A Spectral Origin for the BAO Ruler from Curved Momentum Space

Baryon acoustic oscillations (BAO) provide a robust comoving standard ruler used to constrain the expansion history through galaxy clustering. In the standard picture, the ruler is set by the sound horizon at baryon drag and propagated to late times through linear theory, non-linear damping, and reconstruction. In this paper I develop a complementary momentum-first formulation in which an effective BAO ruler emerges as the configuration-space projection of a low-lying spectral scale of a curved momentum manifold (𝒫, 𝑔). The central object is an elliptic (Riemannian) sector (𝛴, ℎ) ⊂ (𝒫, 𝑔) admitting a Laplace–Beltrami eigenvalue problem; the first non-trivial eigenvalue 𝜆1 selects a distinguished long-wavelength mode 𝑘𝑃 = √𝜆1. A covariant operational dual map 𝛷: ℳ → 𝒫 induces an effective configuration metric 𝑔̃= 𝛷∗𝑔, allowing the spectral scale to be mapped into a comoving length 𝑟 ⋆ on ℳ. In the weak-curvature regime this yields a controlled deformation of the effective BAO ruler, 𝑟𝑑 → 𝑟𝐵𝐴𝑂 = 𝑟𝑑(1 + 𝛿𝑟), which enters standard BAO inference as a shift of dilation parameters. I derive the leading-order mapping to BAO observables, classify non-degenerate signatures (mode splitting, redshift drift, and anisotropic projection), and outline how existing BAO likelihoods can constrain 𝛿𝑟 at the sub- percent level using current DESI precision. [1–3,7,10–15].