An investigation has been carried out on a reconfigured form of the Einstein-Hilbert action, denoted by f(R,Tϕ), where Tϕ represents the energy-momentum tensor trace of the scalar field under consideration. The study has focused on how the structural behavior of the scalar field changes based on the potential’s shape, which has led to the development of a new set of Friedmann equations. In the context of modified theories, researchers have extensively explored the range of gravitational wave polarization modes associated with relevant fields. In addition to the two transverse-traceless tensor modes that are typically observed in general relativity, two additional scalar modes have been identified: a massive longitudinal mode and a massless transverse mode, also known as the breathing mode.
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