Studying Heavy Ion Collisions Using Methods From Cosmic Mi- crowave Background (CMB) Analysis

We present and discuss a framework for studying the morphology of highmultiplicity events from relativistic heavy ion collisions using methods commonly employed in the analysis of the photons from the Cosmic Microwave Background (CMB). The analysis is based on the decomposition of the distribution of the number density of (charged) particles expressed in polar and azimuthal coordinates into a sum of spherical harmonic functions. We present an application of the method exploting relevant symmetries to the study of azimuthal correlations arizing from collective flow among charged particles produced in relativistic heavy ion collisions. We discuss perspectives for eventby-event analyses, which with increasing collision energy will eventually open entirely new dimensions in the study of ultrarelaticistic heavy ion reactions. ae-mail: gardhoje@nbi.dk bOn leave from COMSATS, Islamabad, Pakistan. e-mail: hajrah.tabassam@gmail.com. DOI: 10.1051/ C © Owned by the authors, published by EDP Sciences, 2014 , / 000 (2014) 201 epjconf EPJ Web of Conferences 471000 71 61 61 This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20147100061


Introduction
Many advanced techniques have been developed in the last decade to analyse the single cosmic event that we have at our disposal, our Universe, through the cosmic microwave background (CMB) radiation which carries the imprint of the conditions prevailing in the Universe shortly after the Big Bang. Notably, the study of the CMB radiation and its polarization provides new insigt into the fluctuations in the early Universe before inflation and can shed ligt on the very mechanism underlying the inflationary phase transition.
Ultra-relativistic collisions between heavy ions are often highlighted as a mechanism to produce and study 'mini universes'under controlled laboratory conditions. Indeed, in such collisions a tremendous amount of kinetric energy may be transferred from the colliding heavy ions (of either gold or lead) which have been accelerated in machines like RHIC and LHC to an extremely hot and energydense zone that contains tens of thousands of produced particles in a volume of about the one of the colliding atomic nuclei. We now know that the reaction is characterized by a high degree of transparency [1], [2], in the sense that the nucleons in the colliding nuclei pass though each other and that, after the collison, the space between them is filled with tens of thousands of quark-antiquark pairs that are created from the color field between the colliding partons.
This super-hot piece of matter, which initially has a temperature in excess of 300MeV ( ≈ 4·10 12 K) constitutes an isolated system that expands and cools like a tiny fragment of the early Universe. It is also established that the system initially, and before it hadronizes, is strongly coupled and hence behaves like a fluid [3], [4], [5], [6]. This fluid has been labelled the s-QPP, for 'strongly interacting Quark Gluon Plasma'. It is the state of matter that the Universe was in up to about a microsecond after the Big Bang.
The subsequent evolution of the tiny droplet of s-QGP is complex although shortlived: the system expands, cools and hadronizes and the newly formed hadrons interact and scatter as the system continues to expand until the density has decreased so much that interactions cease (freeze-out) after about 10 f m/c (≈ 10 −22 sec). From this time on, produced hadrons stream to the detectors only affected by possible decays in flight and reinteractions with the surrounding material. In central collisions of lead ions at the LHC the energy density after about 1 f m/c exceeds 16 GeV/ f m 3 and about 17.500 charged particles are produced in a single central collision [12]. The particles that reach the detectors carry the imprint of the early stages of the collision and we may expect, dependent on the reinteractions and degree of equilibration attained during the rapid expansion expansion that fluctuations in the final channel may reflect some of the initial conditions.
In this sense it is tempting to attempt an analysis of the number of particles detected in heavy ion reactions at the single event level using tools developped for the photons emitted from the early Universe. In the following we present such a first study focusing on large scale collective phenomena in heavy ion collisions events.

Spherical harmonic decomposition of heavy ion events.
Data from the CMB sky are usually presented as a temperature variation map as a function of polar angle θ and azimuthal angle φ. It is conventional to use the Mollweide projection depicted in Fig. 1. The azimuthal angle runs along the 'equator' and the polar angle runs from the 'south pole' to the 'north pole'.
For heavy ion collisions spherical coordinates may not seem to be the natural choice, on the one hand, due to the typical cydindrical geometry of present day particle detectors and on the other hand, and more fundamentally, due to the strong peaking of the number of particles particle measured in the laboratory frame of reference per unit solid angle near the direction of the beam of colliding  particles. Indeed, particle production is found to follow a nearly Gaussian distribution as a function of pseudorapidity η = −ln(tan(θ/2)). This translates to the distribution shown in Fig. 2 as a function the polar angle θ. The CMB map of the sky is, in contrast characterized by flatness as a function over the angular range. We shall, however, show that this is not a limitation and, that we may exploit specific symmetries to overcome such particular features in heavy ion distributions and linearize the distribution to be analyzed.
Data analysis of the CMB map of the sky is based on the expansion of the entire two-dimensional where a lm = |a l,m | exp(−imΦ l,m ) are the weight coefficients of the decomposition. Correspondingly, |a l,m | and Φ l,m are the amplitudes and the phases of each l, m component.
The power spectrum is defined by Much of the understanding of CMB data is based on interpretation of the power spectrum up to order l=2500 and comparison to suitable cosmological models that try to predict the multipole spectrum.
In practice, the maximum l-value that is to be considered in the above decompositions depends on the experimentally achievable resolution in the experiment, both in terms of detector (angular) segmentation, but also in terms of detector resolution. In general, low order l-components probe large angle correlations, while high-order l values probe short scale correlations. Roughly speaking Δθ = 2π l . We may, for illustration purposes, use the HIJING generator [7] to generate semi-realistic heavy ion events. HIJING does not contain new physics, e.g. a QGP phase, but treats collisions starting from a Glauber description of the colliding nuclear density distributions in terms of their nucleon copntent and follows particle production and expansion and the hadronic and leptonic level. Fig. 2 shows the charged particle distribution for one small impact parameter (central) event as a function of pseudorapidity eta (top) and polar angle θ (bottom). Fig. 3 shows the corresponding Mollweide representation of this event. Note the large density of counts around the poles in this representation. We may, however, 'linearize' the event by omitting the m=0 modes of the harmonic representation. This is due to the theta-symmetry of the spherical harmonic functions Y lm for m = 0.  In Fig. 4 we plot the power spectrum for this event (eq. 2). We use either the CMB GLESP package [8] for these operations or a recently developped version integrated with the ROOT package. The black histogram shows the power of the l-multipoles otained by a Fast Fourier Transform of the Mollweide distribution of the simulated heavy-ion event. The blue histogram shows the power associated with the m=0 components only for each l-multipole. Finally the red histogram shows the power associated with the m 0 components only of the respective l-multipoles. It is seen that this power spectrum is essentially flat and random as one would expect for an event not carrying any particular microscopic structure (no dominating multipoles) .

