1. Power spectrum of flow fluctuations in Relativistic Heavy-Ion Collisions Ajit M. Srivastava Institute of Physics Bhubaneswar, India

2. Outline: 1. Elliptic Flow and flow fluctuations in relativistic heavy-ion collision experiments (RHICE) 2.Power spectrum of flow fluctuations: Acoustic peaks and superhorizon fluctuations for RHICE, just as for CMBR. Mishra, Mohapatra, Saumia, AMS, PRC77, 064902 (2008); 81, 034903(2009) 3. Hydrodynamics simulations: Confirmation of correspondence with inflationary fluctuations (Digal, Saumia, AMS: present work) 4. Magnetic field in RHICE, effect on power spectrum: Enhancement of flow anisotropies, larger v 2 (larger η/s than AdS/CFT limit?) Mohapatra, Saumia, AMS, MPLA 26, 2477 (2011) 5. Magnetohydrodynamics simulations: Effect of magnetic Field on power spectrum, vorticity generation,…… (Bagchi, Das, Dave, Digal, Saumia, AMS: present work) 6.Vorticity detection by power spectrum at FAIR/NICA (ongoing work : Bagchi, S. De, Das, Dave, S. Sengupta, AMS)

3. 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 Important: Initially no transverse expansion Anisotropic pressure gradient implies: Buildup of plasma flow larger in x direction than in y direction Spectators Collision region Relativistic heavy-ion collision experiments (RHICE):

4. 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 (Note: no odd harmonics) Importantly: mostly discussion on average values of (with the identification of the event plane). Few discussions about fluctuations of for n = 2,4,6

5. First argued by us: Inhomogeneities of all scales present, even in central collisions: Arising from initial state fluctuations These anisotropies were known earlier, however, they were only discussed in the context of determination of the eccentricity for elliptic flow calculations. We argued that due to these initial state fluctuations all flow coefficients will be non-zero in general: All Fourier coefficients are of interest (say, n=1 to 30 -40, including Odd harmonics, these were never discussed earlier, For example, triangular flow is discussed now only as arising from Initial state fluctuations). We emphasized: Learn from CMBR power spectrum analysis: Calculate root-mean-square values of, and NOT their average values.

6. 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: the equilibrated matter will also have azimuthal anisotropies (as well as radial fluctuations) of similar level

7. 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 a reference frame with fixed orientation in the lab system for All Events. So: No need for identifying event plane

8. 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

9. For central events average values of flow coefficients will be zero (same is true even for non-central events if a coordinate frame with fixed orientation in laboratory system is used). Following CMBR analysis,we proposed 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

10. HIJING parton distribution Uniform distribution of partons Errors less than ~ 2% Our earlier predictions: We estimated spatial anisotropies for RHICE using HIJING : Momentum anisotropies were taken to be proportional (0.2x) to spatial anisotropies. Important: Irrespective of its shape, such a plot has important information about the nature of fluctuations (specially initial ones)

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

12. Solid curve: Prediction from inflation Recall: Acoustic peaks in CMBR anisotropy power spectrum Can such a power spectrum be plotted for heavy-ion collisions ? 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 as above.

13. References : 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, and will be non-trivial for all n (up to about 30, including odd n). This applies to all collisions, including central collisions.

14. 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. Tiny fluctuations are stretched by superluminal expansion Horizon size = speed of light c X age of the universe t No physical effect possible for distances larger than this For RHICE: Fluctuations of all length scales are present here at t = 1 fm All fluctuations with sizes larger Than 1 fm are superhorzon No superluminal expansion here

15. 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.

16. 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. It is possible to argue that this freezing out is similar to absence of initial transverse expansion velocity in RHICE. Thus coherence Will be expected to hold for RHICE also.

17. Oscillatory behavior for the fluctuations. Important: Small perturbations in a fluid will always propagate as acoustic waves, hence oscillations are naturally present. Note: The only difference from the universe is the absence of Gravity for RHICE. However, in the universe, the only role of attractive Gravity is to collapse the initial overdensities. Acoustic oscillations happen on top of these fluctuations. One can say that for RHICE one will get harmonic oscillations (for a given mode) while for the Universe one gets oscillations of a forced oscillator. (One can also argue for oscillations of the irregular shape of the boundary of the QGP region.)

18. For small wavelengths, the time scale for the build up of momentum anisotropy will be 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: Momentum anisotropy should show oscillations for short wavelenths (requiring short time scale when radial expansion may be subdominant). This will happen for with large values of n. Later we will see: Hydro simulations confirm this prediction: for large n oscillate first compared to for smaller n.

19. We have argued that sub-horizon fluctuations in RHICE should display oscillatory behavior, as well as some level of coherence just as fluctuations 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. Superhorizon fluctuations for universe do not oscillate (are frozen, as we discussed earlier). Importantly, they also do not grow, That is: they are suppressed compared to the fluctuation which enters the horizon.

