Trajectum: Heavy-Ion Collision Simulator

• Trajectum is a modular heavy-ion collision simulation framework that integrates fluctuating initial conditions, pre-equilibrium dynamics, hydrodynamics, and hadronic afterburner.
• It employs advanced Bayesian calibration with Gaussian-process emulators to infer key transport coefficients and nuclear structure from diverse collision systems.
• The framework excels in reproducing soft-sector observables, guiding systematic tests of initial-state hypotheses and enabling precise postdictions in heavy-ion experiments.

Trajectum is a modular computational framework for ultrarelativistic heavy-ion collisions that couples fluctuating initial conditions, a pre-equilibrium stage, viscous hydrodynamics, particlization, and a hadronic afterburner, and was designed to support Bayesian calibration against experimental data. Across successive studies it has been used for global analyses of Pb–Pb and p–Pb collisions, precision postdictions and predictions for oxygen–oxygen systems, nuclear-structure inference in isobars, hard-probe path-length engineering, thermal photon and dilepton tomography, and persistent-homology analyses of final-state momentum distributions. In the first CMS measurement of charged-hadron pseudorapidity densities in OO collisions at $\sqrt{s_\mathrm{NN}} = 5.36$ TeV, the hydrodynamic model TRAJECTUM provided the best overall description of the centrality dependence, particularly in central collisions (Nijs et al., 2020, Nijs et al., 2021, Collaboration, 1 Jun 2026).

1. Origins and conceptual scope

Trajectum emerged as a self-contained heavy-ion simulation environment with a deliberately modular software structure. In the thesis-level formulation, the framework was organized around two executables: collide, which runs the event-by-event physics simulation, and analyze, which computes observables from the generated events and from hadronic afterburner output. The collide workflow comprises initial conditions, transport coefficients, a hydrodynamics model, a PDE solver, and a hadronizer, with standard C++ polymorphic interfaces between modules, multithreading up to 20 events in parallel, and output in UrQMD-compatible format (Nijs, 2020).

The first full Bayesian-analysis paper cast Trajectum as a 20-parameter model for PbPb and pPb collisions, extending earlier practice by introducing a variable free-streaming velocity $v_{\rm fs}$ and varying three second-order transport coefficients in addition to temperature-dependent shear and bulk viscosities. That formulation was already intended not merely as a forward simulator, but as a statistical inference framework using Gaussian-process emulators, closure tests, and convergence studies to connect parameters to broad observable sets across energies and systems (Nijs et al., 2020).

Subsequent work enlarged both the physics content and the calibration strategy. A higher-precision global analysis introduced a continuous treatment of the constituent number $n_c$ and a centrality-normalization parameter cent_norm, while a later extension generalized the reduced-thickness initialization and replaced a purely free-streaming pre-equilibrium stage by an interpolation toward a holographically inspired fast-hydrodynamizing evolution. This progression shifted Trajectum from a calibrated event generator to a platform for systematic statements about early-time dynamics, transport, and nuclear geometry (Nijs et al., 2021, Nijs et al., 2023).

2. Dynamical structure of the framework

Trajectum implements the standard multi-stage heavy-ion pipeline. Depending on the study, initial conditions are generated from Monte Carlo Glauber, OSU, TRENTo, or deformed Woods–Saxon geometries with subnucleonic constituents; this is followed by pre-equilibrium evolution, second-order viscous hydrodynamics, particlization through Cooper–Frye conversion, and a hadronic afterburner using either SMASH or UrQMD. The framework has been used primarily in 2+1D boost-invariant mode, although later descriptions explicitly state that it implements 2+1D or 3+1D viscous hydrodynamic evolution with temperature-dependent transport coefficients and a switching temperature $T_{\rm sw}$ (Nijs, 2020, Collaboration, 1 Jun 2026).

A central ingredient is the reduced-thickness mapping. In the earlier TRENTo-based implementation, the local deposition is written as

$$\mathcal{T} \;=\; N\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{1/p},$$

with $p$ controlling the generalized mean. The 2023 extension generalized this to

$$\mathcal{T} \;=\; N\,E_\text{ref}^{\,2-2q}\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{q/p},$$

thereby interpolating between different interpretations of the deposited quantity and allowing explicit tests of binary scaling. That analysis found $q = 1.34^{+0.14}_{-0.18}$ at 90% credibility and concluded that binary scaling is incompatible with experimental data (Nijs et al., 2023).

