Equilibration and decoupling of a relativistic gas in a Friedmann-Robertson-Walker spacetime Juan M. Torres-Rincon (Frankfurt Institute for Advanced Studies) in collaboration with J. Tindall, J.-B. Rosé, and H. Petersen Seminar at Fac. CC Fisicas, Universidad Complutense de Madrid June 14, 2017 Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 1
Outline Introduction: Relativistic heavy-ion collisions SMASH: Transport approach for low-energy RHICs Solutions of the Boltzmann equation in a Friedmann-Robertson-Walker spacetime Results and Comparison Toy model for decoupling/freeze-out Conclusions Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 2
Introduction: Relativistic heavy-ion collisions Quantum Chromodynamics: theory of strong interactions Phase diagram of QCD: Finite temperature and net baryon density RHICs create a system which evolves along the phase diagram Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 3
Introduction: Relativistic heavy-ion collisions 1 Nucleus-nucleus collision 2 Local thermalization (τ i 0.5 fm). Quark-gluon plasma 3 Expansion, cooling down and hadronization (τ 5 fm). Hadronic phase: pions, kaons, protons (,, )... 4 Freeze-out: no more collisions. Frozen spectra until particles are detected (τ f 5 10 fm) 2 3: (viscous) hydrodynamics 3 4: kinetic theory (transport models) Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 4
SMASH transport model SMASH (Simulating Many Accelerated Strongly-interacting Hadrons), new transport code for nonequilibrium hadronic evolution in RHICs It solves the Boltzmann equation: k µ f i(x, k) x µ + m i F µ f i(x, k) k µ = C i coll [f i, f j ] Being developed at FIAS by the group of Prof. Hannah Petersen Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 5
SMASH transport model Different projects carried out by students, Ph.D. candidates, and postdocs e.g. strangeness, dileptons, collectivity signatures, jets... Electromagnetic probes: Photon production Dissipative processes: Transport coefficients with Green-Kubo η = V T dt T xy (0)T xy (t) V : volume of box T : temperature T ij : energy-momentum tensor Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 6
SMASH transport model J. Weil et al. Phys.Rev. C94 (2016) no.5, 054905 Application to RHICs is complicated. Many tests required! (Lorentz invariance, causality, equilibration, detailed balance...) E.g. isolated infinite medium should equilibrate (Boltzmann s H theorem) Test the whole evolution? Known solution of Boltzmann equation? Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 7
Exact solution of the Boltzmann equation Bazow, Denicol, Heinz, Martinez, Noronha, PRL116, 022301 (2016) Bazow, Denicol, Heinz, Martinez, Noronha, PRD94, 125006 (2016) Boltzmann equation in a general metric (no forces F α = 0) µ f (x, k) k x µ + k λ k µ Γ λ f (x, k) µi = C coll [f ] k i Exact solution conserving nonlinear structure of C coll [f ] Homogeneous and isotropic gas of massless particles with constant elastic cross section σ Particular metric: Friedmann-Robertson-Walker spacetime ds 2 = dt 2 a 2 (t)δ ij dx i dx j a(t) (scale factor) fixed by Einstein equations Used in cosmology as a model for the universe expansion with rate H(t) = ȧ(t)/a(t) Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 8
Test for SMASH Bazov et al. PRL116, 022301 (2016) [ exp f (t, k) = λ 0 ] ka κ(τ)t 0 κ 4 (τ) [4κ(τ) 3 + κ(τ) = 1 1 ( 4 exp τ ) 6 ka κ(τ)t 0 (1 κ(τ))], τ = τ(t) Details Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 9
Test for SMASH Bazov et al. PRL116, 022301 (2016) [ exp f (t, k) = λ 0 ] ka κ(τ)t 0 κ 4 (τ) [4κ(τ) 3 + κ(τ) = 1 1 ( 4 exp τ ) 6 ka κ(τ)t 0 (1 κ(τ))], τ = τ(t) Details SMASH implementation: k α k µ x α = Γµ αβ k α k β k 1/a (momentum redshift) Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 9
