This is the 2D counterpart to the 3D pulsar-timing-array simulator, and it builds the gravitational-wave background a genuinely different way. The 3D version samples correlated residuals directly from the theoretical Hellings–Downs covariance matrix via Cholesky factorisation — a mathematical shortcut. This version instead literally superposes many independent monochromatic gravitational plane waves arriving from random isotropic sky directions, each with a random polarisation angle and phase, exactly the physical picture of a background made of many unresolved supermassive black-hole binaries:
z_p(t) = Σ_k A_k [ F⁺(p̂,Ω̂_k) cos(ω_k t+φ_k) + Fˣ(p̂,Ω̂_k) sin(ω_k t+φ_k) ]
F^A(p̂,Ω̂) = ½ · (p̂_i p̂_j e^A_ij(Ω̂)) / (1 + Ω̂·p̂) (quadrupole antenna pattern)
Each pulsar's sky position p̂ is a real 3D unit vector, projected here onto a flat polar sky-chart (zenith at the centre, horizon at the rim) instead of an orbiting 3D globe. Averaged over enough random source directions and polarisations, the pairwise correlation of these Earth-term redshifts converges to the exact same Γ(θ) formula the 3D version samples directly — this is in fact how Hellings & Downs originally derived it, as an ensemble average over an isotropic background.
x = (1 − cos θ) / 2
Γ(θ) = 1/2 + (3/2)·x·ln(x) − x/4 (Hellings & Downs, 1983)
- Discrete GW sources — how many independent point sources make up the background. Too few and the correlation is noisy and lumpy (a real, physically-motivated "shot noise" effect from an unresolved-binary background); many sources smooths it toward the isotropic-limit curve.
- GWB strain amplitude / Pulsar timing noise — same roles as the 3D version: the shared signal strength vs. independent per-pulsar clock/instrument noise.
- Trace panel — live residual time series for one nearby pulsar pair (small angular separation, strongly correlated wiggles) and one wide pair (large separation, weakly/negatively correlated) — a direct visual of what the correlation statistic on the right is measuring.
- Correlation panel — each dot is the online Pearson correlation of one pulsar pair's full residual history vs. their true angular separation; the solid line is the theoretical Γ(θ) curve.
Verified numerically offline (Monte-Carlo, 60 simulated years, 30 pulsars, 400 sources, no timing noise): the measured-vs-theory correlation across all pulsar pairs reaches R ≈ 0.91, with the correct sign structure — high near 0°, a dip around 60–100°, and a partial rise back toward 180° — confirming the discrete-source superposition genuinely reproduces the Hellings–Downs signature rather than just visually resembling it.