  We argue that the presence of conformal anomalies in gravitational theories
can lead to observable modifications to Einstein's equations via the induced
anomalous effective actions, whose non-localities can overwhelm the smallness
of the Planck scale. The fact that no such effects have been seen in recent
cosmological or gravitational wave observations therefore imposes strong
restrictions on the field content of possible extensions of Einstein's theory:
all viable theories should have vanishing conformal anomalies. We then show
that a complete cancellation of conformal anomalies in D=4 for both the $C^2$
invariant and the Euler (Gauss-Bonnet) invariant can only be achieved for
N-extended supergravities with $N \geq 5$, as well as for M theory compactified
to four dimensions.
