Generalization of instanton-induced inflation and dynamical compactification

It was shown that Yang-Mills instantons on an internal space can trigger the expansion of our four-dimensional universe as well as the dynamical compactification of the internal space. We generalize the instanton-induced inflation and dynamical compactification to general Einstein manifolds with positive curvature and also to the FLRW metric with spatial curvature. We explicitly construct Yang-Mills instantons on all Einstein manifolds under consideration and find that the homogeneous and isotropic universe is allowed only if the internal space is homogeneous. We then consider the FLRW metric with spatial curvature as a solution of the eight-dimensional Einstein-Yang-Mills theory. We find that open universe (k = −1) admits bouncing solutions unlike the other cases (k = 0, +1).


Introduction
To understand the origin of the universe, we must combine the general relativity with quantum theory, dubbed as quantum gravity.String theory is a leading candidate for the fundamental theory of quantum gravity which requires a six-dimensional internal space in addition to our four-dimensional spacetime [1,2].However, most quantum gravity theories also require unification of forces in which gravity plays an equivalent role with gauge theories.If so, both gravity and gauge theory should play an important role in the early evolution of the universe such as the cosmic inflation [3,4,5,6] (see also reviews [7] and [8]).In particular, this implies that nonperturbative objects in gravity and gauge theories may play an important role for the inflationary epoch of the early universe.These are gravitational instantons in gravity such as Einstein manifolds [9] and Yang-Mills instantons in gauge theories [10].See also reviews [11], [12] and [13].
Instantons are defined as topologically nontrivial and nonsingular solutions of classical equations such as Yang-Mills equations or Einstein equations that minimize the action functional within their topological type [12].Mathematically, an instanton is a self-dual or anti-self-dual curvature in a vector bundle over a four-dimensional Riemannian manifold [14].A nonperturbative effect will be important in the strong coupling regime.Because gravity is also a gauge theory like Yang-Mills theory, gravitational instantons also satisfy the same kind of self-duality equations as Yang-Mills instantons.Indeed it was shown [15,16,17,18,19] that Einstein manifolds can be understood as Yang-Mills instantons for the local Lorentz group SO (4) or Spin (4).An important point is that Yang-Mills instantons formed in extra dimensions provide a quantized energy density in our fourdimensional spacetime [20].Therefore, in order to test the role of instantons in the inflationary epoch of the early universe, one may consider a four-dimensional internal space on which Yang-Mills instantons are supported.Following the motivation in [20], we will study the eight-dimensional Einstein-Yang-Mills theory in a more general setup by considering the FLRW universe with spatial curvature and general compact Einstein manifolds.We explicitly construct Yang-Mills instantons on the internal space that generate the energy-momentum in our four-dimensional spacetime.
In order for both string theory and inflation to be correct, the internal space has to be compactified with a microscopic size and only four-dimensional spacetime remains in a macroscopic scale after the end of cosmic inflation.This picture suggests that the cosmic inflation of our four-dimensional universe would be closely related to the dynamical compactification of extra dimensions.A recent study has found a model realizing such behavior by showing that the cosmic expansion of our fourdimensional spacetime causes a dynamical compactification of extra dimensions simultaneously [20].In this paper we will clarify why such behavior must be generic for the eight-dimensional Einstein-Yang-Mills theory.
An accelerating expansion from extra dimensions and a dynamical compactification of extra dimensions are an old idea explored in many literatures [21]- [38] (see also chapter 11.4 in [7]) and similar ideas using monopoles and instantons in extra dimensions have appeared in [39]- [45].How-ever, their interconnections have not been significantly explored.
This paper is organized as follows.In section 2, we generalize the instanton-induced inflation and dynamical compactification in [20] to a more general eight-dimensional spacetime M 8 where the Lorentzian part of M 8 includes a spatial curvature and the four-dimensional internal space is a general Einstein manifold with positive curvature.We explicitly construct Yang-Mills instantons on all Einstein manifolds under consideration.We find that the homogeneous and isotropic universe is possible only when the internal space is homogeneous, so a nonhomogeneous Einstein manifold must be excluded.We then consider the FLRW metric with spatial curvature as a solution of the eight-dimensional Einstein-Yang-Mills theory.We find that open universe (k = −1) admits bouncing solutions unlike the other cases (k = 0, +1).In section 3, we discuss how our instanton-induced inflation and dynamical compactification can evade no-go theorems in [46,47].We also comment on some more generalizations and possible implications of the instanton-induced inflation, in particular, the dynamical compactification of extra dimensions and the reheating mechanism in inflationary cosmology.Appendix A provides useful algebraic relations for the 't Hooft symbols which are crucially used for the explicit construction of Yang-Mills instantons on a compact Einstein manifold.In appendix B, we present the explicit solutions of Yang-Mills instantons on all Einstein manifolds under consideration and confirm that a nonhomogeneous Einstein manifold does not satisfy the cosmological principle.

