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Equation 2 · Part 8 · Reading the Oldest Light: How a Temperature Map Becomes a Cosmology

superscript

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

What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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 exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the article section

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