For all problems, unless otherwise stated, use \(\mathrm{H}_{0}=\) \(70 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\). Suppose we detect radiation that was emitted by some galaxy far away. In the time the radiation traveled to reach us, its wave- length doubled. What happened to the scale factor of the universe in that time?

The Hubble constant (denoted as \(\text{H}_0\)) is a key concept in cosmology. It represents the rate at which the universe is expanding. Essentially, it measures the speed at which galaxies are moving away from each other. The value of \(\text{H}_0\) is typically given in units of kilometers per second per megaparsec (km/s/Mpc). A common value used is 70 km/s/Mpc. This means that for every megaparsec (about 3.26 million light-years) away a galaxy is, it appears to be moving 70 kilometers per second away from us.
Understanding the Hubble constant is crucial for grasping the broader concept of cosmological expansion. It helps explain why distant galaxies appear to be receding, which is a cornerstone of the Big Bang theory.
In simpler terms, the Hubble constant acts like the 'speedometer' of the universe, showing how fast the cosmic landscape is stretching out over time.

Wavelength redshift refers to the phenomenon where the wavelength of light from distant galaxies is stretched, making it appear more red than when it was emitted. This stretching occurs because the universe is expanding. As light travels through the expanding universe, its wavelength increases, shifting towards the red end of the spectrum (hence the term 'redshift').
The redshift can be quantified using the formula:

• \(\frac{\text{wavelength at detection}}{\text{wavelength at emission}} = 1 + z\),

where 'z' is the redshift.
A redshift of 1 (z=1) means the wavelength has doubled. This is directly related to the scale factor of the universe, which is discussed next. Understanding wavelength redshift helps us determine how much the universe has expanded since the light was emitted. It is a key observational evidence for the expanding universe.

The universe scale factor, often denoted as \(a(t)\), is a measure of how much the universe has expanded over time. It is a dimensionless number that changes as the universe evolves. If we think of the universe as a giant balloon, the scale factor represents how much the balloon has inflated.
The relationship between the scale factor and the wavelength of light is given by the formula:

• \(\frac{\text{wavelength at detection}}{\text{wavelength at emission}} = \frac{a(t_{\text{now}})}{a(t_{\text{then}})}\).

In the context of our exercise, if the wavelength has doubled, then<|[|> \(\frac{a(t_{\text{now}})}{a(t_{\text{then}})} = 2\),\then this means the scale factor now is twice what it was when the light was emitted. Understanding the scale factor is vital for interpreting cosmological data and for understanding the history and future of the universe's expansion.