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\begin{flushright}
HIP-2000-56/TH 
\end{flushright}
\vspace*{26mm}

\title{THE EFFECT OF PRIMORDIAL TEMPERATURE FLUCTUATIONS 
       ON THE QCD TRANSITION\protect\footnotemark[1] \vspace*{8mm} }
\footnotetext[1]{
 To appear in the Proceedings of Strong and Electroweak Matter 2000
 (SEWM2000), ${\mbox{\rm Marseilles}}$, France, 14--17 June 2000.}

\author{J. IGNATIUS \vspace*{1mm} }

\address{Department of Physics,
         P.O. Box 9, FIN-00014 University of Helsinki, Finland\\
E-mail: janne.ignatius@iki.fi  \vspace*{3mm} }

\author{DOMINIK J. SCHWARZ \vspace*{1mm} }

\address{Institut f\"ur Theoretische Physik, 
         TU Wien,\\ 
         Wiedner Hauptstra\ss e 8 -- 10, A-1040 Wien, Austria\\ 
%and\\
%Institut f\"ur Theoretische Physik, Universit\"at Frankfurt,
%         Postfach 111932, D-60054 Frankfurt a.M., Germany\\
E-mail: dschwarz@hep.itp.tuwien.ac.at \vspace*{8mm}  }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\maketitle\abstracts{
We 
analyse                 %N
a new mechanism for the cosmological QCD
first-order             %N
phase transition:
inhomogeneous nucleation. The primordial temperature fluctuations
%
are larger than the tiny temperature 
interval, in which bubbles would form in the standard picture of homogeneous 
nucleation. Thus the bubbles nucleate at cold spots. We find the typical 
distance between bubble centers to be a few meters. This exceeds 
the estimates from homogeneous nucleation by two orders of magnitude. 
The resulting baryon inhomogeneities may affect primordial nucleosynthesis. 
}

\clearpage

\pagestyle{plain}


%A separation of cosmic phases during a first-order QCD transition 
%%\cite{Witten} 
%could give rise to inhomogeneous nucleosynthesis. 
%%\cite{IBBN,Fuller,Mathews,Kainulainen}. 
%%During a 
%%thermal first-order phase transition in a homogeneous medium bubbles nucleate 
%%due to statistical fluctuations (homogeneous nucleation). 
%%Their mean separation 
%The mean separation between bubbles           %N
%at nucleation introduces a scale for isothermal 
%inhomogeneities in the early Universe, which may influence the local
%neutron-to-proton ratio. 
%%providing inhomogeneous initial conditions for nucleosynthesis. 
%The baryon inhomogeneities may survive until the time of 
%neutron freeze-out, if the mean bubble nucleation distance, $d_{\rm nuc}$, 
%exceeds the diffusion length of the proton. Comparing those scales at the 
%time of the QCD transition, assuming a thermodynamic transition temperature
%$T_{\rm c} = 150$ MeV, gives $d_{\rm nuc} > 2$ m \cite{Mathews}. 
%%The causal 
%%scale is set by the Hubble distance at the QCD transition, 
%%$d_{\rm H} \equiv c/H \sim 10$ km.

%The order of the QCD transition and the values of its parameters
%are still under debate. Nevertheless there are 
%indications from lattice QCD calculations 
%\cite{Brown,Iwasaki96,Iwasaki,Lattice,Latticecs2}. For the physical 
%masses of the quarks the order of the transition is still unclear 
%\cite{Brown,Iwasaki96}. Quenched QCD (no dynamical quarks) shows a 
%first-order phase transition with a small latent heat, compared to the 
%bag model, and a small surface tension, compared to dimensional arguments 
%\cite{Iwasaki}. We assume that the QCD transition is of first order and 
%that the values from quenched lattice QCD (scaled appropriately by the 
%number of degrees of freedom) are typical for the physical QCD transition.
%Based on these values and homogeneous bubble nucleation a small supercooling, 
%$\Delta_{\rm sc} \equiv 1- T_{\rm f}/T_{\rm c} \sim 10^{-4}$, and a tiny 
%bubble nucleation distance, $d_{\rm nuc} \sim 1$ cm, follow \cite{Ignatius}. 
%The actual nucleation temperature is denoted by $T_{\rm f}$. 
 