Results 1.2 1 (k) eq f(t,k) / f 0.8 0.6 0.4 0.2 t=0.1 fm, SMASH t=2.0 fm, SMASH t=4.0 fm, SMASH t=10.0 fm, SMASH 0 0 1 2 3 4 5 6 7 8 k a/t 0 t=0.1 fm, Exact t=2.0 fm, Exact t=4.0 fm, Exact t=10.0 fm, Exact ε(t) = ε(0) n(0), n(t) = a 4 (t) a 3 (t) ) 3 (GeV/fm ), ε 3 n (1/fm 0.3 0.25 0.2 0.15 0.1 0.05 ε(0) = 3λ 0T 4 0 π 2 n(0) = λ 0T 3 0 π 2 n, SMASH n, analytical ε, SMASH ε, analytical 0 0 2 4 6 8 10 t (fm) Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 10
Another example: 2-mode initial condition Initial condition f (t = 0, k) = [ 1 1 10 L(2) 3 ( ka T 0 ) + 1 20 L(2) 4 ( )] ka T 0 e ka T 0 with L (i) j the Laguerre polynomials L (β) n (x) = n i=0 ( 1)i( ) n+β x i n i i! 1.2 1 (k) eq f(t,k) / f 0.8 0.6 0.4 0.2-3 t=10 fm, SMASH t=2.0 fm, SMASH t=4.0 fm, SMASH t=8.0 fm, SMASH -3 t=10 0 0 1 2 3 4 5 6 7 8 k a/t 0 fm, Numerical t=2.0 fm, Numerical t=4.0 fm, Numerical t=8.0 fm, Numerical No exact solution. We solve the Boltzmann equation numerically. Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 11
Summary I SMASH is a new transport approach to simulate low-energy heavy-ion collisions We can use for more academical studies (thermal box, spherical expansion...) We have tested SMASH in a non-trivial metric for massless particles The result coincides with an exact analytical solution of the Boltzmann equation in a FRW metric Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 12
Application: Decoupling of particles Scattering rate Γ = nσ v and Hubble rate H are functions of time Equilibrium reached (and maintained) when Γ H If Γ H particles decouple (lack of interactions). Similar idea for freeze-out in RHICs. Can we exploit our system to study RHICs? H identified with a quantity describing expansion like µ u µ Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 13
Toy model for freeze-out Gas of relativistic massive particles in equilibrium, under a Hubble expansion H = 0.01t (model parameter, not obtained from Friedmann equation). When Γ H [ f eq (t, k) = g ] k 2 (2π) 3 exp + m 2 µ(t) T (t) How to fix T (t) and µ(t)? Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 14
Toy model for freeze-out Gas of relativistic massive particles in equilibrium, under a Hubble expansion H = 0.01t (model parameter, not obtained from Friedmann equation). When Γ H [ f eq (t, k) = g ] k 2 (2π) 3 exp + m 2 µ(t) T (t) How to fix T (t) and µ(t)? N V (t) = N V 0 ( a0 a(t) ) 3, ( µ ) T exp K 2 (m/t ) = T 0 exp T S V (t) = S ( ) 3 a0 V 0 a(t) ( µ0 T 0 ) ( a0 ) 3 K 2 (m/t 0 ) a µ(t ) m = µ 0 T + K 1(m/T ) m T 0 K 2 (m/t ) T K 1 (m/t 0 ) T 0 K 2 (m/t 0 ) Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 14
Equilibrium for ultra- and nonrelativistic particles Ultrarelativistic (m T ) and nonrelativistic (m T ) limits: a T 0 0 a(t) for m T, T (t) = ( 2 T a0 0 a(t)) for m T, which are well-known results. Ultrarelativistic particles E k T T 1/a Nonrelativistic particles E = k 2 /2m T T 1/a 2 Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 15
Time evolution of the particle distribution f eq (t, k) = g k 2 +m 2 µ(t) (2π) 3 e T (t) t = 0.1 fm t = 2.5 fm t = 5 fm dn/dp (1/GeV) 14000 12000 10000 8000 6000 4000 Equilibrium SMASH dn/dp (1/GeV) 14000 Equilibrium SMASH 12000 10000 8000 6000 4000 dn/dp (1/GeV) 16000 14000 12000 10000 8000 6000 4000 Equilibrium SMASH 2000 2000 2000 dn/dp (1/GeV) 25000 20000 15000 10000 5000 0 0 1 2 3 4 5 k (GeV) Equilibrium SMASH 0 0 0.5 1 1.5 2 2.5 3 k (GeV) dn/dp (1/GeV) 0 0 1 2 3 4 5 k (GeV) 45000 40000 35000 30000 25000 20000 15000 10000 5000 Equilibrium SMASH 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 k (GeV) t = 10 fm t = 15 fm t = 20 fm dn/dp (1/GeV) 100 0 0 1 2 3 4 5 k (GeV) 80 60 40 20 3 10 Equilibrium SMASH 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 k (GeV) Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 16
