  The vacuum catastrophe is a fundamental puzzle, where the observed scales of
the cosmological constant are many orders of magnitude smaller than the natural
scales expected in the theory. This work proposes a new "bi-world" construction
that may offer an insight into the cosmological constant problem. The model
generally includes a $(3+1)$-dimensional manifold with two different geometries
and matter fields residing on them. In contrast to bimetric or massive theories
of gravity, classically, and in the vacuum limit, the model reduces to two
massless gravity theories, thereby living ghost-free. The diffeomorphism
invariance and causality highly constrain the two metrics to be conformally
related, $\eta_{\mu \nu} = \phi^2 g_{\mu \nu}$. This reduces the theory to a
standard single-world description, but introduces a new inherently geometrical
"moir\'e field," $\phi$. Interestingly, the moir\'e field has the character of
both a dilaton and Higgs field familiar in the conventional theory. Integrating
out the moir\'e field naturally gives rise to the Starobinsky action and
inflationary dynamics. In the framework of the
Friedmann-Lemaitre-Robertson-Walker solution, we reduce an effective action for
the moir\'e field to that of a particle moving in a Mexican hat potential. The
equations of motion are then solved numerically and the moir\'e field is shown
to approach a Mexican-hat minimum in an oscillatory fashion, which is
accompanied by the decay of the Hubble parameter. Under additional reasonable
assumptions, the vacuum energy asymptotically approaches zero in the end of
inflationary evolution. The physics presented here shares similarities with the
moir\'e phenomena in condensed matter and elsewhere, where two similar
structures superimposed upon give rise to a superstructure with low emergent
energy scales compared to the native theories.
