Gravitational-Wave Sky Mapping with Pulsar Timing Arrays: The Full Earth-Pulsar Response and Fundamental Resolution Limits
Pulsar timing arrays (PTAs) are the only means to observe nanohertz gravitational waves (GWs). While current analyses primarily exploit the Earth term, the full detector response encodes additional directional information in the pulsar terms. We develop a GW sky-mapping framework based on the complete Earth--pulsar response and a tensor spherical harmonic decomposition of the GW field. This yields closed-form response functions for an elementary baseline and casts PTA sky reconstruction as a linear inverse problem. We show that a PTA behaves as a diffraction-limited GW observatory whose angular sensitivity is governed by the dimensionless parameter $ωL$, where $ω$ is the GW angular frequency and $L$ is the pulsar distance. The detector response exhibits four distinct regimes: an Earth-term dominated regime, a transition regime, a pulsar-term-dominated regime, and an exponential sensitivity cutoff at $l_{cut}\simeqωL$. This cutoff defines the fundamental angular resolution limit of PTA sky maps. Using Fisher-information and singular-value analyses, we show that the achievable angular resolution is constrained not only by the intrinsic detector response but also by the finite number of pulsars, their sky distribution, and timing noise. In particular, we find that coherent pulsar-term information can improve full-sky gravitational-wave mapping only for PTAs containing of order $N_{trans}\sim10^{11}$ precisely timed pulsars. This result demonstrates that, although the transition to a pulsar-term-sensitive regime exists mathematically, it is inaccessible for realistic PTAs and therefore provides a quantitative justification for the Earth-term approximation adopted in contemporary observations. Finally, we extend the formalism to stochastic GW backgrounds, establishing a unified mathematical framework for PTA sky mapping and anisotropy studies.