Linearized nonequilibrium dynamics in nonconformal plasma
Romuald A. Janik, Grzegorz Plewa, Hesam Soltanpanahi, Michal Spalinski
LLinearized nonequilibrium dynamics in nonconformal plasma
Romuald A. Janik ∗ , Grzegorz Plewa † , Hesam Soltanpanahi ‡ , and Micha(cid:32)lSpali´nski § Institute of Physics, Jagiellonian University, ul. (cid:32)Lojasiewicza 11, 30-348 Krak´ow, Poland National Center for Nuclear Research, ul. Ho˙za 69, 00-681 Warsaw, Poland Physics Department, University of Bia(cid:32)lystok, ul. Lipowa 41, 15-424 Bia(cid:32)lystok, Poland
Abstract
We investigate the behaviour of the lowest nonhydrodynamic modes in a classof holographic models which exhibit an equation of state closely mimicking theone determined from lattice QCD. We compute the lowest quasinormal modefrequencies for a range of scalar self-interaction potentials and find that thedamping of the quasinormal modes at the phase transition/crossover falls off bya factor of around two from conformality after factoring out standard conformaltemperature dependence. The damping encoded in the imaginary part of thefrequencies turns out to be correlated with the speed of sound and is basicallyindependent of the UV details of the model. We also find that the dynamicsof the nonhydrodynamic degrees of freedom remains ultralocal, even to a higherdegree, as we deviate from conformality. These results indicate that the role ofnonhydrodynamic degrees of freedom in the vicinity of the crossover transitionmay be enhanced. ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] § Email: [email protected] a r X i v : . [ h e p - t h ] M a r ontents The quark-gluon plasma produced in relativistic heavy-ion collisions at RHIC andLHC is very successfully described by phenomenological hydrodynamic models [1].Nevertheless, it is quite clear that the hydrodynamic description is applicable only froma certain initial time after an inherently nonequilibrium initial phase of expansion. Theestimation of this so-called ‘thermalization time’ has been at the focal point of muchtheoretical effort.The main fundamental difficulty in studying such a problem from first principles isthat when the plasma is strongly coupled we need a method which would be at thesame time nonperturbative and which would work directly in Minkowski signature.2or these reasons, the methods of the AdS/CFT correspondence have been applied inthis context although predominantly for the case of the conformal N = 4 SYM theory[2, 3]. Despite that, the results obtained within this framework are very encouraging.Firstly, the appearance of a hydrodynamical description of the resulting plasma systemis not an input but rather a result from a much more general dual gravitational de-scription which incorporates genuine nonhydrodynamical degrees of freedom. Thus onecan study the transition to hydrodynamics and its properties. Secondly, it has turnedout that at the transition to hydrodynamics, the components of the energy-momentumtensor in the local rest frame are still significantly anisotropic, meaning that the plasmais then still quite far from thermal equilibrium which indicates that the phrase ‘earlythermalization’ used in this context is really a misnomer. Thirdly, for numerous initialconditions, the plasma behaves hydrodynamically with very good accuracy when thedimensionless product of the proper time and temperature T τ ∼ . − . are typically very important.It is worth mentioning also numerous important results on the dynamics of shock wavecollisions [6, 7, 8] which we do not describe in more detail here.Since the full nonlinear dynamics in the deeply nonequilibrium regime is very compli-cated as it is described by higher dimensional Einstein’s equations and can be studiedessentially only using the methods of numerical general relativity, it was suggested in[9, 10] that it may be useful to incorporate just the lowest, least damped nonhydrody-namic degrees of freedom into the commonly used nonlinear hydrodynamic description.On the dual gravity side these degrees of freedom are the so-called quasinormal modes(QNM) of a finite temperature black hole. The 4D equations involving these nonequi-librium modes proposed in [11] take as an input from the gravitational descriptiononly their real and imaginary frequencies ω QNM = ± ω R + iω I . Moreover, it turns outthat the dependence of these