The conjectured QCD critical point, once discovered, will be an important landmark in the QCD phase diagram. There are dedicated experimental programs around the globe aimed at discovering this critical point and its associated first-order curve. The fluctuations of the conserved densities, that describe the quark-gluon plasma (QGP) formed during these collisions exhibit non-monotonic behavior as the fireball passes close to and away from the critical point. The dynamical QGP is, however, affected by critical slowing down, and hence, the fluctuations of the conserved densities lag behind their equilibrium values. The system cools down and eventually produces a multitude of stable hadrons that get detected at the detectors. In this talk, I'll discuss some effective field theoretical tools to map the final state of hadrons to the dynamical QGP that traversed the phase diagram during the collision. In the first part of my talk, I'll discuss the dynamical evolution of the out-of-equilibrium fluctuations of the QGP within the framework of the extended hydrodynamics, referred to as Hydro+. In the latter part, I'll introduce a novel freeze-out procedure connecting the hydrodynamic evolution of QGP to the subsequent description in terms of observable hadrons. This procedure is a generalization of the half-a-century-old Cooper-Frye freeze-out scheme and allows us to calculate the cumulants of particle multiplicities in heavy-ion collisions that pass close to the critical point. The framework that we introduce allows us to study the effects of critical slowing down and baryon number conservation and the consequent deviation of the observable predictions from equilibrium expectations quantitatively.
