Deciphering the Milky Way disc formation time encrypted in the bar chrono-kinematics

We present a novel method to constrain the formation time of the Milky Way disc using the chrono-kinematic signatures of the inner Galaxy. We construct an O-rich Mira variable sample from the Gaia Long-period Variable catalogue to study the kinematic behaviour of stars with different ages in the inner Galaxy. From the Auriga suite of cosmological zoom-in simulations, we find that the age of the oldest stellar population with imprints of the bar in density and kinematics matches the disc spin-up epoch. This is because stars born before the spin-up show insufficient rotation and are not kinematically cold enough to be efficiently trapped by the bar. We find that the bar kinematic signature disappears for Mira variables with a period shorter than 190 days. Using the period-age relation of Mira variables, we constrain the spin-up epoch of the Milky Way to be younger than $\sim11-12$~Gyr (redshift $\sim3$). We also discuss and compare our method and result to other evidence of the Milky Way spin-up epoch under the context of a realistic age uncertainty. Age uncertainty leads to an overestimation of the disc formation time when performing backward modelling. Our constrain of the spin-up epoch is independent from previous studies because it relies on the kinematics of the inner Galaxy instead of the solar vicinity.