Paper XXXVI - Baryon Acoustic Oscillations and Operational Distance Scales

DOI: 10.5281/zenodo.18010251 Baryon Acoustic Oscillations and Operational Distance Scales Paper XXXVI of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Abstract. We analyze baryon acoustic oscillation (BAO) measurements within the OrderedDynamics Reconstruction Program (ODRP), incorporating them as discrete distance-scale probes acting on operational distance operators constrained by independent temporal and distance observations. Operational time scaling is fixed by supernova temporal structure and strong gravitational lensing, while continuous distance structure is constrained by supernova distance residuals and synthesized in a joint admissible operator space. BAO observables are treated as localized constraints on transverse and radial distance operators, with the sound-horizon scale marginalized as a nuisance parameter. Synthetic injection–recovery tests validate identifiability and robustness. The results are interpreted strictly as refinements of operational distance structure and do not invoke cosmological expansion dynamics, earlyuniverse physics, or dark energy. Contents 1 Program context and objectives 1 2 BAO observables as operational quantities 2 3 Operational BAO distance operators 2 3.1 Separation from expansion dynamics 2 3.2 Operational transverse distance 2 3.3 Operational radial distance 2 3.4 Volume-averaged operational distance 2 3.5 Relation to supernova distances 2 3.6 Sound-horizon normalization 2 3.7 Admissibility 3 4 Datasets and preprocessing 3 5 Likelihood and nuisance parameters 3 6 Synthetic injection–recovery 3 7 Results 3 7.1 Injection–recovery outcomes 3 7.2 Consistency with admissible operator space 3 7.3 Refinement of distance constraints 3 7.4 Robustness checks 4 8 Interpretation 4 9 Conclusion 4 Contents 1 Program context and objectives Paper XXXVI extends the empirical sequence of the Ordered-Dynamics Reconstruction Program by incorporating baryon acoustic oscillation (BAO) measurements as discrete distancescale probes [1]. Previous papers have established: (i) operational time scaling from supernova temporal structure (Paper XXXII), (ii) clock–distance consistency from strong gravitational lensing time delays (Paper XXXIII), (iii) continuous operational distance structure from supernova distance residuals with clocks fixed (Paper XXXIV), and (iv) a synthesized admissible operator space consistent across probes (Paper XXXV) [9–12]. All statistical and validation conventions inherit the standardized ingestion and likelihood discipline of Paper XXXI [8]. The purpose of this paper is to test whether BAO measurements are consistent with that admissible space and to determine how they further restrict operational distance operators at specific redshifts. No assumptions are made regarding cosmological expansion dynamics, early-universe physics, or absolute distance calibration. – 1 – 2 BAO observables as operational quantities BAO measurements probe a characteristic spatial separation imprinted in the matter distribution and recovered statistically from galaxy clustering and related tracers [1–4]. Observationally, this scale is inferred from angular and redshift separations and is commonly reported in terms of transverse, radial, or isotropic distance combinations. Within the ODRP framework, BAO observables are treated as operational constraints on distance mappings at discrete redshifts, without identifying them with expansion rates, metric distances, or specific cosmological parameters. 3 Operational BAO distance operators 3.1 Separation from expansion dynamics Standard BAO analyses express observables in terms of an expansion rate and a comoving distance, implicitly assuming a specific spacetime metric. In contrast, the ODRP treats BAO measurements as constraints on phenomenological distance operators inferred directly from observed separations. 3.2 Operational transverse distance We define an operational transverse distance operator Dop M(z) such that a characteristic transverse separation ℓ⊥satisfies ℓ⊥=Dop M(z) ∆θ. (3.1) 3.3 Operational radial distance We define an operational radial distance operator Dop H(z) such that a characteristic radial separation ℓ∥satisfies ℓ∥=Dop H(z) ∆z. (3.2) No identification with c/H(z) is assumed. 3.4 Volume-averaged operational distance For isotropic BAO measurements, we define Dop V(z) = z Dop M(z)2Dop H(z)1/3.(3.3) 3.5 Relation to supernova distances The operational luminosity distance constrained in Paper XXXIV is related to the transverse distance operator via distance reciprocity: Dop L(z) = (1 + z)2Dop M(z),(3.4) which follows from photon number conservation and reciprocity under minimal assumptions [5, 6]. 3.6 Sound-horizon normalization BAO measurements are reported relative to the sound-horizon scale rs. In this paper, rs is treated strictly as a nuisance normalization parameter. BAO constraints are therefore implemented as measurements of ratios such as Dop M(z)/rs,Dop H(z)/rs, or Dop V(z)/rs. – 2 – 3.7 Admissibility Operational BAO distance operators are required to be positive, smoothly interpolable between BAO redshifts, and compatible with the admissible operator space Adefined in Paper XXXV [12]. 4 Datasets and preprocessing We analyze publicly released BAO measurements spanning multiple redshifts and survey methodologies, including BOSS/eBOSS and newer DESI BAO results [2–4]. Reported covariance matrices and systematic uncertainties are incorporated without modification. No recalibration is performed to enforce agreement with external cosmological models. 5 Likelihood and nuisance parameters The likelihood follows the unified Gaussian form defined in Paper XXXI [8], with data vectors consisting of BAO distance-scale measurements and full covariance matrices. Operational time scaling is fixed by independent constraints [9, 10]. Distance operators and the soundhorizon scale rsare treated as nuisance parameters or constrained only through consistency across redshifts and compatibility with the admissible operator space [12]. 6 Synthetic injection–recovery Synthetic BAO datasets are generated by injecting known operational distance operators evaluated at BAO redshifts, with noise and covariance matched to real data. Null and nonnull injections are recovered without significant bias. Injected variations in rsare absorbed by the nuisance posterior and do not induce spurious distance deformation. Negative controls, including redshift reassignment, observable mismatch, and covariance truncation, fail in the expected manner. 7 Results 7.1 Injection–recovery outcomes Injected operational distance operators are recovered without significant bias, and null injections do not produce spurious deviations. Sound-horizon variation does not correlate with distance deformation. 7.2 Consistency with admissible operator space Observed BAO measurements are compatible with the admissible operator space Adefined in Paper XXXV [12]. No BAO point requires probe-dependent rescaling of distances or clocks. 7.3 Refinement of distance constraints BAO measurements reduce uncertainty in operational distance operators at intermediate redshifts relative to supernova-only constraints, refining but not reshaping the inferred distance structure. – 3 – 7.4 Robustness checks Results are stable under isotropic versus anisotropic BAO treatments, alternative priors on rs, covariance assumptions, and dataset subsets. 8 Interpretation Within the ODRP framework, BAO measurements act as discrete distance anchors that refine operational distance structure already constrained by independent probes. Consistency with supernova and lensing results indicates coherence across continuous and discrete distance observables. The sound-horizon scale is treated purely as a nuisance parameter, and no claims are made regarding early-universe physics, cosmological expansion dynamics, or dark energy. 9 Conclusion We have incorporated baryon acoustic oscillation measurements into the Ordered-Dynamics Reconstruction Program as operational distance-scale probes. With operational time scaling fixed independently and continuous distance structure constrained by supernovae, BAO measurements act to further restrict the admissible distance operator space without invoking cosmological expansion dynamics or early-universe assumptions. Together with Papers XXXII–XXXV, this work advances the ODRP toward a coherent, multi-probe reconstruction of spacetime observables. Future papers will extend the framework to growth-of-structure measurements and early-universe boundary conditions, using the admissible operator space established here as a controlled starting point. References [1] D. J. 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