Top-heavy stellar mass distribution in galactic nuclei inferred from the
  universally high abundance ratio of [Fe/Mg]
  Recent observations of active galactic nuclei (AGNs) have shown a high
Fe~II/Mg~II line-flux ratio in their broad-line regions, nearly independent of
redshift up to $z \gtrsim 6$. The high flux ratio requires rapid production of
iron in galactic nuclei to reach an abundance ratio of ${\rm [Fe/Mg]} \gtrsim
0.2$ as high as those observed in matured galaxies in the local universe. We
propose a possible explanation of rapid iron enrichment in AGNs by massive star
formation that follows a top-heavy initial mass function (IMF) with a power-law
index of $\Gamma$ larger than the canonical value of $\Gamma=-2.35$ for a
Salpeter IMF. Taking into account metal production channels from different
types of SNe, we find that the high value of ${\rm [Fe/Mg]} \gtrsim 0.2$
requires the IMF to be characterized with $\Gamma \gtrsim -1$ ($\Gamma \gtrsim
0$) and a high-mass cutoff at $M_{\rm max} \simeq 100$--$150~{\rm M_\odot}$
$(M_{\rm max} \gtrsim 250~{\rm M_\odot})$. Given the conditions, core-collapse
SNe with $M_\ast \gtrsim 70~{\rm M_\odot}$ and pair-instability SNe give a
major contribution for iron enrichment. Such top-heavy stellar IMFs would be a
natural consequence from mass growth of stars formed in dense AGN disks under
Bondi-like gas accretion that is regulated by feedback at $M_\ast \gtrsim
10~{\rm M_\odot}$. The massive stellar population formed in AGN disks also
leave stellar-mass black hole remnants, whose mergers associated with
gravitational-wave emission account for at most 10 \% of the merger rate
inferred from LIGO/Virgo observations to simultaneously explain the high ${\rm
[Fe/Mg]}$ ratio with metal ejection.
