  In this paper I discuss how to consistently incorporate higher-order
corrections to the bubble-nucleation rate at finite temperature. Doing so I
examine the merits of different approaches, with the goal of reducing
uncertainties for gravitational-wave calculations. To be specific, the region
of applicability and accuracy of the derivative expansion is discussed. The
derivative expansion is then compared to a numerical implementation of the
Gelfand-Yaglom theorem. Both methods are applied to popular first-order phase
transition models, like a loop-induced barrier and a SM-EFT tree-level barrier.
The results of these calculations are presented in easy-to-use parametrizations
that can directly be used in gravitational-wave calculations. In addition,
higher-order corrections for models with multiple scalar fields, such as
singlet/triplet extensions, are studied. Lastly, the convergence and
uncertainty of all calculations are discussed in detail. And I argue that
leading-order results for the Standard Model with a tree-level barrier might be
inaccurate.
