Nonlinear fluctuation spectra for viscous dark matter and heavy ions

Transcription

1 Nonlinear fluctuation spectra for viscous dark matter and heavy ions Nikolaos Tetradis University of Athens N Tetradis University of Athens

2 Motivation The framework Backreaction of perturbations Effective viscosity Nonlinear spectrum Other applications Conclusions Figure: 2MASS Galaxy Catalog (more than 15 million galaxies) N Tetradis University of Athens

3 Figure: 2MASS Galaxy Catalog: Various redshifts N Tetradis University of Athens

4 Sloan Digital Sky Survey (2005) Figure: Galaxy correlation function N Tetradis University of Athens

5 Figure: Linear matter power spectrum N Tetradis University of Athens

6 Figure: Matter power spectrum in the range of Baryon Acoustic Oscillations (BAO) N Tetradis University of Athens

7 Standard framework Basic assumptions: Homogeneity and isotropy of the background: ds 2 = dt 2 + a 2 (t)δ ij dx i dx j (ȧ a ) 2 = 8πG 3 ρ i ä a = 4πG 3 (ρ i + 3p i ) i i Inhomogeneities are treated as small perturbations of this background Under gravitational collapse, the overdensities grow and produce the observed Large Scale Structure The rate of cosmological expansion is currently increasing: Distant supernovae Cosmic microwave background Power spectrum of the galaxy distribution Leading paradgim (ΛCDM): Dominant contributions (not observed directly) to the energy content of the Universe: 1 Dark matter: p = 0 ( 25%) 2 Cosmological constant (Dark energy): p < 0 ( 70%) N Tetradis University of Athens

8 Link with observations Study the formation of structure in the distribution of dark (and baryonic) matter Dark energy does not cluster The evolution of inhomogeneities depends on the cosmological background The matter spectrum at various redshifts reflects the detailed structure of the cosmological model Comparison with observations of the galaxy distribution can differentiate between models Baryon acoustic oscillations: 100 Mpc range Analytical calculation of the matter spectrum beyond the linear level (Crocce, Scoccimarro 2005) N Tetradis University of Athens

9 Perturbation theory 100 Plin 1-loop 2-loop 3-loop k ref =k 3-loop log measure P(k,z=0) [(h/mpc) -3 ] k [h/mpc] Figure: Matter power spectrum in Standard Perturbation Theory (SPT) (Blas,Garny,Konstandin 2014) N Tetradis University of Athens

10 Outline Covariant formulation of the fluid description of dark matter Viscous dark matter Backreaction of shear viscosity on the average expansion Effective viscosity from the short-distance modes The nonlinear power spectrum in an effective viscous theory Power spectrum of density fluctuations in heavy-ion collisions Conclusions N Tetradis University of Athens

11 Assumptions Single-stream approximation (no shell crossing) Smoothing scale: Kinetic theory fluid description ΛCDM Irrotational flow (no vorticity) Ignore the baryons and focus on dark matter N Tetradis University of Athens

12 Covariant hydrodynamic description Work within the first-order formalism, allowing for nonzero viscosity Energy-momentum tensor: T µν = ρu µ u ν + (p + π b ) µν + π µν (1) ρ: energy density p: pressure in the fluid rest frame π b : bulk viscosity π µν : shear viscosity, satisfying: u µ π µν = π µ µ = 0 µν projector orthogonal to the fluid velocity: µν = g µν + u µ u ν New elements: Bulk viscosity: π b = ζ ρ u ρ Shear viscosity tensor: ( 1 π µν = 2ησ µν = 2η 2 ( µρ ρ u ν + νρ ρ u µ ) 1 ) 3 µν ( ρ u ρ ) N Tetradis University of Athens

13 Dynamical equations Einstein equations: G µν = 8πG N T µν, Conservation of the energy momentum tensor ( ν T µν = 0): u µ µ ρ + (ρ + p) µ u µ ζ ( µ u µ ) 2 2ησ µν σ µν = 0 (ρ + p + π b )u µ µ u ρ + ρµ µ (p + π b ) + ρ ν µ π µν = 0 N Tetradis University of Athens

