantitative Benchmarks & Parameters
| Parameter / Observable | Measured Value / Scale | Method & Context |
|---|---|---|
| Present value H(z = 0) | 67.4 to 73.0 km/s/Mpc | Early-universe vs local distance-ladder bounds |
| H(z = 1.0) at t ≈ 5.9 Gyr | ~120 km/s/Mpc | BOSS/DESI BAO radial clustering |
| H(z = 2.3) in Lyα forest | ~227 ± 4 km/s/Mpc | SDSS-IV eBOSS / DESI quasar Lyα BAO |
| Acceleration onset z_acc | z ≈ 0.64 (t ≈ 7.7 Gyr) | Epoch where d²a/dt² = 0 (Ω_m(1+z)³ = 2Ω_Λ) |
Observational Evidence & How Scientists Know
Mapping H(z) and angular diameter distance D_M(z) across 0.1 < z < 2.5 in SDSS and DESI galaxy and quasar surveys confirms the transition from matter deceleration to dark-energy acceleration independently of supernovae.
Across reference datasets (Planck 2018 Cosmological Parameters (PR3), SH0ES Local Distance Ladder Hubble Constant), quantitative benchmarks for hubble parameter h(z) & expansion history are anchored by: Present value H(z = 0) (67.4 to 73.0 km/s/Mpc, via Early-universe vs local distance-ladder bounds); H(z = 1.0) at t ≈ 5.9 Gyr (~120 km/s/Mpc, via BOSS/DESI BAO radial clustering); H(z = 2.3) in Lyα forest (~227 ± 4 km/s/Mpc, via SDSS-IV eBOSS / DESI quasar Lyα BAO); Acceleration onset z_acc (z ≈ 0.64 (t ≈ 7.7 Gyr), via Epoch where d²a/dt² = 0 (Ω_m(1+z)³ = 2Ω_Λ)).
Uncertainties, Systematics & Open Questions
Recent DESI BAO + supernova combinations test whether the dark energy equation-of-state w(z) = w₀ + w_a z/(1+z) evolves slightly away from w = −1, which would modify H(z) at intermediate redshifts.
Real Astronomical Benchmarks & Cross-Domain Links
Redshift & Lookback Explorer
Interactive H(z), lookback time, and distance calculator
Expansion of Space
Metric scale factor a(t) foundations
Baryon Acoustic Oscillations
Standard ruler used to measure H(z) directly
Guided Learning Path
Related Concepts, Methods & Explorers
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 ↗
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