12.3 Hubble's Law and Expansion of the Universe

Hubble's Law and Cosmic Expansion

Hubble's Law describes a surprisingly simple pattern: the farther a galaxy is from us, the faster it's moving away. This discovery revealed that the universe is expanding, which overturned the old assumption of a static cosmos and became one of the strongest pieces of evidence for the Big Bang theory.

Understanding cosmic expansion lets us estimate the age of the universe, trace its history, and predict its future. It's one of the most important ideas in modern cosmology.

Hubble's Law and the Expansion Rate

Hubble's Law is expressed as:

$v = H_0 \, d$

• $v$ is the recessional velocity of the galaxy (how fast it's moving away from us)
• $d$ is the distance to the galaxy
• $H_0$ is the Hubble constant, which quantifies the current expansion rate of the universe

The Hubble constant is measured in units of km/s/Mpc (kilometers per second per megaparsec). A megaparsec is about 3.26 million light-years. So if $H_0 \approx 70 \, \text{km/s/Mpc}$, a galaxy 100 Mpc away recedes at roughly 7,000 km/s.

The law implies that expansion is uniform in all directions: no matter where you stand in the universe, you'd see the same pattern of more distant galaxies receding faster. This doesn't mean Earth is at the center. Every point in the universe sees the same thing, like dots on the surface of a balloon that's being inflated.

By running the expansion backward in time, you can estimate when everything was compressed into a single point. This gives an approximate age of the universe (currently about 13.8 billion years). At very large distances, though, Hubble's Law breaks down because dark energy causes the expansion to accelerate, making the relationship between velocity and distance more complicated.

Implications for Cosmic History and Structure

If the universe is expanding now, it must have been smaller, hotter, and denser in the past. This supports the hot Big Bang model, which predicts things like the cosmic microwave background and the abundances of light elements we actually observe.

The nature of the expansion also determines the universe's fate:

• Big Freeze: expansion continues forever, and the universe gradually cools to near absolute zero
• Big Rip: if dark energy strengthens over time, it could eventually tear apart galaxies, stars, and even atoms
• Big Crunch: if gravity were strong enough to reverse expansion, the universe would collapse back on itself (currently considered unlikely given observations of accelerating expansion)

The concept of cosmic inflation is related but distinct. Inflation refers to an extremely rapid expansion in the first fraction of a second after the Big Bang. It solves problems the standard Big Bang model can't explain on its own, like why the universe looks so uniform in every direction (the horizon problem) and why space is so geometrically flat (the flatness problem).

Redshift and the Expanding Universe

Understanding Redshift in Cosmology

When a light source moves away from you, the light waves get stretched out, shifting toward longer wavelengths (the red end of the spectrum). This is redshift, and it works similarly to how a siren sounds lower-pitched as an ambulance drives away from you.

Cosmological redshift is slightly different from a simple Doppler shift. It's not caused by a galaxy flying through space; it's caused by space itself expanding while the light is in transit. The light's wavelength stretches along with the expanding space it travels through.

The redshift parameter $z$ is defined as:

$z = \frac{\Delta\lambda}{\lambda_0}$

where $\Delta\lambda$ is the change in wavelength and $\lambda_0$ is the original emitted wavelength. A galaxy with $z = 1$ has had its light wavelength doubled since emission, meaning the universe has expanded by a factor of 2 since that light was emitted.

For nearby galaxies, the relationship between redshift and distance is roughly linear, consistent with Hubble's Law. For very distant objects, the relationship becomes non-linear because the expansion rate has changed over cosmic history.

Redshift as a Cosmic Measurement Tool

Redshift is one of the most versatile tools in observational cosmology:

• Distances: By measuring a galaxy's redshift and applying Hubble's Law (with corrections at large distances), astronomers estimate how far away it is.
• Cosmic history: High-redshift objects are seen as they were billions of years ago. Observing galaxies at $z = 6$ or higher means looking back to when the universe was less than a billion years old.
• Large-scale structure: Redshift surveys map out the three-dimensional distribution of galaxies, revealing structures like galaxy clusters, filaments, and voids (the "cosmic web").
• Dark energy: Comparing the brightness of Type Ia supernovae (which have a known intrinsic luminosity) with their redshifts revealed that the expansion of the universe is accelerating. This discovery led to the concept of dark energy.
• Chemical composition: Spectral lines shifted by redshift still carry information about the elements present in distant stars and gas clouds.

