Dynamical Compactification with Matter

We start with the following action:

where $G_{(8)}$ and $g_{\text{YM}}$ are the 8–dimensional Newton constant and the YM coupling, respectively. $\mathcal{L}_{m}$ denotes a matter action that does not couple to the YM field directly. In this work, we will consider $SU(2)$ as the gauge group of the YM fields. Then, the equations of motion are as follows:

where $T^{\text{YM}}_{MN}$ and $T^{m}_{MN}$ denote the energy-momentum tensors from the YM field and the matter described by $\mathcal{L}_{m}$. The explicit form of $T^{\text{YM}}_{MN}$ is given by

Also, $D_{M}$ represents the gauge covariant derivative defined by

where $\nabla_{M}$ is the gravity covariant derivative. The gauge field and the field strength are matrix-valued as follows:

Here $\tau^{i}$’s are the generator of the $SU(2)$ group and they satisfy the corresponding Lie algebra¹¹1The structure constant is $f^{ijk}=-\epsilon^{ijk}$ and $\tau^{i}=\frac{\sigma^{i}}{2i}$, where $\sigma^{i}$’s are the Pauli matrices.:

In addition, the generators are normalized by ${\rm Tr}\,\tau^{i}\tau^{j}=-\delta^{ij}$.

Now, we choose an ansatz for cosmological solutions. The 8–dimensional metric is restricted to the following form:

We use coordinate decomposition given as $X^{M}=\left(x^{\mu},y^{a}\right)$. The indices $M$, $\mu$, and $a$ are running from 0 to 7, from 0 to 3, and from 1 to 4, respectively. We call the spacetime described by $x^{\mu}$ and the space in terms of $y^{a}$, the (toy) universe, and the extra dimensions for convenience. A general form of the Einstein equation based on this metric is presented in Appendix A.

For the gauge field, we will take an (anti)instanton solution in the sub-manifold of the extra dimensions, whose metric is given by $ds_{h}^{2}=h_{ab}dy^{a}dy^{b}$. Thus the gauge one-form is

where $A_{a}(y)$ is the one-(anti)instanton solution. The field strength is (anti)self-dual. This choice simplifies the equations of motion. For instance, (3) is easily satisfied by the Bianchi identity. In addition, some components of the energy-momentum tensor vanish as $T^{\text{YM}}_{ab}=T^{\text{YM}}_{\mu a}=0$. On the other hand, the nontrivial components are given by

where $\rho_{1}(y)$ is the (anti)instanton action density as follows:

To find cosmological solutions, we grant homogeneity and isotropy to the universe. A conformally flat geometry will be taken for the extra dimensions due to the conformal nature of the (anti)instanton solution. Therefore, the metric ansatz becomes

where x denotes the spatial component of $x^{\mu}$ and y is nothing but $y^{a}$, and the inner product “$\cdot$” is defined as a flat space inner product. As an explicit form of gauge field configuration, the action density is given by

where $\textbf{y}_{0}$ is an (anti)instanton position parameter which will be taken as the origin of the sub-manifold in the extra dimensions. So it is convenient to use a radial coordinate $r$ defined by $r^{2}=|\textbf{y}-\textbf{y}_{0}|^{2}$. Here, $\zeta_{s}$ is the size parameter of the instanton. Plugging this action density into the Einstein equation (A), one can notice that an easy way to satisfy the Einstein equation is to take a constant $\rho_{1}(y)$. Thus we can identify

where

Therefore, the metric becomes

with the 4–dimensional sub-manifold, whose metric is

where $\tilde{\textbf{y}}$ is the dimensionless coordinate defined by $\tilde{\textbf{y}}=\zeta_{s}\textbf{y}$. Then, this manifold is a 4–sphere with constant curvature $R_{(4)}=48/\zeta_{4}^{2}$ and volume $\pi^{2}\zeta_{4}^{4}/6$. Therefore the time-dependent volume of the extra dimensions is

We take into account $\mathbb{S}^{4}$ as extra dimensions in this work. This extra-dimension manifold can be extended to other compact manifolds. In KKYH001 , this extension will be studied. The possible compact manifolds with constant curvature give rise to the same dynamics by identifying the volume of the compact manifolds, and the corresponding curvature depends on $\zeta_{4}$.

