Induced moduli oscillation by radiation and space expansion in a higher-dimensional model

We investigate the cosmological expansion of the 3D space in a 6D model compactified on a sphere, beyond the 4D effective theory analysis. We focus on a case that the initial temperature is higher than the compactification scale. In such a case, the pressure for the compact space affects the moduli dynamics and induces the moduli oscillation even if they are stabilized at the initial time. Under some plausible assumptions, we derive the explicit expressions for the 3D scale factor and the moduli background in terms of analytic functions. Using them, we evaluate the transition times between different cosmological eras as functions of the model parameters and the initial temperature.


Introduction
The existence of the extra dimensions is predicted in string theory.Since the experimental constraints on the size of the extra dimensions that the gravity feels are much weaker than those for the standard model particles, there is a wide allowed parameter space for braneworld models with relatively large extra dimensions [1,2,3].This type of models have been considered not only as a solution to the large hierarchy problem, but also as a solution to the cosmological constant problem [4].Recently, it is also pointed out that a micron-size large extra dimension may be predicted by the swampland conjecture [5], which is called the dark dimension scenario [6].If a large extra compact space exists, it affects the cosmological history at early times.In particular, the temperature T tmp in the radiation-dominated era can be higher than the compactification scale m KK since the latter has a small value in such a case.
In our previous papers [7,8], we studied the time evolution of the space and the background values of the moduli in a six-dimensional (6D) model compactified on a sphere S2 by solving the 6D field equations numerically. 1 We found that when the initial temperature of the universe is higher than m KK , the expansion rate for the three-dimensional (3D) non-compact space deviates from that of the usual 4D cosmology.In general, the 3D scale factor e A evolves as t 2/(3(1+w)) , where t is the cosmological time and w is the ratio of the pressure to the energy density.In the radiation-dominated era, w −1 measures the dimensions that the radiation feels.In fact, when T tmp > m KK , the radiation feels the whole five-dimensional (5D) space and e A ∝ t 5/9 .As the universe expands and the temperature goes down, the radiation gradually ceases to feel the compact space, and w −1 approaches to three after T tmp gets lower than m KK .Then, the expansion rate slows down to e A ∝ t 1/2 .We also found that even if the moduli are stabilized at the initial time, the moduli oscillation is induced by the pressure for the two-dimensional (2D) compact space p rad 2 .This effect cannot be discussed in the conventional 4D effective theory approach since p rad 2 is absent in the 4D Einstein equation.When T tmp > m KK , the effect of p rad 2 on the moduli dynamics cannot be neglected.If the lifetime of the moduli is long enough, the induced moduli oscillation eventually dominates the energy density and the 3D space expands as e A ∝ t 2/3 at later times.Therefore, there are the following eras in this setup.
1. 6D radiation-dominated era (e A ∝ t 5/9 ) 2. 4D radiation-dominated era (e A ∝ t 1/2 ) 3. (Induced) moduli-oscillation-dominated era (e A ∝ t 2/3 ) The era 3 will end by the decay of the moduli, and transition into the 4D radiation-dominated era again [15].After that, the universe behaves as the standard cosmology.Let us denote the transition time from the era 1 to the era 2 as t rad , and that from 2 to 3 as t mod .In principle, the spacetime evolution is determined once the model parameters and the initial conditions are provided.However, since these results are obtained by the numerical computations in the previous works, we cannot directly see how the transition times t rad and t mod depend on the initial parameters.Besides, it is difficult to pursue the whole history of the universe due to the limitation of the computational power.
In this paper, we derive approximate expressions for the 3D scale factor, the moduli background values and the transition times in terms of analytic functions by solving the 6D evolution equations under some approximations.Since we can discriminate the eras by the power p, which is defined as e A ∝ t p in each era, we will focus on the change of p during the spacetime evolution.The expressions derived in this paper enable us to pursue the spacetime evolution until much later times than the previous works, and to clarify the dependence of the transition times t rad and t mod on the model parameters and the initial temperature.These results will help make discussions transparent.
The paper is organized as follows.In Sec. 2, we briefly explain our setup and show the evolution equations.In Sec. 3, we derive analytic expressions for various quantities by solving the evolution equations under some plausible approximations.We then define the effective power p and derive its explicit expression using the functions we have defined.In Sec. 4, we discuss the time evolution of p, and estimate the transition times.Sec. 5 devoted to the summary.In Appendix A, brief derivations of the energy density and the pressures for the radiation are shown.In Appendix B, we derive the evolution equation for the temperature from the conservation law of the energy-momentum tensor.In Appendix C, we provide a general solution to the inhomogeneous differential equation that describes the moduli oscillation.

