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% s200.tex
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% paper by P. Kolb, and  R. Rapp 
% on transverse flow and hadrochemistry in Au+Au collisions at 200 GeV
% Final Version, October 15, 2002
%
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\begin{document}


\twocolumn[\hsize\textwidth\columnwidth\hsize\csname @twocolumnfalse\endcsname

\title{Transverse Flow and Hadro-Chemistry 
in Au+Au Collisions at $\sqrt{s_{\rm NN}} = 200$~GeV}   
\author{Peter F.~Kolb$^a$ and Ralf Rapp$^b$}
\address{$^a$Department of Physics and Astronomy, 
             SUNY,
             Stony Brook, NY 11794-3800, USA\\
         $^b$NORDITA, Blegdamsvej 17, DK-2100 Copenhagen {\O}, Denmark
         }
\date{\today}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ABSTRACT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract}

We present a hydrodynamic assessment of preliminary particle spectra
observed in Au+Au collisions at $\sqrt{s_{\rm NN}}=200$~GeV. 
The hadronic part of the underlying equation of state is based on explicit 
conservation of (measured) particle ratios throughout the resonance gas 
stage after chemical freezeout by employing  
chemical potentials for stable mesons, nucleons and anti-nucleons. 
%
We find that under these conditions the data 
(in particular the proton spectra) favor a low freeze-out temperature of 
around $\sim$~100~MeV. 
Furthermore we show that through inclusion of a moderate pre-hydrodynamic 
transverse flow field the shape of the spectra improves with respect to 
the data.
%
The effect of the initial transverse boost on elliptic flow 
and the freeze-out geometry of the system is also elucidated.
%
\end{abstract}
\vspace{0.1in}
]
\begin{narrowtext}
\newpage



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% INTRODUCTION
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\em Introduction.}  
In its second year of operation, RHIC 
(the Relativistic Heavy Ion Collider at Brookhaven National Laboratory) 
has collided $^{197}$Au nuclei at center of mass ($CM$) energies
of 200 GeV per nucleon-pair to create strong interaction 
matter at high energy densities in the laboratory.  
To identify signals of a possible phase transition from low-energy nuclear
matter to deconfined quark-gluon matter, 
a large amount of data was analyzed and recently presented for the first 
time~\cite{QM02}.
%

%
In the present study we investigate single particle spectra of various hadronic
species from the perspective of a hydrodynamic formulation, which
assumes rapid thermalization in the reaction volume and the subsequent 
expansion according to the conservation of energy, momentum, entropy and baryon 
number (for details of the approach which resides on explicit 
longitudinal boost invariance, see~\cite{KSH00}).
%
At $CM$ energies of 130~$A$GeV, 
the shapes of the observed single particle spectra as well as
their azimuthal modulation in non-central collisions have 
validated this approach down to decoupling temperatures of 
$\sim$~130~MeV, at which particles have been assumed to leave 
the compound of strongly rescattering particles~\cite{hydrov2}.
%
Alternatively to the full hydrodynamic description, 
hybrid models have been developed~\cite{BD00,SBD01,TLS01} 
which treat the hadronic stage in sequential scattering models, propagating
the produced hadrons individually.
%
While the momentum space observables in both approaches
are in good agreement with experiments at RHIC, 
the freeze-out geometry, to be discussed below,
persists to be inconsistent with the data in either description.
%
Hydrodynamic calculations appear to be too 
long-lived but too small in radial extent~\cite{HK02}, whereas 
hybrid calculations produce an emission cloud which appears to be 
too large~\cite{BD00,SBD01}.
%

