← Cosmology

Expansion of the Universe

Thursday, April 21, 2022

Doppler Shift

The expansion of the universe is apparent when looking at light from distant galaxies. The Doppler shift of light, from special relativity, is written as

$$\lambda'=\lambda\sqrt{\frac{1+v/c}{1-v/c}}=\lambda\frac{1+v/c}{\sqrt{1-v^2/c^2}}$$

where $v$ is the relative velocity between the source of light and the observer. When looking at light from stars, we see a continuous emission spectrum. However, the gases in the atmosphere absorb particular wavelengths of light which are seen as dark absorption lines. Comparing the emission and absorption lines, we can see Doppler-shifted wavelengths of light.

Some stars in our galaxy are seen to be moving towards us (those stars' light is blue-shifted) and some are moving away from us (whose light is red-shifted). Most stars in our galaxy are not moving fast relative to other galaxies. Even in our local group of galaxies, some are moving towards us and some are moving away.

All galaxies beyond our local group are moving away from us in a way that is not possible from simply random motion of galaxies. The cosmological principle says that the universe must look the same from any vantage point (in other words, Earth is not special), meaning all galaxies appear to be moving away from any point in space.

Hubble's Law

Through observation, Edwin Hubble observed that the Milky Way is composed of hundreds of billions of stars. Hubble also looked at other galaxies and saw some stars whose brightness oscillated with a period of days. Using a method of measuring astronomical distances developed by Henrietta Swan Leavitt and the red-shifts of galaxies, Hubble established a relationship between distance and speed, called Hubble's law:

$$v=H_0d$$

where $H_0$ is the Hubble parameter. Empirically, this value is measured to be about

$$H_0=72~\frac{\text{km/s}}{\text{Mpc}}$$

Hubble's law matches observations that all galaxies appear to recede from all points in space, like points on a balloon moving away from one another or raisins in a loaf of bread rising in an oven.

The actual cause of the red-shifting of light is general relativity, where the fabric of spacetime itself stretches photons. By this principle, the galaxies are not "in motion," as they would be in conventional physics.

The Doppler shift model results in some speeds greater than the speed of light. Instead, we use a better analysis using the stretching model:

$$\frac{\lambda'}{\lambda}=\frac{R_0}{R}$$

where $R_0$ is a size or distance scale factor of the universe at the present time and $R$ is a similar factor at the time the light was emitted.

Consequences of Universal Expansion

Although the expansion of the universe is widely accepted, it results in two possibilities:

1. Long ago, galaxies used to be a lot denser than it currently is (this is the Big Bang hypothesis).
2. As galaxies separate, additional matter is created in the empty space between them, maintaining a constant density of the universe (this is the Steady State hypothesis).

Both hypotheses were popular for a while until about the 1960s, when the cosmic microwave background radiation was discovered, heavily favoring the Big Bang hypothesis.