Feasibility of measuring the speed of sound of the quark-gluon plasma from the multiplicity and mean $p_T$ of ultracentral heavy-ion collisions

Abstract

The mean transverse momentum $⟨p_T⟩$ of hadrons has been observed experimentally and in numerical simulations to have a power-law dependence on the hadronic multiplicity $N$ in ultracentral relativistic heavy-ion collisions: $⟨p_T⟩\propto N^{b_{\mathrm{UC}}}$. It has been put forward that this exponent $b_{\mathrm{UC}}$ is the speed of sound of quark-gluon plasma measured at a temperature determined from $⟨p_T⟩$. We study step by step the connection between (i) the energy and entropy of hydrodynamic simulations and (ii) experimentally measurable observables. We show that an argument based on energy and entropy should yield an exponent equal to the pressure over energy density $P/\epsilon$, rather than the speed of sound $c_s^2$; however, we also observe that $⟨p_T⟩$ and $N$ are not sufficiently accurate proxies for the energy and entropy to make this possible in practice. From simulations, we find that the exponent $b_{\mathrm{UC}}$ is significantly different whether the ''effective volume'' is strictly constant or not, a condition that cannot be enforced experimentally. Additional tests using a modified equation of state find that the exponent $b_{\mathrm{UC}}$ exhibits a variable degree of correlations with the speed of sound and with $P/\epsilon$, but is not an accurate measurement of either quantity in general.

Comments: 8 pages, 4 figures -- v4: version accepted for publication