Appendix Q — Toy Model of Galaxy-Scale Lensing in PBG

(axisymmetric analytic benchmark • full FEM under development)

⚠ Concept benchmark only
This note gives a first-order analytic estimate of light bending by a
flattened galaxy in Phase-Biased Geometry.
It keeps only monopole + quadrupole terms.
A full 3-D Helmholtz/FEM calculation is tracked in
notebooks/gal-lens-FEM.ipynb.

1 Axisymmetric mass model

Sersic-like disc + bulge (cylindrical coordinates $R,z$):

2 Monopole + quadrupole kernel

From Foundations §2, expand the Green function to $\ell =2$:

• $k=\sqrt{\beta /\alpha }=2.37\times {10}^{-4}{\text{Mpc}}^{-1}$
• $N=c^{2}M/(4\pi \alpha )$
• $q$ (oblateness) ${\displaystyle q=\frac{M_{zz}-\frac{1}{3}M}{M}\approx 0.1}$ for the disc

3 Index profile and deflection

Lensing index (see Lensing-from-Coherence appendix):

Straight-line eikonal integral → deflection angle at impact $b$:

Fractional anisotropy between disc plane (θ = 90°) and polar axis:

4 Qualitative data check

Current uncertainties are larger than the predicted 1–2 % signal; more
precise weak-shear data could discriminate.

5 Limitations and roadmap

• Only monopole + single quadrupole term — no radial $q(R)$ variation
• Ignores dipole warp and bar components
• Ray-bundle convergence κ and shear γ await full FEM kernel

Next steps

1. Solve the Helmholtz kernel numerically for the exponential disc.
2. Ray-trace ${10}^4$ photons → κ($R,\theta$), γ($R,\theta$).
3. Compare χ² with DES Year-3 shear maps.

Disc anisotropy in PBG naturally yields percent-level differential
lensing absent in a pure monopole ΛCDM model. A full numeric appendix
will replace this toy estimate after FEM validation.

Appendix P - Orbital Motion | [Index](./Appendix Master) | Appendix R - Modal Entropy