  In this paper, two things are done. (i) Using cohomological techniques, we
explore the consistent deformations of linearized conformal gravity in 4
dimensions. We show that the only possibility involving no more than 4
derivatives of the metric (i.e., terms of the form $\partial^4 g_{\mu \nu}$,
$\partial^3 g_{\mu \nu} \partial g_{\alpha \beta}$, $\partial^2 g_{\mu \nu}
\partial^2g_{\alpha \beta}$, $\partial^2 g_{\mu \nu} \partial g_{\alpha \beta}
\partial g_{\rho \sigma}$ or $\partial g_{\mu \nu} \partial g_{\alpha \beta}
\partial g_{\rho \sigma} \partial g_{\gamma \delta}$ with coefficients that
involve undifferentiated metric components - or terms with less derivatives) is
given by the Weyl action $\int d^4x \sqrt{-g} W_{\a\b\g\d} W^{\a\b\g\d}$, in
much the same way as the Einstein-Hilbert action describes the only consistent
manner to make a Pauli-Fierz massless spin-2 field self-interact with no more
than 2 derivatives. No a priori requirement of invariance under diffeomorphisms
is imposed: this follows automatically from consistency. (ii) We then turn to
"multi-Weyl graviton" theories. We show the impossibility to introduce
cross-interactions between the different types of Weyl gravitons if one
requests that the action reduces, in the free limit, to a sum of linearized
Weyl actions. However, if different free limits are authorized, cross-couplings
become possible. An explicit example is given. We discuss also how the results
extend to other spacetime dimensions.
