Revisiting the Origin of the Universe and the Arrow of Time

Abstract

In this paper, we reconsider two long-standing yet unsolved problems: the origin of the universe and the arrow of time. We show that only a closed universe is free from singularities and that the arrow of time is symmetric with respect to the universe’s maximum scale. The Wheeler–DeWitt equation is explicitly solved to obtain local dynamical times. It is shown that the thermodynamic arrow of time coincides with both the dynamical time and the cosmological (expanding universe) time. This correspondence is explicitly demonstrated in two-dimensional spacetime.

1. Introduction

What is the origin of the universe? Can it be free from a singularity [1,2,3]? What is time? These are long-standing but still unsolved problems. Vilenkin proposed a model in which the universe emerges from “nothing” [4,5,6,7], particularly in the context of a closed universe. This idea appears to align well with the no-boundary proposal [8] and may avoid the initial singularity. Following the birth of the universe, inflation naturally occurs, potentially addressing the question of why the early universe was in a low-entropy state [9]. These concepts are supported by the Wheeler–DeWitt equation [10,11,12,13], which implies that the universe does not include an extrinsic time.

In a closed universe, this gives rise to the problems of how to derive intrinsic time and how to determine the arrow of time during both the expanding and contracting phases. The early universe can only be properly interpreted once these problems are addressed comprehensively and consistently. In our previous works, we discussed the dynamical arrow of time [14] and the thermodynamical arrow of time [15] separately. However, their mutual relationship was not fully clarified, which is found to be crucial for solving the aforementioned problems in the present paper. We also provide comments on alternative views [16].

The outline of this paper is as follows. Section 2 provides a brief review of the birth of the universe in Vilenkin’s scenario. In Section 3, we derive dynamical time from the Wheeler–DeWitt equation. The thermodynamic arrow of time is discussed in Section 4. The topics of Section 3 and Section 4 were treated separately in [14,15], respectively. These different concepts of time are unified in the second half of Section 4, which enables us to present a consistent view of the arrow of time. Section 5 is devoted to discussions.

2. Creation of the Universe from Nothing

This section is a review of the creation of the universe from nothing, proposed by Vilenkin [4,5], which is necessary to understand the motivation of this paper and a key ingredient of it. Let us start with the following system of gravity plus scalar matter:

$$S=\int \sqrt{-g}{d}^{4}x\left[-{\displaystyle \frac{1}{16\pi G}}R-{\displaystyle \frac{1}{2}}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -V\left(\varphi \right)\right].$$

(1)

If we consider the closed Friedman–Robertson–Walker (FRW) universe, the invariant line element is

$$d{s}^2=-d{t}^2+a^2\left(t\right)\left(d{\chi }^2+{sin}^2\chi (d{\theta }^2+{sin}^2\theta d{\phi }^2)\right),$$

(2)

where physical distance $r=a sin\chi$. In this paper we adopt $\hslash =c=1$ units. The Lagrangian of (1) in the above metric takes the form of

$$\mathcal{L}={\displaystyle \frac{3\pi }{4G}}(1-{\dot{a}}^2)a+{\pi }^2{a}^3{\dot{\varphi }}^2-2{\pi }^2{a}^{3}V\left(\varphi \right).$$

(3)

Here, and hereafter, the dot indicates the derivative with respect to t. The Wheeler–DeWitt equation of this system is

$$\mathcal{H}\mathsf{\Psi }=0,$$

(4)

where

$$\mathcal{H}=-{\displaystyle \frac{G}{3\pi a}}{P}_a^2+{\displaystyle \frac{1}{4{\pi }^2{a}^3}}{P}_{\varphi }^2-{\displaystyle \frac{3\pi }{4G}}a\left(1-{\displaystyle \frac{8\pi }{3}}G{a}^{2}V\left(\varphi \right)\right).$$

(5)

$P_a$ and $P_{\varphi }$ are the canonical conjugates of a and $\varphi$, respectively. Then, (4) indicates

$$\left({\displaystyle \frac{{\partial }^2}{\partial a^2}}+{\displaystyle \frac{p}{a}}{\displaystyle \frac{\partial }{\partial a}}-{\displaystyle \frac{1}{a^2}}{\displaystyle \frac{{\partial }^2}{\partial {\tilde{\varphi }}^2}}-U(a,\varphi )\right)\mathsf{\Psi }=0.$$

