Bar instability and formation timescale across Toomre's $Q$ parameter and central mass concentration: slow bar formation or true stability

We investigate the bar formation process using $N$-body simulations across the Toomre's parameter $Q_{min}$ and central mass concentration (CMC), focusing principally on the formation timescale. Of importance is that, as suggested by cosmological simulations, disk galaxies have limited time of $\sim 8$ Gyr in the Universe timeline to evolve secularly, starting when they became physically and kinematically steady to prompt the bar instability. By incorporating this time limit, bar-unstable disks are further sub-divided into those that establish a bar before and after that time, namely the normal and the slowly bar-forming disks. Simulations demonstrate that evolutions of bar strengths and configurations of the slowly bar-forming and the bar-stable cases are nearly indistinguishable prior to $8$ Gyr, albeit dynamically distinct, while differences can be noticed afterwards. Differentiating them before $8$ Gyr is possible by identifying the proto-bar, a signature of bar development visible in kinematical maps such as the Fourier spectrogram and the angular velocity field, which emerges in the former group $1-2$ Gyr before the fully developed bar, whereas it is absent in the latter group until $8$ Gyr and such bar-stable disk remains unbarred until at least $10$ Gyr. In addition, we find complicated interplays between $Q_{min}$ and CMC in regulating the bar formation. Firstly, disk stabilization requires both high $Q_{min}$ and CMC. Either high $Q_{min}$ or high CMC only results in slow bar formation. Secondly, some hot disks can form a bar more rapidly than the colder ones in a specific range of $Q_{min}$ and CMC.