Curvature dependence of quantum gravity

We investigate the phase diagram of quantum gravity with a vertex expansion about constantly-curved backgrounds. The graviton two- and three-point function are evaluated with a spectral sum on a sphere. We obtain, for the first time, curvature-dependent UV fixed point functions of the dynamical fluctuation couplings $g^{*}(R)$, ${\mu }^{*}(R)$, and ${\lambda }_3^{*}(R)$, and the background $f(R)$-potential. Based on these fixed point functions we compute solutions to the quantum and the background equation of motion with and without standard model matter. We have checked that the solutions are robust against changes of the truncation.

Figure 1

Displayed are the diagrammatic representations of the flows of the graviton two- and three-point functions. Double and dashed lines represent dressed graviton and ghost propagators respectively, while filled circles denote dressed vertices. Crossed circles stand for regulator insertions. All quantities are explicit background curvature dependent and carry further background curvature dependence via the spectral value of the respective vertex/propagator.

Figure 2

Fixed point function solution for the system $(g^{*}(r),{\mu }_{\mathrm{eff}}^{*}(r),{\lambda }_{3,\mathrm{eff}}^{*}(r))$ with the boundary condition from the first-order heat kernel. The solutions are stable in the whole investigated region. Note, that the effective couplings according to (37) are displayed.

Figure 3

Displayed are background potentials $f^{*}(r)/r^2$ obtained from the fixed point solution ${\mu }^{*}(r)$ (left and right panel) and from the approximations ${\mu }_{\mathrm{eff}}^{*}(r)={\mu }_0^{*}$ and ${\mu }^{*}(r)={\mu }_0^{*}$ (right panel). All curves are obtained with the condition $f^{*}(r=1.4)=0$. Other conditions just shift the potential $f^{*}(r)/r^2$ by a constant. The full solution does not contain a minimum, it becomes asymptotically flat. The approximation ${\mu }_{\mathrm{eff}}^{*}(r)={\mu }_0^{*}$ is qualitatively very good, see also Fig. 2. The approximation ${\mu }^{*}(r)={\mu }_0^{*}$ corresponds to a pure Einstein-Hilbert computation. Here we find a minimum at $r_0=0.97$.

Figure 4

Fixed point background potential for different constant input values of ${\mu }_{\mathrm{eff}}^{*}(r)$. The minimum that corresponds to the solution of the background equation of motion is at $r>0$ for ${\mu }_{\mathrm{eff}}^{*}(r)\lesssim 0.77$, while for ${\mu }_{\mathrm{eff}}^{*}(r)\gtrsim 0.77$ it is at $r<0$ (left panel). For ${\mu }_{\mathrm{eff}}^{*}(r)\lesssim 0.25$ the minimum vanishes completely, while for ${\mu }_{\mathrm{eff}}^{*}(r)=0.26$ the minimum is located at $r_0=1.1$ (right panel).

Figure 5

Depicted is the fixed point background potential if standard model matter content is included. In the full solution as well as in the Einstein-Hilbert solution we find a minimum at small background curvature, $r_0=0.11$ and $r_{0,\mathrm{EH}}=0.05$, respectively, which corresponds to the solution of the background equation of motion.

Figure 6

Shown is the fixed point function $f_1^{*}(r)$ for different constant input values of ${\mu }_{\mathrm{eff}}^{*}(r)$. The zeros in these functions correspond to solutions to the quantum equations of motion (7). Our best result ${\mu }_{\mathrm{eff}}^{*}(r)={\mu }_0^{*}=-0.38$ has a solution at negative curvature, $r_0=-1.0$.

Figure 7

Visualization of the existence of a solution to the background and quantum equation of motion in dependence on the parameter ${\mu }_{\mathrm{eff}}$. Solutions at positive curvature ($r>0$) and negative curvature ($r<0$) are distinguished.

Figure 8

Comparison of the trace evaluation using different Laplacians and starting with different eigenvalues. In particular we compare the spin-two Laplacian ${\mathrm{\Delta }}_2$ and the spin-zero Laplacian ${\mathrm{\Delta }}_0$ and further we start once from the zero mode and once start from the $l=2$ mode. In the left panel we display the background flow $\mathrm{Tr}[G{\partial }_{t}R]$ of the combined transverse traceless and trace mode, where also the exact solution is computed. In the right panel we display the self-energy diagram of the two-point function, which is the second diagram in Fig. 1. From these results we infer that this particular approximation is qualitatively reliable in the range $r<2$.

Figure 9

Comparison of fixed point functions with initial condition at different curvature values $r_{\text{start}}\in \{0.01,0.03,0.05,0.07\}$. In the left panel we compare the fixed point functions of the Newton’s coupling $g(r)$ and in the right panel the effective graviton mass parameter ${\mu }_{\mathrm{eff}}(r)=\mu (r)+\frac{2}{3}r$. Both fixed point functions show only a small dependence on the initial condition. All initial conditions are determined by $g_i^{*}(r_{\text{start}})=g_{i,0}^{*}+r_{\text{start}}{g}_{i,1}^{*}$, where the zeroth and linear order in $r$ of the couplings are given by (35) and (39).