$\Lambda$CDM model with a scalar perturbation vs. preferred direction of
  the universe
  We present a scalar perturbation for the $\Lambda$CDM model, which breaks the
isotropic symmetry of the universe. Based on the Union2 data, the
least-$\chi^2$ fit of the scalar perturbed $\Lambda$CDM model shows that the
universe has a preferred direction
$(l,b)=(287^\circ\pm25^\circ,11^\circ\pm22^\circ)$. The magnitude of scalar
perturbation is about $-2.3\times10^{-5}$. The scalar perturbation for the
$\Lambda$CDM model implies a peculiar velocity, which is perpendicular to the
radial direction. We show that the maximum peculiar velocities at redshift
$z=0.15$ and $z=0.015$ equal to $73\pm28 \rm km\cdot s^{-1}$ and $1099\pm427
\rm km\cdot s^{-1}$, respectively. They are compatible with the constraints on
peculiar velocity given by Planck Collaboration.
