  Spectroscopic phase curves provide unique access to the three-dimensional
properties of transiting exoplanet atmospheres. However, a modeling framework
must be developed to deliver accurate inferences of atmospheric properties for
these complex data sets. Here, we develop an approach to retrieve temperature
structures and molecular abundances from phase curve spectra at any orbital
phase. In the context of a representative hot Jupiter with a large day-night
temperature contrast, we examine the biases in typical one-dimensional (1D)
retrievals as a function of orbital phase/geometry, compared to two-dimensional
(2D) models that appropriately capture the disk-integrated phase geometry. We
guide our intuition by applying our new framework on a simulated HST+Spitzer
phase curve data set in which the "truth" is known, followed by an application
to the spectroscopic phase curve of the canonical hot Jupiter, WASP-43b. We
also demonstrate the retrieval framework on simulated JWST phase curve
observations. We apply our new geometric framework to a joint-fit of all
spectroscopic phases, assuming longitudinal molecular abundance homogeneity,
resulting in an a factor of 2 improvement in abundances precision when compared
to individual phase constraints. With a 1D retrieval model on simulated
HST+Spitzer data, we find strongly biased molecular abundances for CH$_4$ and
CO$_2$ at most orbital phases. With 2D, the day and night profiles retrieved
from WASP-43b remain consistent throughout the orbit. JWST retrievals show that
a 2D model is strongly favored at all orbital phases. Based on our new 2D
retrieval implementation, we provide recommendations on when 1D models are
appropriate and when more complex phase geometries involving multiple TP
profiles are required to obtain an unbiased view of tidally locked planetary
atmospheres.