%
It was soon realized\cite{BMMRS01} that, 
also at RHIC energies, the measured particle 
ratios reflect a chemical composition of the fireball with 
a temperature close to the expected critical QCD transition 
temperature, $T\simeq 170-180~{\rm MeV} \simeq T_c$.
% 
Thus, in a thermodynamic description of the cooling process from chemical 
to thermal freezeout, the preservation of the relative hadronic abundances 
requires the introduction of (effective) chemical 
potentials~\cite{BGGL92,HS98,AGHS01,Rapp02,Teaney02} for species that are 
stable on the scale of typical fireball lifetimes.  
%
In particular it was pointed out in \cite{Rapp02} that the conservation
of antibaryons plays an important role at collider energies.
Despite their large annihilation cross sections, their production is in
complete agreement with chemical-freezeout systematics (for a possible
microscopic explanation of this fact, based on multi-meson fusion
reactions to maintain detailed balance, see \cite{RS01}). 
This implies the build-up of large antibaryon chemical potentials,  
$\mu_{\bar{\rm N}}^{\rm eff}$, defined via  
$\mu_{\bar{\rm N}}=-\mu_{\rm N}+\mu_{\bar{\rm N}}^{\rm eff}$.
Towards thermal freezeout this, in turn, entails rather large baryon 
chemical potentials ($\mu_N \simeq 350$~MeV), and is at the origin    
of appreciable pion chemical potentials ($\mu_\pi\simeq 80$~MeV). 
%
The influence of meson-chemical potentials 
on the hydrodynamic evolution and the
resulting observables has been investigated in \cite{HT02} for $CM$ 
energies of 130~$A$GeV. 
%
Here, we include (anti-) baryonic chemical potentials along the lines 
of~\cite{Rapp02}. For consistency with previous analyses 
\cite{KSH00,hydrov2,KSH99,KHHET01} we assume  
a phase transition from quark-gluon to hadron matter at  
$T_c=165$~MeV with a latent heat of $e_{{\rm lat}}=1.15$~GeV/fm$^3$.
The resonances in the hadronic phase correspond to our
earlier hadronic equation of state with chemical potentials as generated 
in \cite{Rapp02}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% PARTICLE SPECTRA -- CENTRAL COLLISIONS                                 %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\em Particle spectra -- central collisions.}
%
We compare results of our calculations with (preliminary)\footnote{
          Due to the preliminary character of the data we  
          did not include errorbars.} 
data from Au+Au collisions at 200~$A$GeV \cite{spectra200}. 
%
According to the $\sim$~15\% larger hadronic particle yield per unit
rapidity, $dN/dy$, in central collisions\cite{PHOBOS02,BRAHMS02} 
compared to collisions at 130~$A$GeV, 
we increase the maximum
entropy parameter from $s_0=95$~fm$^{-3}$~\cite{HK02} to 110~fm$^{-3}$
(at fixed equilibration time $\tau_0=0.6$~fm/$c$ to facilitate the 
interpretation of observed changes).
% 
The correct baryon admixture is accounted for by raising $s_0/n_0$ from 
220 to 250 ($s_0$ and $n_0$ are the entropy- and baryon-density 
in the center of the collision). 
%
The thermodynamic fields in the transverse plane are set to scale 
with the number of wounded nucleons and 
binary collisions as described in \cite{HK02,KHHET01},  
thus also allowing for a straightforward geometrical generalization 
of the fields to collisions with finite impact parameter $b$.

It was observed in \cite{HT02} that for the expansion of 
the chemically non-equilibrated hadron gas 
the slopes of the resulting pion spectra are hardly sensitive to the 
decoupling temperature. Proton spectra clearly favor a 
freeze-out at $T\simeq 100$~MeV. 
This is shown in Fig.~\ref{fig:F1}, where we compare preliminary
spectra of $\pi^+$, $K^+$ and protons from central 
collisions~\cite{spectra200} with hydrodynamic results at an 
impact parameter of $b=2.4$~fm (to approximately account 
for the experimental centrality selection of 5~\%). 
The thick solid lines result from freeze-out at $T_{\rm dec}=100$~MeV,
which corresponds to an energy density of roughly 0.075 GeV/fm$^3$ 
(which is about the same freeze-out density as employed 
in previous calculations). 
The thin lines result from decoupling at the phase transition (note
that the multiplicity of the individual particle species is 
independent of freeze-out as a result of the chemical potentials).
%
%%%%%%%%%%%%%%%%%%%%%%%%%%% Fig. 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[t,b,p]
 \epsfig{file=fig1.ps,width=8cm}
 \caption{$\pi^+$, $K^+$ and proton spectra for central collisions at 200 $A$GeV
          ($K^+$ and proton data and calculations scaled by factors of 
           1/10 and 1/100, respectively). 
          The thick lines represent the results for $T_{\rm dec}=100$~MeV, 
          the thin lines for 165~MeV.
          The solid lines assume no 
          initial transverse flow at $\tau_0=0.6$~fm/$c$
          whereas the dashed curves include an 
          initial transverse boost (see text).}
 \label{fig:F1}
 \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
Independent of freeze-out temperature, 
the experimental pion spectra appear {\em flatter} than what 
can be achieved through the flow generated through
the hydrodynamic description (at transverse momenta larger than  
$p_{\rm T}\ge 2$~GeV this is conceivably due to additional perturbative 
hard scattering contributions).
The spectra of heavier particles show some dependence on the freeze-out 
temperature, but the calculations appear slightly too steep even at 100~MeV.

