Test 35 — BAO Standard-Ruler Consistency Test
Part of the 44-test Early-Universe Granularity Test Suite. This test asks whether the baryon acoustic oscillation (BAO) scale — a "ruler" imprinted in the distribution of galaxies today — matches the same sound horizon that the cosmic microwave background (CMB) recorded roughly 380,000 years after the Big Bang.
1. Verdict
The number, both ways
- Number we’re testing
- The baryon-acoustic-oscillation standard ruler — the sound horizon at the drag epoch, r_d, a fixed length frozen into cosmic structure
- Standard cosmology & measured
- Standard cosmology here is the measurement itself — r_drag = 147.09 ± 0.26 Mpc (Planck 2018 base-ΛCDM, CMB-inferred); galaxy-survey distance ratios (D_M/r_d, D_H/r_d) at z = 0.15–2.33 agree with this to ~1–2% · r_d = 147.09 ± 0.26 Mpc — the one ruler, recovered by BOSS DR12 (~1% level), eBOSS DR16, and DESI DR1 (r_d h ≈ 101.8 Mpc, consistent at ~1–2σ per tracer)
- This framework (granularity)
- The independent read: The galaxy-scale ruler is a late-time fossil of the very same acoustic physics that set the CMB peaks — the framework recovers the same r_d, inherited without modification
- Agreement
- BAO distance ratios consistent with the Planck-calibrated r_d = 147.09 Mpc to within the quoted 1–2% measurement errors across the full z = 0.15–2.33 range (consistency check — shared inputs)
Check the source → the calculation shown on this page (Data Used · Calculation Summary)
CLOSED — the granularity interpretation (BAO distances as a late-time, large-scale-structure fossil record of the same acoustic-oscillation physics that set the CMB sound horizon) is consistent with the current standard-cosmology comparison between CMB-inferred and BAO-measured sound-horizon quantities, within reported uncertainties. Secondary note: a small (roughly 1–2σ, non-definitive) tension between the CMB-inferred sound-horizon-times-Hubble-parameter combination and some BAO-based low-redshift determinations exists in the literature (part of the broader Hubble-tension conversation) and is reported here for honesty, but it is a standard-cosmology-level tension under active study, not something this framework introduces, resolves, or is in a position to adjudicate.
2. Tested Claim
The precise granularity claim under test: the acoustic oscillations imprinted in the primordial baryon-photon plasma before recombination froze in a characteristic comoving length scale (the sound horizon at the baryon drag epoch, \(r_d\)); this length scale is a stable, recordable distinction that survives as (a) the CMB acoustic-peak spacing at \(z\approx1089\) and (b) a preferred clustering scale in the distribution of galaxies at much lower redshift (BAO). Both records must therefore recover the same underlying \(r_d\), within measurement uncertainty, unless the model explicitly changes early-universe physics between the two epochs. This framework does not propose any such change to early-universe physics for this test, so the claim reduces to the standard ΛCDM consistency check.
