1. From the Universe to Relativistic Heavy-Ion Collisions: CMBR Fluctuations and Flow Anisotropies Ajit M. Srivastava Institute of Physics Bhubaneswar, India

2. Happy Birthday Joe

3. Outline: 1.Relativistic heavy-ion collision experiments (RHICE): Anisotropies in particle momenta for non-central collisions, Elliptic Flow and flow fluctuations 2. Applying CMBR analysis techniques to RHICE. 3.Deeper correspondence with CMBR anisotropies: Recall – Inflationary superhorizon density fluctuations, coherence and acoustic peaks in CMBR anisotropy power spectrum 4. Effects of magnetic field: on CMBR acoustic peaks, and in RHICE: Enhancement of flow anisotropies, larger v 2 possibility of accommodating a larger value of η/s ? Probing Anisotropic flow with excursion sets (as for CMBR to detect anisotropic expansion of the universe). 5. Concluding remarks

4. Relativistic heavy-ion collision experiments (RHICE): QGP phase a transient stage, lasts for ~ 10 -22 sec. Finally only hadrons detected carrying information of the system at freezeout stages (chemical/ thermal freezeout). Often mentioned: This is quite like CMBR which carries the information at the surface of last scattering in the universe. Just like for CMBR, one has to deduce information about The earlier stages from this information contained in hadrons Coming from the freezeout surface. We have shown that this apparent correspondence with CMBR is in fact very deep There are strong similarities in the nature of density fluctuations in the two cases (with the obvious difference of the absence of gravity effects for relativistic heavy-ion collision experiments). First note: Flow anisotropies and Elliptic flow

5. Recall: Elliptic Flow in RHICE In non-central collisions: central QGP region is anisotropic x y central pressure = P 0 Outside P = 0 Anisotropic shape implies: z collision along z axis Recall: Initially no transverse expansion Anisotropic pressure gradient implies: Buildup of plasma flow larger in x direction than in y direction Spatial eccentricity decreases Spectators Collision region

6. x y Initial particle momentum distribution isotropic : it develops anisotropy due to larger flow in x direction This momentum anisotropy is characterized by the 2nd Fourier coefficient V 2 (Elliptic flow) Earlier discussions mostly focused on a couple of Fourier Coefficients, with n = 2, 4, 6, 8 Importantly: mostly discussion on average values of (with the identification of the event plane). Few discussions about fluctuations of for n = 2,4

7. Contour plot of initial (t = 1 fm) transverse energy density for Au-Au central collision at 200 GeV/A center of mass energy, obtained using HIJING Azimuthal anisotropy of produced partons is manifest in this plot. Thus: reasonable to expect that the equilibrated matter will also have azimuthal anisotropies (as well as radial fluctuations) of similar level First argued by us: inhomogeneities of all scales present, even in central collisions: Arising from initial state fluctuations so, all Fourier coefficients are of interest (say, n=1 to 30 -40 odd, even, both sets of values). However more important to Calculate root-mean-square values, and NOT the average values

8. This brings us to our first proposal for correspondence Between CMBR physics and physics of heavy-ion collisions Recall: for the case of the universe, density fluctuations are accessible through the CMBR anisotropies which capture imprints of all the fluctuations present at the decoupling stage: For RHICE: the experimentally accessible data is particle momenta which are finally detected. Initial stage spatial anisotropies are accessible only as long as they leave any imprints on the momentum distributions (as for the elliptic flow) which survives until the freezeout stage. So: Fourier Analyze transverse momentum anisotropy of final particles (say, in a central rapidity bin) in terms of flow coefficients with n varying from 1 to large value ~ 30,40

9. The most important lesson for RHICE from CMBR analysis CMBR temperature anisotropies analyzed using Spherical Harmonics Now: Average values of these expansions coefficients are zero due to overall isotropy of the universe However: their standard deviations are non-zero and contain crucial information. This gives the celebrated Power Spectrum of CMBR anisotropies Lesson : Apply same technique for RHICE also

