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

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Outline

• 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 RHICEIn non-central collisions: central QGP region is anisotropic

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x

y central pressure = P0

Outside P = 0

y

P

x

P

z

Buildup of plasma flow larger in x directionthan in y direction

Spatial eccentricitydecreases

Spectators

Collisionregion

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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: CorrespondenceCMBR 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 Plot shows variance of various spherical harmonic components harmonic components YYlmlm as a function of as a function of ll. .

Is there any reason to Is there any reason to expect similar features for expect similar features for fluctuations in RHICE ?fluctuations in RHICE ?

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Central Au – Au CollisionCentral Au – Au Collision

C M Energy 200 GeVC M Energy 200 GeV

Azimuthal anisotropy of Azimuthal anisotropy of the produced partons is the produced partons is manifest here.manifest here.

Transverse energy density Transverse energy density fluctuations in a single event at fluctuations in a single event at

1 fm time using HIJING.1 fm time using HIJING.

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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All fluctuations of a given wavelength are phase locked.

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.

Physics of CMBR Peaks

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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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The expected evolution of plasma region.

I

II

III IV

Hydrodynamic simulations only show the saturation of the momentum anisotropy and possibly a turn over part

Acoustic oscillations in RHICE: There is a non-zero pressure in RHICE

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22

2

xy

xy

pp

ppv

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22

xy

xy

V2 starts from 0 (initial momentum distribution isotropic), itincreases to a maximum value, then starts decreasing

ε starts from a large value, decreases to 0, turns over:a possibility of oscillatory behaviour

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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 theφ1 and φ2 directions:momentum anisotropy will build up in a relatively short time.

Spatial anisotropy shouldreverse 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 crossingthe 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 quarksand 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,2/

Hsfr

where Hs

fr is the sound horizon at the freezeout time tfr (~ 10 fm for RHIC)

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sfrH

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 Hs

fr /(λ /2) of the full amplitude.

Sound horizonat freezeout

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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

where R is the average of R(φ).

Fluctuations are represented essentially in terms of fluctuations in the boundary of the initial region.

R

RR

R

R

)(

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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 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 part

HIJING parton distribution

Uniform distribution of partons

with momentum cut-off

no cut-off

Flow coefficients calculated Flow coefficients calculated from HIJING final particle from HIJING final particle momentamomenta.

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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:Important information contained in such a plot:

first peak first peak : freeze-out stage, equation of state. : freeze-out stage, equation of state.

successive peakssuccessive peaks : dissipative factors, nature of : dissipative factors, nature of the phase transition if any.the phase transition if any.

overall shape of the plot overall shape of the plot :: information on the early information on the early stages of evolution of the system and evolution, stages of evolution of the system and evolution, regime of break down of hydrodynamics (look at regime of break down of hydrodynamics (look at higher modes)..higher modes)..

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Summary: Transverse energy density fluctuations in RHICE have Transverse energy density fluctuations in RHICE have

strong similarities with the inflationary density strong similarities with the inflationary density fluctuations present in the universe.fluctuations present in the universe.

They are superhorizon and are expected to be coherent.They are superhorizon and are expected to be coherent.

Larger wavelength modes (eg. elliptic flow) remains Larger wavelength modes (eg. elliptic flow) remains superhorizon at freezeout and are suppressedsuperhorizon at freezeout and are suppressed

by a factor ~ by a factor ~

Shorter wavelength modes may show oscillatory Shorter wavelength modes may show oscillatory behaviour.behaviour.

More to be explored in the correspondence between and More to be explored in the correspondence between and CMBR and RHICE….CMBR and RHICE….

2/

Hsfr