In an attempt to improve on the shape of the spectra we now include a 
pre-hydrodynamic transverse flow field; on the microscopic level, the latter 
might be generated by a separation of the random velocities of particles during
a period of free streaming~\cite{Kolbthesis,HK02}, an earlier 
or more gradual onset of 
hydrodynamics than in our assumptions~\cite{ENRR02}, or novel effects such as    
sphaleron explosions~\cite{Shuryak02}.
The possible mechanisms are essentially pre-equilibrium in character, 
and are likely to produce rather complicated structures
of the generated flow field with modified energy density distributions.
As an exploratory study, we here investigate the effect of
a simplified initial `seed' transverse velocity according to  
$v_{\rm T}(r)=\tanh ( \alpha \, r )$, 
where $r$ is the radial distance from the origin
superimposed on the original fields at $\tau_0$.  
%
For $\alpha=0.02$~fm$^{-1}$ 
the results are represented by the dashed lines in Fig.~\ref{fig:F1}.
%
Starting the hydrodynamic evolution at an even earlier time 
(i.e. $\tau_0=0.19$~fm/$c$ in \cite{ENRR02}) and evolving 
it to $\tau_0=0.6$~fm/$c$ leads to a radial flow profile similar in 
magnitude. 
%
However, as the hydrodynamic transverse flow is generated through the pressure 
gradients and is influenced throughout its creation by the equation of state of the 
expanding matter, this flow profile is more parabolic around the origin and
does not rise monotonically to larger distances as the generation of flow
is suppressed in the mixed phase region. 
%
It should also be noted that stronger transverse flow due to larger transverse 
pressure is expected if the longitudinal expansion is not fully 
thermalized~\cite{HW02}.

%
Compared to particle spectra in standard ({\it i.e.}, che\-mical-equilibrium) 
hydrodynamics we find a better description of the overall curved shape 
of the hadronic spectra, in particular the low-$p_{\rm T}$ region of the 
pions. This is a result of the chemical potential ($\mu_\pi \approx 80$~MeV
 at freeze-out), which amplifies the Bose-statistics effect. In addition, 
the heavy resonances are more populated after inclusion of
chemical potentials which entails larger contributions at low $p_{\rm T}$ 
from their decay products.
At large transverse momenta (1.5-2~GeV), the hydrodynamic calculations 
deviate from the data which is suggestive for  
the onset of hard scattering regime. High energy partons 
evolving within a hydrodynamic background can be introduced to study the
particle spectra beyond the collective behavior~\cite{HN02}.
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% PARTICLE SPECTRA - NON-CENTRAL COLLISIONS                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\em Particle spectra -- non-central collisions.}
In Fig.~\ref{fig:F2} we compare preliminary spectra of positive 
pions~\cite{spectra200} to our hydrodynamic results in 3 different 
centrality bins 
    (`central',      $b=2.4$~fm, $N_{\rm part}=343.8$; 
     `semi-central', $b=7  $~fm, $N_{\rm part}=170.8$;
     `peripheral',   $b=9.6$~fm, $N_{\rm part}= 76.6$). 
Again, we display calculations with an initial transverse boost by 
dashed lines.

