Conditions for the cosmological viability of $f(R)$ dark energy models

We derive the conditions under which dark energy models whose Lagrangian densities $f$ are written in terms of the Ricci scalar $R$ are cosmologically viable. We show that the cosmological behavior of $f(R)$ models can be understood by a geometrical approach consisting of studying the $m(r)$ curve on the $(r,m)$ plane, where $m\equiv R{f}_{,RR}/f_{,R}$ and $r\equiv -R{f}_{,R}/f$ with $f_{,R}\equiv \mathrm{d}f/\mathrm{d}R$. This allows us to classify the $f(R)$ models into four general classes, depending on the existence of a standard matter epoch and on the final accelerated stage. The existence of a viable matter-dominated epoch prior to a late-time acceleration requires that the variable $m$ satisfies the conditions $m(r)\approx +0$ and $\mathrm{d}m/\mathrm{d}r>-1$ at $r\approx -1$. For the existence of a viable late-time acceleration we require instead either (i) $m=-r-1$, $(\sqrt{3}-1)/2<m\le 1$ and $\mathrm{d}m/\mathrm{d}r<-1$ or (ii) $0<m\le 1$ at $r=-2$. These conditions identify two regions in the $(r,m)$ space, one for the matter era and the other for the acceleration. Only models with an $m(r)$ curve that connects these regions and satisfies the requirements above lead to an acceptable cosmology. The models of type $f(R)=\alpha R^{-n}$ and $f=R+\alpha R^{-n}$ do not satisfy these conditions for any $n>0$ and $n<-1$ and are thus cosmologically unacceptable. Similar conclusions can be reached for many other examples discussed in the text. In most cases the standard matter era is replaced by a cosmic expansion with scale factor $a\propto t^{1/2}$. We also find that $f(R)$ models can have a strongly phantom attractor but in this case there is no acceptable matter era.

Figure 1

The effective equation of state $w_{\mathrm{eff}}$ for $P_6$ as a function of $m$. The point is stable and accelerated in the grayed regions. In region (A) $m<-(\sqrt{3}+1)/2$ the point is always a nonphantom ($w_{\mathrm{eff}}>-1$); in region (B) $-1/2<m<0$ it is strongly a phantom ($w_{\mathrm{eff}}<-7.6$); in region (C) $m\ge 1$ it is slightly phantom ($-1.07<w_{\mathrm{eff}}\le -1$); and in region (D) it is a nonphantom ($w_{\mathrm{eff}}>-1$). In all the other regions $P_6$ is either decelerated or unstable. Notice the gap between $w_{\mathrm{eff}}=-1.07$ and $-7.6$.

Figure 2

The $(r,m)$ plane for the four classes of $f(R)$ models. In all panels, the straight diagonal line is the critical line $m=-r-1$. In the dotted ranges $P_6$ is not accelerated or is unstable, if we assume $m_6^{\prime }>-1$. In the thick ranges labeled by A, B, and C, $P_6$ is accelerated and stable, again assuming $m_6^{\prime }>-1$ (we omit region D for clarity except for the Class IV panel). The gray triangles represent the forbidden directions near the critical points. The dashed green lines are hypothetical $m(r)$ curves, intersecting the critical line in the critical points $P_5$ and $P_6$. The intersection at $(r,m)=(-1,0)$ (light gray triangles) corresponds to the standard matter epoch $P_5^{(0)}$. In Class I models, the $m(r)$ curve does not intersect $(r,m)=(-1,0)$ and therefore there is no standard matter era. In Class II models, the point $(r,m)=(-1,0)$ is connected to the $P_1$ de Sitter point (along the segment $0<m\le 1$ at $r=-2$) and therefore represents a viable cosmological solution. The two additional critical points in the regions A and C are unstable since the curve enters the forbidden triangles and are therefore not acceptable as final accelerated stages. In Class III models the $m(r)$ line with a slope $m^{\prime }>-1$ intersects the critical line at a negative $m$ in the strongly phantom range (B). Note that the curves with $m_5^{\prime }<-1$ which are attracted by $P_6$ in the regions (A), (B), (C) are possible, but such cases are not viable because of the absence of a prolonged matter era for $m<0$. In Class IV models, the $m(r)$ curve connects the matter era with $m_5^{\prime }>-1$ to the region (D) with a derivative $m_6^{\prime }<-1$ and therefore represents a viable cosmology with a matter era followed by a stable acceleration ($w_{\mathrm{eff}}>-1$). No single trajectory can cross the critical line $m=-r-1$: each solution is trapped between two successive roots on the critical line.

Figure 3

This figure illustrates several possible $m(r)$ curves (thick dashed line). Only the $f=R\log R$ and the $f=R\exp (1/R)$ models show an acceptable connection between the matter point $(r,m)=(-1,0)$ and the de Sitter point $P_1$ along the dashed segment at $r=-2$. In all other cases, there is either no intersection of the $m(r)$ curves with the critical line $m=-r-1$ near $(r,m)=(-1,0)$ or the $m(r)$ curve enters the forbidden direction regions (the gray triangles). In all panels we show the forbidden regions for three points in the (A), (B), (C) ranges of $P_6$, even when there are no critical points in one of those regions. For clarity we omit the range (D).

