Resurrecting the Power-law, Intermediate, and Logamediate Inflations in
  the DBI Scenario with Constant Sound Speed
  We investigate the power-law, intermediate, and logamediate inflationary
models in the framework of DBI non-canonical scalar field with constant sound
speed. In the DBI setting, we first represent the power spectrum of both scalar
density and tensor gravitational perturbations. Then, we derive different
inflationary observables including the scalar spectral index $n_s$, the running
of the scalar spectral index $dn_s/d\ln k$, and the tensor-to-scalar ratio $r$.
We show that the 95\% CL constraint of the Planck 2015 T+E data on the
non-Gaussianity parameter $f_{{\rm NL}}^{{\rm DBI}}$ leads to the sound speed
bound $c_{s}\geq0.087$ in the DBI inflation. Moreover, our results imply that,
although the predictions of the power-law, intermediate, and logamediate
inflations in the standard canonical framework ($c_s=1$) are not consistent
with the Planck 2015 data, in the DBI scenario with constant sound speed
$c_s<1$, the result of the $r-n_s$ diagram for these models can lie inside the
68\% CL region favored by the Planck 2015 TT,TE,EE+lowP data. We also specify
the parameter space of the power-law, intermediate, and logamediate inflations
for which our models are compatible with the 68\% or 95\% CL regions of the
Planck 2015 TT,TE,EE+lowP data. Using the allowed ranges of the parameter space
of the intermediate and logamediate inflationary models, we estimate the
running of the scalar spectral index and find that it is compatible with the
95\% CL constraint from the Planck 2015 TT,TE,EE+lowP data.
