STR

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Acoustic peaks in CMBR and Relativistic Heavy
Ion Collisions Discussions
Ajit M. Srivastava Institute of Physics
Bhubaneswar
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Recall Important Points of our model 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). 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 argued that this apparent
correspondence with CMBR is in
fact much deeper 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).
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Consider Central collisions Same
considerations apply for
non-central collisions also
It has been noticed that even in central
collisions, due to initial state fluctuations,
one can get non-zero anisotropies in particle
distribution (and hence in final particle
momenta) in a given event. These will be
typically much smaller in comparison to the
non-central collisions, and will average out to
zero when large number of events are
included. For a given central event azimuthal
distribution of particles and energy density in
general anisotropic due to fluctuations of
nucleon coordinates also due to localized
nature of parton production during initial
nucleon collisions.
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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
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
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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. Thus,
inhomogeneities, especially anisotropies with
wavelengths larger than the thermalization scale
should be necessarily present at the
thermalization stage when the hydrodynamic
description is expected to become applicable.
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.
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First note Relevant experimental
observables 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
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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.
Lesson Apply same technique for RHICE also
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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
These values may 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. Straightfo
rward Fourier series expansion of particle momenta
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Acoustic peaks in CMBR anisotropy power spectrum
(Resulting from coherence and acoustic
oscilations of density fluctuations).
Solid curve Prediction from inflation
Proposal for RHICE Plot of for
large values of n will give important
information about initial density fluctuations.
It may also reveal non-trivial structure like for
CMBR, as we argue
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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, Hs cs t , where cs
is the sound speed, is smaller than 1 fm at t
1 fm. With the nucleon size being about 1.6 fm,
the equilibrated matter will necessarily have
density inhomogeneities with superhorizon
wavelengths at the equilibration stage.
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We have argued that in RHICE also, coherence and
acoustic oscillations may be present for flow
anisotropies. Coherence resulting from the fact
that the transverse velocities are zero to begin
with. Acoustic oscillations seem natural, for
small wavelengths Due to unequal initial
pressures in the two directions f1 and f2
momentum anisotropy will rapidly build up in
these two directions in relatively short time.
Expect Spatial anisotropy should reverse
sign in time of order l/(2cs) 2 fm (radial
expansion may still not be dominant).
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Now 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. We have argued that flow
anisotropies for superhorizon fluctuations in
RHICE should be suppressed by a factor
where Hsfr is the sound horizon at the freezeout
time tfr
Note Scaling of V_2 with c_s(t - t_0) is known
(Bhalerao et al
PLB 627, 49
(2005)) Our model implies that similar scaling
(by above factor) should apply to all
superhorizon modes
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Suppression of superhorizon anisotropies
Interface
Sound horizon at freezeout
When l gtgt Hsfr , then by the freezeout time
full reversal of spatial anisotropy is not
possible The relevant amplitude for oscillation
is only a factor of order Hsfr /(l /2) of the
full amplitude.
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We incorporate the above analysis in the
estimates of spatial anisotropies for RHICE using
HIJING event generator We calculate initial
anisotropies in the fluctuations in the spatial
extent R(f) (using initial parton distribution
from HIJING) R(f) represents the energy density
weighted average of the transverse radial
coordinate in the angular bin at azimuthal
coordinate f. We calculate the Fourier
coefficients Fn of the anisotropies in
where R is the
average of R(f). 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
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For elliptic flow we know Momentum anisotropy
v2 0.2 spatial anisotropy e. For simplicity,
we use same proportionality constant for all
Fourier coefficients Note This does not affect
any peak structures 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, as
we will see, averages of Fn (and hence of vn)
will vanish when large number of events are
includedin the analysis. However, the root
mean square values of Fn , and hence of vn , will
be non-zero in general and will contain
non-trivial information.
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Results
HIJING parton distribution
Errors less than 2
uniform distribution of partons
Include superhorizon suppression
Include oscillatory factor also
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From HIJING final particle momenta.
HIJING Parton distribution
Uniform distribution of partons
with momentum cut-off
no momentum cut-off
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Paul Sorensen Searching for Superhorizon
Fluctuations in Heavy-Ion
Collisions, nucl-ex/0808.0503
Left panel p_T autocorrelations derived from the
ltp_Tgt fluctuation scale dependence in AuAu
collisions at sqrts_NN 200 GeV. Sinusoidal
modulations associated with v_2 have been
subtracted. Right panel p_T autocorrelations
plotted in cylindrical coordinates. The positive
near-side peak is subtracted revealing a valley.
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Paul Sorensen, nucl-ex/0808.0503
The power spectrum from p_T fluctuations in
heavy-ion collisions. C_l are calculated at
midrapidity with theta0
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One important difference in favor of RHICE
For CMBR, for each l, only 2l1 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.
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Plots of may reveal important
information

