$Λ$CDM model with a scalar perturbation vs. preferred direction of the universe

We present a scalar perturbation for the $Λ$CDM model, which breaks the isotropic symmetry of the universe. Based on the Union2 data, the least-$χ^2$ fit of the scalar perturbed $Λ$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 $Λ$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.

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