Spectral Grade Decomposition of Pulsar Timing Residuals: Separating Gravitational Waves from Intrinsic Noise

Pulsar timing arrays (PTAs) detect the stochastic gravitational wave background (GWB) through correlated timing residuals across a network of millisecond pulsars. The key detection statistic — the Hellings-Downs angular correlation — must be separated from intrinsic pulsar noise, spatially correlated red noise, and solar system ephemeris errors. We apply the eigenvalue-conditioning method and grade decomposition from the Latent framework to the inter-pulsar correlation matrix \(\Gamma_{ab}\) and show that gravitational wave contributions occupy a low-grade subspace with a characteristic eigenvalue signature, while noise sources occupy orthogonal higher-grade components.

Specifically, we decompose the \(N_p \times N_p\) correlation matrix (where \(N_p\) is the number of pulsars) into eigenvalue-conditioned components:

\[\Gamma = \sum_{k=1}^{K} \lambda_k \mathbf{v}_k \mathbf{v}_k^T + \Gamma_{\text{residual}}\]

where the \(K\) dominant eigenmodes carry the GWB signal and \(\Gamma_{\text{residual}}\) contains noise. The Latent theorem predicts \(K^ = \Theta(\log(1/\varepsilon)/\log \rho_{\text{GW}})\) where \(\rho_{\text{GW}}\) is determined by the smoothness of the GWB angular power spectrum. For a power-law GWB spectrum (as expected from supermassive black hole binaries), \(\rho_{\text{GW}} \approx 2.5\)–\(4\), giving \(K^ \approx 3\)–\(5\) dominant modes — consistent with the observation that the Hellings-Downs curve is well-described by the first few Legendre multipoles.

The method provides: (1) a principled truncation criterion for PTA data analysis pipelines, (2) optimal noise separation without assuming Gaussian noise statistics, and (3) a direct bridge to portfolio risk decomposition in quantitative finance, where the identical eigenvalue-conditioning method decomposes correlated asset returns.