Cosmological aspects of a hyperbolic solution in $f(R,T)$ gravity

This article deals with a cosmological scenario in $f(R,T)$ gravity for a flat FLRW model of the universe. We consider the $f(R,T)$ function as $f(R)+f(T)$ which starts with a quadratic correction of the geometric term $f(R)$ having structure $f(R)=R+\alpha R^2$, and a linear matter term $f(T)=2\lambda T$. To achieve the solution of the gravitational field equations in the $f(R,T)$ formalism, we take the form of a geometrical parameter, i.e. scale factor $a(t)= sinh^{\frac{1}{n}}(\beta t)$ \cite{cha}, where $\beta$ and $n$ are model parameters. An eternal acceleration can be predicted by the model for $0<n<1$, while the cosmic transition from the early decelerated phase to the present accelerated epoch can be anticipated for $n\geq 1$. The obtained model facilitate the formation of structure in the Universe according to the Jeans instability condition as our model transits from radiation dominated era to matter dominated era. We study the varying role of the equation of state parameter $\omega$. We analyze our model by studying the behavior of the scalar field and discuss the energy conditions on our achieved solution. We examine the validity of our model via Jerk parameter, Om diagnostic, Velocity of sound and Statefinder diagnostic tools. We investigate the constraints on the model parameter $n$ and $H_0$ (Hubble constant) using some observational datasets: $SNeIa$ dataset, $H(z)$ (Hubble parameter) dataset, $BAO$ (Baryon Acoustic Oscillation data) and their combinations as joint observational datasets $H(z)$ + $SNeIa$ and $H(z)$ + $SNeIa$ + $BAO$. It is testified that the present study is well consistent with these observations. We also perform some cosmological tests and a detailed discussion of the model.