  Pulsar timing arrays (PTAs) are currently the only experiments directly
sensitive to gravitational waves with decade-long periods. Within the next five
to ten years, PTAs are expected to detect the stochastic gravitational-wave
background (SGWB) collectively sourced by inspiralling supermassive black hole
binaries. It is expected that this background is mostly isotropic, and current
searches focus on the monopole part of the SGWB. Looking ahead, anisotropies in
the SGWB may provide a trove of additional information both on known and
unknown astrophysical and cosmological sources. In this paper, we build a
simple yet realistic Fisher formalism for anisotropic SGWB searches with PTAs.
Our formalism is able to accommodate realistic properties of PTAs, and allows
simple and accurate forecasts. We illustrate our approach with an idealized PTA
consisting of identical, isotropically distributed pulsars. In a companion
paper, we apply our formalism to current PTAs and show that it can be a
powerful tool to guide and optimize real data analysis.
