(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 (MT~10^(8.9-11.5) M_solar and MB~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 MT, MB, 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 MT, 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 MT. 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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