(6)

Here, p is an order one parameter relevant to the order of a and $P_a$, ${\tilde{\varphi }}^2\equiv 4\pi G{\varphi }^2/3$, and

$$\begin{array}{ccc}\hfill U(a,\varphi )& =& {\left({\displaystyle \frac{3\pi }{2G}}\right)}^2{a}^2\left(1-{\displaystyle \frac{8\pi }{3}}G{a}^{2}V\left({\varphi }_0\right)\right)\hfill \\ & =& {\left({\displaystyle \frac{3\pi }{2G}}\right)}^2{a}^2(1-H^2{a}^2)\hfill \end{array}$$

(7)

with $H^2\equiv {\displaystyle \frac{8\pi G}{3}}\left|V\left({\varphi }_0\right)\right|$. $V\left(\varphi \right)$ is

$$V\left(\varphi \right)={\rho }_v-{\displaystyle \frac{1}{2}}{m}^2{\varphi }^2$$

(8)

with $V^{\prime }\left({\varphi }_0\right)=0$, having a spontaneous symmetry breaking with a wrong mass sign. The scale factor satisfies the Friedman equation:

$${\dot{a}}^2-H^2{a}^2=-k,(k=1,0,-1\mathrm{for} \mathrm{closed}, \mathrm{flat}, \mathrm{and} \mathrm{open} \mathrm{universes}, \mathrm{respectively})$$

(9)

with k = 1 in this case.

This potential is depicted in Figure 1. It is very important that, in the region $0<a<1/H$ (region I), ${\dot{a}}^2<0$ and the metric is Euclidean (time parameter t is imaginary) and that, in the region $a>1/H$ (region II), the metric is Lorenzian.

Before further discussion on this potential energy, let us make a comment on the implication of the closed universe. We can consider the case of an open FRW universe simply by changing $a\to ia$ in Equation (7) or $k=-1$ in Equation (9). Its invariant length is

$$d{s}^2=-d{t}^2+a^2\left(t\right)\left(d{\chi }^2+{sinh}^2\chi (d{\theta }^2+{sin}^2\theta d{\phi }^2)\right).$$

(10)

In this case, the physical distance $r=a sinh\chi$ increases without limit. As is easily seen from Equation (7), the whole region is Lorenzian and the universe shrinks into singularity in the early stage [1,2]. We now proceed to discuss the closed FRW universe in more detail. From (9), the Lorenzian region is restricted in the $a>1/H$ region. In region I, the time is imaginary ($t=i\tau$), which means that the $a=1/H$ region is connected with $a=0$ point by tunneling [4]. $a=0$ has no spacetime and no energy, called “nothing”. From (9),

$$a=H^{-1}cos\left(H\tau \right).$$

(11)

After tunneling to the real time world, the scale factor a behaves as

$$a=H^{-1}cosh\left(Ht\right),$$

(12)

which implies the inflation. Thus, in this scenario, the universe is free from singularities and has no boundary. The initial singularity is avoided via quantum tunneling from “nothing.” On the other hand, the Hartle–Hawking proposal [8] avoids the singularity through a smooth Euclidean geometry. Although these two approaches are different, it is noteworthy that both employ an imaginary time. Due to the potential of (8), this inflation phase ends when $\varphi$ reaches ${\displaystyle \frac{m}{H}}\varphi \approx 1$ and the Big Bang begins. This also explains why the early universe is in a low-entropy state [9]. However, a serious problem remains unresolved. We have assumed the direction of time to correspond to increasing a or increasing t, but this choice is ad hoc, as there is no extrinsic notion of time.

3. The Dynamical Arrow of Time

We have reviewed the notion that a closed universe can be free from a singularity. However, this raises another problem: whether $t=0$ truly represents the beginning of the universe throughout its entire history. This leads to the question of which direction the arrow of time points in the contracting phase. If the arrow of time follows the direction in which the coordinate time t increases, then the universe would once again approach the point $1/H$ during contraction. At that stage, it might undergo a bounce or, more rarely, tunnel back to $a=0$. In such a case, it becomes unclear whether our present universe represents the genuine first stage of expansion or merely one cycle in a series of repeated universes. In the latter case, we cannot claim that the “early” universe is in a low-entropy state. So far, t is really a parameter. Equation (4) indicates that there is no time variable outside of the dynamical system, the universe. We must introduce a “time” variable intrinsic to the dynamical system. In this and next sections, we consider two different time arrows: the dynamical time arrow and thermodynamic one. These two are mutually related as we will show.

