  Using the harmonic superspace approach, we construct, at the linearized
level, $\mathcal{N}=2$ supersymmetric curvatures generalizing scalar curvature,
Ricci curvature and Weyl tensor. These supercurvatures are the building blocks
of various linearized $4D, \, \mathcal{N}=2$ Einstein supergravity invariants.
The supercurvatures involving the scalar and Ricci curvatures are analytic
harmonic ${\cal N}=2$ superfields, while the Weyl supertensor is a chiral
$\mathcal{N}=2$ superfield. As the basic distinguished feature of our
construction, all these objects are expressed through the fundamental analytic
gauge prepotentials $h^{++M}, M= (\alpha\dot\alpha, +\alpha, +\dot\alpha, 5)$.
The related characteristic features are the heavy use of harmonic derivatives
and harmonic zero-curvature equations. On a number of instructive examples, we
describe the component reduction of the superfield invariants constructed.
