Know that the current estimate for H_0 is 2.2 × 10^-18 per second

6.2.3 The Universe

Learning Objective

Students should be able to state that the current best estimate for the Hubble constant is

\$H_0 = 2.2 \times 10^{-18}\ \text{s}^{-1}\$

What is the Hubble Constant?

The Hubble constant (\$H_0\$) describes the rate at which the Universe is expanding. It relates the recession velocity (\$v\$) of a distant galaxy to its distance (\$d\$) from us:

\$v = H_0 \, d\$

where:

• \$v\$ is the galaxy’s recession speed (usually expressed in km s⁻¹).
• \$d\$ is the distance to the galaxy (commonly in megaparsecs, Mpc).
• \$H_0\$ is the proportionality constant – the Hubble constant.

Units and Conversions

In cosmology the Hubble constant is often quoted in units of km s⁻¹ Mpc⁻¹. The relationship between the two common units is:

\$\$1\ \text{km s}^{-1}\text{Mpc}^{-1} = \frac{1\ \text{km}}{1\ \text{s}\times 1\ \text{Mpc}}

= \frac{10^3\ \text{m}}{1\ \text{s}\times 3.086\times10^{22}\ \text{m}}

= 3.2408\times10^{-20}\ \text{s}^{-1}\$\$

Therefore, the current estimate can also be expressed as:

\$\$H_0 = \frac{2.2 \times 10^{-18}\ \text{s}^{-1}}{3.2408\times10^{-20}\ \text{s}^{-1}\,\text{(km s}^{-1}\text{Mpc}^{-1})}

\approx 68\ \text{km s}^{-1}\text{Mpc}^{-1}\$\$

How is \$H_0\$ Measured?

1. Standard Candles (e.g., Cepheid variables, Type Ia supernovae): Measure the intrinsic brightness, compare with observed brightness to obtain distance, then use red‑shift to get velocity.
2. Cosmic Microwave Background (CMB) fitting: Use observations of the CMB power spectrum (e.g., from the Planck satellite) and fit cosmological models to infer \$H_0\$.
3. Gravitational‑wave “standard sirens”: Measure the distance from the waveform of merging neutron stars and combine with red‑shift from host galaxies.

Current \cdot alue and Uncertainty

Different methods give slightly different results, leading to the so‑called “Hubble tension”. The value quoted here, \$2.2 \times 10^{-18}\ \text{s}^{-1}\$ (≈ 68 km s⁻¹ Mpc⁻¹), represents a consensus average of recent measurements.

Example Calculation

Suppose a galaxy is observed to have a red‑shift corresponding to a recession speed of \$v = 10\,200\ \text{km s}^{-1}\$. Using \$H_0 = 68\ \text{km s}^{-1}\text{Mpc}^{-1}\$, its distance is:

\$d = \frac{v}{H_0} = \frac{10\,200\ \text{km s}^{-1}}{68\ \text{km s}^{-1}\text{Mpc}^{-1}} \approx 150\ \text{Mpc}\$

Key Points to Remember

• The Hubble constant quantifies the expansion rate of the Universe.
• Current best estimate: \$H_0 = 2.2 \times 10^{-18}\ \text{s}^{-1}\$ (≈ 68 km s⁻¹ Mpc⁻¹).
• It is derived from multiple independent observational techniques.
• Understanding \$H_0\$ is essential for estimating cosmic distances and the age of the Universe.

Suggested Diagram

Reference Table