Revise Hubble's law and the Big Bang theory for Cambridge 9702 (25.3)

The fundamental evidence for our expanding universe comes from observing the light from distant galaxies. When we analyse their light, we find that the characteristic spectral lines are shifted towards the red end of the electromagnetic spectrum. This phenomenon is called redshift, and it's a consequence of the stretching of spacetime.

Redshift (Z) is quantitatively defined by the fractional change in wavelength, which can be related to recessional velocity (v) for non-relativistic speeds:

$Z = \frac{\Delta \lambda}{\lambda_{original}} \approx \frac{v}{c}$

Where:

• $\Delta \lambda$ is the change in wavelength.
• $\lambda_{original}$ is the original wavelength emitted by the source.
• $v$ is the recessional velocity.
• $c$ is the speed of light in a vacuum ($3.00 \times 10^8 \text{ m s}^{-1}$).

It's important to distinguish between Doppler redshift and cosmological redshift. Doppler redshift is caused by an object's motion through space away from an observer. Cosmological redshift, which is what we observe for distant galaxies, is caused by the expansion of space itself. As space expands, the wavelength of light travelling through it gets stretched, leading to redshift. The raisin bread analogy illustrates this: the raisins aren't moving through the dough, but the dough (spacetime) is expanding and carrying them apart.

In the late 1920s, astronomer Edwin Hubble made a groundbreaking discovery: the speed at which a galaxy is moving away from us is directly proportional to its distance. This relationship, known as Hubble's Law, is a cornerstone of modern cosmology.

Hubble's Law is expressed as:

$v = H_0 d$

Where:

• $v$ is the recessional speed of the galaxy.
• $H_0$ is Hubble's constant.
• $d$ is the distance to the galaxy.

To apply Hubble's Law, we need to know the distances to galaxies, which are vast. Astronomers use special objects called standard candles to determine these distances. A standard candle is essentially an astronomical object whose true intrinsic brightness, or luminosity, is known.

The inverse square law for radiation relates observed flux ($F$) to intrinsic luminosity ($L$) and distance ($d$):

$F = \frac{L}{4\pi d^2}$

Rearranging to find distance:

$d = \sqrt{\frac{L}{4\pi F}}$

The universal observation of redshift and the expansion of the universe strongly suggest that all matter was once much closer together. Tracing this expansion backward in time leads us to the Big Bang Theory, which describes the origin of the observable universe.

Besides galactic redshift, the most compelling piece of evidence for the Big Bang is the Cosmic Microwave Background (CMB) radiation. This is faint, uniform radiation that fills all of space. It is interpreted as the leftover heat from the Big Bang, a 'fossil' from the early, hot, dense universe.

Hubble's Law provides a simple way to estimate the age of the universe. If we assume the expansion has been constant, we can trace it back to a time when all galaxies were at the same point. The time taken for a galaxy to travel a distance d at a speed v is $t = d/v$. From Hubble's Law, $v = H_0 d$, so we can substitute for v.

The age of the universe ($t$) can be estimated as the reciprocal of Hubble's constant:

$t \approx \frac{1}{H_0}$

This gives a rough estimate known as the Hubble time. To get the age in seconds, $H_0$ must be converted to SI units (s⁻¹).