# What is a 'square' Kelvin degree? μ$K^2$? In terms of the cosmic microwave background's (CMB's) temperature fluctuations?

From what I have read and seen, the minute temperature fluctuations in the CMB are measured in microKelvin, or μK.

However, many charts and graphs show μK2, or 'microKelvin-squared'.

Do they simply mean 'square degrees', as in the 41,252.96° in a sphere?

An example is seen in this WMAP power spectrum: • I have a hunch that it's variance like $\sigma^2$.
– uhoh
Jul 11, 2022 at 5:17
• Also this question would profit from exemplary citation and link to a source where this is seen or used Jul 11, 2022 at 5:44
• Quoting out of context makes it difficult to provide an exact answer. The "general" answer is that this unit represents, as @uhoh said, $\sigma$ since it's described as a 'fluctuation". Otherwise it might belong to a specific equation within which temperature probably appears more than once. Jul 11, 2022 at 12:41
• I voted to re-open, since I think it's rather clear that the question is about the reason for the unit of the power spectrum of the CMB.
– pela
Jul 24, 2022 at 9:37

A way of thinking about it is that a point in the plotted "angular power spectrum" represents an averaged sum over the whole sky of the temperature fluctuation in one part multiplied by the temperature fluctuation in another part separated by some angle from the first - represented on the x-axis of the graph by the multiple moment, where $$\theta \simeq 180^\circ/l$$ (and is shown along the top x-axis).

This means that if the fluctuations have a characteristic angular scale, this reveals itself as a peak in the power spectrum at the corresponding multiple moment.

Since the units of the temperature fluctuations from the mean are Kelvins, then the power spectrum, which is a sum over the product of fluctuation pairs is in Kelvin$$^2$$.

The units of $$\mu$$K$$^2$$ are used because the fluctuations are of order $$10^{-5}$$ K and so their product is of order thousands of $$\mu$$K$$^2$$.

The $$l(l+1)/2\pi$$ are just unitless normalising factors to make the power spectrum roughly horizontal.

The square of CMB temperature fluctuations represented in spherical coordinates is proportional to $$C_l$$, where $$C_l$$ is the angular power spectrum (reference).

And according to this review paper, $$C_l$$ usually absorbs the $$T_\mathrm{CMB}$$ and the extra quantites are due to "historical reasons", which I imagine are just factors from spherical harmonics.