10. For central events average values of flow coefficients will be zero (same is true even for non-central events if a laboratory fixed coordinate system is used). Following CMBR analysis, we propose to calculate root-mean-square values of these flow coefficients using a lab fixed coordinate system, And plot it for a large range of values of n = 1, 30-40 These values will be generally non-zero for even very large n and will carry important information Important: No need for the identification of any event plane So: Analysis much simpler. Straightforward Fourier series expansion of particle momenta We will see that such a plot is non-trivial even for large n ~ 30-40

11. We estimate spatial anisotropies for RHICE using HIJING event generator We calculate initial anisotropies in the fluctuations in the spatial extent R (using initial parton distribution from HIJING) R represents the energy density weighted average of the transverse radial coordinate in the angular bin at azimuthal coordinate . We calculate the Fourier coefficients F n of the anisotropies in where R is the average of R(). Note: We represent fluctuations essentially in terms of fluctuations in the boundary of the initial region. May be fine for estimating flow anisotropies, especially in view of thermalization processes operative within the plasma region

12. Contour plot of initial (t = 1 fm) transverse energy density for Au-Au collision at 200 GeV/A center of mass energy, obtained using HIJING

13. We use: Momentum anisotropy v 2 ~ 0.2 spatial anisotropy  For simplicity, we use same proportionality constant for all Fourier coefficients: Just for illustration of the technique Note: we are just modeling the momentum anisotropy using Spatial anisotropies Proper values for momentum anisotropies will come from hydrodynamics simulations Important: In contrast to the conventional discussions of the elliptic flow, we do not try to determine any special reaction plane on event-by-event basis. A fixed coordinate system is used for calculating azimuthal anisotropies. Thus: This is why, averages of F n (and hence of v n ) vanishes when large number of events are included in the analysis. However, the root mean square values of F n, and hence of v n, will be non-zero in general and will contain non-trivial information.

14. HIJING parton distribution uniform distribution of partons Errors less than ~ 2% Results: Just like power spectrum for CMBR, non-zero values are obtained from HIJING for all n = 1- 30

15. Would like to point out: The suggestion to plot this “power spectrum” for relativistic Heavy-ion collision experiments was first given by us (Mishra, Mohapatra, Saumia, AMS) in the following papers : 1)Super-horizon fluctuations and acoustic oscillations in relativistic heavy-ion collisions: PRC 77, 064902 (2008) 2) Using CMBR analysis tools for flow anisotropies in relativistic heavy-ion collisions: PRC 81, 034903 (2010) It was emphasized in these papers that such a plot will have Valuable information about nature of fluctuations, especially initial fluctuations, of all wavelengths, down to the scale where hydrodynamics description breaks down. This applies to all collisions, including central collisions.

16. b =0 b = 5 fm b = 8 fm For non-central colisions, elliptic flow can be obtained directly From such a plot

17. Solid curve: Prediction from inflation Recall: Acoustic peaks in CMBR anisotropy power spectrum So far we discussed: Plot of for large values of n will give important information about initial density fluctuations. We now discuss: Such a plot may also reveal non-trivial structure like acoustic peaks for CMBR: (Mishra et al. PRC 77, 064902 (2008)

18. We have noted that initial state fluctuations of different length scales are present in Relativistic heavy-ion collisions even for central collisions The process of equilibration will lead to some level of smoothening. However, thermalization happens quickly (for RHIC, within 1 fm) No homogenization can be expected to occur beyond length scales larger than this. This provides a natural concept of causal Horizon Thus, inhomogeneities, especially anisotropies with wavelengths larger than the thermalization time scale should be necessarily present at the thermalization stage when the hydrodynamic description is expected to become applicable. As time increases, the causal horizon (or, more appropriately, the sound horizon) increases with time. Note: we are discussing Causal horizon in transverse direction as relevant for flow anisotropies. Earlier discussions of causal horizon only referred to longitudinal directions.

