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Abstract

In this thesis we investigate the locally scale-invariant theories of conformal and
Weyl quadratic gravity, and the ensuing relationship between them. Using a
Yang-Mills inspired gauge theory perspective, we derive the actions describing
these theories and discuss their phenomenologies. Particular focus is put on the
physically propagating degrees of freedom in each theory, as well as the cosmological
implications of conformal gravity and the dimensional transmutation that
occurs naturally in Weyl quadratic gravity. The issue of Ostrogradsky instabilities
that arise from the fourth-order derivatives present in both theories is also
discussed. Finally, using a process known as Ricci gauging, we find that conformal
gravity can be viewed as gauged-fixed Weyl quadratic gravity, and we speculate
on the existence of an alternative gauge fixing procedure that leads to a new
theory which is dual to conformal gravity.