tokens: et al. 2016a ) in the standard canonical inflationary scenario ( c s = 1), as has been explicitly shown in Rezazadeh et al. ( 2015 , 2016 , 2017a ). In the present paper, we aim to refine the power-law, intermediate, and logamediate inflationary models in light of the current observational results by considering them in the DBI scalar field setting. Motivated by Spaliński ( 2008 ) and Tsujikawa et al. ( 2013 ), we assume in our work that the sound speed is constant during inflation. We examine the viability of each scale factor in light of the Planck 2015 results (Ade et al. 2016a ) and also present some observational constraints on them. The outline of this paper is as follows. In Section 2 , we review briefly the basic equations governing the background cosmology in the DBI inflation. In addition, we also represent the power spectra of the primordial scalar and tensor perturbations. Then, in Sections 3 – 5 , we respectively investigate the power-law, intermediate, and logamediate DBI inflationary models with constant sound speed. Finally, in Section 6 , we summarize our conclusions. 2. DBI Inflation Within the framework of Einstein’s gravity, the action of non-canonical scalar field models is given by (Armendáriz-Picón et al. 1999 ; Garriga Mukhanov 1999 )where g and R are the determinant of the background metric and the Ricci scalar, respectively. Throughout this paper, we work in units where the Planck reduced mass is taken equal to unity, = 1, for the sake of simplicity. In the present work, we concentrate on the DBI non-canonical scalar field with the Lagrangian density (Alishahiha et al. 2004 ; Silverstein Tong 2004 )where is the warp factor, while , and are the canonical kinetic term and potential of the DBI scalar field φ , respectively. For the DBI Lagrangian ( 2 ), the energy density and pressure of the non-canonical scalar field φ are given by where the subscript, “,” indicates the partial derivative with respect to X . Moreover,is the parameter determining the relativistic limit of brane motion in a warped background. Note that the parameter γ is defined in analogy of the Lorentz factor appearing in relativistic particle dynamics. The sound speed is generally defined bywhich specifies the propagation speed of the inflaton fluctuations relative to the homogeneous background. In order for the sound speed to be physical, it should be real and subluminal, (Franche et al. 2010 ). Applying Equations ( 3 )–( 5 ), the sound speed in DBI inflation takes the form For the homogeneous and isotropic background spacetime, we consider the spatially flat Friedmann–Robertson–Walker (FRW) metric. Therefore, varying the action ( 1 ) with respect to the metric yields the Friedmann equations in the DBI inflation as follows: Here, is the Hubble parameter, and a is the scale factor a of the universe. The dot denotes the derivative with respect to the cosmic time t . Considering the FRW metric, the canonical kinetic term is simplified to . Taking the
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