We use the lensing potential map from Planck CMB lensing reconstruction analysis and the “ Public Cosmic Void Catalog ” to measure the stacked void lensing potential .
In this profile , four parameters are needed to describe the shape of voids with different characteristic radii R _ { V } .
However , we have found that after reducing the background noise by subtracting the average background , there is a residue lensing power left in the data .
The inclusion of the environment shifting parameter , \gamma _ { V } , is necessary to get a better fit to the data with the residue lensing power .
We divide the voids into two redshift bins : cmass1 ( 0.45 < z < 0.5 ) and cmass2 ( 0.5 < z < 0.6 ) .
Our best-fit parameters are \alpha = 1.989 \pm 0.149 , \beta = 12.61 \pm 0.56 , \delta _ { c } = -0.697 \pm 0.025 , R _ { S } / R _ { V } = 1.039 \pm 0.030 , \gamma _ { v } = ( -7.034 \pm 0.150 ) \times 10 ^ { -2 } for the cmass1 sample with 123 voids and \alpha = 1.956 \pm 0.165 , \beta = 12.91 \pm 0.60 , \delta _ { c } = -0.673 \pm 0.027 , R _ { S } / R _ { V } = 1.115 \pm 0.032 , \gamma _ { v } = ( -4.512 \pm 0.114 ) \times 10 ^ { -2 } for the cmass2 sample with 393 voids at 68 % C.L .
The addition of the environment parameter is consistent with the conjecture that the Sloan Digital Sky Survey voids reside in an underdense region .