frequencies on the momentum k is very mild and can beneglected in a first approximation. This property leads to a certain ‘ultralocality’ ofthe dynamics of the nonequilibrium modes on top of a hydrodynamic flow.All the above investigations were performed in the context of the conformal N = 4SYM theory which, by its very definition does not exhibit any kind of phase transi-tion or crossover behaviour. It is thus very interesting to study what modificationsto the above picture appear for a nonconformal theory. There are various ways to In the present paper we use this term to denote all nonhydrodynamical degrees of freedom in theplasma. .In this paper we will concentrate on the dynamics of the lowest nonhydrodynamicdegrees of freedom in the nonconformal setting. In particular we will investigate howthe damping of these modes changes when we approach the crossover temperature T c (defined more precisely later). This answers an important question whether the roleof nonhydrodynamic degrees of freedom becomes more important or less importantcloser to the phase transition. Secondly, we will investigate whether the ‘ultralocality’property observed for N = 4 SYM nonequilibrium degrees of freedom (QNM) stillholds in the nonconformal case, especially close to the crossover/phase transition.The plan of this paper is as follows. In section 2 we will review the family of gravita-tional backgrounds that we will consider. In sections 3 and 4 we will give some detailson their explicit numerical construction and on our procedure for finding quasinormalmode frequencies. As far as we know such a procedure has not been employed so farin the literature and may be useful also in other contexts. There we will also discussthe relation of scalar QNM with the metric ones for these backgrounds and proceed,in section 5, to describe our results. We close the paper with conclusions. In this paper we will study a family of black hole backgrounds which follow from anaction of gravity coupled to a single scalar field with a specific self interaction potential: S = (cid:90) d x √ g (cid:20) R −
12 ( ∂φ ) + V ( φ ) (cid:21) (1)This family has been introduced by Gubser and collaborators in a series of papers[12, 13, 14] and used to mimic the QCD equation of state by a judicious choice of thepotential. These backgrounds were then used to study bulk viscosity (which identically A paper investigating the complementary top-down approach appeared simultaneously [20]. c s for the potentials V − V correspond to deformationsof the theory by operators of dimensions 3.93, 3.67, 3.55, 3.10, 3.00 respectively givenin table 1, together with Lattice QCD data from [16].vanishes in the conformal case) and quite recently in [15], where second order hydrody-namic transport coefficients have been calculated. Ref. [12] provided an approximate but quite accurate formula for computing the equation of state (or more precisely thespeed of sound c s ), c s = d log Td log S (2)directly in terms of the scalar potential V ( φ ). Thus the scalar potential parametrizesthe physics of the particular type of gauge theory plasma.The family of scalar potentials that we consider is V ( φ ) = −
12 cosh( b φ ) + c φ + c φ + c φ (3)The quadratic terms in φ (mass term) determine, according to the standard AdS/CFTdictionary, the dimension ∆ of the operator ∆ (∆ −
4) = m L , where L is the AdSradius which we fix it to one.In this paper we will mostly concentrate on a set of parameter choices in (3) whichapproximately reproduce the equation of state of QCD plasma as determined by theBudapest-Wuppertal group in [16]. The resulting potentials are listed in table 1 andthe corresponding speed of sound is shown in Fig. 1. In that figure we show thespeed of sound extracted from a numerical construction of the corresponding blackhole solution, as described in the following section, together with the lattice QCD datafor c s . Note that in each case we are free to choose the units of temperature. Here,5otential b c c c ∆ V V V V V V in table 2 wasfirst constructed in [12], while V was recently used in [15].following [15] we fix this freedom so that the temperature corresponding to the lowestdip in c s coincides for all the potentials. This will also be our provisional definitionof the critical temperature T c . According to [15] this value should be 143.8 MeV forQCD. Note finally that