find a pseudo-critical phase transition temperature T ≈ 155 MeV, where hadrons melt to form a deconfined soup of quarks and gluons dubbed a quark-gluon plasma (QGP) T ~ 155 MeV Baryon Density μ [GeV] Temperature T [MeV] critical point? quark-gluon plasma early universe hadron gas nuclear collisions J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 1 / 24
what conditions is it formed in a nuclear collision? How does it recombine to form colorless hadrons? Equation of state? Relations between thermal quantities, e.g. P = P( ) Transport properties? shear/bulk viscosity, probe energy loss, etc J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 2 / 24
L(x) → L(x1, ..., xn) Curse of dimensionality Typically interested in marginalized probabilities L(x1, ..., xn) easy to calculate, hard to integrate. Solution Monte Carlo integration, e.g. importance sampling MCMC importance sampling: 1. large number of walkers in {x1, ..., xn } space 2. update walker positions 3. accept new x with prob P ∼ Lnew/Lold Marginalize by histogramming over flattened dimensions J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 6 / 24
for MCMC varies greatly not enough better better still Several of the published results in this talk use Nsample > 106 If model is slow, e.g. 1 CPU hour per likelihood evaluation ...good luck J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 7 / 24
to normally-distributed outputs • Specified by mean and covariance functions As a model emulator: • Non-parametric interpolation • Predicts probability distributions • Narrow near training points, wide in gaps • Fast surrogate to actual model −2 −1 0 1 2 Output Random functions 0 1 2 3 4 5 Input −2 −1 0 1 2 Output Conditioned on data Mean prediction Uncertainty Training data J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 8 / 24
{x} → {x } Bayesian posterior y = f (x) L(y, yexp) ...after many steps histogram {x} to visualize J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 9 / 24
parameter(s) General relativity gravitational waves black hole masses Relativistic hydro particle yields & corr. transport coefficients Analogue time: 0 fm/c 20 fm/c Hydro framework imposes local energy and momentum conservation. Clearly breaks in dilute limit. Should apply with care. J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 11 / 24
parameter(s) General relativity gravitational waves black hole masses Relativistic hydro particle yields & corr. transport coefficients Analogue Hydro for heavy-ion collisions not trusted on same level as e.g. GR for gravitational waves • Posterior results always subject to framework credibility J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 11 / 24
to relativistic heavy-ion collisions: simultaneous characterization of the initial state and QGP medium, Bernhard, Moreland, Bass, Liu, Heinz PRC 94 (2016) 024907 Generational improvements • New TRENTo initial condition model: absorbs initial state uncertainties into several free parameters • Full event-by-event hydro with hadronic afterburner • Calculate observables exactly as experiment • Bulk and shear viscous corrections • More experimental observables Physics insights η/s min = 0.07 ± 0.05 non-zero bulk viscosity 0.15 0.20 0.25 0.30 Temperature [GeV] 0.0 0.2 0.4 0.6 /s KSS bound 1/4 Prior range Posterior median 90% CI J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 16 / 24
for bulk viscous corrections at freezeout • Less obvious parametric form for (ζ/s)(T) • Hydro cavitates if bulk is too large T/Tpeak band = 1 sigma 0.4 0.8 1.2 1.6 0 0.1 0.2 0.3 0.4 0.5 LHC RHIC z/s(T) 0.08 0.12 0.16 0.20 Temperature [GeV] 0.00 0.04 0.08 ζ/s Prior range Denicol, Paquet, Gale, Jeon, Shen Bernhard, Moreland, Bass J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 19 / 24
8 4 0 4 8 x [fm] 8 4 0 4 8 y [fm] 8 4 0 4 8 x [fm] Initial energy density (3D) x η x η x η Figure credit: Schenke, Schlichting J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 20 / 24
parameter(s) Hydrodynamics QGP viscosity: η/s, ζ/s Langevin transport charm diffusion coefficient: Ds,p QGP shear viscosity 0.15 0.20 0.25 0.30 Temperature [GeV] 0.0 0.2 0.4 0.6 /s KSS bound 1/4 Prior range Posterior median 90% CI Charm diffusion coefficient 0.1 0.2 0.3 0.4 0.5 0.6 T [Ge V] 0 5 10 15 20 Ds 2πT p=0Ge V/c median value prior 90% C.R A data driven analysis for the temperature and momentum dependence of the heavy quark diffusion coefficient in relativistic heavy-ion collisions Xu, Bernhard, Bass, Nahrgang, Cao (in preparation) J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 23 / 24
with multiple correlated parameters • Rigorous accounting of errors and effect on quantities of interest • Global analysis can promote and kill models J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 24 / 24