Hubble's Law in Heavy Ion Collisions

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.

💡 Research Summary

The manuscript investigates the time evolution of a “microscopic” Hubble parameter (H) that characterizes the expansion of the hot, dense fireball created in relativistic heavy‑ion collisions. Using the Parton‑Hadron‑String‑Dynamics (PHSD) transport approach, the authors simulate Au+Au collisions at a center‑of‑mass energy per nucleon pair √sₙₙ = 7.8 GeV for four impact parameters (b = 2.5, 5.0, 7.5, and 10.0 fm). The study focuses on two particle species—nucleons and pions—because they dominate the final‑state composition.

A central methodological innovation is the extraction of H from the simulated velocity field without relying on a simple linear fit of velocity profiles. After discretizing the fireball on a 1 fm lattice (Δx = Δy = γΔz = 1 fm) and defining a cell as “matter” when its energy density exceeds 0.05 GeV/fm³, the authors compute the Eckart velocity v(x,t) = J(x,t)/J⁰(x,t) from the particle four‑flow. They then form the probability‑density distribution of the directional derivatives ∂ᵢvᵢ (i = x, y, z). In a typical expanding fireball this distribution exhibits two peaks: a low‑value peak associated with non‑linear edge effects and a higher‑value peak corresponding to the bulk Hubble‑like flow. By discarding the low‑value peak and fitting the remaining histogram with a Gaussian, they obtain the average value of ∂ᵢvᵢ, which directly yields the microscopic Hubble parameter via H = ⟨∂ᵢvᵢ⟩. This statistical‑histogram technique reduces event‑by‑event fluctuations (the authors average over ~10⁶ configurations) and provides a robust way to separate bulk expansion from peripheral distortions.

The results are presented separately for the longitudinal (z) direction, aligned with the beam axis, and the two transverse directions (x along the impact‑parameter vector and y perpendicular to the reaction plane).

Longitudinal expansion: After the nuclei have fully passed each other (the “last‑touch” time t_LT ≈ 5 fm/c), the inverse Hubble parameter 1/H_z grows linearly with time for both nucleons and pions. A linear fit 1/H_z = C (t – t₀) yields C ≈ 1.00 fm⁻¹·c and t₀ ≈ 4.07 fm/c, indicating a ballistic regime where particle velocities become essentially constant (v = const) and the expansion follows v = r/t. This behavior is independent of impact parameter and matches earlier findings that the bulk flow in the central region of the fireball is irrotational. In the early stage (t < t_FT ≈ 1.8 fm/c) a more complex transition is observed: the nucleon and pion curves differ, and the pion data are well described by a third‑order polynomial. All extrapolated curves intersect near t ≈ 1.7–2.5 fm/c, suggesting a common “fireball emergence” time.

Transverse expansion: In contrast, the transverse inverse Hubble parameters 1/H_x and 1/H_y display a rapid decrease at early times, followed by an exponential‑type low‑time asymptotics. This indicates that the transverse expansion is not purely ballistic; instead, strong pressure gradients and edge effects dominate, leading to a non‑linear velocity profile near the fireball surface. The transverse Hubble parameter shows only a weak dependence on impact parameter, reinforcing the view that the overall geometry of the collision does not strongly affect the bulk transverse expansion at this beam energy.

Quantitatively, the extracted microscopic Hubble parameters are many orders of magnitude larger than the cosmological Hubble constant H₀ ≈ 70 km s⁻¹ Mpc⁻¹; the authors estimate a factor of ~10⁴⁰. This stark difference underscores that the “little bang” created in heavy‑ion collisions is a fundamentally different physical system, despite the formal analogy in the expansion law.

The paper also discusses limitations. The current PHSD implementation cannot reliably probe times earlier than ~2 fm/c because the medium definition (energy‑density threshold) becomes ambiguous and the lattice resolution limits precision. Consequently, the early‑time behavior of H remains uncertain, and future work should aim at finer spatial discretization, improved initial‑state modeling, and perhaps alternative definitions of the fireball boundary.

In summary, the authors provide the first systematic, species‑resolved determination of a microscopic Hubble parameter in non‑central heavy‑ion collisions at intermediate energies. Their histogram‑based method offers a new tool for separating bulk Hubble‑like flow from peripheral non‑linearities. The findings confirm ballistic longitudinal expansion after nuclear separation, reveal a non‑ballistic, exponentially damped transverse expansion, and demonstrate that the expansion dynamics are largely insensitive to impact parameter. These results enrich our understanding of the space‑time evolution of the fireball and strengthen the conceptual bridge between cosmology and relativistic nuclear physics.