  The bispectrum, the three-point correlation in Fourier space, is a crucial
statistic for studying many effects targeted by the next-generation galaxy
surveys, such as primordial non-Gaussianity (PNG) and general relativistic (GR)
effects on large scales. In this work we develop a formalism for the bispectrum
in the Spherical Fourier-Bessel (SFB) basis - a natural basis for computing
correlation functions on the curved sky, as it diagonalizes the Laplacian
operator in spherical coordinates. Working in the SFB basis allows for
line-of-sight effects such as redshift space distortions (RSD) and GR to be
accounted for exactly, i.e without having to resort to perturbative expansions
to go beyond the plane-parallel approximation. Only analytic results for the
SFB bispectrum exist in the literature given the intensive computations needed.
We numerically calculate the SFB bispectrum for the first time, enabled by a
few techniques: We implement a template decomposition of the redshift-space
kernel $Z_2$ into Legendre polynomials, and separately treat the PNG and
velocity-divergence terms. We derive an identity to integrate a product of
three spherical harmonics connected by a Dirac delta function as a simple sum,
and use it to investigate the limit of a homogeneous and isotropic Universe.
Moreover, we present a formalism for convolving the signal with separable
window functions, and use a toy spherically symmetric window to demonstrate the
computation and give insights into the properties of the observed bispectrum
signal. While our implementation remains computationally challenging, it is a
step toward a feasible full extraction of information on large scales via a SFB
bispectrum analysis.