The hydrodynamic core uses relativistic viscous fluid dynamics in the Landau frame. Its constitutive tensor is written as

$$T^{\mu\nu} = \rho\,u^\mu u^\nu - \big(P + \Pi\big)\,\Delta^{\mu\nu} + \pi^{\mu\nu},$$

with $\Pi$ the bulk pressure and $v_{\rm fs}$0 the shear-stress tensor. Across the Trajectum literature, the evolution is formulated with Israel–Stewart-type or DNMR-like second-order equations, a lattice-QCD-based hybrid equation of state, temperature-dependent $v_{\rm fs}$1 and $v_{\rm fs}$2, and selected second-order coefficients such as $v_{\rm fs}$3 and $v_{\rm fs}$4 (Nijs, 2020, Nijs et al., 2020, Nijs et al., 2023).

The pre-equilibrium stage is one of the distinctive parts of the framework. In the original Bayesian implementation, matter free streams from $v_{\rm fs}$5 to $v_{\rm fs}$6 with tunable velocity $v_{\rm fs}$7. Later work replaced this with an interpolation between free streaming and a holographically inspired, fast-hydrodynamizing initialization through a parameter $v_{\rm fs}$8, with $v_{\rm fs}$9 corresponding to free streaming and $n_c$0 to the holographic limit. The posterior in the extended analysis is strongly peaked at $n_c$1, and $n_c$2 fm/$n_c$3 is preferred, although weakly constrained (Nijs et al., 2020, Nijs et al., 2023).

3. Bayesian calibration and uncertainty propagation

A defining feature of Trajectum is that inference is built into the framework rather than appended externally. The Bayesian analyses use Gaussian-process emulators trained on principal components of high-dimensional observable vectors. In the 2020 study, Latin-hypercube designs of 1000 PbPb points with 6000 events each and 2000 pPb points with 40,000 events each were used; the emulator was trained on the first 25 principal components per system, and the posterior was sampled with emcee using 600 walkers and 15,000 steps. Closure tests showed that percentiles of the true values in the recovered posteriors were approximately uniform (Nijs et al., 2020).

The 2021 precision analysis pushed this program further. For Pb+Pb it used 750 design points with 15k events each, while for O+O it used 1500 points with 40k events each. High-statistics postdictions and predictions were then produced by drawing 20 parameter sets from the posterior and running 0.5M events per set for Pb+Pb and 1–2M for O+O. That study also introduced a new analysis-level centrality rescaling parameter, cent_norm, to reconcile experimental forward-detector centrality estimators with model centrality definitions; its posterior peaked near 100%, indicating consistency between model and data anchor points (Nijs et al., 2021).

The 2023 extension refined both the statistics and the emulator engineering. It calibrated a 23-dimensional parameter space against 653 data points from ALICE Pb–Pb at 2.76 and 5.02 TeV and the CMS p–Pb inelastic cross section. The design comprised 1200 Latin-hypercube points, with 60k initial conditions, about 15k hydrodynamic events, and five SMASH samplings per point. By organizing the emulator into separate observable classes, the average emulator uncertainty was reduced from $n_c$4 to $n_c$5, and the Pb–Pb hadronic cross-section uncertainty dropped from 3.14% to 0.24% (Nijs et al., 2023).

Posterior propagation has also been used beyond soft-hadron observables. In the topological-data-analysis study, 20 parameter choices were randomly drawn from the posterior and roughly 400k events were simulated for each; systematic uncertainty was extracted via

$n_c$6

An analogous 20-sample strategy was used for thermal photons and dileptons, where systematic bands were defined as the RMS across samples after quadratically subtracting statistical uncertainty. This makes Trajectum not only a calibration framework, but also a vehicle for uncertainty-aware theory predictions in observables that were not part of the fit (Capellino et al., 2 Sep 2025, Massen et al., 2024).

4. Observable program and validation

The core validation set of Trajectum comprises the canonical soft-sector observables of relativistic heavy-ion physics: $n_c$7, $n_c$8, identified yields, $n_c$9 spectra, mean transverse momentum, $T_{\rm sw}$0 fluctuations, integrated and differential flow cumulants, and event-plane correlations. Already in the thesis implementation, the framework reproduced the general magnitude of multiplicity, mean $T_{\rm sw}$1, $T_{\rm sw}$2 fluctuations, and $T_{\rm sw}$3, $T_{\rm sw}$4, $T_{\rm sw}$5, $T_{\rm sw}$6 at the LHC. The 2020 and 2021 analyses then showed detailed agreement with ALICE and ATLAS data for identified $T_{\rm sw}$7, ultracentral flow, and event-plane correlators, while also diagnosing tensions in proton yields and in high-$T_{\rm sw}$8 tails, especially in peripheral bins where hard components become relevant (Nijs, 2020, Nijs et al., 2020, Nijs et al., 2021).

