Assume that the most distant galaxies have a redshift \(z=10\) The average density of normal matter in the universe today is \(4 \times 10^{-28} \mathrm{kg} / \mathrm{m}^{3} .\) What was its density when light was leaving those distant galaxies? (Hint: Keep in mind that volume is proportional to the cube of the scale factor.)

Redshift is a critical concept in cosmology that helps us understand how the universe expands. When we observe distant galaxies, we notice that the light coming from them is shifted towards the red end of the spectrum, known as redshift. This phenomenon occurs because the universe is expanding, causing wavelengths of light to stretch. The redshift (\(z\)) is defined as: \(z = \frac{\text{observed wavelength}}{\text{emitted wavelength}} - 1\).
For example, a redshift of \(z=10\) means the universe has expanded 11 times since the light left those very distant galaxies (as \(1+z=11\)).
Understanding redshift allows us to measure the rate of expansion of the universe and observe objects from different times in cosmic history, giving us a glimpse into the universe’s past.

The scale factor (\(a\)) is a mathematical concept used in cosmology to describe the size of the universe at a particular time relative to its current size. It is directly linked to redshift with the relation: \(a = \frac{1}{1+z}\).
For instance, if a galaxy has a redshift of \(z = 10\), the scale factor at that time would be \(a = \frac{1}{11}\). This tells us that at the moment the light left that galaxy, the universe was 1/11th of its current size.
The scale factor is essential in translating redshift into a more intuitive understanding of the universe's expansion over time. It also plays a key role when calculating distances and relating them to the density of the universe at different points in time.

The density of the universe refers to the amount of matter per unit volume. It is crucial for understanding how the universe evolves. By knowing the current density (\(\rho_0\)), we can use the scale factor to determine the density at an earlier time.
Given that volume changes with the cube of the scale factor, the density of the universe at a past time can be calculated using: \(\rho = \rho_0 \times \frac{1}{a^3}\).
Take the problem where the current density is \(4 \times 10^{-28} \text{ kg/m}^3\) and the redshift is \(z = 10\). First, compute the scale factor: \(a = \frac{1}{1+10} = \frac{1}{11}\).
Then use the formula to find the past density: \(\rho = 4 \times 10^{-28} \times \frac{1}{(\frac{1}{11})^3}\), resulting in \(\rho = 5.324 \times 10^{-25} \text{ kg/m}^3}\).
This calculation shows that the universe was much denser when the light we now observe was initially emitted from those distant galaxies.