  Background: Studies of low-energy fusion of light nuclei are important in
astrophysical modeling, with small variations in reaction rates having a large
impact on nucleosynthesis yields. Due to the lack of experimental data at
astrophysical energies, extrapolation and microscopic methods are needed to
model fusion probabilities.
  Purpose: To investigate deep sub-barrier $^{12}$C+$^{12}$C fusion cross
sections and establish trends for the $S$ factor.
  Method: Microscopic methods based on static Hartree-Fock (HF) and
time-dependent Hartree-Fock (TDHF) mean-field theory are used to obtain
$^{12}$C+$^{12}$C ion-ion fusion potentials. Fusion cross sections and
astrophysical $S$ factors are then calculated using the incoming wave boundary
condition (IWBC) method.
  Results: Both density-constrained frozen Hartree-Fock (DCFHF) and
density-constrained TDHF (DC-TDHF) predict a rising $S$ factor at low energies,
with DC-TDHF predicting a slight damping in the deep sub-barrier region
($\approx1$~MeV). Comparison between DC-TDHF calculations and maximum
experimental cross-sections in the resonance peaks are good. However the
discrepancy in experimental low energy results inhibits interpretation of the
trend.
  Conclusions: Using the fully microscopic DCFHF and DC-TDHF methods, no $S$
factor maximum is observed in the $^{12}$C+$^{12}$C fusion reaction. In
addition, no extreme sub-barrier hindrance is predicted at low energies. The
development of a microscopic theory of fusion including resonance effects, as
well as further experiments at lower energies must be done before the deep
sub-barrier behavior of the reaction can be established.