In these two sections we consider the two-dimensional gravity, called Jackiw–Teitelboim (JT) gravity [17,18] which is amenable but reserves the universality of the FRW metric in four dimensions

$$S_{grav}=\int \sqrt{-g}(R-2\mathrm{\Lambda })N{d}^{2}x.$$

(13)

Here, R and $2\mathrm{\Lambda }$ are the scalar curvature and the cosmological constant, respectively. It should be remarked that R is total derivative in two dimensions. This action was formulated as $O(2, 1)$ gauge theory by Fukuyama–Kamimura [19]. The usual four-dimensional gravity is formulated as $O(4, 1)$ or $O(3, 2)$ gravity, which will be argued in the Section 5. In general, $g_{\mu \nu }$ is expressed as

$$g_{\mu \nu }=e^{2\chi }\left(\begin{array}{cc}{\eta }_1^2-{\eta }_{\perp }^2& {\eta }_1\\ {\eta }_1& 1\end{array}\right).$$

(14)

Then, $S_{grav}$ reads

$$S_{grav}=\int (P_{\chi }\dot{\chi }+P_N\dot{N}-{\eta }_{\perp }{\mathcal{H}}_{\perp }-{\eta }_1{\mathcal{H}}_1)d^{2}x$$

(15)

with

$${\mathcal{H}}_{\perp }=P_N{P}_{\chi }+N^{\prime }{\chi }^{\prime }-N^{\prime \prime }-\mathrm{\Lambda }N{e}^{2\chi },$$

(16)

and a diffeomorphism constraint,

$${\mathcal{H}}_1=P_{\chi }{\chi }^{\prime }+P_N{N}^{\prime }-P_{\chi }^{\prime }.$$

(17)

Thus, ${\eta }_{\perp }$ and ${\eta }_1$ act as Lagrange multipliers and we have the Hamiltonian constraint (the Wheeler–DeWitt equation) ${\mathcal{H}}_{\perp }\approx 0$ and the diffeomorphism constraint ${\mathcal{H}}_1\approx 0$.

We consider two-dimensional spacetime since it makes possible to solve the Wheeler–DeWitt equation explicitly and to induce the “time” variable, as will be shown [14]. We diagonalize ${\mathcal{H}}_{\perp }$ by using

$${\sigma }_{\pm }\equiv {\displaystyle \frac{\chi \pm N}{\sqrt{2}}},$$

(18)

and their canonical conjugates

$${\pi }_{\pm }={\displaystyle \frac{P_{\chi }\pm P_N}{\sqrt{2}}}$$

(19)

as

$${\mathcal{H}}_{\perp }={\displaystyle \frac{1}{2}}({\pi }_{+}^2-{\pi }_{-}^2)+{\displaystyle \frac{1}{2}}({\sigma }_{+}^{\prime 2}-\sqrt{2}{\sigma }_{+}^{\prime \prime })-{\displaystyle \frac{1}{2}}({\sigma }_{-}^{\prime 2}-\sqrt{2}{\sigma }_{-}^{\prime \prime })-{\displaystyle \frac{\mathrm{\Lambda }}{\sqrt{2}}}({\sigma }_{+}-{\sigma }_{-})e^{\sqrt{2}({\sigma }_{+}+{\sigma }_{-})}.$$

(20)

Here, $P_{\chi }$ and $P_N$ are variables canonically conjugate to $\chi$ and N, respectively. Dot (dash) implies the derivative with respect to $t\left(x\right)$. However, if we use the FWR metric

$$d{s}^2=d{t}^2-a^2\left(t\right)d{x}^2,$$

(21)

then the metrics are reduced to

$${\eta }_{\perp }=e^{-\chi }\equiv {\displaystyle \frac{1}{a}},{\eta }_1=0.$$

(22)

Here,

$$\chi =\chi \left(t\right),N=N\left(t\right),$$

(23)

being independent of x, and (17) $\approx 0$ is automatically satisfied. Equations of motion are

$$\begin{array}{ccc}\hfill & & \ddot{\chi }+\dot{\chi }=\mathrm{\Lambda },\hfill \end{array}$$