3. Data Used
| Quantity | Value | Source |
|---|---|---|
| Sound horizon at drag epoch, \(r_{drag}\) (Planck base-ΛCDM, CMB-inferred) | 147.09 ± 0.26 Mpc | Planck Collaboration 2018 (Aghanim et al. 2020), A&A 641, A6, TT,TE,EE+lowE+lensing |
| \(\Omega_b h^2\) | 0.02237 ± 0.00015 | Planck 2018, as above |
| \(T_{CMB}\) | 2.7255 ± 0.0006 K | Fixsen 2009 (COBE/FIRAS), as adopted by Planck 2018 |
| BOSS DR12 consensus BAO: \(D_V/r_d\), \(D_M/r_d\), \(D_H/r_d\) at \(z=0.38, 0.51, 0.61\) | consistent with Planck ΛCDM \(r_d\) at the ∼1% level | Alam et al. 2017 (BOSS DR12), MNRAS 470, 2617 |
| eBOSS DR16 final consensus BAO+RSD (quasars, LRGs, ELGs, Lyα), \(z=0.15\text{–}2.33\) | joint fit consistent with Planck ΛCDM within quoted errors | Alam et al. 2021 (eBOSS DR16), Phys. Rev. D 103, 083533 |
| DESI DR1 BAO (2024): \(r_d h\) and per-tracer \(D_M/r_d\), \(D_H/r_d\) | \(r_d h \approx 101.8\) Mpc; consistent with Planck ΛCDM at the ∼1–2σ level per tracer | DESI Collaboration 2024, arXiv:2404.03002 (DR1 BAO cosmology paper) |
| \(n_s\) (scalar spectral index, context) | 0.9649 ± 0.0042 | Planck 2018, as above |
4. Calculation Summary
Step 1 — sound horizon at drag epoch, model value. The comoving sound horizon at the baryon drag epoch is
\[ r_d = \int_{z_d}^{\infty} \frac{c_s(z)}{H(z)}\,dz, \qquad c_s(z) = \frac{c}{\sqrt{3\left(1+\frac{3\rho_b}{4\rho_\gamma}\right)}} \]evaluated with standard ΛCDM background parameters (\(\Omega_b h^2\), \(\Omega_c h^2\), \(T_{CMB}\), \(N_{eff}=3.044\)). This framework does not modify \(H(z)\), \(\rho_b\), \(\rho_\gamma\), or \(N_{eff}\) prior to recombination for this test, so the model value of \(r_d\) is the standard Planck 2018 value, \(r_{drag}=147.09\pm0.26\) Mpc. There is no distinctive alternative computation being checked against this — per the Minimum-Calculations guardrail (item 3 in the README), this test's rate/threshold check is: is the sound-crossing time before baryon drag \((z_d\approx1060)\) short enough, relative to the Hubble time at that epoch, for the oscillation to complete and freeze in a well-defined comoving scale? Standard cosmology answers yes (the plasma is tightly coupled by Compton scattering down to \(z_d\), giving a well-defined, calculable \(r_d\)); this framework inherits that answer without modification.
Step 2 — compare with BAO measurements across redshift. Galaxy-survey BAO measurements (BOSS DR12, eBOSS DR16, DESI DR1) report the ratio of transverse/radial comoving distances to \(r_d\) (\(D_M/r_d\), \(D_H/r_d\), or the isotropic \(D_V/r_d\)) at several effective redshifts between \(z=0.15\) and \(z=2.33\). When these ratios are combined with the Planck-inferred \(r_d=147.09\) Mpc, the implied absolute distances \(D_M(z)\), \(D_H(z)\) match the distances predicted by the Planck-calibrated flat-ΛCDM expansion history to within the quoted 1–2% measurement errors, across the full redshift range surveyed. This is exactly the "BAO distances and sound horizon are mutually consistent with CMB/BBN constraints" pass criterion from the test suite.
Step 3 — check consistency with CMB-derived sound horizon. Because this framework does not invoke early-universe physics that changes \(r_d\) (no modified expansion rate before recombination, no extra relativistic species beyond the standard \(N_{eff}=3.044\), no altered baryon density) for this test, Step 4 of the procedure ("if early physics changes \(r_d\), propagate effects into CMB peaks and BBN") does not apply — there is nothing to propagate. The comparison is therefore a direct inheritance of the standard ΛCDM CMB–BAO consistency result, which is a well-established, actively monitored result in the cosmology literature (DESI 2024; eBOSS 2021).
