  Cosmological analyses with Type Ia Supernovae (SNe Ia) have traditionally
been reliant on spectroscopy for both classifying the type of supernova and
obtaining reliable redshifts to measure the distance-redshift relation. While
obtaining a host-galaxy spectroscopic redshift for most SNe is feasible for
small-area transient surveys, it will be too resource intensive for upcoming
large-area surveys such as the Vera Rubin Observatory Legacy Survey of Space
and Time, which will observe on the order of millions of SNe. Here we use data
from the Dark Energy Survey (DES) to address this problem with photometric
redshifts (photo-z) inferred directly from the SN light-curve in combination
with Gaussian and full p(z) priors from host-galaxy photo-z estimates. Using
the DES 5-year photometrically-classified SN sample, we consider several
photo-z algorithms as host-galaxy photo-z priors, including the Self-Organizing
Map redshifts (SOMPZ), Bayesian Photometric Redshifts (BPZ), and
Directional-Neighbourhood Fitting (DNF) redshift estimates employed in the DES
3x2 point analyses. With detailed catalog-level simulations of the DES 5-year
sample, we find that the simulated w can be recovered within $\pm$0.02 when
using SN+SOMPZ or DNF prior photo-z, smaller than the average statistical
uncertainty for these samples of 0.03. With data, we obtain biases in w
consistent with simulations within ~1$\sigma$ for three of the five photo-z
variants. We further evaluate how photo-z systematics interplay with
photometric classification and find classification introduces a subdominant
systematic component. This work lays the foundation for next-generation fully
photometric SNe Ia cosmological analyses.
