Bayesian parameter estimation for heavy-ion collisions: inferring - - PowerPoint PPT Presentation

Bayesian parameter estimation for heavy-ion collisions: inferring properties of the quark-gluon plasma

• J. Scott Moreland—Duke U.

XLVII International Symposium on Multiparticle Dynamics September 14, 2017

Lattice predicts existence of a quark-gluon plasma

Lattice QCD calculations 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

n u c l e a r c

• l

l i s i

• n

s

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 1 / 24

What are the quark-gluon plasma bulk properties?

How and under 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(ǫ) Transport properties? shear/bulk viscosity, probe energy loss, etc

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 2 / 24

Formulating an inverse problem

MODEL-TO-DATA COMPARISON (IN AN IDEAL WORLD) Model A Model B Model C Model D Model E

• Exp Data
• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 3 / 24

Formulating an inverse problem

REALISTIC MODEL-TO-DATA COMPARISON Model A Model B Model C Model D Model E

? ?

Exp Data

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 3 / 24

I) BAYESIAN PARAMETER ESTIMATION

Formulating an inverse problem

PARAMETRIZE THEORY LANDSCAPE Model A Model B Model C Model D Model E Exp Data

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 4 / 24

Formulating an inverse problem

BAYESIAN PARAMETER ESTIMATION continuous model parameter: x Exp Data P(x⋆|model, data) BAYES’ THEOREM: P(x⋆|model, data)

• posterior

∝ P(model, data|x⋆)

• likelihood

P(x⋆)

prior

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 4 / 24

Formulating an inverse problem

BAYESIAN PARAMETER ESTIMATION continuous model parameter: x Exp Data P(x⋆|model, yexp) BAYES’ THEOREM: P(x⋆|model, data)

• posterior

∝ P(model, data|x⋆)

• likelihood

P(x⋆)

prior

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 4 / 24

Formulating an inverse problem

BAYESIAN PARAMETER ESTIMATION continuous model parameter: x Exp Data P(x⋆|model, yexp) BAYES’ THEOREM: P(x⋆|model, data)

• posterior

∝ P(model, data|x⋆)

• likelihood

P(x⋆)

prior

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 4 / 24

Formulating an inverse problem

YIELDS POSTERIOR DISTRIBUTION ON x⋆

4 2 2 4

x P(x |model, data)

Includes uncertainty in “best-fit value”

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 4 / 24

Multiple observables

posterior = likelihood × prior

More than one observable f : x → (y1, ..., yn)? No problem, calculate likelihood using multivariate Gaussian

Log-likelihood

ln(L) = −1 2(ln(|Σ|) + (y − yexp)TΣ−1(y − yexp) + k ln(2π)) Σ = Σmodel + Σstat

exp + Σsys exp

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 5 / 24

Multiple model parameters

posterior = likelihood × prior

Likelihood function 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

• ver flattened dimensions
• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 6 / 24

MCMC and evaluating the likelihood

Number of likelihood samples needed 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

Training an emulator

Gaussian process:

• Stochastic function: maps inputs

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 1 2

Output

Random functions

1 2 3 4 5

Input

−2 −1 1 2

Output

Conditioned on data

Mean prediction Uncertainty Training data

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 8 / 24

Workflow Physics model Emulated model MCMC update

update walkers {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

Bayesian parameter estimation in physics

LIGO EXPERIMENT

10 20 30 40 50 60

m1(M ⊙)

10 20 30 40

m2(M ⊙)

GW150914 LVT151012 GW151226 GW170104

Average Effective Precession Full Precession

• est. black hole masses

PRL 118.221101

• PLANCK COLLABORATION 2015:

constraints on inflation

• Astron. Astrophys. 594 (2016)
• CKM parameters
• Eur. Phys. J. C21 (2001)
• GALAXY FORMATION
• Astron. Astrophys. 409 (2003)

...and many more examples not listed here

Adapt machinery to relativistic heavy-ion collisions?

