1 Introduction↩︎

The effects of quantum gravity are crucial for establishing a consistent cosmological model spanning from the early to late universe. Developing a consistent and predictive theory of quantum gravity to achieve this is a significant challenge in theoretical physics. The perturbative approach in quantum gravity has been limited due to the negative mass dimension of the gravitational constant. However, in recent years, the asymptotically safe gravity has emerged as a promising scenario. The presence of a nontrivial fixed point renders the theory safe from UV divergences [1], [2]. The use of the functional renormalization group (FRG) within the context of asymptotically safe gravity has provided new avenues to explore phenomena in black hole physics and cosmology, extending from the UV scale to the IR scale [1], [3]–[5]. This approach enables the examination of consistent cosmological models across the entire UV to IR range.

Building on the framework of the FRG, the central element in the asymptotically safe gravity is the effective average action, ${\mathrm{\Gamma }}_k[g_{\mu \nu }]$. This action is constructed to describe gravitational phenomena at a momentum scale $k$, accounting for the effects of quantum loops. To define the effective average action, a regulator term $R_k$ is introduced, which suppresses the contributions from momentum modes $p$ below $k$. As a result, quantum fluctuations below the FRG scale $k$ are excluded from the effective action, while modes with $p>k$ are fully integrated out. More explicitly the FRG equation is $$\begin{array}{}\text{(1)}& \begin{array}{r}\frac{d{\mathrm{\Gamma }}_k}{dt}=\frac{1}{2}\mathrm{T}\mathrm{r}{[{\mathrm{\Gamma }}_k^{(2)}+R_k]}^{-1}\frac{d{R}_k}{dt},\end{array}\end{array}$$ where ${\mathrm{\Gamma }}_k^{(2)}$ is the second variation of the effective average action and $t=\log (k/k_0)$ with $k_0$ being an arbitrary initial value.

However, solving fully the FRG equation in a realistic theory is challenging. To make progress, the Einstein-Hilbert truncation is often employed. The action is defined as $$\begin{array}{r}S=\int d^{4}x\sqrt{-g}\frac{1}{16\pi G}(R-2\mathrm{\Lambda }),\end{array}$$ where $G$ is the Newton constant and $\mathrm{\Lambda }$ is the cosmological constant. The FRG ensures that the effective average action interpolates between the fundamental action, devoid of quantum corrections in the UV regime, and the quantum effective action as $k\to 0$. This interpolation makes the coupling constants scale dependent, and their evolution is governed by the FRG equation [1], [6], [7]. We show in the appendix 6 that the FRG equations yield scale-dependent expressions for these coupling constants: $$\begin{array}{}\text{(2)}& \begin{array}{rlr}G(k)& =& G_0[1-\omega G_0{k}^2+{\omega }_1{G}_0^2{k}^4+\mathcal{O}\left(G_0^3{k}^6\right)],\\ \mathrm{\Lambda }(k)& =& {\mathrm{\Lambda }}_0[1-\mu G_0{k}^2+{\mu }_1{G}_0^2{k}^4+\mathcal{O}\left(G_0^3{k}^6\right)]+G_0{k}^4[\nu +{\nu }_1{G}_0{k}^2+\mathcal{O}(G_0^2{k}^4)],\end{array}\end{array}$$ where $G_0$ and ${\mathrm{\Lambda }}_0$ are the Newton coupling and cosmological constant at $k=0$. The dimensionless parameters $\omega ,{\omega }_1$ etc. depend explicitly on the choice of regularization scheme, and we give the results for two schemes, the optimized and exponential cutoffs there. However the results are qualitatively similar, and both $G(k)$ and $\mathrm{\Lambda }(k)$ may be written compactly to allow the ${\mathrm{\Lambda }}_0=0$ solution by simply setting ${\mathrm{\Lambda }}_0=0$ in 2 . The solutions correspond to two distinct branches. The first branch, known as the Type IIa trajectory or the separatrix, assumes ${\mathrm{\Lambda }}_0=0$ [7]–[9]. It extends to $k\to 0$, smoothly approaching the Gaussian fixed point in the IR regime. In contrast, the second branch, referred to as Type IIIa trajectories, corresponds to ${\mathrm{\Lambda }}_0\ne 0$. This branch may not extend to $k\to 0$ due to the emergence of a singularity in the IR regime, limiting its validity to a finite momentum scale. However, as discussed in [10], the nonlocal extension of the Einstein-Hilbert truncation can potentially resolve the infrared singularity in the RG trajectories with a positive cosmological constant. It is crucial to note that the solutions for ${\mathrm{\Lambda }}_0=0$ and ${\mathrm{\Lambda }}_0\ne 0$ are derived separately, and we will see that these give distinct quantum corrections for the two branches. Nevertheless, the coefficients may be written uniformly like 2 . We note that the quantum correction to the cosmological constant begins at the $k^4$ order for ${\mathrm{\Lambda }}_0=0$, while it starts at the $k^2$-order for ${\mathrm{\Lambda }}_0\ne 0$, which differs from previous reports in the literature [8], [11]. This distinction shows the need to consider quantum-improved cosmological solutions for the two cases separately.

Quantum effects can be incorporated into cosmological studies at various levels. In this work, we incorporate quantum corrections into the Einstein equations by replacing the ordinary gravitational constant and cosmological constant with their scale-dependent counterparts. This approach allows us to explore the impact of quantum gravity on cosmology. A crucial step in this setup is the identification of the IR momentum scale $k$ with a suitable physical cutoff. Unlike black hole physics, where the cutoff scale can often be determined from physical mechanisms, cosmology lacks a direct method for such identification. Existing literature addresses this challenge by expressing $k$ in terms of physical quantities such as particle momenta or the spacetime curvature. Most natural and common choice in cosmology is to identify $k$ with the Hubble parameter, which is an almost unique physical scale. Various studies have explored Friedmann–Lemaı̂tre–Robertson–Walker (FLRW) cosmology choosing different cutoff scales [12]–[21]. In this work, we aim to determine a suitable cutoff scale motivated by the classical solutions at late times. In general, different choices for the cutoff scale can lead to varying cosmic evolution scenarios. This raises a critical question: how can we identify which cutoff choices are more viable? To address this, we investigate the quantum-corrected late-time behaviour of the Hubble parameter and the scale factor. Since quantum corrections are expected to be small at late times, the quantum-corrected solutions should closely resemble the classical solutions with minor quantum improvements. This implies that the perturbations introduced by quantum corrections must not be so large as to significantly alter the form of the solutions at late times. Through this analysis, we can identify a way to assess the viability of different cutoff scale choices.

We study the FLRW universe using the FRG-improved Einstein equations. This approach leads to a set of differential equations involving the Hubble parameter, energy density, the scale-dependent Newton’s gravitational constant, the cosmological constant, and a cutoff function. An important input for defining these equations is the requirement that the left-hand side of the Einstein equations is covariantly conserved, resulting in a modified continuity equation. This equation establishes a relationship between the matter density, the time-dependent gravitational constant, and the cosmological constant. Many cosmological studies have employed this modified continuity equation approach [16], [20], [22], [22]–[24]. It is important to note that this method differs from the consistency approach, which enforces the covariant conservation of both the energy-momentum tensor and the left-hand side of the Einstein equations [6], [12], [25]. In contrast, the modified continuity equation approach focuses on the conservation of the left-hand side of the Einstein equations only. Using this framework, we investigate the cosmological evolution for different cutoff choices across two trajectory branches: one with ${\mathrm{\Lambda }}_0=0$ and the other with ${\mathrm{\Lambda }}_0\ne 0$. For the ${\mathrm{\Lambda }}_0=0$ branch, we consider cutoff identifications where $k$ is either the classical Hubble parameter or a function of the quantum Hubble parameter, and find that the choice of the classical Hubble parameter gives reasonable cosmology, and another choice of quantum Hubble parameter gets some constraint. For the ${\mathrm{\Lambda }}_0\ne 0$ branch, the Hubble parameter becomes constant in late time, so it is not suitable to describe the time development. We find that the choice of the inverse cosmic time successful for ${\mathrm{\Lambda }}_0=0$ is not suitable. Instead, we consider a time-dependent cutoff scale proportional to the inverse of the classical scale factor, $k\propto 1/a_{\mathrm{cl}}(t)$, which is suggested by the scalar curvature for compact space. We find that this choice ensure consistent solutions with the classical Hubble parameter and scale factor.

This paper is organized as follows. In sect. 2, we present the improved field equations within the FRG framework. Section 3 focuses on FLRW cosmology with ${\mathrm{\Lambda }}_0=0$. We first summarize the classical solution for ${\mathrm{\Lambda }}_0=0$ in subsect. 3.1. In subsect. 3.2, we use the identification of the momentum scale with the classical Hubble parameter $H_{\mathrm{c}\mathrm{l}}$ and find reasonable behavior of the late-time cosmology with quantum corrections. In subsect. 3.3, we modify the identification to use the quantum Hubble parameter, discuss how the solution is different. In sect. 4, we consider FLRW cosmology with ${\mathrm{\Lambda }}_0\ne 0$. We then summarize the classical solution in subsect. 4.1. In this case, we show in subsect. 4.2 that the same choice of the identification as the ${\mathrm{\Lambda }}_0=0$ case, namely the inverse of the cosmic time does not give good late-time behavior, which does not reproduce the classical behavior in the classical limit. So it is not a viable identification. Then in subsect. 4.3, we show that good identification is obtained by using the scale factor, ensuring consistent solutions at late times. Finally, in sect. 5, we summarize our results.

2 The Improved Field Equation↩︎

In this section, the FRG improvement is applied at the “equation level” by replacing the ordinary Newtonian gravitational constant and cosmological constant with the scale-dependent running coupling constants $G(k)$ and $\mathrm{\Lambda }(k)$ in the Einstein equation. We take the viewpoint that these running coupling constants capture quantum effects at the leading order. The quantum improved Einstein equation can thus be written as $$\begin{array}{}\text{(3)}& \begin{array}{r}{R}_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }=-\mathrm{\Lambda }(t)g_{\mu \nu }+8\pi G(t)T_{\mu \nu }.\end{array}\end{array}$$ In writing down this equation, $G(k)$ and $\mathrm{\Lambda }(k)$ are replaced with $G(t)$ and $\mathrm{\Lambda }(t)$ to facilitate the study of cosmology, where the infrared cutoff is identified as $k=k(t)$ with the cosmic time $t$.

