Looking for non-Gaussianity in all the right places: A new basis for
  non-separable bispectra
  Non-Gaussianity in the distribution of inflationary perturbations, measurable
in statistics of the cosmic microwave background (CMB) and large scale
structure fluctuations, can be used to probe non-trivial initial quantum states
for these perturbations. The bispectrum shapes predicted for generic
non-Bunch-Davies initial states are non-factorizable ("non-separable") and are
highly oscillatory functions of the three constituent wavenumbers. This can
make the computation of CMB bispectra, in particular, computationally
intractable. To efficiently compare with CMB data one needs to construct a
separable template that has a significant similarity with the actual shape in
momentum space. In this paper we consider a variety of inflationary scenarios,
with different non-standard initial conditions, and how best to construct
viable template matches. In addition to implementing commonly used separable
polynomial and Fourier bases, we introduce a basis of localized piecewise
spline functions. The spline basis is naturally nearly orthogonal, making it
easy to implement and to extend to many modes. We show that, in comparison to
existing techniques, the spline basis can provide better fits to the true
bispectrum, as measured by the cosine between shapes, for sectors of the theory
space of general initial states. As such, it offers a useful approach to
investigate non-trivial features generated by fundamental properties of the
inflationary Universe.
