Super-Eddington accreting massive black holes explore high-$z$
  cosmology: Monte-Carlo simulations
  In this paper, we simulate Super-Eddington accreting massive black holes
(SEAMBHs) as the candles to probe cosmology for the first time. SEAMBHs have
been demonstrated to be able to provide a new tool for estimating cosmological
distance. Thus, we create a series of mock data sets of SEAMBHs, especially in
the high redshift region, to check their abilities to probe the cosmology. To
fulfill the potential of the SEAMBHs on the cosmology, we apply the simulated
data to three projects. The first is the exploration of their abilities to
constrain the cosmological parameters, in which we combine different data sets
of current observations such as the cosmic microwave background from {\it
Planck} and type Ia supernovae from Joint Light-curve Analysis (JLA). We find
that the high redshift SEAMBHs can help to break the degeneracies of the
background cosmological parameters constrained by {\it Planck} and JLA, thus
giving much tighter constraints of the cosmological parameters. The second uses
the high redshift SEAMBHs as the complements of the low redshift JLA to
constrain the early expansion rate and the dark energy density evolution in the
cold dark matter frame. Our results show that these high redshift SEAMBHs are
very powerful on constraining the early Hubble rate and the evolution of the
dark energy density; thus they can give us more information about the expansion
history of our Universe, which is also crucial for testing the $\Lambda$CDM
model in the high redshift region. Finally, we check the SEAMBH candles'
abilities to reconstruct the equation of state of dark energy at high redshift.
In summary, our results show that the SEAMBHs, as the rare candles in the high
redshift region, can provide us a new and independent observation to probe
cosmology in the future.