• The overall shape of these plots should contain
  non-trivial information about the early stages of
  the system and its evolution.
• For CMBR, anisotropy power spectrum plots
  reveal crucial
• information about detailed nature of initial
  density fluctuations
• (e.g. non-Gaussianity effects), So Calculate
  3-pt. functions etc.
• Here, for RHICE also, these plots will
  directly relate to distribution
• of density fluctuations of the initial produced
  matter
• Important to note This is true irrespective of
  the validity of the
• physics of coherence and acoustic oscillations
  of our model.
• This only follows from applying the very
  successful tools of
• analysis of CMBR anisotropy power spectrum for
  the case RHICE

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If any of the peaks shown in the plots are
observed
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). We have
checked, using HIJING, that the peak shifts to
higher n for larger center of mass energies, and
for heavier nuclei. Increasing speed of sound
shifts the peak to smaller n as sound horizon
becomes larger Effects of changing initial time
of equilibration t_0 are nontrivial increasing
t_0 lowers value of n where plot flattens
implying changeover in nature of fluctuations
happening at large scales
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• 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).
• We plot vn up to n 30, which corresponds to
  wavelength of fluctuation of order 1 fm.
  Fluctuations with wavelengthssmaller than 1 fm
  cannot be treated within hydrodynamical
  framework. A changeover in the plot of vn 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

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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.
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Full Hydrodynamical simulations can check these
possibilities, we plan to do that. Meanwhile
check the evolution of (complex scalar) field
with non-trivial boundaries. (Polyakov loop
order parameter) Transverse expansion
(radial flow) is visible below
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Evolve field with initial configuration as shown
below
Any surface waves (Edge states)?
Expect similar behavior as for fluid expansion
Check for any oscillations. For QGP,
hydrodynamical evolution simulates plasma. What
about Polyakov loop condensate background ?
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Quark-Hadron phase transition Spontaneous
breaking of Z(3) symmetry in the QGP phase
For the confinement-deconfinement phase
transition in a SU(N) gauge theory, the
Polyakov Loop Order Parameter is defined as
Here, P denotes path ordering, g is the coupling,
b 1/ T, with T being the temperature, A0
(x,t) is the time component of the vector
potential at spatial position x and Euclidean
time t.
Under a global Z(N) transformation, l(x)
transforms as
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The expectation value of the Polyakov loop l0
is related to the Free energy F of a test quark
0 exp(-F/ T)
l
l0 is non-zero in the QGP phase corresponding to
finite energy of quark, and is zero in the
confining phase.
Thus, it provides an order parameter for the QCD
transition, (with N 3) As l0 transforms
non-trivially under the Z(3) symmetry, its
non-zero value breaks the Z(3) symmetry
spontaneously in the QGP phase. The symmetry is
restored in the Confining phase. Thus, there are
Z(3) domain walls in the QGP phase Quarks lead
to explicit breaking of Z(3) symmetry Domain
walls move away from the true vacuum
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Let us first discuss the properties of these
walls, and a new string like structure in the QGP
phase.
For numerical estimates, we use the following
Lagrangian for l(x), proposed by Pisarski (no
explicit symmetry breaking)
Here, V(l) is the effective potential. Values
of various parameters are fixed by making
correspondence with Lattice results b
2.0, c 0.6061 x 47.5/16, a(x)
(1-1.11/x)(10.265/x)2 (10.300/x)3 0.487
where, x T/ Tc
The value of Tc used is 182 MeV.
With suitable re-scaling,
Note b term gives cos 3q , leading to Z(3)
vacuum structure
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Plot of V(l) in units of TC4 for T 185 MeV,
l l exp(iq)
q 0
l l0
Note relative heights of barriers
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Compare Axionic wall/string case
Polyakov loop case