Superhorizon Fluctuations And Acoustic
Oscillations In Heavy Ion Collisions
ref: Phys. Rev. C 77, 064902 (2008)
P.S. Saumia
Collaborators: A. P. Mishra, R. K. Mohapatra, A. M. Srivastava
Institute of Physics, Bhubaneswar-751005
18th September 2008, ICHIC, Goa
Outline
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Introduction
CMBR and RHIC: Correspondence
Our model
Results
Summary
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One of the important results from RHIC: elliptic flow.
Early thermalization and collective flow of the partons produced in
relativistic heavy ion collision experiments (RHICE).
Elliptic flow is a result of initial spatial anisotropy of the thermalized
region in non-central collisions.
Elliptic Flow in RHICE
In non-central collisions: central QGP region is anisotropic
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y
central pressure = P0
Outside
P=0
Collision
region
x
z
Spectators
Buildup of plasma flow larger in x direction
than in y direction
P P

x y
Spatial
eccentricity
decreases
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Initial spatial anisotropies are present even in central collisions in a
given event (which will average out to zero for large number of
events)
They arise :
As a result of the nucleon distribution in the nucleus and localized
nature of parton production during initial nucleon interactions.
We will argue :
There are strong similarities in the nature of the density fluctuations
in the two cases (even with the obvious absence of gravity in RHICE).
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CMBR and RHICE: Correspondence
Information on the early stages of heavy ion collisions from hadrons:~
Information on the early stages of the universe from the cosmic
microwave background radiation (CMBR).
Surface of last scattering in the universe:~
Freeze-out surface in RHICE. This has been discussed earlier.
However:
We argue that there is a much deeper correspondence between
the physics of RHICE and that of the early universe
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Inflationary density fluctuations and CMBR anisotropies:
In the universe, density fluctuations with wavelengths of superhorizon
scale have their origin in the inflationary period.
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.
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CMBR Acoustic Oscillations
Plot shows variance of various spherical
harmonic components Ylm as a function of l.
Is there any reason to
expect similar features for
fluctuations in RHICE ?
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Transverse energy density
fluctuations in a single event at
1 fm time using HIJING.
Central Au – Au Collision
C M Energy 200 GeV
Azimuthal anisotropy of the
produced partons is manifest
here.
Equilibrated matter is expected to have azimuthal anisotropies of
similar level.
Process of equilibration may smooth this out.
But equilibration time scale < 1fm, no smoothing beyond this length
scale is possible.
Superhorizon fluctuations
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Why Superhorizon?
Nucleon size ~ 1.6 fm
horizon at thermalization ~ 1fm
There are fluctuations of wavelengths larger than thermalization scale
at thermalization
Sound horizon, Hs = cs t , where cs is the sound speed,
is smaller than 1 fm at t = 1 fm.
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Experimentally measurable quantities:
For universe, density fluctuations at the surface of last
scattering accessible through the temperature fluctuations
imprinted on CMBR
For RHICE, initial density anisotropies may get transferred to
momentum anistropies: If they survive until freeze-out stage
they will leave imprints on final particle momenta.
Look for the momentum anisotropies (i.e. essentially different
flow coefficients) existing at the freezeout stage.
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Physics of CMBR Peaks
Two most crucial aspects of the inflationary density fluctuations which
gave rise to remarkable acoustic peaks in CMBR: coherence and
acoustic oscillations
Inflationary density fluctuations when stretched out to superhorizon
scales are frozen out dynamically.
Later when they re-enter the horizon and start growing due to gravity and
subsequently start oscillating due to radiation pressure, the fluctuations
start with zero velocity.
For oscillation, it means that only cos(ωt) term survives.
All fluctuations of a given wavelength are phase locked.
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In summary: Crucial requirement for coherence:
fluctuations are essentially frozen out until they re-enter the horizon
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Acoustic peaks in CMB:
Oscillations:
Gravity
Radiation pressure
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Coherence in RHICE:
The transverse velocity of the fluid to start with is zero.
Initial state fluctuations in parton position and momenta may give rise to
some residual velocities at earlier stages
But for wavelengths larger than the nucleon size, due to averaging, it is
unlikely that the fluid will develop any significant velocity at thermalization.
Larger wavelength modes those which enter (sound) horizon at times
much larger than equilibration time may get affected due to the build up
of the radial expansion.
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(wt) term at the stage
when the fluctuation is entering the sound horizon.
So the fluctuations are expected to be coherent.
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Acoustic oscillations in RHICE: There is a non-zero pressure in RHICE
III
I
II
IV
The expected evolution of plasma region.
ε starts from a large value, decreases to 0, turns over:
a possibility of oscillatory behaviour
p y  px
2
p y  px
2
2
v2 
2

