Reconstruction of $f(R)$ lagrangian from a massive scalar field

We present a reconstruction of the Lagrangian for $f(R)$ gravity by using a massive scalar field. The scalar field is minimally coupled to the action of $f(R)$ gravity. We demonstrate the use of a theorem based on invertible point transformation of anharmonic oscillator equation that has only recently been applied to gravitational physics. This avenue gives us a direct way to solve the field equations for a few well-known self-interaction potential of the scalar field. A few cases which do not fall in the regime of the theorem are also discussed. The $f(R)$ models are discussed in the context of recent cosmological observations and viability issues.

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