(24)

$$\begin{array}{ccc}& & \ddot{N}+\dot{\chi }\dot{N}-2\mathrm{\Lambda }N=0.\hfill \end{array}$$

(25)

When

$$\mathrm{\Lambda }=-{\alpha }^2<0,$$

(26)

with the initial condition,

$$a(t=0)=0,$$

(27)

Equation (24) gives the closed universe solution

$$a=a_{max}sin\left(\alpha t\right),$$

(28)

and (25) produces

$$N=Acos\left(\alpha t\right).$$

(29)

Here, A is an integral constant which is assumed to be positive. It should be remarked that N, the overall factor of the Lagrangian, changes its signature at the maximum radius ($a_{max}$) and that a variable with a negative (positive) metric is transformed to that with a positive (negative) metric. Corresponding to it, the dynamical time variable is induced locally in phases and not globally. An amenable point of two-dimensional gravity is that the Wheeler–DeWitt equation in the FRW metric,

$${\displaystyle \frac{{\partial }^2\psi }{\partial N\partial \chi }}+\mathrm{\Lambda }N{e}^{2\chi }\psi =0,$$

(30)

can be solved explicitly as

$$\psi =c_{0}exp\left({\displaystyle \frac{c_1}{2}}{N}^2+{\displaystyle \frac{{\alpha }^2}{2c_1}}{e}^{2\chi }\right),$$

(31)

where $\mathrm{\Lambda }=-{\alpha }^2$. $c_0$ and $c_i$ are complex numbers.

Then, we have the current conservation:

$${\partial }_{-}{j}^{-}+{\partial }_{+}{j}^{+}=0.$$

(32)

Here,

$$j^{-}\equiv i({\psi }^{\ast }{\partial }_{-}\psi -\psi {\partial }_{-}{\psi }^{\ast }),j^{+}\equiv -i({\psi }^{\ast }{\partial }_{+}-\psi {\partial }_{+}{\psi }^{\ast }),$$

(33)

and

$${\partial }_{-}\equiv {\displaystyle \frac{\partial }{\partial {\sigma }_{-}}}\mathrm{and}{\partial }_{+}\equiv {\displaystyle \frac{\partial }{\partial {\sigma }_{+}}}.$$

(34)

Writing $c_1\equiv -\beta +i\gamma$, (31) becomes

$$\psi =c_0\exp \left[{\displaystyle \frac{i\gamma }{2}}\left(N^2-{\displaystyle \frac{{\alpha }^2}{|c_1{|}^2}}{e}^{2\chi }\right)-{\displaystyle \frac{\beta }{2}}\left(N^2+{\displaystyle \frac{{\alpha }^2}{|c_1{|}^2}}{e}^{2\chi }\right)\right].$$

(35)

This equation implies that the width of the wave packet is

$$\Delta N\approx {\displaystyle \frac{1}{\sqrt{\beta }}}$$

(36)

in the early universe and

$$\Delta e^{\chi }\approx {\displaystyle \frac{|c_1|}{\alpha \sqrt{\beta }}}$$

(37)

around $a_{max}$. Equations (36) and (37) indicate that the spacetime relation is

$${\sigma }_{+}-{\sigma }_{-}\approx {\displaystyle \frac{1}{\sqrt{\beta }}}$$

(38)

and

$${\sigma }_{+}+{\sigma }_{-}\approx ln\left({\displaystyle \frac{|c_1|}{\alpha \sqrt{\beta }}}\right),$$

(39)

respectively. These asymptotic behaviours are depicted in Figure 2.

