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

TL;DR

This work develops a theory-space framework for f(R) gravity that frames cosmological evolutions as flows in a two-dimensional space of dimensionless variables $r=\frac{R f'}{f}$ and $m=\frac{R f''}{f'}$, enabling stability and viability assessments without explicit $f(R)$ reconstruction. By kinematically specifying GR-like solutions through cosmographic parameters, notably the jerk $j$ and deceleration $q$, the authors study two prototypical backgrounds: a ΛCDM-like solution with $j=1$ and a toy phantom-crossing form $j=1+3\varepsilon(q-1/2)$. For ΛCDM-mimicking models, they find that many viable $f(R)$ theories exist but do not show a generic GR past/future attractor and often harbor instabilities (ghost or tachyonic) along the evolution, challenging the feasibility of exact $j=1$ realizations within healthy $f(R)$ gravity. The phantom-crossing constructions, while able to reproduce DESI-like hints, inevitably suffer tachyonic instabilities ($f''(R)<0$) in the underlying theories, and theory-space trajectories tend to diverge toward unstable regions. Overall, the theory-space perspective reveals structural constraints on viable $f(R)$ realizations of realistic cosmologies and provides a robust, reconstruction-free lens to assess stability and naturalness across modified gravity models.

Abstract

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.

Figures (13)