Since our model is defined in 8–dimensional spacetimes, the starting point is the 8–dimensional Einstein frame in this paper. It is notable that this frame is different from the 4-dimensional Einstein frame. In other words, our analyses after compactification are focused on the 4–dimensional Jordan frame. The appendix A contains the relation between the two frames.

So far, we have described the ansatz for the geometry and the gauge field. If there is no more matter other than the gauge field, i.e., $T_{MN}^{m}=0$, the resultant solutions are the same as those discussed in Kim:2018mfv . Thus, we specify a matter on this geometry. As a matter in our study, we employ a perfect fluid given in the following form:

where $u_{M}$ is the unit timelike velocity field in this background (16). $P_{MN}^{(3)}$ and $P_{MN}^{(4)}$ are projections associated with the 3–dimensional space (described by the coordinates x’s) and the extra dimensions (described by coordinates y’s), respectively. $\mathcal{E}$ is the energy density, and $\mathcal{P}_{(3)}$ and $\mathcal{P}_{(4)}$ are pressures associated with the 3–dimensional space in the universe and the extra dimensions.

Including the perfect fluid, the Einstein equation (2) becomes

where dotted functions denote time-derivatives of the functions, and $\zeta_{0}$ is defined by $\zeta_{0}^{2}\equiv\zeta_{4}^{4}g_{YM}^{2}/(384\pi G_{(8)})$. Since there are three equations for the dynamical functions $h$ and $f$, one of these equations must be a constraint. A combination of these equations, which reads $\tilde{E}_{4}=\dot{\tilde{E}}_{1}+\left(4\dot{f}+3\dot{h}\right)\tilde{E}_{1}-\dot{h}\tilde{E}_{2}-4\dot{f}\tilde{E}_{3}$, should also vanish to satisfy the other Einstein equation.

For homogeneity, the energy density and the pressures are required as functions of the time only. While we allow only time dependence for the energy density and pressures, the vanishing condition $\tilde{E}_{4}=0$ is

This is nothing but the Bianchi identity corresponding to energy-momentum conservation. It is easy to see that one of the possible perfect fluids satisfying this conservation law is the cosmological constant. The following subsection is focused on the investigation of this case.

Now, let us take a convenient redefinition of fields and the time scaling to simplify the equations of motion (2.2). This is achieved by taking the following transformation:

Then, the equations of motion and the energy-momentum conservation law becomes

where the inverse size of the extra dimensions $X$ is given by $X=e^{-2f}$, and the time-dependent Hubble parameter $H$ is defined by $H\equiv\dot{h}$.

To find $H$ and $f$, we first need to solve the constraint equation, $E_{1}=0$. This equation is quadratic in $X$, so we have two roots given by

where $X_{+}$ describes the extra dimensions whose size $e^{2f}$ is larger than 1. On the other hand, the solutions with $X_{-}$ correspond to the geometries with the smaller size, i.e., $e^{2f}<1$. If we recover the original volume of the extra dimensions by inverting (22), $X_{+}$ or $X_{-}$ solutions describe the case where $\sqrt{\mathcal{V}_{4}}$ is larger or smaller than $\pi\sqrt{\frac{2}{3}}\frac{16\pi G_{(8)}}{g_{\text{YM}}^{2}}$, respectively. If we replace the coupling constant with length scales given by $16\pi G_{(8)}=2\,l_{p}^{6}$ and $g_{\text{YM}}^{2}=l_{\text{YM}}^{4}$, the 4-dimensional volume becomes

If we naively apply the length scales based on string theory, this volume size becomes the order of the string scale. Even though the two scales, $l_{p}$ and $l_{\text{YM}}$, are in a similar order, the solution set associated with $e^{-2f}=X_{-}$ describes the extra dimensions smaller than the Planck scale volume. Therefore, this set of solutions is hard to believe as a classical description. For this reason, we abandon the $X_{-}$ root and set $X=X_{+}$²²2For completeness, we summarize a different slicing of solution sets which shows the dynamics near $X=1$ in the Appendix B..

