This thesis discusses results in theoretical cosmology related to the primordial evolution of cosmological perturbations with the help of numerical analysis. Primordial fluctuations are thought to be the seeds of large-scale structure, therefore modelling their initial conditions and evolution is key to understanding how structure forms. Recent tensions in the inferred values of cosmological parameters have raised interest in models that deviate from the currently accepted standard model of cosmology, the Λ cold dark matter model, but these alternative models can form the computational bottleneck of cosmological inference. I first present a numerical method (and associated open-source software) for solving a class of highly oscillatory ordinary differential equations efficiently, which speeds up the forward-modelling step of cosmological inference significantly by enabling fast numerical evolution of primordial fluctuations. I discuss other uses of the numerical routine in the physical sciences and report on its latest application to more accurately constrain the universe's spatial curvature.

The evolution of primordial perturbations cannot be fully determined without initial conditions. I therefore inspect popular methods for setting initial conditions from the perspective of their behaviour under canonical transformations, and find only one set of initial conditions invariant under such transformations. I demonstrate the possible observational consequences of canonical non-invariance of the initial conditions and argue that an invariant set should be used in models that retain memory of the initial conditions. I discuss preliminary investigations into generalising the canonically invariant initial conditions to universes with non-zero spatial curvature before concluding with a summary of future research avenues I view as worthy of exploration.

Chapters 3 and 6 of this thesis are based on the publications titled Efficient method for solving highly oscillatory ordinary differential equations with applications to physical systems, in Physical Review Research [1], and (py)oscode: fast solutions of oscillatory ODEs, published in The Journal of Open Source Software [2]. Chapter 4 is based on work published as a pre-print under the title Dense output for highly oscillatory numerical solutions on the arXiv [3]. Finally, Chapter 7 is based on the manuscript Quantum initial conditions for inflation and canonical invariance published in Physical Review D [4]. I am the sole or leading author and contributor to all of the above publications.