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Phantom crossing, equation-of-state singularities, and local gravity constraints in f(R) models

Luca Amendola, Shinji Tsujikawa

TL;DR

The paper investigates whether a broad class of $f(R)$ gravity models can replicate $\Lambda$CDM while producing late-time acceleration. It employs a phase-space analysis using the deviation parameter $m=\frac{R f_{,RR}}{f_{,R}}$ and examines fixed points and viable trajectories, finding a matter epoch at $m(r=-1)\approx0$ and a late-time attractor for $0<m<1$, with the dark-energy EOS $w_{\rm DE}$ diverging at $z_c$ and phantom-crossing at $z_b<z_c$. Using SNIa and CMB data, the authors constrain $m$ to be $\lesssim \mathcal{O}(0.1)$, but local gravity constraints enforce $m\ll1$ in high-density regions via a chameleon mechanism, strongly restricting model freedom. Models like Hu–Sawicki and Starobinsky can still exhibit detectable deviations from $\Lambda$CDM around the present epoch while remaining viable under LGC, whereas many other $f(R)$ constructions are pushed toward near-LCDM behavior. The work highlights distinctive features such as phantom crossing and EOS singularities as potential observational signatures to distinguish these theories from standard cosmology.

Abstract

We identify the class of f(R) dark energy models which have a viable cosmology, i.e. a matter dominated epoch followed by a late-time acceleration. The deviation from a LambdaCDM model (f=R-Lambda) is quantified by the function m=Rf_{,RR}/f_{,R}. The matter epoch corresponds to m(r=-1) simeq +0 (where r=-Rf_{,R}/f) while the accelerated attractor exists in the region 0<m<1. We find that the equation of state w_DE of all such ``viable'' f(R) models exhibits two features: w_DE diverges at some redshift z_{c} and crosses the cosmological constant boundary (``phantom crossing'') at a redshift z_{b} smaller than z_{c}. Using the observational data of Supernova Ia and Cosmic Microwave Background, we obtain the constraint m<O(0.1) and we find that the phantom crossing could occur at z_{b}>1, i.e. within reach of observations. If we add local gravity constraints, the bound on m becomes very stringent, with m several orders of magnitude smaller than unity in the region whose density is much larger than the present cosmological density. The representative models that satisfy both cosmological and local gravity constraints take the asymptotic form m(r)=C(-r-1)^p with p>1 as r approaches -1.

Phantom crossing, equation-of-state singularities, and local gravity constraints in f(R) models

TL;DR

The paper investigates whether a broad class of gravity models can replicate CDM while producing late-time acceleration. It employs a phase-space analysis using the deviation parameter and examines fixed points and viable trajectories, finding a matter epoch at and a late-time attractor for , with the dark-energy EOS diverging at and phantom-crossing at . Using SNIa and CMB data, the authors constrain to be , but local gravity constraints enforce in high-density regions via a chameleon mechanism, strongly restricting model freedom. Models like Hu–Sawicki and Starobinsky can still exhibit detectable deviations from CDM around the present epoch while remaining viable under LGC, whereas many other constructions are pushed toward near-LCDM behavior. The work highlights distinctive features such as phantom crossing and EOS singularities as potential observational signatures to distinguish these theories from standard cosmology.

Abstract

We identify the class of f(R) dark energy models which have a viable cosmology, i.e. a matter dominated epoch followed by a late-time acceleration. The deviation from a LambdaCDM model (f=R-Lambda) is quantified by the function m=Rf_{,RR}/f_{,R}. The matter epoch corresponds to m(r=-1) simeq +0 (where r=-Rf_{,R}/f) while the accelerated attractor exists in the region 0<m<1. We find that the equation of state w_DE of all such ``viable'' f(R) models exhibits two features: w_DE diverges at some redshift z_{c} and crosses the cosmological constant boundary (``phantom crossing'') at a redshift z_{b} smaller than z_{c}. Using the observational data of Supernova Ia and Cosmic Microwave Background, we obtain the constraint m<O(0.1) and we find that the phantom crossing could occur at z_{b}>1, i.e. within reach of observations. If we add local gravity constraints, the bound on m becomes very stringent, with m several orders of magnitude smaller than unity in the region whose density is much larger than the present cosmological density. The representative models that satisfy both cosmological and local gravity constraints take the asymptotic form m(r)=C(-r-1)^p with p>1 as r approaches -1.

Paper Structure

This paper contains 4 sections, 36 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Viable cosmological trajectories
  3. Observational constraints
  4. Conclusions

Figures (3)

  • Figure 1: 4 trajectories in the $(r,m)$ plane. Each trajectory corresponds to (i) $\Lambda$CDM, (ii) $f(R)=(R^{b}-\Lambda)^{c}$, (iii) $f(R)=R-\alpha R^{n}$ with $\alpha>0,0<n<1$, and (iv) $m(r)=-C(r+1)(r^{2}+ar+b)$. Here $P_{M}$, $P_{A}$ and $P_{B}$ are matter, de-Sitter and non-phantom accelerated points, respectively.
  • Figure 2: Evolution of the DE equation of state $w_{{\rm DE}}$ for the model $f(R)=(R^{1/c}-\Lambda)^{c}$ with parameters $c=1.01,1.1,1.8$. As $c$ approaches 1, the critical value $z_{c}$ gets larger. In the limit $c\to1$ ($\Lambda$CDM model) we have $z_{c}\to\infty$.
  • Figure 3: Evolution of the variable $m$ in terms of $z$ for the models: (A1) $f=(R^{1/c}-\Lambda)^{c}$, (A2) $f=R-\alpha R^{n}~(0<n<1)$, (A3) $m=-C(r+1)(r+2.1)$ and (B1) $m=-C(r+1)(r^{2}+r+1)$. Each curve shows the maximal $m$ that still satisfies SNIa and CMB constraints.