General CMB and Primordial Trispectrum Estimation
  We present trispectrum estimation methods which can be applied to general
non-separable primordial and CMB trispectra. We present a general optimal
estimator for the connected part of the trispectrum, for which we derive a
quadratic term to incorporate the effects of inhomogeneous noise and masking.
We describe a general algorithm for creating simulated maps with given
arbitrary (and independent) power spectra, bispectra and trispectra. We propose
a universal definition of the trispectrum parameter $T_{NL}$, so that the
integrated bispectrum on the observational domain can be consistently compared
between theoretical models. We define a shape function for the primordial
trispectrum, together with a shape correlator and a useful parametrisation for
visualizing the trispectrum. We derive separable analytic CMB solutions in the
large-angle limit for constant and local models. We present separable mode
decompositions which can be used to describe any primordial or CMB bispectra on
their respective wavenumber or multipole domains. By extracting coefficients of
these separable basis functions from an observational map, we are able to
present an efficient estimator for any given theoretical model with a
nonseparable trispectrum. The estimator has two manifestations, comparing the
theoretical and observed coefficients at either primordial or late times. These
mode decomposition methods are numerically tractable with order $l^5$
operations for the CMB estimator and approximately order $l^6$ for the general
primordial estimator (reducing to order $l^3$ in both cases for a special class
of models). We also demonstrate how the trispectrum can be reconstructed from
observational maps using these methods.
