Detecting Primordial Gravitational Waves Signal from BICEP2 and {\it
  Planck} HFI $353$GHz Dust Polarization
  The dust polarization is parameterized as a power law form of the multipole
$l$: $D^{XX}_{l}=A^{XX}l(l+1)l^{\alpha_{XX}}/(2\pi)$ ($XX$ denotes $BB$ or
$EE$), where $A^{XX}$ is its amplitude with the ratio $A^{BB}/A^{EE}=0.52\pm
0.02$ and $\alpha_{BB,EE}=-2.42\pm 0.02$. Extrapolating to $150$GHz from
$353$GHz yields a value of $D^{BB}_{l=80}=(1.32\pm 0.29)\times 10^{-2}\mu K^2$
(and an additional uncertainty $(+0.28,-0.24)\times 10^{-2}\mu K^2$) over the
range $40<l<120$. Based on these data, we report the tensor-to-scalar ratio
$r=A_{t}/A_{s}$ defined at $k_0=0.05 \text{Mpc} ^{-1}$ by joining the
BICEP2+{\it Planck}2013+WMAP9+BAO+HST and {\it Planck} HFI $353$GHz dust
polarization and its implication to the detection of the primordial
gravitational waves. Considering the $\Lambda$CDM+$r$ model, we found $r<0.108$
at $95\%$ confidence level with $\sigma_{stat}=0.29$ and $r<0.129$ at $95\%$
confidence level with $\sigma_{stat+extr}=0.29+0.28$. The results imply no
significant evidence for the primordial gravitational waves in $1\sigma$
regions. However the post probability distribution of $r$ peaks at a small
positive value. And $r$ moves to larger positive values when the extrapolation
error bars are included. This might imply a very weak signal of the primordial
gravitational waves. It also implies the crucial fact in calibrating the
amplitude of the dust polarizations in detecting the primordial gravitational
waves in the future. When the running of the scalar spectral tilt is included,
we found $r<0.079$ at $95\%$ confidence level with $\sigma_{stat}=0.29$ and
$r=0.091_{-0.069}^{+0.042}$ at $95\%$ confidence level with
$\sigma_{stat+extr}=0.29+0.28$. The later one implies the detection of the
primordial gravitational waves in $1\sigma$ regions at the cost of decreasing
the value of $D^{BB}_{l=80}$ to $0.67_{-0.25}^{+0.25}$.
