Know that the equation d / v = 1 / H_0 represents an estimate for the age of the Universe and that this is evidence for the idea that all the matter in the Universe was present at a single point
represents an estimate for the age of the Universe and that this is evidence for the idea that all the matter in the Universe was present at a single point.
Key Concepts
Hubble’s Law: The recession velocity v of a galaxy is proportional to its distance d from us, expressed as v = H₀ d.
Hubble Constant (H₀): The proportionality constant in Hubble’s Law. Current measurements give values in the range 67–74 km s⁻¹ Mpc⁻¹.
Age of the Universe: By rearranging Hubble’s Law, the time taken for a galaxy to travel its current distance at its present speed is t = d/v = 1/H₀. This provides a first‑order estimate of the Universe’s age.
Big Bang Evidence: If all matter originated from a single point, the Universe would have expanded uniformly. The finite age derived from 1/H₀ supports this expansion model.
Derivation of the Age Estimate
Starting from Hubble’s Law:
\$v = H_{0} d\$
Rearrange to isolate the ratio d/v:
\$\frac{d}{v} = \frac{1}{H_{0}}\$
The left‑hand side has units of time (distance ÷ speed). Therefore, 1/H₀ gives an estimate of the time elapsed since the expansion began – i.e., the age of the Universe.
Numerical Example
Assume a Hubble constant of H₀ = 70 km s⁻¹ Mpc⁻¹. Convert to SI units:
1 Mpc = 3.09 × 10²² m
70 km s⁻¹ Mpc⁻¹ = 70 000 m s⁻¹ ÷ 3.09 × 10²² m ≈ 2.27 × 10⁻¹⁸ s⁻¹
Thus, the simple estimate gives an age of about 14 billion years, close to the modern value of ≈13.8 billion years.
Table: Hubble Constant \cdot alues and Corresponding Ages
H₀ (km s⁻¹ Mpc⁻¹)
H₀ (s⁻¹)
1/H₀ (seconds)
Age (billion years)
67
2.17 × 10⁻¹⁸
4.61 × 10¹⁷
14.6
70
2.27 × 10⁻¹⁸
4.40 × 10¹⁷
14.0
74
2.40 × 10⁻¹⁸
4.17 × 10¹⁷
13.2
Interpretation and Significance
The inverse relationship between distance and velocity (Hubble’s Law) implies that the Universe is expanding.
If the expansion is extrapolated backward, all galaxies converge to a single point, supporting the Big Bang model.
The age derived from 1/H₀ provides a simple, model‑independent check on more sophisticated cosmological calculations.
Discrepancies between different measurements of H₀ (e.g., cosmic microwave background vs. supernovae) are an active research area, but the basic concept remains a cornerstone of modern cosmology.
Suggested diagram: A plot of recession velocity (v) versus distance (d) for galaxies, showing a straight line whose slope is the Hubble constant H₀. Include a point illustrating the inverse of the slope as the estimated age of the Universe.
Summary
The equation d/v = 1/H₀ gives a first‑order estimate of the Universe’s age by treating the current expansion rate as if it had been constant since the beginning. This simple calculation yields a value of roughly 14 billion years, providing strong evidence that the Universe began from a highly dense, hot state—a single point—consistent with the Big Bang theory.