NEW EXACT AND PERTURBTIVE SOLUTIONS OF RELATIVISTIC HYDRO A COLLECTION OF RECENT RESULTS

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1 NEW EXACT AND PERTURBTIVE SOLUTIONS OF RELATIVISTIC HYDRO A COLLECTION OF RECENT RESULTS MÁTÉ CSANÁD (EÖTVÖS THOR LISBON MEETING, JUNE 13, T. CSÖRGŐ, G. KASZA, Z. JIANG, C. YANG, B. KURGYIS, M. NAGY,

2 2/27 PHASES OF QUARK MATTER An evolution throughout many phases Modeling possible with hydrodynamics (?)

3 3/27 EXACT HYDRO HISTORY & BASICS Relativistic hydrodynamics: established by Landau (for p+p!) Exact, analytic solutions important: connect initial and final state Famous solutions by Landau&Khalatnikov and Hwa&Bjorken L. D. Landau, Izv. Akad. Nauk Ser. Fiz. 17, 51 (1953) I.M. Khalatnikov, Zhur. Eksp. Teor. Fiz. 27, 529 (1954) R. C. Hwa, Phys. Rev. D 10, 2260 (1974) J. D. Bjorken, Phys. Rev. D 27, 140 (1983) Discovery of sqgp: many new solutions See e.g. this review: de Souza, Koide, Kodama, Prog. Part. Nucl. Phys. 86, 35 (2016) Analytic solutions capture many features of data MCs, Vargyas, Eur. Phys. J. A 44, 473 (2010) MCs, Szabo, Phys. Rev. C 90, (2014) Still lacking: non-spherical 3D, accelerating, realistic solutions Linearized hydro: perturbations Kurgyis, MCs, Universe 3 (2017) no.4, 84 Shi, Liao and Zhuang, Phys.Rev. C90 (2014) no.6,

4 Log source density M. Csanád (Eötvös THOR Lisbon Meeting 4/27 A NEW CHALLENGE PHENIX observing Lévy sources: L α, R; r = 1 d 3 qe iqr e 1 qr α 2π 3 2 Shape parameter: Gauss if α = 2, power-law tail if α < 2 Cauchy Lévy (a=1.2) Gauss How to reconcile with hydro? Exponential cutoff exp p μ u μ /T in Boltzmann-Jüttner? Rescattering? Gauss (α=2.0) Lévy (α=1.2) Distance

5 5/27 THE PERTURBATIVE METHOD Method: perturbed equations for a known solution Linearized hydro equations: Need a specific base solution Example: standing fluid κ 0 δp + κ + 1 p μ δu μ = 0 κ + 1 p 0 δu μ Q μν μ δp = 0 with Q μν = δ μ1 δ ν1 g μν Result: waves 0 2 δp = c s 2 Δp Method similar to Shi, Liao and Zhuang Phys.Rev. C90 (2014) no.6, [arxiv: ]

6 6/27 A NEW CLASS OF PERTURBATIVE SOLUTIONS Hubble-flow: u μ = xμ τ, p = p 0 τ 0 τ 3+ 3κ, n = n 0 τ 0 τ 3 N(s), u μ μ s = 0 Describes observables, including HBT and higher order flow MCs, Szabo, Phys. Rev. C 90, (2014), MCs, Vargyas, Eur. Phys. J. A 44, 473 (2010) Perturbative solution on top of Hubble-flow possible: Kurgyis, MCs, Universe 3 (2017) no.4, 84, arxiv:

7 7/27 RESCRITIONS FOR PERTURBATIVE SOLUTIONS Flow profile χ(s), pressure profile π(s), density profile ν(s) Auxiliary functions F τ, g x ν, h x ν These are related to each other as: Left side: only depends on scale variable S! Many solutions possible, various scaling variables and profiles

8 8/27 TIME EVOLUTION OF PERTURBATIONS An example perturbation from the class of solutions: density n(x) pressure p(x) flow u(x) Kurgyis, MCs, Universe 3 (2017) no.4, 84, arxiv:

9 9/27 PERTURBATIONS OF THE OBSERVABLES Observables calculable via usual Jüttner-Boltzmann source w/ Cooper-Fry Spectra and correlations obtain a perturbative component Hubble-flow observables stable against small perturbations Kurgyis, MCs, Universe 3 (2017) no.4, 84, arxiv:

