Testing the Completeness of the SDSS Colour Selection for Ultramassive,
  Slowly Spinning Black Holes
  We investigate the sensitivity of the colour-based quasar selection algorithm
of the Sloan Digital Sky Survey to several key physical parameters of
supermassive black holes (SMBHs), focusing on BH spin ($a_{\star}$) at the high
BH-mass regime ($M_{BH} \geqslant10^9\, M_{\odot}$). We use a large grid of
model spectral energy distribution, assuming geometrically-thin,
optically-thick accretion discs, and spanning a wide range of five physical
parameters: BH mass $M_{BH}$, BH spin $a_{\star}$, Eddington ratio $L /
L_{Edd}$ , redshift $z$, and inclination angle $inc$. Based on the expected
fluxes in the SDSS imaging ugriz bands, we find that $\sim 99.8\%$ of our
models with $M_{BH} \leqslant 10^{9.5}\, M_{\odot}$ are selected as quasar
candidates and thus would have been targeted for spectroscopic follow-up.
However, in the extremely high-mass regime, $\geqslant 10^{10} M_{\odot}$, we
identify a bias against slowly/retrograde spinning SMBHs. The fraction of SEDs
that would have been selected as quasar candidates drops below $\sim50\%$ for
$a_{\star} <0$ across $0.5<z<2$. For particularly massive BHs, with $M_{BH}
\simeq 3\times10^{10}\, M_{\odot}$, this rate drops below $\sim20\%$, and can
be yet lower for specific redshifts. We further find that the chances of
identifying any hypothetical sources with $M_{BH} = 10^{11}\, M_{\odot}$ by
colour selection would be extremely low at the level of $\sim 3\%$. Our
findings, along with several recent theoretical arguments and empirical
findings, demonstrate that the current understanding of the SMBH population at
the high-$M_{BH}$, and particularly the low- or retrograde-spinning regime, is
highly incomplete.