14 Subhorizon scales Ansatz for the metric: ds 2 = a 2 (τ) [ (1 + 2Ψ(τ, x)) dτ 2 + (1 2Φ(τ, x)) dx dx ] The potentials Φ and Ψ are weak Their difference is governed by the shear viscosity We can take Φ Ψ 1 The four-velocity u µ = dx µ / ds 2 can be expressed through the coordinate velocity v i = dx i /dτ and the potentials Φ and Ψ: u µ 1 = a (1, v) 1 + 2Ψ (1 2Φ) v 2 N Tetradis University of Athens

15 Continuity equation: ρ + v ρ ( + (ρ + p) 3 ȧ a + ) v [ = ζ a 3 ȧ a + ] 2 v + η a [ i v j i v j + i v j j v i 23 ( v) 2] Take the spatial average: 1 a ρ + 3H ( ρ + p 3 ζh) = D, with Hubble parameter H = ȧ/a 2 and D = 1 a 2 η [ i v j i v j + i v j j v i 2 3 iv i j v j ] + 1 a 2 ζ[ v] a v (p 6ζH) The average of the trace of Einstein s equations R = 8πG N T µ µ reads: ä a 3 = 1 aḣ + 2H2 = 4πG N ( ρ ) 3 p + 9 ζh 3 N Tetradis University of Athens

16 Backreaction S Floerchinger, N T, U Wiedemann arxiv: [gr-qc], Phys Rev Lett 114: 9, (2015) Ignore bulk viscosity: ζ = 0 Define p = ŵ ρ and the deceleration parameter q = 1 Ḣ/(aH2 ) One finds: For dq d ln a + 2(q 1) ( q 1 2 (1 + 3ŵ)) = 4πG ND(1 3ŵ) 3H 3 4πG N D 3H 3 > 1 + 3ŵ 1 3ŵ there is a fixed point with acceleration: q < 0 N Tetradis University of Athens

17 deceleration parameter q dq d ln a + 4 G ND 3H 3 Figure: Graphical representation of the evolution equation for the deceleration parameter q for vanishing pressure, ŵ = 0 N Tetradis University of Athens

18 Estimate of backreaction Assume that typical fluid-velocity gradients are of order H: η i v j i v j + i v j j v i 2 3 iv i j v j /a 2 ηh 2 This corresponds to realistic peculiar velocities of the order of 100 km/s on distances of 1 Mpc A large shear viscosity arises in systems of particles with a long mean free path (eg an additional component to CDM): η ρ R τ R ρ R : energy density of these particles, τ R : mean free time Accelerating expansion would result for 4πG N D 3H 3 ρ Rτ R H 2ρ cr 1 For τ R < 1/H one needs ρ R ρ cr Realistic model, consistent with large-scale structure? Difficult, but worth pursuing N Tetradis University of Athens

19 Convergence of perturbation theory 100 linear one-loop two-loop 10 P / Mpc 3 1 0,1 0,001 0,01 0, k / Mpc -1 Figure: Linear power spectrum and the one- and two-loop corrections in standard perturbation theory (SPT) at z = 0 N Tetradis University of Athens

20 Loop corrections dominate for k > h/mpc The theory becomes strongly coupled for k 1 h/mpc The deep UV region is out of the reach of perturbation theory For large k, higher loops involve powers of the dimensionless variance of the linear density field σ 2 l = 4π km 0 dq q 2 P L (q), with k m the UV cutoff and P L (k) the linear density spectrum For k m 05 h/mpc we have (4π/3) kmp 3 L (k m ) 1 For small k, the one-loop correction depends on the dimensionful scale σd 2 = 4π dq P L (q) 3 0 The complete expression is (Blas, Garny, Konstandin 2013) ) P 1 loop ab (k) = k 2 σd 2 PL (k) ( N Tetradis University of Athens

21 Effective description D Blas, S Floerchinger, M Garny, N T, U Wiedemann work in progress Introduce an effective low-energy description in terms of an imperfect fluid (Baumann, Nicolis, Senatore, Zaldarriaga 2010, Carrasco, Foreman, Green, Senatore 2014) Use the Wilsonian approach: Integrate out the modes with k > k m and replace them with effective couplings (viscosity, pressure) in the low-k theory For a spectrum that scales as 1/k 3, define σ 2 dm (η) = 4π 3 k m dq P L (q, η) 1 2 [ ] 4π D 3 PL (k m, 0) km 3 2 L (η) km 2, with η = ln a = ln(1 + z) and D L (η) the linear growth factor The deep UV does not contribute significantly N Tetradis University of Athens