19. This brings us to the most important correspondence between the universe and relativistic heavy-ion collisions: It is the presence of fluctuations with superhorizon wavelengths. Recall: In the universe, density fluctuations with wavelengths of superhorizon scale have their origin in the inflationary period. Horizon size = speed of light c X age of the universe t No physical effect possible for distances larger than this

20. Inflationary Density Fluctuations: We know: Quantum fluctuations of sub-horizon scale are stretched out to superhorizon scales during the inflationary period. During subsequent evolution, after the end of the inflation, fluctuations of sequentially increasing wavelengths keep entering the horizon. The largest ones to enter the horizon, and grow, at the stage of decoupling of matter and radiation lead to the first peak in CMBR anisotropy power spectrum. We have seen that superhorizon fluctuations should be present in RHICE at the initial equilibration stage itself. Note: sound horizon, H s = c s t here, where c s is the sound speed, is smaller than 1 fm at t = 1 fm. At time t from the birth of the plasma, physical effects cannot propagate to distances beyond H s With the nucleon size being about 1.6 fm, the equilibrated matter will necessarily have density inhomogeneities with superhorizon wavelengths at the equilibration stage.

21. Recall: Two crucial aspects of the inflationary density fluctuations leading to the remarkable signatures of acoustic peaks in CMBR: Coherence and Acoustic oscillations. Note: Coherence of inflationary density fluctuations essentially results from the fact that the fluctuations initially are stretched to superhorizon sizes and are subsequently frozen out dynamically. Thus, at the stage of re-entering the horizon, when these fluctuations start growing due to gravity, and subsequently start oscillating due to radiation pressure, the fluctuations start with zero velocity. X(t) = A cos(t) + B sin(t) = A’ cos(t +  where is the phase of oscillation. Now, the velocity is: dX(t)/dt = -A sin(t) + B cos(t) = 0 at t = 0 B = 0, So only cos(t) term survives in oscillations or, phase = 0 for all oscillations, irrespective of amplitude. So: all fluctuations of a given wavelength () are phase locked. This leads to clear peaks in CMBR anisotropy power spectrum

22. Phase locked oscillations, with varying initial amplitudes Strong (acoustic) peaks in the power spectrum analysis (strength of different l modes of spherical harmonics)

23. In summary: Crucial requirement for coherence (acoustic peaks): fluctuations are essentially frozen out until they re-enter the horizon This should be reasonably true for RHICE Oscillations with varying phases, and initial amplitudes Peaks get washed out in the power spectrum

24. Note: for RHICE we are considering transverse fluctuations. Main point: Transverse velocity of fluid to begin with is expected to be zero. Though, with initial state fluctuations due to fluctuations in nucleon (hence partons) positions and momenta, there may be some residual transverse velocities even at the earliest stages. However, due to averaging, for wavelengths significantly larger than the nucleon size, it is unlikely that the fluid will develop any significant velocity at the thermalization stage. For much larger wavelengths, those which enter (sound) horizon at times much larger than equilibration time, build up of the radial expansion will not be negligible. However, our interest is in oscillatory modes. For oscillatory time dependence even for such large wavelength modes, there is no reason to expect the presence of sin(t) term at the stage when the fluctuation is entering the sound horizon. In summary: For RHICE also fluctuations may be coherent

25. Let us now address the possibility of the oscillatory behavior for the fluctuations. In the universe, attractive forces of gravity and counter balancing forces from radiation pressure (with the coupling of baryons to the radiation) lead to acoustic oscillations. For RHICE, there is no gravity, but there is a non-zero pressure present in the system. First let us just follow the conventional analysis as for elliptic flow. As the spatial anisotropy decreases in time by the buildup of momentum anisotropy (starting from isotropic momentum distribution), it eventually crosses zero and becomes negative. This forces momentum anisotropy to saturate first (when spatial anisotropy becomes zero), and then start decreasing.