Generalization of instanton-induced inflation and dynamical compactification
We will generalize the basic idea in [20] by considering a more general eight-dimensional spacetime M 8 where the Lorentzian part of M 8 includes a spatial curvature and the four-dimensional internal space is a general Einstein manifold with positive curvature.Consider an eight-dimensional spacetime M 8 whose metric is given by where where A = A i M (X)τ i dX M = A µ (x, y)dx µ , A α (x, y)dy α is a connection one-form of the SU(2)bundle E and τ i i = 1, 2, 3 are Lie algebra generators with a normalization Trτ i τ j = −δ ij and obey the commutation relation In order to investigate the dynamical behavior of the eight-dimensional spacetime M 8 induced by Yang-Mills instantons on the internal space X 4 , let us consider the eight-dimensional Einstein-Yang-Mills theory with the action The gravitational field equations read with the energy-momentum tensor given by The action (2.5) also leads to the equations of motion for Yang-Mills gauge fields If X 4 is an Einstein manifold with a positive cosmological constant, that is, R αβ = λh αβ with λ > 0, then X 4 is a closed compact manifold with vol(X 4 ) = X 4 d 4 y √ h finite [9].We are interested in the gauge field configuration on the Einstein manifold X 4 described by for which the Yang-Mills action reduces to For the gauge field configuration (2.9), the equations of motion (2.8) thus take the simple form Suppose that the gauge fields (2.9) are Yang-Mills instantons on X 4 satisfying the self-duality equation [12,13] (2.12) Then, Eq. (2.11) is automatically satisfied and the Yang-Mills instanton solves the equations of motion (2.8) even in a warped spacetime with the metric (2.1).
An important point is that Yang-Mills instantons on the internal space X 4 generate a nontrivial energy-momentum tensor given by T µα = 0, where g µν (x) = e −4f (x) g µν (x) and, for convenience, all indices are raised and lowered with the product metric (2.1) with f (x) = 0. Let us denote the energy-momentum tensor T µν as the form where ρ n (y) = − 1  4 TrF αβ F αβ is the instanton density on X 4 which is uniform along the fourdimensional spacetime M 3,1 .The coupling in Eq. (2.14) implies an interesting behavior of fourdimensional spacetime M 3,1 .The energy-momentum tensor T µν depends on the size of internal space X 4 characterized by the warp factor e 2f (x) .For a fixed instanton density, to be specific, if the internal space becomes small, i.e., f (x) decreasing, then the energy-momentum tensor T µν becomes large, i.e., M 3,1 more expanding, or vice versa.This behavior leads to a dynamical mechanism for the compactification of an internal space through the inflation of our four-dimensional spacetime.Recently, Ref. [48] generalized the eight-dimensional Einstein-Yang-Mills theory by including coupling with a perfect-fluid matter that admits several interesting solutions such as bouncing universes and oscillatory solutions converging to a de Sitter spacetime.
Suppose that the solution of an SU(2) Yang-Mills instanton on an Einstein manifold X 4 is known.In appendix B, we explicitly construct a Yang-Mills instanton on a general Einstein manifold X 4 with positive curvature.Then, it is enough to solve the Einstein equations (2.6) for the warped product geometry (2.1) which take the form where R µ is a covariant derivative with respect to the metric g µν (x).A close inspection on the dependence of M 3,1 and X 4 in Eq. (2.16) shows that a solution exists only if the internal space must be a space with a constant Ricci scalar, i.e., an Einstein manifold.With this condition, Eq. (2.15) further requires that its consistent solution exists only if the instanton density ρ n (y) is constant.This means that the instantons are uniformly smeared out on the internal space X 4 .Note that the internal space X 4 in our case is a compact Einstein manifold.It is known [9] that an Einstein manifold with a positive cosmological constant must be compact and four examples are known: S 4 , CP 2 , S 2 × S 2 and Page where the Page [49] is an inhomogeneous Einstein manifold on the nontrivial product as a twisted S 2 bundle over S 2 .We will show that the Page space does not admit a constant instanton density, so it cannot be a solution of the above equations.It is interesting to note that a homogeneous and isotropic universe cannot be obtained from an inhomogeneous internal space such as the Page space.This result suggests that the uniformity of the internal space is closely related to the cosmological principle of our universe.
To solve the above equations, let us take the specific ansatz [7] ds 2 = −dt 2 + e 2h(t) dr 2 1 − kr 2 + r 2 dΩ 2 + e 2f (t) h αβ (y)dy α dy β , (2.17 where k is normalized as −1, 0, or 1 and dΩ 2 is the metric of unit sphere S 2 . 1 The Einstein equations (2.15) and (2.16) for the metric (2.17) leads to the following differential equations where 8πG 8 ρn .Note that new terms proportional to k have been appeared.In general relativity, the metric component g 00 is not a dynamical variable but a Lagrange multiplier.Thus E 1 = 0 derived from the variation δg 00 is not an evolution equation but a (Hamiltonian) constraint that initial conditions for the evolution equations, E 2 = 0 and E 3 = 0, must satisfy.Indeed, one can show that Ė1 + 4 ḟ (t) + 3 ḣ(t) E 1 = ḣ(t)E 2 + 4 ḟ (t)E 3 [20].Therefore, if the evolution equations are solved with initial conditions chosen to be consistent with E 1 = 0, the constraint equation will always be satisfied.As a result, the constraint equation will give us an important information for the time evolution of an eight-dimensional spacetime M 8 .
The solution space is most easily understood by studying the constraint equation (2.18).Let us define a graph for the constraint Y := E 1 = 0 as where The graph (2.21) is thus described by a concave parabola with the symmetric axis X s = − b 2a = λζ 2 I > 0. Then the constraint equation can be understood as finding X -intercepts for given velocity data ( ḣ, ḟ ) 1 In our metric ansatz (2.17), h(t) = ln a(t) where a(t) is a usual scale factor of the four-dimensional FLRW universe, so ḣ = ȧ a corresponds to the Hubble parameter.Accordingly, which determines the y-intercept c.One can easily infer the behavior of a solution from Fig. 1.The quadratic equation Y(X ) = 0 has two (in general, complex) roots X L and X R given by From our definition X = e −2f > 0, these X -intercepts must be positive.The symmetric axis X s > 0 is positive and solely determined by the ratio 12 . Hence it does not depend on velocities ( ḣ, ḟ ).It means that the parabola in Fig. 1 simply moves up or down along the symmetric axis X = X s depending on velocities ( ḣ, ḟ ).Note that the X -intercept X L becomes positive only when the yintercept c is negative.