22 One finds η δp ab (k, η) = 2 ( k 2 km 2 P L (k, η) ) [ ] [ ] 4π 3 PL (k m, 0) km 3 dd L (η) D L dη This is equivalent to the contribution of effective pressure and viscosity terms to the linear spectrum N Tetradis University of Athens

23 Parameters of the effective theory ρ(τ, x) = ρ 0 (τ) + δρ(τ, x) p(τ, x) = 0 + δp(τ, x) δ = δρ ρ 0 θ = v H = ȧ a H = 1 a H cs 2 (τ) = δp δρ = α H 2 s ν(τ) = η ρ 0 a = 3 4 α H ν with α s, α ν = O(1) k 2 m k 2 m N Tetradis University of Athens

24 We rely on a hierarchy supported by linear perturbation theory for subhorizon perturbations 1 We treat δ and θ/h as quantities of order 1, v as a quantity of order H/k and Ψ, Φ as quantities of order H 2 /k 2 2 We assume that a time derivative is equivalent to a factor of H, while a spatial derivative to a factor of k 3 We assume that c 2 s, νh are of order H 2 /k 2 m Keeping the dominant terms, we obtain δ + v + ( v )δ + δ v = 0 v + H v + ( v ) v + Φ + cs 2 (1 δ) δ ( ν(1 δ) 2 v + 1 ) ( 3 v) = 0 2 Φ = 3 2 Ω mh 2 δ N Tetradis University of Athens

25 Use Fourier-transformed quantities to obtain δ k + θ k + d 3 p d 3 q δ(k p q) α 1 (p, q) δ p θ q = 0 θ k + (H + 43 ) ( ) 3 νk 2 θ k + 2 Ω mh 2 cs 2 k 2 δ k + d 3 p d 3 q δ(k p q) ) (β 1 (p, q) δ p δ q + β 2 (p, q) θ p θ q + β 3 (p, q) δ p θ q = 0, with α 1 (p, q) = (p + q)q q 2 β 1 (p, q) = c 2 s (p + q)q β 2 (p, q) = (p + q)2 p q 2p 2 q 2 β 3 (p, q) = 4 3 ν(p + q)q N Tetradis University of Athens

26 Define the doublet ϕ 1 (η, k) ϕ 2 (η, k) = δ k (τ) θ k(τ) H, where η = ln a(τ) The evolution equations take the form η ϕ a (k) = Ω ab (k, η)ϕ b (k)+ d 3 p d 3 q δ(k p q)γ abc (p, q, η) ϕ b (p) ϕ c (q), where Ω(k, η) = ( 3 2 Ω m + α s k 2 k 2 m H H + α ν k 2 k 2 m ) and a prime denotes a derivative with respect to η The nonzero elements of γ abc are expressed in terms of α 1 (p, q), β 1 (p, q), β 2 (p, q), β 3 (p, q) N Tetradis University of Athens

27 The linearized part of the evolution equations has the form ( ) η ϕ a (k) = Ω L (k, η) + δω(k, η) ϕ b (k), ( Ω L (k, η) = Ω m 1 + H H ) δω(k, η) = ( α s k 2 k 2 m We can treat the term δω as a perturbation to the leading linear evolution in the absence of pressure and viscosity Define ϕ a = ϕ L a + δϕ a Then 0 0 α ν k 2 k 2 m ) η δϕ a (k, η) = Ω L ab (k, η) δϕ b(k, η) δω ab (k, η) ϕ L b (k, η) For Ω L ab (k, η) growing with η, the second term is the dominant one The first one gives small corrections N Tetradis University of Athens

28 The correction to the ideal-dm spectrum becomes η δp ab (k, η) = k 2 ( 0 α ν Pδδ L + α spδθ L α ν Pδδ L + α spδθ L 2α ν Pδθ L + 2α spθθ L k 2 m ) Keeping properly the growing mode gives η δp ab (k, η) = k 2 ( 0 α ν + α s /f α ν + α s /f 2 (α ν + α s /f ) k 2 m ) P L (k, η), with f = 1 D L d D L dη Assuming comparable α ν, α s /f and comparing with the one-loop UV contribution in the ideal-dm theory, gives [ ] [ ] 4π α ν (η) α s (η)/f 3 k m 3 P L dd L (η) (k m, 0) D L 1 DL 2 dη f N Tetradis University of Athens