We consider a spatially flat, homogeneous, and isotropic cosmological model described by the FLRW spacetime: $$\begin{array}{}\text{(4)}& \begin{array}{r}d{s}^2=-d{t}^2+a^2(t)(d{x}^2+d{y}^2+d{z}^2),\end{array}\end{array}$$ where $a(t)$ is the scale factor. The associated geometric quantities for this spacetime are given by $$\begin{array}{}\text{(5)}& R_{tt}=-3\frac{\ddot{a}}{a},R_{ii}=a\ddot{a}+2{\dot{a}}^2,R=6(\frac{\ddot{a}}{a}+\frac{{\dot{a}}^2}{a^2}).\end{array}$$ Moreover, The cosmic matter is assumed to be a perfect fluid, described by the energy-momentum tensor $$\begin{array}{}\text{(6)}& \begin{array}{r}{T}^{\mu }{}_{\nu }=\mathrm{diag}(-\rho ,p,p,p),\end{array}\end{array}$$ where $\rho$ is the energy density, $p$ is the pressure and they are related by the equation of state $$\begin{array}{}\text{(7)}& \begin{array}{r}p=w\rho .\end{array}\end{array}$$ From these, the first modified Friedmann equation can be written as $$\begin{array}{}\text{(8)}& \begin{array}{rl}{H}^2=\frac{{\dot{a}}^2}{a^2}& =\frac{8\pi }{3}G(t)\rho +\frac{\mathrm{\Lambda }(t)}{3}.\end{array}\end{array}$$ In addition, the left-hand side of the Einstein equation is covariantly conserved, expressed as $D^{\mu }(R_{\mu \nu }-R{g}_{\mu \nu }/2)=0$, which is a mathematical identity in Riemannian geometry known as the Bianchi identity. Consequently, the entire right-hand side of the Einstein equation must also be covariantly conserved $D^{\mu }(-\mathrm{\Lambda }{g}_{\mu \nu }+8\pi G{T}_{\mu \nu })=0$. This leads to the continuity equation: $$\begin{array}{}\text{(9)}& \begin{array}{rl}\dot{\rho }+3H(p+\rho )& =-\frac{8\pi \rho \dot{G}+\dot{\mathrm{\Lambda }}}{8\pi G(t)}.\end{array}\end{array}$$ Then there are two possible standpoints at this stage.

(i) Consistency condition

In the absence of quantum effects, in particular if the Newton coupling $G$ and cosmological constant $\mathrm{\Lambda }$ are independent of time, the energy-momentum tensor is conserved $D^{\mu }{T}_{\mu \nu }=0$. So it may appear natural to require this. This leads to the ordinary continuity equation, and both the LHS and RHS of 9 should vanish [6], $$\begin{array}{}\text{(10)}& \begin{array}{r}\dot{\rho }+3H(p+\rho )=0,8\pi \rho \dot{G}+\dot{\mathrm{\Lambda }}=0.\end{array}\end{array}$$ However in this case, the second equation gives $$\begin{array}{r}\rho =-\frac{\dot{\mathrm{\Lambda }}}{8\pi \dot{G}}.\end{array}$$ As we will discuss later, if we identify the momentum scale $k$ with inverse of the cosmic time $t$, this is translated into $$\begin{array}{r}\rho =-\frac{{\partial }_k\mathrm{\Lambda }}{8\pi {\partial }_{k}G}.\end{array}$$ We see from the low energy expansions in Eq. 2 , the leading term of ${\partial }_{k}G$ is negative (for $\omega >0$). So the sign of the energy density $\rho$ depends entirely on the sign of the coefficient of the leading quantum correction to the cosmological constant. For ${\mathrm{\Lambda }}_0\ne 0$, it is negative, giving negative $\rho$. For ${\mathrm{\Lambda }}_0=0$, it is positive (for $\nu >0$), resulting in positive $\rho$. The case of ${\mathrm{\Lambda }}_0=0$ is often considered, and discussed in detail especially in Ref. [6]. However, in our case, there is another possibility, which we employ to identify the cutoff scale in both cases.

(ii) Modified continuity equation

When we consider the quantum effects, $G$ and $\mathrm{\Lambda }$ depend on the cosmic time. This arises because of the quantum effects of gravity. It is then expected that they also contribute to the energy-momentum. It must be $G$ in front of the energy-momentum tensor and $\mathrm{\Lambda }$ that give such contribution in the present formulation. This suggests that we should include the time derivative of $G(t)$ and $\mathrm{\Lambda }(t)$ in Eq. 9 as the energy-momentum from the gravity, instead of making its both sides separately vanish. This modified equation has been employed in various studies of cosmology within the asymptotic safety framework [16], [20], [22], [22]–[24]. This is weaker condition than (i) and gives interesting cosmology for ${\mathrm{\Lambda }}_0\ne 0$. In this paper, we take this condition.

Let us write the modified continuity equation as $$\begin{array}{}\text{(11)}& \begin{array}{r}8\pi {\partial }_t[G(t)\rho +\frac{\mathrm{\Lambda }(t)}{8\pi }]=-24\pi (1+w)HG(t)\rho .\end{array}\end{array}$$ where the equation of state 7 has been used. Substituting $G(t)\rho$ from Eq. 8 into the above equation, we obtain $$\begin{array}{}\text{(12)}& \begin{array}{r}\dot{H}=-\frac{1}{2}(3+3w)[H^2-\frac{1}{3}\mathrm{\Lambda }(t)]=-\alpha (H^2-\frac{\mathrm{\Lambda }}{3}),\end{array}\end{array}$$ where $\alpha \equiv 3(1+w)/2$. Furthermore, the energy density $\rho (t)$ can be expressed in terms of the solution for $H(t)$ and the cosmological constant $\mathrm{\Lambda }(t)$: $$\begin{array}{}\text{(13)}& \begin{array}{r}\rho =\frac{3}{8\pi G}(H^2-\frac{\mathrm{\Lambda }}{3}).\end{array}\end{array}$$ We shall solve the system of differential equations given by 12 and 13 for $H(t)$ and $\rho (t)$, under the condition that $H(t)\ne 0$ for the prescribed form of $G(t)$ and $\mathrm{\Lambda }(t)$. It is important to note that to solve Eq. 12 for the Hubble parameter $H(t)$, only the form of the external function $\mathrm{\Lambda }(t)$ is required. The running gravitational constant $G(t)$ only appears in Eq. 13 when solving for $\rho (t)$ in terms of $H(t)$ and $\mathrm{\Lambda }(t)$. To determine $\mathrm{\Lambda }(t)$, we have to make identification of the infrared momentum scale $k$ with the physical cutoff scale. Existing literature often expresses $k$ in terms of scales, such as particle momenta, field strengths, or the curvature of spacetime [6]. However, there is no satisfactory principles to directly correlate the FRG scale $k$ with the physical scale in cosmology, similar to the consistent thermodynamics for quantum improved black holes [26], [27]. For the FLRW universe, homogeneity and isotropy imply that the only physical sacle is the Hubble parameter which is determined by the cosmic time. Thus we have $$\begin{array}{}\text{(14)}& \begin{array}{r}G(t)\equiv G(k=k(t)),\mathrm{\Lambda }(t)\equiv \mathrm{\Lambda }(k=k(t)).\end{array}\end{array}$$ A natural choice of the cutoff identification is in terms of the Hubble parameter $H(t)$. We still have the possibility of using classical or quantum Hubble parameters. In the next section, we first consider FLRW cosmology for ${\mathrm{\Lambda }}_0=0$ using the identification with the classical Hubble parameter, and then the quantum Hubble. In the latter case, we consider a general form of $k(H)=\xi G_0^{(\beta -1)/2}{H}^{\beta }$, where $\beta$ is constant.

3 FLRW Cosmology for ${\mathrm{\Lambda }}_0=0$↩︎

In this section, we discuss the classical and quantum-corrected late-time solutions for $H(t)$, which, in turn, provide the solution for the scale factor $a(t)$ for ${\mathrm{\Lambda }}_0=0$. This corresponds to a unique trajectory that reaches the Gaussian fixed point as $k\to 0$, with ${\mathrm{\Lambda }}_0=0$; this is commonly referred to in the literature as the Type IIa or separatrix trajectory [7], [8]. The second branch is discussed in sect. 4.

3.1 Classical solution↩︎

The classical solution for the Hubble parameter, denoted as $H_{\mathrm{cl}}(t)$, in a single component universe specifically for radiation $(w=1/3)$ and matter ($w=0$) can be obtained from Eq. 12 by setting $G(t)=G_0$ and $\mathrm{\Lambda }(t)={\mathrm{\Lambda }}_0$. The classical solution for the Hubble parameter with ${\mathrm{\Lambda }}_0=0$ is given by $$\begin{array}{}\text{(15)}& H_{\mathrm{cl}}(t)=\frac{1}{\alpha (t-c^{\prime })},\end{array}$$ where $c^{\prime }$ is an integration constant. Because the system has the time translation symmetry in Einstein equation, the free time parameter $c^{\prime }$ does not carry physical significance and we can set $c^{\prime }=0$ or absorb it into the definition of the cosmic time $t$: $$\begin{array}{}\text{(16)}& H_{\mathrm{cl}}(t)=\frac{1}{\alpha t}\end{array}$$ By this, we fix the time origin in the system. The classical scale factor $a_{\mathrm{cl}}(t)$ is then derived by integrating the Hubble parameter as $$\begin{array}{}\text{(17)}& a_{\mathrm{cl}}(t)=a_0{t}^{1/\alpha }.\end{array}$$ The energy density can then be expressed using the classical Hubble parameter solution from Eq. 15 as follows: $$\begin{array}{}\text{(18)}& {\rho }_{\mathrm{cl}}(t)=(3/8\pi G_0)/(\alpha t{)}^2.\end{array}$$

