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\begin{document}
\title{Magnetic Wall From Chiral Phase Transition and CMBR correlations}
\author{Leonard S. Kisslinger$^\dagger$\\
    Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213}
\maketitle
%pacs
\indent
\begin{abstract} Possible CMBR correlations are estimated with a model in 
which a Hubble-size magnetic wall is formed during the QCD chiral phase
transition. Measureable polarization correlations are found for $l$-values
greater than 1000. It is also found that metric perturbations from the wall 
could give rise to observable CMBR correlations for large $l$.
  
\end{abstract}

\vspace{0.5 in}

\noindent
PACS Indices:98.80.Cq,98.80Hw,12.38.Lg,12.38.Mh \\
\vspace{2 mm}
$^\dagger$ email: kissling@andrew.cmu.edu

\newpage
\section{Introduction}

\hspace{.5cm}

   In the present paper the possibility is explored that large-scale 
magnetic structures created during the QCD (Quantum Chromodynamic) 
chiral phase transition might lead to observable CMBR  (cosmic microwave 
background radiation) correlations . This work is motivated by improved CMBR
observations in progress, which promise measurements of polarization and
temperature correlation measurements with multipoles $l$ in the thousands.
These magnetic structures could also be primoidal seeds of galactic and 
extra-gallactic magnetic fields. A long-standing problem of astrophysics 
is the origin of the large-scale galactic and extra-gallactic magnetic 
fields which have been observed. Of particular importance for cosmology is 
the possible seeding of these magnetic structures by primoidal, 
early-universe, magnetic structures. For a recent review see Ref\cite{gr}.
We do not, however, investigate gallactic or extra-gallactic
magnetic structure seeded by the QCD phase transition in the present work, 
but center on CMBR polarization and gravitational wave correlations,

   In most of the theoretical treatments, including inflationary models,
the magnetic fields arise from electrically charged particle motion.
Considerations of nucleosynthesis and CMBR and the galactic magnetic 
fields have been used to constrain the magnitudes of the primoidal tangled
(random) magnetic fields to values of about 10$^{-9}$
Gauss, although in a recent model\cite{dfk} fields of two orders of magnitude
larger are consistent with CMBR observations. Constraints on homogeneous 
primoidal magnetic fields from CMBR have also been determined\cite{bfs}.

Early universe phase transitions are of great interest. The electroweak and 
QCD chiral phase transitions are of particular interest as common ground
for particle physics and astrophysics. For the electroweak phase transition,
using an Abelian Higgs model\cite{kv} with the QED lagrangian included, 
magnetic field generation has been calculated\cite{ae,cst} in bubble 
collisions. For the QCD phase transition primoidal magnetic fields 
generated from charged currents at the bubble surfaces during the nucleation 
have been estimated\cite{co,soj} to be as large as 10$^8$ Gauss, and still
be consistent with observed values of gallactic and extra-galactic fields.
There has also been a recent calculation\cite{kl} of the CMBR power spectrum 
from density perturbations caused by promoidal magnetic fields that is
similar to the calculation of Ref\cite{dfk}, but for scalar perturbations.

The present paper is motivated by our QCD instanton model of bubble walls
formed during the QCD chiral phase transition\cite{lsk} and by the recent 
work of Forbes and Zhitnitsky, who have used a QCD domain wall\cite{fz1}
as a mechanism for generating magnetic fields which could evolve to 
large-scale galactic fields\cite{fz2}. Considering classical theory of bubble 
collisions, in which walls with the same surface tension as the colliding 
bubbles are formed within the merged bubbles, it was conjectured\cite{lsk} 
that hubble-scale instanton walls might be formed, with lifetimes sufficient 
to form magnetic walls. Since the instanton walls are similar to those 
modelled in Ref\cite{fz2}, we use the magnitude of the magnetic wall
conjectured in that work to study possible consequent CMBR correlations.
In section II the description of the QCD phase transition in our instanton
model and the possible resulting magnetic wall are discussed. In section III
the CMBR polarization correlations are derived, and in section IV the
correlations from metric perturbations are derived. In section V we give
our conclusions.