10 10/27 THE BJORKEN-ESTIMATE The original idea: energy density based on de/dy QGP critical ε c ~ 1 GeV/fm 3 (from ε c = (6 8) T c 4 ) Result (~2000x cited) E = N de dy Δy = N de dy ε Bj = 1 R 2 πτ 0 de dη = Boost invariant flow Phys.Rev. D27 (1983) Needs correction! 1 2d 2 t = εad E dn R 2 πτ 0 dη

11 11/27 AN ANALYTIC SOLUTION WITH ACCELERATION The CNC solution in 1 + d dimensions Csörgő, Nagy, Csanád, Phys.Lett. B663 (2008) Nagy, Csörgő, Csanád, Phys.Rev. C77 (2008) τ λd κ+1 v = tanh λη, p = p 0 κ 0 τ σ = σ 0 ν s p p 0 Classes of solutions: κ κ+1, T = T 0 ν s p p 0 cosh η 2 κ κ+1 d 1 φ λ λ d κ φ λ Hwa-Bjorken 1 R R 0 Fixed acceleration, any dim. 2 R d 0 d = 1, κ = 1, any acceleration R Fixed deceleration 1/2 R 1 (κ + 1)/κ Fixed acceleration 3/2 R (4d 1)/3 (κ + 1)/κ

12 12/27 AN ADVANCED ENERGY DENSITY ESTIMATE Fact: dn/dy not flat Finiteness & acceleration Acceleration parameter l Corrections needed: y η & η final η initial Work done by pressure Corrected estimate for κ = 1 ε = ε Bj 2λ 1 τ f τ i λ 1, τ = λτ Bj = λ m T T f R long Björken estimate: only for κ = (dust EoS) Will come back to this soon

13 13/27 THE PSEUDORAPIDITY DENSITY FROM CNC dn dy N 0 cosh α 2 1 y α exp m T f cosh α y α Main parameter: α = 2λ 1 λ 1 Particle mass m, Freeze-out temp. T f Measure acceleration from rapidity distributions Extension to more complex flows?

14 14/27 INITIAL ENERGY DENSITY AT RHIC Bjorken estimate from BRAHMS: ε Bj = Advanced estimate: ε = ε Bj 2λ 1 τ f /τ i λ 1 E dn 5 GeV/fm3 R 2 πτ 0 dη Correction: 2-3x, result ~15 GeV/fm3, QCD agreement! Corresponds to Tini 2Tc 340 MeV, confirmed by g spectra BRAHMS dn/dη

15 15/27 PSEUDORAPITY DENSITIES IN A+A Described well from RHIC to LHC Jiang, Yang, MCs, Csörgő, Phys. Rev. C 97, , 2018

16 16/27 ENERGY DENSITIES IN AA, RHIC TO LHC Effect of acceleration and conjectured effect of Equation of State Jiang, Yang, MCs, Csörgő, Phys. Rev. C 97, , 2018 Effect of EoS: important to understand analytically! What about p+p?

17 17/27 BJORKEN ENERGY DENSITY ESTIMATE IN PP Rough estimate via the Bjorken formula:ε Bj = Number of particles at midrapidity: Average energy: m t = E = GeV Transverse size of the system R 2 π = σ 2 tot /4σ el = 9.8 fm 2 Formation time τ 0 = 1 fm/c (conservative estimate) E dn R 2 πτ 0 dη Energy density from this: ε Bj 7 TeV = 1 R 2 πτ 0 de dη = ε Bj 8 TeV = 1 R 2 πτ 0 de dη = E dn R 2 πτ 0 dη E dn R 2 πτ 0 dη MCs, Csörgő, Jiang, Yang, Universe 3 (2017) no.1, 9,arXiv: = π = π This is at average multiplicity; compare to ε crit 1 GeV fm 3 GeV GeV = fm3 fm 3 GeV GeV fm3 = fm 3

18 18/27 ENERGY DENSITY IN P+P Data from p+p well described 7 TeV: λ = 1.073, ε corr = GeV fm 3 8 TeV: λ = 1.067, ε corr = GeV fm 3 13 TeV: λ = 1.065, ε corr = GeV fm 3 Multiplicity dependence Energy density above 1 GeV/fm 3 for multiplicites above ~10! EoS dependence? MCs, Csörgő, Jiang, Yang, Universe 3 (2017) no.1, 9, arxiv: manuscript in preparation

