Theory space and stability analysis of General Relativistic cosmological solutions in modified gravity

Some aspects of two General Relativistic cosmological solutions, an exact $\Lambda$CDM-like cosmological solution $j=1$ ($j$ is cosmographic jerk parameter), and a specifically designed toy cosmological solution $j=1+3\varepsilon(q-1/2)$ ($q$ is cosmographic deceleration parameter, $0<|\varepsilon|<1$) that is capable of accommodating a phantom crossing scenario as suggested by DESI DR2, are studied within the context of $f(R)$ gravity, by portraying them as a \emph{flow} in the 2-dimensional \emph{theory space} spanned by the quantities $r=\frac{R f'}{f}, m=\frac{R f''}{f'}$. For the $f(R)$ theories exactly reproducing a background $\Lambda$CDM-like expansion history $j=1$, it is shown by means of a \emph{cosmographic} reconstruction approach that the curvature degree of freedom need not necessarily behave like an effective cosmological constant, and that cosmologies under different possible such theories lead to different possible values of $\Omega_{m0}$. With the theory space analysis, it is also shown that $\Lambda$CDM-mimicking $f(R)$ cosmologies that asymptote to General Relativistic $\Lambda$CDM in the limit $q\to1/2$, are prone to instability under small homogeneous and isotropic perturbation, casting a doubt on achieving an exact $\Lambda$CDM-like cosmological solution $j=1$ within $f(R)$ gravity. Regarding the toy cosmological solution $j=1+3\varepsilon(q-1/2)$ that is capable of accommodating a phantom crossing scenario, it is shown that possible underlying $f(R)$ theories that admit it as a solution are inevitably plagued by tachyonic instability ($f''(R)<0$). All the above physically interesting conclusions are derived without explicitly reconstructing, even numerically, the functional form of the underlying $f(R)$, which demonstrates the edge of the $r$-$m$ theory space analysis over the traditional explicit reconstruction approach.