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Equation 2 · Part 5 · Reading the Oldest Light: How a Temperature Map Becomes a Cosmology

fraction

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

What this part means

Divide the expression above the line by the one below it.

Its job in the formula

The expression above the fraction bar is divided by the complete expression below it. The denominator must not be zero.

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 fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the article section

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