CMB Peak Fregenucy (Planck's law vs. Wein's law)

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The Cosmic Microwave Background Radiation (CMBR) exhibits a thermal black body spectrum at a temperature of 2.725 K, with peak frequencies differing based on the law used for measurement. According to Planck's law, the peak frequency is 160.2 GHz, while Wien's law indicates a peak at 283 GHz. This discrepancy arises from the need to transform the differential elements when switching between frequency and wavelength measurements. The transformation affects the peak values, highlighting the importance of the measurement method. Understanding these differences is crucial for accurate interpretations of CMBR data.
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wikipedia says:
"The CMBR has a thermal black body spectrum at a temperature of 2.725 K, which peaks at the microwave range frequency of 160.2 GHz, corresponding to a 1.873 mm wavelength. This holds if measured per unit frequency, as in Planck's law. If measured instead per unit wavelength, using Wien's law, the peak is at 1.06 mm corresponding to a frequency of 283 GHz."

How come there is a difference in the peak frequencies, between using Planck's law vs. Wein's law?
 
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In short, the difference is because you have to transform the differential element d\lambda to d \nu (or vice-versa) rather than just substituting in the corresponding expression from the dispersion relation.
 

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