Flat and closed universes
For k = 0 and k = +1 (i.e., flat and closed universes), the condition that c = ḣ2 + 2 ḟ 2 + 4 ḟ ḣ + ke −2h is negative requires ḟ ḣ < 0 for any solution of the X L -branch.Thus there are two possibilities for the solution: The right-moving solution describes an eight-dimensional spacetime where our four-dimensional spacetime is expanding and the internal space is contracting while the left-moving solution does the opposite.We will show later that open universe (k = −1) exhibits a completely different behavior; it admits bouncing solutions unlike the other cases (k = 0, +1).
A simple argument shows that f (t) in the X L -branch is a (strictly) monotonic function of t. 2 Suppose that the function f (t) in the X L -branch is not monotonic.It means that the function f (t) can pass through the region where ḟ = 0 at some time.Since the X L -branch does not allow c > 0, ḣ = 0 there and so c = 0 (i.e., X L = 0).Therefore ḟ = 0 in the X L -branch also enforces ḣ = 0, in which the eight-dimensional spacetime becomes M 8 = R 3,1 ×X 4 with vol(X 4 ) → ∞.The evolution equations, E 2 = E 3 = 0, then imply that this spacetime must continue thereafter.This behavior is an exceptional case possible only in a particular contracting solution but cannot appear in generic cases, especially in expanding solutions.Therefore, the warp factor f (t) in the X L -branch must be a (strictly) monotonic function of t for generic solutions.It leads to an interesting picture [22] that the cosmic expansion of our four-dimensional spacetime necessarily accompanies a dynamical compactification of extra dimensions.However, the X R -branch admits a solution satisfying ḟ ḣ > 0, so it does not necessarily require a similar behavior as the X L -branch.Now we will argue that the X R -branch corresponds to a quantum branch so that it should be discarded within a classical approximation.
A notable point is that the size of internal space depends on the warp factor e 2f (t) = X −1 whose value is determined by solving the constraint equation Y = E 1 = 0. First, note that the graph in Fig. 1 is a concave parabola, so the parabola cannot move down below the X -axis since the parabola below the X -axis means that there is no real solution of Eq. (2.21).This implies that the internal space has a minimum size for an X L -branch solution while a maximum size for an X R -branch solution, whose size is set by the value X s of the symmetric axis of the parabola.In order to estimate the explicit value of X s , let us use the relation, ) where l s characterizes a fundamental scale as the size of string.Using the explicit result ρ 1 = 3 4R 4 for the single instanton on S 4 , we get 3 where R = ζc 2 is the radius of the internal space S 4 .Since the size of internal space is given by Re f (x) , we see that (Re f ) ≥ √ 2πl s ≈ 2.5l s for the X L -branch whereas 0 ≤ (Re f ) ≤ √ 2πl s ≈ 2.5l s for the X R -branch.In particular, the X R -branch travels at most 2.5l s distances.Moreover, when X → ∞, i.e. e f → 0, the X R -branch hits the spacetime (internal) singularity.If the fundamental scale l s is comparable to the Planck length l P = 10 −35 m, the Planck distance travel cannot be described by a classical geometry.Hence it is reasonable that the X R -branch can be ignored within the classical approximation.
We emphasize that the existence of the minimum size of the internal space is due to the Yang-Mills instanton smeared over the compact space X 4 .Without the instanton, the graph in Eq. (2.21) is replaced by a linear equation, i.e., Y = bX + c with the same c = ḣ2 + 2 ḟ 2 + 4 ḟ ḣ + ke −2h [30].In this case, there is only a single branch which can travel the entire positive range, X > 0, but still requires the condition (2.23) to have a real root.Even in this case, f (t) must be a monotonic function of t since ḟ = 0 enforces c > 0 for generic cases, where X = e −2f becomes negative, physically not allowed.The monotonicity of the function f (t) implies that there is no bounce.A similar behavior is also expected for the case with Yang-Mills instantons.Actually the same reasoning implies that the bounce X L ( ḟ < 0) ⇄ X L ( ḟ > 0) is not allowed either for any solution of the differential equations 3 There was a minor error in Eq. (3.19) in [20].Considering the difference between SU (2) generators in gravity and gauge theory, it should read as ρ 1 (y) = 12 See the footnote 2 in [19] for the explanation of this difference.

