Equation 1 · Reading the Oldest Light: How a Temperature Map Becomes a Cosmology
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Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol theta_mathrm s
thetathrm s is part of the quantity the equation computes from the expression on the right.
Symbol r_mathrm s
athrm s occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol z_ast
st is an input to the expression that computes the quantity on the left.
Symbol D_mathrm A
athrm A occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
Denominator: D_mathrm A(z_ast)
The complete quantity below the fraction bar; it must be nonzero for this division.
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.
What the article says around this equation
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: . 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 surrounding passage
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: . 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 that happened afterwards along the line of sight, including spatial curvature and the late-time expansion history. This single ratio is why an angular measurement can constrain geometry, and it is also the origin of the degeneracies that make CMB-only constraints on some parameters much weaker than the headline error bars suggest.
Sources cited in the article section
These citations give research context. Read each source to check which claims it supports.
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