On de Sitter solutions in asymptotically safe $\boldsymbol {f(R)}$ theories

Inflation is the theory of the universe for which space-time undergoes a phase of accelerated expansion. There are strong observational indications for inflationary phases, both during the early-time [52, 53] and the late-time cosmological evolution of the universe [54]. During an inflationary era of the universe the spacetime metric is approximately that of a de Sitter universe. Simple models of inflation include single scalar field theories coupled to gravity. Inflation may also be generated purely gravitationally. In either case, accelerated expansion sets in provided the scalar curvature R becomes nearly constant so that the dynamical evolution of the universe is very similar to that of a de Sitter universe. To be specific, we consider a FRW universe with scale parameter $a(t)$ and a gravitational action of the $f(R)$ type. The scalar curvature is then related to the Hubble parameter $H=\dot{a}/a$ by $\newcommand{\eps}{\epsilon} \newcommand{\e}{{\rm e}} R=12(H^2-\epsilon)$, where $\newcommand{\eps}{\epsilon} \newcommand{\e}{{\rm e}} \epsilon=-\dot{H}/H^2$ is the slow-roll parameter, and an overdot denotes a derivative wrt cosmological time. For sufficiently small , $\newcommand{\eps}{\epsilon} \newcommand{\e}{{\rm e}} \dot\epsilon$ an exponential expansion is observed with $a\propto e^{H\, t}$. In this case, the equation of motion becomes

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{EofM} R f'(R) - 2 f(R) = 0 \nonumber \end{align} \tag{ 2.1 }$$

whose solutions $R=R_{\rm dS}$ determine the Hubble parameter. An effective $f(R)$ theory could have two different types of de-Sitter solutions: One which would end after a finite time with $R_{\rm dS}= R_{\rm infl}$ and $f(R) \propto R^2_{\rm infl}$ leading to inflation in the early universe [76], and a second one for which $R_{\rm dS} = 4 \Lambda$ (where Λ is the cosmological constant) giving rise to accelerated expansion when dominating the contributions to the energy density of the universe.

From the perspective of quantum gravity, effective actions of the $f(R)$ type can arise due to quantum fluctuations of the metric field. A possible window into the form of such actions comes from the asymptotic safety scenario for quantum gravity, which stipulates that the UV action should be a fixed point of the renormalisation group. Here we shall assume that the form of such an action is determined by the existence of an asymptotically safe fixed point for a quantum version of gravity. The main new effect is that the gravitational couplings become running couplings depending on the renormalisation group scale k. The most widely studied actions which have been considered for asymptotic safety are those of the $f(R)$ form [12–14, 20, 21]. For lower energies and curvatures the action should approximate the Einstein–Hilbert form with a small but possibly non-zero cosmological constant which could provide for late time acceleration.

A suitable de Sitter fixed point leading to a viable inflationary model has been found in an $f(R)$-type approximations not including a cosmological constant [67]. Additional scalar matter fields have been taken into account in [64, 65, 69–73]. In this paper, we will instead include a cosmological constant and we will search for de Sitter points based on the high order results in [20, 21]. In this way, our analysis also covers earlier approaches on cosmological models with asymptotic safety based on pure gravity as studied in [56–66].

Next we recall how a function $f(R)$ is obtained from quantum gravity effects. We begin with gravitational effective actions in four-dimensional euclidean space-time which are functions of the Ricci scalar

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \displaystyle \label{S} \Gamma_k=\int {\rm d}^4x \sqrt{|\det g_{\mu\nu}|} \ F_k(\bar R), \nonumber \end{align} \tag{ 2.2 }$$

where $\bar R=\bar R(g_{\mu\nu})$ denotes the Ricci scalar and $g_{\mu\nu}$ the metric field. The index k denotes the renormalisation group (RG) momentum scale. Effective actions which are generic functions of the curvature scalar are of interest for cosmological model building and dark energy, see [77]. In our setup, however, the form of the function F_k is not an input: Rather, its shape is entirely dictated by quantum gravity and must be determined by explicit computation. In more concrete terms, the RG scale dependence of (2.2) is governed by its functional RG flow [2, 78],

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Ga}{\Gamma} \newcommand{\Tr}{{\rm Tr}} \newcommand{\re}{{\rm Re}} \displaystyle \label{dGamma} \partial_t\Gamma_k=\frac12 \Tr\frac{1}{\Gamma^{(2)}_k+R_k}\partial_t R_k. \nonumber \end{align} \tag{ 2.3 }$$

Here, R_k denotes a suitably defined Wilsonian momentum cutoff [79–81], $t = \ln k$, and $\newcommand{\Ga}{\Gamma} \Gamma^{(2)}_k$ the variation of the effective action with respect to the propagating fields. The gravitational effective action (2.2) interpolates between an asymptotically safe fixed point action $\newcommand{\Ga}{\Gamma} \Gamma_*$ in the limit $k\to\infty$ and a semi-classical low-energy effective action $\newcommand{\Ga}{\Gamma} \Gamma_0$ in the limit $k\to 0$. For the latter, the effective action should fall back onto the Einstein–Hilbert action

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle F_0(\bar R)\approx \frac{\Lambda}{8\pi\,G}-\frac{1}{16\pi G} \bar R+\cdots, \nonumber \end{align} \tag{ 2.4 }$$

possibly up to higher order corrections in the Ricci scalar. Here, Λ denotes the cosmological constant, $G=1/(8 \pi\, M_{\rm Pl}^2)$ is Newton's constant with $M_{\rm Pl}=2.4.35 \times10^{18}$ GeV the reduced Planck mass, and ${\Lambda}/(8\pi\, G)\approx 10^{-47}~{\rm GeV}^4$ the vacuum energy. The perturbative iterative solution of (2.3) reproduces the conventional loop expansion [82, 83]. In this limit, corresponding to weak gravitational interactions $G \cdot k^2 \ll 1$, the perturbative non-renormalisability of gravity is recovered. A key feature of functional renormalisation with (2.3) is that it does not necessitate weak coupling. Rather, it allows systematic approximations even at strong coupling where gravitational interactions become of order unity. The main physics novelty to be exploited below is that the running Newton coupling g(k)  =  G(k)k² does not diverge in the UV, but rather takes a finite value $g\to g_*$ owing to its own quantum fluctuations. This is the phenomenon of asymptotic safety. The gravitational fluctuations also dictate the shape of the function F_k in (2.2).

