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    {\small\rm\zihao{6}\songti\thepage\hfill
    \quad\quad ACTA ASTRONOMICA SINICA \hfill
    \hspace{2mm} Vol.44\vspace{2mm}}}} %%%%%%%
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    {\small\rm \zihao{6}\songti suppl.\hfill X. P. You et al.: Modeling
    the Evolving Cosmological
    Magnetic Fields %%�����ߣ���Ŀ���ƣ�
    \vspace{1.5mm}
    \hfill\thepage}}} %%%%%%%
    \normalsize\rm
\zihao{5}\songti

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    \small\rm
    \zihao{6}\songti
    \noindent
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    Vol.44~ Suppl.
    \hfill\hspace{8mm} %%% if ``October/January'' then \hspace{8/10 mm} July
    \small ACTA ASTRONOMICA SINICA
    \hfill
      Feb.,~2003 \\[-1.5 true mm] %%January
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\vs\vs\vs \bc {\Large\bf Modeling
    the Evolving Cosmological
    Magnetic Fields    %%�����⣩
%\\[1ex]
%$\!\!^*$
}

\vs\vs{\normalsize X. P. You\up{1,2},\hs\hs J. L.
Han\up{1}\hs\&\hs
Yang Chen\up{2}}  %%���������ߣ�

\vs
(\footnotesize 1~ \nao)\\
Email: xpyou@bao.ac.cn, hjl@bao.ac.cn\\
 (\footnotesize 2~ \nua)\\
  ygchen@nju.edu.cn
\vs\vs \ec

\normalsize
\begin{quotation}

 \no{\bf A{\scriptsize\bf BSTRACT }}~~~\small
We try to use the change of variance of residual rotation
measures (RRMs) of 43 extragalactic radio sources at Galactic
latitudes $|b|>70\deg\ $with their redshift to explore the
evolution of cosmological magnetic fields. Three models have been
considered to fit our data: the evolving Friedmann (EF) model, the steady
state (SS) model and the ionized cloud (IC) model. We find that the
EF+IC model can fit our data best, which means cosmological
magnetic fields probably evolve with time and evolve more in the
intercloud medium, though we can not exclude the other models.

 %\vn
 %\no{\bf Key words}~~  pulsar-mode  change-spectrum
\end{quotation}

\vs\vn Rotation measures (RMs) of extragalactic radio sources
contain information of large-scale cosmological magnetic fields.
The observed rotation measures contain two contributions: Galactic
rotation measure (GRM) and residual rotation measure (RRM). In
other directions of our Galaxy, the Galactic contribution is very
uncertain, while near the two Galactic poles, the GRM is very
small, about $\pm3\mbox{ rad m}^{-2}$ (Han, Manchester \& Qiao
1999) .

We select all objects of Galactic latitudes $|b|>70\deg\
$and use the change of variance of RRMs with redshift to
explore the evolution of cosmological magnetic fields. In this sky
area, some sources have RM values but do not have redshift
measured, which had to be discarded in our analysis. We
exclude a few objects with redshift $z>2.5$. These objects with
rotation measure $|RM|>100\mbox{ rad m}^{-2}$ are ruled out too,
because the large rotation measure is most probably intrinsic to
the source. Finally, we have 43 objects with observed
rotation measures and redshift values.

\begin{figure}[h]
\parbox[b]{6.5cm}
{\hspace{3.5mm}\includegraphics[bb=0 61 408
488,scale=0.4]{RMz_variance.eps}
    {\footnotesize
    Fig.\,1~~Estimated variance of residual rotation measure $V_{RRM}$
        versus redshift $z$
        for objects near two Galactic poles}
 }
 \end{figure}
 \begin{figure}[h]
 \vspace{-77mm}
 \ \hfill \
\parbox[b]{6.5cm}
{\hspace{3mm}\includegraphics[bb=0 61 408 488,scale=0.4]{wpk.eps}
    {\footnotesize
    Fig.\,2~~The relation between variance of residual rotation measure
        $V_{RRM}$ and redshift
        $z$ in WPK's data}
 }
 \end{figure}


%\begin{figure}
 %  \begin{minipage}[t]{.5\textwidth}
%        \centering
 %       \psfig{file=RMz_variance.ps,height=7cm,angle=-90}
 %       \caption{Estimated variance of residual rotation measure $V_{RRM}$
 %       versus redshift $z$
  %      for objects near two Galactic poles}
  %      \par\vspace{0pt}
  % \end{minipage}%
  % \hfill
  % \begin{minipage}[t]{.5\textwidth}
  %      \centering
  %      \psfig{file=wpk.ps,height=7cm,angle=-90}
  %      \caption{The relation between variance of residual rotation measure
  %      $V_{RRM}$ and redshift
   %     $z$ in WPK's data}
  %      \par\vspace{0pt}
  %  \end{minipage}
%\end{figure}
We now consider three models of intergalactic Faraday rotation.

