  The possibility has been recently demonstrated to manufacture
(nonrelativistic, Hamiltonian) many-body problems which feature an isochronous
time evolution with an arbitrarily assigned period $T$ yet mimic with good
approximation, or even exactly, any given many-body problem (within a large,
physically relevant, class) over times $\tilde{T}$ which may also be
arbitrarily large (but of course such that $\tilde{T}<T$). Purpose and scope of
this paper is to explore the possibility to extend this finding to a general
relativity context. For simplicity we restrict our consideration to the case of
homogeneous and isotropic metrics and show that, via an approach analogous to
that used for the nonrelativistic many-body problem, a class of homogeneous and
isotropic cyclic solutions of Einstein's equations may be obtained. For these
solutions the duration of the cycles does not depend on the initial conditions,
so we call these models isochronous cosmologies. We give a physical
interpretation of such metrics and in particular we show that they may behave
arbitrarily closely, or even identically, to the Friedman-Robertson-Walker
solutions of Einstein's equations for an arbitrarily long time (of course
shorter than their period, which can also be assigned arbitrarily), so that
they may reproduce all the satisfactory phenomenological features of the
standard cosmological $\Lambda $-CDM model in a portion of their cycle; while
these isochronous cosmologies may be geodesically complete and therefore
singularity-free.