Even for peripheral collisions we obtain good agreement of
theory and experiment up to $p_{\rm T}\simeq 1$~~GeV, which
accounts for more than 96\% of the emitted particles.
Of course, the prerequisites for hydrodynamic calculations, especially 
strong rescattering and a sufficiently large system size, break down 
at large impact parameters and
for high-momentum particles, which rapidly escape the fireball.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%% Fig. 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[!htb]
 \epsfig{file=fig2.ps,width=8cm}
 \caption{Centrality dependence of positive pion spectra at mid-rapidity
         ($y=0$) in terms of most central, semi-central (scaled by 1/2)
          and peripheral collisions (scaled by 1/3).}
 \label{fig:F2}
 \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

Fig.~\ref{fig:F3} shows experimental and calculated 
proton spectra which are of particular interest in the present context as 
they acquire the largest chemical potentials (e.g., 
around thermal freeze-out $\mu_{\rm N}=380$~MeV and $\mu_{\bar {\rm N}}=343$~MeV
implying $\mu_{\bar{\rm N}}^{{\rm eff}}=723$~MeV,  
which yields a  antiproton-to-proton ratio of 0.72 consistent with
experiment~\cite{ppbarratio200}). 
Again we find good agreement of calculations and experiment at a freeze-out
temperature of 100 MeV (solid lines).
The additional transverse `kick' in the initial state as described above
(dashed lines) extends the close agreement of the hydrodynamic spectra with 
data in central collisions to at least $p_{\rm T}=3$~GeV.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%% Fig. 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[h,t,b,p]
 \epsfig{file=fig3.ps,width=8cm}
 \caption{Proton spectra at mid-rapidity ($y=0$) for the most central, 
          semi-central (scaled by 1/2) 
          and peripheral collisions (scaled by 1/3).}
 \label{fig:F3}
 \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
Even though particle densities in the hadronic stage of the expansion
are moderate, we conclude that rescattering is still strong enough
to allow for a hydrodynamic description down to thermal decoupling. 
The correct chemical composition of the hadronic gas is maintained by 
the generation of large chemical potentials, which (at given temperature) 
provide a larger number
of scattering partners with large cross-sections than in a chemically
equilibrated environment.
 
We have focused here on positively charged pions and protons.
The corresponding results for $\pi^-$, $K^+$, $K^-$ and antiprotons in
non-central collisions are of similar quality. 
The multiplicities and mean transverse momenta of the particles discussed above
are collected in Table 1.

%
%%%%%%%%%%%%%%%%%%%%%%%%%% Table 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\vspace*{2mm}
 \begin{tabular}{|cc|ccc|ccc|} \hline
            &               &     &$\alpha=0.00$&   &  
\multicolumn{3}{c|} {\hspace*{2mm}$\alpha=0.02$ fm$^{-1}$ }  \\
            &               &   1   &  2    &  3    & 1     & 
\hspace{4mm} 2 \hspace{4mm}   & 3    \\ \hline
            &$\frac{dN}{dy}$& 280.8 & 134.3 & 57.84 & 282.3 & 134.6 & 57.82\\
\raisebox{1.5ex}[-1.5ex] {$\pi^+$}
         &$\la p_{\rm T}\ra$& 0.398 & 0.392 & 0.375 & 0.419 & 0.405 & 0.383\\ \hline
            &$\frac{dN}{dy}$& 50.18 & 23.99 & 10.33 & 50.43 & 24.05 & 10.33\\
\raisebox{1.5ex}[-1.5ex] {$K^+$}
         &$\la p_{\rm T}\ra$& 0.619 & 0.608 & 0.572 & 0.660 & 0.634 & 0.589\\ \hline
            &$\frac{dN}{dy}$& 28.08 & 13.44 & 5.798 & 28.13 & 13.44 & 5.794\\
\raisebox{1.5ex}[-1.5ex] {$p$}
         &$\la p_{\rm T}\ra$& 0.880 & 0.861 & 0.802 & 0.949 & 0.906 & 0.831\\ \hline
 \end{tabular}
\end{center}
{\small   Table 1: Particle multiplicities and mean transverse momenta (in GeV) 
          of different particle species for the three centrality selections 
          (central to peripheral -- 1 to 3) at $y=0$. The antiproton to
          proton ratio is 0.72; $\la p_{\rm T} \ra$ of antiprotons is 
          within 1\% of the proton value. 
        }
\vspace*{5mm}
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%            ELLIPTIC FLOW                                            %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\em Elliptic flow.}
For the same impact parameters as employed above we
proceed by studying the azimuthal anisotropies of particle spectra, 
{\it i.e.}, the momentum dependence of elliptic flow defined as 
$v_2(p_{\rm T};\,b)= \la \cos (2\phi) \ra$, 
where the average is taken over the angular dependence of the 
particle distribution $dN/dy p_{\rm T} dp_{\rm T} d\phi$.

