A population synthesis fitting of the $Gaia$ resolved white dwarf binary
  population within 100 pc
  The $Gaia$ mission has provided an unprecedented wealth of information about
the white dwarf population of our Galaxy. In particular, our studies show that
the sample up to 100\,pc from the Sun can be considered as practically
complete. This fact allows us to estimate a precise fraction of
double-degenerate ($1.18\pm 0.10$\%) and white dwarf plus main-sequence stars
($6.31\pm0.23$\%) among all white dwarfs through comoving pairs identification.
With the aid of a detailed population synthesis code we are able to reproduce
synthetic white dwarf populations with nearly identical fractions as the
observed ones, thus obtaining valuable information about the binary fraction,
$f_{\rm b}$, initial mass ratio distribution, $n(q)$, and initial separation
distribution, $f(a)$, among other parameters. Our best-fit model is achieved
within a $1\sigma$ confidence level for $f(a)\propto a^{-1}$, $n(q)\propto
q^{n_q}$, with $n_q=-1.13^{+0.12}_{-0.10}$ and $f_{\rm b}=0.32\pm 0.02$. The
fraction of white dwarf mergers generated by this model is $9\sim16\%$,
depending on the common-envelope treatment. As sub-products of our modelling we
find that around $1\sim3\%$ of the white dwarf population are unresolved
double-degenerates and that only $\sim1\%$ of all white dwarfs contain a
He-core. Finally, only a mild kick during white dwarf formation seems to be
necessary for fitting the observed sky separation of double-degenerate systems.
