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The centres of ellipticals and bulges are formed dissipationally, via gas inflows over short time‐scales – the ‘starburst’ mode of star formation. Recent work has shown that the surface brightness profiles, kinematics and stellar populations of spheroids can be used to separate the dissipational component from the dissipationless ‘envelope’ made up of stars formed over more extended histories in separate objects, and violently assembled in mergers. Given high‐resolution, detailed observations of these ‘burst relic’ components of ellipticals (specifically their stellar mass surface density profiles), together with the simple assumptions that some form of the Kennicutt–Schmidt law holds and that the burst was indeed a dissipational, gas‐rich event, we show that it is possible to invert the observed profiles and obtain the time‐ and space‐dependent star formation history of each burst. We perform this exercise using a large sample of well‐studied spheroids, which have also been used to calibrate estimates of the ‘burst relic’ populations. We show that the implied bursts scale in magnitude, mass and peak star formation rate (SFR) with galaxy mass in a simple manner, and provide fits for these correlations. The typical burst mass  M  burst  is  ∼ 10 per cent  of the total spheroid mass, the characteristic starburst time‐scale implied is a nearly galaxy‐mass‐independent   t  burst ∼ 10 8  yr , the peak SFR of the burst is  ∼ M  burst / t  burst   and bursts decay subsequently in power‐law fashion as . As a function of time, we obtain the spatial size of the starburst; burst sizes at peak activity scale with burst mass in a manner similar to the observed spheroid size–mass relation, but are smaller than the full galaxy size by a factor of ∼10; the size grows in time as the central, most dense regions are more quickly depleted by star formation as   R  burst ∝ t  0.5  . Combined with observational measurements of the nuclear stellar population ages of these systems – i.e. the distribution of times when these bursts occurred – it is possible to re‐construct the dissipational burst contribution to the distribution of SFRs and infrared (IR) luminosity functions (LFs) and luminosity density of the Universe. We do so and show that these burst LFs agree well with the observed IR LFs at the brightest luminosities, at redshifts   z ∼ 0–2 . At low luminosities, however, bursts are always unimportant; the transition luminosity between these regimes increases with redshift from the ultraluminous infrared galaxy threshold at   z ∼ 0  to hyper‐luminous infrared galaxy thresholds at   z ∼ 2 . At the highest redshifts   z ≳ 2 , we can set strict upper limits on starburst magnitudes, based on the maximum stellar mass remaining at high densities at   z = 0 , and find tension between these and estimated number counts of sub‐millimetre galaxies, implying that some change in bolometric corrections, the number counts themselves or the stellar initial mass function may be necessary. At all redshifts, bursts are a small fraction of the total SFR or luminosity density,  ∼5–10 per cent , in good agreement with estimates of the contribution of merger‐induced star formation.

A new empirical method to infer the starburst history of the Universe from local galaxy properties