$f(R)$ gravity from the renormalization group

We explore the cosmological dynamics of an effective $f(R)$ model constructed from a renormalization group improvement of the Einstein-Hilbert action, using the nonperturbative beta functions of the exact renormalization group equation. The resulting $f(R)$ model has some remarkable properties. It naturally exhibits an unstable de Sitter era in the ultraviolet (UV), dynamically connected to a stable de Sitter era in the IR, via a period of radiation and matter domination, thereby describing a nonsingular universe. We find that the UV de Sitter point is one of an infinite set, which makes the UV renormalization group fixed point inaccessible to classical cosmological evolution. In the vicinity of the fixed point, the model behaves as $R^2$ gravity, while it correctly recovers general relativity at solar system scales. In this simplified model, the fluctuations are too large to be the observed ones, and more ingredients in the action are needed.

Figure 1

A viable RG trajectory (blue, continuous) on the $g\mathrm{\text{−}}\lambda$ plane in the Einstein-Hilbert truncation for the choice $\rho =1$ and the initial conditions (37). It spirals around the UV RG fixed point and evolves towards the IR as curvature $R$ decreases. The intersection of the phase curve with the de Sitter line (black, continuous) corresponds to a de Sitter point in the cosmological evolution, while intersection with the dashed (black) one is where for the slow roll parameter ${ϵ}_V=1$. The regions where $m_{\mathrm{eff}}^2>0$ ($m_{\mathrm{eff}}^2<0$) are separated by the dotted lines, with $m_{\mathrm{eff}}^2$ the Jordan frame mass squared, defined in (16). The dotted curve consists of two separate curves (green and red dots) corresponding to the vanishing of the numerator (denominator) of $m_{\mathrm{eff}}^2$. They join at the upper part of the dotted “ellipsis,” where $m_{\mathrm{eff}}^2$ remains finite and nonzero. Along the lower part of the ellipsis (green) $m_{\mathrm{eff}}^2$ vanishes. The dotted curves outside the ellipsis (red) correspond to $m_{\mathrm{eff}}^2\to \infty$. Notice that beyond the red dot at ${\lambda }_{*}\simeq 0.27$ on the $\lambda$ axis, $m_{\mathrm{eff}}^2$ becomes negative, and therefore de Sitter space unstable too.

Figure 2

The derivative of the $f(R)$ model for the initial conditions (37) (in Planck units). For large values of $R$, the model effectively behaves as $R^2$ gravity, since $f_R(R)\sim R$. For smaller $R$, the evolution enters the classical regime near the GFP. The Einstein-Hilbert term dominates, $f_R(R)$ becomes nearly constant, with a small positive slope reflecting the positivity of the scalaron mass squared. At very small $R$ (not shown) there is a departure from the Einstein-Hilbert action due to the IR divergence in the beta functions. This part of the action is never encountered as the system freezes at the IR de Sitter point.

Figure 3

The Einstein frame scalar potential (in Planck units), described by relations (43, 44), for $\rho =0.8$ (red, dashed line), $\rho =1$ (back, solid line), and $\rho =1.15$ (blue, dotted line), respectively. Cosmological evolution starts from the maximum of the potential, which corresponds to the unstable UV de Sitter point, and evolves towards smaller values of the field $\Phi$.

Figure 4

The limiting de Sitter (solid line) and slow roll lines (dashed line), respectively. Red corresponds to $\rho \to \infty$ (“bell” shaped curves), and green to $\rho \to 0$, respectively. For $\rho \to \infty$ both de Sitter and slow roll line go to zero as $\lambda \to 0$.

Figure 5

The $m\mathrm{\text{−}}r$ plane for $\rho =1$ and the set of initial conditions (37), with $m(r)$ and $r$ given by relations (67, 68), respectively. Point A corresponds to the unstable UV de Sitter point, point B to the saddle matter point, while C to the stable IR de Sitter, respectively, as described in Sec. 5c. The dashed lines correspond to $r=-2$ and $m=-r-1$, respectively.

Figure 6

The cosmological trajectory described by the dynamical system (62, 63, 64, 65), in the space of the coordinates $(x_1,x_2,x_3)$, leaving the matter point and evolving towards the IR de Sitter. In particular, it spirals around the unstable matter point and then evolves towards the stable de Sitter in the IR. The initial conditions chosen are $({\lambda }_0,g_0)=({10}^{-2},{10}^{-5})$ and $(x_{10},x_{20},x_{30})=(x_{10(\mathrm{m})}+{10}^{-5},x_{20(\mathrm{m})}-{10}^{-6},x_{30(\mathrm{m})}+{10}^{-5})$, with $(x_{10(\mathrm{m})},x_{20(\mathrm{m})},x_{30(\mathrm{m})})$ denoting the coordinates of the matter fixed point given in (75), and $m_0$ is evaluated as $m_0({\lambda }_0,g_0)$ using (67, 20, 21). The above initial conditions give $r_0\simeq -1.02$, $m_0\simeq 1.24\times {10}^{-4}$. We also assumed that $x_4=0$. The amplitude of the oscillation along the $x_1$ axis is of the order ${10}^{-5}$.

Figure 7

The effective index $w_{\mathrm{eff}}$ and the slow roll parameter $ϵ$ [relations (69, 90), respectively] from the UV de Sitter to matter domination for initial conditions: $(x_{10},x_{20},x_{30},{\lambda }_0,g_0)=({10}^{-2},-1-{10}^{-3},2-{10}^{-5},0.26,0.02)$ and $\ln (a_0)=-30$, and $ϵ$ reexpressed as $ϵ=2-x_3$.

Figure 8

Upper row: The scalar (left) and gravitational (right) fluctuation power spectrum, as given by relations (100, 101) respectively, as a function of the couplings $\lambda$, $g$, and setting $\rho =1$. The scalar power spectrum peaks along the de Sitter line, as on a de Sitter point it is ${\mathcal{P}}_s\to \infty$. Lower row: The corresponding contour plots of the scalar (left) and gravitational (right) spectrum of upper row. In the scalar power spectrum the dotted line corresponds to the de Sitter line, along which the power spectrum diverges. Higher values correspond to lighter shaded areas.