29 The evolution of the spectrum Define the spectra, bispectra and trispectra as φ a (k, η)φ b (q, η) δ D (k + q)p ab (k, η) φ a (k, η)φ b (q, η)φ c (p, η) δ D (k + q + p)b abc (k, q, p, η) φ a (k, η)φ b (q, η)φ c (p, η)φ d (r, η) δ D (k + q)δ D (p + r)p ab (k, η)p cd (p, η) +δ D (k + p)δ D (q + r)p ac (k, η)p bd (q, η) +δ D (k + r)δ D (q + p)p ad (k, η)p bc (q, η) +δ D (k + p + q + r)q abcd (k, p, q, r, η) Essential approximation: Neglect the effect of the trispectrum on the evolution of the bispectrum N Tetradis University of Athens

30 In this way we obtain (Pietroni 2008) η P ab (k, η) = Ω ac P cb (k, η) Ω bc P ac (k, η) + d 3 q [ γ acd (k, q, q k)b bcd (k, q, q k) +γ bcd (k, q, q k)b acd (k, q, q k) ], η B abc (k, q, q k) = Ω ad B dbc (k, q, q k) Ω bd B adc (k, q, q k) Ω cd B abd (k, q, q k) +2 d 3 q [ γ ade (k, q, q k)p db (q, η)p ec (k q, η) +γ bde ( q, q k, k)p dc (k q, η)p ea (k, η) +γ cde (q k, k, q)p da (k, η)p eb (q, η) ] Solve with the effective couplings cs 2, νh and a UV cutoff k m in the momentum integrations N Tetradis University of Athens

31 P k,z 0, Α Ν D 2 f 1, Α s D 2 f 2 1, k m 04h Mpc 14 P k,z P lin k,z k h Mpc Figure: One-loop spectrum in the effective theory at z = 0 N Tetradis University of Athens

32 P k,z 0375, Α Ν D 2 f 1, Α s D 2 f 2 1, k m 04h Mpc 14 P k,z P lin k,z k h Mpc Figure: One-loop spectrum in the effective theory at z = 0375 N Tetradis University of Athens

33 P k,z 0833, Α Ν D 2 f 1, Α s D 2 f 2 1, k m 04h Mpc P k,z P lin k,z k h Mpc Figure: One-loop spectrum in the effective theory at z = 0833 N Tetradis University of Athens

34 Other applications Coupled quintessence F Saracco, M Pietroni, N T, V Pettorino, G Robbers arxiv: [astro-ph], Phys Rev D 82: (2010) Variable equation of state N Brouzakis, N T arxiv: [astro-ph], JCAP 1101: 024 (2011) Growing neutrino quintessence N Brouzakis, V Pettorino, N T, C Wetterich arxiv: [astro-ph], JCAP 1103: 047 (2011) Heavy-ion collisions S Floerchinger, N T, U Wiedemann arxiv: [hep-ph], Phys Rev D 91: 6, (2015) N Tetradis University of Athens

35 Bjorken model of heavy-ion collisions Bjorken 1983: In ultra-relativistic nuclear collisions particle production is almost flat in rapidity y = arctanh(x 3 / x 0 ) The evolution preserves this independence Use coordinates (τ, x 1, x 2, y) with proper time: τ = x0 2 x 3 2 Metric: g µν = diag( 1, 1, 1, τ 2 ) Assume translational and rotational invariance in the transverse plane This yields the one-dimensional Bjorken model with τ ϵ + w ( 1 4ν ) = 0, τ 3τ w = ϵ + p = st (enthalpy), ν = η/w (kinematic viscosity) For ν/τ 1 and and ϵ = 3p we have ϵ Bj (τ) (τ 0 /τ) 4/3 For normalized viscosity η/s 1/(4π) (Kovtun,Son,Starinets 2005) and T Bj (τ 0 ) 1/fm at τ 0 1 fm ν Bj (τ) τ ( η ) = s 1 ( τ0 T Bj (τ 0 ) τ 0 τ ) 2/3 1 N Tetradis University of Athens

