Mapping gravitational-wave background (GWB) anisotropy with pulsar timing arrays (PTAs) is affected by harmonic space mode suppression and mode coupling arising from an array’s nonuniform sky response. Due to computational limitations, one must truncate spherical harmonic expansions at a finite multipole for recovery, ${\ell }_{\max }^{\mathrm{rec}}$, often chosen to equal ${\ell }_{\max }^{N_{\mathrm{pair}}}\equiv \mathrm{int}[\sqrt{N_{\mathrm{pair}}}-1]$, where $N_{\mathrm{pair}}\equiv N_{\mathrm{psr}}(N_{\mathrm{psr}}-1)/2$ is the number of distinct pulsar pairs in an array comprised of $N_{\mathrm{psr}}$ pulsars. This choice is motivated by the counting argument that cross-correlations provide at most $N_{\mathrm{pair}}$ independent pieces of information for a single frequency bin. Here, we explicitly obtain the multipole value (denoted ${\ell }_{\max }^{\mathrm{res}}$) that approximates the maximum informative angular scale of an array of pulsars. This value is defined by the requirement that a spherical harmonic expansion out to ${\ell }_{\max }^{\mathrm{res}}$ approximately span the space of “observable skies” extracted by the array, which is encoded in the $N_{\mathrm{pair}}$ eigenmaps of the Fisher information matrix. The value of ${\ell }_{\max }^{\mathrm{res}}$ depends on specifics of the PTA configuration. We also explicitly show that GWB power contained in multipoles $\ell \gtrsim {\ell }_{\max }^{\mathrm{res}}$ do not significantly affect analyses that use expansions out to ${\ell }_{\max }^{\mathrm{res}}$, due to the mode suppression (i.e., low-pass filtering) induced by the PTA response to the GWB. However, truncating spherical harmonic expansions at ${\ell }_{\max }^{\mathrm{rec}}<{\ell }_{\max }^{\mathrm{res}}$ leads to leakage of small-scale angular power from multipoles ${\ell }_{\max }^{\mathrm{rec}}<\ell \le {\ell }_{\max }^{\mathrm{res}}$. Nonetheless, even if we use ${\ell }_{\max }^{\mathrm{res}}$ for recovery, the standard frequentist estimator of the angular power spectrum components $C_{\ell }$ is still biased due to modes that are not observable by the array. Although we can partially debias the standard estimator—improving its agreement with an injected angular power spectrum—this reduction in bias comes at the expense of an increase in variance, with especially large variances arising for poorly constrained modes, $\ell \gg {\ell }_{\mathrm{eff}}$. In summary, the results of our analyses strongly advocate for: (i) using ${\ell }_{\max }^{\mathrm{res}}$ for PTA analyses involving spherical harmonic expansions, and (ii) using the debiased standard estimator for $C_{\ell }$ recovery, but only out to multipoles $\ell <{\ell }_{\mathrm{eff}}(\ll {\ell }_{\max }^{\mathrm{res}})$ corresponding to sufficiently constrained modes.

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• Gravitational wave detection
• Gravitational waves
• Large scale structure of the Universe