$f(R)$ gravity theories in Palatini formalism: Cosmological dynamics and observational constraints

We make a systematic study of the cosmological dynamics for a number of $f(R)$ gravity theories in Palatini formalism, using phase space analysis as well as numerical simulations. Considering homogeneous and isotropic models, we find a number of interesting results: (i) models based on theories of the type (a) $f(R)=R-\beta /R^n$ and (b) $f(R)=R+\alpha \ln R-\beta$, unlike the metric formalism, are capable of producing the sequence of radiation-dominated, matter-dominated, and de Sitter periods, and (ii) models based on theories of the type (c) $f(R)=R+\alpha R^m-\beta /R^n$ can produce early as well as late accelerating phases but an early inflationary epoch does not seem to be compatible with the presence of a subsequent radiation-dominated era. Thus, for the classes of models considered here, we have been unable to find the sequence of all four dynamical epochs required to account for the complete cosmological dynamics, even though three out of four phases are possible. We also place observational constraints on these models using the recently released supernovae data by the Supernova Legacy Survey as well as the baryon acoustic oscillation peak in the Sloan Digital Sky Survey luminous red galaxy sample and the cosmic microwave background shift parameter. The best-fit values are found to be $n=0.027$, $\beta =4.63$ for the models based on (a) and $\alpha =0.11$, $\beta =4.62$ for the models based on (b), neither of which are significantly preferred over the $\Lambda \mathrm{CDM}$ model. Moreover, the logarithmic term alone is not capable of explaining the late acceleration. The models based on (c) are also consistent with the data with suitable choices of their parameters. We also find that some of the models for which the radiation-dominated epoch is absent prior to the matter-dominated era also fit the data. The reason for this apparent contradiction is that the combination of the data considered here does not place stringent enough constraints on the cosmological evolution prior to the decoupling epoch, which highlights the importance of our combined theoretical-observational approach to constrain models.

Figure 1

Evolutions of the variables $y_1$ and $y_2$ for the model $f(R)=R-\beta /R^n$ with $n=0.02$, together with the effective equation of state $w_{\mathrm{eff}}$. Initial conditions were chosen to be $y_1={10}^{-40}$ and $y_2=1–{10}^{-5}$. This demonstrates that the sequence of ($P_{r1}$) radiation-dominated ($w_{\mathrm{eff}}=1/3$), ($P_m$) matter-dominated ($w_{\mathrm{eff}}=0$), and ($P_d$) de Sitter acceleration ($w_{\mathrm{eff}}=-1$) eras occur for these models.

Figure 2

Evolution of $y_1$ and $y_2$ together with the effective equation of state $w_{\mathrm{eff}}$ for the model $f(R)=R+\alpha R^m-\beta /R^n$ with $m=1.9$, $n=1$, $\alpha =1$, $\beta ={10}^{-60}$. The initial conditions are $y_1=1.71\times {10}^{-6}$ and $y_2=1–2.09\times {10}^{-6}$. The system starts from a nonstandard radiation-type phase (corresponding to the $P_{r2}$ point with $w_{\mathrm{eff}}=-2/3+1/m=-0.14$) followed by an inflationary phase (corresponding to the $P$ point with $w_{\mathrm{eff}}=-1+1/m=-0.47$) and then followed by a matter-dominated phase (corresponding to the $P_m$ point with $w_{\mathrm{eff}}=0$). The system finally approaches a de Sitter point $P_d$ because of the presence of the $\beta /R^n$ term.

Figure 3

The 68.3% and 95.4% confidence levels for the model based on the theory $f(R)=R-\beta /R^n$, constrained by SNLS data only.

Figure 4

The 68.3% and 95.4% confidence levels for the model based on the theory $f(R)=R-\beta /R^n$, constrained by SNLS, BAO, and CMB data.

Figure 5

Evolution of the effective equation of state $w_{\mathrm{eff}}$, for the model based on the theory $f(R)=R-\beta /R^n$, with the best-fit value $\beta =4.63$ and $n=0.027$. For very large $z$, $w_{\mathrm{eff}}=1/3$, which then decreases and approaches 0 at redshift of a few tens. The accelerated expansion occurs around $z=1$ after which $w_{\mathrm{eff}}$ becomes smaller than $-1/3$. In the future, $w_{\mathrm{eff}}$ asymptotically approaches $-1$.

Figure 6

The 68.3% and 95.4% confidence levels for the model $f(R)=R+\alpha R^2-\beta /R$ constrained by SNLS, BAO, and CMB data.

Figure 7

Evolution of $w_{\mathrm{eff}}$ for the model $R+\alpha R^2-\beta /R$ with the best-fit value $\alpha =2.69$ and $\beta =-0.008$. There is no radiation-dominated epoch prior to the matter-dominated era ($w_{\mathrm{eff}}=0$). Although $w_{\mathrm{eff}}$ can be smaller than $-1/3$ around the present epoch, the late-time evolution does not correspond to a de Sitter universe.

Figure 8

68.3% and 95.4% confidence level for the theory $f(R)=R+\alpha \ln R-\beta$ constrained by SNLS, BAO, and CMB data. The models without a cosmological constant ($\beta =0$) are excluded by the data.