In order to see the physical implications of “time” variable, we study the semiclassical limit:

$$\psi =c{e}^{iS},$$

(40)

where S is the classical action (13) in the FRW metric. From Equation (32) and Figure 2, we can take ${\sigma }_{-}$ as the dynamical time at the early stage of the universe and $j^{-}$ takes the form

$$j^{-}=-{2\left|c\right|}^2{\partial }_{-}S=2{\left|c\right|}^2{e}^{\chi }{\dot{\sigma }}_{-}.$$

(41)

Positive definiteness of the probability density is thus equivalent to that of ${\dot{\sigma }}_{-}(=-{\partial }_{-}S)$. In the contracting phase (from A to the point $t=\pi$), we should replace ${\partial }_{-}$ by ${\partial }_{+}$ in $j^{-}$,

$$j^{(-)}\equiv j^{-}({\partial }_{-}\to {\partial }_{+})=-{2\left|c\right|}^2{\partial }_{+}S=-2{\left|c\right|}^2{e}^{\chi }{\dot{\sigma }}_{+}.$$

(42)

Considering the behaviour observed in Figure 2, the positive definiteness of $j^{(-)}$ is satisfied in the region from $t=0$ point to the point A. Over the point A, ${\dot{\sigma }}_{-}<0$; then, “time” is replaced by ${\sigma }_{+}$ there. Thus, the dynamical time is defined locally. In the next section, it is shown that this local time concept leads to the law of increasing entropy.

4. Thermodynamic Arrow of Time

We have examined the thermodynamic arrow of time [15] within the same framework as in (13). We assume that the universe was governed by quantum theory near its birth, whereas at present, there appears to be no observable quantum correlations that would, if present, significantly undermine the deterministic interpretation of cosmological observations. There must have been a transition, during the course of cosmological evolution, from an era that was fully quantum mechanical to one in which quantum coherence is no longer present. We refer to this direction from the quantum to the classical regime as the thermodynamic arrow of time. More concretely, this direction corresponds to the decrease in the dispersion (i.e., the width of quantum coherence), denoted by ${\sigma }^2$ of the density matrix.

$$\overline{\rho }[a_{+},{\eta }_{\perp +},{\varphi }_{+};a_{-},{\eta }_{\perp -},{\varphi }_{-}]\equiv \psi (a_{+},{\eta }_{{\perp }_{+}},{\varphi }_{+}){\psi }^{\ast }(a_{-},{\eta }_{\perp -},{\varphi }_{-}).$$

(43)

Here, $g_{\mu \nu }$ of (14) is parameterized as

$$g_{\mu \nu }=a^{2}diag(-{\eta }_{\perp }^2,1)$$

(44)

and $\varphi$ is a scalar matter. Based on the influence functional method [20,21,22], the reduced density matrix is given by

$$\rho [a_{+},{\eta }_{\perp +};a_{-},{\eta }_{\perp -}]\equiv \int d{\varphi }^{+}\int d{\varphi }^{-}\overline{\rho }[a_{+},{\eta }_{\perp +},{\varphi }_{+};a_{-},{\eta }_{\perp -},{\varphi }_{-}]\delta ({\varphi }_{+}-{\varphi }_{-}).$$

(45)

Then, the dispersion $\sigma$ is given by

$$\rho [a_{+};a_{-}]\propto exp[-{\displaystyle \frac{{(a_{+}-a_{-})}^2}{2{\sigma }^2}}].$$

(46)

The detailed calculations are given in [15]. The dispersion as a function of the scale factor a is shown in Figure 3. We observe that the dispersion decreases during the expanding phase but begins to increase again during the contracting phase. Since the thermodynamic arrow of time is defined by the direction of decreasing dispersion, this implies that the arrow of time reverses direction at the point of maximum expansion $a_{max}$. In other words, the arrow of time is symmetric with respect to $a_{max}$. Then it appears to decrease again beyond the point A as the parameter t increases. However, it is fictitious since in this region, ${\dot{\sigma }}_{-}<0$, and the dynamical time should be replaced by ${\sigma }_{+}$ instead of ${\sigma }_{-}$, as we explained at the end of the previous section. It should be remarked that in the contracting phase, (${\displaystyle \frac{\pi }{2}}<t<\pi$).

$${\sigma }_{+}\left(t\right)={\sigma }_{-}(\pi -t),{\sigma }_{-}\left(t\right)={\sigma }_{+}(\pi -t),$$

(47)

which implies that the direction of the dynamical time changes the signature. Then, the gradient of the dispersion changes its sign correspondingly and the dispersion increases in the region $AO$ as the time ${\sigma }^{+}$ increases.

