Chameleon bigravity and spin-2 dark matter

Final project, AI-assisted research (Claude Code + Codex) · Guadalupe Ahumada Acuña, DF–FCEN–UBA · September 2026

Can the massless gravitons of the metric \(g\) mix with those of a second dynamical metric \(f\) and produce massive spin-2 particles that behave as dark matter? I took my own bigravity calculation and used two agents to extend it to the chameleon case, \(\beta_n\to\beta_n(\phi)\), with a general relative lapse \(c(t)\) and background ratio \(\xi(t)\). For every result I recorded where it comes from and checked it along a path that shares no code with the one that produced it.

Report (PDF, in Spanish) Provenance How to reproduce it with an agent Checks My notebook (clean run) All material

The model

\[ S=\frac{M_g^2}{2}\!\int\!\sqrt{-g}R[g]+\frac{M_f^2}{2}\!\int\!\sqrt{-f}R[f]+M_g^2m^2\!\int\!\sqrt{-g}\sum_{n=0}^{4}\beta_n(\phi)\,e_n\!\big(\sqrt{g^{-1}f}\big)-\tfrac12\!\int\!\sqrt{-g}(\partial\phi)^2+S_m\!\left[A^2(\phi)g\right] \]

The backgrounds are \(ds_g^2=-dt^2+a^2d\vec x^2\) and \(ds_f^2=\xi^2(t)\,[-c^2(t)dt^2+a^2d\vec x^2]\). My starting notes cover standard bigravity with \(c=1\) and constant \(\xi\). The goal is the stress-energy tensor at second order in perturbations.

How the work was done

My notesstandard theory + Next steps

Claudewrites Secs. 7–11 of the notes and 55 sympy checks

Codexaudits the notes 4 times (11 → 16 → 20 → 17 issues)

My model codexPand, \(c=1\), constant \(\xi\)

A/Bsame prompt to both agents: rules for \(R_{\mu\nu}[f]\)

Codexre-derives 1711.04655 and audits my notebook

My general codecodigo/Perturbaciones_Chameleon.nb

Independent checksverificaciones/run_all.sh

Claudesign fix, \(c(t)\), tensors with \(c\neq1\), clean run of my notebook

Timeline of the work by lane: me, Claude, Codex — Timeline, reconstructed from file modification dates.

Results, and how much I trust each one

Notation: \(J=\beta_1+2\beta_2\xi+\beta_3\xi^2\), \(Z=\beta_1+(1+c)\beta_2\xi+c\beta_3\xi^2\), \(\kappa=M_f^2/M_g^2\), \(\Delta=H-\xi H_f\).

Result	Who produced it	Independent check	Status
Backgrounds of \(g\) and \(f\), Klein–Gordon, conservation	me (xPand), with the \(f\) geometry designed with the agents	re-derived from the action in components; with the old sign of \(\dot\rho\) the constraint fails (negative control)	verified
Generalised Bianchi constraint	me · Claude's notes · 1702.04490	the three forms are identical (sympy, 7/7 PASS)	verified
\(c(t)\)	me · Claude's notes	from the action (Mathematica, no xAct) and inside my notebook (residual 0)	verified
Linear tensor sector with \(c\neq1\), \(\xi(t)\), and \(m_{\rm M}^2\)	me (`EqTg`) + component block in my notebook	`derivacion_TT_general.m`: 10 PASS; matches `EqTg` and the notes' mass	verified
Vectors, scalar no-ghost	Claude's notes; Codex re-derived 1711.04655	171 checks by Codex; Claude copied \(\kappa_1,\Sigma\) from the paper	partial
Dark matter from \(\mathcal G\to\mathcal M\) conversion	only Claude's notes	4 audits, 7 issues still open	not verified

Constraint. With \(\beta_n(\phi)\), the branch dichotomy \(J(H-\xi H_f)=0\) of standard bigravity is replaced by an inhomogeneous constraint. The Hubble-rate mismatch is driven by the rolling chameleon, not by matter:

\[ J\,(H-\xi H_f)=-\tfrac13\,\dot\phi\,\big(\beta_1'+3\beta_2'\xi+3\beta_3'\xi^2+\beta_4'\xi^3\big),\qquad c-1=\frac{\mathcal P-\Delta\,\xi H_f}{\widetilde W} \]

Tensor sector for general \(c(t)\), \(\xi(t)\). The \(f\)-graviton propagates on the light cone of \(f\), and the mass terms close on \(h-\gamma\):

\[ \ddot\gamma+\Big(3H+2\tfrac{\dot\xi}{\xi}-\tfrac{\dot c}{c}\Big)\dot\gamma+c^2\tfrac{k^2}{a^2}\gamma+m_\gamma^2(\gamma-h)=0,\qquad m_{\rm M}^2=m_h^2+m_\gamma^2=-\frac{m^2Z\,(c+\kappa\xi^2)}{\kappa\xi} \]

For \(c\neq1\) the gradient matrix \(\mathrm{diag}(1,c^2)\) does not commute with the mass matrix. So no constant basis decouples \(\mathcal G\) and \(\mathcal M\): there is a mixing channel \(\propto k^2(c^2-1)\) on top of the one driven by \(\dot\theta\). This is the starting point of the second-order calculation.

What I learned about the agents

Issues per audit and per topic — Background issues disappear over the audits. The total does not go down: the issues migrate to phenomenology.

Cross-auditing helps but does not converge by itself. Every fix brings new claims.
Copying instead of deriving: one of Claude's notebooks loads formulas from the paper and then "checks" against them.
Numerics where they do not belong: checks on a single numerical curve; a plot of two trivial straight lines.
Invented references in the derivation guide.
Silent failures: a patch by Codex failed its own checks and still wrote the file.
What worked: giving the agents my own code as the model; A/B with the same prompt; exact zeros from a clean kernel plus negative controls; re-deriving the key results along a path that shares no code; running the whole notebook from scratch, which exposed three latent bugs.

Reproduce

git clone <this repo> && cd <repo>/verificaciones
bash run_all.sh      # ~3 min: 7/7 + 6/6 + 21/21 + 34/34 + 17/17 PASS, and the c=1 tensor sector closes

Environment: Python 3.12 (sympy 1.12, numpy, scipy, mpmath); Mathematica 12.1.1 for the two component derivations, with no xAct needed. The notebooks use xAct (xTensor 1.2.0, xPert 1.0.6, xPand 0.4.3).

Next steps

Vector and scalar sectors in my notebook, with the exact \(\delta\sqrt{g^{-1}f}\) (Sylvester equation).
\(T_{\mu\nu}\) at second order on top of the verified tensor equations.
Only then, conversion and relic abundance for a concrete \(\beta_n(\phi)=\bar\beta_n e^{-\lambda\phi}\).