  Extensions of the Standard Model and general relativity featuring a UV fixed
point can leave observable implications at accessible energies. Although mass
parameters such as the Planck scale can appear through dimensional
transmutation, all fundamental dimension-4 operators can (at least
approximately) respect Weyl invariance at finite energy. An example is the
Weyl-squared term, whose consistency and observational consequences are
studied. This quasi-conformal scenario emerges from the UV complete quadratic
gravity and is a possible framework for inflation. We find two realizations. In
the first one the inflaton is a fundamental scalar with a quasi-conformal
non-minimal coupling to the Ricci scalar. In this case the field excursion must
not exceed the Planck mass by far. An example discussed in detail is hilltop
inflation. In the second realization the inflaton is a pseudo-Goldstone boson
(natural inflation). In this case we show how to obtain an elegant UV
completion within an asymptotically free QCD-like theory, in which the inflaton
is a composite scalar due to new strong dynamics. We also show how efficient
reheating can occur. Unlike the natural inflation based on Einstein gravity,
the tensor-to-scalar ratio is well below the current bound set by Planck. In
both realizations mentioned above, the basic inflationary formulae are computed
analytically and, therefore, these possibilities can be used as simple
benchmark models.
