(a) Suppose the scale factors at times \(t_{1}\) and \(t_{2}\) are \(R_{1}\) and \(R_{2},\) and the background radiation temperatures are \(T_{1}\) and \(T_{2}\). What is the relationship between \(T_{1}\) and \(T_{2}\) in terms of \(R_{1}\) and \(R_{2} ?(b)\) If we know that the back ground temperature now is \(3 \mathrm{K}\), and we let \(R=1,\) write an expression for the back ground temperature at any time when the scale factor is \(\mathrm{R}(t)\).

Cosmic Background Radiation, often called the Cosmic Microwave Background (CMB), is the thermal radiation left over from the time of recombination in Big Bang cosmology. It's essentially the afterglow of the Big Bang. The temperature of the CMB is uniform across the sky, and it is an essential tool for understanding the early universe.

The scale factor, denoted by \( R(t) \), is a dimensionless number that describes how the size of the universe changes with time. It normalizes the expansion of the universe, with the current value typically set to 1. If we imagine the universe as a balloon, the scale factor tells us how much the balloon has inflated since an earlier time. The relationship between the scale factor and temperature of cosmic background radiation is crucial for understanding the universe's evolution.

The evolution of temperature in cosmology shows how the temperature of the universe changes as it expands. Since the temperature of the background radiation is inversely proportional to the scale factor, when the universe was younger and smaller (a smaller scale factor), it was hotter. Conversely, as the universe expands (a larger scale factor), the temperature decreases. This relationship is expressed mathematically as: \[ T \times R = \text{constant} \] where \( T \) is temperature and \( R \) is the scale factor. For example, if the current temperature is \( 3 \text{K} \) and the scale factor is \( 1 \), the temperature at some earlier or later time can be found using the formula \[ T(t) = 3 \times \frac{1}{R(t)} \].

Inverse proportionality in cosmology refers to the relationship where one quantity increases as another decreases. Here, the temperature of the cosmic background radiation \( T \) is inversely proportional to the cosmological scale factor \( R \). This means: \[ T \times R = \text{constant} \] If we know the temperature and scale factor at one time, we can find them at another time using this proportionality. For example, if the current temperature \( (T_{now}) \) is \(3 \text{K} \) and the scale factor \( (R_{now}) \) is 1, then at any time \( t \) we can write: \[ T(t) = 3 \times \frac{1}{R(t)} \]. This equation shows how the temperature decreases as the universe expands (as \( R(t) \) increases).