26. Expected evolution of plasma flow Note: At stage IV spatial anisotropy reverses sign due to larger flow buildup in x direction This implies: By this stage, pressure gradient becomes larger in the y direction than in the x direction Now: flow should accelerate faster in y direction This pattern can continue until thermal equilibrium is broken That is: Until Freezeout. After Freezeout: particles (hadrons) fly out to detectors Final anisotropy in flow leads to anisotropy in particle momenta II III IV I

27. Thus: it may be possible that momentum anisotropy decreases to zero, becoming negative eventually. The whole cycle could then be repeated, resulting in an oscillatory behavior for the spatial anisotropy as well as for the momentum anisotropy, possibly with decreasing amplitude. Unfortunately, the situation is not that favorable. Elliptic flow: hydrodynamic simulations: One does see saturation of the flow, and possibly turn over part, but the momentum anisotropy does not become zero, And there is never any indication of an oscillatory behavior. However: Simulations show that this primarily happens because of the build up of the strong radial flow by the time elliptic flow saturates, and subsequently freezes out. One then would like to know whether the same fate is necessarily true for fluctuations of much smaller wavelength as well.

28. Hydrodynamical simulations: Note: By the time changes sign, radial velocity V T becomes large: suppresses oscillation Question: What about fluctuations of much smaller wavelengths ? Kolb,Sollfrank,Heinz, PRC 62, 054909 (2000)

29. For small wavelengths, if the time scale for the build up of momentum anisotropy is much smaller than freezeout time, then oscillations may be possible before the flow freezes out. Due to unequal initial pressures in the two directions   and   momentum anisotropy will rapidly build up in these two directions in relatively short time. Expect: Spatial anisotropy should reverse sign in time of order c s )~ 2 fm Due to short time scale of evolution here, radial expansion may still not be most dominant and there may be possibility of momentum anisotropy changing sign, leading to some sort of oscillatory behavior.

30. We have argued that sub-horizon fluctuations in RHICE may display oscillatory behavior, as well as some level of coherence just as for CMBR What about super-horizon fluctuations: Recall: For CMBR, the importance of horizon entering is for the growth of fluctuations due to gravity. This leads to increase in the amplitude of density fluctuations, with subsequent oscillatory evolution, leaving the imprints of these important features in terms of acoustic peaks. For RHICE, there is a similar (though not the same, due to absence of gravity here) importance of horizon entering. One can argue that flow anisotropies for superhorizon fluctuations in RHICE should be suppressed by a factor where H s fr is the sound horizon at the freezeout time t fr (~ 10 fm for RHIC)

31. Presence of such a suppression factor is most naturally seen for the case when the build up of the flow anisotropies is dominated by the surface fluctuations of the boundary of the QGP region. When H s fr, then by the freezeout time full reversal of spatial anisotropy is not possible: The relevant amplitude for oscillation is only a factor of order H s fr /(2) of the full amplitude. Same can be argued when oscillation results from pressure gradients boundary Sound horizon at freezeout

32. uniform distribution of partons HIJING parton distribution Include superhorizon suppression Include oscillatory factor also Results: Errors less than ~ 2%

33. Paul Sorensen “Searching for Superhorizon Fluctuations in Heavy-Ion Collisions”, nucl-ex/0808.0503 See, also, youtube video by Sorensen from STAR: http://www.youtube.com/watch?v=jF8QO3Cou-Q

34. One important difference in favor of RHICE: For CMBR, for each l, only 2 l+ 1 independent measurements are available, as there is only one CMBR sky to observe. This limits accuracy by the so called cosmic variance. In contrast, for RHICE: Each nucleus-nucleus collision (with same parameters like collision energy, centrality etc.) provides a new sample event (in some sense like another universe). Therefore with large number of events, it should be possible to resolve any signal present in these events as discussed here.