Nelson (1973) has presented the evolving Friedmann (EF) model. In this
model, both the cloud and the medium between the clouds evolve
with redshift. The distribution of RMs have a variance of
\[V_{EF}=\gamma_{_{EF}}\int^{z_s}_0 \frac{(1+z)^3}{(1+\Omega
z)^{1/2}}dz \qquad\mbox{ , } \qquad
\gamma_{_{EF}}=(0.81n_0B_{\parallel 0})^2\frac{l_0c}{H_0}\] where
$z_s$ is the source redshift, $\Omega$ the density parameter of
the universe, $n$ the electron density (cm$^{-3}$), $B_{||}$ the
line-of-sight field component ($\mu$G), $l$ the size of the cell
(pc) and $H$ the Hubble parameter. The subscript 0 denotes the
present epoch. $c$ is the light speed.

Burman (1974) derived an expression for the variance of RMs in the
steady state (SS) model. In this case, neither the cloud nor the
medium between the clouds evolves with redshift. The electron
density $n_e$, the field component along line-of-sight $B_{||}$ and cell
size $l$ do not vary with epoch. The variance becomes
\[V_{SS}=\frac{\gamma_{_{SS}}}{4}\left[1-(1+z_s)^{-4}\right]
\qquad\mbox{ , } \qquad
\gamma_{_{SS}}=(0.81n_eB_{\parallel})^2\frac{lc}{H}\]

In the ionized cloud (IC) model (Thomson \& Nelson 1982), the
cloud does not evolve with redshift, but the medium between the
clouds does, then the variance is
\[V_{IC}=\frac{\gamma_{_{IC}}}{2}\left[1-(1+z_s)^{-2}\right]
\qquad\mbox{ , } \qquad \gamma_{_{IC}}=(0.81n_eB_{\parallel c
})^2\frac{l_ccf_0}{H_0}\] where $f_0$ is the fraction of space in
present epoch. The subscript c means the cloud.

We can use these three models to fit our RRM data. The
results are shown in figure 1. It is clear that the dispersion
of $|RRM|$ increases with redshift (upper panel). It is also
obvious that none of the three models can fit our data well, but the
combination of the EF model and IC model (see model parameters in table
1) can fit our data better (lower panel).
\bc{\small\bf

 Table 1~~   Best fit model parameter $\gamma$ as estimated
from RRM data
}\\[2mm]

\fns \tabcolsep 4mm
\begin{tabular}{cccccc}
\hline\\[-3.6mm]\hline
 &EF&SS&IC&\multicolumn{2}{c}{EF+IC}\\
 & & & &$\gamma_{\rm EF}$&$\gamma_{\rm IC}$\\
 \hline
$ \gamma $ (rad$^2$\,m$^{-4}$) &     88    &     828     &     650    &    44&442 \\
 \hline\\[-3.6mm]\hline%\\[-4.5mm]\hline
 \end{tabular}
 \ec
%\begin{table}
%\centering \caption{Best fit model parameter $\gamma$ as estimated
%from RRM data} \label{parameter}
%\begin{tabular}{|c|c|c|c|c|c|}\hline
%                          &\raisebox{-2.3ex}[0pt]{EF}&\raisebox{-2.3ex}[0pt]{SS}&\raisebox{-2.3ex}[0pt]{IC}&
 %                         \multicolumn{2}{|c|}{\raisebox{-1.5ex}[0pt]{EF+IC}}\\\cline{5-6}
 %                         & & & &$\gamma_{\rm EF}$&$\gamma_{\rm IC}$\\
 %                         \hline
%$ \gamma $ (rad$^2$ m$^{-4}$) &     88    &     828     &     650    &    44&442 \\
%\hline
%\end{tabular}
%\end{table}

We also use the same method to fit the WPK's RRM data (Welter, Perry
\& Kronberg, 1984). The sources are distributed randomly in sky,
so the GRM is more uncertain, hence error bar of their data is
larger than ours. But the number of their sample is greater than
ours. In figure 2, we can see that the best fitting to their data is
almost the same as ours.

We conclude that the EF+IC model can best fit our and WPK's RRM
data, which indicates that cosmological magnetic fields probably evolve
with time and evolve more in the intercloud medium. But no model
can be excluded from our results because of small sample and large
error bar. We need observe more rotation measures of high
polarized quasars and AGNs to increase sample size and decrease the error
bar, which can improve our results.

%\vs
\vn{\bf ACKNOWLEDGEMENTS}~~ We would like to thank Katia
Ferriere for helpful discussion. This work is supported by
the National Natural Science
Foundation of China (19903003 and 10025313), the National
Key Basic Research Science Foundation of China (G19990754)
and the partner group of MPIfR at NAOC.

\newpage\vspace*{1mm}
\bc {\bf References} \ec

\vspace{-4mm} \no \fns

\no Burman R.R.  PASJ, 1974, 26, 507\\
\no Han J.L., Manchester R.N., Qiao G.J., MNRAS, 1999, 306, 371\\   %��ȱ���
\no Nelson A.H.,  PASJ, 1973, 25, 489\\
\no Thomson R.C., Nelson A.H.,   MNRAS, 1982, 201, 365\\
\no Welter G.L., Perry J.J., Kronberg P.P.,  ApJ, 1984, 279, 19\\

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