Constraints on the primordial gravitational waves with variable sound
  speed from current CMB data
  We make a comprehensive investigation of the observational effect of the
inflation consistency relation. We focus on the general single-field inflation
model with the consistency relation $r=-8c_s n_t$, and investigate the
observational constraints of sound speed $c_s$ by using the Seven-Year WMAP
data, the BICEP tensor power spectrum data, and the constraints on $f_{\rm
NL}^{\rm equil.}$ and $f_{\rm NL}^{\rm orth.}$ from the Five-Year WMAP
observations. We find that the constraints on the tensor-to-scalar ratio $r$ is
much tighter if $c_s$ is small, since a large tilt $n_t$ is strongly
constrained by the observations. We obtain $r<0.37, 0.27$ and 0.09 ($dn_s/d\ln
k=0$) for $c_s$=1, 0.1 and 0.01 models at 95.4% confidence level. When taking
smaller values of $c_s$, the positive correlation between $r$ and $n_s$ also
leads to slightly tighter constraint on the upper bound of $n_s$, while the
running of scalar spectral index $dn_s/d\ln k$ is generally unaffected. For the
sound speed $c_s$, it is not well constrained if only the CMB power spectrum
data is used, while the constraints are obtainable by taking $f_{\rm NL}^{\rm
equil.}$ and $f_{\rm NL}^{\rm orth.}$ priors into account. With the
constraining data of $f_{\rm NL}^{\rm equil.}$ and $f_{\rm NL}^{\rm orth.}$, we
find that, $c_s\lesssim 0.01$ region is excluded at 99.7% CL, and the $c_s=1$
case (the single-field slow-roll inflation) is slightly disfavored at 68.3% CL.
In addition, the inclusion of $f_{\rm NL}^{\rm equil.}$ and $f_{\rm NL}^{\rm
orth.}$ into the analysis can improve the constraints on $r$ and $n_s$. We
further discuss the implications of our constraints on the test of inflation
models.