Setup
We consider a 6D model used in our previous works [7,8].In this section, we briefly review the model and the evolution equations for the universe.The spacetime is compactified on a 2D sphere S2 .As coordinates on S 2 , we choose the spherical ones (x 4 , x 5 ) = (θ, ϕ), where θ and ϕ are the polar and the azimuthal angles, respectively.
The action is given by where M, N = 0, 1, • • • , 5 denote the 6D indices, g (6) is the determinant of the 6D metric tensor, R (6) is the 6D Ricci scalar, σ is a real scalar, gauge field A M , and g gc is the gauge coupling constant.The scalar potential V pot (σ) is given by where m and σ * are positive constants.Except for the second term in (2.2), the action (2.1) can be embedded into a gauged 6D N = (1, 0) supergravity [9,16].We add the second term in order to stabilize the moduli completely.
In this paper, we neglect effects of the 3-branes, one of which the standard model particles live 3 , and assume that the background spacetime has homogeneity and isotropy for 3D noncompact space and a spherical symmetry for S 2 .Thus, we take the following ansatz for the background fields.
In the absence of the radiation in the bulk, the model has the following static solution.
and the Kaluza-Klein (KK) mass scale is given by5 (2.5) In addition to the above field content, we introduce the radiation in the bulk.In 6D N = 1 supergravity, the number of hypermultiplets n H and that of vector multiplets n V are constrained by the anomaly cancellation condition n H − n V = 244 [16,20,21]. 6Therefore, at least 245 hypermultiplets exist in the bulk.Since each hypermultiplet has four bosonic and four fermionic degrees of freedom, we assume that the degrees of freedom for the radiation is g dof = 2000 in this paper.Due to the isometries of the spacetime, the energy-momentum tensor for the radiation has the form of where ρ rad , p rad 3 and p rad 2 are the radiation energy density, the pressures in the non-compact 3D space and in the compact 2D space, respectively.Their explicit forms are listed in Appendix A.
In the presence of the radiation, the static solution (2.4) is no longer a solution of the field equations, and the universe continues to expand.Substituting the background ansatz (2.3) and (2.6) into the 6D Einstein equations and the dilaton field equation, we obtain the evolution equations for the background fields, which are summarized as with the constraint, We have used the relation (A.11).
The energy density and the pressures are expressed as (see Appendix A)7 where β ≡ 1/T tmp is the inverse temperature, the functions Q 1 (x) and Q 2 (x) are defined in (A.6) and (A.8) respectively, and their arguments are e −B β.The evolution equation for β is obtained from the conservation law for the energy-momentum tensor as where Q 3 (x) is defined in (B.6).(See Appendix B.) The profiles of x 2 Q i (x) (i = 1, 2, 3) are shown in Fig. 16.
We consider a situation that the moduli B and σ have already been stabilized at t = 0. Hence, we choose the initial conditions at t = 0 as where β I is a positive constant.The value of Ȧ(0) is determined by the constraint (2.8).
3 Induced moduli oscillation and 3D scale factor As pointed out in our previous work [8], the pressure for the compact space S 2 , p rad 2 , pushes out the moduli from the potential minimum, and induces an oscillation of the moduli around the stabilized values in (2.4).Namely, even in the case that the moduli have been settled at the stabilized point before the radiation-dominated era, they will start to oscillate again.This effect cannot be neglected if the temperature is high enough compared to m KK .
In order to see this behavior, we will see the time evolution of the moduli at early times.We assume that the radiation dominates the energy density at t = 0, and the initial temperature is higher than m KK (i.e., βI ≡ e −B * β I ≪ 1).Since we are interested in the oscillation around the stabilized values in (2.4), we define B ≡ B − B * and σ ≡ σ − σ * .The mass eigenstates are linear combinations of them, which are defined as The evolution equations for them are derived from (2.7) as where the ellipsis denotes higher order terms in φ 1 or φ 2 , and We have neglected terms involving Ä, which are assumed to be small at initial times.When m 2 ≫ m 2 KK , for example, these become λ 1 ≃ m 2 KK , λ 2 ≃ m 2 and θ ≃ λ 1 /λ 2 .In general, it is hard to solve (2.7) analytically because A, B and σ are coupled to each other.However, due to the assumption that the radiation is dominated at initial times, the expansion of the 3D space is determined only by ρ rad , and is almost independent of the moduli.Hence, we can treat the 3D expansion and the moduli oscillation separately.In fact, from (A.13) and (B.7), the energy density and the pressure for the compact space S 2 are approximately written as which are independent of the moduli.Notice that Ȧ ≫ | Ḃ| at the very early times because of the initial condition (2.11).Then, from (2.10) and (B.7), we obtain which is immediately solved as β ≃ β I e 3 5 A .Thus, (3.4) is rewritten as where From (2.8), we have which leads to Therefore, we have where ).This approximation is valid when β ≡ e −B * β < 1.This condition is translated as We have used that βI ≪ 1.Using (3.9), (3.2) is rewritten as From (2.11), the initial conditions at t = 0 are read off as The solution is expressed as Im e Im e Im e Im e (See Appendix C.) The function U q (t; λ, C) is defined by the incomplete gamma function as (C.7).From these and (3.9), we obtain an approximate solution of the moduli evolution equations at early times.We have checked that this approximate solution agrees well with the solution of the full evolution equation (2.7) obtained by the numerical computation.As we will see in the next subsection, it is convenient to define 4. × 10 -7 6. × 10 -7 8. × 10 -7 1. × 10 -6  Since | B(t)| ≪ A(t) except for an early short period 0 , e Ā can be understood as a modified 3D scale factor.The mixing term between Ȧ and Ḃ in the constraint (2.8) is absorbed into Ȧ2 , and (2.8) is rewritten in a similar form to the 4D Friedmann equation. 8 Ȧ2 = ρ mod + ρ rad , ( where represents the energy density of the moduli oscillation.The moduli energy density ρ mod is expressed in terms of φ 1 and φ 2 as where the ellipsis denotes higher order terms in φ 1 or φ 2 .Using 3.13, we can plot e 3 Āρ mod (t) as Fig. 1.From this plot, we can see that e 3 Āρ mod is almost a constant for t > t ref , where We have used (C.8).Namely, ρ mod decays as Here, we comment on the validity of the above approximations.We have used that ρ rad ≫ ρ mod to obtain (3.9).When this condition is satisfied, the ratio of ρ mod to ρ rad is where φ 1 (t) and φ 2 (t) are given by (3.13), and Ȧ(t) = 5 √ C A / 9 1 + √ C A t .Note that the function r m/r (t) is determined when the model parameters m, σ * and the initial condition β I are given.This ratio is plotted in Fig. 2. We can see from the plots that the approximate solution in (3.13) is no longer valid when t ≃ t ref in the case of β I = 10, m = 0.01 and σ * = 14.As a general property, the ratio ρ mod /ρ rad takes a larger value for higher initial temperature or for shallower moduli potential (i.e., smaller m or larger σ * ).When ρ mod approaches to ρ rad , the expansion rate for the 3D space becomes larger than (3.7), and the inhomogeneous term in (3.2) decays more rapidly than (3.11).After the inhomogeneous term becomes negligible, the solution will reduce to a linear combination of two simple harmonic oscillators, and e 3A ρ mod becomes constant.Thus, the constant C mod in (3.20) takes a smaller value than (3.19) if r m/r is close to one before t = t ref .