Open universe
For a numerical analysis, it is convenient to rescale the variables in the constraint equation (2.18) by absorbing the model parameters as The constraint equation (2.18) can then be expressed with the rescaled parameters as where X ≡ e −2f > 0 and Z ≡ e −2h > 0. The contour plot in the first quadrant of (X , Z)-plane is shown in Fig. 2 and each contour line represents c 0 ≡ 4 ḟ ḣ + 2 ḟ 2 + ḣ2 .If we solve Eq. (2.26) in terms of X , the roots are given by The effect of introducing a new term, kZ, is only to shift the Y-intercept in Fig. 1 from c 0 to c ≡ c 0 + kZ, as shown in Fig. 2, but not to change the position of the symmetry axis X s = 1 (or λζ 2 I ).Therefore X − corresponds to a quantum branch for the same reason as the k = 0 case, so we discard the X − -branch.Solutions in the X + -branch are constrained by 0 ≤ X + ≤ 1 as seen from Eq. (2.27).
where H ≡ ḣ is the Hubble parameter and Z is eliminated using the constraint equation (2.26), This set of equations generates three-dimensional flows depending on the data ( ḟ , H, X ).Given the initial condition ḟ , H, X t=t i , these flows show the full-time evolution of metric functions.The numerical analysis is separately done for k = +1 and k = −1 cases by solving the constraint (2.29) for each case.Figures 3 and 4 have been obtained by solving numerically the flow equations (2.28).
It may be instructive to compare the behavior of solutions in the X + -branch for k = −1 case and other cases (k = 0, 1).As was pointed out above, the Y-intercept in the graph defined by Eq. (2.26) for a given Z is given by c = c 0 + kZ, which must be negative to satisfy X + > 0. Therefore, for k = 1, it is necessary to require ḟ ḣ < 0 in order to obey c + ≡ c 0 + Z < 0. Hence the behavior of solutions for k = 1 is similar to that for the k = 0 case.The three-dimensional solution flows for k = +1 are depicted in Fig. 3.Because the behavior of solutions with k = +1 is similar to the case with k = 0, no bounce appears in the k = +1 case for the same reason as the k = 0 case.However, the k = −1 case is very different from the other cases.Since the Y-intercept in this case is given by c − ≡ c 0 − Z, the condition c − < 0 does not necessarily require the condition ḟ ḣ < 0. So the function f (t) needs not be monotonic.In other words, a bounce solution is allowed for the k = −1 case, where a solution makes a smooth transition from a region where ḟ > 0 to a region where ḟ < 0 or vice virsa.Fig. 4 clearly shows this behavior. 4inally, we briefly comment on the four-dimensional Ricci scalar R(g) = g µν R µν for the metric (2.17).Since the (µ, ν)-components of the eight-dimensional Ricci tensor are given by the Ricci scalar reads  after taking into account the rescaling (2.25).In order to obtain a de Sitter-like solution, it is necessary to have R(g) > 0. For k = 1, c + = c 0 + Z < 0 requires ḟ ḣ < 0 and the condition c + < 0 is stronger than that for the k = 0 case, c 0 < 0. Therefore, in order for the four-dimensional FLRW spacetime to have a positive Ricci scalar, the condition ḟ ḣ < 0 is more stringent than the flat case.However, the stringent condition is not necessary for the k = −1 case since, even if ḟ ḣ > 0, R(g) could be positive while obeying c − = c 0 − Z < 0. As a result, bouncing solutions are also allowed in this case as was illustrated by Fig. 4.