35. Other important insights from CMBR physics: 1)Presence of magnetic fields in the plasma in the early universe distorts the power spectrum of CMBR anisotropies (due to Complex dependence of magnetosonic waves on magnetic field And density gradients). We study such effects in relativistic heavy-ion collisions in view of the fact that in non-central collisions there is a large magnetic field of the order of B = 10^15 Tesla present Further, due to induced currents the time scale for this May be several fm. (Question: Turbulance etc. can also amplify the magnetic field?) We argue that magnetic field can enhance flow anisotropies, e.g. elliptic flow can be increased by about 30 % Question: Can this accommodate larger values of viscosity ?

36. Estimate of variation of elliptic flow v 2 with magnetic field (taken as proportional to the impact parameter b) (fm) R.K.Mohapatra, P.S. Saumia, AMS, MPLA26, 2477 (2011).

37. 2)Techniques of investigating anisotropic expansion of the universe can be applied to study elliptic flow using excursion sets. Anisotropic expansion distorts the shapes of fluctuation Patches (the excursion sets). By analyzing this distortion in different directions, the presence of any anisotropy can be detected. This provides independent determination of the event plane and elliptic flow. Mohapatra, Saumia, AMS, arXiv: 1111.2722; 1112.1177

38. sin θ dΦ θ Shape analysis of excursion sets for CMBR

39. Detection of anisotropic expansion of the universe with distribution widths of Excursion sets width Original data Data stretched by a factor 2

40. Effect of Magnetic Field On acoustic peaks of CMBR: Primordial magnetic fields are present in the universe. Plasma will evolve according to the equations of magnetohydrodynamics. In presence of magnetic field, there are three types of waves in the plasma in place of ordinary sound waves. Fast magnetosonic waves: Generalised sound waves with significant contributions from the magnetic pressure. Their velocity is given by c 2 + ~c 2 s + v 2 A sin 2 θ where θ is the angle between the magnetic field B 0 and the wave vector and the Alfven velocity v A =B 0 /√4πρ Slow magnetoacoustic waves: Sound waves with strong magnetic guidance. c 2 - =v 2 A cos 2 θ Alfven waves: Propagation of magnetic field perturbations.

41. Jenni Adams et al. (1996) The magnetic field effects distort the CMBR acoustic peaks. The distortion can be seen as an effect due to the modified sound velocity (fast magnetosonic waves) with some modulation from slow magnetosonic waves. We study the effect of magnetic field on flow Anisotropies in relativistic Heavy-ion collisions

42. Effect of magnetic field in Relativistic Heavy-ion Collisions R.K. Mohapatra, P.S. Saumia, AMS, MPLA26, 2477 (2011). In non-central collisions in RHICE, there is a large magnetic field of the order of B = 10 15 Tesla present (discussed in the context of CP violation due to instantons in RHICE). The relativistic hydrodynamics equations need to be replaced by equations of relativistic magnetohydrodynamics. In this case perturbations evolve with modified velocity Which depends on the angle between the magnetic field and the phase velocity.

43. Note: Direction of group velocity depends on the coefficient of t above, which depends on local pressure (for a given B). Thus: with pressure variations, direction keeps changing, Thus: Complex flow pattern even with radial expansion B n x t v gr

44. First: The time scale of the magnetic field: Note: The magnetic field here arises from the valence charges (quarks) of highly Lorentz contracted nuclei. Peak value of the magnetic field is achieved for very short time of order of 0.2 fm for RHIC, (and shorter for LHC), ~ passing time of nuclei. Subsequently, the field decays as 1/t^2 In such a case, B will decay by a factor of order 1/100, by the time flow development happens (in few fm time). However, due to induced currents (Tuchin PRC 82, 034904 (2010)) via Faraday’s law, a large magnetic field is induced in the same direction as the original field (Lenz’s law, as external B decreases). Magnetic field obeys a Diffusion eqn. (for relevant length/time scales here) leading to typical decay time of B as several fm. B is essentially constant for time of order 2-3 fm, which is precisely the time scale for flow development.