Finally, we are ready to solve the dynamical equation for the general perfect-fluid matter (19). Together with $X=X_{+}$, the flow equation for time evolution is as follows:

where we introduce the equations of state, $\omega_{3}$ and $\omega_{4}$, with $p_{3}=\omega_{3}\epsilon$ and $p_{4}=\omega_{4}\epsilon$. Given a set of initial values $\{\,\dot{f},H,\epsilon\,\}_{t=t_{i}}$ and the equations of state $\omega_{3}(t)$ and $\omega_{4}(t)$, the time-dependent functions, $\{\,\dot{f}(t),H(t),e^{-2f(t)},\epsilon(t)\,\}$, can be obtained by solving (2.2).

When discussing gravitational systems coupled to matter, an energy condition is crucial in justifying physical reality. We adopt the NEC in this work as the criterion for this justification. One may consider the weak energy condition, but some accelerating universe solutions are possible even for negative energy in our model due to extra-dimension dynamics. Thus we don’t demand the positivity of energy density.

The NEC stipulates that the contraction of the energy-momentum tensor for any null vector $n^{M}$ must be greater than or equal to zero. That is, it can be expressed as

for any null vector $n^{M}$. Since the metric (16) contains an isotropic 3–dimensional space ($d{\textbf{x}}\cdot d\textbf{x}$) and 4–extra dimensions ($d{\textbf{y}}\cdot d\textbf{y}$), one may introduce a generic form of the null vector given by $n^{M}=(1,n^{i}_{(3)},n^{a}_{(4)})$. The components of the null vector satisfy $P^{(3)}_{ij}n_{(3)}^{i}n_{(3)}^{j}=\cos^{2}\Theta$ and $P^{(4)}_{ab}n^{a}_{(4)}n^{b}_{(4)}=\sin^{2}\Theta$, where $\Theta$ can be taken as an arbitrary constant and $P^{(3)}_{ij}$ and $P^{(4)}_{ab}$ are the projections appearing in (19). Then the NEC inequality becomes

Adopting (22) again, one may write down this inequality as

Therefore, we arrive at the NEC written as follows:

As a comment on this NEC (30), the first condition is the same as the usual NEC in (3+1)–dimensions. The equation of state $\omega_{3}$ has the minimum corresponding to a cosmological constant, i.e., $\omega_{3}=-1$. However, the other equation of state $\omega_{4}$ of the extra dimensions is bounded from a value below $-1$ for a positive $\epsilon$ due to the extra-dimension contribution. More explicitly, The second condition in (30) can be written as

For the large extra-dimension limit ($X\to 0$), the above condition becomes the usual one whose bound is given by a cosmological constant. On the other hand, for a finite extra dimension, the bound of the equation of state $\omega_{4}$ is less than $-1$. Thus one can say that the extra-dimension volume relaxes the bound of the NEC.

In KKYH001 , it is explained how an expanding universe together with a dynamical compactification is possible by considering the Maldacena–Nunez no–go theorem in the context of our model Maldacena:2000mw . Even though the theorem can not be applied to an accelerating universe, a slight generalization of the theorem is useful to understand our case. The key quantity of the theorem is the scalar curvature of the (3+1)–dimensional universe. In this perfect-fluid case, the curvature is

In the absence of the matter ($\epsilon=0$), a negative value of $H\dot{f}$ is crucial to obtain a positive curvature $R^{(g)}$ that describes an accelerating or expanding universe as in KKYH001 . So $\dot{f}$ is always negative for positive $H$ or positive for negative $H$. To have an expanding three–space in the universe, $\dot{f}$ must be negative. Thus dynamical compactification is necessary for an accelerating universe. In addition, one may also notice that $H=\dot{f}=0$ case corresponds to $R^{(g)}=0$ with $X=0$. This is nothing but a Minkowski spacetime with an infinite size of extra dimensions. This solution doesn’t flow under time evolution and divides the full solution space into positive $\dot{f}$ (or negative $H$) solution set and negative $\dot{f}$ (or positive $H$) solution set.

