Properties of polarized synchrotron emission from fluctuation-dynamo
  action -- I. Application to galaxy clusters
  Using magnetohydrodynamic simulations of fluctuation dynamos, we perform
broad-bandwidth synthetic observations to investigate the properties of
polarized synchrotron emission and the role that Faraday rotation plays in
inferring the polarized structures in the intracluster medium (ICM) of galaxy
clusters. In the saturated state of the dynamo, we find a Faraday depth (FD)
dispersion $\sigma_{\rm FD} \approx 100$ rad m$^{-2}$, in agreement with
observed values in the ICM. Remarkably, the FD power spectrum is qualitatively
similar to $M(k)/k$, where $M(k)$ is the magnetic spectrum and $k$ the
wavenumber. However, this similarity is broken at high $k$ when FD is obtained
by applying RM synthesis to polarized emission from the ICM due to poor
resolution and complexities of spectrum in FD space. Unlike the Gaussian
probability distribution function (PDF) obtained for FD, the PDF of the
synchrotron intensity is lognormal. A relatively large $\sigma_{\rm FD}$ in the
ICM gives rise to strong frequency-dependent variations of the pixel-wise mean
and peak polarized intensities at low frequencies ($\lesssim 1.5\,{\rm GHz}$).
The mean fractional polarization $\langle p \rangle$ obtained at the resolution
of the simulations increases from $<0.1$ at 0.5 GHz to its intrinsic value of
$\sim0.3$ at 6 GHz. Beam smoothing significantly affects the polarization
properties below $\lesssim 1.5\,{\rm GHz}$, reducing $\langle p \rangle$ to
$\lesssim 0.01$ at 0.5 GHz. At frequencies $\gtrsim 5\,{\rm GHz}$, polarization
remains largely unaffected, even when recovered using RM synthesis. Thus, our
results underline the need for high-frequency ($\gtrsim 5\,{\rm GHz}$)
observations with future radio telescopes to effectively probe the properties
of polarized emission in the ICM.
