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We perform a forecast analysis of the capability of the eLISA space-basedinterferometer to constrain models of early and interacting dark energy usinggravitational wave standard sirens. We employ simulated catalogues of standardsirens given by merging massive black hole binaries visible by eLISA, with anelectromagnetic counterpart detectable by future telescopes. We considerthree-arms mission designs with arm length of 1, 2 and 5 million km, 5 years ofmission duration and the best-level low frequency noise as recently tested bythe LISA Pathfinder. Standard sirens with eLISA give access to an intermediaterange of redshift $1\lesssim z \lesssim 8$, and can therefore providecompetitive constraints on models where the onset of the deviation from$\Lambda$CDM (i.e. the epoch when early dark energy starts to benon-negligible, or when the interaction with dark matter begins) occursrelatively late, at $z\lesssim 6$. If instead early or interacting dark energyis relevant already in the pre-recombination era, current cosmological probes(especially the cosmic microwave background) are more efficient than eLISA inconstraining these models, except possibly in the interacting dark energy modelif the energy exchange is proportional to the energy density of dark energy.

Constraining early and interacting dark energy with gravitational wave standard sirens: the potential of the eLISA mission