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Equation 2 · Part 4 · Reading the Oldest Light: How a Temperature Map Becomes a Cosmology

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Cℓ=12ℓ+1∑m=−ℓℓ∣aℓm∣2.C_\ell = \frac{1}{2\ell+1}\sum_{m=-\ell}^{\ell} \left|a_{\ell m}\right|^2 .
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What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

No one fits a map. The quantity carried into a likelihood is the angular power spectrum, the variance of the spherical-harmonic coefficients at each multipole: Cℓ=12ℓ+1∑m=−ℓℓ∣aℓm∣2C_\ell = \frac{1}{2\ell+1}\sum_{m=-\ell}^{\ell} \left|a_{\ell m}\right|^2 . This choice embeds an assumption that deserves to be stated rather than assumed: that the fluctuation field is statistically isotropic and Gaussian, so that the power spectrum is a sufficient statistic. If the primordial field had significant non-Gaussianity, the power spectrum would discard real information; the assumption is tested separately, not granted.

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the article section

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