for high temperatures, the equation of state becomes essentiallyconformal. Similarly other properties such as the quasinormal frequencies also approachthe conformal values characteristic of N = 4 SYM at high temperature.In order to check that the qualitative conclusions are generic, we also considered someother potentials leading to different profiles of c s ( T ). We will discuss them in sec-tion 5.4. This section describes the black hole background solutions for the quasinormal modecalculations. These backgrounds are the same as those in [12], but since our goal isto determine the quasinormal mode frequencies, it will be convenient to express themin Eddington-Finkelstein coordinates, rather than in the coordinates used in [12]. Wewill discuss this in more detail in the following section on quasinormal modes.The Ansatz for these solutions follows from the assumed symmetries: translation in-variance in the Minkowski directions as well as SO (3) rotation symmetry in the spatialpart. This leads to the following form of the line element:d s = g tt d t + g xx d (cid:126)x + g rr d r + 2 g rt drdt (4)where all the metric coefficients appearing in (4) are functions of the radial coordinate r alone, as is the scalar field φ . This form of the field Ansatz (determined so far only bythe assumed symmetries) allows two gauge choices to be made. In [12] a Schwarzschild-6ike gauge was adopted by taking g tr = 0, accompanied by the condition φ = r . Thelatter condition on the scalar field lead to key simplifications which were used to solvethe field equations. The final form of the Ansatz in [12] was thusd s = e A ( − h d t + d (cid:126)x ) + e B h d r (5) φ = r (6)where A , B , and h are functions of r (or, equivalently φ ).For the purpose of computing the quasinormal modes it is very convenient to use adifferent gauge – the Eddington-Finkelstein gauge g rr = 0. It is typically convenientto also impose the gauge choice g tr = 1, but for our purposes it turns out to be veryeffective to use the remaining gauge freedom to set φ = r . Furthermore, if we label themetric components as d s = e A ( − h d t + d (cid:126)x ) − e A + B d t d r (7) φ = r (8)then the field equations take the form A (cid:48)(cid:48) − A (cid:48) B (cid:48) + 16 = 0 (9) h (cid:48)(cid:48) + (4 A (cid:48) − B (cid:48) ) h (cid:48) = 0 (10)6 A (cid:48) h (cid:48) + h (24 A (cid:48) −
1) + 2 e B V = 0 (11)4 A (cid:48) − B (cid:48) + h (cid:48) h − e B h V (cid:48) = 0 , (12)where the prime denotes a derivative with respect to φ .With the assumed labelling of metric coefficients (5), equations (9)-(12) are identical tothose of appearing in [12]. and so they can be solved following the method describedthere (see also [15]). In the remainder of this section we review this procedure forcompleteness.We are interested in solutions possessing a horizon, which requires that the function h should have a zero at some φ = φ H : h ( φ H ) = 0 , (13)It is easy to see that the solutions of Eq. (9)–(12) can be expressed in terms of a single7unction G ( φ ) ≡ A (cid:48) ( φ ): A ( φ ) = A H + (cid:90) φφ H d ˜ φ G ( ˜ φ ) , (14) B ( φ ) = B H + ln (cid:18) G ( φ ) G ( φ H ) (cid:19) + (cid:90) φφ H d ˜ φ G ( ˜ φ ) , (15) h ( φ ) = h H + h (cid:90) φφ H d ˜ φe − A ( ˜ φ )+ B ( ˜ φ ) (16)In the expressions above A H , B H , h H and h denote constants of integration which willbe determined by requiring the appropriate near-boundary behaviour and eq. (13).As in [12] by manipulating the field equations (9)-(12) one finds the nonlinear “masterequation” G (cid:48) G + V / V (cid:48) = dd φ ln (cid:18) G (cid:48) G + 16 G − G − G (cid:48) G + V / V (cid:48) (cid:19) (17)The strategy is to solve this equation numerically by integrating it from the horizon at φ = φ H down toward the boundary at φ = 0. Once G is known, the metric coefficientscan be recovered from Eq. (14)-(16).Solving Eq. (17) requires appropriate boundary conditions, which can be determinedby evaluating (11) and (12) at the horizon and using (13). In this way one finds V ( φ H ) = − e − B ( φ H ) G ( φ H ) h (cid:48) ( φ H ) , V (cid:48) ( φ ) = e − B ( φ H ) h (cid:48) ( φ H ) . (18)From this it follows that G ( φ H ) = − V ( φ H )3 V (cid:48) ( φ H ) , (19)Using (19) and (17) one finds the following near-horizon expansion: G ( φ ) = − V ( φ H )3 V (cid:48) ( φ H ) + 16 (cid:18) V ( φ H ) V (cid:48)(cid:48) ( φ H ) V (cid:48) ( φ H ) − (cid:19) ( φ − φ H ) + O ( φ − φ H ) (20)In particular, the expansion (20) implies that G (cid:48) ( φ H ) = 16 (cid:18) V ( φ H ) V (cid:48)(cid:48) ( φ H ) V (cid:48) ( φ H ) − (cid:19) . (21)To summarize: to find a numerical solution of (17) we can specify a value for φ H and then use the conditions (19) and (21) as boundary conditions for integrating (17).There is however one technical complication in performing the numerical integration8utlined above: Eq. (19) implies that at the horizon G ( φ H ) + V ( φ H ) / (3 V (cid:48) ( φ H )) = 0 (22)which makes some terms of (17) singular. Even though such superficially singularterms cancel, their presence makes numerical computations troublesome. In order tocircumvent this difficulty, instead of φ H one can initialize the integration at a point justoutside the horizon, at φ = φ H − (cid:15) H , where (cid:15) H (cid:28)
1. Then using (21) one can calculate G ( φ H − (cid:15) H ) and G (cid:48) ( φ H − (cid:15) H ) and then use these values as the boundary conditions.One also needs to regularize at the boundary ( φ = 0) by integrating down to a small,but finite value φ = (cid:15) B .Having determined G , one can find the metric from (14)–(16). The constants of inte-gration can be determined following [12]. The result is A H = ln φ H ∆ − (cid:90) φ H d φ (cid:20) G ( φ ) − − φ (cid:21) B H = ln (cid:18) − V ( φ H ) V (0) V (cid:48) ( φ H ) L (cid:19) + (cid:90) φ H d φ G ( φ ) h H = 0 h = 1 (cid:82) φ H d φe − A ( φ )+ B ( φ ) . (23)This way the metric is determined for any given choice of φ H .The Beckenstein-Hawking formula for entropy leads to the following expression for theentropy density s = 2 πκ e A H (24)and the standard argument requiring non-singularity of the Euclidean continuation atthe horizon gives T = e A H − B H | h (cid:48) ( r H ) | π (25)These equations lead to the formula c s = d log T /dφ H d log s/dφ H ≈ − V (cid:48) ( φ H ) V ( φ H ) . (26)where the latter equality, proposed in [12], is only approximate but works surprisinglywell. In the present paper when determining the speed of sound c s , we always usethe exact formula and determine it from d log T /d log s with temperature and entropy9xtracted from the exact numerical solutions. It is convenient and enlightening to formulate the problem of finding quasinormalmodes in terms of gauge invariant variables, which are diffeomorphism invariant linearcombinations of the perturbations. This approach is well known in general relativity,and has been adopted in the holographic context in [10], where the conformal case of N = 4 supersymmetric Yang-Mills theory was considered. The generalisation to thenon-conformal cases was undertaken in [19]. In this section we briefly summarise ourfindings in the context of the models under consideration in this paper.Under an infinitesimal diffeomorphism transformation, the metric and the scalar fieldfluctuations transform as the metric and the scalar field itself, i.e. g µν → g µν − ∇ µ ξ ν − ∇ ν ξ µ , φ → φ − ξ µ ∇ µ φ. (27)By examining linear combinations of the linearized perturbations one finds five gaugeinvariant channels: two shear channels, one scalar channel, one sound channel and onebulk channel [19].We assume the plane wave e − iω t + i k x dependence of the fluctuations on boundary coor-dinates t, x and some nontrivial dependence on the radial coordinate r . In general, theequations for the shear and scalar channels are decoupled from the rest, which have thesame form as the conformal case. However the sound and bulk channels have coupledsecond order equations [19].In the zero momentum limit, k →
0, the equation for the sound mode becomes decou-pled from the bulk mode and at the same time the equations for the other channelsreduce to the equation for the QNM’s of a massless scalar field, except for the equationof the bulk channel which is still coupled to the sound mode. Interestingly, this hastwo advantages. At k = 0 it is enough to find the QNM’s for an external massless The bulk channel is a linear combination of transverse metric fluctuations and massive scalarfluctuation. One reason we call it ”bulk” channel is it leads to non-zero bulk viscosity as well [13].One could refer to it as the ”non-conformal” channel and to the rest, which are already known inconformal case, as conformal channels.