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 10 / 24

II) BAYESIAN PARAMETER ESTIMATION

APPLIED TO HEAVY-ION PHYSICS

Bayesian methodology for heavy-ion collisions

TRUSTED FRAMEWORK EXPERIMENTAL DATA FREE PARAMETER(S) General relativity gravitational waves black hole masses

Relativistic hydro

particle yields & corr. transport coefficients

A n a l

• g

u e

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

Bayesian methodology for heavy-ion collisions

TRUSTED FRAMEWORK EXPERIMENTAL DATA FREE PARAMETER(S) General relativity gravitational waves black hole masses

Relativistic hydro

particle yields & corr. transport coefficients

A n a l

• g

u e

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

Seminal Bayesian works in heavy-ion physics

• Rel. Probability

2.5 5

• Rel. Probability

T dep. of η energy norm. σsat (mb) 30 50 σsat (mb) W.N./Sat. frac. 1 W.N./Sat. frac.

• Init. Flow

0.25 1.25

• Init. Flow

η/s 0.02 0.5 η/s T dep. of η 0.85 1.025 1.2 energy norm. 5 30 40 50 σsat (mb) 0.5 1 W.N./Sat. frac. 0.25 0.75 1.25

• Init. Flow

0.02 0.26 0.5 η/s

• Event-averaged hydro
• Parametric pre-flow
• Parametric initial state
• First Bayesian posterior on

(η/s)(T)

• Omits bulk viscosity
• Two centrality bins

Determining Fundamental Properties of Matter Created in Ultrarelativistic Heavy-Ion Collisions, Novak, Novak, Pratt, Vredevoogd, Coleman-Smith, Wolpert PRC 89 (2014) 034917

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 12 / 24

Seminal Bayesian works in heavy-ion physics

• Equation of state from lattice QCD is very close to

parametric equation of state preferred by simulation

• BNL

Constraining the Eq. of State of Super-Hadronic Matter from Heavy-Ion Collisions, Pratt, Sangaline, Sorensen, Wang, PRL 114 (2015) 202301

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 13 / 24

Seminal Bayesian works in heavy-ion physics

0.0 0.1 0.2 0.3

´=s Glauber 0.08 KLN 0.20

Theoretical biases affect preferred viscosity

• Event-by-event hydro
• MC-Glauber & KLN

initial conditions

• Centrality bins like

experiment

• Constant η/s
• Omits bulk viscosity,

pre-flow

Constraining the Eq. of State of Super-Hadronic Matter from Heavy-Ion Collisions, Bernhard, Marcy, Coleman-Smith, Huzurbazar, Wolpert, Bass, PRC 91 (2015) 054910

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 14 / 24

Initial stages and onset of hydrodynamic flow

Strong coupling limit → hydrodynamics Weak coupling limit → freestreaming

me momentum anisotropy

free streaming h y d r

• d

y n a m i c s free streaming + hydro

τs

Pre-equilibrium dynamics and heavy-ion observables, Heinz, Liu, Nucl. Phys. A956 (2016) 549-552

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 15 / 24

Towards precision extraction of QGP properties

Applying Bayesian parameter estimation 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

Towards precision extraction of QGP properties

Applying Bayesian parameter estimation to relativistic heavy-ion collisions: simultaneous characterization of the initial state and QGP medium, Bernhard, Moreland, Bass, Liu, Heinz PRC 94 (2016) 024907

Model calculations with high-likelihood parameters from Bayesian posterior provide excellent description of bulk observables

100 101 102 103 104

π ± K ± p¹ p Nch × 5

solid: identified dashed: charged

Yields dN/dy, dNch/dη

10 20 30 40 50 60 70

Centrality %

0.8 1.0 1.2

Model/Exp

0.0 0.4 0.8 1.2

π ± K ± p¹ p

Mean pT [GeV]

10 20 30 40 50 60 70

Centrality %

0.8 1.0 1.2 0.00 0.03 0.06 0.09

v2 v3 v4

Flow cumulants vn{2}

10 20 30 40 50 60 70

Centrality %

0.8 1.0 1.2

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 17 / 24

Leveraging data from RHIC beam energy scan

0.8 1.6 2.4

τ0 [fm/c]