\section{QCD Chiral Phase Transition and Magnetic Wall}

\hspace{.5cm}

  In our model\cite{lsk} of the bubbles of the hadronic-phase 
universe nucleating within the quark-gluon phase universe during the QCD
chiral phase transition, which we assume is first order, we use the purely
gluonic QCD Lagrangian density, $ \mathcal{L}^{glue} = G_{\mu\nu}^a
G^{\mu\nu a}/4$, with $G^{\mu\nu a}$ the color field tensor, with $(\mu,\nu)$
= (1...4) and (a,b,...) = (1...3) Dirac indices and color indices, 
respectively. Recognizing that it has been shown that QCD instantons 
can represent the midrange nonperturbative aspects of QCD, we use the 
instanton model\cite{inst} for the color field
\beq
\label{instanton}
 A_\mu^{n,inst}(x)& = & \frac{2 \eta^{-n}_{\mu\nu}x^\nu}{(x^2 + \rho^2)}\\
        G^{n,inst}_{\mu\nu}(x) & = & -\frac{\eta^{-n}_{\mu\nu} 4 \rho^2}
{(x^2 + \rho^2)^2},
\eeq
for the instanton and a similar expression with -n for the anti-instanton, 
where $\rho$ is the instanton size and the  $\eta^{n}_{\mu\nu}$ 
are defined in Ref.\cite{inst}. The energy-momentum tensor of the bubble 
wall is given by
\beq
\label{enmom}
T^{\mu \nu} &=&  G^{\mu\alpha}_a G_{\alpha a}^\nu
-\frac{1}{4}g^{\mu \nu} G^{\alpha \beta}_a G_{\alpha \beta a} ,
\eeq
which gives in the instanton model at finite temperature in Minkowski
space the spatial energy momentum tensor
\beq
\label{momentum}
  T^{ij,inst} & = & \, \left( \frac{4 \overline{\rho}^2 \overline{N}}
{(x^2 +\overline{\rho}^2)^2} \right)^2 \, \delta_{ij},
\eeq
where $\overline{\rho}$ is the instanton size and $\overline{N}$ is the
instanton density at the bubble surface at temperature T=T$_c$, the
temperature of the chiral phase transition, approximately 150 MeV.
In Ref\cite{lsk} the energy density, $T^{0 0}$ was shown to be consistent 
with numerical calculations of the surface tension. $T^{ij}$ can be used 
to calculate bubble collisions.

Recently, effective field models have been used to calculate QCD domain
walls\cite{fz1}, which have a space-time structure very similar to a wall
composed of instantons, with the form given in Eq.(\ref{instanton}). Such 
a wall could interact with nucleons in the hadronic phase to produce 
electromagnetic structures via effective magnetic and electric dipole 
moments\cite{cvvw} of the nucleon field. This model was used\cite{fz2} 
to investigate possible primoidal magnetic fields at the time of the chiral 
phase transition that could lead to large-scale galactic magnetic fields.
As pointed out in Ref.\cite{lsk}, the collision of nucleating bubbles
during the phase transition could lead to an interior gluonic wall. If
the theory of classical bubble collisions can applied, the interior wall
would be similar to the bubble walls, with the same surface tension. In
other words, there would also be an instanton wall with an energy-momentum
tensor given by Eq.(\ref{momentum}). Recognizing that the mathematical form
of the instanton and domain walls are very similar, the arguments of 
Ref.\cite{fz2}, including estimates of the lifetime of the interior QCD
instanton wall, can be applied to estimate the primary electromagnetic wall 
that might have been formed at t$\simeq 10^{-4}$ s, giving equal electric
and magnetic fields:
\beq
\label{wall}
  {\bf B}_W({\bf x}) & = & B_W e^{-b^2(x^2 + y^2)} e^{-M_n^2 z^2},
\eeq
or in momentum space
\beq
\label{wallk}
  {\bf B}_W({\bf k}) & = & \frac{B_W}{2\sqrt{2}b^2M_n} e^{-(k_x^2 + k_y^2)
/4b^2} e^{-k_z^2/4M_n^2},
\eeq
where $b^{-1}$ is of the scale of the horizon size, $d_H$, at the end of the 
chiral phase transition (t $\simeq 10^{-4}$ s), $b^{-1} =d_H \simeq$ few km, 
while $M_n^{-1} \simeq 0.2 fm$. 
Therefore, although $B_W$ is very large, estimated
to be $\simeq 4\times  10^{17}$ Gauss in Ref\cite{fz2}, since the wall 
occupies a very small volume of the universe, such a structure is 
compatible with nucleosynthesis, galaxy structure and the present CMBR 
observations.