General Einstein manifolds with positive curvature
The eight-dimensional Einstein- αβ F (±)αβ .The following formula is useful; Although the equations of motion (2.8) have been reduced to the problem solving the instanton equation (2.12) on a compact Einstein manifold X 4 , it is, in general, difficult to find an instanton solution on the Einstein manifold.Fortunately there exists a general method [15,16] to find an SU(2) Yang-Mills instanton on X 4 if the metric on X 4 is known.Since the explicit instanton construction is a technical justification of the assumption already made in [20], every details will be delegated to appendices.Instead, we summarize the obtained results.In order to realize a homogeneous and isotropic universe, it is necessary to construct Yang-Mills instantons uniformly smeared out over a compact Einstein manifold X 4 = S 4 , CP 2 , S 2 × S 2 and Page = CP 2 ♯CP 2 .We find that such instantons are allowed for all compact Einstein manifolds under consideration except the inhomogeneous Page space.Table 1 provides a summary of the geometric properties of these compact Einstein manifolds, including the Kähler, spin, and homogeneous structures [9,11].The emerging conclusion is that the uniformity of internal space is essential for realizing a homogeneous and isotropic universe.

Discussion
There are no-go theorems stating that there are no nonsingular de Sitter compactifications for large class of gravity theories [46,47].We will clarify why our instanton-induced inflation and dynamical compactification can evade such no-go theorems.Maldacena and Nuñez consider a D-dimensional warped metric [46] ds Using the D-dimensional Einstein equations, they showed that the warping factor should satisfy the equation where R(g) is the scalar curvature of the d-dimensional metric g.A simple manipulation with Eq. (3.2) shows that the de Sitter spacetime with R(g) > 0 is not allowed without a bare positive cosmological constant.
A main reason for our model to evade the Maldacena-Nuñez argument is that the warp factor in Eq. (2.17) is time-dependent whereas that in (3.1) depends only on the coordinates of the internal space. 5Let us clarify this aspect for the eight-dimensional Einstein-Yang-Mills theory (2.5).The Einstein equations (2.6) can be written as the form Contracting (µ, ν)-components leads to (2.31), i.e., R(g Since we are interested in a de Sitter-like solution, we will exclude the open universe (i.e., k = −1) case.(In fact, the no-go theorem in Ref. [46] was also restricted to d-dimensional metrics which correspond to Minkowski or de Sitter space.)If the metric were time-independent, R(g) = −6kZ − 2 ζ 2 I X 2 < 0, so we could not obtain the de Sitter space.However, if the metric is time-dependent, we can obtain R(g) > 0 as long as the term ḟ ḣ is negative and large enough.Therefore, in order to have a de Sitter space including expanding universes, the time-dependence is crucial so that ḟ ḣ has a negative value.Thus the dynamical compactification of an internal space is an inevitable consequence of an inflating universe in our toy model.The no-go theorem in [46] has been generalized in [47] to D-dimensional time-dependent metrics the d-dimensional effective theory is defined in the Einstein frame with the potential where , and χ = 3 (11−d)  d−2 ρ is scaled to have a canonically normalized kinetic term of scalar field, 1 2 (∇χ) 2 .In Eq. (3.8), V R = −R(h) is the scalar curvature of the internal manifold and V G (which is always positive) is the average of the four-form G-flux in the eleven-dimensional supergravity.Note that λ 1 ≤ λ 2 for d ≥ 2. Then one can show that the effective potential (3.8) does not admit a de Sitter vacuum χ 0 where V ′ (χ 0 ) = 0 and V (χ 0 ) > 0. In particular, for the region where V > 0, |∂χV | V is a monotonically decreasing function for both V R > 0 and V R < 0 and satisfies the swampland bound [47].
The metric ansatz (3.5) has the same form as ours in (2.1).We should emphasize that the no-go theorem in [47] is a result derived from the effective potential (3.8), which is defined in d-dimensional Einstein frame with the metric q µν .However, we did not consider any compactification (or dimensional reduction) to four-dimensional spacetime.Instead, we have considered the dynamical evolution of the entire eight-dimensional spacetime, rather than addressing the dynamics in a lower-dimensional effective field theory.Furthermore, we describe the evolution in the Einstein frame of the eightdimensional spacetime rather than the four-dimensional spacetime.As we have noticed, the evolution of the four-dimensional universe and the internal space is deeply interconnected, so in our case, it is necessary to consider the entire evolution of the eight-dimensional spacetime.The description based on the four-dimensional effective field theory is not sufficient to capture the dynamical response of the internal space.As a result, the no-go theorem derived from the lower-dimensional effective potential (3.8) cannot be applied to the eight-dimensional spacetime, where the internal space undergoes dynamical compactification.
It was shown in [47] that the swampland inequality, |∇V | V ≥ λ 1 , related to the no-go theorem discussed above, is a consequence of the fact that the D-dimensional theory satisfies the strong energy condition.It should also hold true in our case.The strong energy condition states that for any timelike vector V M , R M N V M V N ≥ 0, or in any local coordinate systems, R 00 ≥ 0. A simple calculation with the metric (2.17) shows that where Eqs.(2.19) and (2.20) are used again.Substituting (2.18) into (3.9)leads to Therefore, we find that the strong energy condition is satisfied in our model for all k.However, it does not imply that a D-dimensional gravity theory does not admit a de Sitter-like solution which describes an expanding or contracting universe, as we discussed above.
