  The centers of bulges are formed dissipationally via gas inflows over short
timescales: the 'starburst' mode of star formation (SF). Recent work has shown
that detailed observations can be used to separate the stellar mass profile of
these 'burst relic' components in local systems. Together with the assumption
that some Kennicutt-Schmidt law holds, and that the burst was indeed a
dissipational gas-rich event, we show that the observed profiles can be
inverted to obtain the time and space-dependent SF history of each burst.
Performing this with a large sample of well-studied spheroids, we show that the
implied bursts scale in magnitude, mass, peak SFR, and spatial extent with
galaxy mass in simple manner, and provide fits to these correlations. Burst
masses are ~10% the total spheroid mass; timescales a mass-independent ~10^8
yr; peak SFR ~M_burst/t_burst; and they decay in power-law fashion ~t^-2.4.
Sizes are ~0.1 R_e(spheroid), but grow with time. Combined with measurements of
the nuclear stellar population ages of these systems (i.e. burst times), it is
possible to re-construct the burst contribution to the distribution of SFRs and
IR luminosity functions at all redshifts. The burst LF agrees well with
observed IR LFs at the brightest luminosities, at z=0-2. At low-L, bursts are
always unimportant; the transition to their importance increases from ULIRG
luminosities at z~0 to HyLIRG luminosities at z~2. At all redshifts, bursts are
a small fraction (~5-10%) of the total SFR density. We discuss possible
implications of tension between maximum relic stellar mass densities in massive
systems, and estimated number counts of brightest sub-millimeter galaxies.