On the other hand, the existence of matter relaxes the negativity of $H\dot{f}$ to produce an expanding three–space with positive curvature $R^{(g)}$. Therefore, more diverse cases can happen. The following sections discuss various effects of matter on this 8–dimensional spacetime. We also comment on selected examples in terms of the curvature expression (32).

To find the behavior of solution flow, or trajectory, using (34), one can utilize the linear stability theory. In the linear stability theory, a given dynamical system with fixed points can be linearized around its fixed points by the so-called Jacobian matrix Boehmer:2014vea . The eigenvalues of the Jacobian matrix, evaluated at the fixed point, reveal information about stability. If all eigenvalues have negative (positive) real values, the fixed point is regarded as stable (unstable). The trajectories around stable (unstable) points are attracted to (repelled by) such fixed points. If at least one of the eigenvalues has a positive real value, the corresponding fixed point would not be stable hence it is a saddle point, which attracts trajectories in some directions, but repels them along others. The fixed point is regarded as unstable if all eigenvalues have positive real values; such points repel all trajectories around it. In addition, if all eigenvalues are complex-valued, the sign of the real part contains information about stability. If the real parts of all eigenvalues are negative (positive), the fixed point is a stable (unstable) spiral. Thus, in this subsection, we first find the fixed points of the flow equation, and then check the asymptotic stability at those fixed points via linear stability analysis.

In Table 1, we present all possible fixed points of a system determined by (34) and provide their existence conditions, eigenvalues of the Jacobian matrix, and stability information.

The first notable pair of fixed points are the $A_{1}^{\pm}$ points. As the table shows, these points exist only for $\Lambda\geq 0$, which corresponds to the positive cosmological constant. However, the $\Lambda=0$ value, the vanishing cosmological constant, is excluded because the linear stability theory fails to determine stability when at least one of all eigenvalues of the Jacobian matrix becomes zero Boehmer:2014vea ³³3These vanishing eigenvalues denote $\ddot{f}=\dot{H}=0$ with the fixed points $\dot{f}=H=0$, and the corresponding inverse size is given by $X=0$. Therefore, this is the Minkowski limit with large extra dimensions.. For $\Lambda>0$, the system is stable at the $A_{1}^{+}$ fixed point and unstable at the $A_{1}^{-}$ fixed point due to the negative and positive sign of eigenvalues, respectively.

Next, the $A_{2}^{\pm}$ and $A_{3}^{\pm}$ fixed points look similar, but their phenomenological implications are quite different. As for the $A_{2}^{\pm}$ fixed points, the existence condition implies that these fixed points can have real values for $0<\Lambda\leq 3.375$, where the upper bound $\Lambda=3.375$ is determined by the real values of $H$ expression in the fixed point column. Within this range of $\Lambda$ values, one obtains two eigenvalues have opposite signs; one of the eigenvalues is positive while the other one is negative. Thus, the $A_{2}^{\pm}$ fixed points are always saddle points. On the other hand, the $A_{3}^{\pm}$ fixed points exist when the $\Lambda$ takes values within the $3<\Lambda\leq 3.375$ range. As is seen in the table, the existence condition for $A_{3}^{\pm}$ points is tighter than that of the $A_{2}^{\pm}$ points. The system is stable at the $A_{3}^{+}$ point and unstable at the $A_{3}^{-}$ point when $3.357\lesssim\Lambda\leq 3.375$. In the $3<\Lambda<3.357$ range, the eigenvalues are complex-valued hence the sign of their real parts reveals that the $A_{3}^{-}$ point is spirally unstable and the $A_{3}^{+}$ point is spirally stable.

