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 _ { { NL } } ^ { { DBI } } leads to the sound speed bound c _ { s } \geq 0.087 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 .