We add a few technical remarks and refer to [12–14, 16] for more details. The RG flow for F_k is obtained by inserting (2.2) into (2.3), together with suitable gauge fixing and Faddeev–Popov ghost terms. To find explicit expressions, the operator trace on the RHS of (2.3) for the action (2.2) is evaluated using background field techniques. The Hessian is obtained by expanding the second variation of the action around a suitable background metric in the form of $g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}$. Gauge invariance is ensured using the background field method. The operator trace is evaluated using the early-time heat kernel expansion and spherical backgrounds with constant scalar curvature. Intermediate technical steps simplify when using the York decomposition and optimised momentum cutoffs [7, 79, 80]. Analysing the flow equation for F_k within a power series in the curvature scalar shows the existence of an interacting UV fixed point with three relevant couplings related to the cosmological constant, Newton's coupling, and R² interactions. The scaling dimensions for these deviate from Gaussian values owing to large anomalous dimensions. Furthermore, the fluctuations of the metric field are strong enough for gravity to become anti-screening towards shortest distances. Higher order gravitational interactions also take interacting fixed points. For these, near-Gaussian scaling exponents are observed [20, 21]. As a final remark, it is useful to relate these findings with those obtained using conventional perturbation theory. Evaluating the flow equation (2.3) perturbatively, and around a flat background one recovers the results of Stelle [84], including the well-known issues with perturbative unitarity. In our setup, however, issues with massive ghosts are circumvented dynamically [85] owing to the spectral positivity of the perturbative RG flow, see [86, 87]. Also, the notorious Goroff–Sagnotti term which plays a central role for gravity's perturbative non-renormalisability becomes irrelevant once gravity becomes anti-screening [51].

From now on we will investigate the shape of the function F in the deep UV regime of the theory. The explicit form of the fixed point action $\newcommand{\Ga}{\Gamma} \Gamma_*$, and hence the function F, has been derived within high-polynomial orders in the Ricci scalar in [20, 21]. It is the central object of this study. For the remainder, it is convenient to introduce dimensionless fields, and we write

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Rf} \begin{array}{@{}rcl@{}} R&=&\bar R/k^2 \nonumber \\ f(R)&=&16\pi\, F(\bar R)/k^4. \end{array} \nonumber \end{align} \tag{ 2.5 }$$

Following [20, 21], we found it convenient to introduce an extra factor $16\pi$ into the definition of f, ensuring that Newton's coupling is given as $-1/f'$ in units of the RG scale without additional numerical factors, $G\cdot k^2=-1/f'$. Explicit functional flows (2.3) for actions (2.2) in four euclidean dimensions have been given in [12–14], and in [16] based on the on-shell action, also using [7, 79, 80]. To facilitate consistency checks and a comparison with earlier findings we have adopted the approach put forward in [12, 14]. We are interested in the shape of $f(R)$ in the deep UV limit where the theory displays an interacting UV fixed point. At a non-trivial fixed point, the function $f(R)$ becomes scale-invariant,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{dtf} \partial_t f(R)=0. \nonumber \end{align} \tag{ 2.6 }$$

Using the explicit functional RG equations as discussed in the previous section, the fixed point condition (2.6) leads to an explicit, non-linear differential equation determining the function $\newcommand{\R}{R} f(\R)$ [12, 14]

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\R}{R} \displaystyle \frac{df''(R)}{d\R}=&~ \frac{\frac{37}{756} \R^4+\frac{29}{10} \R^3+\frac{121}{5} \R^2+12 \R-216}{\frac{181}{1680}\R^4+\frac{29}{15} \R^3+\frac{91}{10} \R^2-54}\,\frac{f''(\R)}{\R} - \frac{\frac{37}{756} \R^3+\frac{29}{15} \R^2+18 \R+48}{\frac{181}{1680} \R^4+\frac{29}{15} \R^3+\frac{91}{10} \R^2-54} \,\frac{f'(\R)}{\R} \nonumber \\ &+ \frac{(\R-3)^2 f''(\R)+(3-2 \R)\,f'(\R)+2 f(\R)}{\R \left(\frac{181}{1680} \R^4+\frac{29}{15} \R^3+\frac{91}{10} \R^2-54\right)}\times \label{df''} \nonumber \\ &\times \left[\frac{\R \left(-\frac{311}{756} \R^3+\frac{1}{6}\R^2+30 \R-60\right)\,f''(\R)+\left(\frac{311}{756} \R^3 -\frac{1}{3}\R^2-90 \R+240\right)\,f'(\R)}{3 f(\R)-(\R-3)\,f'(\R)} \right. \nonumber \\ & \nonumber \left.-\frac{607 \R^2-360 \R-2160}{15 (\R-4)}-\frac{511 \R^2-360 \R-1080}{30(\R-3)}+48 \pi \big(\R f'(\R)-2 f(\R)\big)\right]. \nonumber \end{align} \tag{ 2.7 }$$

An asymptotically safe interacting fixed point solution f_*(R) with (2.6) and (2.7) has been given in [20, 21]. Its 'UV critical surface' is found to be three-dimensional. The relevant couplings are mainly given by the vacuum energy, Newton's coupling, and the R² coupling, in the sense that adding further invariants to the polynomial approximation does not lead to new relevant directions. Moreover, all higher order polynomial couplings are irrelevant and approach near-Gaussian scaling exponents with increasing canonical mass dimension. Therefore, UV-safe trajectories running out of the fixed point are characterised by three fundamentally free parameters.

In the remaining part of the paper, we are interested in the UV fixed point solutions f_*(R) to (2.7) and their main properties, both within polynomial expansions about vanishing curvature scalar, and within numerical solutions, based on integrations of (2.7) starting from suitable initial conditions. We note that the RHS of (2.7) may become singular at specific points in field space arising through the various denominators in (2.7), specifically

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\R}{R} \displaystyle \label{singular} \begin{array}{@{}rcl@{}} \R&=&-9.99\,855\cdots, \nonumber \\ \R&=&\ \ \,0, \nonumber \\ \R&=&\ \ \,2.00\,648\cdots, \nonumber \\ \R&=&\ \ \,3, \nonumber \\ \R&=&\ \ \,4. \end{array} \nonumber \end{align} \tag{ 2.8 }$$

The potential singularity at R  =  0 is harmless. Those at R  =  3 and 4 are due to the momentum cutoff. In principle, fixed point solutions can be extended across such singularities at the expense of a free parameter, see [19, 25, 88] for recent examples. This also signals the onset of a regime where the derivation of the RG flow based on the early time expansion of the heat kernel may no longer be trusted. The potential singularities of the differential equation at $R=-9.998\, 55$ and $R=2.006\, 48$ arise from the scalar metric fluctuations. Therefore, for polynomial expansions about R  =  0, we may expect from (2.8) that the radius of convergence R_c is bounded from above by R_max given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Rmax} R_c\leqslant R_{\rm max}=2.006\,48\cdots. \nonumber \end{align} \tag{ 2.9 }$$

Extending fixed point solutions beyond (2.9) requires that the polynomial couplings at R  =  R_c fullfill certain continuity conditions.