%We argue that the assumption of homogeneous nucleation is violated in 
%a first-order QCD phase transition in                                  %N
%he early Universe 
%by the inevitable density perturbations from inflation 
%or from other seeds for structure formation. 
%%Those fluctuations in density and temperature have been measured by COBE 
%%\cite{Bennett} to have an amplitude of $\delta T/T \sim 10^{-5}$. 
%%The effect of the QCD transition on density perturbations \cite{SSW,SSW2}
%%and gravitational waves \cite{Schwarz} has been studied previously, while 
%%we investigate the effect of the density perturbations on the QCD phase 
%%transition here. 
%We conclude that a first order QCD transition
%taking place at $T_{\rm c} = 150$~MeV                   %N  
%induces an inhomogeneity scale of a few meters
%\cite{IS}.                                               %N 
%%In comparison with heterogeneous 
%%nucleation via ad hoc dirt \cite{Christiansen}, we do not introduce any new, 
%%unknown objects. Our findings might have interesting implications for 
%%precision measurements of primordial abundances \cite{Mathews,Kainulainen}.

First-order phase transitions normally proceed via nucleation of bubbles 
of the new phase. When the temperature is spatially uniform and no 
%significant 
impurities are present, the mechanism is homogeneous nucleation. 
%The probability to nucleate a bubble of the new phase per time and volume
%is approximated by $\Gamma \approx T_c^4 \exp[-S(T)]$.
%The nucleation action $S$ is the free energy difference of the system 
%with and without the nucleating bubble, divided by the temperature. 
%
%Nucleation is a very rapid process, compared with the extremely slow 
%cooling of the Universe. The duration of the nucleation period, 
%$\Delta t_{\rm nuc}$, is found to be \cite{Fuller,Enqvist} 
%\begin{equation}
%  \Delta t_{\rm nuc} = 
%  - \frac{ \pi^{1/3} }{{\rm d}S / {\rm d}t \left|_{t_f} \right.} .
%\end{equation} 
%The time $t_f$ is defined as the moment when the fraction of space where
%nucleations still continue equals $1/e$. The heat flow preceding the 
%deflagration fronts reheats the rest of the Universe. 
We denote 
by $v_{\rm heat}$ the effective speed by which released latent heat 
propagates in sufficient amounts to shut down nucleations.
%(for details see \cite{IS}).         %N 
%In practice, 
%$v_{\rm def} < v_{\rm heat} < c_s$, where $v_{\rm def}$ is the velocity 
%of the deflagration front and $c_s$ is the sound speed \cite{Kurki-Suonio}.
%In the unlikely case of detonations $v_{\rm heat}$ should be replaced by the 
%velocity of the phase boundary in all expressions that follow.
%
The mean distance between nucleation centers, measured immediately after 
the transition completed, is
\begin{equation}
  d_{\rm nuc,hom} = 2 v_{\rm heat} \Delta t_{\rm nuc}
  \label{dnuceq},
\end{equation}
where $\Delta t_{\rm nuc}$ is the duration of the nucleation period   %N
\cite{IS,Ignatius}.                                                   %N
%This nucleation distance sets the spatial scale for baryon number 
%inhomogeneities.

%Lattice simulations~\cite{Lattice,Latticecs2} imply that in real-world 
%QCD the energy density must change very rapidly in a narrow temperature 
%interval. This can be seen from the microscopic sound speed in the quark 
%phase, $c_s \equiv (\partial p/\partial \varepsilon)_{\rm S}^{1/2}$.
%Lattice QCD indicates that $3 c_{\rm s}^2(T_{\rm c}) = {\cal O}(0.1)$
%\cite{Latticecs2}. Thus, the cosmological time-temperature relation
%is strongly modified already before the nucleations, due to
%\begin{equation}
%  \frac{ {\rm d}T }{ {\rm d}t } = - 3 c_{\rm s}^2 \frac{T}{t_{\rm H}},
%  \label{Tteq}
%\end{equation}
%where $t_H  \equiv  1/H = (3 M_{\rm pl}^2/8\pi\varepsilon_{\rm q})^{1/2}$
%with $\varepsilon_{\rm q}$ being the energy density in the quark phase. 
%This behavior of the sound speed increases the nucleation distance because 
%of the proportionality $\Delta t_{\rm nuc}\propto 1/[3 c_{\rm s}^2 (T_f)]$ 
%\cite{Ignatius}.