Flow anisotropy is generated during the earliest stages of the collision, 
at which the spatial eccentricity of the thermodynamic fields
and the anisotropies in the pressure gradients are the largest. 
The matter is set into anisotropic motion as larger forces are acting in
the short direction. This movement rapidly evens out the spatial anisotropies, 
thereby bringing further generation of anisotropic motion to a stall
\cite{Sorge97,KSH00}. 
If the system evolves in chemical equilibrium, the dominant particle 
species at freeze-out are the light pions, which will carry away 
the generated anisotropy in their momentum distribution. Their 
differential elliptic flow $v_2(p_{\rm T})$ is then almost   
independent of the decoupling temperature $T_{\rm dec}$ \cite{hydrov2}.
Heavier particles on the other hand do exhibit some dependence on 
$T_{\rm dec}$, mainly because of the continually increasing radial flow 
which shifts the generated anisotropy out to larger transverse momenta.
%
In the presence of chemical potentials the 
contribution of protons to the {\em total} anisotropic flow 
(of all particles) is still small, however the contribution
of their number to the particle yield is significant. 
The anisotropic flow  must thus be absorbed by the pions (who due
to their small masses also adjust their momentum distribution easier).
Through this effect their elliptic flow now also becomes
sensitive on the decoupling temperature as seen in
\cite{HT02}. 


In addition, the influence of resonance decays is increased in this 
formulation.  The heavy resonances, which at large transverse 
momentum carry themselves rather large elliptic flow, decay and 
transfer this elliptic flow to the pion yield at relatively 
low transverse momentum.

In Fig.~\ref{fig:F4} we show results of elliptic flow for pions 
(left panel) and protons (right panel) from the hydrodynamic 
calculation under inclusion of chemical potentials. 
Again, an initial transverse boost shifts the anisotropy to larger 
transverse momenta which leads to a reduction of elliptic flow at  
fixed $p_{\rm T}$. 
Additionally, the development of anisotropic flow is hindered 
as it has to form on top of the {\em isotropic} initial boost 
field which we have employed here. The value $\epsilon_p$ at which 
the anisotropy of the energy momentum tensor of the fluid $T^{\mu \nu}$ 
saturates during the evolution~\cite{KSH00}, is about 25\% smaller 
than without the initial `kick'.  

%
%%%%%%%%%%%%%%%%%%%%%%%%%%% Fig. 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[t,b,p]
 \epsfig{file=fig4.ps,width=8cm}
 \caption{Elliptic flow of positively charged pions (left) and protons (right)
          for three different impact parameters.
          The dashed lines include an initial transverse boost as described 
          in the text.}
 \label{fig:F4}
 \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

