Document text
Universal Flow in the First fm/c at RHIC
Joshua Vredevoogd and Scott Pratt
"Department of Physics and Astronomy, Michigan State University
East Lansing, Michigan 48824
Abstract
Q\
We will show how both elliptic and radial flow generated during the first fm/c at RHIC is inde-
pendent of the state of matter and depends only on the initial energy density profile. Descriptions
based on partons or classical fields, thermalized or highly anisotropic, all lead to the same collec-
tive velocity given a few easily satisfied conditions. This significantly narrows the uncertainty
for initializing hydrodynamic prescriptions.
1. Introduction and Definitions
(~*| Colliding heavy ions at the energies available at the Relativistic Heavy Ion Collider have pro-
duced experimental signatures strongly consistent with almost fully thermalized matter. For in-
stance, heavier particles show a larger average transverse energyfl]. In addition, the matter shows
strong anisotropic collcective flow on the order of ten percent[2 |. Each of these observations are
consistent with early predictions from ideal hydrodynamics [3 1. Two particle correlations give
insight into the spatial dimensions of the source (4). These data provide minimal constraints in
^vq isolation, but in conjunction, there appear strong features. Ideal hydrodynamics were quickly
J> shown to describe both the strong anisotropic flow and the relatively small longitudinal size
despite the large velocity gradients required by boost invariance at central rapidity (6j. However,
such models required an extremely rapid thermalization of the matter and under-predicted the
explosiveness of the source. This apparent tension was referred to as the 'HBT puzzle.'
Hydrodynamics, for obvious reasons, is not capable of describing the nuclear matter imme-
diately following the first interactions as the nuclei pass through one another. An early model
is required, but needs only to provide a description of roughly the first one fm/c, while hydro-
dynamics - maybe viscous hydrodynamics - will run until ~ 10 fm/c. It is important to note
however, that early collective, transverse acceleration is preferentially important as compared to
later acceleration. Therefore, there is a certain model dependence in the initial conditions chosen
to instantiate a hydrodynamic simulation with several prospects |7],|H). Each of these models
o
On
O
>
X
can be thought of in terms of a conserved stress-energy tensor, d a T a a = 0, which should hold in
general while the contents of the tensor will vary from model to model. These differences can be
parameterized by k, which we define to be
TXX = Tyy = KToQ (1)
where k would generally be thought of as a transverse stiffness that would vary from one-third
for relativistic hydrodynamics to one-half for longitudinally free-streaming particles to unity for
Preprint submitted to Nuclear Physics A
October 14, 2009
longitudinal coherent fields. The result of this paper will be that the transverse flow generated at
early times, as defined by
Fi - ^, (2)
^oo
is independent of k.
2. Results
Calculating the evolution of flow at early times requires only the application of the conserva-
tion equation fl9). We impose a few simple assumptions:
1. That the stress energy tensor be traceless. This assumption is completely valid for non-
interacting particles or fields as well as for ideal hydrodynamics. At the large energy
densities during the first fm/c, this should be valid at the 10% level. While the transverse
pressures are set by k, this condition specifies that T zz = (1 — 2k)7oo-
2. That the longitudinal dynamics be entirely boost-invariant. For r\ < 1 this has been proved
to be valid at roughly the 10% level as well. This fixes the longitudinal velocity to be
u z = z/t.
3. That the anisotropy in the stress energy tensor by purely time dependent. This is akin to
the same description of the system applying at a given time.
Intuitively, one might expect that an increase in the value of k should lead directly to an
increase in the transverse velocity. For a one-dimensional system with no longitudinal expansion,
the conservation of momentum would require that
d T T 0x = -d x T xx = -Kd x T m (3)
This indicates a linear increase in the transverse flow as one increases the anisotropy. However,
the longitudinal expansion adds an additional term since
d z T„ = d z T 0x z/r = T Qx /t (4)
for small velocities. Using the fact that the energy density falls according to
d T Too = ~-(l-*)7oo (5)
T
Calculating the time derivative of the flow is then straightforward
Q Tq x 9 t Tq x Tq x 3 T r o
o T — = (6)
Too Too Too Too
xdJoo + To.J t (2 - 2k) T Qx
+
which lead directly to
Too T Too
Tox -9 x Tqq
(7)
Too Tqq
This equation is independent of k, meaning that it does not depend on the exact description of
the system's dynamics beyond those stipulated in the assumptions.
2
0.4
4 8
r (fm)
Figure 1: Triangles - ideal hydrodynamics (k = 1/3). Circles - free-streaming particles (k = 1/2). Squares - coherent
fields (k = 1). Each panel contains integration results from r = 0.3,0.6, l.Ofm/c. Bottom panel shows the transverse
velocity as defined in each model, showing vast differences between the models. Middle panel shows the transverse flow
in each model. Top panel shows the transverse velocity as defined in ideal hydrodynamics after a rapid thermalization
from each model. (Color Online)
As shown in Figure 1, while the transverse flow of the system develops identically between
the models, the collective velocities differ. After around one fm/c, the system should thermalize.
