\(\beta_{\rm PPN}\) − 1

Post-Newtonian non-linearity coefficient under the canonical Brans–Dicke + DCT action.

Prediction	\(+1.0 \times 10^{-10}\) (analytic identity)
Closed form	\(\beta - 1 = 1/[(2\omega_0 + 3)(2\omega_0 + 4)] = 1/(100{,}077 \cdot 100{,}078) = 9.985 \times 10^{-11}\)
Measured	LLR Williams 2009 \(\|\beta - 1\| < 1.1 \times 10^{-4}\) (90% CL upper bound)
Forward test	BepiColombo MORE 2027–2028 (Iess 2021), projected \(\sigma_\beta \sim 2 \times 10^{-6}\) — DCT prediction is below threshold (no clean detection)
Status	open — consistent with current bounds; below the next-generation precision floor
Source paper	DCT-PPN-01

Mechanism

In the Brans–Dicke class with quadratic coupling \(\omega(P) = (c_{BD} P^n - 3)/2\) at \(n = 2\), the post-Newtonian coefficient \(\beta\) deviates from unity at second order in the small parameter \(\varepsilon \equiv 1/(2\omega_0 + 3)\). The closed expression at \(P = P_0\) is

\(\beta_{\rm PPN} - 1 = \dfrac{1}{(2\omega_0 + 3)(2\omega_0 + 4)} = \dfrac{1}{c_{BD} P_0^2 \cdot (c_{BD} P_0^2 + 1)}\),

and substituting the master-identity values \(2\omega_0 + 3 = 100{,}077\) and \(c_{BD} P_0^2 = 100{,}077\) gives \(\beta - 1 = 1/(100{,}077 \cdot 100{,}078) = 9.985 \times 10^{-11} \approx 1.0 \times 10^{-10}\).

Derivation chain

Canonical action: Brans–Dicke scalar with the quadratic-in-\(P\) coupling \(\omega(P) = (c_{BD} P^n - 3)/2\), \(n = 2\).
Master identity (DCT-SPI-01): \(2\omega_0 + 3 = c_{BD} P_0^2 = 100{,}077\) with \(c_{BD} = 138{,}189\), \(P_0 = 0.851\).
Standard PPN expansion of the BD field equations to second order in \(1/(2\omega + 3)\), with the quadratic coupling activating the second-derivative term, gives \(\beta - 1 = 1/[(2\omega_0 + 3)(2\omega_0 + 4)]\) (DCT-PPN-01 §II.C, Eq. 16).
Numerical substitution: \(1/(100{,}077 \cdot 100{,}078) = 9.985 \times 10^{-11}\).

Audit notes — abstract vs body discrepancy

The DCT-PPN-01 abstract carries an order-of-magnitude transcription error

The DCT-PPN-01 abstract and its §I summary equation list \(\beta - 1 = +5 \times 10^{-11}\); the paper's own body (§II.C, Eq. 16) and the analytic closed form give \(+1.0 \times 10^{-10}\). The two differ by a factor of two. Independent re-derivation from the master identity confirms the body value: \(1/(100{,}077 \cdot 100{,}078) = 9.985 \times 10^{-11}\), which rounds to \(+1.0 \times 10^{-10}\).

This site uses the canonical body-and-analytic value \(+1.0 \times 10^{-10}\). The cluster paper abstract requires a one-character cluster-paper-text fix (DCT-workspace session).

Python verification

Reproduce the analytic identity from the canonical constants:

omega_0 = 50037
beta_minus_1 = 1.0 / ((2*omega_0 + 3) * (2*omega_0 + 4))
# = 9.985e-11 ≈ 1.0e-10

The general gravity-tests module that exercises this constant alongside the other PPN coefficients is gravity_tests.py.

References

Will 2014, "The Confrontation between General Relativity and Experiment", Living Rev. Rel. 17, 4 — PPN formalism.
Williams, Turyshev, Boggs 2009 — LLR Nordtvedt η bound and \(|\beta - 1|\) inference.
Iess et al. 2021 — BepiColombo MORE radio science, projected \(\sigma_\beta\)
Source paper: DCT-PPN-01
Companion observables: \(\gamma_{\rm PPN}\) − 1, Cassini \(\gamma\), Nordtvedt \(\eta\), master identity.

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