For the dipole anisotropy, the fractional increase in temperature, \(\Delta \mathrm{T} / \mathrm{T}\), is equal to \(-v | c,\) where \(v\) is our speed. Justify this relation using Wien's displacement law and the Doppler shift relation.

Wien's displacement law is a fundamental principle in the study of blackbody radiation. It states that the wavelength at which a blackbody emits radiation most intensely (\(\text{λ}_{max}\)) is inversely proportional to its temperature (\(\text{T}\)). This can be mathematically expressed as \(\text{λ}_{max}\text{T} = b\), where \(b\) is Wien's constant, approximately equal to \(2.897 \times 10^{-3}\) K·m.
This law helps us understand how temperature shifts affect the peak wavelength of emitted radiation.
When the temperature increases, the peak wavelength decreases, meaning the radiation becomes more energetic. Similarly, if the temperature drops, the peak wavelength shifts to longer wavelengths, indicating less energetic radiation.

The Doppler effect describes the change in frequency or wavelength of radiation due to the relative motion between the source and the observer. When considering temperature, it's crucial to understand how motion affects the observed temperature through this effect.
If an observer is moving towards a radiation source, they perceive an increase in the frequency, leading to a higher temperature. Conversely, moving away results in a lower observed frequency and thus a lower temperature.
Mathematically, the observed temperature (\(T'\)) can be related to the original temperature (\(T\)) using the equation:
\(\frac{\text{T}'}{\text{T}} = \frac{\text{1 + (v/c)}}{\text{1 - (v/c)}} \).
For speeds much less than the speed of light (\(v \< \text{c}\)), this equation approximates to \(1 + \frac{v}{c}\).
This relationship indicates that a higher speed (\(\text{v}\)) results in a more significant change in the temperature observed due to the Doppler effect.

Fractional temperature increase is a concept used to describe the small changes in temperature resulting from the motion of an observer in space. This concept is essential for understanding dipole anisotropy in the Cosmic Microwave Background (CMB).
Dipole anisotropy refers to the temperature variations observed in the CMB due to the motion of our solar system relative to the CMB. This motion causes a slight increase in temperature in the direction of motion and a decrease in the opposite direction.
The fractional temperature increase (\(\frac{\text{ΔT}}{\text{T}}\)) can be derived using Wien's displacement law and the Doppler effect relation. It is approximately given by \(\frac{v}{c}\), where \(v\) is the relative speed and \(c\) is the speed of light.
Understanding this helps students grasp why we observe slight differences in the CMB temperature across the sky and how this is directly related to our movement through space.