Evidence for Cosmic Expansion

Observational Evidence Supporting Expansion

Multiple independent lines of evidence confirm that the universe is expanding:

1. Hubble's original observations (1929): Edwin Hubble used Cepheid variable stars as distance markers in nearby galaxies. Cepheids have a known relationship between their pulsation period and luminosity, so measuring the period tells you the true brightness, and comparing that to the apparent brightness gives the distance. Hubble found that nearly all galaxies showed redshifted spectra, and the redshift increased with distance.

2. Cosmic Microwave Background (CMB): The CMB is thermal radiation left over from about 380,000 years after the Big Bang, when the universe cooled enough for atoms to form and light to travel freely. Its existence and detailed properties (a near-perfect blackbody spectrum at about 2.725 K) match predictions of the Big Bang model precisely.

3. Light element abundances: The observed ratios of hydrogen (~75%), helium (~25%), and trace amounts of lithium and deuterium match what Big Bang nucleosynthesis predicts for an expanding, cooling universe. A static universe has no mechanism to produce these specific ratios.

4. Large-scale structure: The distribution of galaxy clusters and superclusters is consistent with models of structure formation in an expanding universe, where small density fluctuations in the early universe grew over billions of years through gravitational attraction.

5. Gravitational lensing: Massive objects bend light from more distant sources. Observations of lensed quasars and galaxies provide independent measurements of distances and expansion rates.

Additional Supporting Evidence

Several more advanced observations reinforce the expansion picture:

• Baryon acoustic oscillations (BAO): Sound waves in the early universe left a characteristic imprint in the spacing of galaxies. Measuring this "standard ruler" at different redshifts traces the expansion history.
• Type Ia supernovae time dilation: Light curves of distant supernovae appear stretched in time by a factor of $(1 + z)$, exactly as predicted if the universe is expanding. A supernova at $z = 0.5$ appears to brighten and fade 1.5 times more slowly than an identical nearby one.
• Lyman-alpha forest: Absorption lines in quasar spectra caused by intervening hydrogen gas clouds trace the expansion and density of the intergalactic medium at different epochs.
• Integrated Sachs-Wolfe effect: Correlations between the CMB and the distribution of galaxies provide evidence for late-time cosmic acceleration driven by dark energy.
• Sunyaev-Zeldovich effect: CMB photons passing through hot gas in galaxy clusters get an energy boost, providing an independent way to measure cluster distances and constrain expansion parameters.

Implications of an Expanding Universe

Cosmological Model Implications

An expanding universe has a finite age, which gives rise to the concept of cosmic time, a universal clock that started ticking at the Big Bang. The discovery in 1998 that the expansion is accelerating (from Type Ia supernovae data) forced cosmologists to introduce dark energy, a mysterious component making up roughly 68% of the universe's total energy content.

This expanding framework replaced the steady-state theory, which proposed that the universe looks the same at all times and that new matter continuously forms to fill expanding gaps. The steady-state model couldn't account for the CMB or the observed evolution of galaxy populations with redshift.

The expansion also provides the scaffolding for understanding how cosmic structures formed. Small density variations in the early universe (visible as tiny temperature fluctuations in the CMB) grew through gravity over billions of years into the galaxies, clusters, and large-scale filaments we observe today.

Philosophical and Practical Implications

The expanding universe raises deep questions about the nature of space and time. If space is expanding, what is it expanding into? (The answer, in general relativity, is that space isn't expanding into anything; the metric describing distances between points is simply changing.)

On the practical side, studying expansion has driven the development of powerful observational tools: space telescopes like Hubble and James Webb, adaptive optics systems for ground-based telescopes, and massive galaxy survey programs. These technologies often find applications beyond cosmology.

The expanding universe also connects to open problems in fundamental physics. Explaining dark energy may require new physics beyond general relativity, potentially linking cosmology to quantum gravity and unification theories. The measured value of the Hubble constant currently shows a tension between different measurement methods (the "Hubble tension"), which could point to unknown physics or systematic measurement errors still being investigated.