  The chemical fingerprints of the first stars are retained within the
photospheres of ancient unevolved metal-poor stars. A significant fraction of
these stellar fossils is represented by stars known as Carbon-Enhanced
Metal-Poor (CEMP), $\rm [C/Fe]>+0.7$ and $\rm [Fe/H]<-2$, which are likely
imprinted by low-energy primordial supernovae. These CEMP stars are largely
observed in the Galactic halo and ultra-faint dwarf galaxies, with values
reaching $\rm [C/Fe]=+4.5$. The Galactic bulge is predicted to host the oldest
stars, but it shows a striking dearth of CEMP stars with $\rm [C/Fe]\gtrsim
+2.0$. Here we explore the possible reasons for this anomaly by performing a
statistical analysis of the observations of metal-poor stars in combination
with the predictions of $\Lambda$CDM models. We suggest that the dearth of CEMP
stars with high $\rm [C/Fe]$ is not due to the low statistics of observed
metal-poor stars but is the result of the different formation process of the
bulge. $N$-body simulations show that the first star-forming halos which end up
in the bulge are characterized by the highest star-formation rates. These rates
enable the formation of rare massive first stars exploding as pair-instability
supernovae (PISNe), which wash out the signature of primordial faint
supernovae. We demonstrate that the mean $\rm [C/Fe]$ of first stars polluted
environments decreases with the increasing contribution of PISNe. We conclude
that the dearth of CEMP stars in the Galactic bulge indirectly probes the
existence of elusive PISNe, and propose a novel method which exploits this lack
to constrain the mass distribution of the first stars.