Although we have generalized the instanton-induced inflation and dynamical compactification in Ref. [20] to general Einstein manifolds with positive curvature and also to the FLRW metric with spatial curvature, we have omitted (or postponed) an important class of generalization: Turning on Yang-Mills gauge fields on M 3,1 as well as those on X 4 .We can consider a more general gauge field configuration instead of Eq. (2.9).However, a time-dependent as well as static finite-energy solution does not exist in pure SU(2) Yang-Mills theory on R 3,1 [53,54].Recently it was shown in [55,56,57] that nontrivial solutions with finite energy and finite action can be constructed in the pure SU(2) Yang-Mills theory if the theory is defined on four-dimensional de Sitter dS 4 and anti-de Sitter AdS 4 spaces.Indeed, if we take h(t) = ln cosh(t) with k = 1 and h(t) = ln | sin(t)| with k = −1 in Eq. (2.17), the FLRW metric describes the de Sitter space dS 4 and the anti-de Sitter space AdS 4 , respectively.In particular, since each of the two four-dimensional Yang-Mills theories is conformally invariant for the ansatz (3.11), it is possible to generalize to any conformally flat spacetime such as FLRW if one can find isotropic Yang-Mills solutions A µ (x) on Minkowski, de Sitter, or anti-de Sitter spacetime [57].Therefore it will be interesting to embed the nontrivial solutions in [55,56,57] into the eightdimensional Einstein-Yang-Mills theory (2.5) with the ansatz (3.11).We hope to address this issue in the near future.
Our toy model can be embedded into string theory by introducing a two-dimensional surface Σ such that ten-dimensional spacetime becomes M 8 × Σ (see [20] for a brief discussion of such an embedding).Then it is natural to expect that the inflation of our four-dimensional spacetime has to simultaneously occur with the dynamical compactification of extra dimensions because only the fourdimensional spacetime in M 8 × Σ remains in a macroscopic scale after the end of cosmic inflation.The internal space must be stabilized in some way with a microscopic size after the inflation.We have observed that the internal space in M 8 has a minimum size.Our model is controlled by two parameters denoted by 8πG 8 ρn .These are basically determined by the curvature of an internal Einstein manifold X 4 and the instanton density on X 4 , respectively.Their ratio turns out to be a fundamental scale which is precisely the minimum size of the internal space.Therefore an interesting question is: When the extra dimensions approach to this scale, what happens?For a solution in the X L -branch moving to the right, the (3+1)-dimensional spacetime is expanding whereas the four-dimensional internal space is contracting.Since the internal space X 4 has the minimum size at X = X s which is roughly the size of Planck scale and the warp factor f (t) must be a (strictly) monotonic function for k = 0, 1, the inflation should end near the minimum size while the internal space would be stabilized there.A crude speculation is that there will be a huge quantum back-reaction from the instanton over the tiny internal space when the extra dimensions approach to the minimum size.Then the quantum back-reaction would excite Yang-Mills gauge fields as well as metric fluctuations in the eight-dimensional Einstein-Yang-Mills theory.It is expected that the inflation energy will be transferred to ubiquitous fluctuations and rapidly decreased by producing a large amount of radiations and matters.Ultimately, the cosmic inflation will end and the extra dimensions would be stabilized around the critical size X s .If so, the dynamical stabilization of the internal space may be closely related to the matter generation in our four-dimensional spacetime.This phenomenon would correspond to a reheating mechanism in inflationary cosmology [7,8].
We showed that the open universe with k = −1 (negatively curved) allows a bounce solution.In four-dimensional Einstein gravity coupled with a scalar field, it is known [58] that the existence of bouncing solution depends on the spatial curvature.For a flat or open universe, the possibility of a bounce is precluded by the null energy condition.For a closed universe, on the other hand, the bounce can take place when the curvature term balances with the total energy of the universe.But the bounce solution appeared in the open universe in our case.There is no violation of the null energy condition.This might be possible due to an energy exchange between the four-dimensional spacetime and the internal space and the open universe might be more favorable for the energy exchange.It should be interesting to clarify a detailed mechanism of the bouncing behavior for the k = −1 case.
Another interesting feature in our model is that the homogeneity of internal space is crucial to realize a homogeneous and isotropic universe.We found that the Page space [49] does not admit such universe that satisfies the cosmological principle.In order to incorporate the Page space within our framework, it would be necessary to consider a generalized Bianchi-type metric with a warp factor depending on the extra dimensions too, which may be responsible for an inhomogeneous and anisotropic universe [59].
≡ η i ab and η (−)i ab ≡ η i ab .If we introduce two families of 4 × 4 matrices defined by the matrices in (A.9) provide two independent spin s = 3 2 representations of su(2) Lie algebra.According to the definition (A.1), they are explicitly given by The relations in (A.6) and (A.7) immediately show that τ i ± satisfy su(2) Lie algebras, i.e.,

