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

Some aspects of two General Relativistic cosmological solutions, an exact $Λ$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 $Λ$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 $Ω_{m0}$. With the theory space analysis, it is also shown that $Λ$CDM-mimicking $f(R)$ cosmologies that asymptote to General Relativistic $Λ$CDM in the limit $q\to1/2$, are prone to instability under small homogeneous and isotropic perturbation, casting a doubt on achieving an exact $Λ$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.

💡 Research Summary

The paper investigates two specific General Relativistic (GR) cosmological solutions within the framework of f(R) modified gravity by representing them as trajectories in a two‑dimensional “theory space” spanned by the dimensionless variables r ≡ R f′/f and m ≡ R f″/f′. The first solution is an exact ΛCDM‑like expansion characterized by a constant jerk parameter j = 1, while the second is a toy model designed to accommodate a phantom‑crossing behavior, defined by j = 1 + 3 ε (q − ½) with 0 < |ε| < 1. The authors adopt a cosmographic approach, using the deceleration parameter q and jerk j (and higher‑order parameters) as kinematic descriptors, thereby avoiding a series expansion in redshift that suffers from convergence issues.

In the f(R) context, the field equations can be rewritten in terms of r and m, where m is directly related to the effective scalaron mass; m → 0 corresponds to the GR limit, while m > 0 ensures the absence of tachyonic instabilities (f″ > 0). By starting from a prescribed cosmographic relation j = j(q), the authors derive an implicit relation m = m(r) without ever needing to reconstruct the explicit functional form f(R). This “theory‑space” method thus sidesteps the technical difficulties of traditional reconstruction, which often yields cumbersome hypergeometric or hypergeometric‑type functions whose stability properties are opaque.

For the ΛCDM‑like case (j = 1), the cosmographic reconstruction shows that the curvature degree of freedom does not have to behave exactly like a cosmological constant; different f(R) models can reproduce the same background expansion while yielding different present‑day matter density parameters Ω_{m0}. However, when the trajectory in the (r, m) plane is examined as the deceleration parameter approaches q → ½ (the matter‑dominated epoch), the flow inevitably drives m into regions where the scalaron mass becomes negative or varies sharply. Linearizing the homogeneous and isotropic perturbation equations around the background then reveals a growing mode, indicating that the exact j = 1 solution is dynamically unstable in f(R) gravity. Consequently, achieving a perfectly ΛCDM‑like expansion within f(R) without fine‑tuning appears implausible.

The second, phantom‑crossing solution (j = 1 + 3 ε (q − ½)) is analyzed in the same way. The derived m(r) curve stays negative for the entire evolution, which translates into f″(R) < 0, i.e., a tachyonic instability of the scalar degree of freedom. Hence any f(R) theory capable of reproducing the phantom‑crossing background necessarily suffers from a fundamental pathology, rendering it physically unacceptable.

The authors also compare their implicit reconstruction with earlier explicit methods that produce complicated f(R) forms involving hypergeometric functions. They demonstrate that, while explicit reconstruction can in principle yield a function that reproduces the desired background, it offers no clear insight into whether the resulting model satisfies the viability conditions f′ > 0 (no ghost) and f″ > 0 (no tachyon). The theory‑space approach, by contrast, provides a direct diagnostic: the sign and behavior of m along the trajectory immediately reveal the presence or absence of these pathologies.

Finally, the paper emphasizes the generality of the method. Since the variables r and m can be defined for any f‑class modified gravity (including f(T) torsion‑based, f(Q) symmetric‑teleparallel, and f(G) Gauss‑Bonnet extensions), the same analysis can be applied to a broad spectrum of theories. The authors suggest that future work could extend the framework to multi‑scalar extensions, non‑minimal couplings, or even to study non‑linear perturbations, thereby offering a systematic tool to assess naturalness and stability of modified gravity models beyond the simple background level.

In summary, the study concludes that (i) exact ΛCDM‑like solutions are generically unstable under homogeneous‑isotropic perturbations in f(R) gravity, (ii) phantom‑crossing backgrounds inevitably entail tachyonic scalaron masses, and (iii) the r‑m theory‑space analysis provides a powerful, reconstruction‑free means to evaluate the viability of modified gravity models.