%In the thin-wall approximation the nucleation action has the following 
%explicit expression:
%\begin{equation}
%\label{C}
%  S(T)  =  \frac{ C^2 }{(1-T/T_c)^2}, \,\,\,\,\,\,
%  C  \equiv  4\sqrt{\frac{\pi}{3}} \frac{\sigma^{3/2}}{l\sqrt{T_c}}\ ,
%\end{equation}
%for small supercooling.
%Assuming further that $c_{\rm s}$ does not change very much during 
%supercooling, the following relation holds for the supercooling and
%nucleation scales:
%\begin{equation}
%  \frac{\Delta t_{\rm sc}}{\Delta t_{\rm nuc}} = 
%  \frac{\Delta_{\rm sc}}{\Delta_{\rm nuc}} = \frac{2}{\pi^{1/3}} \bar{S}
%  \label{Seq}. 
%\end{equation}
%Here we denote by $\Delta$ a relative (dimensionless) temperature interval
%and by $\Delta t$ a dimensionful time interval. $\bar{S} \equiv S(T_f)$ is 
%the critical nucleation action, $\bar{S} = {\cal O}(100)$.

Surface tension and latent heat are provided by lattice
simulations with quenched QCD only, giving the values $\sigma = 0.015
T_c^3$, $l = 1.4 T_c^4$~\cite{Iwasaki}.
Scaling the latent heat for the physical QCD 
leads us to take $l = 3 T_c^4$.
%
%With 
Using              %N
these values 
%for the latent heat and surface tension, 
the amount of
supercooling is $\Delta_{\rm sc} = 2.3 \times 10^{-4}$,
with large error bars.     %N                     
%{}From Eq.~(\ref{Seq}) it follows that 
%$\Delta_{\rm nuc} = 1.5 \times 10^{-6}$. Substituting 
%$3 c_{\rm s}^2 = 0.1$ into Eq.~(\ref{Tteq}), we find
%$\Delta t_{\rm nuc} = 1.5 \times 10^{-5} t_{\rm H}$ for the duration of the
%nucleation period. The nucleation distance depends on the unknown
%velocity $v_{\rm heat}$ in Eq.~(\ref{dnuceq}). 
%With the value 0.1
%for $v_{\rm heat}$, the nucleation distance $d_{\rm nuc,hom}$ would have 
%the value $2.9 \times 10^{-6} d_{\rm H}$. 
%One should take 
%%these values 
%this number            %N
%with caution, due to large uncertainties in $l$ and $\sigma$.
Due to rapid change of energy density near $T_c$, the microscopic sound  %N
speed in the quark phase has a small value,                              %N
$3 c_{\rm s}^2(T_{\rm c}) < 1$                                           %N
(for references see \cite{IS}).                                          %N
%As our reference set of 
%parameters, we take: $\Delta_{\rm sc} = 10^{-4}$, $\Delta_{\rm nuc} = 
%10^{-6}$, $\Delta t_{\rm nuc} = 10^{-5} t_H$.