In the present work we conjecture that the QCD bubble collisions lead
to a magnetic wall given by Eq.(\ref{wall}) at $10^{-4}$s and study the 
effects on CMBR polarization correlations and metric fluctuations.

\section{CMBR Polarization Correlations}

\hspace{.5cm}

  In this section we investigate the polarization correlations arising from
the electric and magnetic fields given in Eq.(\ref{wall}). The primoidal
magnetic wall of the present work is quite different from the tangled 
fields considered previously\cite{sb,ss}, however, the evolution to the last 
scattering surface has many features in common with these studies. 
In addition to the polarization anisitropy which results from the magnetic
wall itself, which we discuss in next subsection, in the subsequent
subsection we also discuss possible effects arising from scattering of the 
background radiation from the nucleon moments at the time of the phase 
transition.

\subsection{Polarization Correlations From Magnetic Wall}

\hspace{.5cm}

  In treating the temperature and polarization matrix for the Stokes
parameters\cite{kks} we use the angular representation of Ref.\cite{hw}
to derive the power spectrum of B-type polarization anisitropies, $C^{BB}_l$,
which arise to a good approximation from the polarization source $P^{(1)}
 = -E^{(1)}_2/\sqrt{6}$, given by
\beq
\label{source}
   E^{(1)}_2 & = & \frac{1}{4}\sqrt{\frac{5}{6\pi}}\int d\Omega 
E(\hat{n},x,\eta)(Y_2^1 - \sqrt{5}Y_1^1),
\eeq
where the electromagnetic wave is propagating with ${\bf k} = k\hat{n}$,
$\eta$ is conformal time, and the $Y_l^m$ are the standard spherical
harmonics.
From our model the source at the wall is $E^{(1)}_2 = 5 B_W/12 \sqrt{2} b^2 
M_n \equiv \mathcal{B}_W$ at the wall. To get the power spectrum we must
evaluate $<B_z({\bf k},\eta)B_z({\bf k'},\eta)>$. Using the fact that
$exp(-k^2 d_H^2) \simeq 1.0$ at the time of the chiral phase transition,
\beq
\label{bcor}
   <B_z({\bf k},\eta)B_z({\bf k'},\eta)> & \simeq & \mathcal{B}_W^2
 \delta(k_x-k'_x) \delta(k_y-k'_y)<e^{-k_z^2/4M_n^2}e^{-k^{'2}_z/4M_n^2}> 
\nonumber \\
       & \simeq & \mathcal{B}_W^2 d_H e^{-k_z^2/4M_n^2} \delta({\bf k-k'}).
\eeq 
This gives for the polarization
power spectrum
\beq
\label{Bpower}
   C^{BB}_l & = & \frac{(l+1)(l+2)}{\pi}\mathcal{B}_W^2 d_H \int dk k^2
 \frac{j_l^2[k(\Delta \eta)]}{k^2(\Delta \eta)^2},
\eeq
where the conformal time integral over the visibility function has been
carried out and $\Delta \eta$ is the conformal time width at the last
scattering. The integral over the spherical Bessel function is carried
out by using the fact that $j_l(x)$ peaks at $l$ and that the integral
$\int dz j_l^2(z) = \pi/(4l)$ for large $l$. Therefore in the range
$ 100 < l < 2000$ we have the approximate result
\beq
\label{cbb}
    C^{BB}_l & \simeq & \frac{25 d_H^5 B_W^2}{1152 M_n^2 \Delta\eta^3}l^2.
\eeq
Using the parameters $M_n\Delta\eta =1.5 \times 10^{39}$ (from 
Refs.\cite{ss,hs}), $d_H = 0.37\times 10^{24} GeV^{-1}$, and $B_W = 
1.0\times10^{17}$ Gauss, 
\beq
\label{cbbf}
    C^{BB}_l & \simeq & 4.25\times10^{-8} l^2
\eeq

The result for the B-type power spectrum is shown in Fig. 1. One can see
that for $l \simeq 1000$ the values of $ C^{BB}_l$ exceed those in
inflationary models, and the $l$ dependence is quite different.
\begin{figure}
\begin{center}
\epsfig{file=cmbrwall1.eps,height=5cm,width=12cm}
%\caption{}
{\label{Fig.1}}
\end{center}
\end{figure}