0.6 1.2 1.8

Wtrans [fm]

0.6 1.2 1.8

Wlong [fm]

0.0 0.1 0.2 0.3

η/s

0.15 0.30 0.45 0.60 0.75

ǫSW [GeV/fm3]

0.8 1.6 2.4

τ0 [fm/c]

0.6 1.2 1.8

Wtrans [fm]

0.6 1.2 1.8

Wlong [fm]

0.15 0.30 0.45 0.60

ǫSW [GeV/fm3]

√sNN = 19 GeV

Left: Bayesian posterior for hydrodynamic model param- eters calibrated at different beam energies √sNN

Revealing the collision energy dependence of η/s in RHIC-BES Au+Au collisions using Bayesian statistics, Auvinen, Karpenko, Bernhard, Bass, QM17 proceedings

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 18 / 24

Leveraging data from RHIC beam energy scan

0.8 1.6 2.4

τ0 [fm/c]

0.6 1.2 1.8

Wtrans [fm]

0.6 1.2 1.8

Wlong [fm]

0.0 0.1 0.2 0.3

η/s

0.15 0.30 0.45 0.60 0.75

ǫSW [GeV/fm3]

0.8 1.6 2.4

τ0 [fm/c]

0.6 1.2 1.8

Wtrans [fm]

0.6 1.2 1.8

Wlong [fm]

0.15 0.30 0.45 0.60

ǫSW [GeV/fm3]

√sNN = 39 GeV

Left: Bayesian posterior for hydrodynamic model param- eters calibrated at different beam energies √sNN

Revealing the collision energy dependence of η/s in RHIC-BES Au+Au collisions using Bayesian statistics, Auvinen, Karpenko, Bernhard, Bass, QM17 proceedings

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 18 / 24

Leveraging data from RHIC beam energy scan

0.8 1.6 2.4

τ0 [fm/c]

0.6 1.2 1.8

Wtrans [fm]

0.6 1.2 1.8

Wlong [fm]

0.0 0.1 0.2 0.3

η/s

0.15 0.30 0.45 0.60 0.75

ǫSW [GeV/fm3]

0.8 1.6 2.4

τ0 [fm/c]

0.6 1.2 1.8

Wtrans [fm]

0.6 1.2 1.8

Wlong [fm]

0.15 0.30 0.45 0.60

ǫSW [GeV/fm3]

√sNN = 62 GeV

Left: Bayesian posterior for hydrodynamic model param- eters calibrated at different beam energies √sNN

Revealing the collision energy dependence of η/s in RHIC-BES Au+Au collisions using Bayesian statistics, Auvinen, Karpenko, Bernhard, Bass, QM17 proceedings

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 18 / 24

Leveraging data from RHIC beam energy scan

0.8 1.6 2.4

τ0 [fm/c]

0.6 1.2 1.8

Wtrans [fm]

0.6 1.2 1.8

Wlong [fm]

0.0 0.1 0.2 0.3

η/s

0.15 0.30 0.45 0.60 0.75

ǫSW [GeV/fm3]

0.8 1.6 2.4

τ0 [fm/c]

0.6 1.2 1.8

Wtrans [fm]

0.6 1.2 1.8

Wlong [fm]

0.15 0.30 0.45 0.60

ǫSW [GeV/fm3]

Beam-energy dependence of η/s

19.6 39 62.4

√sNN [GeV]

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

η/s

Revealing the collision energy dependence of η/s in RHIC-BES Au+Au collisions using Bayesian statistics, Auvinen, Karpenko, Bernhard, Bass, QM17 proceedings

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 18 / 24

Bulk viscosity: a work in progress...