• Figure 1: (\ref{['fig:f(R)_plot']}),(\ref{['fig:f(R)_plot_zoomed_1']}),(\ref{['fig:f(R)_plot_zoomed_2']}) $f(R)$, (\ref{["fig:f'(R)_plot"]}) $f'(R)$, (\ref{['fig:f"(R)_plot']}) $f"(R)$ and (\ref{['fig:Omega_m_plot']}),(\ref{['fig:Omega_m_plot_zoomed_1']}),(\ref{['fig:Omega_m_plot_zoomed_2']}) $\Omega_m$ corresponding to different $\Lambda$CDM-mimicking $f(R)$ models that are obtained from solving equation \ref{['eq:recon_LCDM_2']}. The initial conditions are $f(R)=(-2+10^4\epsilon)\Lambda+R,\,f'(R)=1+10^2\epsilon,\,f"(R)=\frac{\epsilon}{\Lambda}$. $\epsilon$ is a dimensionless smallness parameter parametrizing small deviations from GR at a high redshift of $z_{\rm in}=6.0316$ (at which $q_{\Lambda\rm{CDM}}=0.49$). The red, dashed orange, dotted orange, dashed blue, dotted blue, dashed black, dotted black curves correspond to $\epsilon=0,10^{-7},-10^{-7},10^{-6},-10^{-6},10^{-5},-10^{-5}$ respectively. In the plots, $\Lambda$ denotes the constant quantity $H_0^2(1-2q_0)$, which is identified with the cosmological constant only for GR (the red curve). $\epsilon$ values are taken to be small enough that the $\Lambda$CDM-mimicking $f(R)$s remain very close to GR and the $\Omega_m$ evolution remains almost the same as the General Relativistic $\Lambda$CDM model, as is apparent from Figs.(\ref{['fig:f(R)_plot']}) and (\ref{['fig:Omega_m_plot']}). However, there are actually distinct $\Lambda$CDM-mimicking $f(R)$ solutions, giving rise to distinct evolution of $\Omega_m$. This can be made apparent by zooming more and more into the corresponding figures (Figs.(\ref{['fig:f(R)_plot_zoomed_1']}),(\ref{['fig:f(R)_plot_zoomed_2']}) and Figs.(\ref{['fig:Omega_m_plot_zoomed_1']}),(\ref{['fig:Omega_m_plot_zoomed_2']})). Differences in $f'(R)$ and $f"(R)$ are more prominent, as evident from Figs.(\ref{["fig:f'(R)_plot"]}) and (\ref{['fig:f"(R)_plot']}).
• Figure 2: Panel \ref{['fig:wDE plot']} shows $w_{\rm DE}(z)$ versus $z$ corresponding to different $\Lambda$CDM-mimicking $f(R)$ models that are obtained by numerically solving the reconstruction differential equation \ref{['eq:recon_LCDM_2']}. The red, dashed orange, dotted orange, dashed blue, dotted blue, dashed black, dotted black curves correspond to $\epsilon=0,10^{-7},-10^{-7},10^{-6},-10^{-6},10^{-5},-10^{-5}$ respectively. Panel \ref{['fig:wdf plot']} shows the evolution of the unified dark fluid equation of state $w_{\rm df}(z)$ (\ref{['eq:DDE']}) for different values of $w_0$, that are chosen to coincide with the values of $w_{\rm DE}(z=0)$ for the respective plots of panel \ref{['fig:wDE plot']}.
• Figure 3: Cosmological dynamics of $\Lambda$CDM-mimicking $f(R)$ theories in Figure \ref{['fig:LCDM_mimicking_f(R)s']}, portrayed in the $\{r,m\}$ theory space, where $r=Rf'/f$ and $m=Rf"/f'$. The curves are numerical solutions of the reconstruction equation \ref{['eq:recon_LCDM_2']}, starting at $z_{\rm in}=6.0316$ with initial conditions $f(R)=(-2+10^4\epsilon)\Lambda+R$, $f'(R)=1+10^2\epsilon$, $f"(R)=\epsilon/\Lambda$, and evolving to $z=0$. The red, dashed orange, dotted orange, dashed blue, dotted blue, dashed black, dotted black curves correspond to $\epsilon=0,10^{-7},-10^{-7},10^{-6},-10^{-6},10^{-5},-10^{-5}$ respectively, with arrows indicating the direction of cosmic time. The $\Lambda$CDM model is the line $m(r)=0$. The present-day values ${r_0,m_0}$ are at the curve tips. Only positive $\epsilon$ yields non-pathological mimicking models. Panel \ref{['fig:m-r plot']} is a zoomed-out version of panel \ref{['fig:m-r plot_zoomed']}.
• Figure 4: The dynamics of $\Lambda$CDM-mimicking $f(R)$ theories are shown as parametric curves ${r(q),m(q)}$ in the $m-r$ plane. The GR $\Lambda$CDM model ($m=0$, $r=(3q-3)/(q-2)$) is the central red line. Panels show numerical solutions of the nonautonomous system. In Panel (a), trajectories start from the GR line shifted by $+\epsilon$ in both $r$ and $m$; in Panel (b), from the GR line shifted by $-\epsilon$ in $r$ and $+\epsilon$ in $m$. The red, dashed orange, dotted orange, dashed blue, dotted blue, dashed black, dotted black curves correspond to $\epsilon=0,10^{-7},-10^{-7},10^{-6},-10^{-6},10^{-5},-10^{-5}$ respectively. The evolution runs from $q=0.49$ to $q_0 \approx -0.55$ (today). Panel (c) and (d) are zoomed-in version of the figure (a) and (b), respectively. Zoomed panels reveal that GR is not a generic past attractor. Trajectories can start in the physically viable region ($m>0$) and cross into the theoretically unstable region ($m<0$).
• Figure 5: The curves in the panel \ref{['fig:m-r plot_3']} corresponds to solutions of the nonautonomous system \ref{['eq:nonautonomous_LCDM']} with the initial condition $\lbrace r(-0.55),m(-0.55) \rbrace = \lbrace \frac{3q-3}{q-2}\vert_{q=-0.55} + \epsilon,\epsilon\rbrace$, whereas that in the panel \ref{['fig:m-r plot_4']} corresponds to the initial conditions $\lbrace r(-0.55),m(-0.55) \rbrace = \lbrace \frac{3q-3}{q-2}\vert_{q=-0.55} + \epsilon,-\epsilon\rbrace$. The red, dashed blue, dotted blue, dashed black, dotted black curves correspond to $\epsilon=0,10^{-6},-10^{-6},10^{-5},-10^{-5}$ respectively. It appears that even though the underlying $\Lambda$CDM-mimicking $f(R)$ theories are very close to GR at the present epoch, it is possible for them to be different from GR near the matter-domination.