12.1 Hubble's Law and Distance Measurements

Hubble's Law and Redshift

Hubble's Law and the Expanding Universe

Hubble's Law describes a direct proportionality between a galaxy's distance from us and the speed at which it recedes. The farther a galaxy is, the faster it moves away. Edwin Hubble established this empirical relationship in 1929, building on observations of Cepheid variable stars in what were then called "spiral nebulae" to pin down their distances.

The law is expressed as:

$v = H_0 \times d$

where $v$ is the recession velocity (in km/s), $H_0$ is the Hubble constant, and $d$ is the distance to the galaxy (in Mpc). This linear relationship provided the first strong observational evidence that the universe is expanding. Note that Hubble's Law applies to the large-scale motion of galaxies due to cosmic expansion; local motions (peculiar velocities) can cause individual galaxies to deviate from the relation.

Redshift and the Doppler Effect

When a galaxy moves away from us, the light it emits gets stretched to longer wavelengths. This shift toward the red end of the spectrum is called redshift. You can quantify it with:

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

where $\lambda$ is wavelength. A galaxy with $z = 0.01$, for example, has its spectral lines shifted by 1% toward longer wavelengths.

Two distinct physical mechanisms produce redshift:

• Doppler redshift arises from the relative motion of a source through space, the same effect that changes the pitch of a passing siren. For galaxies at low redshift ($z \ll 1$), the recession velocity is well approximated by $v \approx cz$.
• Cosmological redshift is fundamentally different. It results from the expansion of space itself stretching the wavelength of photons as they travel. At higher redshifts, this distinction matters: the galaxy isn't flying through space at near-light speed; rather, the space between us and the galaxy has expanded while the light was in transit.

For nearby galaxies, both descriptions give the same answer. At cosmological distances, you need general relativity and the cosmological redshift framework.

The Hubble Constant and Its Significance

The Hubble constant ($H_0$) sets the current expansion rate of the universe. Its units, km/s/Mpc, tell you how much faster a galaxy recedes for each additional megaparsec of distance. A value of $H_0 \approx 70 \text{ km/s/Mpc}$ means a galaxy at 100 Mpc recedes at roughly 7,000 km/s.

Why does pinning down $H_0$ matter so much?

• It directly determines the distance scale of the universe. Every cosmological distance estimate that relies on redshift depends on $H_0$.
• Its inverse gives the Hubble time, a first-order estimate of the universe's age.
• It constrains cosmological parameters like the density of dark energy and matter.

Current measurements cluster around two values that don't quite agree. The cosmic microwave background (CMB) measurements from the Planck satellite yield $H_0 \approx 67.4 \text{ km/s/Mpc}$, while local distance-ladder measurements using Cepheids and Type Ia supernovae give $H_0 \approx 73 \text{ km/s/Mpc}$. This discrepancy, known as the Hubble tension, exceeds 4σ and remains one of the most actively investigated problems in cosmology. Whether it signals new physics or unresolved systematics is still an open question.

Hubble Diagram and Time

The Hubble-Lemaître Law and Hubble Diagram

The relationship is now formally called the Hubble-Lemaître Law, recognizing Georges Lemaître's independent derivation of the velocity-distance relation from general relativity in 1927, two years before Hubble's observational paper.

The Hubble diagram is the key visualization: it plots recession velocity (y-axis, in km/s) against distance (x-axis, in Mpc). For a uniformly expanding universe, this plot yields a straight line whose slope equals $H_0$.

• At low redshifts ($z \lesssim 0.1$), the relationship is cleanly linear.
• The slope of the best-fit line through the data gives the measured value of $H_0$.
• Extending the diagram to higher redshifts requires reliable distance indicators that reach far into the Hubble flow, which is where Type Ia supernovae become essential.

Interpreting the Hubble Diagram

Scatter in the Hubble diagram comes from two main sources: uncertainties in distance measurements and peculiar velocities, the random motions galaxies have on top of the Hubble flow (typically a few hundred km/s). For nearby galaxies, peculiar velocities can be comparable to the Hubble flow velocity, which is why the diagram is cleaner at larger distances.

The most consequential feature of the modern Hubble diagram appears at high redshift. In the late 1990s, two teams (the Supernova Cosmology Project and the High-z Supernova Search Team) used Type Ia supernovae to extend the diagram and found that distant supernovae were fainter than expected for a universe expanding at a constant rate. This deviation from linearity at large distances provided the first direct evidence for the accelerating expansion of the universe, driven by dark energy.

So the Hubble diagram isn't just a confirmation of expansion. At different distance scales, it encodes the entire expansion history of the universe and constrains the cosmological model.

Hubble Time and the Age of the Universe

The Hubble time is defined as:

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

This gives the age the universe would have if it had always expanded at its current rate. For $H_0 = 70 \text{ km/s/Mpc}$:

$t_H = \frac{1}{70 \text{ km/s/Mpc}} \approx 14.0 \text{ Gyr}$

(To get this number, you convert megaparsecs to kilometers: $1 \text{ Mpc} \approx 3.086 \times 10^{19} \text{ km}$, then invert and convert seconds to years.)

The Hubble time is only an approximation. The actual age depends on the full expansion history:

• Gravity from matter and radiation decelerates expansion, which would make the true age less than $t_H$.
• Dark energy accelerates expansion, which pushes the true age closer to or slightly beyond $t_H$.

These effects partially cancel. The current best estimate for the age of the universe is $13.8 \pm 0.02$ Gyr (from Planck CMB data combined with other probes), which is close to but slightly less than the Hubble time. The near-coincidence is not an accident; it reflects the particular mix of matter and dark energy in our universe.