On the contrary, Hawking et al. asserted that the universe does not change its arrow of thermodynamic time in the contracting phase, asymmetric with respect to the maximal radius, $a_{max}$ [16]. In their work, they explicitly showed that one of the dynamical freedoms, massive scalar amplitude $a_n$, continues to increases over the $a_{max}$. They also showed that the scalar mode $f_n$ is asymmetric about the time of maximum expansion. However, it is not clear whether it necessarily leads to the one-way direction of the time arrow. Also the asymmetry comes from the initial condition of the regularity of the scalar perturbation, which explicitly breaks time-reversal symmetry [23]. On the other hand, we can solve the Wheeler–DeWitt equation explicitly and can define the intrinsic time variable in Section 3, and the obtained thermodynamic arrow of time is changed in terms of this intrinsic time variable in this section. It should be noted in our case, the dispersion ${\sigma }^2$ is decreasing again in the $AO$ region in Figure 3, seemingly leading to the same result as that of [16], except for the region $BA$. However, as we have mentioned above, it is fictitious. If we correctly replace the time variable from ${\sigma }_{-}$ to ${\sigma }_{+}$ in the region $AO$, the dispersion ${\sigma }^2$ continues to increase as in the region of $BA$. The global use of the “time”, ${\sigma }_{-}$, in the whole region leads to the negative probability in the region $AO$, as was discussed in (41) and (42). The decoherence is asymmetric about the time of maximum expansion (Figure 3) but the time arrow is changed. Thus, three different arrows of times—dynamical, thermo-dynamical, and cosmological ones—coincide with each other.

5. Discussions

In Section 2, matter field was essential for tunneling. The intrinsic (dynamical) time in Section 3, however, was determined independently of matter. Nevertheless, matter was essential for the arrow of thermodynamical time, as it is determined by the decoherence of matter presented in Section 4. It is remarkable that even in the case of pure gravity, a scalar field appears implicitly. Specifically, we have the condition

$$Z_A^2=l^2.$$

(48)

where $A=1,2,3$ in two dimensions and l has a length scale. The spin connection and tetrad are defined via

$$D_{\mu }{Z}^A={\partial }_{\mu }{Z}^A-{{{\omega }_{\mu }}^A}_B{Z}^B.$$

(49)

$Z_A$ is, in general, an $n+1$-plet scalar in an n-dimensional spacetime and resembles a Higgs-like field. However, unlike the Higgs field, it leaves no imprint on matter fields. For further details, refer to [14,24]. Thus, we have shown in Section 3 that the dynamical time variables are defined locally. In Section 4, we derived the thermodynamical arrow of time as emerging from the direction of decoherence of the quantum state. This thermodynamic arrow of time was shown to be symmetric around $a_{max}$, based on the arguments presented in Section 3. Thus far, these discussions have been conducted within a two-dimensional spacetime framework, which permits explicit solutions to the Wheeler–DeWitt equation without resorting to approximations. The arguments in Section 2, based on the minisuperspace model, are likewise effectively two-dimensional. However, to fully understand the cosmological arrow of time, we ultimately need to extend these arguments to four-dimensional spacetime. JT gravity is not so peculiar in two dimensions. In four dimensions, the JT gravity of (13) is generalised to [24,25]:

$${\mathcal{L}}_{gravity}=\pm {ϵ}^{abcd}{ϵ}^{\mu \nu \rho \sigma }{\widehat{R}}_{\mu \nu ab}{\widehat{R}}_{\rho \sigma cd}/\left(16g^2\right).$$

(50)

Here, Latin and Greek letters indicate world and local Lorentz coordinates, respectively. ${ϵ}^{abcd}$ is a fully anti-symmetric tensor, and

$${\widehat{R}}_{\mu \nu ab}=R_{\mu \nu ab}+e_{[\mu a}{e}_{\nu ]b}/l^2,$$

(51)

where

$$R_{\mu \nu ab}={\partial }_{[\mu }{\omega }_{\nu ]ab}-{\omega }_{[\mu a^c}{\omega }_{\nu ]cb}.$$

(52)

Here ${\omega }_{\mu ab}$ and $e_{\mu a}$ are the spin connection and tetrad, respectively, and $e_{[\mu a}{e}_{\nu ]b}\equiv e_{\mu a}{e}_{\nu b}-e_{\nu a}{e}_{\mu b}$. g and l are the gauge coupling constant and the length scale characterizing (anti-) de-Sitter gauge group, respectively. The details are shown in [24]. Substituting (51) and (52) into (50), we obtain

$${\mathcal{L}}_{gravity}={\partial }_{\mu }{K}^{\mu }-{\displaystyle \frac{1}{16\pi G}}e\left(R+{\displaystyle \frac{6}{l^2}}\right),$$

