ZTF SN Ia DR2: Improved SN Ia colors through expanded dimensionality with SALT3+

Type Ia supernovae (SNe Ia) are a key probe in modern cosmology, as they can be used to measure luminosity distances at gigaparsec scales. Models of their light-curves are used to project heterogeneous observed data onto a common basis for analysis. The SALT model currently used for SN Ia cosmology describes SNe as having two sources of variability, accounted for by a color parameter c, and a "stretch parameter" x1. We extend the model to include an additional parameter we label x2, to investigate the cosmological impact of currently unaddressed light-curve variability. We construct a new SALT model, which we dub "SALT3+". This model was trained by an improved version of the SALTshaker code, using training data combining a selection of the second data release of cosmological SNe Ia from the Zwicky Transient Facility and the existing SALT3 training compilation. We find additional, coherent variability in supernova light-curves beyond SALT3. Most of this variation can be described as phase-dependent variation in g-r and r-i color curves, correlated with a boost in the height of the secondary maximum in i-band. These behaviors correlate with spectral differences, particularly in line velocity. We find that fits with the existing SALT3 model tend to address this excess variation with the color parameter, leading to less informative measurements of supernova color. We find that neglecting the new parameter in light-curve fits leads to a trend in Hubble residuals with x2 of 0.039 +/- 0.005 mag, representing a potential systematic uncertainty. However, we find no evidence of a bias in current cosmological measurements. We conclude that extended SN Ia light-curve models promise mild improvement in the accuracy of color measurements, and corresponding cosmological precision. However, models with more parameters are unlikely to substantially affect current cosmological results.