In Figure 1, we plot the phase portraits of our dynamical system described by (34) in a $\dot{f}$ vs. $H$ plane. To present all fixed points listed in Table 1 in one plane, we set $\Lambda=3.25$. The brown- and red-shared regions indicate the parameter space excluded by the $X$ to be positive and real-valued, respectively. The red, blue, green, and black dots represent the $A_{1}^{\pm}$, $A_{2}^{\pm}$, $A_{3}^{\pm}$, and $A_{4}^{\pm}$ fixed points, respectively. As we discussed earlier, the $A_{1}^{-}$ fixed point, or the small red dot, is the unstable fixed point hence trajectories around it are repelled by this point, see the outgoing arrows in the figure. The $A_{1}^{+}$ fixed point, the large red dot, conversely, is the stable fixed point, which attracts trajectories around it as indicated by ingoing arrows. Table 1 also shows that the $A_{2}^{\pm}$ fixed points are always the saddle points. The blue dots in the figure represents the $A_{2}^{\pm}$ points, and, as saddle points, they attract trajectories in some directions, but repel them along others. The chosen value $\Lambda=3.25$ falls in the range $3<\Lambda\lesssim 3.357$. Thus, the $A_{3}^{-}$ fixed point, the small green dot, is an unstable spiral, while the $A_{3}^{+}$ fixed point, the large-green dot, is a stable spiral. For the unstable spiral, the arrows of trajectories toward outward, while the arrows of trajectories toward inward for the stable spiral, see middle and right panels in Figure 1, respectively.

The time-dependent Hubble parameter $H(t)$ and the cosmological constant $\Lambda$, in this case, are independent due to the choice of the initial condition, while those are directly related to each other in a de Sitter spacetime without extra dimensions. Thus these two quantities are useful observables to show the role of extra dimensions. To see how they are related to physical observables, we need to express the physical Hubble parameter and the physical cosmological constant in terms of $\Lambda$ and $H(t)$ using (22). If we use the length scales given by $16\pi G_{(8)}=2l_{p}^{6}$ and $g_{\text{YM}}^{2}=l_{\text{YM}}^{4}$, the physical cosmological constant $\Lambda_{p}$ of 8–dimensional spacetime can be written as

Also, the physical Hubble parameter function in the unit of the Planck scale can be written by

Thus the ratio of the Hubble parameter to the cosmological constant is determined by

In a usual de Sitter spacetime without extra dimension, this ratio is just a constant. As opposed to this pure de Sitter case, the ratio is determined by the dynamical field $H(t)$ based on the flow equation (34). We will discuss this quantity in a specific example, which is a de Sitter fixed point, later. The following two subsections are devoted to some intriguing cosmological evolutions. One is the bouncing universe solution, which is ubiquitous in the solution flow, and the other is the (spirally) stable de Sitter solution ($A_{3}^{+}$) appearing in Table 1.

Our model is based on the interplay between the universe and extra dimensions through the Einstein-matter theory. The volume-preserving nature of Einstein gravity is expected to give rise to seesaw-like dynamics of the universe and the extra dimensions as a very common time evolution. In the universe view or the extra dimension view, these dynamics can be regarded as bouncing universes or bouncing extra dimensions. This subsection presents bouncing solutions and conditions for the bouncing behavior.