Next we analyse the ultraviolet fixed point solution of polynomial $f(R)$ gravity of [20, 21] in four dimensions. Following [75] we identify the location of a convergence-limiting pole (or cut) in the plane of complexified Ricci scalar, which is exploited to estimate the radius of convergence. In [20, 21], we have computed the fixed point coordinates in a polynomial approximation up to order $N=N_{\rm max}=35$ in the expansion

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\R}{R} \displaystyle \label{expansion0} f(\R)=\sum_{n=0}^{N-1}\lambda_n \R^n. \nonumber \end{align} \tag{ 2.10 }$$

Numerical results for the fixed point couplings up to the order $N\leqslant N_{\rm max}=35$ are summarized in table 1, showing the fixed point coordinates for selected orders of the approximation. Notice that the signs of the couplings follow, approximately, an eight-fold periodicity in the pattern [20]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{8} (++++----). \nonumber \end{align} \tag{ 2.11 }$$

Table 1. Coordinates of the ultraviolet fixed point $\lambda_n(N)$ in a polynomial base (2.10) for selected orders in the expansion. Note that $g_*=-1/\lambda_1$. We observe the approximate eight-fold periodicity pattern (2.11) in the signs of couplings. Notice that the couplings $\lambda_{2+4i}$ at approximation order N₁  =  3  +  4i (the order at which they first arise) always come out with the 'wrong' sign. To achieve the desired accuracy in the subsequent analysis, we have retained at least 50 significant digits for the couplings. The data for N  =  7 and N  =  11 agrees with earlier findings in [12] and [61], respectively.

N	35	31	27	23	19	15	11	7
$\lambda_{0}$	0.255 62	0.255 55	0.255 60	0.255 46	0.255 59	0.255 22	0.255 77	0.253 88
$\lambda_{1}$	−1.0272	−1.0276	−1.0276	−1.0286	−1.0281	−1.0309	−1.0289	−1.0435
$\lambda_{2}$	0.015 67	0.015 49	0.015 39	0.014 98	0.014 90	0.013 69	0.013 54	0.007 106
$\lambda_{3}$	−0.441 58	−0.446 87	−0.439 97	−0.449 46	−0.434 55	−0.457 26	−0.402 46	−0.512 61
$\lambda_{4}$	−0.364 53	−0.368 02	−0.366 84	−0.374 07	−0.369 81	−0.389 66	−0.371 14	−0.480 91
$\lambda_{5}$	−0.240 57	−0.232 32	−0.245 84	−0.231 88	−0.259 27	−0.228 42	−0.316 78	−0.180 47
$\lambda_{6}$	−0.027 17	−0.026 24	−0.022 86	−0.019 49	−0.015 64	−0.002 072	−0.003 987	0.123 63
$\lambda_{7}$	0.151 86	0.138 58	0.158 94	0.136 20	0.177 02	0.126 49	0.236 80
$\lambda_{8}$	0.230 14	0.234 41	0.224 65	0.229 04	0.216 09	0.213 50	0.236 00
$\lambda_{9}$	0.216 10	0.238 20	0.209 17	0.249 18	0.188 30	0.284 60	0.127 56
$\lambda_{10}$	0.084 84	0.082 07	0.092 099	0.095 052	0.095 688	0.137 22	${\bf -0.041\, 490}$
$\lambda_{11}$	−0.145 51	−0.177 74	−0.133 48	−0.194 44	−0.097 057	−0.255 27
$\lambda_{12}$	−0.325 05	−0.332 44	−0.332 42	−0.362 05	−0.318 12	−0.464 76
$\lambda_{13}$	−0.296 99	−0.255 44	−0.324 10	−0.242 39	−0.395 20	−0.167 35
$\lambda_{14}$	−0.056 08	−0.040 49	−0.056 33	−0.000 217	−0.112 04	0.167 62
$\lambda_{15}$	0.224 83	0.163 47	0.269 44	0.143 17	0.373 36
$\lambda_{16}$	0.363 15	0.340 00	0.377 95	0.286 11	0.509 97
$\lambda_{17}$	0.340 98	0.444 88	0.281 38	0.501 87	0.171 99
$\lambda_{18}$	0.185 36	0.239 41	0.152 07	0.350 74	${\bf -0.119\, 01}$
$\lambda_{19}$	−0.163 04	−0.320 36	−0.075 88	−0.417 33
$\lambda_{20}$	−0.614 57	−0.731 33	−0.537 76	−0.951 76
$\lambda_{21}$	−0.753 46	−0.538 75	−0.889 29	−0.412 30
$\lambda_{22}$	−0.251 60	−0.057 46	−0.437 56	0.299 53
$\lambda_{23}$	0.557 01	0.229 98	0.730 65
$\lambda_{24}$	0.933 92	0.609 48	1.3116
$\lambda_{25}$	0.706 08	1.2552	0.542 66
$\lambda_{26}$	0.357 10	0.988 91	${\bf -0.311\, 79}$
$\lambda_{27}$	−0.091 06	−0.928 72
$\lambda_{28}$	−1.1758	−2.3752
$\lambda_{29}$	−2.2845	−1.1315
$\lambda_{30}$	−1.4145	0.647 46
$\lambda_{31}$	1.6410
$\lambda_{32}$	3.5054
$\lambda_{33}$	1.7098
$\lambda_{34}$	${\bf -0.668\, 83}$

Four consecutive fixed point couplings $\lambda_{3+4i}-\lambda_{6+4i}$ come out negative (positive) for even (odd) integer $i\geqslant 0$. Periodicity patterns such as this one often arise due to convergence-limiting singularities (poles or cuts) of the fixed point solution $\newcommand{\R}{R} f(\R)$ in the complex $\newcommand{\R}{R} \R$-plane, away from the real axis. If the singularity nearest to the origin in field space arises at R  =  R₀ (which can and will be complex), then the radius of convergence for a field expansion about vanishing field is given by $R_{\rm c}=|R_0|$. This is well-known from scalar theories at criticality where 2n-fold periodicities are encountered regularly [75, 89–92]. Below we determine R_c to establish properties of the interacting fixed point including the existence or not of de Sitter solutions.

