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Conformal Invariant Teleparallel Cosmology

We study teleparallel gravitational theories with are invariant under the conformal transformations. Wide family of the gravitational Lagrangians that are invariant under conformal transformations have investigated. Cosmological solutions inspired by the observational data for a flat Universe in the vacuum has been found. To add matter fields to the cosmological models by preserving the conformal symmetry there are significant limitations. Friedmann-Robertson-Walker (FRW) equations based on the effective terms of energy density and pressure have been rewriting and continuity equation for the effective quantities are derived. In the vacuum and in the absence of any matter field,the FRW equation has an exact solution for Hubble parameter which is consistent with the cosmological data,specially given analytical solution is in good agreement with $Λ$CDM model in the present time. Then the scalar field-Torsion models in the cosmological FRW background investigated. Massless scalar field equations are very complex with an exact analytical solution in special limits. We have shown that the Lagrangian scalar field with self interaction $V(ϕ)=\frac{1}{4!}μϕ^4$ can be written that the conformal symmetry is preserved.