\subsection{Polarization Correlations From Magnetic Dipole Scattering}

\hspace{.5cm}

In this subsection we discuss another possible source of polarization
correlation arising from the horizon-size instanton wall created during
the chiral phase transition and leading to an orientation of nucleon
magnetic dipole moments. As the nucleon moments are alligned along the
wall the background radiation will scatter from these moments producing
scattered waves and magnetic structure which would give polarization at
the last scattering surface. Taking the thickness of the instanton wall
as the scale for the volume associated with the magnetic dipole moments,
The density of moments, n, is obtained from the strength of the magnetic
field:
\beq
\label{bscat}
     B_z & = & \frac{2n\mu_n}{r_W^3},
\eeq
where $\mu_n$ is the neutron magnetic dipole moment and $r_W$ is the wall
thickness. Making use of the torque of the magnetic moment in a magnetic
field ${\bf B}$, $d{\bf m}/dt = \mu_n^2 {\bf s} \times {\bf B}$, with
${\bf s}$ the spin of the neutron, one obtians the magnitude of the scattered
B-field at time t=$10^{-4}$ s: $B_{scat} = n^2 \mu^2/2 cos(\theta_s) k E_0
exp(ikr)/r$, with $\theta_s$ the angle between the neutron spin and the
direction of the incident radiation and $ E_0$ the E-field of the incident
radiation. From this one obtains the Stokes parameter U, and in the notation
of Ref\cite{hw} the polarization source
\beq
\label{b2scat}
   B_2^{0}({\bf k},\eta) & = & \frac{k^2 E_0^2}{2\sqrt{6}}(n\mu_n^2)^2.
\eeq

  Without a detailed calculation one sees that the resulting polarization
correlations will be very small, since the extra $k^2$ dependence from the
scattering from the moments introduces a factor of $ (l^2/\Delta\eta)^2$.
Therefore the scattering from the nucleon moments at the time of the
phase transition is negligible.
 
\section{Metric Perturbations From Magnetic Wall and CMBR Power Spectrum}

\hspace{.5cm}

  In this section we derive the power spectrum from the gravitational waves
arising from the magnetic wall of Sec.2. The calculation is very similar
to that of Ref.\cite{dfk}, in which the power spectra was derived
for tangled magnetic fields such as those considered in Refs.\cite{sb,ss}, 
with various scenarios for the scale dependence. Although our model of the 
narrow hubble-size magnetic structure is quite different, we can make use 
of much of the formalism of Ref.\cite{dfk}.

