Density and Goldstone emission of an RP³ ≃ SO(3) global-string network relative to U(1), and a massive-bath systematic of the emission estimator

We simulate networks of global ℤ₂ strings whose vacuum manifold is RP³ ≃ SO(3), produced by a real symmetric-tensor order parameter, and compare them with U(1) axion-string networks evolved in the same code, with the same Goldstone decay constant and core mass, at 576³ (U(1) control also at 768³, where it passes preregistered scaling gates).

The string-density ratio is Rξ ≃ 1.6 for every protocol tested: sparse and dense starts, with and without draining the initial massive radiation, and networks started late and dense that approach the attractor from above. They give nearly the same ratio, Rξ = 1.57 (from above) and 1.68 (from below), at ℓ = ln(m_r/H) = 5.61.

We identify a systematic of the standard kinetic Goldstone-emission estimator: the conversion into Goldstones of massive radiation left by the initial state is counted as string emission. It is 19 % (RP³) and 28 % (U(1)) of the measured emission for sparse starts, and more than half for dense ones. For U(1), two independent methods agree: a drain of the massive modes, and a direct measurement of the radial-to-Goldstone transfer.

With the bath drained, the emission ratio is RΓ = 1.56 ± 0.09 (stat) ± 0.08 (sampling), with drained sparse and dense starts 7 % apart at 512³ (two seeds each). Networks started late and dense, still transient in our window, give RΓ = 1.30 ± 0.14, but per unit string length the emission is within 13 % of U(1) from both sides of the attractor.

The spectral slope of the RP³ emission at ℓ = 5 is q = 0.87 ± 0.06 with our estimator. An independent re-implementation shows that the convention used to map the physical fitting window onto lattice momenta shifts it by up to 0.04, and a fit weighted by the spectrum gives 0.94 ± 0.04; we therefore quote q as compatible with 1.

All verdict-bearing analyses were preregistered (the snapshot-sampling systematic was adopted after the data) and are reproducible from the released data.

Code and data: https://github.com/indigenica/whalefall-rp3-string-networks. It contains:

• the lattice kernel;
• one JSON summary per network;
• the dated preregistrations and reports (in Spanish);
• the independent audit;
• a script that recomputes every number in the paper from the stored results.