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We study mimetic gravity in the presence of a DBI-like term which is anon-canonical setup of the scalar field's derivatives. We consider two generalcases with varying and constant sound speeds and construct the potentials forboth the DBI and Mimetic DBI models. By considering the power-law scale factoras $a=a_{0}\,t^{n}$, we seek for the observational viability of these models.We show that, the Mimetic DBI model in some ranges of the parameters space isfree of ghost and gradient instabilities. By studying $r-n_{s}$ and$\alpha_{s}-n_{s}$ behavior in confrontation with Planck2018 data, we find someconstraints on the model's parameters. We show that for the case with varyingsound speed, although power-law DBI inflation is not consistent with Planck2018TT, TE, EE+low E+lensing data, but the Mimetic DBI inflation is consistent withPlanck2018 TT, TE, EE+low E+lensing data at 95$\%$ CL, in some ranges of themodel's parameters space as $40\leq n \leq 55$ where the model isinstabilities-free in these ranges of parameters too. For the constant soundspeed, by adopting some sample values of $c_{s}$, we study both DBI and MimeticDBI model numerically and find $n\sim 10^{2}$ for DBI model and $n\sim 10$ forMimetic DBI model. We also compare the results with Planck2018 TT, TE, EE+lowE+lensing+BK14+BAO data and see that the DBI and Mimetic DBI model with varyingsound speed are ruled out with these joint data. However, these models withconstant sound speed are consistent with Planck2018 TT, TE, EE+lowE+lensing+BK14+BAO data with $n\sim 10^{2}$ for DBI model and $n\sim 10$ forMimetic DBI model. In this case, we find some tighter constraints on thecorresponding sound speed.

Mimetic DBI Inflation in Confrontation with Planck2018 data