In the real Universe the local temperature of the radiation 
fluid fluctuates. 
%We decompose the local temperature $T(t,{\bf x})$ 
%into the mean temperature $\bar{T}(t)$ and the perturbation 
%$\delta T(t,{\bf x})$. 
The temperature contrast is denoted by $\Delta
\equiv \delta T/ \bar{T}$. On subhorizon scales in the radiation dominated 
epoch, each Fourier coefficient $\Delta(t,k)$ oscillates with constant 
amplitude
%, which we denote by 
$\Delta_T(k)$. Inflation predicts a Gaussian 
distribution,
\begin{equation}
\label{pd}
p(\Delta){\rm d}\Delta = {1\over \sqrt{2\pi}\Delta_T^{\rm rms}}
\exp\left( - \frac12 {\Delta^2\over (\Delta_T^{\rm rms})^2}\right) 
{\rm d}\Delta \ .
\end{equation}
We find \cite{norm} for the COBE normalized \cite{Bennett} rms temperature 
fluctuation of the radiation fluid (not of cold dark matter)  
$\Delta_T^{\rm rms} = 1.0 \times 10^{-4}$ for a primordial Harrison-Zel'dovich
spectrum. The change of the equation of state prior to the QCD transition  
modifies the temperature-energy density relation, 
$\Delta = c_s^2 \delta\varepsilon /(\varepsilon + p)$. 
We may neglect the pressure $p$ 
near the critical temperature since 
$p \ll \varepsilon_{\rm q}$ at $T_{\rm c}$. 
On the other hand the drop of the sound speed enhances the amplitude of 
the density fluctuations proportional to $c_s^{-1/2}$ 
%\cite{SSW2}.
\cite{SSW}.             %N
Putting all
those effects together and allowing for a tilt in the power spectrum, 
the COBE normalized rms temperature fluctuation reads
\begin{equation}
\Delta_T^{\rm rms} \approx 10^{-4} (3 c_s^2)^{3/4} 
\left({k\over k_0}\right)^{(n-1)/2} \ .
\end{equation}
%where $k_0 = (aH)_0$. 
For a Harrison-Zel'dovich spectrum 
%($n=1$) 
and
$3 c_s^2 = 0.1$, we find $\Delta_T^{\rm rms} \approx 2 \times 10^{-5}$. 

A small scale cut-off in the spectrum of primordial temperature fluctuations
comes from collisional damping by neutrinos 
%\cite{Weinberg,SSW2}.
\cite{SSW}.                      %N 
%The 
%interaction rate of neutrinos is $\sim G_{\rm F}^2 T^5$. This has to be 
%compared with the angular frequency $c_s k_{\rm ph}$ of the acoustic
%oscillations. At the QCD transition neutrinos travel freely on scales 
%$l_{\nu} \approx 4 \times 10^{-6} d_{\rm H}$. Fluctuations below the 
%diffusion scale of neutrinos are washed out,
%\begin{equation}
%\label{diff}
%l_{\rm diff} = \left[\int^{t_c}_0\!\! l_{\nu}(\bar t) {\rm d} \bar t
%\right]^{\frac{1}{2}}\!\! 
%\approx 7\!\times\! 10^{-4} d_{\rm H} \label{nueq}\, .
%\end{equation}
%In Ref.~\cite{SSW2} the damping scale from collisional damping by neutrinos
%has been calculated to be $k^{\rm ph}_{\nu} = 10^4 H$ at $T=150$ MeV. The 
%estimate (\ref{diff}) is consistent with this damping scale.  
%We assume $l_{\rm smooth} = 10^{-4} d_{\rm H}$.
For $l_{\rm smooth}$, the length over which temperature is constant,    %N
we take the value $10^{-4} d_{\rm H}$ \cite{IS}.                        %N 
%The compression timescale for a homogeneous volume $\sim l_{\rm smooth}^3$
%is $\delta t = \pi l_{\rm smooth}/c_s \sim 10^{-3} t_{\rm H}$. Since 
%$\delta t \gg \Delta t_{\rm nuc}$ 
The temperature fluctuations are 
frozen with respect to the time scale of nucleations. 
%As long as 
%$l_{\rm smooth}$ exceeds the Fermi scale homogeneous 
%bubble nucleation applies within these small homogeneous volumes.
%This is a crucial difference to the scenario of heterogeneous nucleation 
%\cite{Christiansen}, where bubbles nucleate at ad hoc impurities.

Let us now investigate bubble nucleation in a Universe with spatially
inhomogeneous temperature distribution. 
%Bubble 
Nucleation effectively takes place while the 
temperature drops by the tiny amount $\Delta_{\rm nuc}$. To determine the 
mechanism of nucleation, we compare 
$\Delta_{\rm nuc}$ with the rms 
temperature fluctuation $\Delta_T^{\rm rms}$:

1. If $\Delta_T^{\rm rms} < \Delta_{\rm nuc}$, the probability to
      nucleate a bubble at a given time is {\em homogeneous} in space. This is
      the case of homogeneous nucleation. 