In view of the results described in the previous section, we turn to exploring the effectsof conformal symmetry breaking by considering the QNM of a massless scalar field Ψ inthe background (7). As discussed earlier, the equation obtained for this case containsall the essential elements for QNM perturbations of the background.The field equation for a massless scalar is simply the wave equation ∇ A ∇ A Ψ = 0 (28)Quasinormal modes are solutions of the formΨ = e − iωt + ikx ψ ( φ ) (29)which satisfy the ingoing condition at the horizon, which in the Eddington-Finkelsteincoordinate system reduces to regularity there. Substituting (29) into Eq. (28) and usingthe form of the background metric given in Eq. (7) leads to the following equation forthe amplitude ψ ( φ ):(3 G V (cid:48) + V ) ψ (cid:48)(cid:48) − e − A − B G (cid:48) (cid:0) e A + B V (cid:48) + 2 iω (cid:1) ψ (cid:48) + 3 e − A − B G (cid:48) (cid:0) k e B − iωe A G (cid:1) ψ = 0 . (30)where primes denote derivatives with respect to φ . Note that only the functions A and B appear here – the function h drops out. Since we are imposing two boundaryconditions, this is an eigenvalue problem which can be solved only for specific valuesof the complex QNM frequency ω . 11 .3 Numerical approach Quite generally, the chief advantage of using the Eddington-Finkelstein coordinate sys-tem for finding quasinormal mode frequencies is twofold. Firstly, the ingoing boundarycondition at the horizon gets translated just to ordinary regularity of the solution atthe horizon. Secondly, due to the special form of the temporal part of Eddington-Finkelstein metric, the dependence on the mode frequency of the relevant differentialequation is linear . Hence the problem of finding the quasinormal frequencies amountsto solving a linear ODE of the form ˆ L Ψ = ω ˆ L Ψ (31)where ˆ L and ˆ L are specific differential operators, with Ψ satisfying essentially Dirich-let boundary conditions at the boundary and being regular at the horizon. Whilemany approaches to the problem of finding quasinormal modes have been described inthe literature [17], we believe that the approach we describe here is very effective inconjunction with the spectral representation in terms of Chebyshev polynomials [18].This representation reduces the task of solving Eq. (31) to a set of linear equations.The differential operators appearing in (31) are represented as matrices, and due to thelinear dependence of this equation on ω this reduces to a generalized matrix eigenvalueproblem which can be solved very efficiently.In the case we are studying, the relevant equation is Eq. (30). This equation is indeedlinear in ω . We have implemented the strategy outlined in the previous paragraphand verified the stability of the resulting solutions when varying the number of gridpoints in the Chebyshev discretization. The results of these numerical calculations arepresented in section 5. We are interested in the dependence of the QNM frequencies on temperature and k in the vicinity of k = 0. Therefore, in line with the discussion in section 4.1, wefocus on the quasinormal modes of an external massless scalar field. By use of theterm “external” we wish to emphasize that this scalar is distinct from the scalar φ appearing in the background geometry, as QNM modes of the latter are mixed with12igure 2: The imaginary parts of the lowest quasinormal mode at k = 0 for thepotentials from table 1 (left). The imaginary part for potential V together with the“phenomenological” according to Eq. (33) (right).the metric perturbations . In the left panel of figure 2 we show the imaginary parts of the QNM frequencies inunits of temperature which is the natural scale in the problem i.e.Im ω π T (32)We observe that the