20. 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) This is because spatial variations of density in RHICE are not directly detected, in contrast to the Universe where one directly detects the spatial density fluctuations in terms of angular variations of CMBR. For RHICE, spatial fluctuation of a given scale (i.e. a definite mode) has to convert to fluid momentum anisotropy of the corresponding angular scale. This will get imprinted on the final hadrons and will be experimentally measured. As we argued earlier, this conversion of spatial anisotropy to Momentum ansitropy (via pressure gradients) is not effective for Superhorizon modes. Conclusion: Superhorizon modes will be suppressed in RHICE also

21. Presence of such a suppression factor can also be 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. boundary Sound horizon at freezeout

22. uniform distribution of partons HIJING parton distribution Include superhorizon suppression Include oscillatory factor also Results: (modeling only, no hydrodynamical simulation here yet) Errors less than ~ 2% Note: Dissipation, e.g. from viscosity, diffusion, will damp higher n modes

23. 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

24. 1st peak at n = 3 2nd peak at n = 9 1st peak at n = 5 1 st dip at n = 7 1 st dip At n = 7 2nd peak at n = 9 Focus on dip for low n below First peak

25. We have developed two independent codes for hydrodynamic simulation for RHICE. We compare the results of both simulations and cross-check for consistency. One simulation is 3-D simulation with random Gaussian fluctuations of specific width on a Woods-Saxon background corresponding to QGP with initial temperature of 500 MeV. Power spectrum are obtained for the central rapidity region. Second one is a 2-D simulation with Bjorken scaling model. Initial density profile is obtained from Glauber initial conditions for Pb-Pb collision at 200 GeV energy. We study power spectrum of spatial anisotropies, momentum anisotropies, time evolution of power spectrum and the peak structure in the power spectrum. Our results show startling correspondence with evolution of fluctuations in the universe and CMBR power spectrum, in complete agreement with our earlier predictions. Relativistic Hydrodynamics Simulations: (Digal, Saumia, AMS)

26. Changes in the location of peaks with energy-matter density of the Universe, (apparent horizon size changes)

27. Plots of Vn(rms) vs. n for Gaussian fluctuations of width s Blue plot: s = 0.4 fm Red plot: s = 0.8 fm Green plot: s = 1.6 fm Woods-Saxon density profile, 2 fm radius with 10 Gaussian fluctuations, To=500 MeV Note: Here peak position gives information about length scale of fluctuations Just as for CMBR, where first peak location directly gives size of largest fluctuation at last scattering surface Relativistic Hydrodynamics Simulations: (Digal, Saumia, AMS)

28. Evidence for Superhorizon suppression Power spectrum of spatial anisotropies Power spectrum of momentum anisotropies Superhorizon suppression

29. Evidence for Superhorizon suppression Power spectrum of momentum anisotropies Power spectrum of spatial anisotropies

30. Development of acoustic oscillations: Larger n oscillate first, lower n oscillate later n=30 n=28 n=24 time

31. n=20 n=18 n=16 Development of acoustic oscillations: Larger n oscillate first, lower n oscillate later

32. 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 can be several fm. (Tuchin): Caution: needs further analysis of pre-equilibration stage for magnetic flux trapping We argue that magnetic field can enhance flow anisotropies, e.g. elliptic flow can be increased by about 30 % Can this accommodate larger values of viscosity ? Relativistic Magneto-Hydrodynamics: Motivation

33. Effect of Magnetic Field on plasma waves: A conducting plasma in the presence of magnetic field 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.

34. 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 

35. Our earlier work: The effect of magnetic field on v n was modeled 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, now as 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.

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

37. Enhancement of elliptic flow v 2 with magnetic field (up to 30 %) (fm) R.K.Mohapatra, P.S. Saumia, AMS, MPLA26, 2477 (2011). Subsequent analysis by Tuchin: Results in agreement with this K. Tuchin, J.Phys. G 39, 025010 (2012)

38. Relativistic Magneto-Hydrodynamics Simulations: (Bagchi, Das, Dave, Digal, Saumia, AMS) Equations of relativistic MHD: (follow formalism from: Mignone and Bodo, Mon. Not. R. Astron. Soc. (2005) We solve these equations using Leapfrog algorithm of second order. For large velocities, accuracy not enough, so Keep evolution for short times and for small gradients. Initial Conditions: Distribution of energy density, magnetic Field, velocities (taken to be zero) Equation of state: relativistic ideal gas, zero chemical potential

39. B = 20 m^2, t = 1.3 fm, with fluctuations, X-Y plane Energy density Distribution of By: Taken to be proportional to energy density, conductivity decreases with decreasing temperature (nontrivial temperature dependence) Bx Note: initial Bx, Bz = 0, at to = 0.3 fm Woods-Saxon density profile, 2 fm radius with 10 Gaussian fluctuations of width 0.8 fm, To = 500 MeV

