Asymptotically free and safe quantum gravity scenarios consistent with Hubble, laboratory, and inflation scale physics

What possibly happened at the earliest time of the universe evolution from the cosmological point of view? There are strong implications to support the scenario that the universe had likely experienced an accelerated expanding era [1], and the quantum gravity theory is expected to play a significant role in revealing the dynamics of early universe.

One of the promising frameworks to understand the inflation mechanism is the asymptotic safety scenario of quantum gravity, also simply called the asymptotic safety [2]. This is based on Wilsonian renormalization group techniques. The core idea of asymptotic safety is that gravity is renormalizable in the sense that the quantum nature of gravity is controlled by a non-Gaussian fixed point: the renormalizable theory can be defined if the theory has a non-Gaussian fixed point to which the number of relevant couplings is finite. Since the discovery of a non-Gaussian fixed point of gravity the so-called Reuter fixed point [3, 4], further evidence for the scenario has been collected (for a recent review see Ref. [5] and references therein).

Solutions of the renormalization group equations are represented in the form of trajectories, and the set of all possible trajectories constitute the renormalization group flow. Each point of the renormalization group flow corresponds to an energy scale-dependent effective action, which we denote by $\Gamma_{k}$. This effective action is calculated from the fundamental quantum action by partially integrating out the excitations with momenta larger than the energy scale $k$ [6]. Thus, the effective action $\Gamma_{k}$ well describes physics at the typical energy scale $k$ in the tree-level approximation. On the other hand, we have to include sufficient loop corrections of $\Gamma_{k}$ to describe physics at much lower energy scales. In this way, the effective action $\Gamma_{k}$ is supposed to be connecting the low energy and the high energy physics, including the Hubble and the Planck scale physics.

The energy scale-dependence of the theory plays an important role in cosmology, because the physical parameters for the gravitational theories are obtained from the observations at different energy scales: the cosmological constant is estimated from the Hubble scale studies [7], the Newton constant from the laboratory experiments [8], and the parameters related to inflationary dynamics from the studies at much higher energy scales [1]. Then, one expects to find out a trajectory which relates all the parameters at given different energy scales, and study quantum gravity at much higher energy scales, keeping the consistency with lower energy scale dynamics.

In Ref. [9], Gubitosi et al. studied the renormalization group flow of the (Euclidean) $f(R)$ gravity up to the second order,

where $S_{k}$ is the tree-level approximation of gravitational quantum action $\Gamma_{k}$. We denote by $x$ the four dimensional space-time coordinate, by $g$ the determinant of the metric tensor $g_{\mu\nu}$, and by $R$ the scalar curvature. The action is parameterized by the three energy scale-dependent coupling parameters, $U_{k}$, $F_{k}$, and $C_{k}$. They numerically analyzed the renormalization group equations (RGEs) of Eq. (1) and found a trajectory which satisfies the observational constraints on the cosmological constant, the Newtonian coupling, and $C_{k}$, assuming the Starobinsky model of inflation.

In this paper, we extend the work of Gubitosi et al. [9] by studying the RGEs of gravitational action that includes a term proportional to the “square” of Weyl tensor, $C_{\mu\nu\rho\lambda}^{2}=C_{\mu\nu\rho\lambda}C^{\mu\nu\rho\lambda}$. The RGEs for this kind of truncated action was derived non-perturbatively by Sen, Wetterich, and Yamada in Ref. [10]. We numerically integrate these RGEs to fill all the boundary conditions introduced by Gubitosi et al. in Ref. [9] with additional conditions for $D_{k}$, the coupling of the square of Weyl term.

We find two types of trajectories which satisfy the above boundary conditions: i) one goes to asymptotically free regime at the Planck scale, and ii) one goes to asymptotically safe regime at the above Planck scale. We will call these trajectories as the asymptotically free and safe trajectories, respectively. To determine the Planck scale physics from our approach, one needs an observational constraints on the parameter of the square of Weyl tensor $D_{k}$ from beyond isotropic and homogeneous cosmological studies.

The rest of this paper is organized as follows. In Section II, we introduce the non-perturbative RGEs derived in Ref. [10] and boundary conditions obtained from the observations. In Section III, basic assumptions of our numerical calculation are explained. In Section IV, we present our results and Section V contains the conclusion.

In this study, we analyze the renormalization group equations for coupling parameters in the following quadratic pure gravitational effective action:

where $D_{k}$ is the running coupling parameter of the square of Weyl tensor. The contribution from the Gauss-Bonnet term is defined by

where $E_{k}$ is a coupling parameter for the Gauss-Bonnet term.