damping significantly decreases (by a factor of 2) close to thetransition. This shows that in the nonconformal case nonequilibrium dynamics becomemore important close to T c . Moreover we find that the plots basically lie on top of eachother for the various potentials from table 1. This indicates that the QNM frequenciesare not sensitive to the fine details of the potentials but are essentially dependent juston the equation of state (speed of sound c s ( T )), which was the common denominatorof all the potentials from table 1.In order to parameterize the dependence of the damping on deviation from conformality,we propose a phenomenological formula expressing this as a linear combination of c s − and T ddT c s ( T ). Specifically, we positIm ω − Im ω conf πT = γ (cid:18) c s ( T ) − (cid:19) + γ (cid:48) T ddT c s ( T ) (33)where γ, γ (cid:48) are phenomenological parameters and Im ω conf π T = − .
373 is the conformal See the discussion in section 4.1 k = 0 for the potentialsfrom table 1 (left) and the ratio of the real to the imaginary part (right).limit value. These parameters can be fitted to the numerically calculated difference ofthe damping w.r.t the conformal case. For the potential V in table 1 we got γ = − . , γ (cid:48) = 0 .
452 (34)In figure 2(right) we show a plot of (Im ω − Im ω conf ) / (2 πT ) together with the fit. Thistwo parameter fit is surprisingly good and may be thus used phenomenologically toestimate the damping in a nonconformal theory with the QCD equation of state. Sincethe quasinormal frequencies for the family of potentials we used in the left panel offigure 2 basically coincide, the single choice of parameters given in Eq. (34) works wellfor all of them. It is worth noting that similarly as for the imaginary part of the quasinormal fre-quencies, the real parts of the frequencies corresponding to the various potentials fromtable 1 are also very close to each other (see figure 3(left)). This signifies that theQNM frequencies are basically insensitive to differences in the UV (since the variouspotentials correspond to different ∆’s) and are governed by IR physics i.e. essentiallythe equation of state.In figure 3(right), we also see an enhancement of the real part of the frequencies slightlybelow T c . 14igure 4: The curvature at k = 0 of the damping frequency, overlaid with α (cid:39) . β (cid:39) .
342 for the potential V in table 1. As indicated in the introduction, an interesting property of the dispersion relation forthe nonhydrodynamic degrees of freedom in the conformal case of N = 4 SYM theoryis the very mild dependence of the frequencies ω I and ω R on the momentum k . In thissection we show that for nonconformal theories this property holds to an even higherdegree. Interestingly, the curvature of the damping i.e.Im ω (cid:48)(cid:48) ( k = 0)2 π T (35)follows to a surprising accuracy (at least until T ∼ T c , then it starts to deviate) justthe speed of sound c s , up to an overall numerical factor determined in the conformalhigh temperature limit (i.e. equivalently in N = 4 SYM). The relevant plot is shownin figure 4. The plot for the other potentials in the same family are basically the same. As discussed in section 2, the choice of scalar potential translates into a specific equationof state for the QCD-like gauge theory. In this subsection we argue how the resultsdiscussed above apply to some additional cases listed in table 2.The potentials V and V have been introduced in [12]. The former was used to mimicthe QCD equation of state using holography while the latter have a second-order phasetransition at T = T c . Fixing b = 0 .
606 corresponds to c s = 0 .
15 in the infrared [12].The variety of potentials leads to a range of conformal weights 3 ≤ ∆ ≤ .
