← Mathematical compendium

Published equation contexts

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

Why this formula appears here

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…

Read the full article-specific guide →

Read the representative guide

θs\theta_{\mathrm s}

Symbol theta_mathrm s

thetama_mathrm s is part of the quantity the equation computes from the expression on the right.

Read this term in its guide →
rsr_{\mathrm s}

Symbol r_mathrm s

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

Read this term in its guide →
DAD_{\mathrm A}

Symbol D_mathrm A

DmD_mathrm A occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Read this term in its guide →
DA(z∗)D_{\mathrm A}(z_\ast)

Denominator: D_mathrm A(z_ast)

The complete quantity below the fraction bar; it must be nonzero for this division.

Read this term in its guide →

How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

Research cited beside this formula

Learn the mathematics behind it

Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

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

Equation 1 · Cosmology

Reading the Oldest Light: How a Temperature Map Becomes a Cosmology

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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…

Equation guide → · Article →