Context. A tantalizing enigma in extragalactic astronomy concerns the chronology and driving mechanisms of the build-up of late-type galaxies (LTGs). The standard scenario envisages two formation routes, with classical bulges (CBs) assembling first in a quick and violent quasi-monolithic episode followed by gradual disk assembly, and pseudo-bulges (PBs) progressively forming over gigayear-long timescales through gentle gas inflow from the disk and in situ star formation. The expectation from this antagonistic rationale is the segregation of present-day LTG bulges into two evolutionary distinct groups, which is in sharp contrast with recent observations.

Aims. The present study aims for a thorough investigation of the star formation history (SFH) of LTGs with its ultimate goal being to outline a coherent framework for the formation and evolution of spiral galaxies and their main stellar components.

Methods. Using population spectral synthesis models, we analyse the spatially resolved SFH of bulges and disks of 135 LTGs from the CALIFA survey, covering the relevant range in LTG mass. Complementarily, characteristic physical properties of bulges and disks, such as mean colours, mass- and light-weighted stellar age and metallicity, and EW(Hα), were contrasted with predictions from evolutionary synthesis models, by adopting exponentially declining SFHs with e-folding times τ between 0.1 and 20 Gyr.

Results. Analysis of the SFH of roughly half a million spaxels consistently reveals that the main physical and evolutionary properties of both bulges and disks are continuously distributed across present-day total stellar mass ℳ_⋆, T. The τ in spiral galaxies with log(ℳ_⋆, T) > 10 increases from the centre to the periphery, suggesting that these systems grow in an inside-out fashion. Quite importantly, the radial gradient of τ in an individual galaxy increases with increasing ℳ_⋆, T, which is consistent with a high bulge-to-disk age contrast in high-mass spirals, while lower-mass LTGs display roughly the same τ throughout their entire radial extent, with intermediate mass galaxies in between. Predictions obtained through evolutionary synthesis are overall consistent with observed properties. Finally, bulges and disks of higher mass galaxies exhibit shorter formation timescales as compared to their lower mass counterparts.

Conclusions. Collectively, the obtained results evince a coherent and unified picture for the formation and evolution of LTGs, in which PBs and CBs denote extremities of a continuous mass sequence. Our findings are consistent with the framework where bulges are assembled jointly with their parent disks by gradual inside-out growth, at a pace that is regulated by the depth of the galactic potential. This postulate is further supported by the fact that the revealed correlations are entirely devoid of a bimodality, as it would be expected if CBs and PBs were to emerge from two distinct formation routes.