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Equation 1 · Part 8 · Reading the Oldest Light: How a Temperature Map Becomes a Cosmology

Numerator: r_mathrm s(z_ast)

θs=rs(z∗)DA(z∗).\theta_{\mathrm s} = \frac{r_{\mathrm s}(z_\ast)}{D_{\mathrm A}(z_\ast)}.
rs(z∗)r_{\mathrm s}(z_\ast)

What this part means

The complete quantity above the fraction bar.

Its job in the formula

rmr_mathrm s(zaz_ast) occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

The consequence is a harmonic series in the sky. Modes that happened to be at maximum compression or maximum rarefaction at last scattering show the largest temperature contrast; modes caught at their zero crossing show the least. Because the phase depends on wavenumber times the sound horizon, the peaks fall at multiples of a fundamental scale. The physical length of that scale is the comoving sound horizon at last scattering, and the angle it subtends is a ratio of two lengths: θs=rs(z∗)DA(z∗)\theta_{\mathrm s} = \frac{r_{\mathrm s}(z_\ast)}{D_{\mathrm A}(z_\ast)}. The numerator is set by pre-recombination physics — the expansion rate and sound speed before decoupling, which depend on the baryon and radiation densities. The denominator is set by everything…

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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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