The hydrodynamic description is not expected to hold for particles with 
arbitrarily large transverse momenta, as they do not rescatter sufficiently 
in the fireball. Indeed, the experimentally observed elliptic flow reaches a 
limiting maximum value as a function of transverse momentum.
The PHENIX collaboration has pointed out~\cite{spectra200} that 
this saturation is reached at smaller transverse momenta for pions 
than for protons, and that the saturation value appears to be larger
for the latter.
From the hydrodynamic point of view this reflects the earlier breakdown of
the strong rescattering assumption for pions, which for protons 
remains valid up to higher $p_{\rm T}$ due to larger 
scattering cross sections ($\bar \sigma_{\pi N} > \bar \sigma_{\pi\pi}$).  
This is corroborated by the description of the single-particle spectra which 
extents to larger $p_{\rm T}$ for the (anti-)protons.
The different saturation momentum of mesonic and baryonic elliptic flow
is also consistent with the formation of hadrons via 
quark-coalescence~\cite{Voloshin02}. Within this picture one similarly 
expects a larger saturation value of $v_2$ for protons than for pions.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       FREEZE-OUT GEOMETRY                                         %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\em Freeze-out geometry.}
%
In \cite{Teaney02} it was pointed out that the relation between 
energy-density and pressure, $e(p)$, of the hadronic equation of state 
is hardly modified by the introduction of chemical potentials. 
Therefore, the space-time evolution of the system, 
which is largely driven by this relation, 
is not substantially modified either. 
A large change, however,  does occur in the relation
of temperature and energy density, $T(e)$, 
which thus influences the construction of the 
freeze-out hypersurface and the thermal properties of the fluid on this 
surface. 
E.g., in chemical equilibrium the energy density at $T=130$~MeV
corresponds to a temperature of only 100~MeV in the presence of large chemical
potentials, since the latter increase particle and energy 
densities approximately by pertinent fugacity factors ${\rm e}^{\mu /T}$.  
%
Therefore the freeze-out hypersurface of the hydrodynamic calculations out
of chemical equilibrium is not much different from the 
hypersurface of previous calculations if freeze-out is performed 
at a similar energy density (or the freeze-out temperature is adapted 
accordingly). In both cases, the fireball freezes out at about 15~fm/$c$ 
after equilibration (in central collisions) 
and has about the same spatial extent. 
Only after inclusion of the initial radial flow profile is the life-time 
shortened by $\sim$~15\% and the transverse expansion increases by
about the same percentage. For observables, this entails smaller
longitudinal correlation radii (which reflect the system's lifetime) 
but only slightly larger sideward radii. This effect works in the right direction 
(on the order of a few percent) to cure the RHIC HBT-puzzle~\cite{HK02}, 
but is not sufficient by itself. 
Additional effects like viscosity~\cite{viscosity02}, 
large partonic cross sections in the early phases~\cite{LKP02}, 
or a refined treatment of hadronic rescattering~\cite{SBD01} and 
freeze-out~\cite{SBHP02} (including a large $\rho$-meson 
width as predicted in \cite{Ra02}), might be necessary to 
fully resolve the discrepancy.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       SUMMARY                                                     %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\em Summary.}
Based on a resonance gas equation of state which explicitly 
incorporates hadro-chemical freezout by employing chemical potentials 
for (stable) mesons and baryons in the later hadronic stages, we have 
performed hydrodynamic simulations of heavy ion-collisions at full 
RHIC energy.  
We have compared the results for pion, kaon, and proton spectra to data 
from 200~$A$GeV Au+Au collisions at different centralities. Our 
investigations indicate the necessity of an initial (pre-hydrodynamic) 
transverse flow to better account for the slopes of the observed 
spectra. 
Good agreement with preliminary data for transverse momentum spectra 
in central collisions is obtained up to $\sim$~1.5-2~GeV for pions, and 
up to at least 3~GeV for protons.  
We further studied the influence of hadrochemistry and initial flow 
on elliptic flow and source-geometry. The former have been presented as 
a prediction for pions and protons for upcoming experimental analyses.
For the latter, some improvement with respect to discrepancy
between model and data has been found, but additional effects remain
mandatory.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ACKNOWLEDGMENTS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\em Acknowledgments:}
We thank U. Heinz and E.V. Shu\-ryak for  fruitful discussions and 
critical remarks on the manuscript. 
This work was supported in part by the U.S. Department of Energy under
grant No. DE-FG02-88ER40388.
PFK acknowledges support from the Alexander von Humboldt Foundation
through a Feodor-Lynen Fellowship.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%              REFERENCES                                                  %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\bibitem{hydrov2}
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\end{thebibliography}

\end{narrowtext}

\end{document}

