Abstract
In this work, we examine solutions of the system of equations obtained by applying the Noether gauge symmetry (NGS) and its conserved quantity for the standard general relativity (GR) and the non-minimal derivative coupling (NMDC) cosmological model. We discover two salient features of the solutions. The first one is a(t)propto t^{1/3} for a kinetic-dominant phase which may emerge before inflationary period at very early time for GR case. The second one is a new form of scalar field phi (t) govern by the exponential cosmological solution for NMDC case, phi (t)=(c_{1}+c_{2}t)e^{-lambda t}+c_{3}.
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