B Yang-Mills instantons from Einstein manifolds
In this appendix, we show in detail how to construct the explicit solution of Yang-Mills instantons on Einstein manifolds with positive curvature, namely X 4 = S 4 , CP 2 , S 2 × S 2 and Page.We will also check whether these manifolds admit a constant instanton density.Let us briefly recapitulate the method in [15,16] and apply it to find the instanton solution on a general Einstein manifold X 4 .Consider an Einstein manifold X 4 which may be one of S 4 , CP 2 , S 2 × S 2 and Page.The metric on X 4 takes the form Using the metric, one can calculate the spin connection ω ab = ω abα dy α and curvature tensor R ab = 1 2 R abαβ dy α ∧ dy β by solving the structure equations [11,12] T a = de a + ω ab ∧ e b = 0, (B.2) The underlying idea is that Einstein gravity can be formulated as a gauge theory of the Lorentz group, where spin connections play a role of gauge fields and Riemann curvature tensors correspond to their field strengths.Another important point is that Riemann curvature tensors, denoted as R ab , are so(4)valued two-forms in Ω 2 (X 4 ) = Λ 2 T * X 4 .These facts are combined with the well-known theorems [14]: Self-duality.On an orientable Riemannian four-manifold, the 2-forms decompose into the vector spaces of self-dual and anti-self-dual 2-forms, defined by the ±1 eigenspaces of the Hodge star operator * : Ω 2 → Ω 2 .Lie group isomorphism.There is a global isomorphism for the four-dimensional Lorentz group, i.e., SO(4) = SU(2) + ⊗ SU(2) − /Z 2 .It also leads to the splitting of the Lie algebra Indeed, these two decompositions are deeply related to each other due to the canonical vector space isomorphism between the Clifford algebra Cl(4) in four dimensions and the exterior algebra We can apply these two decompositions to the spin connection and curvature tensor.First let us apply the decomposition (B.5) to spin connections: where η i ab and η i ab are the 't Hooft symbols satisfying the self-duality relation See appendix A for the algebraic identities for the 't Hooft symbols.Thus the spin connections are split into a pair of SU(2) + and SU(2) − gauge fields.Note that the index i = (1, 2, 3) refers to the su(2) ± Lie algebra index.Accordingly the Riemann curvature tensors R ab are also decomposed into a pair of SU(2) + and SU(2) − field strengths: where ab e a ∧ e b = dA (±)i − ε ijk A (±)j ∧ A (±)k .The second decomposition (B.4) is that the six-dimensional vector space of two-forms splits canonically into the sum of three-dimensional vector spaces of self-dual and anti-self-dual two forms: where each component satisfies the self-duality equation according to Eq. (B.7).Recently, the decomposition (B.9) has been used significantly in [51] to study the scalar invariants of the curvature tensor.One can show [16] that, if X 4 is an Einstein manifold obeying the equation R αβ = λh αβ with λ a cosmological constant, That is, SU(2) + and SU(2) − field strengths in the Riemann curvature tensor (B.9) satisfy the selfduality equations Using the Riemannian metric over X 4 in Eq. (B.1), the self-duality equations (B.11) can be expressed in a curved frame as which is exactly the same as the instanton equation (2.12).

B.1 S 4
The de Sitter metric on X 4 = S 4 with radius R = ζc 2 is The orthonormal vierbeins e a (a = 1, 2, 3, 4) in the metric (B.13) are defined by where σ i (i = 1, 2, 3) are left-invariant 1-forms on the group manifold SU(2) ∼ = S 3 and satisfy the exterior algebra It is straightforward to calculate the spin connection by solving the torsion-free condition (B.2).They read where and f ′ = df dr .Using the definition (B.6), the corresponding SU(2) gauge fields can easily be read off from the spin connections and they are given by where we have used the explicit representation of the 't Hooft symbols.The SU(2) field strengths are determined by the gauge fields (B.17) as It is straightforward to check the self-duality (B.11).
It may be instructive to point out that the SU(2) gauge fields in (B.17) coincide with the instanton solution on R 4 .(See Remark 2, p. 296, in [12].)Using the coordinates on R 4 , the left-invariant 1-forms σ i on S 3 are represented by [52] and then the gauge fields A (+)i in (B.17), for example, are equal to where the diameter 2R of S 4 is identified with the instanton size ζ c .These gauge fields precisely correspond to the gauge fields of a self-dual SU(2) instanton on R 4 [10,13].It is not merely a coincidence.The self-duality equation (2.12) in four dimensions is conformally invariant, extending to the conformal compactification S 4 = R 4 ∪ {∞}.Consequently, the solution on S 4 is the same as the finite-action solution on R 4 .This is also related to the fact that the single instanton bundle is the Hopf fibration of S 7 [12].
For the instanton field strengths in Eqs.(B.18) and (B.19), we get This result exactly coincides with Eq. (3.19) in [20] (see footnote 3).So we confirm that the Yang-Mills instanton obtained from the spin connection of S 4 admits a constant density.

