Cosmology: Hubble's Law and the Expanding Universe

Understanding the universe’s large-scale structure and history is one of cosmology’s greatest triumphs, and it begins with a deceptively simple relationship: the farther a galaxy is, the faster it moves away from us. This observation, formalized as Hubble's Law, is the cornerstone of the modern picture of an expanding universe, leading directly to the Big Bang theory and our estimates for the age and fate of the cosmos. Mastering this concept allows you to interpret the most fundamental evidence for our universe’s origin and evolution.

The Discovery and Formulation of Hubble's Law

In the 1920s, astronomer Edwin Hubble, building on earlier work by Vesto Slipher, made a revolutionary discovery. By measuring both the distances to and the spectral shifts of galaxies outside our own Milky Way, he found a clear, linear correlation: a galaxy’s recessional velocity increases in direct proportion to its distance from us. This is expressed in the iconic equation of Hubble's Law: $$v=H_{0}d$$ Here, $v$ is the galaxy's recession velocity (typically in km/s), $d$ is its proper distance (in megaparsecs, Mpc), and $H_0$ is the Hubble constant. The Hubble constant is not a true "constant" over cosmic time, but its present-day value, $H_0$, sets the current expansion rate of the universe. For example, if a galaxy is 100 Mpc away and $H_0=70\text{ km/s/Mpc}$, its recessional velocity is $v=70\times 100=7000$ km/s. It is crucial to understand that this recession is not due to galaxies moving through space, but rather to the expansion of space itself between them.

Cosmological Redshift as the Key Evidence

The primary method for measuring a galaxy's recessional velocity is through cosmological redshift. As light travels through expanding space, its wavelength is stretched. This is analogous to drawing a wave on a deflated balloon and then inflating it; the wavelength of the drawn wave increases as the balloon's surface expands. Astronomers observe this by identifying known spectral lines (like those from hydrogen) in a galaxy's light and noting how much they are shifted toward the red (longer wavelength) end of the spectrum. The redshift $z$ is quantified as: $$z=\frac{{\lambda }_{\text{observed}}-{\lambda }_{\text{rest}}}{{\lambda }_{\text{rest}}}$$ where ${\lambda }_{\text{rest}}$ is the wavelength emitted by the source and ${\lambda }_{\text{observed}}$ is the wavelength we measure. For velocities much less than the speed of light $c$, the redshift is directly proportional to velocity: $v\approx cz$. This observed redshift is the direct, observable evidence for the expansion of the universe.

Estimating the Age of the Universe

A profound implication of Hubble's Law is that it allows for a simple, first-order estimate of the age of the universe. If all galaxies have been flying apart from each other, then running the expansion backward in time suggests they were all together at a single point. The Hubble time $t_H$ gives this estimate: $$t_H=\frac{1}{H_0}$$ Note that $H_0$ must be in appropriate units. If $H_0=70\text{ km/s/Mpc}$, we first convert it to units of inverse time: $$H_0=70\text{km/s/Mpc}=70\times \frac{1}{3.086\times 10^{19}\text{km}}{\text{s}}^{-1}\approx 2.27\times 10^{-18}{\text{s}}^{-1}$$ The inverse is then: $$t_H=\frac{1}{2.27\times 10^{-18}}\text{s}\approx 4.4\times 10^{17}\text{s}\approx 13.9\text{billion years}.$$ This calculation assumes a constant expansion rate. In reality, gravity has slowed the expansion over time, so the true age of the universe is slightly less than the Hubble time for a matter-dominated model. Current precise measurements, incorporating data from the cosmic microwave background, place the age at about 13.8 billion years.

Standard Candles and the Accelerating Universe

To measure vast distances and apply Hubble's Law, astronomers need reliable standard candles—objects of known intrinsic brightness. The most important for modern cosmology are Type Ia supernovae. These are thermonuclear explosions of white dwarf stars that reach a very consistent peak luminosity because they occur at a specific mass threshold. By comparing their observed apparent brightness to their known intrinsic brightness, their distance can be determined with high precision.

In the late 1990s, teams studying distant Type Ia supernovae made a Nobel Prize-winning discovery: the supernovae were fainter (and thus farther away) than predicted for a universe whose expansion was slowing down due to gravity. The only consistent explanation was that the expansion of the universe is accelerating. This required the introduction of a repulsive force, now termed dark energy. This discovery fundamentally changed our model of the universe's contents and its ultimate fate, showing that normal and dark matter constitute only about 30% of the total energy density, with dark energy making up the remaining 70%.

Common Pitfalls

1. Confusing Recession with Motion Through Space: A common mistake is to think galaxies are rocketing away through a static space, like shrapnel from an explosion. The correct interpretation is that space itself is expanding, carrying galaxies apart. This distinction is critical for understanding cosmological redshift and the geometry of the universe.
2. Misapplying the Redshift-Velocity Formula: The simple relation $v=cz$ is only valid for small redshifts ($z<0.1$). For larger distances, the relativistic formula must be used. Applying $v=cz$ to a high-redshift quasar will give a velocity incorrectly exceeding the speed of light.
3. Treating the Hubble Constant as Truly Constant: Students often forget that $H_0$ is the current value of the Hubble parameter $H(t)$. The parameter changes over cosmic time—it was larger in the past. When using $v=H_{0}d$, you are using today's rate to relate today's distance to today's observed velocity.
4. Misunderstanding the Hubble Time: The calculation $t=1/H_0$ gives the Hubble time, which is only an estimate of the universe's age. It assumes a constant expansion rate with no gravity or dark energy. The actual age is derived by integrating the Friedmann equation with the correct densities of matter and dark energy, which modifies this simple result.

Summary

• Hubble's Law, $v=H_{0}d$, establishes a linear relationship between a galaxy's distance and its recession velocity due to the expansion of space.
• Cosmological redshift—the stretching of light's wavelength as space expands—is the observable proof of this expansion, with $v\approx cz$ for nearby galaxies.
• The inverse of the Hubble constant provides a Hubble time, a first-order estimate (~13.9 Gyr) for the age of the universe, which is refined by models accounting for gravity and dark energy to ~13.8 Gyr.
• Type Ia supernovae, used as supremely valuable standard candles, revealed that the universe's expansion is accelerating, leading to the discovery of dark energy as the dominant component of the cosmos.