Hubble's Law in Heavy Ion Collisions

TL;DR

The paper investigates the time evolution of the microscopic Hubble parameter in heavy-ion collisions by simulating Au+Au events at $\sqrt{s_{NN}}=7.8$ GeV with the PHSD transport framework. A novel method analyzes the distributions of velocity-field derivatives to extract $H$ from the expanding fireball, revealing ballistic longitudinal expansion after nucleus separation and rapid exponential growth in the transverse directions. The longitudinal behavior follows $H_z(t) \approx C/(t-t_0)$ with $C\approx1$ and $t_0\approx4.07$ fm/$c$, while transverse expansions exhibit $H_i(b,t)=C(b) e^{k t}$ with $k\approx0.824$ fm$^{-1}$c$^{-1}$ and species-dependent scale factors. Post-separation $H$ values lie in $0.02$--$0.20$ fm$^{-1}$c$^{-1}$, vastly exceeding the cosmological Hubble constant, illustrating the strong, non-cosmological expansion dynamics in the fireball and offering a route to experimentally access microscopic expansion parameters through femtoscopy and related observables.

Abstract

The evolution of the "microscopic" Hubble parameter related to the expansion of matter born in heavy-ion collisions was obtained for nucleons and pions. The calculations were carried out within the parton-hadron-string dynamics (PHSD) transport model. Au+Au collisions with $\sqrt{s_{NN}} = 7.8$ GeV at $b = 2.5,\ 5.0,\ 7.5$, and $10.0$ fm were considered. A new method for determining the "microscopic" Hubble parameter from simulated data was used. The ballistic motion was obtained for the longitudinal direction after the separation of the nuclei. In earlier times, the evolution of the "microscopic" Hubble parameter in this direction was more complicated. For transverse directions, an exponential low-time asymptotics of the Hubble parameter was observed. The obtained values of the "microscopic" Hubble parameter are about 40 orders of magnitude higher than the cosmological Hubble constant.

Figures (6)

• Figure 1: Density of spectators projected on the reaction plane (Au+Au collision at $\sqrt{s_{NN}} = 7.8$ GeV, $b = 7.5$ fm).
• Figure 2: A longitudinal velocity profile and the corresponding statistical distribution of $\partial_z v_z$ (Au+Au collision at $\sqrt{s_{NN}} = 7.8$ GeV, $b = 7.5$ fm).
• Figure 3: The inverse Hubble parameter in the longitudinal direction. The solid straight line shows the linear fit; the solid curves correspond to polynomial fit given by Eq. (\ref{['eq::cube']}); the dashed curves correspond to calculation by Eq. (\ref{['eq::H_Bondorf']}). Two thick crosses show the estimated positions of the intersection points of the fit curves for pions and nucleons. Two vertical dashed lines are for the maximal overlapping moment of time and for the last touch moment. The inset subplot shows zoomed evolution before the linear growth.
• Figure 4: The Hubble parameter in the transverse directions. The slanted lines show exponential fits. Two vertical dashed lines are for the maximal overlapping moment of time and for the last touch moment.
• Figure 5: The scale factor $C(b)$ from Eq. (\ref{['eq::exp']}) for pions and nucleons.