2. If $\Delta_T^{\rm rms} > \Delta_{\rm nuc}$, the probability to
      nucleate a bubble at a given time is {\em inhomogeneous} in space. 
      We call this inhomogeneous nucleation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
%\epsfig{figure=ihnfig1.ps,width=0.7\linewidth}
\epsfig{figure=itfig1.eps,width=0.325\linewidth}          %N
\epsfig{figure=itfig2.eps,width=0.325\linewidth}          %N 
\epsfig{figure=itfig3.eps,width=0.325\linewidth}          %N
\end{center}
\caption{\label{fig1}
Sketch of a first-order QCD transition in the inhomogeneous Universe. 
At $t_1$ the first hadronic bubbles (H) nucleate at the coldest spots
(light gray), while most of the Universe remains in the quark phase (Q). 
At $t_2$ the bubbles inside the cold spots have merged and have grown 
to bubbles as large as the temperature fluctuation scale. At $t_3$ the 
transition is almost finished. The last quark droplets are found in the 
hottest spots (dark gray).}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The quenched lattice QCD data and a COBE normalized flat spectrum 
lead to 
the values $\Delta_{\rm nuc} \sim 10^{-6}$ and $\Delta_T^{\rm rms} \sim 
10^{-5}$. We conclude that the cosmological QCD transition may proceed
via inhomogeneous nucleation. A sketch of inhomogeneous nucleation is shown 
in Fig.~\ref{fig1}. The basic idea is that temperature inhomogeneities 
determine the location of bubble nucleation. Bubbles nucleate first in the 
cold regions.

%The temperature change at a given point is governed by the Hubble 
%expansion and by the temperature fluctuations. 
For the fastest 
%changing 
fluctuations, with angular frequency $c_s/l_{\rm smooth}$, we find
\begin{equation}
\label{coolingrate}
{{\rm d}T(t,{\bf x})\over {\rm d} t} = {\bar T\over t_{\rm H}}
\left[- 3 c_s^2 + 
{\cal O}\left(\Delta_T {t_{\rm H}\over \delta t}\right)\right]\ .
\end{equation}
The Hubble expansion is the dominant contribution, as typical values
are $3 c_s^2 = 0.1$ from quenched lattice QCD and $\Delta_T^{\rm rms}
t_{\rm H}/\delta t \approx 0.01$. 
%from the discussion above. 
This means that the local temperature does never increase, except by 
the released latent heat during bubble growth.

To gain some insight in the physics of inhomogeneous nucleation,
let us first inspect a simplified case. We have some randomly 
distributed cold spheres of diameter $l_{\rm smooth}$ with equal and uniform 
temperature, which is by the amount $\Delta_T^{\rm rms}T_c$ smaller than 
the again uniform temperature in the rest of the Universe. When the 
temperature in the cold spots has dropped to $T_{\rm f}$,
homogeneous nucleation takes place in them. Due to the
Hubble expansion the rest of the Universe would need the time
%\begin{equation}
%\Delta t_{\rm cool} = {\Delta^{\rm rms}_T \over 3 c_s^2} t_H
%\end{equation}
$\Delta t_{\rm cool} = t_H \Delta^{\rm rms}_T  / 3 c_s^2$          %N
to cool down to $T_{\rm f}$. 
%Inside each cold spot there is a large number 
%of tiny hadron bubbles, assumed to grow as deflagrations. 
%They merge within $\Delta t_{\rm cool}$ if  
%$\Delta_{\rm nuc} < (v_{\rm def}/v_{\rm heat}) \Delta^{\rm rms}_T$.
%This condition should be clearly 
%fulfilled for our reference set of parameters. Thus 
The cold spots have fully 
been transformed into the hadron phase while the rest of the Universe 
still is in the quark phase. The latent heat released in a cold spot 
propagates in all directions, which provides the length scale
\begin{equation}
\label{lheat}
  l_{\rm heat} \equiv 2 v_{\rm heat} \Delta t_{\rm cool}. 
\end{equation}
If the typical distance from the boundary of a cold spot to the boundary
of a neighboring cold spot is less than 
$l_{\rm heat}$, then no hadronic bubbles can nucleate in the intervening 
space. 
%In this case the nucleation process is totally dominated by the 
%cold spots, and the average distance between their centers gives the 
%spatial scale for the resulting inhomogeneities. 
%In the following analysis for a more realistic scenario we 
%concentrate in this case, $l_{\rm heat} > l_{\rm smooth}$.