3.2 The identification with the classical Hubble parameter↩︎

We begin by discussing the quantum-corrected solutions and their implications. The unique physical scale in this case is the Hubble parameter. Here we consider the classical Hubble parameter in the identification. So we take $$\begin{array}{}\text{(19)}& \begin{array}{r}k={\xi }^{\prime }{H}_{\mathrm{c}\mathrm{l}}(t)=\frac{\xi }{t}\propto a_{\mathrm{c}\mathrm{l}}^{-\alpha }(t),\end{array}\end{array}$$ where ${\xi }^{\prime }$ and $\xi (={\xi }^{\prime }/\alpha )$ are dimensionless parameters of order 1. By substituting this cutoff scale into Eq. 2 , the running gravitation coupling and the cosmological coupling can be written as a function of time: $$\begin{array}{}\text{(20)}& \begin{array}{rlr}G(t)& =& G_0[1-\tilde{\omega }{G}_0{t}^{-2}+{\tilde{\omega }}_1{G}_0^2{t}^{-4}+\mathcal{O}\left(G_0^3{t}^{-6}\right)],\\ \mathrm{\Lambda }(t)& =& t^{-2}[\tilde{\nu }{G}_0{t}^{-2}+{\tilde{\nu }}_1{G}_0^2{t}^{-4}+\mathcal{O}\left(G_0^3{t}^{-6}\right)],\end{array}\end{array}$$ where $\tilde{\omega }\equiv \omega {\xi }^2,{\tilde{\omega }}_1\equiv {\omega }_1{\xi }^4$ and $\tilde{\nu }\equiv \nu {\xi }^4,{\tilde{\nu }}_1={\nu }_1{\xi }^6$. Note that the time translation symmetry is no longer there by these parameters.

Substituting the cosmological constant 20 into 12 , and taking the ansatz for $H(t)$ as $$\begin{array}{}\text{(21)}& H(t)=\frac{1}{\alpha t}(1+\frac{c}{t}+\frac{c_1}{t^2}+\frac{c_2}{t^3}+\frac{c_3}{t^4}+\cdots ),\end{array}$$ we can determine the the above constants by comparing terms order by order in $1/t$. We then obtain $$\begin{array}{}\text{(22)}& H(t)=\frac{1}{\alpha t}[(1+\frac{c}{t}+\frac{c^2}{t^2}+\frac{c^3}{t^3}+\frac{c^4}{t^4}+\cdots )-\frac{{\alpha }^2\tilde{\nu }{G}_0}{3t^2}(1+\frac{c}{t}+\frac{c^2}{t^2}+\frac{c^2}{3t^2}-\frac{{\alpha }^2\tilde{\nu }{G}_0}{9t^2}+\cdots )-\frac{{\alpha }^2{\tilde{\nu }}_1{G}_0^2}{9t^4}(1+\cdots )],\end{array}$$ where $c$ is a new constant parameter. The appearance of one undetermined parameter $c$ is expected, as Eq. 12 is a first-order differential equation, which naturally provides one integration constant. We see the terms in the first bracket sums up to $1/(t-c)$, quite similar to the classical solution. However we cannot remove this parameter by absorbing it to the cosmic time $t$ because it changes the form of the cosmological constant term, in contrast to classical case. But in the absence of the quantum corrections ($\tilde{\nu }={\tilde{\nu }}_1=0$), $c$ becomes identical with the classical parameter.

We now show that this new parameter $c$ has physical meaning. This point was not discussed in [24]. The solution, up to fourth order in $1/t$, can be re-expressed as $$\begin{array}{}\text{(23)}& H(t)=\frac{1}{\alpha t}[1-\frac{{\alpha }^2\tilde{\nu }{G}_0}{3t_{c}t}(\frac{t_c}{t}-1)-{\left(\frac{{\alpha }^2\tilde{\nu }{G}_0}{3t_{c}t}\right)}^2(\frac{t_c}{t}-1)-{\left(\frac{{\alpha }^2\tilde{\nu }{G}_0}{3t_{c}t}\right)}^3(\frac{t_c}{t}-1)+\mathcal{O}\left(t^{-4}\right)],\end{array}$$ where the critical time scale $t_c$ is defined by $$t_c={\alpha }^2\tilde{\nu }{G}_0/3c.$$ The quantum parameter $\tilde{\nu }$, which defines the critical time, is always positive for both the optimized and exponential cutoffs: specifically, $\tilde{\nu }={\xi }^4/8\pi$ for the former case, as defined in Eq. 55 and $\tilde{\nu }={\xi }^4\zeta (3)/2\pi$ for the latter in Eq. 54 . Therefore, this positive critical time scale indicates that the leading quantum correction has an attractive effect (AdS-phase) for which the expansion rate of the FLRW universe slows down compared with the speed of classical expansion, $H(t)<H_{\mathrm{cl}}(t)=1/\alpha t$, for $t<t_c$. Conversely, it becomes repulsive (dS-phase) and makes the expansion rate bigger than the classical expansion rate when $t>t_c$. The factor $(t_c/t-1)$ is not universal, but appears up to $t^{-3}$-order and the remaining terms are much smaller. So the critical time scale, which distinguishes the AdS and dS phases, is qualitatively correct.

The value of the parameter $c$ can be estimated by physical observation data. We can get approximated idea about the value of $c$ is by setting the Hubble parameter $H(t_0)=H_0$ at present time $t_0$, with $\alpha =3/2$ ($w=0$ matter dominated era) in the solution $H(t_0)=1/(\alpha (t_0-c))$ with the leading correction from Eq. 22 . From the condition that this should be the current time Hubble parameter $H_0$ at $t_0$, we get $$\begin{array}{}\text{(24)}& c=t_0-\frac{1}{\alpha H_0},t_0=(1.373\pm 0.012)\times {10}^{10}\mathrm{years},1/H_0=1.45\times {10}^{10}\mathrm{years}.\end{array}$$ This means that in general $c$ is a huge positive number of the same order or one order smaller than $t_0$, and thus the value of $t_c$ is very small. This implies the quantum correction always provides repulsive effect in late time of FLRW cosmology $t\gg t_c$. Note that the late time solution 22 is valid for $t>c$, implying 24 is a consistent condition. The quantum corrected scale factor $a(t)$ can be calculated integrating the Hubble parameter $H(t)=\dot{a}(t)/a(t)$ as follows: $$\begin{array}{}\text{(25)}& \begin{array}{rl}a(t)& =a_0{t}^{1/\alpha }[1-\frac{c}{\alpha t}-\frac{(\alpha -1)c^2}{2{\alpha }^2{t}^2}+\frac{\alpha \tilde{\nu }{G}_0}{6t^2}+\mathcal{O}\left(t^{-3}\right)]\\ & =a_0{t}^{1/\alpha }[1-\frac{\alpha \tilde{\nu }{G}_0}{3t_{c}t}-\frac{{\alpha }^2(\alpha -1){\tilde{\nu }}^2{G}_0^2}{18t_c^2{t}^2}+\frac{\alpha \tilde{\nu }{G}_0}{6t^2}+\mathcal{O}\left(t^{-3}\right)].\end{array}\end{array}$$ Substituting $\mathrm{\Lambda }(t)$ from Eq. 20 and the Hubble parameter $H(t)$ from Eq. 22 in Eq. 13 , we obtain the quantum corrected energy density to be $$\begin{array}{}\text{(26)}& \rho (t)=\frac{3}{8\pi {\alpha }^2{G}_0{t}^2}[1+\frac{2c}{t}+\frac{3c^2}{t^2}+\frac{\tilde{\omega }{G}_0-{\alpha }^2\tilde{\nu }{G}_0}{t^2}+\mathcal{O}\left(t^{-3}\right)].\end{array}$$ Note that, by setting $\tilde{\nu }=\tilde{\omega }=0$ and $c=0$, we recover the classical result of energy density in FLRW cosmology. Here we can have a further physical understanding for the existence of the critical time $t_c$. The leading-order quantum correction to the cosmological constant, $\mathrm{\Lambda }\simeq \tilde{\nu }{G}_0/t^4>0$, introduces a repulsive force. However, the integration constant $c$ increases the energy density $\rho (t)$ which provides an attractive force. The entire effect is the competition of these two opposite contributions, with the critical time $t_c$ characterizing the balance point between them, as can be seen in Eq. 23 .