Trajectum has also been used to study observables not usually included in global fits. In a path-length-engineering analysis at $T_{\rm sw}$9 TeV Pb–Pb, 2.2 million hydrodynamic events were generated and non-interacting probe trajectories were propagated through the medium. The study defined flow-weighted and temperature-plus-flow-weighted path-length proxies such as

$$\mathcal{T} \;=\; N\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{1/p},$$0

and showed that event-shape engineering combined with in-plane versus out-of-plane selection can produce maximal effective path-length ratios up to about 2.5 in peripheral collisions with high $$\mathcal{T} \;=\; N\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{1/p},$$1, while the difference is about 1.1 in ultracentral collisions (Beattie et al., 2022).

Electromagnetic probes have been treated within the same Bayesian-calibrated medium. For thermal photons and dileptons in Pb–Pb at $$\mathcal{T} \;=\; N\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{1/p},$$2 TeV, Trajectum was used to compute emission rates over the full space-time evolution and to extract effective temperatures and elliptic flow. The analysis concluded that thermal photon slopes yield $$\mathcal{T} \;=\; N\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{1/p},$$3–300 MeV and are almost independent of centrality because of blue shift from radial flow, whereas thermal dileptons are much better probes of the QGP temperature because the invariant-mass slope is not blue shifted; by varying $$\mathcal{T} \;=\; N\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{1/p},$$4 and $$\mathcal{T} \;=\; N\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{1/p},$$5 cuts, average emission times from $$\mathcal{T} \;=\; N\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{1/p},$$6 fm/$$\mathcal{T} \;=\; N\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{1/p},$$7 down to $$\mathcal{T} \;=\; N\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{1/p},$$8 fm/$$\mathcal{T} \;=\; N\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{1/p},$$9 were isolated (Massen et al., 2024).

A different extension used persistent homology on 2D $p$0 point clouds of final-state charged hadrons. There, Trajectum produced predictions for Betti curves and persistence distributions in Pb–Pb and O–O collisions using alpha complexes and GUDHI. The study found that these topological observables largely reflect known phenomenology, including multiplicity ordering, radial-flow mass ordering, and azimuthal anisotropies, but do not show enhanced sensitivity to the model’s tunable parameters compared to conventional observables. A plausible implication is that Trajectum can serve as a testing ground for novel observables even when they do not outperform standard soft-hadron diagnostics (Capellino et al., 2 Sep 2025).

5. Geometry, system size, and nuclear structure

One of the most consequential uses of Trajectum has been the study of geometry-dominated observables across nuclei and system sizes. In a global analysis of Pb+Pb and O+O, the framework used TRENTo with nucleon substructure, second-order viscous hydrodynamics, and UrQMD, and generated oxygen nuclei from a library of 6000 $p$1-clustered configurations with realistic two- and three-body correlations. That study predicted identified yields, $p$2, $p$3 fluctuations, event-plane correlations, and $p$4 for O+O at RHIC and the LHC, and argued that O+O should be especially sensitive to pre-equilibrium parameters and initial-state granularity (Nijs et al., 2021).

The 2026 CMS OO multiplicity measurement supplied the most direct benchmark of Trajectum in a light-ion system. Charged-hadron pseudorapidity densities were measured with a rigorously corrected tracklet method in $p$5, yielding a centrality-integrated midrapidity density $p$6 (syst) and a 0–5% central value $p$7 (syst). In that comparison, Trajectum provided the best overall description among the models considered, predicting the OO $p$8 centrality trend within about 10% in peripheral bins and $p$9 in the most central collisions; calculations based on the NLEFT and PGCM oxygen densities were nearly identical (Collaboration, 1 Jun 2026).

Trajectum has also been used to infer nuclear structure from isobar data. In Ru+Ru and Zr+Zr at RHIC, the framework forward-simulated several Woods–Saxon parameterizations and showed that STAR’s multiplicity and flow ratios uniquely favor a DFT-based case with separate proton and neutron densities, a neutron skin in $$\mathcal{T} \;=\; N\,E_\text{ref}^{\,2-2q}\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{q/p},$$0Zr, $$\mathcal{T} \;=\; N\,E_\text{ref}^{\,2-2q}\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{q/p},$$1, $$\mathcal{T} \;=\; N\,E_\text{ref}^{\,2-2q}\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{q/p},$$2, and a nonzero $$\mathcal{T} \;=\; N\,E_\text{ref}^{\,2-2q}\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{q/p},$$3 in Zr smaller than 0.202. The robustness of the Ru/Zr ratios against viscosity variations was central to that inference (Nijs et al., 2021).