The polynomial expansion (2.10) has a finite radius of convergence R_c, which is expected to arise due to a singularity or a cut of the fixed point solution for $f(R)$ in the complex field plane. Its location can be estimated from the convergence behaviour of the fixed point coordinates. Standard convergence tests such as the root test or the ratio test fail due to the eight-fold periodicity in the couplings, and a high-accuracy computation of R_c requires many orders in the expansion. Therefore we adopt the following strategies. Firstly, we make a low-parameter fit of the fixed point couplings from table 1 onto a form which reflects the observed periodicity pattern. This provides us with an estimate for R_c. Secondly, we then cross-check the findings with more sophisticated convergence tests such as a modified ratio test and the Mercer-Roberts test [75]. We begin with a very simple four-parameter ansatz

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{cos} \lambda_n= A\,\frac{\cos(n\,\phi+ \Delta)}{(R_c)^n} \nonumber \end{align} \tag{ 2.12 }$$

for the couplings, motivated by the sign pattern observed in the couplings. Also, expansions such as (2.12) arise naturally provided that the fixed point displays a singularity in the complex plane. We observe that the periodic pattern starts roughly from n  =  5 onwards, and hence we must omit the first few couplings for the fit. We also omit a few of the highest couplings $\lambda_n$ with n  >  26. Typically, the last cycle of eight highest couplings has not yet properly settled on their asymptotic values. For these reasons we fit (2.12) using the most reliable values of the best solution $(N=35)$ given by approximately 2.5 cycles of data points (22 consecutive values from $\lambda_5$ up to $\lambda_{26}$). For the amplitude, the radius, the angle and the shift, we find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{fit1} \begin{array}{@{}rl@{}} A&=\ \ \,0.1172 \nonumber \\ R_c&=\ \ \,0.9182 \nonumber \\ \phi&=\ \ \, 0.7863 \nonumber \\ \Delta&=-0.2919. \end{array} \nonumber \end{align} \tag{ 2.13 }$$

Notice that the angle ϕ is very close to $\pi/4\approx 0.7854$, as expected due to the eight-fold periodicity pattern. Comparing with the exact fixed point values, we observe good agreement even beyond the fitted domain, see figure 1.

Zoom In Zoom Out Reset image size

A first cross-check is done using the Mercer-Roberts test based on the same data set. It estimates the angle as $\phi=0.7736\pm 3\%$, confirming that (2.13) provides a good estimate for the angle ϕ under which the fixed point solution displays a singularity in the complex field plane, see figure 2. The radius of convergence should be reliable on the 5–10% level. Neither the root test, the ratio test nor the Mercer-Roberts test provide stable results for the radius of convergence. As a cross-check for the radius, we therefore adopt a generalised ratio test according to which the radius is estimated via the limit

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{root:1} R_c=\lim_{n\to\infty}\, \left|\frac{\lambda_n}{\lambda_{n+m}}\right|^{1/m} \nonumber \end{align} \tag{ 2.14 }$$

provided it exists, irrespective of the free parameter m. It turns out that if m is taken to be the underlying periodicity or larger, $m\geqslant 8$, the ratios $\left|{\lambda_n}/{\lambda_{n+m}}\right|^{1/m}$ depend only weakly on the parameter m, and converge well with increasing n. Since our data sets are finite, the limit $1/n\to 0$ can only be performed approximately. We obtain a mean value for R by computing the ratios for various m, and then averaging over all m. In this manner, the estimate is as stable as it gets, and largely insensitive to the choice for m. We find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{root} R_c\approx 0.91\pm 5\%, \nonumber \end{align} \tag{ 2.15 }$$

and the statistical error of approximately a standard deviation is due to the variation with m.

Zoom In Zoom Out Reset image size

Notice that this procedure is insensitive to the angle ϕ. It is therefore interesting that the estimates (2.15) and (2.13) agree. The smallness of the statistical error indicates that the radius is achieved for most of the data points except for a few where the radius comes out smaller. In this light, a conservative 'lower bound' for the radius is then obtained by projecting onto the smallest estimate

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\R}{R} \newcommand{\RLB}{R_{L}} \displaystyle \RLB(m)\equiv \min_n \left|\frac{\lambda_n}{\lambda_{n+m}}\right|^{1/m} \nonumber \end{align} \tag{ 2.16 }$$

from the most advanced data set $(N=35)$, and for each admissible parameter m $(8\leqslant m\leqslant N-m)$. We then average these lower bounds over m to estimate the conservative lower bound as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\R}{R} \newcommand{\RLB}{R_{L}} \displaystyle \label{RL} \RLB\, \approx 0.82\, \pm\, 5\% \nonumber \end{align} \tag{ 2.17 }$$

where, again, the statistical error of approximately a standard deviation is due to the variation with m. The smallness of the statistical error reflects that the value (2.17) is achieved for essentially all $m\geqslant 8$. For illustration, we show in figure 3 the fixed point solution as a function of R to order N  =  31 and N  =  35. Both solutions visibly part each other's ways at fields of the order of (2.17), supporting our rationale.

Zoom In Zoom Out Reset image size

We now relate our findings with earlier studies of the radius of convergence based on approximations up to order N  =  11 [61]. There, a larger radius of convergence has been found to be

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{N11} R_c\approx 0.99 \nonumber \end{align} \tag{ 2.18 }$$

adopting a procedure different from ours. Error estimates have not been given. Had we restricted our procedure to the first 11 fixed point couplings (by using either the N  =  11 data, or the first 11 entries from the N  =  35 data set), our analysis leads to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{N11new} R_c\approx 1.0\pm 20\%. \nonumber \end{align} \tag{ 2.19 }$$

Within errors, (2.19) is consistent with the more accurate estimates (2.15) and (2.17). It is also consistent with the earlier estimate (2.18). The slight over-estimation for R_c at low orders (2.19) can be understood in terms of the eight-fold periodicity pattern underlying the data. A first full cycle of eight is completed at approximation order N  =  11; see table 1. At approximation order N  =  35, four such cycles have been completed, offering a more accurate estimate of the radius of convergence.

We also note that the various estimates for R_c (2.13) and (2.15) the lower bound R_L (2.17) are substantially smaller than the formal upper bound R_max identified in (2.9). Hence, the theoretically achievable radius of convergence is not realised by the fixed point solution. Rather, a singularity or cut in the plane of complex Ricci scalar imposes more stringent limits.