3.3 The identification by the Hubble parameter with quantum correction↩︎

Here we consider some general type of cutoff identification as mentioned before, defined as a function of the quantum Hubble parameter $H(t)$ in the form $$\begin{array}{r}k(H)=\zeta G_0^{(\beta -1)/2}{H}^{\beta },\end{array}$$ where $\zeta$ is a dimensionless constant. To ensure that $\zeta$ in the cutoff scale is dimensionless, unlike in [24], we introduce $G_0$ within the cutoff scale. Three specific values of $\beta =1/4,3/4,1$ were considered in [24], and we study which of these values are viable. By substituting this cutoff choice into 2 with ${\mathrm{\Lambda }}_0=0$, we obtain the leading correction to the cosmological constant as $$\begin{array}{}\text{(27)}& \mathrm{\Lambda }=3\gamma H^{4\beta }+\cdots ,\end{array}$$ where $\gamma ={\zeta }^4\nu G_0^{2\beta -1}/3$. Putting 27 in 12 , the Einstein equation 12 for $H(t)$ becomes $$\begin{array}{}\text{(28)}& \frac{dH}{dt}=-\alpha (H^2-\gamma H^{4\beta }).\end{array}$$ Using the relation $H^{-1}(dH/dt)=a(dH/da)$, we can rewrite the above equation in the form $$\begin{array}{}\text{(29)}& -\frac{H^{2-4\beta }}{H(\gamma -H^{2-4\beta })}dH=-\alpha \frac{da}{a}.\end{array}$$ Integrating the above equation, we obtain the solution of $H(t)$ in terms of scale factor $a(t)$ as $$\begin{array}{}\text{(30)}& H=a^{-\alpha }{(\gamma a^{-\alpha (4\beta -2)}-C_0)}^{-1/(4\beta -2)}.\end{array}$$ where $C_0$ is the integration constant. To determine this, we require the scale factor $a(t_0)=1$ and $H(t_0)=H_0$ at present time, yielding $$\begin{array}{}\text{(31)}& C_0=\gamma -H_0^{2-4\beta }.\end{array}$$ By substituting the integration constant $C_0$ into Eq. 30 , we get the final form of the quantum corrected Hubble parameter $H(t)$: $$\begin{array}{}\text{(32)}& H=H_0{a}^{-\alpha }{[1+\gamma H_0^{4\beta -2}(a^{-\alpha (4\beta -2)}-1)]}^{-1/(4\beta -2)}.\end{array}$$ We will examine the asymptotic behaviour of the Hubble parameter at $t\to \mathrm{\infty }$. Our aim is to determine if there exists a critical value of the cutoff scale exponent $\beta$ at which the entropy transitions from constant to divergent occurs at $t\to \mathrm{\infty }$, as previously discussed [24]. In the limit of infinitely large time, the scale factor $a$ becomes $\mathrm{\infty }$ for spatially flat FLRW universe. We then observe from Eq. 32 two distinct behaviours of $H(t)$ depending on the value of $\beta$: $$H\approx \{\begin{array}{cl}{\tilde{H}}_0{a}^{-\alpha },& \beta >1/2,\\ \tilde{\gamma },& \beta <1/2,\end{array}$$ where ${\tilde{H}}_0=H_0(1-\gamma H_0^{4\beta -2}{)}^{-1/(4\beta -2)}$ and $\tilde{\gamma }={\gamma }^{-1/(4\beta -2)}$. Furthermore, the scale factor $a(t)$ takes the form $$\begin{array}{}\text{(33)}& a(t)\approx \{\begin{array}{cl}(\alpha {\tilde{H}}_0{)}^{1/\alpha }{t}^{1/\alpha },& \beta >1/2,\\ {\mathrm{e}}^{\tilde{\gamma }t},& \beta <1/2,\end{array}.\end{array}$$ From the asymptotic behaviour of $H(t)$ and the scale factor $a(t)$, it is clear that $\beta =1/2$ is a critical value which changes the scale factor behaviour from a power law ($\beta >1/2$) to exponential ($\beta <1/2$).

In order to find the higher order terms in the solution of the Hubble parameter $H(t)$ at late time, we expand $H(t)$ as $$\begin{array}{}\text{(34)}& H(t)=\{\begin{array}{cl}{\tilde{H}}_0{a}^{-\alpha }{[1+\mathrm{\Delta }{a}^{-\alpha (4\beta -2)}]}^{-1/(4\beta -2)}={\tilde{H}}_0{a}^{-\alpha }[1-\frac{\mathrm{\Delta }}{4\beta -2}{a}^{-\alpha (4\beta -2)}+\cdots ],& \beta >1/2\\ \tilde{\gamma }{[1+{\mathrm{\Delta }}^{-1}{a}^{-\alpha (2-4\beta )}]}^{-1/(4\beta -2)}=\tilde{\gamma }[1-\frac{1}{\mathrm{\Delta }(4\beta -2)}{a}^{-\alpha (2-4\beta )}+\cdots ],& \beta <1/2\end{array},\end{array}$$ where $\mathrm{\Delta }=({\tilde{H}}_0/H_0{)}^{4\beta -2}-1$. Integrating the above equation, we obtain the late time behaviour of the scale factor $a(t)$ with leading order correction as $$\begin{array}{}\text{(35)}& a(t)=\{\begin{array}{cl}(\alpha {\tilde{H}}_0{)}^{1/\alpha }{t}^{1/\alpha }[1+\frac{\mathrm{\Delta }(4\beta -1)}{\alpha (4\beta -2{)}^2}(\alpha {\tilde{H}}_0{)}^{-(4\beta -2)}{t}^{-(4\beta -2)}+\cdots ],& \beta >1/2\\ \exp [\tilde{\gamma }t-\frac{1}{\mathrm{\Delta }\alpha (2-4\beta {)}^2}{\mathrm{e}}^{-\alpha \tilde{\gamma }(2-4\beta )t}+\cdots ],& \beta <1/2\end{array}.\end{array}$$ The exponential solution for the cutoff choice with $\beta <1/2$ suggests that this choice is not viable. For $\beta <1/2$, large momentum scales $k$ cause significant perturbations, leading to exponential expansion which is completely different from the classical solution. On the other hand, for $\beta >1/2$, the behaviour follows a power law similar to the classical case, as the perturbations remain small enough to maintain this expected behaviour with the small quantum corrections. Thus, the cutoff identification is only valid for $\beta >1/2$ for late time.

For the sake of completeness, let us examine the cosmic evolution when the exponent of the momentum cutoff scale is exactly $\beta =1/2$. This gives $k=\zeta G_0^{-1/4}{H}^{1/2}=\zeta \sqrt{m_{\mathrm{Pl}}}{H}^{1/2}$ and the leading order cosmological constant becomes $\mathrm{\Lambda }={\zeta }^4\nu H^2$. The Friedmann equation then takes the form $$\begin{array}{}\text{(36)}& \frac{dH}{dt}=-\alpha (1-\frac{{\zeta }^4\nu }{3})H^2.\end{array}$$ Solving this equation, we find the Hubble parameter as a function of cosmic time: $$H(t)=\frac{1}{\alpha (1-{\zeta }^4\nu /3)(t-c)},$$ where $c$ is an arbitrary integration constant. Integrating $H(t)$, the scale factor can be written as $$a(t)=a_0(t-c{)}^{\frac{1}{\alpha (1-{\zeta }^4\nu /3)}}.$$ We observe that the solution maintains a power-law behaviour, similar to the classical case, unlike the solutions for $\beta <1/2$. Although the cutoff scale $k$ is large, scaling with the Planck mass, the running cosmological constant does not depend on the Planck mass and remains small. Thus, $\beta \ge 1/2$ is a viable cutoff choice and $\beta =1/2$ represents the critical value, below which $\beta$ are not good choices for late time.

4 FLRW cosmology for ${\mathrm{\Lambda }}_0>0$↩︎

In this section, we begin by summarizing the classical solutions for $H(t)$, $a(t)$ and $\rho (t)$ for the branch of trajectories where $\mathrm{\Lambda }={\mathrm{\Lambda }}_0>0$.

4.1 Classical solution↩︎

To calculate the classical solution for the Hubble parameter $H_{\mathrm{cl}}(t)$ in a universe dominated by a cosmological constant, the classical Einstein equation 12 with $\mathrm{\Lambda }={\mathrm{\Lambda }}_0>0$ yields: $$\begin{array}{}\text{(37)}& H_{\mathrm{cl}}(t)=\sqrt{{\mathrm{\Lambda }}_0/3}\frac{1+\overline{c}\exp (-2\alpha \sqrt{{\mathrm{\Lambda }}_0/3}t)}{1-\overline{c}\exp (-2\alpha \sqrt{{\mathrm{\Lambda }}_0/3}t)},\end{array}$$ where $\overline{c}$ is the integration constant. This exact solution has two branches, depending on the sign of the free parameter $\overline{c}$. Again using the time translation symmetry to shift the cosmic time $t$, we can set $|\overline{c}|=1$. Depending on the sign of $\overline{c}$, the classical solution then becomes $$\begin{array}{}\text{(38)}& H_{\mathrm{cl}}(t)=\{\begin{array}{ll}\sqrt{{\mathrm{\Lambda }}_0/3}\tanh \left(\alpha \sqrt{{\mathrm{\Lambda }}_0/3}t\right),& \mathrm{sign}(\overline{c})=-1,\\ \sqrt{{\mathrm{\Lambda }}_0/3}\coth \left(\alpha \sqrt{{\mathrm{\Lambda }}_0/3}t\right),& \mathrm{sign}(\overline{c})=1.\end{array}\end{array}$$ At late times, both branches asymptotically approach a constant value of $\sqrt{{\mathrm{\Lambda }}_0/3}$, consistent with the expected behaviour of a cosmological constant dominated universe. However, we select the branch with $\mathrm{sign}(\overline{c})=-1$, as it provides a continuous solution at $t=0$, unlike the $\mathrm{sign}(\overline{c})=1$ branch, which has a discontinuity at $t=0$. For $\mathrm{sign}(\overline{c})=-1$, the exact late-time behaviour is written as $$\begin{array}{}\text{(39)}& H_{\mathrm{cl}}(t)=\sqrt{{\mathrm{\Lambda }}_0/3}(1+2\sum _{j=1}^{\mathrm{\infty }}(-1{)}^j{\mathrm{e}}^{-2j\alpha \sqrt{{\mathrm{\Lambda }}_0/3}t}).\end{array}$$ The classical scale factor for the $\mathrm{sign}(\overline{c})=-1$ branch in Eq. 38 is $$\begin{array}{}\text{(40)}& a_{\mathrm{cl}}(t)=a_0{\cosh }^{1/\alpha }\left(\alpha \sqrt{{\mathrm{\Lambda }}_0/3}t\right)\approx a_0\exp \left(\sqrt{{\mathrm{\Lambda }}_0/3}t\right).\end{array}$$ Finally, the energy density is obtained from Eq. 13 as $$\begin{array}{}\text{(41)}& {\rho }_{\mathrm{cl}}(t)=-({\mathrm{\Lambda }}_0/8\pi G_0){\cosh }^{-2}\left(\alpha \sqrt{{\mathrm{\Lambda }}_0/3}t\right)\approx -({\mathrm{\Lambda }}_0/2\pi G_0)\exp (-2\alpha \sqrt{{\mathrm{\Lambda }}_0/3}t)\approx 0,.\end{array}$$ Thus, for $\mathrm{\Lambda }>0$, $a_{\mathrm{cl}}$ at very late times ($t\to \mathrm{\infty }$) grows exponentially and ${\rho }_{\mathrm{cl}}$ effectively vanishes.

4.2 The identification using the cosmic time↩︎

We now analyze how the cosmic evolution of the FLRW universe differs in the different branches of FRG trajectories with ${\mathrm{\Lambda }}_0>0$ compared to the branch with ${\mathrm{\Lambda }}_0=0$.