(53)

where

$$e=det{e}_{\mu a}=\sqrt{-g},R=e^{\mu a}{e}^{\nu b}{R}_{\mu \nu ab},16\pi G=g^2{l}^2.$$

(54)

The first quadratic term of $R_{\mu \nu ab}$ in (50) is the topological invariant and a total derivative. Thus, gravity is formulated as the quadratic field-strength, like the other conformal gauge theories, but dynamically survives the linear Einstein’s action and cosmological constant due to the topological invariant. The scale parameter l or, equivalently, the cosmological constant $l^2/6$, comes from the breaking of the conformal gauge symmetry $O(4, 2)$ [24]. Thus, this Lagrangian leads us to the model of Section 2 in the FRW metric (3) with the effective cosmological constant due to the homogeneous Higgs potential and a strict constant $\mathrm{\Lambda }$. This cosmological constant makes the universe inflate and becomes the reason why the universe was born in the low-entropy state [9]. Here, the cosmological constant is exactly “constant”, as the breaking scale of the $O(4, 2)$ symmetry. Anyhow, we have not shown explicitly in this paper that the arguments of Section 3 and Section 4 are valid in four-dimensional spacetime. However, it is suggestive that the spectral dimension of the quantum universe as function of the diffusion time moves from two to four continuously under the definite arrow of time [26].

Secondly, let us consider the homogeneous and isotropic FRW metric in three spatial dimensions:

$$d{s}^2=d{t}^2-a^2\left(t\right){\gamma }_{ij}d{x}^{i}d{x}^j.$$

(55)

In this case, the Hubble constant behaves as

$$\dot{H}=-4\pi G(\rho +p)+{\displaystyle \frac{\kappa }{a^2}}.$$

(56)

Here, $\kappa \le 0(=1)$ is an open (closed) universe. Here, the null energy condition [27]

$$T_{\mu \nu }{n}^{\mu }{n}^{\nu }>0$$

(57)

for the null vector, $g_{\mu \nu }{n}^{\mu }{n}^{\nu }=0$, which indicates

$$\rho +p\ge 0.$$

(58)

Thus, if the time arrow is one-way, the bouncing universe is possible if the energy density and pressure grow slower than $a^{-2}$ as the universe shrinks [28]. This may appear improbable. However, it does not imply that the universe is asymmetric with respect to $a_{max}$, as this conclusion is based on the assumption that time t flows unidirectionally—a premise that must itself be examined within the framework of quantum cosmology and is, in fact, a central theme of this work.

Finally, we give two additional comments. The first comment is on the so-called Aharanov–Bohm time variable [29],

$$T={\displaystyle \frac{1}{2}}M\left(a{\displaystyle \frac{1}{P_a}}+{\displaystyle \frac{1}{P_a}}a\right),$$

(59)

which is the canonically conjugate to the Hamiltonian,

$$\mathcal{H}={\displaystyle \frac{P_a^2}{2M}}$$

(60)

with the total mass of the universe, $M={\displaystyle \frac{3\pi a_{max}}{2G}}$. This “time” variable has been discussed from a motivation different from that of this paper and leading to the weak measurement [30], which contradicts the usual Copenhagen interpretation of quantum mechanics [31]. Despite this, it is suggestive that this “time” is not smoothly behaved at $P_a\approx 0$ at $a_{max}$, unlike the asymmetric time with respect to $a_{max}$.

The second comment is on Conformal Cyclic Cosmology [32]. It was claimed that the ${\mathcal{I}}^{-}$ of the next cyclic universe can be immersed in the past ${\mathcal{I}}^{+}$ by making a conformal, rescaling the latter ${\mathcal{I}}^{+}$. However, such conformal transformation is not the coordinate transformation and conformally transformed spacetime is different from the original one. The problem of apparently too early formation of the primordial supermassive black holes at high z, discovered by the James Webb Space Telescope [33], is solved by the self-attraction of ultralight axions [34] without damaging the hot Big Bang model. It is very interesting that such an ultralight axion explains the origin of the stochastic nano-Hz gravitational wave [35,36,37,38].