Given the good convergence of fixed point couplings, our results should not vary strongly if the polynomial expansion is pushed to higher orders, including the asymptotic limit $N\to \infty$. Furthermore, as we argued, the polynomial expansion has a finite radius of convergence, and pushing the expansion towards $N\to \infty$ will not provide access to the regime |R|  >  R_c. Therefore, in order to go beyond all polynomial orders, we must find the fixed point solution beyond $\newcommand{\R}{R} \R_c$ by integrating the fixed point condition numerically with initial data provided by the polynomial approximation, see figure 3. Since the fixed point condition is a third order differential equation for f, (2.7), we need to give three initial conditions. At $\newcommand{\R}{R} \R =0$ this reduces to two initial conditions since one condition is 'used up' in order to avoid a divergence at the origin. This leaves us with the two free parameters $\lambda_0$ and $\lambda_1$. To numerically integrate (2.7) into positive (negative) values of $\newcommand{\R}{R} \R$ we take initial conditions for f, $f'$ and $f''$ from our highest polynomial approximation $(N=35)$ at field values $\newcommand{\R}{R} \R = 0.1$ ($\newcommand{\R}{R} \R = -0.1$) deeply within the radius of convergence. We have checked that our results for f are independent of these technical choices.

In figure 3 we compare the full numerical integration (thick red curve) with the polynomial approximations at order N  =  34 and N  =  30 (thin gray curves) and note that we are able to compute f outside the radius of convergence $\newcommand{\R}{R} \R_L$. The polynomial and numerical solution coincide within the radius of convergence, as they must. The validity of the full numerical solution extends substantially beyond R_L. However, we cannot integrate the fixed point up to infinite field due to technical singularities of the differential equation (2.7) at intermediate or large curvature scalar. With increasing $\newcommand{\R}{R} \R$, the first integrable pole is located at $\newcommand{\R}{R} \R\approx 2.006$, see (2.8). On the negative curvature axis, the closest integrable pole is located at $\newcommand{\R}{R} \R\approx-9.99$. Both of these can in principle be dealt with using ideas discussed in eg. [19]. Our numerical integration becomes unreliable close to $R\approx 2.006$ and close to $R\approx -2.541$, which prevents the solution being continued to larger values of $|R|$. For asymptotically large R, the approximations adopted in (2.7) are less reliable, and we can expect corrections in a more complete treatment.

The results of the previous sections show that the polynomial fixed point solution is bounded to the regime of small fields. However, one may wonder whether resummation techniques could be used to extrapolate results reliably towards larger values for the scalar curvature. We begin by exploring whether the eightfold periodicity in the result (2.13) can be used to resum the polynomial series (2.10) to infinite order. To that end we write

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{resum} f_{\rm resum.}(R)= \sum_{n=0}^{m-1}\lambda_n\,R^n+\sum_{n=m}^\infty\lambda_n\,R^n. \nonumber \end{align} \tag{ 2.20 }$$

Figure 1 indicates that the eightfold periodicity pattern sets in at about $n\approx 5$. Therefore we take m  =  5 in (2.20). We then leave the first sum as it is, and denote it as $f_{\rm trunc}(R)$. The second sum can be evaluated in closed form if we assume that the couplings $\lambda_n$ are well-approximated by (2.12) for all n  >  4. This leads to the resummed expression

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{resum2} \sum_{n=5}^\infty\lambda_n\,R^n=A\,\frac{R^4}{R_c^4}\,\frac{R\,R_c\cos(\Delta+5\phi)-R^2\,\cos(\Delta+4\phi)}{R^2-2\,R\,R_c\cos\phi+R_c^2} \nonumber \end{align} \tag{ 2.21 }$$

in terms of four parameters $A, R_c, \phi$ and Δ which are, approximately, given by (2.13). Owing to the split (2.20), the resummed expression behaves $\propto R^4$ asymptotically.

In figure 4, we compare the resummation (2.20) and (2.21) (short dashed blue line) with the polynomial approximation at high orders ($N=30, 34$: thin gray lines), the truncation $f_{\rm trunc.}$ (N  =  5: long dashed black line), and the full numerical integration (full red line). The difference between the long-dashed and the short dashed line corresponds to the resummed terms (2.21). While f_trunc starts deviating from the exact result already for R below the radius of convergence $R_L\approx 0.82$, we observe that $f_{\rm resum.}$ provides a much better approximation for $f(R)$ even beyond R_L of the polynomial expansion. The difference $f_{\rm resum}(R)-f_{\rm trunc}(R)$ is only sensitive to the location of the singularity nearest to the origin: the information initially encoded in all the higher couplings $\lambda_n$ with n  >  4 has been reduced to four parameters (concretely, here, 26 parameters have been reduced to four). This input appears to be enough to bring f_resum quite close to the solution from the full numerical integration, up to values of the curvature $|R|\approx 1.1$, and clearly beyond the domain of validity $R_L\approx 0.82$ of the polynomial approximation. We conclude that the simple four-parameter resummation (2.21) offers a substantial improvement.

Zoom In Zoom Out Reset image size

Motivated by the findings of the previous section, we now adopt the more sophisticated technique of Padé approximants. A Padé approximant of the function $f(R)$ in (2.10) is a rational function—the ratio of two unique polynomials in R of degree M and K—which we denote as [M/K]. The integers $M\geqslant 0$ and $K\geqslant 1$ are at our disposal. The first M  +  K  +  1 Taylor coefficients of $f(R)$ serve as input, and, by construction, agree with those of the approximant [M/K]. Unlike in (2.21), the number of parameters is not reduced, and, consequently, we expect that a successful Padé resummation leads to a further improvement upon $f_{\rm resum.}(R)$.

In practice, and in order to exploit the maximum number of Taylor coefficients $N_{\rm max}=35$, we take M close to and below $N_{\rm max}/2$. This is combined with $K=M+1, M-1, M-2, M-3, M-4$ to account for different large-R asymptotics. Concretely, in figure 5 we show a selection of best fits corresponding to Padé approximants $[{M}/{K}]$ with M  =  16, K  =  M  +  1 (short-dashed line); M  =  16, K  =  M  −  1 (full line); M  =  15, K  =  M  −  2 (long-dashed line); $M=19, K=M-3$ (dashed-dotted line); and $M=20, K=M-4$ (dotted line).

Zoom In Zoom Out Reset image size

Three comments are in order. Firstly, it is quite noteworthy that all Padé approximants not only match each other within the radius of convergence of the polynomial expansion (2.17), but also in the much wider regime $\newcommand{\lta}{\lesssim} -2.5\lta R\lta 2$, substantially beyond the radius of convergence of the polynomial approximation. Secondly, the Padé approximants also coincide with the full numerical solution of figure 3 within the same range $\newcommand{\lta}{\lesssim} -2.5\lta R\lta 2$. The domain of validity of the resummed result is more than twice as large as the original one estimated in (2.17). Finally, for even larger fields the different resummations deviate from each other owing to the differences in the assumed asymptotic behaviour. These set in earlier, and are more pronounced, for positive than for negative curvature. Presumably this is so because the nearest singularity on the real axis of the fixed point differential equation (2.7) arises first for positive curvature, see (2.8).