The first problem is how to determine the identification of the momentum scale. If we follow the standard common sense, it is natural to use the Hubble scale. However it becomes constant in the late time and is not suitable for identification with the momentum scale $k$. So let us consider the same cutoff scale identification $$\begin{array}{r}k=\frac{\xi }{t},\end{array}$$ which was successful for ${\mathrm{\Lambda }}_0=0$. By inserting this cutoff scale in Eq. 2 , we can express the running gravitational coupling and cosmological coupling as functions of time as $$\begin{array}{}\text{(42)}& \begin{array}{rlr}G(t)& =& G_0[1-\tilde{\omega }{G}_0{t}^{-2}+{\tilde{\omega }}_1{G}_0^2{t}^{-4}+\mathcal{O}\left(G_0^3{t}^{-6}\right)],\\ \mathrm{\Lambda }(t)& =& {\mathrm{\Lambda }}_0[1-\tilde{\mu }{G}_0{t}^{-2}+{\tilde{\mu }}_1{G}_0^2{t}^{-4}+\mathcal{O}\left(G_0^3{t}^{-6}\right)]+t^{-2}[\tilde{\nu }{G}_0{t}^{-2}+{\tilde{\nu }}_1{G}_0^2{t}^{-4}+\mathcal{O}\left(G_0^3{t}^{-6}\right)],\end{array}\end{array}$$ where $\tilde{\omega }\equiv \omega {\xi }^2,{\tilde{\omega }}_1\equiv {\omega }_1{\xi }^4,\tilde{\mu }\equiv \mu {\xi }^2,{\tilde{\mu }}_1\equiv {\mu }_1{\xi }^4$ and $\tilde{\nu }\equiv \nu {\xi }^4,{\tilde{\nu }}_1={\nu }_1{\xi }^6$.

We can use the Einstein equation 12 to obtain the quantum-corrected solution for $H(t)$ as the inverse power series in the cosmic time: $$\begin{array}{rlr}H(t)& =& \sqrt{\frac{{\mathrm{\Lambda }}_0}{3}}[1-\frac{\tilde{\mu }{G}_0}{2}\frac{1}{t^2}-\frac{\sqrt{3}\tilde{\mu }{G}_0}{2\alpha \sqrt{{\mathrm{\Lambda }}_0}}\frac{1}{t^3}+(\frac{\tilde{\nu }{G}_0}{2{\mathrm{\Lambda }}_0}-\frac{\tilde{\mu }{G}_0(18+{\alpha }^2\tilde{\mu }{G}_0{\mathrm{\Lambda }}_0)}{8{\alpha }^2{\mathrm{\Lambda }}_0}+\frac{{\tilde{\mu }}_1{G}_0^2}{2})\frac{1}{t^4}\\ & & +\sqrt{\frac{3}{{\mathrm{\Lambda }}_0^3}}(\frac{\tilde{\nu }{G}_0}{\alpha }-\frac{\tilde{\mu }{G}_0(9+{\alpha }^2\tilde{\mu }{G}_0{\mathrm{\Lambda }}_0)}{2{\alpha }^3}+\frac{{\tilde{\mu }}_1{G}_0^2{\mathrm{\Lambda }}_0}{\alpha })\frac{1}{t^5}+\mathcal{O}\left(t^{-6}\right)],\end{array}$$ which leads to the scale factor $$\begin{array}{rlr}a(t)& =& a_0\exp [\sqrt{\frac{{\mathrm{\Lambda }}_0}{3}}t+\sqrt{\frac{{\mathrm{\Lambda }}_0}{3}}\frac{\tilde{\mu }{G}_0}{2}{t}^{-1}+\frac{\tilde{\mu }{G}_0}{4\alpha }{t}^{-2}-\sqrt{\frac{3}{{\mathrm{\Lambda }}_0}}(\frac{\tilde{\nu }{G}_0}{18}-\frac{\tilde{\mu }{G}_0(18+{\alpha }^2\tilde{\mu }{G}_0{\mathrm{\Lambda }}_0)}{72{\alpha }^2}+\frac{{\tilde{\mu }}_1{G}_0^2{\mathrm{\Lambda }}_0}{18})t^{-3}+\mathcal{O}\left(t^{-4}\right)]\\ & =& a_0\exp \left(\sqrt{{\mathrm{\Lambda }}_0/3}t\right)[1+\sqrt{\frac{{\mathrm{\Lambda }}_0}{3}}\frac{\tilde{\mu }{G}_0}{2}{t}^{-1}+\frac{\tilde{\mu }{G}_0}{4}(\frac{1}{\alpha }+\frac{\tilde{\mu }{G}_0{\mathrm{\Lambda }}_0}{6})t^{-2}+\mathcal{O}\left(t^{-3}\right)].\end{array}$$ By setting the quantum parameters to zero, we should recover the leading-order classical term of the Hubble parameter. However, the classical solution shows the exponential expansion, which can never reproduced from the power expansion in $t$. Another important observation is that, unlike the ${\mathrm{\Lambda }}_0=0$ case, this solution lacks an undetermined constant. This absence is unusual because the Einstein equation is a first-order differential equation for $H(t)$, which typically permits a constant of integration. All of these suggest that the cutoff identification $k=\xi /t$ is not a physically viable choice.

4.3 The identification using the scale factor↩︎

In pursuit of a consistent identification of the cutoff scale, we notice the scale factor is another physical quantity that determines the scale of our universe. So we propose to use it for the identification. Define $$\tau =\frac{1}{2\alpha \sqrt{{\mathrm{\Lambda }}_0/3}}\exp \left(2\alpha \sqrt{{\mathrm{\Lambda }}_0/3}t\right)\propto a_{\mathrm{c}\mathrm{l}}^{2\alpha }(t),$$ where we have used the cosmological constant to introduce dimension, and we use this for identification $$\begin{array}{}\text{(43)}& \begin{array}{r}k=\frac{\xi }{\tau }\propto a_{\mathrm{c}\mathrm{l}}^{-2\alpha }(t),\end{array}\end{array}$$ where $\xi$ is a constant of order 1. Here we have put the power $\alpha$ on the scale factor in analogy with 19 . The Einstein equation 12 takes the form $$\begin{array}{}\text{(44)}& \sqrt{{\mathrm{\Lambda }}_0/3}\tau \frac{dH}{d\tau }=-\frac{1}{2}(H^2-\mathrm{\Lambda }/3).\end{array}$$ In the literature, this cutoff was employed in [6] within the consistency condition approach, where the energy-momentum tensor is conserved along with the right-hand side of the Einstein equation. In that approach, however, it was shown that no consistent solution exists for a flat FLRW universe, except in the presence of exotic matter.

Here we demonstrate that this cutoff choice is a viable option for the case ${\mathrm{\Lambda }}_0>0$ if we use the modified continuity equation, in contrast to the consistency condition approach. It is also important to note that we select ${\mathrm{\Lambda }}_0$ to match the dimension of the momentum of the cutoff scale $k$, unlike the ${\mathrm{\Lambda }}_0=0$ case, where $G_0$ is chosen. This is because the quantum effects in the solution is expected to become strong around the order of Planck scale: If we had used $G_0$ with positive power (for dimensional reason) in $\tau$, the Planck scale in the quantum effects would cancel out from Eq. 2 , in contradiction to this expectation.

With the choice of cutoff 43 , the scale-dependent cosmological constant 2 becomes $$\begin{array}{}\text{(45)}& \mathrm{\Lambda }(\tau )={\mathrm{\Lambda }}_0[1-\tilde{\mu }{G}_0{\tau }^{-2}+{\tilde{\mu }}_1{G}_0^2{\tau }^{-4}+\mathcal{O}\left(G_0^3{\tau }^{-6}\right)]+{\tau }^{-2}[\tilde{\nu }{G}_0{\tau }^{-2}+{\tilde{\nu }}_1{G}_0^2{\tau }^{-4}+\mathcal{O}\left(G_0^3{\tau }^{-6}\right)].\end{array}$$ By substituting this into Eq. 44 , we obtain the solution for the Hubble parameter in terms of $\tau$ as $$\begin{array}{}\text{(46)}& H(\tau )=\sqrt{{\mathrm{\Lambda }}_0/3}[1+\frac{2c^{\prime }}{\tau }+\frac{2c^{\prime 2}}{{\tau }^2}+\frac{2c^{\prime 3}}{{\tau }^3}+\frac{2c^{\prime 4}}{{\tau }^4}+\frac{\tilde{\mu }{G}_0}{2{\tau }^2}+\frac{\tilde{\mu }{G}_0{c}^{\prime }}{2{\tau }^3}+\frac{2\tilde{\mu }{G}_0{c}^{\prime 2}}{3{\tau }^4}+\frac{({\tilde{\mu }}^2-4{\tilde{\mu }}_1)G_0^2}{24{\tau }^4}-\frac{\tilde{\nu }{G}_0}{6{\mathrm{\Lambda }}_0{\tau }^4}+\mathcal{O}\left({\tau }^{-5}\right)].\end{array}$$ As discussed before for ${\mathrm{\Lambda }}_0=0$, the time translation symmetry is broken here due to the time dependence of the cosmological term in the Einstein equation. Consequently, the new parameter $c^{\prime }$ plays a significant role in obtaining the quantum-corrected solution for the Hubble parameter. However, as mentioned for the ${\mathrm{\Lambda }}_0=0$ case, to recover the classical solution, we should set the quantum parameters $\tilde{\mu }$, $\tilde{\nu }$ and ${\tilde{\mu }}_1$ to zero. This allows us to absorb the free parameter within time or set it to zero. This point will become clearer in the following steps.

