  We formulate new conformal models of inflation and dark energy which
generalise the Higgs-Dilaton scenario. We embed these models in unimodular
gravity whose effect is to break scale invariance in the late time Universe. In
the early Universe, inflation occurs close to a maximum of both the scalar
potential and the scalar coupling to the Ricci scalar in the Jordan frame. At
late times, the dilaton, which decouples from the dynamics during inflation,
receives a potential term from unimodular gravity and leads to the acceleration
of the Universe. We address two central issues in this scenario. First we show
that the Damour-Polyalov mechanism, when non-relativistic matter is present
prior to the start of inflation, sets the initial conditions for inflation at
the maximum of the scalar potential. We then show that conformal invariance
implies that matter particles are not coupled to the dilaton in the late
Universe at the classical level. When fermions acquire masses at low energy,
scale invariance is broken and quantum corrections induce a coupling between
the dilaton and matter which is still small enough to evade the gravitational
constraints in the solar system.