What happened between t = 5 fm and t = 10 fm? Absence of collisions equilibrium cannot be maintained. Non-equilibrium distribution! Distribution after decoupling? For any t > t D ( f dec (t, k) = f eq t D, k a ) a D exp [ ] (ka/ad ) 2 + m 2 T D UR and NR limits Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 17
Freeze-out time Fit at t = 20 fm to [ ] k 2 (a(t)/a D (t D )) f dec (t, k) exp 2 + m 2 T D (t D ) (in fact, only one independent parameter) dn/dp (1/GeV) 100 80 60 40 20 3 10 Fit (non-equilibrium) 0.9 fm SMASH t D = 5.5 +0.8 t D = 5.2 +0.8 0.9 fm t D = 5.3±0.6 fm 0 0 0.1 0.2 0.3 0.4 0.5 0.6 k (GeV) Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 18
Universal vs sequential freeze-out Chemical freeze-out occurs when particle multiplicities are fixed particle ratio -1 10-2 10 Pb-Pb s NN =2.76 TeV Preliminary -3 10 Data: ALICE, 0-20% (preliminary) Model calc. with parameters: T=148 MeV, ( = 1 MeV fixed) b T=164 MeV, = 1 MeV b -4 10 K + / + - - + K / p/ - - + p / / + - - + / / + - / ALICE Collaboration Strange baryons seem to have a larger chemical freeze-out temperature. Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 19
Toy model for freeze-out Simple argument for a sequential freeze-out in our model: 10 10 8 8 t D (fm) 6 4 t D (fm) 6 4 2 2 0 0 1 2 3 4 5 6 m (GeV) 0 20 40 60 80 100 σ (mbarn) Things to improve: Hadronic mixture + physical interactions Modelize a(t) for physical RHICs e.g. Blast-wave model T D for different species Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 20
Freeze-out condition Simple models for RHICs assumed freeze-out at Γ = H e.g. with H = µ u µ. More refined studies adopted ξγ = H with ξ = O(1), but unknown for a long time. We obtain ξ 0.4 ξ 1 for m 0 Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 21
Summary II 1 The FRW spacetime admits a well-defined decoupling process. Straightforward extraction of freeze-out times 2 We obtain indications of a natural sequential decoupling 3 A more refined model can provide insights on the freeze-out mechanism in RHICs 4 Freeze-out occurs systematically earlier than expected from the condition Γ = H Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 22
Equilibration and decoupling of a relativistic gas in a Friedmann-Robertson-Walker spacetime Juan M. Torres-Rincon (Frankfurt Institute for Advanced Studies) in collaboration with J. Tindall, J.-B. Rosé, and H. Petersen Seminar at Fac. CC Fisicas, Universidad Complutense de Madrid June 14, 2017 Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 23
Backup slides Backup slides Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 24
Exact Solution Bazow et al. PRL116, 022301 (2016) & Bazow et al. PRD94, 125006 (2016) Initial condition f (t = 0, k) = 256 ( 243 λ ka 0 exp 4 ) ka T 0 3 T 0 Exact Solution [ exp f (t, k) = λ 0 ] ka κ(τ)t 0 κ 4 (τ) κ(τ) = 1 1 ( 4 exp τ ) 6 [4κ(τ) 3 + ka (1 κ(τ))] κ(τ)t 0 [, τ = 2 ( 1 1 + 0.1 t 0.1 σn 0 ) 1/2 ] σ: total cross section, λ 0 : fugacity, T 0 : initial temperature, n 0 : initial number density Go back Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 25
Decoupling of particles Cosmology Part III: Mathematical Tripos. lectures by D. Baumann http://www.damtp.cam.ac.uk/user/db275/cosmology/lectures.pdf Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 26
Decoupling of ultrarelativistic and nonrelativistic particles Distribution after decoupling ( f dec (t, k) = f eq t D, k a ) [ ] (ka/ad ) exp 2 + m 2 a D T D Ultrarelativistic (m T ) and nonrelativistic (m T ) limits: ( ) exp ka T f dec (t, k) D a D for m T ( ) exp k 2 2m for m T. a 2 T D a 2 D Invisible freeze-out Single species of massless/nonrelativistic particles: decoupled distribution looks like equilibrium Go back Juan M. Torres-Rincon (FIAS) Equilibration and decoupling of a relativistic gas in a FRW spacetime 27