40. B = 10 m^2, t = 4 fm, no fluctuations Velocity distribution, X-Y plane, Z = 0, similar in other planes

41. By = 50 m^2, t = 4 fm, no fluctuations Velocity distribution, X-Z plane, Y = 0

42. By = 50 m^2, t = 4 fm, no fluctuations Velocity distribution, X-Y plane, Z = 0

43. By = 0.01 m^2, t = 1.3 fm, with fluctuations Velocity distribution, X-Y plane, similar in other planes Note: radial Flow only, Originating From highest Density region

44. By = 20 m^2, t = 1.3 fm, with fluctuations Velocity distribution similar in other planes Nontrivial Flow distribution Note: Vortex-antivortex Pair formation This is only due To B field (for small B field No nontrivial Flow seen) Recall: initial Velocity zero everywhere

46. By = 0.01,5,10 m^2, t = 1.3 fm, with fluctuations Plots of Vn(rms) vs. n - Full Power spectrum

47. By = 0.01,5,10,20 m^2, t = 1.3 fm, with fluctuations Plots of Vn(rms) vs. n - Full Power spectrum Note: marked suppression of odd harmonics for large B: Distinctive Signal for presence of strong magnetic field at early times

48. Work in progress: Vorticity generation and its detection at FAIR/NICA Using power spectrum of flow fluctuations For low energy as well as high energy collisions: (Bagchi, Das, Dave, De, Sengupta, AMS) High density transition to exotic phase, superfluidity, Superconductivity. Generation of vorticity and Circulating currents. Qualitative effects on the power spectrum, clean signal of Vorticity generation. Work in progress

49. Concluding remarks: Plot of (the power spectrum) may reveal important information: 1) The first peak contains information about the freezeout stage, and the dominant scale of fluctuations. 2)Most important feature in the power spectrum: Bending down of curve for low n values: Signal of superhorizon suppression 3)Detailed analysis of the peak structure, time evolution of power spectrum and development of acoustic oscillations shows very close correspondence with inflationary density fluctuations and CMBR power spectrum. 4)Magnetic field leads to complex flow patterns, vorticity generation, distinctive signature: suppression of all odd harmonics.

50. Future prospects: This interplay of understanding of density fluctuations in the Universe, and in relativistic heavy-ion collisions has very rich Possibilities. Cosmology has been transformed by precision CMBR data and its detailed analysis using power spectrum. We have seen the startling correspondence of this with power spectrum of flow fluctuations for RHICE using hydro simulations. Detailed analysis of power spectrum from data can lead to deeper insights into the early stages of QGP evolution. Further insights from CMBR analysis may be very useful for RHICE. Understanding of plasma evolution, especially for superhorizon modes, at RHICE under controlled laboratory experiments, may be used to get insight into evolution of inflationary density fluctuations. Note: no controlled experiments possible for early universe

51. Thank You

52. Backup slides

53. 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.

54. 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

55. 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.

56. Note: 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 coordinate system with fixed orientation in the Lab frame is used for calculating azimuthal anisotropies for all the events. 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. Our earlier predictions: We estimated spatial anisotropies for RHICE using HIJING Note: we are just modeling the momentum anisotropy using Spatial anisotropies. Later we will discuss results of relativistic hydrodynamics simulations for power spectrum

57. 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

58. 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

59. 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 Note: For small amplitudes of fluctuations, each harmonic will evolve independently. So, proportionality of spatial anisotropy harmonic to momentum anisotropy harmonic for each mode is reasonable.

60. 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.

61. Note: for RHICE we are considering transverse fluctuations. Main point: Transverse velocity of fluid to begin with is zero. Transverse velocity (anisotropic part for us) arises from pressure gradients. However, for a given mode of length scale, pressure gradient is not effective for times t < c s. In other words, until this time, the mode is essentially frozen, just as in the universe. (Note: This is just the condition > acoustic horizon size c s t) For large 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 all fluctuations with scales larger than 1 fm should be reasonably coherent

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

63. 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

64. 2-D simulation, Bjorken scaling Glauber model density profile and fluctuations Pb-Pb, 200 GeV

65. Messenger of Initial Fluctuation ATLAS-CONF-2011-074 1st peak 2 nd peak (Note: these are Vn’s, but peak structure will be same

66. 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.

67. 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 studied the effect of magnetic field on flow Anisotropies in relativistic Heavy-ion collisions

68. By = 0.01,5,10 m^2, t = 1.3 fm, with fluctuations Plots of Vn(rms) vs. n (Cosine term only) Effect of magnetic field on power spectrum of flow fluctuations

69. By = 0.01,5,10,20 m^2, t = 1.3 fm, with fluctuations Plots of Vn(rms) vs. n (Cosine term only)