Figure 4

Phase space in the plane $(x_1,x_2)$ in Poincaré coordinates for the model $f(R)=\alpha R^{0.9}$ in the absence of radiation. Here and in the following plot, the dotted lines correspond to trajectories at the early stage, the continuous lines to those at the final stage. The circles represent critical points. The solutions approach either the $\varphi \mathrm{MDE}$ point $P_2$ or the phantom point $P_6$. The point $P_5$ is a saddle, but the trajectories do not approach this point if we take into account radiation. The point $P_3$ is an unstable node.

Figure 5

Phase space projected on the plane $(x_1,x_2)$ in Poincaré coordinates for the model $f(R)=R+\alpha /R$ in the absence of radiation. For the initial conditions $x_2>0$ there are two solutions: either (i) the solutions directly approach the accelerated attractor $P_6$ or (ii) they first approach the saddle $\varphi \mathrm{MDE}$ point $P_2$ and then reach the attractor $P_6$. When $x_2<0$ initially, the trajectories move toward $x_2\to -\infty$. Note that the point $P_3$ is unstable.

Figure 6

The cosmic evolution of various quantities is shown for the model $f(R)=R+\alpha /R^2$ with $\alpha =-{\mu }^6$, $\mu /H_0=11.04$. The standard matter era is replaced by the $\varphi \mathrm{MDE}$ which corresponds to $a\propto t^{1/2}$, $w_{\mathrm{eff}}=1/3$ and ${\Omega }_{\mathrm{m}}=2$. The redshift $z_a$ at which acceleration starts is $z_a=0.4$ and we have asymptotically in the future ${\Omega }_{\mathrm{DE}}=1$ and $w_{\mathrm{eff}}=w_{\mathrm{DE}}=-0.82$ [see Eq. (63)].

Figure 7

Projected phase space in Poincaré coordinates for the model $f(R)=R+\alpha R^{0.9}$ in the absence of radiation. The final attractor is the de Sitter point $P_1\mathrm{\text{:}}(x_1,x_2,x_3)=(0,-1,2)$. Note that neither $P_2$ nor $P_6$ is stable unlike the model $f(R)=\alpha R^{0.9}$. In the left panel, $\alpha >0$: here $P_5$ corresponds to $m<0$ with a large eigenvalue and therefore is unstable. In the right panel, $\alpha <0$: now the point $P_5$ is a saddle with positive $m$, so it is possible to have a sequence of an oscillating matter phase followed by the late-time acceleration. We plot a single curve for clarity.

Figure 8

The cosmic evolution of various quantities for the model $f(R)=R\log \alpha R$ with initial conditions $x_1={10}^{-5}$, $x_2=-{10}^{-10}$, $x_3=1.01\times {10}^{-10}$, and $x_4=0.999$ at the redshift $z=1.1\times {10}^{17}$, corresponding to $r=-1.01$. In this case the matter era is too long relative to the standard cosmology. In fact, the energy fraction of the radiation at the present epoch is ${\Omega }_{\mathrm{rad},0}=2.8\times {10}^{-15}$, which is much smaller than the standard value ${\Omega }_{\mathrm{rad},0}\approx {10}^{-4}$.

Figure 9

The cosmic evolution of various quantities for the model $f(R)=R\log \alpha R$ with initial conditions $x_1={10}^{-10}$, $x_2=-{10}^{-7}$, $x_3=1.019\times {10}^{-7}$, and $x_4=0.999$ at the redshift $z=3.15\times {10}^6$, corresponding to $r=-1.019$. In this case we have ${\Omega }_{\mathrm{m},0}\approx 0.3$ and ${\Omega }_{\mathrm{rad},0}\approx {10}^{-4}$ at the present epoch, but the matter era is practically absent.

Figure 10

The cosmic evolution of various quantities for the model $f(R)=R\exp (q/R)$ with initial conditions $x_1=0$, $x_2=2.13\times {10}^{-20}$, $x_3=5.33\times {10}^{-21}$, and $x_4=0.99$ at the redshift $z=3\times {10}^5$. We see that the matter era is absent and is replaced by the $\varphi \mathrm{MDE}$.

Figure 11

The cosmic evolution of various quantities for the model $m(r)=-0.2(1+r)(3.2+0.8r+r^2)$ with initial conditions $x_1={10}^{-10}$, $x_2=-{10}^{-7}$, $x_3=1.000007\times {10}^{-7}$, and $x_4=0.999$ at the redshift $z=3.5\times {10}^6$. The model has an approximate matter-dominated epoch followed by a nonphantom accelerated universe with $w_{\mathrm{eff}}\approx -0.935$.