36 Fields and parameters Rescaled time: t = 3 τ 4/3 4 τ 1/3 0 Energy density or temperature fluctuation: ( ) T d ln = 14 ( T Bj (τ) ln 1 + δϵ ϵ Bj ) Fluid velocity: u µ = (u τ, u 1, u 2, u y ), with u µ u µ = 1 Rescaling of the fluid dynamic fields: δϵ 4 ϵ Bj (2) ( ) 1/3 τ u j = v j, d τ τ 0 Further approximations: ( τ0 ) 2/3 d, ν0 = ν τ ( τ0 ) 1/3, (3) τ 1 Neglect fluctuations in the y-direction (suppressed by 1/τ) 2 Irrotational flow: v m = mf, with m = 1, 2 3 Statistical homogeneity and isotropy in the transverse plane: Meaningful for distances smaller than the typical transverse extension of a heavy-ion collision ( 8 fm) N Tetradis University of Athens

37 Equations of motion d k = 1 2t d ( τ0 k + 12t + ) 1 2 k 2 fk + d 2 p d 2 q δ(k q p) α 1 (p, q) d p fq d 2 p d 2 q δ(k q p) α 2 (p, q) f p fq, d 2 p d 2 q δ(k q p) β(p, q) f p fq, f k = 4ν 0 3 k 2 fk d k + where α 1 (p, q) = pq, α 2 (p, q) = 2ν 0 ((pq) 2 13 ) 3 p2 q 2, β(p, q) = 1 (p + q) 2 [ (pq) p2 q (pq) ( q 2 + p 2)] N Tetradis University of Athens

38 Define the combined field ϕ a (t, k), a = 1, 2 with ( ) ( ) ϕ1 dk ϕ a (t, k) =, fk and the matrix Ω(k, t) = ( 1 ϕ 2 2t ( τ 0 12t 1 4ν 0 3 k 2 ) 1 2 k 2 Then ϕ a (k) = Ω ab (k)ϕ b (k)+ d 2 p d 2 q δ(k p q)γ abc (p, q)ϕ b (p)ϕ c (q), ) with γ abc γ 121 (p, q) = γ 112 (p, q) = α 1(p, q), 2 γ 122 (p, q) = α 2 (p, q), γ 222 (p, q) = β(p, q) N Tetradis University of Athens

39 Expectation values and spectra Expectation value: ϕ a (k) = ϕ a δ(k) Spectrum: ϕ a (k)ϕ b (p) = P ab (k) δ(k + p) + ϕ a ϕ b δ(k)δ(p) Bispectrum: ϕ a (k)ϕ b (p)ϕ c (q) =B abc (k, p, q) δ(k + p + q) Trispectrum: + ϕ a P bc (p) δ(k)δ(p + q) [3 perm] + ϕ a ϕb ϕc δ(k)δ(p)δ(q) ϕ a (k)ϕ b (p)ϕ c (q)ϕ d (r) = Q abcd (k, p, q, r) δ(k + p + q + r) + ϕ a B bcd (p, q, r) δ(k)δ(p + q + r) [4 perm] +P ab (k)p cd (q) δ(k + p)δ(q + r) [3 perm] + ϕ a ϕ b P ab (q) δ(k)δ(p)δ(q + r) [6 perm] + ϕ a ϕ b ϕ c ϕ d δ(k)δ(p)δ(q)δ(r) We set: Q abcd (k, p, q, r) = 0 N Tetradis University of Athens

40 Evolution equations t ϕ a = Ω ab (0) ϕ b + γ abc (0, 0) ϕ b ϕ c + d 2 q γ abc ( q, q)p bc (q) t P ab (k) = Ω ac (k)p cb (k) Ω bc ( k)p ac (k) + d 2 p [ γ acd (p, k p)b bcd ( k, p, k p) +γ bcd (p, k p)b acd (k, p, k p) ] t B abc (k, p, q) k+p+q=0 = Ω ad (k)b dbc (k, p, q) Ω bd (p)b adc (k, p, q) Ω cd (q)b abd (k, p, q) +2 [ γ ade ( p, q)p bd (p)p ce (q) + γ bde ( q, k)p +γ cde ( k, p)p ad (k)p be (p) ] Ω ab (k) = Ω ab (k) γ acb (0, k) ϕ c γ abc (k, 0) ϕ c N Tetradis University of Athens

