Large-Scale Structure and BAO

Galaxy clustering shows a two-point correlation function, a power spectrum with a turnover, and—most sharply—a baryon-acoustic-oscillation bump near 150 Mpc, the standard ruler. Broad clustering can form by deficit-coupled gravity in a static medium, and a follow-up simulation shows the sharp BAO feature arises from a fixed inflow-shell length. Deriving that length is the one open input.

Three statistics summarise large-scale structure: the galaxy two-point correlation function ξ(r), which measures the excess probability of finding pairs at separation r; the matter power spectrum P(k), which rises and turns over near k ≈ 0.02 h Mpc⁻¹; and the baryon-acoustic-oscillation bump at a comoving separation of about 150 Mpc.

The raw facts to be explained

Three statistics summarise large-scale structure. The galaxy two-point correlation function ξ(r) measures the excess probability of finding pairs at separation r and is positive on small scales. The matter power spectrum P(k) rises, turns over near k≈0.02hMpc⁻¹, and falls, with an amplitude conventionally summarised by σ₈≈0.8. Superimposed on ξ(r) is a single sharp bump at r≈150Mpc—the baryon acoustic oscillation, the imprint in the matter distribution of the same acoustic physics that produces the microwave peaks of the previous chapter. We separate these into a part the framework can plausibly reach and a part it cannot.

Broad clustering: plausibly reproduced

The broad shape of ξ(r) and P(k) reflects gravitational clustering of matter from small initial inhomogeneities. The framework has a gravitational mechanism: vacuum deficits attract (Chapter 8), with the radial-acceleration relation automatic, and a static medium imposes no 13.8Gyr limit on the time available for clustering to grow. The turnover scale of P(k) near k≈0.02hMpc⁻¹ could map to a lattice or attenuation scale. To this extent the broad clustering is plausibly reproduced, matching the standard growth picture in shape, though the growth-rate normalisation σ₈≈0.8 is not derived.

The BAO ruler: the sharp conflict

The difficulty is the bump. In the standard model the BAO scale is fixed by the comoving sound horizon at the drag epoch,

the maximum distance a baryon–photon pressure wave travels before decoupling, frozen into the matter distribution as a preferred pair separation. A static medium with no recombination and no sound horizon has no native mechanism to imprint a sharp preferred scale at 150Mpc; deficit-coupled clustering produces a smooth correlation function with no reason for a bump there. This is the same acoustic-scale conflict as the microwave peaks, now in the galaxy distribution—the sharp conflicting item of the chapter, and the reason its headline grade is conflicting.

Simulation: making the target explicit

The simulation again fixes the target without claiming to meet it. Simulation A records the BAO scale of Eq. (lss-bao) as the feature to be explained, and Simulation B grows a toy deficit-coupled density field and shows that it develops broad clustering but no 150Mpc bump unless such a scale is inserted by hand:

# A: target standard ruler (what must be explained)
r_BAO = 150 Mpc      (standard: comoving sound horizon at drag epoch)
# B: toy deficit-coupled growth in a static medium
broad correlation xi(r) grows  (clustering: plausibly reproduced)
sharp bump at r=150 Mpc: ABSENT unless a scale is seeded (BAO: conflicting)

The toy field clusters broadly but shows no acoustic bump, confirming that the broad power spectrum is within reach while a bump must, in this first toy, be inserted by hand. A deeper simulation closes part of that gap.

A fixed inflow length supplies the bump (deeper simulation)

The follow-up simulation shared with Chapter 14 (sim_acoustic_peaks.py) tests whether the 150Mpc ruler can arise statically. If each mass concentration carries an overdensity shell at a fixed inflow radius Rₛ—a standing inflow feature, with no recombination and no synchronized clock—then the center–shell correlation develops a clean bump exactly at r=Rₛ=150Mpc, and the shell power spectrum |sin(kRₛ)/(kRₛ)|² carries oscillation zeros at k=nπ/Rₛ, the observed BAO wiggle spacing. So the BAO bump and its wiggle scale do follow from a single fixed length; what the simulation does not do is derive the value Rₛ=150Mpc from the lattice constants. Notably the standard model is in a comparable position—it does not obtain the sound horizon from pure first principles either, but from the measured baryon and matter densities Ω_bh²,Ωₘh²—so a single undetermined length is a fair, bounded debt rather than a refutation.

The honest verdict is therefore upgraded: large-scale clustering is plausibly reproduced (degenerate with ΛCDM), and the 150Mpc BAO ruler is no longer a bare conflict but an open mechanism with one missing input—the same fixed length Chapter 14 needs for the microwave peak spacing. Recorded as an exploratory research program (\textsf{HYP}) with the success criterion: derive the inflow length L≈150Mpc from the lattice constants. Until that single number is derived the ruler is reproduced conditionally (mechanism-shown), not unconditionally (value-derived)—the volume's own distinction. A dedicated audit shared with Chapter 14 (derive_acoustic_length.py) confirms the verdict from the other side: it computes every natural length the framework provides and finds none within tuning-free reach of 150Mpc—the inflow clock lives near 10³km and the only cosmological scale is the Hubble length ≈4300Mpc—so the missing input is specifically a cosmological cavity length. The two companion sub-criteria of the program (the standing-wave peak heights and the γ-ray dispersion check) close there; this single underived length does not. A second diagnostic (probe_subhorizon_desert.py) sharpens why: 150Mpc lies in an empty 10-decade gap between the framework's microscopic and cosmological scales, and in the standard model it is a sound-horizon integral over an expansion history up to recombination—two ingredients (history, early epoch) a present-emission lattice rejects by design—so the BAO ruler's missing length is a structural object, a present-day sub-horizon correlation length, not a number awaiting a lucky guess. A capping interpretation (acoustic_length_hypothesis_interpretation.py) reads that length, at hypothesis level, as a 2D-rotational→3D-isotropic crossover—the anisotropic-web-to-homogeneity transition—built by the companion volume's selected-length mechanism; the mechanism and its forced prefactor 2π√2 close, while the absolute value is recorded as an empirical normalisation, not a derived number.