(abridged) This study revolves around dmB, a new distance- and extinction-independent measure of the contribution by stellar populations older than 9 Gyr to the mean r-band surface brightness of the bulge component in 135 late-type galaxies (LTGs) from the CALIFA survey, spanning a range of 2.6 dex and 3 dex in total and bulge stellar mass (M*T~10^(8.9-11.5) M_solar and M*B~10^(8.3-11.3) M_solar, respectively). The main insight from this study is that LTG bulges form a continuous sequence of increasing dmB with increasing M*T, M*B, stellar mass surface density S* and mass-weighted age and metallicity: high-dmB bulges are the oldest, densest and most massive ones, and vice versa. Furthermore, we find that the bulge-to-disk age and metallicity contrast, as well as the bulge-to-disk mass ratio increase with M*T, raising from, respectively, ~0 Gyr, 0 dex and 0.25 to ~3 Gyr, ~0.3 dex and 0.67 across the mass range covered by our sample. Whereas gas excitation in lower-mass bulges is invariably dominated by star formation (SF), LINER- and Seyfert-specific emission-line ratios were exclusively documented in high-mass, high-S* bulges. The continuity both in the properties of LTG bulges themselves and in their age and metallicity contrast to their parent disks suggests that these components evolve alongside in a concurrent process that leads to a continuum of physical and evolutionary characteristics. Our results are consistent with a picture where bulge growth in LTGs is driven by a superposition of quick-early and slow-secular processes, the relative importance of which increases with M*T. These processes, which presumably combine in situ SF in the bulge and inward migration of material from the disk, are expected to lead to a non-homologous radial growth of S* and a trend for an increasing Sersic index with increasing galaxy mass.
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