13.3 Expansion of the universe and Hubble's law

Understanding Hubble's Law and the Expanding Universe

Hubble's Law and Mathematical Formulation

In 1929, Edwin Hubble observed that nearly every galaxy he measured was moving away from us, and that more distant galaxies were receding faster. This wasn't a coincidence or a local effect. It was evidence that the fabric of space itself is stretching, carrying galaxies along with it.

Hubble's Law captures this relationship in a simple equation:

$v = H_0 \times d$

• $v$ is the recessional velocity of the galaxy (how fast it's moving away from us), in km/s
• $d$ is the galaxy's distance from us, measured in megaparsecs (Mpc)
• $H_0$ is the Hubble constant, which quantifies the current expansion rate

The best current measurements place $H_0$ at approximately 70 km/s/Mpc. That means a galaxy 1 Mpc away (about 3.26 million light-years) recedes at roughly 70 km/s. A galaxy 10 Mpc away recedes at about 700 km/s.

One subtlety worth noting: this law doesn't mean we're at the center of the expansion. Every observer in the universe would see the same pattern. Think of dots on a balloon being inflated: every dot sees every other dot moving away, with more distant dots moving faster.

Redshift and Universe Expansion

As space expands, it stretches the wavelength of light traveling through it. This shifts the light toward the red end of the spectrum, a phenomenon called cosmological redshift. It's related to, but distinct from, a standard Doppler shift. A Doppler shift comes from an object's motion through space. Cosmological redshift comes from space itself expanding while the light is in transit.

The redshift $z$ is defined as:

$z = \frac{\lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}}$

• $\lambda_{emitted}$ is the wavelength of light when it was originally produced (at the source galaxy)
• $\lambda_{observed}$ is the wavelength we measure when the light arrives

A redshift of $z = 1$ means the observed wavelength is twice the emitted wavelength. Greater redshift corresponds to faster recession and greater distance. This is how astronomers determine that the universe is expanding: virtually every distant galaxy shows redshifted spectral lines.

Universe Age and Size Calculations

Hubble's Law gives a rough way to estimate the age of the universe. If you imagine "rewinding" the expansion at a constant rate, all galaxies would converge to a single point after a time called the Hubble time:

$t_H = \frac{1}{H_0}$

With $H_0 \approx 70$ km/s/Mpc, this gives $t_H \approx 14$ billion years. The actual accepted age of the universe is about 13.8 billion years, which is close but not identical because the expansion rate hasn't been constant. It decelerated early on (when matter's gravity dominated) and has been accelerating more recently (driven by dark energy).

Similarly, the Hubble radius estimates the size of the observable universe:

$R_H = \frac{c}{H_0}$

where $c$ is the speed of light. This yields roughly 14.4 billion light-years. The actual radius of the observable universe is much larger (about 46.5 billion light-years) because space has been expanding throughout the time light has been traveling toward us.

Implications of Cosmic Expansion

The expansion of the universe has profound consequences for its long-term fate. Which scenario plays out depends on the balance between gravity (pulling things together) and dark energy (pushing things apart):

1. Big Freeze: Expansion continues indefinitely. Stars burn out, galaxies drift apart, and the universe approaches thermodynamic equilibrium, a state of maximum entropy sometimes called "heat death." This is currently the most favored scenario given observations of accelerating expansion.
2. Big Crunch: If gravity were strong enough to overcome expansion, the universe would eventually stop expanding and collapse back on itself. Current evidence makes this unlikely.
3. Big Rip: If dark energy strengthens over time, expansion could accelerate so dramatically that it tears apart galaxies, solar systems, atoms, and eventually spacetime itself.

Dark energy is the key variable here. Observations of distant Type Ia supernovae in 1998 revealed that the expansion of the universe is accelerating, not slowing down. Dark energy accounts for roughly 68% of the total energy content of the universe, yet its fundamental nature remains one of the biggest open questions in physics.

Other consequences of ongoing expansion:

• The cosmic microwave background (CMB) has cooled from thousands of kelvin to its current temperature of about 2.7 K as the universe expanded and stretched those ancient photons to microwave wavelengths.
• Intergalactic distances keep growing, which means galaxy merger rates will decrease over time.
• Galaxies currently near the edge of our observable universe will eventually cross beyond the cosmic horizon and become permanently undetectable, even in principle.