If a photon from the cosmic microwave background had wavelength \(\lambda_{0}\) when it was emitted at redshift \(z\), its wavelength today is \(\lambda=\lambda_{0} /(1+z)\). (a) Let \(T\) be the symbol for the temperature of the cosmic microwave background today. Explain why the radiation temperature was \(T_{0}=T(1+z)\) at redshift \(z\). (b) What was the radiation temperature at \(z=1\) ? (c) At what redshift was the radiation temperature equal to \(293 \mathrm{~K}\) (a typical room temperature)?

The concept of redshift is central to understanding the expansion of the universe and the behavior of light traveling through it. It essentially refers to the way light stretches out, increasing in wavelength, as it zips across the cosmos. This is analogous to how the pitch of a siren lowers as an ambulance speeds away from you, a phenomenon known as the Doppler Effect.

In the context of the universe, as galaxies move away from us, the light they emit also stretches or 'shifts' towards the red end of the light spectrum, which has a longer wavelength. This is not due to the Doppler Effect, but rather to the expansion of space itself which 'stretches' the light. Scientists can measure this redshift to deduce just how far away a galaxy is and how fast it's moving.

The higher the redshift, the farther away and the earlier in time we are observing. Redshift is often represented by the variable 'z' and helps us unlock the history of the universe, such as understanding events like the cosmic microwave background radiation (CMB).

Radiation temperature is the measure of the thermal energy of radiation, which includes light. In our context, it specifically refers to the temperature of cosmic microwave background (CMB) radiation. This is the afterglow of the Big Bang, and it bathes the entire universe in a faint glow of microwave radiation.

The temperature of the CMB today is a chilly 2.7 Kelvin. But as we look back in time, to a universe that was smaller, denser, and hotter, the radiation temperature was higher. As learned from the universal gas law, volume and temperature are related: think of the universe as a giant gas that cools as it expands. Therefore, at any given redshift 'z', we can determine the temperature of the CMB -- it was higher by a factor of (1+z).

This relationship allows us to map out the thermal history of the universe. The radiation temperature helps us gauge the state of the universe at different epochs and is critical for understanding the early universe's structure formation.

The wavelength of a photon is a fundamental property describing the distance between successive peaks of a light wave. It determines the light's color and energy; shorter wavelengths equate to more energy and vice-versa. In the context of the cosmic microwave background (CMB), as the universe expands, so does the wavelength of photons that constitute the CMB.

This stretching is a direct consequence of the redshift—the increase in wavelength corresponds to a journey through time, from the early, denser universe to the present day. The shifting of wavelength can also be tied to changes in radiation temperature because of the physics that dictates light's interaction with matter. For example, when the universe was much younger and hotter, the wavelengths of CMB photons were shorter and corresponded to a higher radiation temperature.

Understanding these properties is crucial for cosmologists, as they can use the observed wavelengths to backtrack and understand the physical conditions of the universe throughout different stages of its evolution.