A theory under finite model theory is not necessarily a first-order theory. Finite model theory is concerned with studying logical theories and their properties in the context of finite models. While first-order logic is one of the most commonly studied logics in finite model theory, there are other logics, such as monadic second-order logic and fixed-point logics, that are also studied in this framework. Whether a theory under finite model theory is a first-order theory or not depends on the specific logic being used to express the theory.