  Some barred galaxies, including the Milky Way, host a boxy/peanut/X-shaped
bulge (BPX-shaped bulge). Previous studies suggested that the BPX-shaped bulge
can either be developed by bar buckling or by vertical inner Lindblad resonance
(vILR) heating without buckling. In this paper, we study the observable
consequence of an BPX-shaped bulge built up quickly after bar formation via
vILR heating without buckling, using an N-body/hydrodynamics simulation of an
isolated Milky Way-like galaxy. We found that the BPX-shaped bulge is dominated
by stars born prior to bar formation. This is because the bar suppresses star
formation, except for the nuclear stellar disc (NSD) region and its tips. The
stars formed near the bar ends have higher Jacobi energy, and when these stars
lose their angular momentum, their non-circular energy increases to conserve
Jacobi energy. This prevents them from reaching the vILR to be heated to the
BPX region. By contrast, the NSD forms after the bar formation. From this
simulation and general considerations, we expect that the age distributions of
the NSD and BPX-shaped bulge formed without bar buckling do not overlap each
other. Then, the transition age between these components betrays the formation
time of the bar, and is testable in future observations of the Milky Way and
extra-galactic barred galaxies.