Step 4 — honesty on the residual tension. A small, well-documented tension exists between early-time (CMB-calibrated) and late-time (local distance-ladder / some BAO combinations) determinations of quantities entangled with \(H_0\) and \(r_d\) — this is the broader "Hubble tension" family of results. DESI's 2024 BAO analysis reports mild (\(\sim\)1–2\(\sigma\)) internal tensions in some individual tracer bins and a preference for evolving dark energy (\(w_0w_a\)) at the \(\sim2\text{–}3\sigma\) level in some combinations, which is a live, unresolved research question in standard cosmology — it is not something this framework causes or can currently adjudicate, and it does not rise to the level of "the model fits BAO by breaking CMB or early thermal history" (the stated failure criterion), because no version of this framework's early physics is being altered to produce that fit.
5. Granularity Interpretation
What becomes distinguishable/recordable at this epoch, in the framework's interpretive language: the sound horizon \(r_d\) is a single physical scale that gets recorded twice, in two different physical media, at two very different epochs — once as an angular pattern in the CMB photon temperature/polarization field at \(z\approx1089\), and once as a preferred clustering separation in the 3-D distribution of galaxies from \(z\sim2.3\) down to \(z\sim0.1\). Under the master doctrine ("cosmic history is the history of increasing recordable distinction"), the BAO signal is read as a late-time, degraded-but-still-legible fossil copy of an early-time record — the same distinction (one comoving length) surviving through radically different physical substrates (relativistic photon-baryon plasma → nonrelativistic collapsed large-scale structure). The test checks the internal consistency of treating both as records of the same underlying distinction rather than two independent, coincidentally similar numbers. Standard cosmology already treats it this way; this framework inherits, and does not modify, that treatment for this test.
6. Gate Routing
Acoustic fossil record gate. Ledger entry:
Acoustic fossil record gate -> agrees with standard cosmology (consistent with standard cosmology)
-> supporting calculation: Planck 2018 r_drag = 147.09 +/- 0.26 Mpc vs.
BOSS DR12 / eBOSS DR16 / DESI DR1 BAO distance ratios, consistent
within 1-2% across z = 0.15-2.33
-> open gap: none distinctive; standard-cosmology H0/w0wa
tension noted but not resolved or introduced by this framework
This closure does not prove any part of the full theory. It only establishes that the acoustic-record consistency constraint does not currently block the framework, because the framework does not touch the physics this constraint depends on.
7. Failure Mode
Not applicable in the "the test failed to close" sense — the test closed. Flagged honestly rather than a failure: (a) this is a shared closure, not a distinctive derivation — no number here was produced natively by the framework's geometry or calibration pipeline; standard ΛCDM background cosmology was used unmodified. (b) The residual \(\sim\)1–2\(\sigma\) tensions reported by DESI (2024) in some tracer combinations are real and unresolved in the field at large; they are noted as a secondary caveat per the Allowed Gate Outcomes, not swept under the verdict.
8. Next Action
Data lookup / monitoring, not derivation. This test does not currently require this framework to perform new calculation, because it does not modify the relevant early-universe physics. The open action item is to monitor the standard-cosmology literature (DESI DR2/DR3 releases, upcoming Euclid and Rubin/LSST BAO measurements) for whether the mild DESI-2024 tensions harden into a genuine early-vs-late-time sound-horizon discrepancy. If a future release of this framework proposes a mechanism that changes early-universe expansion history (e.g., to address \(H_0\) tension), this test would need to be re-run as DISTINCTIVE, with the modified \(r_d\) propagated into both CMB and BAO predictions per Step 4 of the procedure, and checked against BBN as well.
Where this agrees with standard cosmology
Every number used in this test's calculation — \(r_{drag}\), \(\Omega_b h^2\), \(T_{CMB}\), the BAO distance ratios — is a standard measured or standard-model-computed cosmological quantity, sourced from Planck, BOSS/eBOSS, and DESI collaboration papers. This framework contributes an interpretive frame (BAO as a recordable-distinction fossil) but no new mechanism, no new free parameter, and no number this framework's own geometry or calibration produced natively. Per the site-wide honesty rule: a qualitative match to the granularity doctrine motivates the route, it does not constitute proof, and it is not credited as a distinctive prediction.