Collective flow
Two important physics signals underpin the discovery of the s-QGP [3], [4], [5], [6]. One is the quenching of jets that propagate through the dense medium, which is understood as due to energy loss of scattered partons due to stimulated gluon emission in an environment characterized by a high density of color charges. The other is the presence of significant anisotropic azimuthal flow in semiperipheral nuclear collisions at high energy. This can be understood in terms of hydrodynamical streaming. Hydrodymanics models are able to reproduce observations using short reinteraction times and significant coupling between the partons.
Due to its importance we focus in this section on anistropic azimuthal flow. The present exposition is a much condensed account of ref. [10]. to which we refer the reader for full details. Analysis of experimental flow data typically relies on a Fourier decomposition of the number of particles that are measured as a function of azimuthal angle for a given (typically semi-peripheral) collision centrality. Often, in selected pseudorapidity regions, the analysis can be made as a function of the transverse  momentum of the particles and/or for idenfied particles. Analysis techniques have progressed from measuring the distribution of particles relative to an estimated event-plane, to using advanced multiparticle correlation techniques that are less sensitive to efects that mimic collective flow (non-flow) but stem from other non-collective processes (f.ex. two body decays) [13], [14]. In all cases the analysis is carried out by averaging over the particles in each event and further averaging over a number of events. For recent reviews see, for example, refs. [15], [16].
In this study we have added flow (elliptic, e.g. of order n=2) to a set of simulated HIJING events. In Fig. 6 we show the resultant power spectrum. It is obvious that additional power is present in the l=2 and l=4 multipoles. In ref. [10] we explain why the l=4 modes also contribute (see also Fig. 7).
For each heavy ion collision the angular distribution of particles is given by the following 'stochastic equation': where S (θ, φ) is the map of particles in an event with flow v n and, f (θ, φ) is the map of a random distribution of particles, without flow, which is characterized by a uniform random distribution in the azimuthal direction. The flow v n can be deterministic or even a random function with a distribution function that differs from the p.
where b l,m are the coefficients of the spherical harmonic decomposition for S (θ, φ). In ref. [10] we describe a systematic analysis of simulated events with flow of various magnitudes (and orders) and varying symmetry plane angle. We also establish relationships between the b lm and the v n and φ n coefficients.
Accumulating statistics for each event over a large number of simulated events yields the distributions shown in figs. 8 and 9. It is seen that it is in principle possible to determine with good precision EPJ Web of Conferences 00061-p.6 both the flow amplitude v n and the symmetry plane angle for every event. The precision is naturally controlled by the number of particles that can be included in the sample for every event. In the next section we elaborate on this.  A main message of this talk is, apart from drawing attention to the method outlined in ref. [10], [11] to point out that the experimental conditions at the LHC at CERN combined with powerful experiments such as ALICE [9] will, in the near future make it possible to collect very significant statistics for each event in f.ex. Pb+Pb collisions. In fact, we may estimate from the systematics presented in ref. [12] infer that at the top LHC energy for Pb+Pb collisions, which is expected to be reached in Y2015 or Y2016, the number of charged particles per single 0-5% central event wil be of the order of. 23.000 particles. For semi-pheripheral collisions, e.g. 40-50 % centrality, which shows ICNFP 2013 00061-p.7 maximal elliptic flow, the particle production per event is expected to decrease by a factor of about 4, resulting in about 6000 particles per event of this class. Figure 9. The correlation between the reaction plane angle Ψ R that was an input for each of 100 HIJING events with fixed flow v 2 = 0.07 is plotted as a function the phase angle φ 2,2 obtained in the harmonic decomposition analysis for each event.The insert shows the distribution of the symmetry planes found by this event-by-event analysis.
In Fig. 10 we show our estimates of the simultaneous distributions of flow amplitudes and symmetry plane angles at the event-by-event level for collections of events of different average multiplicities of charged particles.

Conclusions and perspectives.
It is not given that the methodology outlined here will eventually prove superior to the sophisticated techniques that have been developped in recent years to study the azimuthal collective motion in selected rapidity regions (essentially a 1D analysis, even if it involves multiparticle correlation techniques). We suggest, however, that the method discussed here, being intrinsical multidimensional, (image analysis) may prove useful in studying other fluctuation phenomena at the single event level, and that the mulipole amplitudes may be useful in finding and characterizing such modes. The significantly increase in multiplicity (by ≈ 30%) expected at the single event level at the LHC running at full energy √ s n n = 5.5T eV will open up a new doorway. A future collider operating at an order of magnitude higher energy, e.g. the VLHC which may be an option for at future hadron collider at CERN, would lead to total charged particle production of about 60000 per central event (extrapolating from present systematics) and seriously open up for high precision event-by-event Little Big Bang physics exploration.