y 2  x2
y 2  x2
V2 starts from 0 (initial momentum
distribution isotropic), it
increases to a maximum value, then starts
decreasing
Hydrodynamic simulations only show the saturation of the momentum anisotropy
and possibly a turn over part
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Hydrodynamical simulations:
Kolb, Sollfrank and Heinz, PRC 62, 054909 (2000)
By the time ε changes sign, radial velocity VT
becomes large: suppresses oscillation
What about fluctuations of much smaller wavelengths ?
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Consider a fluctuation with small wavelength, say 2 fm.
Unequal initial pressures in the
φ1 and φ2 directions:
momentum anisotropy will build up
in a relatively short time.
Spatial anisotropy should
reverse sign in time of order
l/(2cs)~ 2 fm
Due to short time scale of evolution here, radial expansion may still not
be most dominant.
Possibility of momentum anisotropy changing sign, leading to some sort
of oscillatory behavior.
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Another reason to expect oscillatory behavior:
Note: The QGP region is surrounded by the confining vacuum.
If crossing this QGP region corresponds to crossing
the phase boundary through the first order transition line
(e.g. with high chemical potential) then:
an interface will be present at the boundary of the QGP region,
its surface tension will induce oscillatory evolution given initial
fluctuations present.
For a relativistic domain wall such fluctuations propagate with speed of
light.
However, here the interface bounds a dense plasma of quarks
and gluons. Hence expect that perturbations on the interface will
evolve with speed of sound.
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Suppression of superhorizon modes
Acoustic horizon and development of flow:
In RHICE, the azimuthal spatial anisotropies are detected only when
they are transferred to momentum anisotropies of particles.
The anisotropies of larger wavelength compared to sound horizon at
freezeout are not expected to build up completely by freeze out.
So the momentum anisotropies corresponding to these wave lengths
should be suppressed by a factor,
H sfr
/2
where Hsfr is the sound horizon at the
freezeout time tfr (~ 10 fm for RHIC)
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H frs
Sound horizon
at freezeout
When λ >> 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 /(λ /2) of the full
amplitude.
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model calculations:
The spatial anisotropies for RHICE are
estimated using HIJING event generator.
We calculate initial anisotropies in the
fluctuations in the spatial extent R(φ)
(using initial parton distribution from
HIJING).
R(φ) represents the energy density weighted average of the transverse
radial coordinate in the angular bin at azimuthal coordinate φ.
Fourier coefficients Fn of the anisotropies are calculated as
R R( )  R

R
R
where R is the average of R(φ).
Fluctuations are represented essentially in terms of fluctuations
in the boundary of the initial region.
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For elliptic flow we know:
Momentum anisotropy v2 ~ 0.2 spatial anisotropy Є.
For simplicity, we use same proportionality constant for all Fourier
coefficients: This does not affect any peak structures
Important: In contrast to the conventional discussions of the elliptic
flow, we do not need to determine any special reaction plane on
event-by-event basis.
A fixed coordinate system is used for calculating azimuthal
anisotropies.
This is why, as we will see, averages of Fn (and hence of vn) will vanish
when large number of events are included
in 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
Uniform distribution of partons
with momentum cut-off
no cut-off
Flow coefficients calculated
from HIJING final particle
momenta.
HIJING part
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peak at n~5,
higher modes are
suppressed.
No cosmic variance limitation on accuracy!
Universe: Only one CMBR sky: 2l+1 independent measurement
for each l mode.
RHICE: Each nucleus-nucleus collision with same parameters
provides a new sample. Accuracy limited only by the number of events.
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Important information contained in such a plot:
first peak :
freeze-out stage, equation of state.
successive peaks : dissipative factors, nature of the
phase transition if any.
overall shape of the plot : information on the early
stages of evolution of the system and evolution, regime
of break down of hydrodynamics (look at higher modes)..
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Summary:
Transverse energy density fluctuations in RHICE have strong
similarities with the inflationary density fluctuations
present in the universe.
They are superhorizon and are expected to be coherent.
Larger wavelength modes (eg. elliptic flow) remains
superhorizons at freezeout and are suppressed
by a factor ~ H fr
/2
Shorter wavelength modes may show oscillatory behaviour.
More to be explored in the correspondence between and
CMBR and RHICE….
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