Our Universe

Cosmological Principle

The cosmological principle states that the universe is homogeneous and isotropic on large scales. This means that, on average, the universe looks the same in all directions (isotropic) and at all locations (homogeneous).

For example, a dartboard can be considered isotropic when viewed from the center, but it isn’t homogeneous, as the center is a special point on the surface. Similarly, an infinite striped pattern can be considered homogeneous since the pattern would look the same from everywhere, but not isotropic because the stripes have a preferred direction.

Observations

Since the universe is so vast and so old, we would expect to encounter a star in any direction we look. Hence, the night sky, instead of being dark, should be fully bright, filled with stars. This is known as Olbers’ Paradox. The resolution to this is that the universe is not static and infinite, but rather dynamic and expanding. We observe a dark night sky because the universe has a finite age and is not infinitely old.

Another observation is Hubble’s Law (or Hubble-Lemaître Law), which states that the velocity at which a galaxy is receding from us is proportional to its distance. Galaxies that are farther from us are receding faster. Hubble’s Law states that

$$v = H_0 d$$

where $H_0$ is the Hubble constant and $d$ is the distance to the galaxy. The Hubble constant is a measure of the rate of expansion of the universe.

Redshift

Redshift is defined as

$$z = \frac{\lambda_{\text{obs}} - \lambda_{\text{emit}}}{\lambda_{\text{emit}}}$$

where $\lambda_{\text{obs}}$ is the wavelength of light observed and $\lambda_{\text{emit}}$ is its wavelength when it was emitted. Redshift can be caused by multiple reasons, such as the Doppler effect, gravitational redshift, or cosmological redshift. If the object emitting light is moving at non-relativistic speeds, its speed can be related to the redshift by

$$v = cz$$

A positive redshift indicates that the object is moving away from us, while a negative redshift indicates that it is moving towards us.

According to Hubble’s law, the redshift of a galaxy is directly related to its distance from us (for low redshifts).

Cosmic Microwave Background Radiation

In 1965, Penzias and Wilson discovered the Cosmic Microwave Background Radiation (CMB), which is the remnant radiation from the Big Bang. The CMB is a nearly uniform background radiation that fills the universe and has a temperature of $T_0 = 2.725 \pm 0.001 \, \mathrm{K}$. The CMB is isotropic and homogeneous, which supports the cosmological principle. Given the temperature of the CMB, we can find its energy density and number density:

$$u_0 \approx 0.260 \, \mathrm{MeV \, c^{-2}}$$

$$n_0 \approx 411 \times 10^6 \, \mathrm{m^{-3}}$$

Observations of the universe show that the oldest objects contain about 25% helium by mass, the second most abundant element in the universe after hydrogen.