  We present an analysis of the B-band and V-band rise-time distributions of
nearby Type Ia supernovae (SNe Ia). We use a two-stretch template-fitting
method to measure the rise and decline of BV light curves. Our analysis of 61
SNe with high-quality light curves indicates that the longer the time between
explosion and maximum light (i.e., the rise time), the slower the decline of
the light curve after maximum. However, SNe with slower post-maximum decline
rates have a faster rise than would be expected from a single-parameter family
of light curves, indicating that SN Ia light curves are not a single-parameter
family of varying widths. Comparison of the B-band rise-time distribution for
spectroscopically normal SNe Ia to those exhibiting high-velocity spectral
features indicates that high-velocity (HV) SNe Ia have shorter B-band rise
times compared to their spectroscopically normal counterparts. After
normalising the B-band light curves to Dm15(B)= 1.1 mag (i.e., correcting the
post-maximum decline to have the same shape as our template), we find that
spectroscopically normal SNe Ia have a rise time of 18.03 +/- 0.24 d, while HV
SNe have a faster B-band rise time of 16.63 +/- 0.29 d. Despite differences in
the B band, we find that HV and normal SNe Ia have similar rise times in the V
band. The initial rise of a SN Ia B-band light curve follows a power law with
index 2.20 +0.27 -0.19, consistent with a parabolic rise in flux predicted by
an expanding fireball toy model. We compare our early-time B-band data to
models for the predicted signature of companion interaction arising from the
single-degenerate progenitor scenario. There is a substantial degree of
degeneracy between the adopted power-law index of the SN light-curve template,
the rise time, and the amount of shock emission required to match the data.
