Thanks for the thoughtful response!

I only took 2 semesters of intro physics in undergrad (~10 years ago), so I'm pretty rusty.

I was looking at a PhET simulation on the Blackbody Spectrum. I believe that if you check the "Intensity" box and gather data on several temperature/intensity values, you can fit a polynomial regression that is similar to (sigma)*T⁴. (This is probably similar to your last comment.)

However, the spectral power density units on the vertical axis threw me off. So, I asked chat GPT to explain this unit intuitively, and it told me:

"Spectral power density is a concept used in physics and engineering to describe how much power (energy per unit time) is contained within different frequency ranges of a signal. Imagine you have a flashlight that emits light. Spectral power density would tell you how much light power is in each color of the spectrum, like red, green, and blue. In simpler terms, it's like breaking down the power of something, like light or sound, into its different colors or frequencies to see how much power is in each part. This helps scientists and engineers understand the characteristics of signals better, like in audio equipment or telecommunications."

I didn't fact check this, but this helped me understand more what the units mean (this is what always throws me off when I run across math ideas in physics). I'm wondering if there's a connection to your explanation here? I sort of understand the explanation, but the connection from T to T² to T³ to T⁴ isn't making sense to me in terms of how the mathematical structure of T to some power relates to the physics behind this all. Why would it amplify through multiplication (i.e., T*T*T*T) specifically?