Background BAO

Photon comoving scales under \(\tilde g_{\mu\nu} = P\,g_{\mu\nu}\) — geometry revision.

Math / derivation	DCT-BAO-01 (revised): null radial geodesics in \(ds^2 = P(t)(-\mathrm dt^2 + a^2\,\mathrm d\chi^2)\) give \(\mathrm d\chi/\mathrm dt = 1/a\); homogeneous \(P\) cancels.
Python verification	`bao_conformal_null_check.py` (toy FLRW), `dct_bd_frame_dictionary.py` (conventions)
Measurement	DESI Year-1 BAO \(D_M/r_d\), \(D_H/r_d\) (Adame et al. 2024)
Prediction (homogeneous \(P\) only)	Standard \(\chi(z) = \int c\,\mathrm dz'/H(z')\) unchanged at background level — no uniform \(1/\sqrt{P_0}\) rescaling of photon comoving distances.
Legacy audit artifact	\(\Delta\chi^2 \approx +33.6\) from `dct_desi_bao_test.py` reproduced a mistaken \(D_M\) map (documented in script header) — not a falsification of homogeneous conformal DCT.
Status	neutral (homogeneous background); live tension tests move to perturbation-level / inhomogeneous / cluster channels.
Methodology note	See DCT-BAO-01.

Formal vocabulary. The 33-facet audit (audit_fails.py; scorecard) counts background BAO as PASS under the revised treatment: homogeneous \(P(t)\) cancels from radial null \(\chi_{\rm null}\), so there is no corpus FAIL from that geometry. On the predictions table, the same homogeneous-background row uses the pill neutral to mean “no uniform percent shift in standard comoving \(\chi(z)\) versus \(\Lambda\)CDM at matched \(H(z)\)” — neither a sharp HIT nor a falsifying MISS. What was retracted is the legacy \(D_M\) rescaling that produced a large \(\Delta\chi^2\) in an old audit branch (dct_desi_bao_test.py), not null propagation. Falsifiable late-universe interfaces remain the perturbation-level kernels (\(\mu_b,\mu_{\rm DM},\Sigma\)) and cluster \(M_{\rm lens}/M_{\rm dyn}\), as in DCT-BAO-01.

Mechanism

For spatially homogeneous \(P(t)\) multiplying both \(-\mathrm dt^2\) and \(a^2(t)\,\mathrm d\chi^2\), radial photon nulls obey \(0 = P(-\mathrm dt^2 + a^2\,\mathrm d\chi^2)\), so \(P\) divides out: \(\mathrm d\chi/\mathrm dt = 1/a(t)\). The comoving distance used in conventional BAO analyses is therefore the same map as in the inner metric \(-\mathrm dt^2 + a^2\,\mathrm d\chi^2\). A separate prescription that multiplied \(D_M^{\Lambda\mathrm{CDM}}\) by \(1/\sqrt{P_0}\) (or \(\sqrt{P_0}\)) inserted \(P\) into a pipeline where geometry forbids it.

Operational statements such as \(H_{\rm phys} = H_E/\sqrt{P_0}\) for late-universe matter ladders must be derived from proper time on \(\tilde g = Pg\) and must not be reverse-engineered into the null \(\chi\) integral. See the methodology note and revised DCT-BAO-01.

Live framework

The perturbation-level programme on a \(\Lambda\)CDM background (\(\mu_b = 1/P\), \(\mu_{\rm DM} = 1/[P(1+\beta)]\), \(\Sigma = 1/\bar P\)) and cluster-scale \(M_{\rm lens}/M_{\rm dyn}\) remain the primary falsifiable interfaces with data. Background BAO under pure homogeneous conformal \(P\) does not, by itself, force a percent-level shift in \(\chi(z)\).

Python verification

bao_conformal_null_check.py — toy check that \(\chi\) depends only on \(a(t)\).
dct_bd_frame_dictionary.py — retired reciprocal branches + prose.
dct_desi_bao_test.py — legacy \(\chi^2\) reproduce for audit; read header for geometry caveats.

References

DESI Collaboration 2024 (Year-1 BAO)
Cluster paper (revised): DCT-BAO-01.tex

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