The real Universe consists of smooth patches of typical linear size 
$l_{\rm smooth}$, their temperatures given by the distribution~(\ref{pd}).
%As discussed above, the merging of tiny bubbles within a cold spot
%can here be treated as an instantaneous process. 
    %N
At time $t$ heat, 
coming from a cold spot which was transformed into hadron phase at time 
$t'$, occupies the volume
$V(t,t') = (4\pi/3) [l_{\rm smooth}/2 + v_{\rm heat}(t-t')]^3$.
The fraction of space that is not reheated by the released latent heat 
(and not transformed to hadron phase), is given at time $t$ by
\begin{equation}
\label{fihn}
f(t) \approx 1 - \int_0^{t}\Gamma_{\rm ihn}(t') V(t,t') {\rm d}t',
\end{equation}
where we neglect overlap and merging of heat fronts. 
%The other factor in Eq.~(\ref{fihn}), 
$\Gamma_{\rm ihn}$ is the volume 
fraction converted into the new phase, per physical time and volume as a 
function of the mean temperature $T=\bar{T}(t)$. 
%$\Gamma_{\rm ihn}$ is 
%proportional to the fraction of space for which temperature is in the 
%interval $[T_{\rm f}, T_f(1 + d\Delta)]$. This fraction of space is given 
%by Eq.~(\ref{pd}) with $\Delta = T_{\rm f}/T - 1$. Rewriting ${\rm d}\Delta$ 
%by means of 
%%Eq.~(\ref{Tteq}) 
%Eq.~(\ref{coolingrate})         %N 
%leads to the expression
We find                         %N
\begin{equation}
\label{gammai}
\Gamma_{\rm ihn} = 3 c_s^2{T_f\over T}{1\over t_{\rm H} {\cal V}_{\rm smooth}}
p(\Delta = \frac{T_{\rm f}}{T} - 1) , 
\end{equation}
where the relevant physical volume is ${\cal V}_{\rm smooth} = 
(4\pi/3) (l_{\rm smooth}/2)^3.$

The end of the nucleation period, $t_{\rm ihn}$, is defined through 
the condition $f(t_{\rm ihn}) = 0$. We 
introduce the variables $N \equiv (1 - T_{\rm f}/T)/\Delta_T^{\rm rms}$ and 
${\cal N} \equiv N(t_{\rm ihn})$. 
%Since $c_s$ may be assumed to be constant 
%during the tiny temperature interval where nucleations actually take place, 
%we find from 
%%Eq.~(\ref{Tteq}): 
%Eq.~(\ref{coolingrate}):               %N
%$1 - t/t_{\rm ihn} \approx 2/(3c_s^2) 
%\Delta_T^{\rm rms} (N - {\cal N})$. 
Putting everything together we determine 
${\cal N}$ from
\begin{equation}
{l_{\rm heat}^3\over l_{\rm smooth}^3} \int_{\cal N}^\infty {\rm d} N 
\frac{e^{-\frac12 N^2}}{\sqrt{2\pi}} 
\left(\frac{l_{\rm smooth}}{l_{\rm heat}} + N - {\cal N}\right)^3 = 1 .
\end{equation} 
%The COBE normalized spectrum gives $l_{\rm heat}/l_{\rm smooth} = 
%2 v_{\rm heat}(3 c_s^2)^{-1/4} (k/k_0)^{(n-1)/2}$.
For $l_{\rm heat}/l_{\rm smooth} = 1, 2, 5, 10$ we find 
${\cal N} \approx 0.8, 1.4, 2.1, 2.6$, respectively.

