  We generalise Einstein's formulation of the traceless Einstein equations to
$f(R)$ gravity theories. In the case of the vacuum traceless Einstein
equations, we show that a non-constant Weyl tensor leads via a conformal
transformation to a dimensionally homogeneous (`no-scale') theory in the
conformal frame with a scalar field source that has an exponential potential.
We then formulate the traceless version of $f(R)$ gravity, and we find that a
conformal transformation leads to a no-scale theory conformally equivalent to
general relativity and a scalar field $\phi $ with a potential given by the
scale-invariant form: $V(\phi )=\frac{D-2}{4D}Re^{-\phi }$, where $\phi
=[2/(D-2)]\ln f^{\prime }(R)$. In this theory, the cosmological constant is a
mere integration constant, statistically distributed in a multiverse of
independent causal domains, the vacuum energy is another unrelated arbitrary
constant, and the same is true of the height of the inflationary plateau
present in a huge variety of potentials. Unlike in the conformal equivalent of
full general relativity, flat potentials are found to be possible in all
spacetime dimensions for polynomial lagrangians of all orders. Hence, we are
led to a novel interpretation of the cosmological constant vacuum energy
problem and have accelerated inflationary expansion in the very early universe
with a very small cosmological constant at late times for a wide range of
no-scale theories. Fine-tunings required in traceless general relativity or
standard non-traceless $f(R)$ theories of gravity are avoided. We show that the
predictions of the scale-invariant conformal potential are completely
consistent with microwave background observational data concerning the
primordial tilt and the tensor-to-scalar ratio.