19 19/27 A NEW CLASS OF EXACT SOLUTIONS How to reconcile the Bjorken estimate (κ = ) with hydro? Search for new solutions with flow u μ = cosh Ω η, sinh Ω η Energy and Euler equations become: η Ω + κ τ τ + tanh Ω η η ln T = 0 η ln T + tanh Ω η τ τ ln T + η Ω = 0 A new class of solutions emerges, if one relaxes self-similarity These will be implicit, introducing η H = Ω H H Ω H = λ λ 1 κ λ atan κ λ tanh H λ 1 Csörgő, Kasza, MCs, Jiang, Universe 2018, 4(6), 69 arxiv:

20 20/27 THE NEW CLASS OF SOLUTIONS u μ = cosh Ω H, sinh Ω H σ τ, H = σ 0 τ 0 τ λ ν s 1 + κ 1 λ 1 sinh2 H λ/2 T τ, H = T 0 τ 0 τ λ κ 1 ν s 1 + κ 1 λ 1 sinh2 H λ/2κ s τ, H = τ 0 τ λ 1 sinh H 1 + κ 1 λ 1 sinh2 H λ/2 Quantities given parametrically as (η H, Ω H ) Simplification: limit the solution in η where η Ω univalent (functional) Not self-similar: Coordinate dependence not only via scaling variable s Explicit and exact solution Csörgő, Kasza, MCs, Jiang, Universe 2018, 4(6), 69 arxiv:

21 21/27 TEMPERATURE EVOLUTION Recall: limited domain in η, where (η H, Ω H ) relation functional Strong dependence on EoS, analytic understanding Csörgő, Kasza, MCs, Jiang, Universe 2018, 4(6), 69 arxiv:

22 22/27 OBSERVABLES Rapidity density calculable in saddle-point approximation dn dy N 0 cosh α(κ) 2 1 y α(1) exp m cosh α κ y T f α 1 1 where α κ = 2λ κ λ κ Normalization: N 0 = R2 πτ f 2πħ 3 3 2πT f m λ(2λ 1) exp was introduced m T f Pseudorapidity density as parametric η y dn y curve dη Using Jacobian: dy = p t y cosh η y dη m 2 + p t y 2 cosh 2 η y dn dy dn dη and p t y = 1+ T f 2 +2mT f α κ 2α 1 T f +m T f +2m y2 Csörgő, Kasza, MCs, Jiang, Universe 2018, 4(6), 69 arxiv:

23 23/27 COMPARISON TO DATA Description valid in limited η interval only Result very close to CNC solution Csörgő, Kasza, MCs, Jiang, Universe 2018, 4(6), 69 arxiv:

24 24/27 WHAT ABOUT THE ENERGY DENSITY? Bjorken estimate: ε Bj = E dn R 2 πτ 0 dη Valid only for dust EoS, κ = CNC solution, finiteness and acceleration (only these effects), valid for κ = 1 Correction factor: 2λ 1 τ f τ 0 λ 1 Work done by pressure (without acceleration, just the expansion) Correction factor: τ f τ 0 CKCJ solution, exact EoS dependent result Correction factor: 2λ 1 τ 1 1+ f κ λ 1 τ 0 Energy density: λ ε = dn dη Csörgő, Kasza, WPCF2018&private comm. E 2λ 1 τ 1 1+ f κ λ 1 R 2 πτ 0 τ 0

25 25/27 DETERMINING THE INITIAL STATE Dependence on EoS: from direct photons and/or lattice QCD Dependence on multiplicity: plug in measured value Dependence on final/initial time: largest source of uncertainty What about the effect of viscosity? Kasza, WPCF2018 & private comm.

26 26/27 A NEW VISCOUS SOLUTION A new analytic solution with bulk viscosity u μ = xμ τ n = n 0 τ 0 τ p = p 0 τ 0 τ T = T 0 τ 0 τ 3 ν s 3 κ+1 κ + ζ κ 3 τ 3 κ + ζ 3 κn 0 τ 3 3 κ+1 τ τ κ 1 Viscous heating at late stages 3 T(s) 3 3 κ+1 κ 1 Note: shear viscosity cancels for Hubble-flow! New shear viscous analytic solutions in preparation Jiang, Yang, Csörgő, Kasza, Nagy, MCs, in preparation

27 27/27 SUMMARY Many new results in exact/analytic hydro Perturbations on top of Hubble-flow Allows to introduce complicated anisotropies To be expanded to other solutions Advanced energy density estimates THANK YOU FOR YOUR ATTENTION! Ceterum censeo Carthaginem esse delendam if you are interested in these subjects, come to: Björken estimate: no acceleration, no pressure Advanced estimates based on exact solutions High energy densities reached in LHC p+p New accelerating families of solutions Arbitrary acceleration, arbitrary EoS EoS dependent energy density estimate

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