The effective nucleation distance in inhomogeneous nucleation is
defined from the number density of 
those cold spots that acted as nucleation centers, 
$d_{\rm nuc,ihn} \equiv n^{- 1/3}$. 
We find
\begin{eqnarray}
d_{\rm nuc,ihn} & \approx &
\left[\int_0^{t_{\rm ihn}} \Gamma_{\rm ihn}(t) {\rm d} t\right]^{-1/3} \\
& = & 
[\frac{3}{\pi} (1 - {\rm erf}({\cal N}/\sqrt{2})]^{-1/3} \, l_{\rm smooth}. 
\end{eqnarray}
With the above values $l_{\rm heat}/l_{\rm smooth} = 1, 2, 5, 10$ we get 
$ d_{\rm nuc,ihn} = 1.4, 1.8, 3.0,
$ $            %N
4.8 \times l_{\rm smooth}$, where
$l_{\rm smooth} \approx 1 \mbox{\ m}$. 

For a COBE normalized spectrum without any tilt and with 
a tilt of $n-1 = 0.2$,
%(where $(k_{\rm smooth}/k_0)^{0.1} \approx 23$),
together with $3c_s^2 = 0.1$ and $v_{\rm heat} = 0.1$, we find the estimate 
$l_{\rm heat}/l_{\rm smooth} \approx 0.4$ and 9, correspondingly. 
Notice that the values of $v_{\rm heat}$ and $3 c_s^2$ are 
in principle unknown.
Anyway, we can conclude that the case $\l_{\rm heat} > l_{\rm smooth}$
is a realistic possibility. 

With $2 v_{\rm heat}(3c_s^2)^{-1/4} (10^{-4}d_{\rm H}/l_{\rm smooth}) < 1$
and without positive tilt
we are in the region $l_{\rm heat} < l_{\rm smooth}$, where the geometry is 
more complicated and the above quantitative analysis does not apply.
In this situation nucleations take place in 
the most common cold spots (${\cal N} 
\sim 1$), which are very close to each other. We expect a structure of 
interconnected baryon-depleted and baryon-enriched layers with typical surface
$l_{\rm smooth}^2$ and thickness 
$l_{\rm def} \equiv v_{\rm def}\Delta t_{\rm cool}$. 
%In between $d_{\rm nuc,hom}$ 
%would be the relevant length scale of inhomogeneities. 
%An accurate analysis of this case requires 
%computer simulations, which is beyond the scope of the present work. However,
%it is clear that the result will be different compared with 
%homogeneous nucleation. 

We emphasize that inhomogeneous and heterogeneous
nucleation~\cite{Christiansen} are genuinely different mechanisms, although 
they give the same typical scale of a few meters by chance. 
If latent heat and surface tension of QCD would turn out to reduce  
$\Delta_{\rm sc}$ to, e.g., $10^{-6}$, instead of $10^{-4}$,
the maximal heterogeneous nucleation distance would fall to the 
centimeter scale, whereas on the distance in 
inhomogeneous nucleation this would have no effect.

We have shown that inhomogeneous nucleation during the QCD transition 
can give rise to an inhomogeneity scale exceeding the proton diffusion
scale
(2~m at 150~MeV).                  %N
The resulting baryon inhomogeneities could provide inhomogeneous initial 
conditions for nucleosynthesis. Observable deviations from the element 
abundances predicted by homogeneous nucleosynthesis seem to be 
possible in that case~\cite{MathewsKainulainen}.

In conclusion, we found that inhomogeneous nucleation leads to 
nucleation distances that exceed by two orders of magnitude 
estimates based on homogeneous nucleation. We 
%emphasize
point out         %N
that this new 
effect appears for the (today) most probable range of cosmological and 
QCD parameters.


\section*{Acknowledgments}
We acknowledge Willy the Cowboy for valuable encouragement. We thank 
K.~Rummukainen for crucial help,
K.~Jedamzik for discussions,                 %N
and J.~Madsen for correspondence.
J.I.~would like to thank the Academy of Finland and
D.J.S.~the 
%Alexander von Humboldt foundation and the 
Austrian Academy of Sciences for financial support.
We are grateful to the organizers of the Strong and Electroweak Matter 
meetings; the research presented here was initiated at SEWM-97 in Eger, 
Hungary.


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\end{thebibliography}

\end{document}

