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The $Gaia$ mission has provided an unprecedented wealth of information aboutthe white dwarf population of our Galaxy. In particular, our studies show thatthe sample up to 100\,pc from the Sun can be considered as practicallycomplete. This fact allows us to estimate a precise fraction ofdouble-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 reproducesynthetic white dwarf populations with nearly identical fractions as theobserved ones, thus obtaining valuable information about the binary fraction,$f_{\rm b}$, initial mass ratio distribution, $n(q)$, and initial separationdistribution, $f(a)$, among other parameters. Our best-fit model is achievedwithin a $1\sigma$ confidence level for $f(a)\propto a^{-1}$, $n(q)\proptoq^{n_q}$, with $n_q=-1.13^{+0.12}_{-0.10}$ and $f_{\rm b}=0.32\pm 0.02$. Thefraction of white dwarf mergers generated by this model is $9\sim16\%$,depending on the common-envelope treatment. As sub-products of our modelling wefind that around $1\sim3\%$ of the white dwarf population are unresolveddouble-degenerates and that only $\sim1\%$ of all white dwarfs contain aHe-core. Finally, only a mild kick during white dwarf formation seems to benecessary for fitting the observed sky separation of double-degenerate systems.

A population synthesis fitting of the $Gaia$ resolved white dwarf binary population within 100 pc