Asymptotic safety of quantum gravity beyond Ricci scalars

We investigate the asymptotic safety conjecture for quantum gravity including curvature invariants beyond Ricci scalars. Our strategy is put to work for families of gravitational actions which depend on functions of the Ricci scalar, the Ricci tensor, and products thereof. Combining functional renormalization with high order polynomial approximations and full numerical integration we derive the renormalization group flow for all couplings and analyse their fixed points, scaling exponents, and the fixed point effective action as a function of the background Ricci curvature. The theory is characterized by three relevant couplings. Higher-dimensional couplings show near-Gaussian scaling with increasing canonical mass dimension. We find that Ricci tensor invariants stabilize the UV fixed point and lead to a rapid convergence of polynomial approximations. We apply our results to models for cosmology and establish that the gravitational fixed point admits inflationary solutions. We also compare findings with those from $f(R)$-type theories in the same approximation and pin-point the key new effects due to Ricci tensor interactions. Implications for the asymptotic safety conjecture of gravity are indicated.

Figure 1

The accuracy of the fixed point solution, showing the absolute value of (5.6) for different orders in the polynomial approximation $N$ with colour-coding detailed in the inset. We observe that the accuracy in field space continues to grow with the approximation order $N$ as long as the scalar curvature remains within the radius of convergence of the polynomial expansion (5.13).

Figure 2

Shown are the first 19 fixed point couplings ${\lambda }_m$ ($m=0,\dots ,18$ from bottom to top) with increasing approximation order $N$, normalized to their value at the highest order in the approximation $N_{\max }=21$. The shift by $m$ is added for better display (see text). We observe a fast convergence with approximation order.

Figure 3

The rate of convergence of the first few fixed point couplings ${\lambda }_m$ ($m=0,\dots ,18$ from left to right) with increasing approximation order $N$, normalized to their value at the highest order in the approximation $N_{\max }=21$. The accuracy in the couplings increases roughly by a decimal place for every $N\to N+2$.

Figure 4

Shown is the nonperturbative “fixed point functional” $f_{*}(R^2/4)+R{z}_{*}(R^2/4)$ given in (5.7), (5.8) as a function of the dimensionless scalar Ricci curvature $R$ and the polynomial approximations from $N=3$ up to $N=19$ (in steps of 2, thin black lines). The highest approximation order (thick red line) also coincides with the numerical integration of (5.18), (5.19). Notice that the radius of convergence, indicated by the shaded area, is maximal, and given by $R_p$ in (5.22).

Figure 5

The universal eigenvalues ${\vartheta }_m(N)$ (real part if complex) for specific values of $m$ as functions of the order of the polynomial approximation $N$ (see main text).

Figure 6

The rate of convergence of the real part of universal eigenvalues ${\theta }_m^{\prime }$ ($m=0,2,3,5,\dots 13$) with increasing approximation order $N$ toward their best values at $N_{\max }=21$ (see main text).

Figure 7

Tomography of the angles ${\varphi }_n$ defined in (6.4), as functions of the approximation order $N$. The non-trivial angles ${\varphi }_n$ are connected by lines ($n=0$, 3, 5, 7, 9, 11, 13, 15 from right to left, respectively). With increasing $N$, we observe a rapid convergence. With increasing $n$, we observe that the imaginary parts are increasingly small compared to the real parts.

Figure 8

The bootstrap test for asymptotic safety: from left to right, each line $D_i$ shows the $i$th largest eigenvalues from all approximation orders $N\ge i$, connected by a line, and $x(N)=N+1+i$. At fixed $i$, and with increasing approximation order $N$ corresponding to increasing $x$, we observe that all curves $D_i$ consistently grow with $N$. This establishes that invariants with a larger canonical mass dimension lead systematically to larger scaling exponents (see main text).

Figure 9

The universal scaling exponents ${\vartheta }_n(N)$ (real parts if complex) are shown for all approximation orders $N$, in comparison with classical exponents (straight line). We observe three relevant eigenvalues. All other eigenvalues are increasingly irrelevant, approaching Gaussian values from above.

Figure 10

Origin of near-Gaussian scaling exponents. Shown are the scaling exponents ${\overline{\vartheta }}_m$, which, for each approximation order $N$, are given by the largest eigenvalue ${\vartheta }_{m=N-1}(N)$ (full dot) or its real part if complex (crossed dot). We observe that with increasing $m$ (or $N$) the largest eigenvalue approach Gaussian values.

Figure 11

The approach towards near-Gaussian scaling: shown is the relative distance ${\overline{v}}_m(N)$ (7.5) of scaling exponents ${\vartheta }_n$ from Gaussian values ${\vartheta }_{G,n}$, as a function of the approximation order $N$. The approach to Gaussian values is well-approximated by an exponential (dashed line).

Figure 12

Shown is the equation of motion (8.2) including the (anti)-de Sitter solutions, corresponding to zeros of (8.3). The three solutions $R_{-}<0<R_{+}<R_{++}$, indicated with arrows, correspond to two de-Sitter solutions at positive $R$ and an anti-de Sitter solutions at negative $R$. For numerical values, see Table 5.

Figure 13

Roots of the equations of motion in the complexified field plane. The three arrows indicate the locations of the real (anti-) de Sitter solutions $R_{-}<0<R_{+}<R_{++}$, which appear as accumulation points with increasing approximation order $N$. The color coding of approximations orders is indicated in the inset. The black circles indicate the radius $R_c\approx 2.006$ (inner circle) and $R_s\approx 2.4$ (outer circle).

Figure 14

Comparison of the equations of motions of the present theory (full line) with results from $f(R)$ quantum gravity (dashed line) from [68], as functions of the scalar Ricci curvature. Both studies use the exact same set of regularization and gauge fixings. We observe that the fluctuations of the Ricci tensor ultimately generate de Sitter solutions $R_{\text{dS}}>0$ which otherwise would be absent for small curvature (see text).