The relation between gas density and velocity power spectra in galaxy
  clusters: qualitative treatment and cosmological simulations
  We address the problem of evaluating the power spectrum of the velocity field
of the ICM using only information on the plasma density fluctuations, which can
be measured today by Chandra and XMM-Newton observatories. We argue that for
relaxed clusters there is a linear relation between the rms density and
velocity fluctuations across a range of scales, from the largest ones, where
motions are dominated by buoyancy, down to small, turbulent scales:
$(\delta\rho_k/\rho)^2 = \eta_1^2 (V_{1,k}/c_s)^2$, where $\delta\rho_k/\rho$
is the spectral amplitude of the density perturbations at wave number $k$,
$V_{1,k}^2=V_k^2/3$ is the mean square component of the velocity field, $c_s$
is the sound speed, and $\eta_1$ is a dimensionless constant of order unity.
Using cosmological simulations of relaxed galaxy clusters, we calibrate this
relation and find $\eta_1\approx 1 \pm 0.3$. We argue that this value is set at
large scales by buoyancy physics, while at small scales the density and
velocity power spectra are proportional because the former are a passive scalar
advected by the latter. This opens an interesting possibility to use gas
density power spectra as a proxy for the velocity power spectra in relaxed
clusters, across a wide range of scales.