41 Solution in closed form dt dt t ϕ a (t) = d 2 q g ab (0, t, t )γ bcd ( q, q)p cd (q, t ) t 0 P ab (k, t) = g ac (k, t, t 0 ) g bd (k, t, t 0 )P cd (k, t 0 ) t + d 2 q g ae (k, t, t )g bf (k, t, t ) t 0 [γ ecd ( q, q k) B fcd (k, q, q k; t ) + γ fcd ( q, q k) B ecd (k, q, q k; t )] B = F [g, γ P P, B] Linear propagator: ϕ L a(k, t) = g ab (k, t, t )ϕ L b (k, t ) It satisfies t g ab (k, t, t ) = Ω ac (k, t)g cb (k, t, t ), g ab (k, t, t) = δ ab N Tetradis University of Athens

42 Initial conditions The transverse enthalpy density w is assumed to originate from localized sources at positions independently and uniformly distributed in the transverse plane (Glauber model) For a single event, the density field d is d(x) = A 4N N f (x x j ) j=1 N: number of sources A: transverse area By construction: d(x) = 0 In practice: f (k) = 1 e σ 2 k 2 (2π) 2 2 P d (k) = 1 A 2 (2π) 2 16N e σ k 2 B d (k 1, k 2, k 3 ) = 1 A 2 σ 2 (2π) 4 64N 2 e 2 (k 2 1 +k 2 2 +k 3) 2 N Tetradis University of Athens

43 Figure: The density-density correlator P 11 (k) at O(γ 0 ) (blue, dashed line) and O(γ 1 ) (purple, solid line) The evolution was initialized at τ 0 = 1 fm, for an initial kinematic viscosity ν 0 = 004 fm corresponding to η/s = 1/(4π) and an initial temperature T MeV The initial number density of sources was N/A = 02 fm 2 Sources are chosen with a Gaussian profile of width σ = 04 fm, that amounts to cut-off initial fluctuations at momentum scales above 3 fm 1 Results are shown for times t = 15, 25, 35, 45 fm (curves from top right to bottom left) (All quantities measured in fm) N Tetradis University of Athens

44 B 111 k,k,k k Figure: B 111 (k, k, k) at O(γ 0 ) (blue, dashed line) and O(γ 1 ) (purple, solid line) for τ 0 = 1 fm, ν 0 = 004 fm, σ = 04 fm at t = 75 fm (All quantities measured in fm) N Tetradis University of Athens

45 P 11 0 nonlinear P 11 0 linear t Figure: The ratio P 11 (0) nonlinear /P 11 (0) linear at order O(γ 1 ) (purple, solid line) and O(γ 2 ) (green, dot-dashed line) as a function of t, for τ 0 = 1 fm, ν 0 = 004 fm, σ = 04 fm (All quantities measured in fm) N Tetradis University of Athens

46 d Τ t Figure: Backreaction of the spectrum on the expectation value of the density field, for τ 0 = 1 fm, ν 0 = 004 fm, σ = 04 fm (All quantities measured in fm) N Tetradis University of Athens

47 Conclusions The nature of dark matter is still unknown It is reasonable to consider possibilities beyond an ideal, pressureless fluid The first-order formalism includes bulk and shear viscosities In the presence of significant shear viscosity, the fluctuations of the cosmological fluid can backreact on the average energy density The backreaction can accelerate the cosmological expansion The growth of large-scale structure is also affected The short-distance cosmological perturbations cannot be described reliably through perturbative methods It is possible to integrate them out in order to obtain an effective description of the long-distance modes This description must allow for nonzero speed of sound and viscosity The form and time-dependence of the effective terms can be determined with reasonable precision The nonlinear spectrum computed through the effective theory is in good agreement with results from N-body simulations Perturbation theory seems to converge quickly for the effective theory if the UV cutoff is taken around 05 h/ Mpc N Tetradis University of Athens

48 We found numerical evidence in support of the assumption that a perturbative expansion applies to the calculation of the spectra and bispectra of fluctuations of realistic size in the quark-gluon plasma produced in heavy-ion collisions The calculated two- and three-point functions of fluid dynamic fields are not directly measurable in heavy-ion collisions A hadronization prescription is needed The measurable momentum correlations of hadrons provide only convoluted information about spatial correlators The calculated spectra and bispectra provide information about how an initial spectrum of fluctuations evolves as a function of time and how perturbations dissipate (ie weaken in strength) Any attempt of constraining the scale and nature of fluctuations at the initial time τ 0 via experimental measurements of fluctuations at freeze-out relies on understanding the dynamic evolution of the scale and size of fluctuations N Tetradis University of Athens