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

The mean transverse momentum <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mo>〈</a:mo> <a:msub> <a:mi>p</a:mi> <a:mi>T</a:mi> </a:msub> <a:mo>〉</a:mo> </a:mrow> </a:math> of hadrons has been observed experimentally and in numerical simulations to have a power-law dependence on the hadronic multiplicity <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>N</b:mi> </b:math> in ultracentral relativistic heavy-ion collisions: <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mrow> <c:mo>〈</c:mo> <c:msub> <c:mi>p</c:mi> <c:mi>T</c:mi> </c:msub> <c:mo>〉</c:mo> </c:mrow> <c:mo>∝</c:mo> <c:msup> <c:mi>N</c:mi> <c:msub> <c:mi>b</c:mi> <c:mi>UC</c:mi> </c:msub> </c:msup> </c:mrow> </c:math> . It has been put forward that this exponent <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:msub> <d:mi>b</d:mi> <d:mi>UC</d:mi> </d:msub> </d:math> is the speed of sound of quark-gluon plasma measured at a temperature determined from <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mo>〈</e:mo> <e:msub> <e:mi>p</e:mi> <e:mi>T</e:mi> </e:msub> <e:mo>〉</e:mo> </e:mrow> </e:math> . 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 <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>P</f:mi> <f:mo>/</f:mo> <f:mi>ɛ</f:mi> </f:mrow> </f:math> , rather than the speed of sound <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:msubsup> <g:mi>c</g:mi> <g:mi>s</g:mi> <g:mn>2</g:mn> </g:msubsup> </g:math> ; however, we also observe that <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mo>〈</h:mo> <h:msub> <h:mi>p</h:mi> <h:mi>T</h:mi> </h:msub> <h:mo>〉</h:mo> </h:mrow> </h:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi>N</i:mi> </i:math> are not sufficiently accurate proxies for the energy and entropy to make this possible in practice. From simulations, we find that the exponent <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:msub> <j:mi>b</j:mi> <j:mi>UC</j:mi> </j:msub> </j:math> 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 <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:msub> <k:mi>b</k:mi> <k:mi>UC</k:mi> </k:msub> </k:math> exhibits a variable degree of correlations with the speed of sound and with <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:mi>P</l:mi> <l:mo>/</l:mo> <l:mi>ɛ</l:mi> </l:mrow> </l:math> , but is not an accurate measurement of either quantity in general.