If we set $\tilde{\mu }$, $\tilde{\nu }$ and ${\tilde{\mu }}_1$ to zero, the expansion of the Hubble parameter 46 becomes $$\begin{array}{}\text{(47)}& \begin{array}{rlr}{H}_{\mathrm{cl}}& =& \sqrt{\frac{{\mathrm{\Lambda }}_0}{3}}[1+\frac{2c^{\prime }}{\tau }+\frac{2c^{\prime 2}}{{\tau }^2}+\frac{2c^{\prime 3}}{{\tau }^3}+\frac{2c^{\prime 4}}{{\tau }^4}+\cdots ]\\ & =& \sqrt{\frac{{\mathrm{\Lambda }}_0}{3}}(1+\frac{c^{\prime }}{\tau })(1+\frac{c^{\prime }}{\tau }+\frac{c^{\prime 2}}{{\tau }^2}+\frac{c^{\prime 3}}{{\tau }^3}+\frac{c^{\prime 4}}{{\tau }^4}+\cdots )=\sqrt{\frac{{\mathrm{\Lambda }}_0}{3}}\frac{1+c^{\prime }/\tau }{1-c^{\prime }/\tau }.\end{array}\end{array}$$ Redefining a new free parameter $c$ $$\begin{array}{}\text{(48)}& c=\frac{\ln \left(2\alpha \sqrt{{\mathrm{\Lambda }}_0/3}|c^{\prime }|\right)}{2\alpha \sqrt{{\mathrm{\Lambda }}_0/3}}\Rightarrow 2\alpha \sqrt{{\mathrm{\Lambda }}_0/3}{c}^{\prime }=\mathrm{sign}(c^{\prime }){\mathrm{e}}^{2\alpha \sqrt{{\mathrm{\Lambda }}_0/3}c},\end{array}$$ we obtain the exact solution for the Hubble parameter: $$H_{\mathrm{cl}}=\sqrt{\frac{{\mathrm{\Lambda }}_0}{3}}\frac{1+\mathrm{sign}(c^{\prime }){\mathrm{e}}^{-2\alpha \sqrt{{\mathrm{\Lambda }}_0/3}(t-c)}}{1-\mathrm{sign}(c^{\prime }){\mathrm{e}}^{-2\alpha \sqrt{{\mathrm{\Lambda }}_0/3}(t-c)}}=\{\begin{array}{ll}\sqrt{\frac{{\mathrm{\Lambda }}_0}{3}}\tanh (\alpha \sqrt{{\mathrm{\Lambda }}_0/3}(t-c)),& \mathrm{sign}(c^{\prime })=-1,\\ \sqrt{\frac{{\mathrm{\Lambda }}_0}{3}}\coth (\alpha \sqrt{{\mathrm{\Lambda }}_0/3}(t-c)),& \mathrm{sign}(c^{\prime })=1.\end{array}$$ Here the parameter $c$ can be either absorbed into the cosmic time or set to $c=0$. However, in the case of the quantum-corrected solution, the free parameter $c$, or more precisely $|c^{\prime }|$, acquires a physical meaning due to quantum corrections through $\mathrm{\Lambda }(t)$. This becomes evident when re-expressing the quantum-corrected solution as $$\begin{array}{}\text{(49)}& H(\tau )=\sqrt{{\mathrm{\Lambda }}_0/3}[1+\frac{2|c^{\prime }|}{\tau }(\frac{{\tau }_c}{\tau }+\mathrm{sign}(c^{\prime }))+\frac{2|c^{\prime }{|}^2}{{\tau }^2}(\frac{{\tau }_c}{\tau }\mathrm{sign}(c^{\prime })+1)+\cdots ],\end{array}$$ where the parameter $|c^{\prime }|$ gives a critical scale determined by $${\tau }_c=\tilde{\mu }{G}_0/4|c^{\prime }|.$$ If $c^{\prime }>0$, which implies $\mathrm{sign}(c^{\prime })=1$, the quantum correction enhances the Hubble parameter eternally. On the other hand, if $c^{\prime }<0$, meaning $\mathrm{sign}(c^{\prime })=-1$, the infintely large late-time expansion rate remains constant at $\sqrt{{\mathrm{\Lambda }}_0/3}$ as the quantum corrections become negligible. However, the value of the constant is still affected by the quantum correction terms at late time. Since the leading-order quantum correction is stronger than the subleading orders, we observe from 49 that late-time cosmic expansion is suppressed due to quantum corrections compared to the very late-time classical behavior when $\tau >{\tau }_c$, and the opposite occurs for $\tau <{\tau }_c$.

In reality, based on Eq. 48 , $|c^{\prime }|$ is found to be very large since the leading-order parameter $c$ is of the order of the present time, making ${\tau }_c$ unreasonably small. As a result, the $\tau >{\tau }_c$ case dominates. While the quantum corrections are small, they produce an effect opposite to what is expected from classical behavior. Specifically, while the present universe is accelerating, the quantum-corrected Hubble parameter appears to be smaller than the classical Hubble parameter. Even in the case of ${\mathrm{\Lambda }}_0=0$, the quantum-corrected Hubble parameter increases, contrary to the classical expectation of a decelerating universe.

Since the Hubble parameter is defined in terms of $\tau$ as $H(\tau )=a^{-1}(da/d\tau )(d\tau /dt)$, the scale factor can be expressed as $$a(\tau )={\tau }^{1/2\alpha }[1-\frac{c^{\prime }}{\alpha \tau }-(\frac{4c^{\prime 2}+\tilde{\mu }{G}_0}{8\alpha }-\frac{c^{\prime 2}}{2{\alpha }^2})\frac{1}{{\tau }^2}+\mathcal{O}\left({\tau }^{-3}\right)].$$ The energy density can be computed straightforwardly from Eq. 13 as $$\rho (\tau )=\frac{{\mathrm{\Lambda }}_0{c}^{\prime }}{2\pi G_0\tau }+\frac{{\mathrm{\Lambda }}_0(4c^{\prime 2}+\tilde{\mu }{G}_0)}{4\pi G_0{\tau }^2}+\frac{3{\mathrm{\Lambda }}_0{c}^{\prime }(4c^{\prime 2}+\tilde{\mu }{G}_0+4\tilde{\omega }{G}_0/3)}{8\pi G_0{\tau }^3}+\mathcal{O}\left({\tau }^{-4}\right).$$ As expected, the entire contribution to the energy density at late times arises from quantum corrections. For $c^{\prime }<0$, the critical scale ${\tau }_c$ emerges from the competition between the first term and the term ${\mathrm{\Lambda }}_0\tilde{\mu }{G}_0/(4\pi G_0{\tau }^2)$ which appears in the parenthesis of the second term.

It is worth noting that the quantum-corrected solution for the Hubble parameter under this cutoff choice addresses the issue of requiring an additional integration constant. Furthermore, this solution reproduces exponential subleading terms in $H(t)$ for late times, in contrast to the power expansion terms obtained with the $k=\xi /t$ choice. Therefore, it can be concluded from the quantum-corrected solution for the Hubble parameter that the late-time behaviour aligns with the classical solution, with minor quantum improvements. This demonstrates the viability of the proposed cutoff choice, unlike the earlier $k=\xi /t$ choice.

5 Summary and Conclusions↩︎

In this work, we have studied a spatially flat FLRW universe in late time, taking quantum gravitational effects into account, within the framework of asymptotically safe gravity. The FRG flow of the effective average action in asymptotically safe gravity has resulted in the running of the gravitational constant and the cosmological constant under the Einstein-Hilbert truncation. We have analyzed the cosmic evolution of the FLRW cosmology at late times by improving the Einstein equations to include the scale dependence of Newton’s constant and the cosmological constant. Specifically, we have considered the conservation of the entire right-hand side of the Einstein equations, which has led to a modified continuity equation. The main focus of this work has been to identify a viable cutoff scale, $k$, that can yield a consistent FLRW cosmology.

In this context, we have examined cosmological solutions for two branches of FRG trajectories. The first branch has involved a power series expansion of the dimensionful cosmological constant, $\mathrm{\Lambda }(t)$, beginning with a $k^4$ term, with ${\mathrm{\Lambda }}_0=0$. This trajectory has approached the Gaussian fixed point as $k\to 0$ and has been referred to in the literature as the “separatrix" [7], [8]. The second branch, which may be realized in nature, starts with a $k^2$ term in the expansion of the cosmological constant and is valid up to a certain non-zero momentum scale, where the $\beta$-function becomes singular as the dimensionless cosmological constant $\lambda (k)$ comes to a”singular line” near $1/2$.

For studying cosmology in these two branches, we have considered the cutoff scale $k$ in terms of either classical or quantum Hubble parameter for the ${\mathrm{\Lambda }}_0=0$ case. We have chosen $k\sim H_{\mathrm{c}\mathrm{l}}(t)$, which measures the curvature of the FLRW spacetime. Before incorporating quantum corrections through the Einstein equations, there was the time translation symmetry, implying that the integration constant in the classical solution does not carry any physical significance. However, in the quantum-corrected solution, a new constant comes into the solution with physical significance through the critical time scale $t_c$ which defines two distinct phases of the quantum-corrected Hubble parameter: the repulsive phase (dS phase) and the attractive phase (AdS phase). In the attractive phase, the leading quantum correction has introduced an attractive effect, causing the expansion rate of the FLRW universe to slow down compared to the classical expansion rate, $H(t)<H_{\mathrm{cl}}(t)=1/(\alpha t)$ for $t<t_c$. In the repulsive phase (dS phase), it has enhanced the expansion rate beyond the classical rate when $t>t_c$.

Considering another possibility that the cutoff scale $k$ may have a simple functional relationship with the quantum Hubble parameter rather than with the classical Hubble parameter, we examined the cutoff choice $k(H)=\zeta G_0^{(\beta -1)/2}{H}^{\beta }$, where $\beta$ is a constant. In earlier work [24], three specific values of $\beta$ namely, $\beta =1/4,3/4$ and 1 were analyzed, and the corresponding entropy generation due to quantum effects was investigated. It was observed that entropy generation diverges at late times for $\beta =1/4$, indicating that this cutoff choice is not suitable for studying cosmology. On the other hand, for $\beta =3/4$ and $\beta =1$, entropy generation converges to a constant and thus these viable cutoff choices are viable. This suggests the existence of a critical value of $\beta$ that determines the appropriateness of a cutoff choice. By studying the asymptotic behaviour of the Hubble parameter at large times, we have demonstrated that the critical value is $\beta =1/2$. This value marks a transition in the scale factor’s behaviour from power-law growth ($\beta \ge 1/2$) to exponential growth ($\beta <1/2$). From the exponential behaviour of the scale factor for $\beta <1/2$, we infer that such cutoff choices are not viable, as the large momentum scale $k$ introduces significant perturbations that entirely disrupt the classical power-law behaviour for the ${\mathrm{\Lambda }}_0=0$ trajectory. On the other hand, for $\beta \ge 1/2$, the behaviour follows a power law similar to the classical case. Here the perturbations remain small enough to preserve this expected behaviour with minor quantum corrections. Therefore, the cutoff identification is valid for $\beta \ge 1/2$ making it suitable for cosmological studies.