B.2 CP 2
The metric on CP 2 takes the form where f (r) = 1 + λr 2 6 and the orthonormal vierbeins are defined by The spin connections read as The corresponding SU(2) gauge fields can be read off from the spin connections as The field strengths are determined from the above gauge fields as In particular, the self-dual sector is an Abelian gauge field.The self-duality (B.11) is manifest.
In order to check whether the instanton density, ρ  The field strengths are determined by the above gauge fields as , (B.30) .

(B.31)
The self-duality (B.11) requires that R 1 = R 2 .For the above field strengths, we obtain Thus we get a constant instanton density even if

B.4 Page
The final example is the Page space X 4 = CP 2 ♯CP 2 which is an S 2 bundle over S 2 [49].It is constructed by attaching the complex projective space and its complex conjugate, removing a 4-ball from each manifold, and gluing them together along the resulting S 3 boundaries.The metric on the Page space takes on a complicated form where and ν is the positive root of ν 4 + 4ν 3 − 6ν 2 + 12ν − 3 = 0 which works out to be 0.2817.The orthonormal vierbeins are defined by Here we choose λ = 3 1+ν 2 3+ν 2 for simplicity.After a little algebra, we get the spin connections The corresponding SU(2) gauge fields are given by It is straightforward (but a bit tedious) to calculate the SU(2) field strengths:

B.5 Squashed S 4
In this example, we aim to understand the importance of the homogeneity of an internal space X 4 in achieving a constant instanton density on X 4 .We examine another nonhomogeneous space obtained by deforming a round sphere S 4 whose deformation is parameterized by three axis-length parameters (r, l, l).The metric of squashed 4-sphere is given by ds 2 = e a ⊗ e a with the vierbeins [60]  ).The field strengths in Eq. (B.38) are neither self-dual nor anti-self-dual, so the squashed foursphere is not an Einstein manifold.According to the result SU(2) gauge fields constructed from a four-manifold X 4 become self-dual or anti-self-dual only if X 4 is Einstein, i.e., obeying the vacuum Einstein equation, R αβ = λh αβ [16].As a result, SU(2) field strengths in Eq. (B.38) do not satisfy the self-duality (B.11).Therefore the squashed four-sphere would be relevant as an internal manifold only when matters are coupled with gravity.It is straightforward to see how the instanton density behaves for the squashed four-sphere.The instanton density for the SU(2) field strengths in (B.38) exhibits a nontrivial distribution as expected: where e 1 ∧ e 2 ∧ e 3 ∧ e 4 = √ hd 4 y is the volume form on the squashed four-sphere.When l = l = r, it becomes the round four-sphere S 4 which is an Einstein manifold.In this case, the SU(2) gauge fields in (B.37) become instanton connections and the instanton density (B.39) becomes constant, i.e., F (±)i ∧ F (±)i = ± 3 2r 4 √ hd 4 y which is consistent with (B.21) in the unit R = r.But the squashed S 4 has a non-uniform distribution.This result supports our claim that the instanton density on a nonhomogeneous manifold is not uniform in general.

. 31 )Figure 3 :
Figure 3: Solution flows with k = 1.The middle and right figures show the projection onto constants X = 0 and X = 0.5, respectively.The size of arrow head in the left figure denotes the rapidity of time evolution.

Figure 4 :
Figure 4: Solution flows with k = −1.The middle and right figures show the projection onto constants X = 0 and X = 0.5, respectively.These flows contain bouncing solutions.
y)dy α dy β .(3.5) Starting with the eleven-dimensional supergravity (D = 11), the effective potential in d dimensions has been obtained by the compactification to (11 − d)-dimensional curved internal manifolds.After the Weyl scaling g

3 2 l 2 l2 r 4 f 6 g 6 √
hd 4 y, (B.39) Yang-Mills theory consists of the Einstein equations (2.6) and the Yang-Mills equations (2.8).For the ansatz (2.1) and (2.9), the Yang-Mills equations (2.8) reduce to (2.11) and the Einstein equations (2.15) depend only on the instanton density which must be constant for the existence of a solution.If one can construct an SU(2) Yang-Mills instanton on a general Einstein manifold X 4 that obeys Eq. (2.12), the Yang-Mills equations (2.11) are automatically satisfied.
Therefore, it is enough to check that the SU(2) Yang-Mills instanton on an Einstein manifold X 4 provides a constant instanton density ρ (±)