To describe the bouncing stage, we plot the $H$ vs. $\dot{f}$ phase diagram in the upper panel and the time evolution of $H(t)$ and $\dot{f}(t)$, as well as the sizes of the three space of the universe – $e^{2h(t)}$ and the extra dimensions – $e^{2f(t)}$, in the bottom panel of Figure 2, respectively, for $\Lambda=1$. The shaded regions are the same as Figure 1. Also, the red and blue dots represent the $A_{1}^{\pm}$ and $A_{2}^{\pm}$ fixed points in Table 1, respectively. The magenta lines indicate particular bouncing solutions, where a trajectory with $H<0$, initially, evolves to $H>0$ by crossing zero at $t=t_{b}$. The bouncing time $t_{b}$ is different for different solutions. At the same time as the $H$ value increases from a negative to a positive value, the $\dot{f}$ value decreases from a positive to a negative value. Thus, for the bounce to occur, there must at least be a trajectory that crosses the origin, $\{\dot{f},H\}=\{0,0\}$, as is shown in the bottom panel of Figure 2. In addition to the time evolution of $H$ and $\dot{f}$, we also present the time evolution of $e^{2f}$ and $e^{2h}$. The figure shows the size of the extra dimension (red) grows in time and reaches the maximum size at $t=t_{b}$. After the bouncing, the size of the extra dimension shrinks, whereas the size of the three spaces (magenta) increases as expected.

The growing $H$ parameter also indicates the $\dot{H}>0$ during the bounce stage. More explicitly, the solution flow at the origin ($\dot{f}=H=0$) is

and the slope at the origin is given by $dH/d\dot{f}=-2$ with $X=1-\sqrt{1-\Lambda/3}$. Thus, for $0<\Lambda<3$, there exists always bouncing flows.

When $\Lambda=3$, no bounce stage occurs because the two separate red regions in the figure intersect only at the origin $\{\dot{f},H\}=\{0,0\}$. A solution with $\Lambda=3$ results in vanishing $\dot{H}$ and $H$ as well as $\ddot{f}$ and $\dot{f}$, so this solution does not evolve. Therefore, this solution is isolated and describes $\mathbb{R}^{1,3}\times\mathbb{S}^{4}$. The stability of such a solution was studied in Randjbar-Daemi:1983xth  ⁴⁴4See the curvature expression (35). This geometry has $X=1$. As we discussed earlier, a quantum gravity effect must be considered. Our argument and this study are achieved without any assumption on $g_{\text{YM}}^{2}$..

In most cases of bouncing models in cosmology, it is hard to obtain non-singular time evolution without violating the NEC Novello:2008ra ; Battefeld:2014uga ; Lilley:2015ksa ; Brandenberger:2016vhg . However, this bouncing dynamics fulfills the NEC and gives rise to regular time evolutions. A difference from other bouncing models is that our model has flexible extra dimensions which relax the NEC (30).

Let us discuss another interesting solution given by stable fixed points. When $\Lambda>0$, the stable fixed points are the $A_{1}^{+}$ and $A_{3}^{+}$ in Table 1. However, at the $A_{1}^{+}$ point, the size of an extra dimension is infinite, i.e., $X=0$; therefore, the total 8–dimensional spacetime undergoes a decompactification process approaching the $A_{1}^{+}$ point. Thus, we exclude the $A_{1}^{+}$ point from our interest and focus more on the $A_{3}^{+}$ point instead.

At the $A_{3}^{+}$ fixed point, the size of the extra dimensions is stabilized, and the universe becomes a de Sitter spacetime ⁵⁵5Since there is a positive cosmological constant $\Lambda$, this situation can evade the Maldacena–Nunez no–go theorem Maldacena:2000mw .. The corresponding $X$ becomes

at this fixed point. Since this inverse size is close to 1 ($1>X>3/4$), the quantum gravity effect should correct this value. Leaving such a concern as future work, we continue our discussion on this scale. The 4–dimensional Newton constant and the corresponding Planck scale are given by

where we used the scaling (22). Employing this quantity, the Hubble parameter becomes

where we used (3.1). In the unit of the 4–dimensional Planck scale, this Hubble parameter depends on the ratio of the coupling-length scales and $H^{2}X^{2}$. Plugging the fixed point $A_{3}^{+}$ into $H$ and $X$, we obtain the following expression:

This is a monotonically increasing function of $\Lambda$, which ranges from $0$ to $27/256\sim 0.105$. When the spacetime approaches $\Lambda=3$ and $X=1$, this function is closed to zero.