For trajectories with ${\mathrm{\Lambda }}_0>0$, the Hubble parameter is not suitable for the identification because it becomes constant quickly. Following the successful case for ${\mathrm{\Lambda }}_0=0$, we tried to use the cutoff scale $k=\xi /t$ in terms of the cosmic time, but found that it does not provide a consistent solution because the quantum corrections to the Hubble parameter introduce power expansion terms that become more significant than the classical subleading exponential terms and cannot reproduce the classical solution in the classical limit. Moreover, unlike the ${\mathrm{\Lambda }}_0=0$ case, this solution with this cutoff lacks an undetermined constant term, despite the fact that the Einstein equation is a first-order differential equation for $H(t)$. All of these suggest that the cutoff identification $k=\xi /t$ may not represent a physically viable choice. In contrast, we can resolve this issue with $k=\xi /\tau$ which indicates that the quantum solution resembles the classical solution, along with quantum improvements in the same form. We expect that the perturbations introduced by quantum corrections should not be too large to significantly alter the form of the solutions at late times. Through this analysis, we have demonstrated a method to evaluate the viability of different cutoff scale choices for two branches of the trajectories ${\mathrm{\Lambda }}_0=0$ and ${\mathrm{\Lambda }}_0>0$.

We would like to thank Akihiro Ishibashi for valuable discussions at the early stage of this work. The work of C.M.C. was supported by the National Science and Technology Council of the R.O.C. (Taiwan) under the grant NSTC 113-2112-M-008-027. The work of R.M. was supported by the National Science and Technology Council of the R.O.C. (Taiwan) under the grant NSTC 113-2811-M-008-046. The work of N.O. was supported in part by the Grant-in-Aid for Scientific Research Fund of the JSPS (C) No. 20K03980.

6 Perturbative Solution of Running Couplings↩︎

The FRG equations for running couplings $G(k)$ and $\mathrm{\Lambda }(k)$ for Einstein-Hilbert truncation are [1] $$\begin{array}{rlr}k{\partial }_{k}G(k)& =& {\eta }_{N}G(k),\\ k{\partial }_k\mathrm{\Lambda }(k)& =& {\eta }_N\mathrm{\Lambda }+\frac{k^{4}G(k)}{2\pi }[10{\mathrm{\Phi }}_2^1(-2\lambda )-8{\mathrm{\Phi }}_2^1(0)-5{\eta }_N{\tilde{\mathrm{\Phi }}}_2^1(-2\lambda )],\end{array}$$ where ${\eta }_N$ is the anomalous dimension of $\sqrt{g}R$: $${\eta }_N=\frac{g{B}_1(\lambda )}{1-g{B}_2(\lambda )},$$ in which the dimensionless running couplings are defined as $g(k)=G(k)k^2,\lambda (k)=\mathrm{\Lambda }(k)/k^2$ and $B_1(\lambda ),B_2(\lambda )$ are abbreviations of $$B_1(\lambda )=\frac{1}{3\pi }[5{\mathrm{\Phi }}_1^1(-2\lambda )-18{\mathrm{\Phi }}_2^2(-2\lambda )-4{\mathrm{\Phi }}_1^1(0)-6{\mathrm{\Phi }}_2^2(0)],B_2(\lambda )=-\frac{1}{6\pi }[5{\tilde{\mathrm{\Phi }}}_1^1(-2\lambda )-18{\tilde{\mathrm{\Phi }}}_2^2(-2\lambda )].$$ The threshold functions that appeared here are defined as $${\mathrm{\Phi }}_n^p(w)=\frac{1}{\mathrm{\Gamma }(n)}{\int }_0^{\mathrm{\infty }}dz{z}^{n-1}\frac{R^{(0)}(z)-z{\partial }_z{R}^{(0)}(z)}{{[z+R^{(0)}(z)+w]}^p},{\tilde{\mathrm{\Phi }}}_n^p(w)=\frac{1}{\mathrm{\Gamma }(n)}{\int }_0^{\mathrm{\infty }}dz{z}^{n-1}\frac{R^{(0)}(z)}{{[z+R^{(0)}(z)+w]}^p},$$ which depend on the choice the cutoff function $R^{(0)}(z)$.

The FRG equations can be further reexpressed as $$\begin{array}{}\text{(50)}& k{\partial }_{k}G(k)& =& \frac{k^2{G}^2(k)B_1(\lambda )}{1-k^{2}G(k)B_2(\lambda )},\text{(51)}& k{\partial }_k\mathrm{\Lambda }(k)& =& -\frac{k^{2}G(k)\{k^{4}G(k)[A_1(\lambda )B_2(\lambda )+A_2(\lambda )B_1(\lambda )]-A_1(\lambda )k^2-B_1(\lambda )\mathrm{\Lambda }(k)\}}{1-k^{2}G(k)B_2(\lambda )},\end{array}$$ with two additional abbreviations $$A_1(\lambda )=\frac{1}{\pi }[5{\mathrm{\Phi }}_2^1(-2\lambda )-4{\mathrm{\Phi }}_2^1(0)],A_2(\lambda )=\frac{5}{2\pi }{\tilde{\mathrm{\Phi }}}_2^1(-2\lambda ).$$

For small value of $k$, appropriate to late time behavior, the running couplings are assumed as $$\begin{array}{}\text{(52)}& G(k)=\sum _{j=0}{G}_j{k}^j,\mathrm{\Lambda }(k)=\sum _{j=0}{\mathrm{\Lambda }}_j{k}^j.\end{array}$$ In what follows, we solve the FRG equations perturbatively.

6.1 Exponential Cutoff↩︎

In [6], the low energy expansions of running couplings were derived with the exponential cutoff $$R^{(0)}(z)=\frac{z}{\exp (z)-1}.$$

In order to solve the FRG equations perturbatively, one should expand the factor $(Z-2\mathrm{\Lambda }(k)/k^2{)}^{-p}$ with $Z=z+R^{(0)}$ in power of $k$. For ${\mathrm{\Lambda }}_0=0$, we write $${(Z-\frac{2\mathrm{\Lambda }(k)}{k^2})}^{-p}=Z^{-p}[1+\frac{2p{\mathrm{\Lambda }}_4}{Z}{k}^2+(\frac{2p{\mathrm{\Lambda }}_6}{Z}+\frac{2p(p+1){\mathrm{\Lambda }}_4^2}{Z^2})k^4+\cdots ],$$ and then the threshold functions ${\mathrm{\Phi }}_n^p(-2\lambda )$ can be expanded as $${\mathrm{\Phi }}_n^p(-\frac{2\mathrm{\Lambda }(k)}{k^2})={\mathrm{\Phi }}_n^p(0)+2p{\mathrm{\Lambda }}_4{\mathrm{\Phi }}_n^{p+1}(0)k^2+[2p{\mathrm{\Lambda }}_6{\mathrm{\Phi }}_n^{p+1}(0)+2p(p+1){\mathrm{\Lambda }}_4^2{\mathrm{\Phi }}_n^{p+2}(0)]k^4+\cdots ,$$ and similarly for ${\tilde{\mathrm{\Phi }}}_n^p(-2\lambda )$ as $${\tilde{\mathrm{\Phi }}}_n^p(-\frac{2\mathrm{\Lambda }(k)}{k^2})={\tilde{\mathrm{\Phi }}}_n^p(0)+2p{\mathrm{\Lambda }}_4{\tilde{\mathrm{\Phi }}}_n^{p+1}(0)k^2+[2p{\mathrm{\Lambda }}_6{\tilde{\mathrm{\Phi }}}_n^{p+1}(0)+2p(p+1){\mathrm{\Lambda }}_4^2{\tilde{\mathrm{\Phi }}}_n^{p+2}(0)]k^4+\cdots .$$ By solving the FRG equation order by order, we determine the leading terms of the solutions 52 as $$\begin{array}{rlr}& & G_2=\frac{{\mathrm{\Phi }}_1^1(0)-24{\mathrm{\Phi }}_2^2(0)}{6\pi }{G}_0^2,\\ & & G_4=\frac{2{[{\mathrm{\Phi }}_1^1(0)-24{\mathrm{\Phi }}_2^2(0)]}^2-[{\mathrm{\Phi }}_1^1(0)-24{\mathrm{\Phi }}_2^2(0)][5{\tilde{\mathrm{\Phi }}}_1^1(0)-18{\tilde{\mathrm{\Phi }}}_2^2(0)]+3{\mathrm{\Phi }}_2^1(0)[5{\mathrm{\Phi }}_1^2(0)-36{\mathrm{\Phi }}_2^3(0)]}{72{\pi }^2}{G}_0^3,\\ & & {\mathrm{\Lambda }}_4=\frac{{\mathrm{\Phi }}_2^1(0)}{4\pi }{G}_0,{\mathrm{\Lambda }}_6=\frac{[{\mathrm{\Phi }}_1^1(0)-24{\mathrm{\Phi }}_2^2(0)][3{\mathrm{\Phi }}_1^1(0)-10{\tilde{\mathrm{\Phi }}}_2^1(0)]+30{\mathrm{\Phi }}_2^1(0){\mathrm{\Phi }}_2^2(0)}{72{\pi }^2}{G}_0^2,\end{array}$$ with the specific values $$\begin{array}{}\text{(53)}& \begin{array}{rlr}& & {\mathrm{\Phi }}_1^0(0)=\frac{{\pi }^2}{3},{\mathrm{\Phi }}_1^1(0)=\frac{{\pi }^2}{6},{\mathrm{\Phi }}_2^1(0)=2\zeta (3),{\mathrm{\Phi }}_1^2(0)=1,{\mathrm{\Phi }}_2^2(0)=1,{\mathrm{\Phi }}_2^3(0)=\frac{1}{2},\\ & & {\tilde{\mathrm{\Phi }}}_1^1(0)=1,{\tilde{\mathrm{\Phi }}}_2^1(0)=1,{\tilde{\mathrm{\Phi }}}_1^2(0)=\ln 2,{\tilde{\mathrm{\Phi }}}_2^2(0)=\frac{1}{2}.\end{array}\end{array}$$ The solutions agree with the results in [6] as, in the notation of 2 : $G_2=-\omega G_0^2,G_4={\omega }_1{G}_0^3,{\mathrm{\Lambda }}_4=\nu G_0,{\mathrm{\Lambda }}_6={\nu }_1{G}_0^2$, $$\begin{array}{}\text{(54)}& \omega =\frac{4}{\pi }(1-\frac{{\pi }^2}{144}),\nu =\frac{\zeta (3)}{2\pi },{\omega }_1={\omega }^2-\frac{\omega }{3\pi }-\frac{13\nu }{6\pi },{\nu }_1=-\omega \nu +\frac{5\omega }{6\pi }+\frac{5\nu }{3\pi }.\end{array}$$

