6.3 Redshift and its relationship to expansion

Redshift and Cosmological Expansion

Redshift: Causes and Definition

Redshift happens when light from a distant object arrives at Earth with a longer wavelength than it had when it was emitted. The spectral lines you'd normally see at specific wavelengths get shifted toward the red end of the electromagnetic spectrum. That means the observed wavelength is longer, and the observed frequency is lower, than what the source originally produced.

There are two main causes of redshift:

• Doppler effect — This occurs when a light source is physically moving away from the observer. The relative motion stretches the wavelength of the emitted light. Think of how an ambulance siren drops in pitch as it drives away from you; light behaves similarly, just with wavelength instead of sound pitch.
• Cosmological redshift — This is caused by the expansion of space itself. As the universe expands, it stretches the wavelength of light that's already traveling through it. The source and observer don't need to be moving relative to each other in any traditional sense. A common analogy: imagine dots drawn on a rubber sheet. As you stretch the sheet, the dots move apart not because they're sliding across the surface, but because the surface itself is expanding.

The distinction matters. Doppler redshift comes from an object's motion through space. Cosmological redshift comes from space expanding while light is in transit.

Redshift-Distance Relationship

Objects that are farther away from us tend to have greater redshifts. A distant galaxy cluster like Abell 2744 (roughly 3.5 billion light-years away) shows a much larger redshift than a relatively nearby galaxy. This pattern isn't random; it reflects the expansion of the universe.

Hubble's law formalizes this relationship:

$v = H_0 \times d$

• $v$ = the galaxy's recessional velocity (how fast it's moving away from us)
• $H_0$ = the Hubble constant, which describes the current expansion rate of the universe (approximately 70 km/s/Mpc)
• $d$ = the distance to the galaxy

The logic is straightforward: because the universe is expanding, more distant galaxies have had more space expand between them and us, so they recede faster and show greater redshifts.

Galaxy Recessional Velocity Calculations

To find how fast a galaxy is receding, you first calculate its redshift from observed spectral lines:

$z = \frac{\Delta \lambda}{\lambda_0} = \frac{\lambda_{obs} - \lambda_0}{\lambda_0}$

• $z$ = redshift (dimensionless)
• $\lambda_{obs}$ = the wavelength you actually observe
• $\lambda_0$ = the wavelength the light was emitted at (the rest wavelength)
• $\Delta \lambda$ = the difference between observed and rest wavelengths

Step-by-step example: Suppose a hydrogen spectral line normally emitted at $\lambda_0 = 486$ nm is observed at $\lambda_{obs} = 510$ nm.

1. Find the change in wavelength: $\Delta \lambda = 510 - 486 = 24$ nm

2. Calculate redshift: $z = \frac{24}{486} \approx 0.049$

3. Since $z \ll 1$, use the non-relativistic approximation: $v = c \times z$

4. $v = (3 \times 10^5 \text{ km/s}) \times 0.049 \approx 14{,}700$ km/s

Which formula to use for velocity:

• For small redshifts ($z \ll 1$), the non-relativistic Doppler formula works well:

$v = c \times z$

• For large redshifts ($z \geq 1$), you need the relativistic Doppler formula to avoid getting velocities that exceed the speed of light:

$v = c \times \frac{(1 + z)^2 - 1}{(1 + z)^2 + 1}$

In both formulas, $c$ is the speed of light and $z$ is the redshift.

Evidence for Universe Expansion

Edwin Hubble's 1929 discovery that galaxies show a systematic relationship between distance and redshift was the first direct observational evidence for an expanding universe. Farther galaxies consistently recede faster. Deep surveys like the Hubble Deep Field confirmed this pattern across billions of light-years.

This observation rules out a static universe. In a universe that isn't expanding, there would be no reason for redshift to correlate with distance in any systematic way.

Hubble's discovery became the observational foundation for the Big Bang theory. Several independent lines of evidence further support expansion:

• Cosmic microwave background radiation (CMB) — Residual thermal radiation from roughly 380,000 years after the Big Bang, detected uniformly in all directions. Its existence and properties match predictions of a universe that expanded and cooled from an extremely hot, dense state.
• Primordial nucleosynthesis — The observed abundances of light elements (hydrogen, helium, deuterium, lithium) match what models predict would form in the first few minutes of a hot, expanding universe.
• Large-scale structure — The distribution of galaxy clusters and superclusters across the cosmos is consistent with a universe that has been expanding and evolving over billions of years.

Together, these observations form a coherent picture: the universe has been expanding since its origin, and redshift is one of the most direct ways to measure that expansion.