CHALLENGES

• Different methods 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 0.1 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

Studying the QGP fireball in 3D

Initial energy density (2D)

8 4 4 8

x [fm]

8 4 4 8

y [fm]

8 4 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

Studying the QGP fireball in 3D

Constraints on rapidity-dependent initial conditions from charged particle pseudorapidity densities and two-particle correlations, Ke, Moreland, Bernhard, Bass (in prep)

Optimization problem

Find initial energy density that evolves into final single particle distribution

• Parametrize initial

longitudinal energy profile with moment-generating function

• Constrain form using charged

particle rapidity distributions

−8 −4 4 8

η

50 100 150

dNch/dη (arb. units) Unregulated

−8 −4 4 8

η Regulated

γ = 0.0 γ = 3.0 γ = 6.0 γ = 9.0 −8 −4 4 8

y [fm] Pb+Pb Pb+Pb

−8 −4 4 8

x [fm]

−3 3

y [fm] p+Pb

−8 −4 4 8

ηs p+Pb

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 21 / 24

Studying the QGP fireball in 3D

Bayesian analysis Initial entropy profile

TB = 0.2

rel-skew abs-skew

TB = 1.0 TB = 1.8 TA = 0.2 TB = 2.6 TA = 1.0 TA = 1.8 −5 0 5 −5 0 5 −5 0 5 −5 0 5 TA = 2.6

η ds/dη (arb. units)

Final particle distribution

−5.0 −2.5 0.0 2.5 5.0

η

250 500 750 1000 1250 1500 1750 2000

dN/dη

ALICE, 2.76 TeV −2 2

η

10 20 30 40 50 60 70 80

dN/dη

ATLAS, 5.02 TeV

• Trust in hydro and Bayesian statistical machinery lets us

deconvolve complex system evolution

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 22 / 24

QGP hard probes: open heavy-flavour

A n a l

• g

u e Theory framework Free 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

.1 .2 .3 .4 .5 .6 T [Ge V] 5 1 1 5 2 Ds2 πT p=0Ge V/c

m e dian 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

Summary

Virtues of Bayesian parameter estimation

• Works for models 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

BACKUP SLIDES

Seminal Bayesian works in heavy-ion physics

dNπ /dy (RHIC, 0-5\%) <pt >π (RHIC, 0-5\%) <pt >K (RHIC, 0-5\%) <pt >p (RHIC, 0-5\%) Rout (RHIC, 0-5\%) Rside (RHIC, 0-5\%) Rlong (RHIC, 0-5\%) dNπ /dy (RHIC, 20-30\%) <pt >π (RHIC, 20-30\%) <pt >K (RHIC, 20-30\%) <pt >p (RHIC, 20-30\%) Rout (RHIC, 20-30\%) Rside (RHIC, 20-30\%) Rlong (RHIC, 20-30\%) v2 (RHIC, 20-30\%) dNπ /dy (LHC, 0-5\%) <pt >π (LHC, 0-5\%) <pt >K (LHC, 0-5\%) <pt >p (LHC, 0-5\%) Rout (LHC, 0-5\%) Rside (LHC, 0-5\%) Rlong (LHC, 0-5\%) dNπ /dy (LHC, 20-30\%) <pt >π (LHC, 20-30\%) <pt >K (LHC, 20-30\%) <pt >p (LHC, 20-30\%) Rout (LHC, 20-30\%) Rside (LHC, 20-30\%) Rlong (LHC, 20-30\%) v2 (LHC, 20-30\%) 0.0 0.1 Zǫ (RHIC) 0.0 0.1 Zǫ (LHC) 0.0 0.1 σsat (RHIC) 0.0 0.1 σsat (LHC) 0.0 0.1 fwn (RHIC) 0.0 0.1 fwn (LHC) 0.0 0.1 τ′

xx (RHIC)

0.0 0.1 τ′

xx (LHC)

0.0 0.1 F0 (RHIC) 0.0 0.1 F0 (LHC) 0.0 0.1 (η/s)0 0.0 0.1 η′ 0.0 0.1 EoSX 0.0 0.1 EoS R

Sensitivity of experimental observables to model parameters Towards a Deeper Understanding of How Experiments Constrain the Underlying Physics of Heavy-Ion Collisions, Sangaline, Pratt, PRC 93 (2016) 024908

• J. S. Moreland (Duke U.)

Bayesian parameter estimation for HIC 1 / 1