We conclude that suitably adapted resummation techniques are powerful tools to increase the domain of validity for polynomial fixed point solutions. The polynomial couplings $\lambda_n$ appear to encode information about $f(R)$ beyond R_L, possibly up to the maximal range (2.9). The fixed point data is not sufficient to infer the large-field asymptotic behaviour of the function $f(R)$. This should not come as a surprise: heat kernel expansions have been used in the first place to obtain the fixed point coordinates which is a very good approximation for small, but less so for large scalar curvature.

Given the RG fixed point to high order in the polynomial expansion, we now turn to the effective action which we write in terms of $f(R)$ as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \displaystyle \label{Gammaf} \Gamma_k= \frac{k^4}{16 \pi}\int d^4x\sqrt{|\det g|}\,f_k(R) . \nonumber \end{align} \tag{ 3.1 }$$

Notice that the explicit RG scale dependence arises because we have used dimensionless fields and the dimensionless function f. Although we have computed $\newcommand{\Ga}{\Gamma} \Gamma_k$ on four-spheres, stationary points of constant curvature can be found on different topologies by analytical continuation of the metric and use of the appropriate boundary conditions. For Lorentzian metrics stationary points of constant (negative) positive curvature correspond to (anti-)de Sitter solutions to the quantum equations of motion for $g_{\mu \nu}$. Such solutions are given by the condition

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\R}{R} \displaystyle \label{deSitter} \begin{array}{@{}rl@{}} E(\R)&\equiv 2 f_k(\R) -\R f_k'(\R)=0 \end{array} \nonumber \end{align} \tag{ 3.2 }$$

where we have introduced $\newcommand{\R}{R} E(\R)$ to denote the equation of motion related to the action (3.1). When evaluated on four-spheres however, the volume integration can be performed yielding

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\de}{\delta} \displaystyle \int {\rm d}^4x\sqrt{|\det g|}= \frac{384\pi^2}{R^2}\frac{1}{k^4}. \nonumber \end{align} \tag{ 3.3 }$$

Accordingly, we find

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Ga}{\Gamma} \displaystyle \label{Gammak} \Gamma_k=24\pi\frac{f_k(R)}{R^2}\equiv 384\pi^2\frac{F_k(\bar R)}{\bar R^2}, \nonumber \end{align} \tag{ 3.4 }$$

where we have re-introduced the action in terms of the dimensionful fields in the second equation. In this form, and for fixed $\newcommand{\R}{R} \R$ or $\bar R$, the sole RG scale dependence originates from the implicit k-dependence of the function f or F, respectively.

Strictly speaking the function of the curvature given by (3.4) is valid only on four-spheres. We also note that $\newcommand{\Ga}{\Gamma} \Gamma_*$ diverges at R  =  0, because the fixed point solution obeys $f(R=0)> 0$ whereas the volume diverges as 1/R². To analyse the extremal points, we therefore must seperately discuss the cases of positive and negative curvature, and Euclidean and Lorentzian signatures. However, for curvatures $R\neq 0$ and $1/R\neq 0$, the stationarity for (3.4)

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\R}{R} \newcommand{\Ga}{\Gamma} \displaystyle \label{condition1} \frac{\partial\Gamma_k}{\partial \R}=0 \nonumber \end{align} \tag{ 3.5 }$$

translates directly to the condition (3.2). Therefore, assuming the validity of analytical continuation, we can use (3.5) to find solutions to the quantum equations of motion. Positive real solutions to (3.2) are termed de Sitter solutions $\newcommand{\Rds}{R_{{}_{\rm dS}}} \newcommand{\R}{R} \R = \Rds$ in the literature. In a slight abuse of language, we will refer to any solution of (3.2) as $\newcommand{\Rds}{R_{{}_{\rm dS}}} \newcommand{\R}{R} \R = \Rds$. The stationarity condition (3.2) remains algebraically the same at or away from fixed points⁵, and it holds equally for dimensionful fields and the dimensionful function F after trivial replacements. It is required that the Ricci scalar is finite for solutions to (3.2). If no such solutions were to be found, the action would take its extremal values at the boundaries of the definition domain, R  =  0, and $1/|R|=0$, without necessarily obeying (3.2).

We can look for solutions to (3.2) at each order N in the approximation by plotting the LHS of the equation and looking for stable zeros for all approximation orders N (see figure 6). Prior to this, we need to estimate the radius of convergence for $E(R)$ accurately, to ensure solutions are in the physical domain. To that end, we expand the stationarity condition (3.2) as a polynomial in the Ricci scalar,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\R}{R} \displaystyle \label{zeta} E(\R)=\sum_n\xi_n R^n, \nonumber \end{align} \tag{ 3.6 }$$

and search for a fit for the expansion coefficients $\xi_n$ as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{desittercos} \xi_n= A\,\frac{\cos(n\,\phi+ \Delta)}{(R_c)^n} . \nonumber \end{align} \tag{ 3.7 }$$

This provides us with an estimate for the radius of convergence for $E(R)$. Fitting the four free parameters in (3.7) to the data, we find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{fitdeSitter} \begin{array}{@{}rl@{}} A&=0.5282 \nonumber \\ R_c&=0.8584 \nonumber \\ \phi&=0.8226 \nonumber \\ \Delta&=2.1334 \end{array} \nonumber \end{align} \tag{ 3.8 }$$

for the amplitude, the radius, the angle and the shift, respectively. Figure 7 shows that (3.8) provides a good fit to the data. The angle is similar to the one found in (2.13) but the radius is slightly smaller. We additionally adopt the generalised ratio test. Since R² is a 'zero-mode' of (3.2) there will be no terms proportional to $\newcommand{\R}{R} \R^2$ in (3.6). Therefore we take $n \geqslant 3$ when determining $\newcommand{\R}{R} \newcommand{\e}{{\rm e}} \R_c(m)\equiv \min_n[\R_{c, m}(n)]$, and average m over values $8 \leqslant m \leqslant 31$, see section 2.4. Using this method we obtain the lower bound estimate

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\R}{R} \newcommand{\RLB}{R_{L}} \displaystyle \label{deSitterLB} \RLB \approx 0.77 \pm 5\% \nonumber \end{align} \tag{ 3.9 }$$

which is lower than the corresponding lower-bound estimate obtained from the polynomial expansion of $\newcommand{\R}{R} f(\R)$. The reason for this is that the equation of motion contains a derivative of $\newcommand{\R}{R} f(\R)$ which is more sensitive to the polynomial approximation, lowering the expected radius of convergence. A summary of the various radii of convergence including error estimates is given in table 3.