Since our approach is a toy model, we comment that we do not know any fundamental theory which can be related to our model. If, however, we naively consider a relation $G_{(8)}=g_{\text{YM}}^{2}l_{s}^{2}$ between the scales appearing in (42), where $l_{s}$ is an unknown fundamental scale, then the Hubble parameter can be estimated as follows:

where we set $H^{2}X^{2}=0.1$ close to the maximum value 27/256 of $H^{2}X^{2}$. If we also assume that $l_{p}$ and $l_{s}$ are in the same or similar order, the Hubble parameter gets the small value, in terms of the 4–dimensional Planck length, because of the smallness of the numerical factors and the smallness of $H^{2}X^{2}$. Since the inverse size of the extra dimensions $X$ is close to 1, it would be interesting to see how this quantity $H^{2}_{p}l_{p,4}^{2}$ changes by a quantum gravity effect. We leave this as a future study. This estimation can explain the toy version of the cosmological constant problem in our model. It is notable that $H_{p}^{2}l_{p,4}^{2}$ is extremely small while the 8–dimensional cosmological constant $\Lambda_{p}l_{p}^{2}=\Lambda(l_{p}/l_{s})^{2}/(4\pi)\sim 0.25(l_{p}/l_{s})^{2}$ is the order of one.

Now let us find the dynamics before arriving at the de Sitter fixed point. We numerically integrate (34) to plot the dynamical evolution of the $H(t)$ and the size of an extra dimension $e^{2f}=X^{-1}$ in Figure 3. Two different values of $\Lambda$, namely $\Lambda=3.25$ (upper) and $\Lambda=3.37$ (lower), are considered for plotting the figure, where we show different dynamics towards the different stable fixed points. The full geometry settles at the $A_{3}^{+}$ fixed point, which can be either the stable or stable spiral fixed point according to the $\Lambda$ value. The $A_{3}^{+}$ point is a spirally stable fixed point if $3<\Lambda\lesssim 3.357$ and is a stable fixed point if $3.357\lesssim\Lambda\leq 3.375$, see Table 1. Thus, when $\Lambda=3.25$, both the $H(t)$ and $X^{-1}(t)$ oscillate several times before settling at the constant values. In contrast, there is less oscillation that occurs when $\Lambda=3.37$. The value of both $\dot{f}$ and $H$ is constant at $A_{3}^{+}$ fixed point, and the size of the extra dimension is finite after all. The oscillation dynamics can be considerable since it can generate another matter production, such as a reheating process. Even though it is interesting, we plan to study it after understanding a quantum gravity correction.

As a first try for a nontrivial matter, we consider a matter satisfying the NEC bound in (30). Thus, in this subsection, we take $\omega_{3}=-1$ and $\omega_{4}=-1-c_{0}X^{2}/\epsilon$, where $c_{0}\leq 12$ is an arbitrary constant, into consideration. After substituting these expressions into the energy-momentum conservation law, we solve (2.2) to obtain

where $\epsilon_{0}>0$ is the constant energy density when the size of the extra dimensions is infinite, i.e., $X=0$. To guarantee the positive energy density, one has to impose $\epsilon_{0}\geq c_{0}X^{2}$ by considering the validity of classical solutions. However, negative energy also satisfies the NEC. After plugging this energy density into (2.2), the equation of motion becomes

where the first equation is again a constraint equation. The constraint equation has only one root for $X$ when $c_{0}=3$ and two roots when $c_{0}\neq 3$.

Let us consider the $c_{0}\neq 3$ case first, for which we find two roots, but the root we are interested in is

and the other root is excluded because it gives a negative value. From (49), the physical conditions read as $1>X\geq 0$ and $(c_{0}-3)(-12\dot{f}H-6\dot{f}^{2}-3H^{2}+\epsilon_{0})/9+1\geq 0$, which can be reduced to

The first inequality corresponds to $X<1$. Using this solution, the flow equation is given by

We provide the fixed points of the flow equation (51) in Table 2 and present them in the phase space diagram in the upper left panel of Figure 3, where $\epsilon_{0}=1$ and $c_{0}=12$.