If the transition is smooth in time, then our constraints on k allow for a smooth transition of
anisotropy into ideal hydrodynamics. In the case that this process occurs suddenly and simulta-
neously on a hypersurface parameterized by the four- vector n a - ( 1 , 0, 0, 0), the elements T a o are
conserved. In the spirit of the Rankine-Hugeniot equations, one can integrate the conservation
equation across a shock as a model of thermalization:
= [d T T 0a + d i T ia ]=T 0a (T + 8T)-T 0a (T-8T) (8)
Jt-St
Integration across a general hypersurface would have resulted in the conservation of npT a P, and
for a space-like surface (e.g. n = (0, 0, 0, 1))) would result in the usual Rankine-Hugeniot equa-
tions. However, for any transition such that n 2 = +1, To a is conserved exactly in some frame.
The top panel of Figure 1 shows that such a transition leads to identical velocity profiles as the
initial conditions of a hydrodynamic simulation.
As a corollary, we point out that this precisely means that anisotropic flow develops as well.
This follows clearly from eq. [7] - the flow develops independent of k but proportional to the
gradient of the energy density. That is to say, along transverse directions in which the gradients
are larger, there will still develop stronger flow. Now, in general, the elliptic flow developed in
hydrodynamics is measured using the quantity
where the brackets indicate an integral over the entire plane. In hydrodynamics, prior to freeze-
out, the quantity is directly proportional to elliptic flow. However, since our models vary in what
3
0.04
iTO.02
0.00
^0.02
0.00
0.0 0.5 1.0
T (fm/c)
Figure 2: Line - ideal hydrodynamics (k = 1/3). Circles - free-streaming particles (*• = 1/2). Squares - coherent fields
(a: = 1). The anisotropy in momentum space as a function of r. Bottom panel shows that ideal hydrodynamics develops
stronger anisotropic flow. Top panel shows that rapid thermalization immediately restores the agreement of the models.
(Color Online).
is meant by T xx and T yy , it is not clear that a change in e p (or lack thereof) results in a changed (or
unchanged) elliptic flow as measured in the final state. The bottom panel of Figure 2 shows that
while hydrodynamics develops a strong signal in e p , other models develop little or no signal. But
the system has yet to thermalize and the flow developed is not wasted. The top panel of Figure 2
shows that anisotropic flow is indeed developing and is only shown after thermalization.
We have shown that early collective velocity as strong as those developed in hydrodynam-
ics are created by any model within some mild constraints. In addition, elliptic flow begins to
develop before a hydrodynamic model can be applied. Using initial conditions developed from
principles outlined here, which we expect to be valid at the w 10% level, particle spectra and HBT
radii can described at roughly the same level using a viscous hydrodynamic simulation coupled
to a resonance cascade. ifTUl
Acknowledgments
Support was provided by the U.S. Department of Energy, Grant No. DE-FG02-03ER41259.
References
[1] J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. 92, 1 12301 (2004).
[2] B. I. Abelev et al. [STAR Collaboration], Phys. Rev. C 77, 054901 (2008), arXiv:nucl-ex/0801.3466.
[3] P. F. K olb and U. W. Heinz, Quark Gluon Plasma III, Eds. R.C Hwa and X.-N. Wang, World Scientic, 634 714
(2003), |arXiv:nucl-fh/ 0305084
[4] M. A. Lisa, S. Pratt, R. Soltz and U. Wiedemann, Ann. Rev. Nucl. Part. Sci. 55, 357 (2005).
[5] U.W. Heinz and P. F. Kolb, Proc. 18th Winter Workshop on Nuclear Dynamics (R. Bellwied, J. Harris, and W.
Bauer, eds.), pp. 205-216, EP Systema, Debrecen, Hungary, 2002, arXiv:hep-ph/0204061 /l.
[6] J. D. Bjorken, Phys. Rev. D 27, 140 (1983).
[7] L. D. McLerran and R. Venugopalan, Phys. Rev. D 49, 2233 (1994).
[8] S. A. Bass, B. Muller and D. K. Srivastava, J. Phys. G 30, S1283 (2004).
[9] J. Vredevoogd and S Pratt, arXiv.org, nucl-th 0810.4325 (2008).
[10] S. Pratt and J. Vredevoogd, Phys. Rev. C 78, 054906 (2008) [Erratum-ibid. C 79, 069901 (2009)] |arXiv:0809.0516
[nucl-th]].
4