Table 3. Summary of estimates for the radius of convergence based on the action and the equations of motion, also comparing different approximation orders. Errors are statistical, roughly a standard deviation.

	approximation	radius estimate	info
(a)	N  =  11	0.99	eff action, (2.18)—from [61]
(b)	N  =  11	$1.00\pm 20\%$	eff action, root test (2.19)
(c)	N  =  35	$0.91\pm 5\%$	eff action, fit and root tests, (2.13), (2.15)
(d)	N  =  35	$0.82\pm 5\%$	eff action, lower bound, (2.17)
(e)	N  =  35	0.86	eq of motion, fit and root tests, (3.8)
(f)	N  =  35	$0.77\pm 5\%$	eq of motion, lower bound, (3.9)

Zoom In Zoom Out Reset image size

With these results at hand we now return to the search for zeros of (3.6). In fact, real solutions to (3.2) can be found at some orders in the approximation. In figure 8, we display all real de Sitter solutions for each and every approximation order N, indicated by crossed circles, and the estimate for the radius of convergence (3.9), indicated by the shaded area. Numerical results are shown in table 2. These solutions may be considered as physical provided they arise within the radius of convergence of the expansion and persist to high orders in the expansion. Some solutions accumulate close to $\newcommand{\Rds}{R_{{}_{\rm dS}}} \newcommand{\R}{R} \Rds\approx 0.8-0.9$ and $\newcommand{\Rds}{R_{{}_{\rm dS}}} \newcommand{\R}{R} \Rds\approx -(0.9-1.0)$, and appear to be at the boundary of the radius of convergence (2.15) and (2.17) as determined from $f(R)$. Most notably, we find that de Sitter solutions occur within the radius of convergence, as determined from $E(R)$, but only low orders in the approximation, without persisting to higher orders. We conclude that the data does not offer evidence for a real de Sitter solution at small Ricci curvature.

Zoom In Zoom Out Reset image size

At this point it is useful to make contact with the results of [61]. While our results up to order N  =  11 agree with [61], we are reaching the exact opposite conclusion. The reason for this is a subtle one, related to the slow convergence of the polynomial expansion. The eight-fold periodicity pattern in the couplings, (2.11), entails de Sitter solutions around $\newcommand{\Rds}{R_{{}_{\rm dS}}} \newcommand{\R}{R} \Rds \approx 0.77$ for nine of the first eleven approximation orders considered in [61], see table 2, suggesting that these are viable. The underlying eightfold periodicity only became visible starting from approximation order N  =  11  −  14 onwards, based on the sign pattern of couplings (table 1) or their magnitude (figure 1), respectively. Then, for all N  >  11  −  14, a de Sitter candidate arises in four out of eight cases. However, with increasing N, the de Sitter candidates no longer reside within the domain of validity. This follows from the various estimates for the radii of convergence summarised in table 3. Firstly, using the polynomial approximation for $f(R)$, the uncertainty in the radius of convergence at order N  =  11 comes out large, about $20\%$, table 3$(b)$. This is going down to the 5% range at N  =  35, tables 3$(c)$ and $(d)$. The positive de Sitter solutions of [61] sit close to the boundary of this error estimate. Secondly, once the polynomial approximation for $E(R)$ is analysed to estimate the domain of validity, the radius comes out smaller by about 5%, see table 3$(e)$ and table 3$(\,f)$. Combining the decrease of the radius of convergence and its error bar (with increasing N) with the slight growth of real de Sitter solutions $R_{\rm dS}>0$ (see table 2), it becomes evident that these solutions slip out of the domain of validity.

For completeness, we check whether de Sitter solutions exist within the radius of convergence for complex Ricci scalar. Within the polynomial approximation to order N, the de Sitter condition (3.2) will naturally have up to N  −  1 different solutions in the complex plane. In figure 9, we display all of these solutions for all orders N. We find that most solutions are outside the estimated radius of convergence. Those with positive real part accumulate close to the distance R_c (3.8). Interestingly, within the radius of convergence (3.9), there exist exactly two solutions with $\newcommand{\Rds}{R_{{}_{\rm dS}}} \newcommand{\R}{R} |\Rds|<R_L$, a complex conjugate pair, for each approximation order as soon as $N\geqslant 5$. The complex de Sitter solution has a positive real and a small imaginary part,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\R}{R} \newcommand{\Rds}{R_{{}_{\rm dS}}} \displaystyle \label{RdS} \begin{array}{@{}rl@{}} {\rm Re}\,\Rds&=\ \ \,0.5647\pm0.02\% \nonumber \\ {\rm Im}\,\Rds&=\pm0.2417\pm0.02\% \nonumber \\ |\Rds|&=\ \ \, 0.6143\pm0.02\%. \end{array} \nonumber \end{align} \tag{ 3.10 }$$

Zoom In Zoom Out Reset image size

Notice that the solution is particularly stable from order to order, reflected by the tiny standard deviation. Quantitatively, our results are summarized in table 2, where the smallest positive, negative, and the smallest complex solution per order are shown.

We have established that the fixed point theory does not offer a stationary point with a real de Sitter solution, at least in the regime for small scalar curvature. On the other hand, a near-stationary solution is located close to where $\newcommand{\Ga}{\Gamma} \partial\Gamma/\partial R$ is smallest. In fact, solving $\min_R(2f-Rf')$ for R one finds the near-stationary solution

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Rnear} R_{\rm ns}\approx 0.602, \nonumber \end{align} \tag{ 3.11 }$$

whose location is dictated by the nearby exact complex root (3.10) with $\newcommand{\Rds}{R_{{}_{\rm dS}}} \newcommand{\R}{R} R_{\rm ns}\approx |\Rds|$. For illustration, the left panel of figure 10 displays the near-stationary solution, showing that it corresponds to a nearly flat effective fixed point action $\newcommand{\Ga}{\Gamma} \Gamma_*(R)$ starting from curvatures R around and above (3.11). We may conclude that the near-stationarity leads to near-de Sitter behaviour in the deep UV for scalar curvatures around (3.11).

