Can $f(R)$ modified gravity theories mimic a $\Lambda \mathrm{CDM}$ cosmology?

We consider $f(R)$ modified gravity theories in the metric variation formalism and attempt to reconstruct the function $f(R)$ by demanding a background $\Lambda \mathrm{CDM}$ cosmology. In particular we impose the following requirements: (a) A background cosmic history $H(z)$ provided by the usual flat $\Lambda \mathrm{CDM}$ parametrization though the radiation ($w_{\mathrm{eff}}=1/3$), matter ($w_{\mathrm{eff}}=0$), and de Sitter ($w_{\mathrm{eff}}=-1$) eras. (b) Matter and radiation dominate during the “matter” and “radiation” eras, respectively, i.e. ${\Omega }_m=1$ when $w_{\mathrm{eff}}=0$ and ${\Omega }_r=1$ when $w_{\mathrm{eff}}=1/3$. We have found that the cosmological dynamical system constrained to obey the $\Lambda \mathrm{CDM}$ cosmic history has four critical points in each era which correspondingly lead to four forms of $f(R)$. One of them is the usual general relativistic form $f(R)=R-2\Lambda$. The other three forms in each era, reproduce the $\Lambda \mathrm{CDM}$ cosmic history but they do not satisfy requirement (b) stated above.

Figure 1

The effective equation of state $w_{\mathrm{eff}}(N)$ imposed on the dynamical system (obtained from (2.22) using (2.24)). The long-dashed red line starts deep in the radiation era while the short-dashed blue line starts in the matter era and ignores radiation (${\Omega }_{0r}=0$).

Figure 2

The evolution of the variables $x_1(N)$, $x_2(N)$, $x_3(N)$, and $x_4(N)$ for standard radiation era initial conditions (dotted red line) and standard matter era initial conditions (dashed blue line). The perturbed trajectories are rapidly dragged by the stable attractors of each era. The numerically obtained evolution along the standard saddle points of Table  is also shown (continuous green line). The instabilities of this trajectory are bypassed by using the constrained system (2.30, 2.31) instead of the full system (2.15, 2.20, 2.21).

Figure 3

The phase space trajectories on the $x_1\mathrm{\text{−}}{x}_2$ plane [Fig. 3a] and $x_3\mathrm{\text{−}}{x}_4$ plane [Fig. 3b] for standard radiation era initial conditions (dotted red line) and standard matter era initial conditions (dashed blue line). The trajectory corresponding to the numerically obtained evolution along the saddle points of Table  is also shown (continuous green line). The points $A_1$, $A_2$, $A_3$ correspond to the attractors of each era (radiation, matter, and de Sitter, respectively) while the points $B_1$, $B_2$, $B_3$ correspond to the standard critical points of each era (see Tables ). Notice that on the projection of Fig. 3b the attractor points $A_2$, $A_3$ coincide with the standard critical points $B_2$, $B_3$.

Figure 4

The evolution of ${\Omega }_m$ (long-dashed line), ${\Omega }_{\mathrm{rad}}$ (dotted line), and ${\Omega }_{\mathrm{DE}}$ (short-dashed line) parameters for standard radiation era initial conditions [Fig. 4a] and standard matter era initial conditions [Fig. 4b]. In both cases the attractor critical points of Table  rapidly take over and drag the trajectories.

Figure 5

The form of $\log (F(N))$ in the numerical reconstruction (dashed lines) and its analytical approximation using both the interpolating steps 1–3 (dotted lines) and the application of the analytical expression (3.5) valid in each era (thick green lines). The agreement between the three approaches is very good. 5a: standard radiation era initial condition, 5b: standard matter era initial condition.

Figure 6

The form of $f(R)$ in the numerical reconstruction (dashed lines) and its analytical approximation using both the interpolating steps 1–3 (dotted lines) and the application of the analytical expression (3.4) valid in each era (continuous green lines). 6a: standard radiation era initial condition with the continuous green lines corresponding to radiation era (larger $R$) and matter era (smaller $R$). The de Sitter era is not shown since it corresponds to a single point (constant $R$). 6b: standard matter era initial condition with the continuous green line corresponding to matter era. The de Sitter era is not shown since it corresponds to a point (constant $R$).