  The stress-energy tensor\cite{jackson} for our model magnetic wall, which
has only 33 spacial components, is
\beq
\label{t33}
   T_{33}({\bf k}) & = & \frac{1}{8\pi} \int d^3q B_3({\bf q})B_3({\bf k-q})
 \\ \nonumber
          & = & \frac{2\pi^3\sqrt{\pi}}{M_n}B_W^2 e^{-\frac{3k_3^2}{8M_n^2}}
\eeq
at the time of the chiral phase transition. Note that we are assuming that
the wall is of hubble size in the x-y directions and have omitted the
parts of the expression for the $k_x,k_y$ dependence shown in Eq.(\ref{wallk}).
From Eq.(\ref{t33}) we obtain the B-wall power spectrum
\beq
\label{bpower}
  < T_{33}({\bf k},\eta) T_{33}({\bf k'},\eta)> & = & \int d^3q d^3q'
 < B_3({\bf q}) B_3({\bf k-q}) B_3({\bf q'}) B_3({\bf q}) B_3({\bf k'-q'})>.
\eeq
In evaluating Eq.(\ref{bpower}) we use
\beq
\label{length}
    < e^{-\frac{k^2}{8M_n^2}} e^{-\frac{k^{'2}}{8M_n^2}}\delta(k_x)\delta(k'_x)
 \delta(k_y)\delta(k'_y) & = &  e^{-\frac{k^{'2}}{4M_n^2}}d_H 
 \delta({\bf k - k')}).
\eeq
Using the notation of Ref\cite{dfk} with a tensor projection,
\beq
\label{bpower1}
     < T_{33}({\bf k},\eta) T_{33}({\bf k'},\eta)> & = & \frac{4}{a^8} 
  f^2(k^2) \delta({\bf k - k')}),
\eeq
with
\beq
\label{f}
    f^2(k^2) & = & \frac{2^3 \pi^9}{M_n^2}d_H B_W^4.
\eeq
With $h_{ij}$ the tensorial perturbations of the Friedman universe,
$ds^2 = a^2[-d\eta^2 +(\delta_{ij} + 2h_{ij})dx^idx^j]$. The Einstein 
equations, using the representation  $h_{ij}= 2HQ^{2}_{ij}$ of Ref\cite{hw},
with $\nabla^2Q^(2)_{ij}=-k^2Q^(2)_{ij}$  are
\beq
\label{einstein}
   \ddot{H}+2\frac{\dot{a}}{a}  \dot{H}+k^2H & = & 8\pi G \frac{4}{a^8}
  f^2(k^2).
\eeq
The power spectrum for the tensor metric fluctuations are given by
\beq
\label{mpower}
 <\dot{h}_{ij}({\bf k'},\eta)\dot{h}_{ij}({\bf k},\eta)> & = & 
4|\dot{H}({\bf k)})|^2 \delta({\bf k - k')}).
\eeq
Since Eq(\ref{einstein}) was investigated in Ref\cite{dfk}, except with a
different magnetic stress tensor, we use the solutions that they obtained.
Assuming that the magnetic wall is formed at t=$10^{-4}$ s in the radiation
dominated epoch, at redshift $z_{in}$, and neglecting perturbations created
after the time of matter-radiation equilibration $\eta_{eq}$, for 
$\eta > \eta_{eq}$ the approximate solution for $\dot{H}$ is
\beq
 \label{H}
   \dot{H}(k,\eta) & = & 4\pi G \eta_o^2 z_{eq}ln(\frac{z_{in}}{z_{eq}})
 \frac{j_2(k\eta)}{\eta}f(k).
\eeq
Carrying out integrals over Bessel functions, the solution
for the metric fluctuation power spectrum is for $l>>1$
\beq
\label{mcl}
  C_l & = & \left[\frac{14}{25}Gz_{eq} ln\frac{z_{in}}{z_{eq}}\right]^2 l^5 
 \eta_o^2 \int dz \frac{1}{z^4}f^2(z/\eta_o)J_{l+3}(z).
\eeq
Using the form of f(z) given in Eq(\ref{f}), the integral in Eq(\ref{mcl})
is approximately
\beq
 \label{integral}
     \int dz \frac{1}{z^4}f^2(z/\eta_o)J_{l+3}(z) & \simeq & 
 \frac{8 \pi^9 d_H}{M_n^2}B_W^4\frac{.1061}{l^4}
\eeq
for $M_n \eta_o >> l >> 1$. Taking $t_{in}$ at $10^{-4}$ s or 
$z_{in} = 1.17 \times 10^8 z_{eq}$ and $d_H = 0.371 \times 10^{24} GeV^{-1}$,
the power spectrum is ($l >> 1$)
\beq
\label{power}
 l(l+1)C_l & \simeq & 6.9 \times 10^{-7} l^3
\eeq
for $B_W = 10^{17}$ Gauss. Therefore, the metric perturbations from the
QCD-induced wall result in CMBR effects large compared to other tensor 
perturbations that have been estimated for for $l$ values of the order
of 1000.

\section{Conclusions}

\hspace{.5cm}

  We conlcude that if the QCD chiral phase transition produces a gluonic
wall of domain size and nucleon thickness at the time of the final
collision of nucleating bubbles, and produces a magnetic wall as that
found in the domain wall model of Refs\cite{fz1,fz2}, it would result
in polarization correlations which have a $l$-dependence for $l > 1000$
different than other cosmological predictions and which should be measurable 
with the next generation of CMBR measurements. Also, there would be 
distinguishable temperature correlations arising from metric fluctuatons.

\vspace{2mm}

\centerline{\bf Acknowledgements}
\bigskip
The author would like to acknowledge helpful discussions with Ariel
Zhitnitsky on his domain wall model, and helpful discussions with
Ho-Meoyng Choi, Ernest Henley, Pauchy Hwang and Mikkel Johnson.
This work was supported in part by NSF grant 8 and by the 
Taiwan CosPA Project, Taiwan Ministry of Education 89-N-FA01-1-3,
and in part by the DOE's Institute of Nuclear Theory
at the University of Washington while the author was in residence.

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\end{document}

