  Within the next several years pulsar timing arrays (PTAs) are positioned to
detect the stochastic gravitational-wave background (GWB) likely produced by
the collection of inspiralling super-massive black holes binaries, and
potentially constrain some exotic physics. So far most of the pulsar timing
data analysis has focused on the monopole of the GWB, assuming it is perfectly
isotropic. The natural next step is to search for anisotropies in the GWB. In
this paper, we use the recently developed PTA Fisher matrix to gain insights
into optimal search strategies for GWB anisotropies. For concreteness, we apply
our results to EPTA data, using realistic noise characteristics of its pulsars.
We project the detectability of a GWB whose angular dependence is assumed to be
a linear combination of predetermined maps, such as spherical harmonics or
coarse pixels. We find that the GWB monopole is always statistically correlated
with these maps, implying a loss of sensitivity to the monopole when searching
simultaneously for anisotropies. We then derive the angular distributions of
the GWB intensity to which a PTA is most sensitive, and illustrate how one may
use these "principal maps" to approximately reconstruct the angular dependence
of the GWB. Since the principal maps are neither perfectly anisotropic nor
uncorrelated with the monopole, we also develop a frequentist criterion to
specifically search for anisotropies in the GWB without any prior knowledge
about their angular distribution. Lastly, we show how to recover existing EPTA
results with our Fisher formalism, and clarify their meaning. The tools
presented here will be valuable in guiding and optimizing the computationally
demanding analyses of pulsar timing data.