Let us introduce dimensionless coupling parameters $u_{k}$ and $w_{k}$ as

In Ref. [10], Sen, Wetterich, and Yamada non-perturbatively derived the RGEs for the coupling parameters of $u_{k}$, $w_{k}$, $C_{k}$, $D_{k}$, and $E_{k}$, which describe the energy scale dependence of the gravitational action of Eq. (2), written in the form of

where $\partial_{t}=k\partial_{k}$ is the dimensionless scale derivative. The r.h.s. of Eqs. (5)–(9) are called beta functions. The explicit forms of the RGEs (5)–(9) are listed in Appendix A. We note that $E_{k}$ does not enter in the beta functions of all of the couplings, including the beta function of $E_{k}$ itself. Since the Gauss–Bonnet term does not contribute to dynamics in the four dimensional space-time, we will drop Eq. (9) from our consideration.

To find out a trajectory which is consistent with the universe evolution, we impose boundary conditions for the coupling parameters based on the observational data. In this study, we assume the isotropic Friedmann universe evolution and treat the Weyl coupling $D_{k}$ as a free parameter, since the square of Weyl tensor term does not contribute to the dynamics in the isotropic Friedmann universe. For $u_{k}$, $w_{k}$, and $C_{k}$, we use the boundary conditions introduced by Gubitosi et al. in Ref. [9]. Their constraints are expressed as

where $\Lambda_{k}$ is the running cosmological constant, and $G_{k}$ is the running Newtonian constant.

Here, we comment on the above constraints. In Eq. (10), it is assumed that the late-time universe evolution is governed by the standard cosmology, i.e., the late-time universe is effectively described by the $2\Lambda-R$ action. The cosmological constant $\Lambda_{k}$ is evaluated at the current Hubble scale, $k_{\rm Hub}=10^{-33}~{}{\rm eV}$.

In Eq. (11), we impose the condition from the laboratory-scale experiments for the Newtonian coupling parameter. The characteristic length scale of the experiments is about $10^{-1}~{}{\rm m}$, which corresponds to $k_{\rm lab}=10^{-5}~{}{\rm eV}$.

In Eq. (12), we presume that the Starobinsky model of inflation [11, 12, 13] effectively describes the early-time universe evolution. Since the effective action of the Starobinsky model takes the form of $-R+BR^{2}$, the running cosmological constant is assumed to be subdominant at $k_{\rm inf}=10^{22}~{}{\rm eV}$, the approximate Hubble length scale at the inflationary era of the Starobinsky model [9].

Notice that the observational constraints Eqs. (10)–(12) are given at different energy scales. To find out a trajectory which is consistent with all of these constraints, in our numerical study, we shall obtain a set of initial conditions to fill all of these conditions. In the following discussion, a relation $A\simeq B$ for given physical values $A$ and $B$ holds if and only if $|(A-B)/A|\ll 1$ holds.

Since quantum effects of gravity would be negligible at much lower energy scales than the Planck scale, we assume that the running of $C_{k}$ and $D_{k}$ are negligible below the inflationary scale:

From Eqs. (10)–(14), we demand

The numerical integration of RGEs (5)–(8) can lead to singular effects in the infrared and the ultraviolet regimes. The beta functions, the r.h.s. of RGEs (5)–(8), can be written by the linear combinations of $l^{2n}_{p}(x)$, which is defined as

where $n$ and $p$ are integers and $x$ is $\tilde{m}^{2}_{t}$ or $\tilde{m}^{2}_{\sigma}$, defined as

where

Thereby, from Eq. (19), the beta functions have a pole at $x=-1$, and we have to check the validity of the RGEs (5)–(8) while the numerical calculations are performed. As discussed in Ref. [10], the beta functions are valid for $\tilde{m}^{2}_{t}>-1$, although more complex for $\tilde{m}^{2}_{\sigma}$. Yet, we will check whether $\tilde{m}^{2}_{t}>-1$ and $\tilde{m}^{2}_{\sigma}>-1$ hold or not, and we will take the numerical calculations are valid as far as there is no singular behavior in $\tilde{m}^{2}_{t}$, $\tilde{m}^{2}_{\sigma}$, and all of the coupling parameters.

We exhibit two types of trajectories which connect the observed universe with i) asymptotically free quantum gravity, and ii) asymptotically safe quantum gravity.

Keeping the consistency with infrared physics and the Starobinsky model, we have found two types of trajectories flowing into i) asymptotically free regime and ii) asymptotically safe regime. These two trajectories predict the same physics under the inflationary scale, with the assumption that the universe is isotropic Friedmann universe. Although the square of Weyl tensor term does not contribute to the homogeneous and isotropic universe dynamics, the boundary condition of $D_{k}$ has a key roll for the prediction of the Planck scale physics of gravity in the asymptotically safe quantum gravity. Thus, the determination of the boundary condition of $D_{k}$ from cosmological study is important.