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    {\small\rm\thepage\hfill
    \quad\quad ACTA ASTRONOMICA SINICA \hfill
    \hspace{2mm} Vol.44\vspace{2mm}}}} %%%%%%%
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    {\small\rm  suppl.\hfill W. Zhao \& Y. Zhang: Multi-Pulsar Cross-Correlation
     for Cosmic Gravitational Waves  %%�����ߣ���Ŀ���ƣ�
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\zihao{5}\songti

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    \zihao{6}\songti
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    Vol.44~ Suppl.
    \hfill\hspace{8mm} %%% if ``October/January'' then \hspace{8/10 mm} July
    \small ACTA ASTRONOMICA SINICA
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      Feb.,~2003 \\[-1.5 true mm] %%January
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\vs\vs\vs \bc {\Large\bf Multi-Pulsar Cross-Correlation Method
     for \\[2mm] Detecting Cosmic Gravitational Waves    %%�����⣩
%\\[1ex]
%$\!\!^*$
}

\vs\vs{\normalsize Wen Zhao\hs\&\hs Yang Zhang}  %%���������ߣ�

\vs
 (\footnotesize \ust)
\vs\vs \ec

\normalsize
\begin{quotation}

 \no{\bf A{\scriptsize\bf BSTRACT }}~~~\small
The method of cross-correlation of timing residuals of many
pulsars to detect the cosmic gravitational waves is studied. An
analytic expression has been derived for the mean squared
cross-correlation residuals in terms of the energy density of
cosmic gravitational waves for a general multi-pulsar case.

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

%%%% ����
  \vs\vs The existence of the gravitational waves predicted in the
theory of General Relativity has been observed from the binary
pulsars PSR B1913+16 indirectly long time ago. Over the last two
decades astronomers have been trying to use millisecond pulsars as
a highly accurate timing tool to detect directly the cosmic
gravitational waves that may have left over from the early
Universe. These studies are important not only in the detection of
cosmic gravitational waves itself, but also in determining the
matter content and the structure formation of the Universe. In
this paper we study  physical concepts involved in the timing
residuals of the radio signals from millisecond pulsars and derive
an analytic expression of the mean squared cross-correlation
residuals in terms of the energy density of cosmic gravitatioanl
waves for the multi-pulsar observations. We also discuss briefly
the advantage of using the multi-pulsar cross-correlation
residuals.

The method using the single-pulsar radio signal residuals to
detect the gravitational waves has long been well known. In the
observations of the radio signals from a millisecond pulsar both
the observer and the pulsar may be subject to the influences of
the comic gravitational waves. The formula for the energy density
of the cosmic gravitational waves in terms of the auto-correlation
of the signal residuals from a single-pulsar  has been given by
Detweiler~(1979). Assuming that the cosmic gravitational wave
spectrum is flat with a constant energy density within the
frequency interval $[\frac{1}{2}f, \,\, \frac{3}{2}f]$ and zero
outside, one has the auto-correlation of timing residuals
\be
<R_0^2(t)>=\frac{208G\rho}{243\pi^3 f^4}\,, \ee where $R_0(t)$ is
the residual of the pulsar signals, $G$ is the gravitational
constant, $ f$  is central frequency of the cosmic gravitational
waves, and $\rho$ is the energy density of the gravitational
waves. By measuring $R_0$, we can obtain an upper limit of $\rho$.
This method has also been used by Thorsett \& Dewey~(1996) to
process the seven-year accumulated data of the two single
millisecond pulsars PSR B1937+12 and PSR B1855+09. The upper limit
has been obtained as being $\rho/\rho_c\leq  10^{-8}h^{-2}$ of the
energy density of the cosmic gravitational  waves, where $\rho _c$
is the critical energy density.

Just like in any astronomical observations,
there is always random noise accompanying the radio signals from pulsars.
This will mix up with the cosmic gravitational waves.
By using the multi-pulsar cross-correlation method
one can reduce the level of noise and yield a better detection
of a possible common gravitational wave signal~(Hellings \& Downs 1983).