45. In quasi-static limit (field variation time scale large for the System length/time scales), one can neglect the Maxwell’s Displacement current. Then ( Tuchin, PRC 82, 034904 (2010)) : Diffusion equation Green’s function for the diffusion eqn. (with ): Using this, we can calculate the time evoultion for B for a given initial profile of B. Initial profile of B at t = t_0 is taken as: is the transverse coordinate, and R ~ 5-6 fm the system size

46. The solution for the magnetic field is found to be: Note: for time up to about, the magnetic field is essentially constant. Lattice data for 0.3 /fm for T ~ 200 MeV, (H.-T. Ding et al. arXiv:1012.4963, S. Gupta, PLB 597, 57 (2004)) With R ~ 5 - 6 fm, we get  ~ 2 - 3 fm This is the same time scale during which substantial flow develops. Thus, for this time scale, reasonable to take constant B with peak magnitude of about 10 15 Tesla.

47. We simulate the effect of magnetic field on v n as follows: (For elliptic flow we know that v 2 / ~ c s ) Generate initial density fluctuations from HIJING Calculate F n (the Fourier coefficients of spatial anisotropy) v n is calculated from F n but now the proportionality constant is proportional to group velocity which in turn is a function of the azimuthal angle. v gr B n x t Magnetic field B is taken to be proportional to the impact parameter b, with maximum value B = 10 15 Tesla at b = 10 fm. (from literature on CP violation due to magnetic field in RHICE)

48. Results: Root-mean square values of flow coefficients b = 10 fm b = 6 fm b = 8 fm b = 2 fm

49. b = 10 fm Results: Average values of flow coefficients

50. Results: variation of elliptic flow v 2 with magnetic field (taken as proportional to the impact parameter b) (fm) R.K.Mohapatra, P.S. Saumia, AMS, MPLA26, 2477 (2011).

51. Concluding Remarks All this needs to be checked with hydrodynamics simulations: Work in progress with P.S. Saumia Plots of may reveal important information: 1)The overall shape of these plots should contain non-trivial information about the early stages of the system and its evolution. E.g. distribution of density fluctuations of the initial produced matter. 2) The first peak contains information about the freezeout stage. Being directly related to the sound horizon it contains information about the equation of state at that stage (just like the first peak of CMBR).

52. 3)Like for CMBR, oscillatory peaks here should contain information about dissipative effects and coupling of different species with each other (by plotting flow coefficients of different particles). 4)We plot v n up to n = 30, which corresponds to wavelength of fluctuation of order 1 fm. Fluctuations with wavelengths smaller than 1 fm cannot be treated within hydrodynamical framework. A changeover in the plot of v n for large n will indicate applicable regime of hydrodynamics. 5) One important factor which can affect the shapes of these curves, especially the peaks, is the nature and presence of the quark-hadron transition. Clearly the duration of any mixed phase directly affects the freezeout time and hence the location of the first peak. More importantly, any softening of the equation of state near the transition may affect locations of any successive peaks and their relative heights

53. 6) Possibility of checking aspects of Inflationary physics in Lab? If one does see even the first peak for RHICE then one very important issue relevant for CMBR can be studied with controlled experiments. It is the issue of horizon entering. For example, by changing the nuclear size and/or collision energy, one can arrange the situation when first peak occurs at different values of n. Theoretical understanding of horizon entering of superhorizon fluctuations can be studied experimentally.

54. Especially: v 2 is larger now Important question: Is it possible to accommodate a larger value of η/s ? Full Magnetohydrodynamical simulations needed 7) The presence of large magnetic fields (as expected in RHICE) will modify the equations of hydrodynamics Effective sound velocity is larger, and varies azimuthally Recall: direction of wave velocity depends on local pressure (for a given magnetic field). Thus, even the radial expansion will lead to complex flow patterns: Generation of Vorticity possible Larger sound velocity: enhancement of flow anisotropies.