A more specialized geometry application concerned beam transmutation in the 2025 LHC light-ion run. There Trajectum was extended with a spectator-clustering model and coupled to GEMINI to estimate stable-fragment production from $$\mathcal{T} \;=\; N\,E_\text{ref}^{\,2-2q}\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{q/p},$$4O–$$\mathcal{T} \;=\; N\,E_\text{ref}^{\,2-2q}\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{q/p},$$5O and $$\mathcal{T} \;=\; N\,E_\text{ref}^{\,2-2q}\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{q/p},$$6Ne–$$\mathcal{T} \;=\; N\,E_\text{ref}^{\,2-2q}\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{q/p},$$7Ne beams. The analysis found that circulating isotope production is dominated by $$\mathcal{T} \;=\; N\,E_\text{ref}^{\,2-2q}\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{q/p},$$8He and deuterons, and quoted a combined relative contamination of about 0.3% after $$\mathcal{T} \;=\; N\,E_\text{ref}^{\,2-2q}\left(\frac{\mathcal{T}_A^p + \mathcal{T}_B^p}{2}\right)^{q/p},$$9 hours of OO collisions. It also showed that transmuted systems such as $q = 1.34^{+0.14}_{-0.18}$0O–$q = 1.34^{+0.14}_{-0.18}$1He have lower multiplicity than $q = 1.34^{+0.14}_{-0.18}$2O–$q = 1.34^{+0.14}_{-0.18}$3O but characteristically higher $q = 1.34^{+0.14}_{-0.18}$4 at fixed multiplicity (Nijs et al., 2 Jul 2025).

6. Tensions, criticism, and open directions

Trajectum is not uniformly successful across all initial-state-sensitive observables. A particularly important counterexample is the ALICE measurement of correlations between event-mean transverse momentum and anisotropic flow in Pb–Pb and Xe–Xe collisions. In that comparison, Trajectum’s TRENTo-based predictions exhibited a strong centrality dependence, underestimated $q = 1.34^{+0.14}_{-0.18}$5 by more than 50% for centralities above 30%, became negative above 40%, and predicted $q = 1.34^{+0.14}_{-0.18}$6 to be negative already above 10% in Pb–Pb. The Letter interprets these failures primarily as deficiencies in the initial geometry, especially the size–shape correlations encoded by nucleon width and the absence of IP-Glasma-like subnucleonic structure in the corresponding Trajectum setup (Collaboration, 2021).

Several other limitations recur across the literature. The 2020 and 2021 global analyses noted tensions in proton yields and in high-$q = 1.34^{+0.14}_{-0.18}$7 spectral tails, suggesting that adding spectra to future fits and improving hard components would sharpen the framework. The thesis version and many later applications rely on 2+1D boost invariance, with 3+1D evolution described as a future direction. In electromagnetic probes, early-time uncertainty is dominated by the hydrodynamization time and by the omission of pre-equilibrium photon and dilepton production. In topological observables, persistent homology is viable and robust but does not out-perform standard soft observables in parameter sensitivity (Nijs, 2020, Nijs et al., 2021, Massen et al., 2024, Capellino et al., 2 Sep 2025).

The 2023 generalized-hydrodynamization analysis sharpened one major theoretical conclusion: the posterior strongly favors a fast-hydrodynamizing pre-equilibrium stage and strongly disfavors binary scaling in the initial condition. This suggests that Trajectum’s long-term significance lies less in any single parametrization than in its ability to adjudicate between dynamical hypotheses within a common inference framework (Nijs et al., 2023).

The near-term program proposed in the cited work is correspondingly concrete. It includes incorporating the new $q = 1.34^{+0.14}_{-0.18}$8–$q = 1.34^{+0.14}_{-0.18}$9 correlations into Bayesian calibration, refining forward-versus-midrapidity centrality mapping, harmonizing Glauber implementations, exploring forward–midrapidity correlations in light ions, extending thermal probes to include pre-equilibrium emission, and using OO and NeNe runs to constrain substructure fluctuations and the temperature dependence of $$T^{\mu\nu} = \rho\,u^\mu u^\nu - \big(P + \Pi\big)\,\Delta^{\mu\nu} + \pi^{\mu\nu},$$0 and $$T^{\mu\nu} = \rho\,u^\mu u^\nu - \big(P + \Pi\big)\,\Delta^{\mu\nu} + \pi^{\mu\nu},$$1. Taken together, these directions present Trajectum not as a finalized model, but as an evolving inference platform for bulk QCD matter, nuclear geometry, and the early-time approach to hydrodynamics (Collaboration, 2021, Massen et al., 2024, Collaboration, 1 Jun 2026).