Theories of massive gravity with one or two dynamical metrics genericallylack stable and observationally-viable cosmological solutions that aredistinguishable from $\Lambda$CDM. We consider an extension to trimetricgravity, with three interacting spin-2 fields which are not plagued by theBoulware-Deser ghost. We systematically explore every combination with two freeparameters in search of background cosmologies that are competitive with$\Lambda$CDM. For each case we determine whether the expansion historysatisfies viability criteria, and whether or not it containsbeyond-$\Lambda$CDM phenomenology. Among the many models we consider, there areonly three cases that seem to be both viable and distinguishable from standardcosmology. One of the models has only one free parameter and displays acrossing from above to below the phantom divide. The other two provide scalingbehavior, although they contain future singularities that need to be studied inmore detail. These models possess interesting features that make themcompelling targets for a full comparison to observations of both cosmologicalexpansion history and structure formation.
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