Usually two effects will cause a shift in the received radio
signal timing from pulsars. One is the gravitational waves and the
other is the noise. So the fractional frequency shift  in the
signal  of the pulsar $i$ may be written as
\be
 \frac{\Delta\nu_i(t)}{\nu_i}=\alpha_i h(t)+N_i(t)\,,
 \ee
 where
$N_i(t)$ represents all random noise sources unique to the pulsar
$i$ (including the random gravitational waves passing through the
pulsar $i$), $h(t)$ is the gravitational wave amplitude at the
Earth, $\alpha_i$ gives the angle factor
$\frac{1}{2}\cos2\phi(1-\cos\theta)$ for the $i$th pulsar,
$\theta$ is the angle between Earth-pulsar distance and the wave
propagation direction ($z$-axis), and $\phi$ is the angel between
a principle polarization vector of the wave and the projection of
the pulsar position on the transverse plane ($x-y$ plane). We
notice that the random cosmic gravitational waves occurring at the
two pulsars have been absorbed in the noise term in Eq(2) and will
be cancelled out in the cross-correlation in the following. The
residual of $i$-th pulsar by definition is
$R_i(t)=\int^t_0\frac{\Delta\nu_i}{\nu_i}dt'$. We define the
Two-Pulsar Cross-Correlation Function (TCF):
\be
\begin{array}{ccc}
R^2_{ij} \equiv\frac{1}{2}
  <R_i(t)R_j(t+\tau)+R_i(t)R_j(t-\tau)>          \\
  =\frac{1}{2T}  \int^{T/2}_{-T/2}
  [\oint d\Omega [\int_0^t \int_0^{t+\tau}+\int_0^t \int_0^{t-\tau}   ]
  (\alpha_i h\alpha_j h +\alpha_i h N_j+\alpha_j h N_i +N_i N_j) dt'dt{''}
   ]dt\,, %\,\,
\end{array}
\ee where $T$ is the time duration the pulsars are monitored, the
terms containing  $N_i$ and/or $N_j$ are the noise terms. Since
the noise is uncorrelated, the noise terms go to zero after
integration. It is usually assumed that $h(t)$ is due to an
isotropic background and is continuous and stochastic, so that the
term of $<h  h>$ can be moved outside the angle integration, and
we obtain, after some calculation
\be
 R^2_{ij}(t)\approx    \frac{104G\rho}{81\pi^3 f^4}
 \alpha_{ij}\,,
 \ee
where the mean angle factor $\alpha_{ij}$ is
given by
 \be \alpha_{ij} =  \frac{1}{4\pi}  \oint \alpha_i\alpha_j
d\Omega  =
   \frac{1-\cos\gamma_{ij}}{2} \ln(\frac{1-\cos\gamma_{ij}}{2})-
  \frac{1-\cos\gamma_{ij}}{12}+\frac{1}{3}\,.
 \ee

For the multi-pulsar observations we define the Multi-Pulsar
Cross-Correlation Function (MCF) as
\be
 R^2_{12...n}\equiv \sum^n_{i\neq j}  R^2_{ij}\,,
 \ee
where $R^2_{ij}$ is given Eq.(3). Similar calculation yields the
result
\be
 R^2_{12...n}=\frac{104G\rho}{81\pi^3 f^4} \sum^n_{i\neq
j}\alpha_{ij}\,.
 \ee
 In this formula $R^2_{12...n}$ and
$\sum\alpha_{ij}$ are  obtained from the measurement.

The method of multi-pulsar cross-correlation that we presented above
is much better than that of single pulsar auto-correlation
in order to detect the cosmic gravitational waves.
The more pulsars, the better.
This is because the noise occurring at each of the multi-pulsars
under consideration are cancelled out in the formulation of cross-correlation.
Therefore, we advocate a multi-pulsar monitoring project
with  regard to detection of gravitational waves whenever it is possible.

\vspace*{1mm}\vs\vn{\bf ACKNOWLEDGEMENTS}~~  This work has been
supported by the Chinese NSF
 under the number10173008,
 and by the 973 Projects of "Formation and Evolution of Galaxies".
 We thank the organizers for inviting us to
 the Sino-German Radio Astronomy Conference.

\bc {\bf References} \ec

\vspace{-4mm} \no \fns

 \no Detweiler S., ApJ, 1979, 234, 1101-1104\\
 \no Thorsett S.E., Dewey R.J., Phys.Rev.D, 1996, 53, 3468\\
\no Hellings R.W., Downs G.S., ApJ, 1983, 265, L39\\

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