6.4 The Hubble-Lemaître law and its implications

The Hubble-Lemaître Law

The Hubble-Lemaître Law describes a direct relationship between how far away a galaxy is and how fast it's moving away from us. This single equation gives cosmologists a way to estimate distances to remote galaxies, approximate the age of the universe, and provide key observational evidence for the Big Bang theory.

Hubble-Lemaître Law Formulation

The law states that a galaxy's recessional velocity is directly proportional to its distance from us:

$v = H_0 \cdot d$

• $v$ = recessional velocity of the galaxy (km/s)
• $d$ = distance to the galaxy (Mpc, or megaparsecs)
• $H_0$ = the Hubble constant (km/s/Mpc)

This proportional relationship means that if you double the distance, you double the recessional velocity. A galaxy 200 Mpc away recedes twice as fast as one at 100 Mpc.

Hubble Constant Interpretation

$H_0$ quantifies the current expansion rate of the universe. Its units, km/s/Mpc, tell you how much additional recessional velocity you get for each additional megaparsec of distance.

For example, if $H_0 = 70$ km/s/Mpc, then for every 1 Mpc farther away a galaxy is, its recessional velocity increases by 70 km/s. A galaxy at 100 Mpc would be receding at $70 \times 100 = 7{,}000$ km/s.

The precise value of $H_0$ is still debated. Measurements from the cosmic microwave background (via the Planck satellite) give roughly 67.4 km/s/Mpc, while measurements using nearby supernovae and Cepheid variables give closer to 73 km/s/Mpc. This discrepancy is known as the Hubble tension and is an active area of research.

Galaxy Distance Calculations

If you know a galaxy's recessional velocity (measured from its redshift) and the value of $H_0$, you can rearrange the law to find its distance:

$d = \frac{v}{H_0}$

Worked example:

1. A galaxy has a measured recessional velocity of $v = 50{,}000$ km/s.
2. Assume $H_0 = 70$ km/s/Mpc.
3. Plug into the equation: $d = \frac{50{,}000}{70} \approx 714$ Mpc.

So the galaxy is roughly 714 megaparsecs away. Keep in mind this method works best for galaxies far enough away that the overall cosmic expansion dominates over local motions (called peculiar velocities). Nearby galaxies can have peculiar velocities large enough to distort the result.

Implications and Relation to the Expanding Universe

Implications of the Hubble-Lemaître Law

The law carries several major consequences for our understanding of the cosmos.

Age of the universe: The inverse of the Hubble constant, $1/H_0$, gives an estimate called the Hubble time. This represents how long the universe has been expanding, assuming the expansion rate has been constant. With $H_0 = 70$ km/s/Mpc, converting units gives a Hubble time of roughly 14 billion years. The actual age (about 13.8 billion years from more detailed models) differs slightly because the expansion rate has not been constant: it decelerated due to gravity for most of cosmic history and has been accelerating more recently due to dark energy.

Size of the observable universe: The law implies a finite observable universe. There's a maximum distance from which light has had time to reach us since the Big Bang. Beyond that horizon, objects exist but their light hasn't arrived yet.

Fate of the universe: What happens in the long run depends on the density of matter and energy compared to a critical density:

• If the total density exceeds the critical value, gravity could eventually halt and reverse the expansion (a "Big Crunch").
• If the density is below the critical value, expansion continues forever, leading to a "Big Freeze" or "Heat Death" as the universe cools toward maximum entropy.
• Current observations suggest the expansion is actually accelerating due to dark energy, making continued expansion the most likely outcome.

The Expanding Universe Concept

The Hubble-Lemaître Law provides direct observational evidence that the universe is expanding. An important distinction: the galaxies aren't flying through space away from us. Instead, the space between galaxies is stretching. Every galaxy sees all other distant galaxies receding, so there's no special center of expansion.

This expansion is a central prediction of the Big Bang theory, which describes the universe beginning in an extremely hot, dense state and expanding and cooling ever since. Several independent lines of evidence support this picture:

• Cosmic microwave background (CMB): Faint thermal radiation left over from about 380,000 years after the Big Bang, detected uniformly in all directions.
• Light element abundances: The observed ratios of hydrogen, helium, and lithium match predictions from nucleosynthesis in the first few minutes of the universe.
• Large-scale structure: The distribution of galaxy clusters, filaments, and voids across the cosmos matches models of how matter clumps together in an expanding universe over billions of years.

Together with the Hubble-Lemaître Law, these observations form a consistent and well-tested framework for understanding cosmic history.