93 (table 2).15otential b c c c ∆ V V V V V V / √ c s (left) and the imaginary parts of the lowest quasinormalmodes at k = 0 (right) for the potentials from table 2. The one with cusp in the c s (left) corresponds to the most decreased in damping (right).For all these cases the qualitative conclusions discussed earlier in this section still hold.In figure 5 we plot the speed of sound (left) and the imaginary parts of the lowestdamped QNM’s (right) for the potentials V − V . Note that in all cases there is acritical temperature, corresponds to the lowest value of c s , which might be consideredas the cross over/phase transition point (related to the potentials). Damping of thelowest QNM’s decreases close to the T = T c by a factor of 2 − namely c s (cid:39) .
114 Im ω (cid:48)(cid:48) ( k = 0)2 π T (36)is valid for this class of potentials too (again untill T ∼ T c ). The coefficient 1 .
114 is the same as ” α ” introduced in figure 4. k = 0 for the damping frequency for potentials from table 2(left) and the ratio of the real to the imaginary part (right) for the same potentials.The ratio of real parts to the imaginary parts of the lowest QNM’s in figure 6 (right)shows a decreasing at T ∼ T c i.e. the lower c s at T = T c the bigger decreasing inRe ω/ Im ω .Surprisingly, even the potential V with the second-order phase transition exhibitsqualitatively the same behaviour as the other potentials discussed in the present paper. In this paper we carried out a study of the lowest quasi-normal modes in a class ofnonconformal holographic models which exhibit an equation of state very similar tothe one obtained using lattice QCD. This class of models was introduced in [12] andincorporates 5-dimensional gravity coupled to a scalar field with a given self-interactionpotential which parametrizes the model.The frequencies of the lowest quasinormal modes provide a scale for the importanceof nonhydrodynamic degrees of freedom, thus their determination is of a definite phe-nomenological interest. Our main observations are the following.Firstly we found that, within the class of considered models, the imaginary part ofthe QNM frequency is strongly correlated with the speed of sound characteristic of theequation of state, once we factor out the trivial conformal temperature dependence. Inparticular, it decreases by a factor of around two at the point of the QCD crossovertransition. This means that nonequilibrium effects will become more pronounced closerto the QCD phase transition/crossover. This seems to be a robust characteristic of thisclass of models and persists also for models with other equations of state considered17or completeness in section 5.4. We provided a phenomenological formula (33) linkingthe damping with the speed of sound. It is important to emphasize, that although thenumerical values of the coefficients in (33) are specific only to the models mimickingthe QCD equation of state, similar fits, with coefficients of the same sign and similarorder of magnitude, work also for other models considered in section 5.4.Secondly, we found that the quasinormal frequencies practically coincide for a wholeclass of models (potentials) which lead to the same equation of state (or more preciselyto the same speed of sound c s ( T ) as a function of temperature). In particular, theyseem to be quite independent of the particular UV properties of the concrete potentialsuch as the anomalous dimension of the operator deforming the theory.Thirdly, we found that the property of ultralocality found in [11], namely the verymild dependence of the quasinormal modes on the momentum k around k = 0 persistsaway from conformality. Even more so, it becomes more pronounced. We also noticedan intriguing feature that the curvature of the imaginary part of the QNM frequenciesaround k = 0 follows surprisingly well the speed of sound squared c s .We believe that the above observations should be of phenomenological interest, espe-cially as they indicate a more pronounced role of nonhydrodynamic degrees of freedomclose to the QCD phase transition/crossover. It would be very interesting to gain someanalytical understanding of these properties as well as to investigate directly nonlineardynamical evolution in such models. Acknowledgements.
We thank Alex Buchel, Micha(cid:32)l Heller and Rob Myers for shar-ing a draft of their preprint prior to submission. HS and MS would like to thank theorganizers of the CERN-CKC TH Institute on Numerical Holography, where a part ofthis research was carried out. RJ and HS wish to thank Galileo Galilei Institute forTheoretical Physics for hospitality and the INFN for partial support during the pro-gram
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