The expansion of the factor $(Z-2\mathrm{\Lambda }(k)/k^2{)}^{-p}$ for ${\mathrm{\Lambda }}_0\ne 0$ is written as $$\begin{array}{rlr}{(Z-\frac{2\mathrm{\Lambda }(k)}{k^2})}^{-p}& =& \frac{(-1{)}^p}{2^p{\mathrm{\Lambda }}_0^p}{k}^{2p}-\frac{(-1{)}^{p}p{\mathrm{\Lambda }}_1}{2^p{\mathrm{\Lambda }}_0^{p+1}}{k}^{2p+1}+\frac{(-1{)}^p}{2^{p+1}}(\frac{p(p+1){\mathrm{\Lambda }}_1^2-2p{\mathrm{\Lambda }}_0{\mathrm{\Lambda }}_2}{{\mathrm{\Lambda }}_0^{p+2}}+\frac{pZ}{{\mathrm{\Lambda }}_0^{p+1}})k^{2p+2}\\ & +& \frac{(-1{)}^p}{2^{p+1}}(\frac{p(p+1)(p+2){\mathrm{\Lambda }}_1^3+6p{\mathrm{\Lambda }}_0{\mathrm{\Lambda }}_3-6p(p+1){\mathrm{\Lambda }}_0{\mathrm{\Lambda }}_1{\mathrm{\Lambda }}_2}{3{\mathrm{\Lambda }}_0^{p+3}}+\frac{p(p+1){\mathrm{\Lambda }}_{1}Z}{{\mathrm{\Lambda }}_0^{p+2}})k^{2p+3}+\cdots ,\end{array}$$ and then it leads to the expansion of ${\mathrm{\Phi }}_n^p(-2\lambda )$ $$\begin{array}{rlr}{\mathrm{\Phi }}_n^p(-\frac{2\mathrm{\Lambda }(k)}{k^2})& =& \frac{(-1{)}^p}{2^p{\mathrm{\Lambda }}_0^p}{\mathrm{\Phi }}_n^0(0)k^{2p}-\frac{(-1{)}^{p}p{\mathrm{\Lambda }}_1}{2^p{\mathrm{\Lambda }}_0^{p+1}}{\mathrm{\Phi }}_n^0(0)k^{2p+1}+\frac{(-1{)}^p}{2^{p+1}}(\frac{p(p+1){\mathrm{\Lambda }}_1^2-2p{\mathrm{\Lambda }}_0{\mathrm{\Lambda }}_2}{{\mathrm{\Lambda }}_0^{p+2}}{\mathrm{\Phi }}_n^0(0)+\frac{p}{{\mathrm{\Lambda }}_0^{p+1}}{\mathrm{\Phi }}_n^{-1}(0))k^{2p+2}\\ & +& \frac{(-1{)}^p}{2^{p+1}}(\frac{p(p+1)(p+2){\mathrm{\Lambda }}_1^3+6p{\mathrm{\Lambda }}_0{\mathrm{\Lambda }}_3-6p(p+1){\mathrm{\Lambda }}_0{\mathrm{\Lambda }}_1{\mathrm{\Lambda }}_2}{3{\mathrm{\Lambda }}_0^{p+3}}{\mathrm{\Phi }}_n^0(0)+\frac{p(p+1){\mathrm{\Lambda }}_1}{{\mathrm{\Lambda }}_0^{p+2}}{\mathrm{\Phi }}_n^{-1}(0))k^{2p+3}+\cdots ,\end{array}$$ and similarly for ${\tilde{\mathrm{\Phi }}}_n^p(-2\lambda )$. The FRG equations lead to $0=G_1=G_3=\cdots ,0={\mathrm{\Lambda }}_1={\mathrm{\Lambda }}_3=\cdots$ and $$\begin{array}{rlr}& & G_2=-\frac{2{\mathrm{\Phi }}_1^1(0)+3{\mathrm{\Phi }}_2^2(0)}{3\pi }{G}_0^2,G_4=\frac{{[2{\mathrm{\Phi }}_1^1(0)+3{\mathrm{\Phi }}_2^2(0)]}^2}{9{\pi }^2}{G}_0^3-\frac{5{\mathrm{\Phi }}_1^0(0)}{24\pi }\frac{G_0^2}{{\mathrm{\Lambda }}_0},\\ & & {\mathrm{\Lambda }}_2=-\frac{2{\mathrm{\Phi }}_1^1(0)+3{\mathrm{\Phi }}_2^2(0)}{3\pi }{G}_0{\mathrm{\Lambda }}_0,{\mathrm{\Lambda }}_4=\frac{{[2{\mathrm{\Phi }}_1^1(0)+3{\mathrm{\Phi }}_2^2(0)]}^2}{9{\pi }^2}{G}_0^2{\mathrm{\Lambda }}_0-\frac{5{\mathrm{\Phi }}_1^0(0)+24{\mathrm{\Phi }}_2^1(0)}{24\pi }{G}_0.\end{array}$$ With the specific values 53 we have $$\begin{array}{rlr}G(k)& =& G_0[1-\frac{9+{\pi }^2}{9\pi }{G}_0{k}^2+[{\left(\frac{9+{\pi }^2}{9\pi }\right)}^2-\frac{5\pi }{72}\frac{1}{G_0{\mathrm{\Lambda }}_0}]G_0^2{k}^4+\mathcal{O}(k^6)],\\ \mathrm{\Lambda }(k)& =& {\mathrm{\Lambda }}_0[1-\frac{9+{\pi }^2}{9\pi }{G}_0{k}^2+{\left(\frac{9+{\pi }^2}{9\pi }\right)}^2{G}_0^2{k}^4+\mathcal{O}(k^6)]+G_0{k}^4[-\frac{5{\pi }^2+144\zeta (3)}{72\pi }+\mathcal{O}(k^2)].\end{array}$$ This result gives the coefficients in Eq. 2 for the exponential cutoff.

6.2 Optimized Cutoff↩︎

For the optimized cutoff [7], the cutoff function is chosen as $$R^{(0)}(z)=(1-z)\mathrm{\Theta }(1-z),$$ then the integration of threshold functions can be carried out $${\mathrm{\Phi }}_n^p(w)=\frac{(1+w{)}^{-p}}{\mathrm{\Gamma }(n+1)},{\tilde{\mathrm{\Phi }}}_n^p(w)=\frac{(1+w{)}^{-p}}{\mathrm{\Gamma }(n+2)}.$$ The four abbreviations in FRG equations 50 and 51 can be simplified $$A_1=\frac{k^2+8\mathrm{\Lambda }}{2\pi (k^2-2\mathrm{\Lambda })},A_2=\frac{5k^2}{12\pi (k^2-2\mathrm{\Lambda })},B_1=-\frac{11k^4-18k^2\mathrm{\Lambda }+28{\mathrm{\Lambda }}^2}{3\pi (k^2-2\mathrm{\Lambda }{)}^2},B_2=\frac{k^4+10k^2\mathrm{\Lambda }}{12\pi (k^2-2\mathrm{\Lambda }{)}^2}.$$

For the case ${\mathrm{\Lambda }}_0=0$, the leading orders of equations lead to ${\mathrm{\Lambda }}_1={\mathrm{\Lambda }}_2={\mathrm{\Lambda }}_3=0$ and the solutions are $$\begin{array}{}\text{(55)}& G(k)=G_0[1-\frac{11}{6\pi }{G}_0{k}^2+\frac{217}{72{\pi }^2}{G}_0^2{k}^4+\mathcal{O}\left(G_0^3{k}^6\right)],\mathrm{\Lambda }(k)=G_0{k}^4[\frac{1}{8\pi }+\frac{7}{54{\pi }^2}{G}_0{k}^2+\mathcal{O}\left(G_0^2{k}^4\right)].\end{array}$$

The solutions for the case ${\mathrm{\Lambda }}_0\ne 0$ are $$\begin{array}{rlr}G(k)& =& G_0[1-\frac{7}{6\pi }{G}_0{k}^2+(\frac{49}{36{\pi }^2}-\frac{5}{24\pi G_0{\mathrm{\Lambda }}_0})G_0^2{k}^4+\mathcal{O}\left(G_0^3{k}^6\right)],\\ \mathrm{\Lambda }(k)& =& {\mathrm{\Lambda }}_0[1-\frac{7}{6\pi }{G}_0{k}^2+\frac{49}{36{\pi }^2}{G}_0^2{k}^4-(\frac{29}{72\pi G_0^2{\mathrm{\Lambda }}_0^2}+\frac{343}{216{\pi }^3})G_0^3{k}^6+\mathcal{O}\left(G_0^4{k}^8\right)]\\ & & +G_0{k}^4[-\frac{17}{24\pi }+\frac{119}{144{\pi }^2}{G}_0{k}^2+\mathcal{O}\left(G_0^2{k}^4\right)].\end{array}$$

Note that these two cases are “disconnected”, i.e. the coefficients in the case ${\mathrm{\Lambda }}_0=0$ are not the ${\mathrm{\Lambda }}_0\to 0$ limit of the related coefficients in the case ${\mathrm{\Lambda }}_0\ne 0$.

References↩︎