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Equation 2 · Part 3 · Reading the Oldest Light: How a Temperature Map Becomes a Cosmology

Symbol a_ell m

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

What this part means

aea_ell m is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

aea_ell m is one of the signed contributions combined to compute the quantity on the left.

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

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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Sources cited in the article section

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