Binary tidal evolution as a sculptor of circumbinary planet architectures
Circumbinary planets (CBPs) are expected to form in circumbinary discs and migrate towards the disc inner cavity, often near the binary stability boundary. Yet the observed population is not strictly marginally stable: excluding extreme systems, the typical offset is $Δ\equiv a_{\rm p}/a_{\rm crit}=1.22^{+0.18}_{-0.07}$, suggesting post-formation evolution may shape present-day architectures. We test whether secular evolution of close binaries can contribute to this displacement. Using high-accuracy $N$-body simulations, we follow coplanar CBPs initially placed near $a_{\rm crit}$ while prescribing binary contraction and eccentricity damping along a track that approximately conserves orbital angular momentum. We quantify how the retreat of $a_{\rm crit}(t)$ affects planetary offsets and survival across binary mass ratios. The moving stability boundary filters the near-boundary population, adding a contraction-driven loss component dominated by ejections. Survivors remain quasi-fossilised, with median changes in $a_{\rm p}$ and $e_{\rm p}$ close to zero and semimajor-axis drifts of only a few per cent. For all valid synthetic survivors, the final offset has median $Δ_f=1.86^{+0.18}{-0.26}$. Applying the same damping prescription to the observed sample shifts the median from $Δ{\rm now}=1.22^{+0.18}{-0.07}$ to $Δ{\rm damp}=1.44^{+0.32}_{-0.10}$. Near-equal-mass binaries undergo stronger filtering, with enhanced ejections and collisions. Binary tidal evolution can therefore shape non-marginal circumbinary architectures. By removing planets close to marginal stability while leaving survivors nearly fossilised, the retreat of $a_{\rm crit}$ can widen the inner gap and increase offsets from the present-day stability boundary. This links binary orbital evolution, dynamical filtering, and transit detectability in a multi-channel interpretation of the CBP desert.