Stellar Velocity Dispersion versus Age: Consistency across Observations and Simulations, with the Milky Way as an Outlier

Within disk galaxies, the velocity dispersion, $\sigma$, of stars increases with age, $\tau$, as measured in the Milky Way (MW) and nearby galaxies. This relation provides a key window into galactic formation history, tracing both the kinematics of stars at birth and the dynamical heating of stars after birth. We compile and compare observational measurements of the MW, M31, M33, and 16 galaxies from the PHANGS survey. The MW exhibits significantly colder stellar kinematics, with 2-3 times lower $\sigma(\tau)/v_{\phi,0}$ at a given age, than all but one other observed galaxy. Therefore, the MW is a kinematic outlier. To assess how measurement effects influence $\sigma(\tau)$, we analyze the FIRE-2 cosmological simulations, quantifying the impact of uncertainties in stellar age, aperture size, galactocentric radius, and galaxy inclination. Aperture size and galactocentric radius affect $\sigma(\tau)$ by up to a factor of $\approx2$ for stars younger than 100 Myr, with milder effects on older stars. Age uncertainties up to 40\% change the \textit{value} of $\sigma(\tau)$ at a given age by $\lesssim20\%$ but can reshape the relation with age and erase merger signatures. We compare $\sigma(\tau)/v_{\phi,0}$ in FIRE-2 simulations with observations. FIRE-2 agrees well with M31 and M33 at all measured ages, and with PHANGS for stars older than $\approx500$ Myr. The average $\sigma(\tau)/v_{\phi,0}$ in FIRE-2 is about two times higher than the MW at most ages, but the youngest stars show better agreement. The velocity ratios ($\sigma_{\phi}/\sigma_{R}$, $\sigma_{Z}/\sigma_{\phi}$, $\sigma_{Z}/\sigma_{R}$) in FIRE-2 broadly agree with the MW. We conclude that $\sigma(\tau)$ in FIRE-2, and most cosmological zoom-in simulations, reasonably matches observed nearby galaxies, but matching the MW is rare, because it is a kinematic outlier.