One Horizon, Two Tables: The Sound Horizon, Priced as a Closure Defect

Some lengths in physics can be read off two completely different tables, and the honest question is whether the two readings agree as well as their own error bars say they should. That is the closure test: one scale, two tables, and a defect — the difference between the tables divided by the error the two tables jointly allow. This paper executes the test on a scale the test has not been run on before: the sound horizon at the drag epoch, r_d, the distance sound could travel in the early universe before baryons broke free of the photons, about 147 million parsecs. It is the ruler printed into both the microwave background sky and the galaxy map, so both skies get a vote. Table one is the microwave background's inference, from the Planck satellite: r_d = 147.09 ± 0.26 Mpc. Table two is a measurement built without the microwave background at all — the angular baryon-acoustic scale in galaxies, combined with nucleosynthesis and cosmic chronometers: r_d = 150.4 Mpc, with errors of about 3 Mpc. The defect between the two tables is +3.31 Mpc, +2.25 percent, at a significance of 1.1 standard deviations. The horizon closes. But the closure is a property of which late table you pick, and that is the paper's second finding: a newer late table, built on DESI galaxy data anchored by gravitational-lens time delays, reads 138.2 ± 3.9 Mpc — a defect of −8.89 Mpc against the microwave table, at 2.3 standard deviations, which does not close. The two late tables disagree with each other (12.2 Mpc, 2.5 standard deviations) more than either one disagrees with the sky they were both checking. The ruler is fine. The readings of it are not yet one reading.