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Two-pion interferometry measurements are used to extract the Gaussian radii$R_{{\rm out}}$, $R_{{\rm side}}$, and $R_{{\rm long}}$, of the pion emissionsources produced in Cu$+$Cu and Au$+$Au collisions at several beam collisionenergies $\sqrt{s_{_{NN}}}$ at PHENIX. The extracted radii, which are comparedto recent STAR and ALICE data, show characteristic scaling patterns as afunction of the initial transverse size $\bar{R}$ of the collision systems andthe transverse mass $m_T$ of the emitted pion pairs, consistent withhydrodynamiclike expansion. Specific combinations of the three-dimensionalradii that are sensitive to the medium expansion velocity and lifetime, and thepion emission time duration show nonmonotonic $\sqrt{s_{_{NN}}}$ dependencies.The nonmonotonic behaviors exhibited by these quantities point to a softeningof the equation of state that may coincide with the critical end point in thephase diagram for nuclear matter.

Beam-energy and system-size dependence of the space-time extent of the pion emission source produced in heavy ion collisions