Zoom In Zoom Out Reset image size

Up to now we have discussed a model of quantum gravity with high-order interactions in the Ricci scalar. In a more extended study one should retain other curvature invariants as well. If the UV fixed point is a feature of the full theory, we expect that the impact of further curvature invariants is to change the fixed point couplings. To mimick the influence of (neglected) curvature invariants, and to check how this is influencing the existence (or not) of de Sitter solutions, we have introduced fudge factors by varying the values of the first few couplings $\lambda_0$, $\lambda_1$, $\lambda_3$ and $\lambda_4$ by  ±$50\%$. (Notice that de Sitter solutions are insensitive to the coupling $\lambda_2$). One finds that variations of $\lambda_3$ and $\lambda_4$ do barely change the nature of the de Sitter solution. Instead, varying $\lambda_0$ and $\lambda_1$ has a more substantial influence. For example, reducing $\lambda_1$ by a fudge factor of c  >  1 without changing g_* nor the other couplings, leads to a real de Sitter solution once $c\simeq 1.25$, see figure 10 (left panel). This can also be understood from figure 11: lowering $\lambda_0$ lowers the entire curve without changing its shape, which can lead to real de Sitter solutions. We conclude that stationary solutions may exist close to the near-stationary solution of the fixed point theory described by (2.10).

Zoom In Zoom Out Reset image size

Interestingly, if a real de Sitter solution exists, it is already well-approximated by the leading order estimate

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{dSapprox} R_{\rm dS}\simeq -2\,\frac{\lambda_0}{\lambda_1}\equiv 4\lambda_*. \nonumber \end{align} \tag{ 3.12 }$$

If the root is complex, then (3.12) offers a fair approximation to its real part. In either case, the impact of higher order couplings on the numerical value for R_dS appears to be small. The right panel of figure 10 compares the approximate solution (3.12) (blue dashed-dotted line) with the full, real or complex, solution (full or dashed red lines, respectively). We conclude that if a real de Sitter points exists, its value is fixed primarily by the dimensionless cosmological constant $\lambda_*$.

Further real de Sitter solutions may exist for larger Ricci curvature and outside the radius $\newcommand{\R}{R} \R_L,$ but within the region where a full numerical or resummed result for $E(R)$ exist. To investigate these cases, we analyse the equation of motion $E(R)$ using numerical integration in the regime $\newcommand{\R}{R} -2.5 \leqslant \R \leqslant 2$, together with Padé approximants, of various kinds, for the polynomial expansion of $E(R)$. Our numerical integration becomes unreliable close to $R\approx 2.006$ and close to $R\approx -2.541$, which prevents the solution being continued to larger values of $|R|$. The numerical solution does not show any de Sitter solutions satisfying the equation of motion within the region $\newcommand{\R}{R} -2 \leqslant \R \leqslant 2$. However, the function $E(R)$ becomes very small in the regime $R\approx 1$, see figure 11. To confirm our result, we additionally approximate $E(R)$ through Padé approximants as we did in section 2.7 for $f(R)$. Specifically, in figure 11, we display the Padé approximants [20/16] (dotted line), [17/14] (dashed-dotted), [14/12] (long-dashed), [16/15] (thin full), and [16/17] (short dashed), together with the numerical integration (full red line). The shaded area indicates the radius of convergence of the underlying polynomial approximation, (3.9).

We find very good agreement between Padé approximants and the numerical solution beyond the radius of convergence R_L, up to the range $\newcommand{\lta}{\lesssim} -1.5\lta R\lta 1.2$, and in the regime where $E(R)$ becomes small numerically. Once $\newcommand{\lta}{\lesssim} 1.2 \lta R$, numerical differences arise. The equation of motion $E(R)$ is more sensitive to the large-R behaviour of the Padé approximants than the fixed point action $f(R)$. The numerical result agrees best with Padé approximants whose asymptotic behaviour is given by $\propto R^2$ or $\propto R^3$. In all cases, no de Sitter solution shows up for $\newcommand{\lta}{\lesssim} R\lta 2$. Within the uncertainties of our study we cannot exclude de Sitter solutions for even larger values of the curvature.

For negative scalar curvature $\newcommand{\lta}{\lesssim} R\lta -1.5$, we also find that differences between Padé approximants become significant. An anti-de Sitter solution is found from the numerical solution at

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{AdS} R_{\rm AdS} \approx - 2.361\,38. \nonumber \end{align} \tag{ 3.13 }$$

This result should be cross-checked with Padé approximants, as these allow extrapolation to the regime where the numerical integration is no longer applicable. The Padé approximants also predict a de Sitter solution, roughly in the range $R_{\rm AdS} \approx - (2.5 - 3.5)$, although the location is clearly not determined accurately. We may conjecture that an anti-de Sitter solution at scalar curvature $|R|$ of the order of a few exists within the present approximation.

We have established the absence of real de Sitter solutions in the fixed point regime for small Ricci scalar |R|  <  R_L, based on high orders in the polynomial approximation for the fixed point. In addition, we have also found evidence for the absence of de Sitter solutions within the wider range $\newcommand{\lta}{\lesssim} -2\lta R\lta 1.5$. Here, resummation techniques and numerical integration have been used to establish the result. We have not explored the possibility of de Sitter solutions for large curvature scalar beyond the domain of validity of our study, but it is conceivable that a de Sitter solution for finite R exists. The anti-de Sitter solution close to (3.13) gives a first indiction of this. Further work is needed to obtain flow equations valid for all values of R where de Sitter solutions may be found. In [19] it has been shown, albeit in a different RG scheme, that a valid fixed point solution may display a power-law behaviour asymptotically, modulo oscillating logarithmic corrections. The existence of de Sitter solutions then depends on the coefficients of this asymptotic behaviour. Similar results may apply to our model and it remains to be seen whether or not they would offer de Sitter solutions (3.2) for asymptotically large $R\gg 1$. If no solutions to the de Sitter condition (3.2) for finite R can be found, the action takes its extremal values at the boundaries of vanishing or infinite curvature. In the limit $|R|\to 0$, the action becomes maximal because of $f(R=0)>0$, suggesting that a global minima is achieved either for large and finite R outside the range considered here, or at asymptotically large $|R|=\infty$.

The ultraviolet fixed point displays a stable complex de Sitter solution with positive real part, close to the real axis, for each and every approximation order starting at N  =  5; see (3.10). The relevancy of this solution for cosmological applications is unclear: it would seem to imply that a 'near de Sitter' phase with inflationary expansion in the fixed point regime of $f(R)$ gravity could exist, as long as the imaginary part is sufficiently small (as it is here). Small variations in the fixed point couplings show that the complex de Sitter solution becomes real once other curvature invariants or matter are taken into account (figure 10). A first example for this has been given in [97]. There, it has been established that Ricci tensor fluctuations such as in quantum gravity with action $\propto F(R^2)+R\cdot Z(R^{\mu\nu}R_{\mu\nu})$ induce de Sitter solutions even at an asymptotically safe UV fixed point.