Smoothed 3D scale factor
In the usual 4D cosmology, the 3D scale factor evolves as e A(t) ∝ t 1/2 in the radiationdominated era and as e A(t) ∝ t 2/3 in the matter-dominated era.Thus, it is convenient to define the effective power p as p ≡ t Ȧ.Then, p = 1/2 (2/3) in the radiation-(matter-) dominated era.However, as we pointed out in Ref. [8], this quantity oscillates due to the effect of the moduli oscillation (see Fig. 3).Thus, we modify the definition of p as where Ā is defined by (3.14).The constant t c is chosen so that p is almost independent of t at early times.We will show how to choose t c in Sec.3.4.As we can see from Fig. 3, this modification removes the effect of the moduli oscillation [8].
For t ≥ t ref , the radiation energy density in (2.9) is approximated as where β ≡ e −B * β = m KK β is the inverse temperature in the unit of the KK mass m KK , and The evolution equation for β (2.10) is approximated as where  where Āref and βref denote the values at t = t ref , and The profile of F(x) is shown in Fig. 5.

Expression of effective power p
Here, we derive an explicit expression for the effective power p.From (3.15) and (3.20), we have 9 C mod e −3 Ā + ρ rad , (3.29) 9 We focus on the positive solution of Ȧ since we are interested in an expanding 3D universe.which leads to where We have used (3.27) at the last step.As shown in Fig. 6, R(x) is well approximated as a linear function.
Thus, the above expression can be approximated as

Choice of t c
As we mentioned, the constant t c is determined so that p is almost independent of t at early times where .
(3.37) Fig. 7 shows the profiles of G(x)/x 2 and H(x)/x 3 .Thus, when t ∼ t ref , (3.35) is approximated as  Then, (3.38) becomes For β ≪ 1, the functions we have defined behave as β4 . (3.41) By assumption, β ≪ βmod .Thus, p( β) behaves as10 which is independent of β (or t).Hence, the choice of t 0 in (3.39) is appropriate.Using this choice of t 0 , (3.35) becomes Combining this with (3.33), we can plot p as a function of t.

Time evolution of 3D space
In this section, we discuss the expansion of the 3D space by evaluating the time evolution of the effective power p.

Parameter dependence of effective power p
When β ≪ βmod , (3.33) and (3.43) reduce to From Fig. 8, we can see that the power p changes its value from 9/5 to 1/2 during the period 2 < β < 10.If βmod ≫ 15, the 3D space expands as in the 4D radiation-dominated era until β approaches to βmod .
When β ≫ βmod , on the other hand, the contribution of the moduli oscillation dominates the energy density, and p( β) in (3.43) can be estimated as where we have used that for x ≫ 1.Thus, if we define the power p changes from 9/5 to 1/2 around t = t rad , and from 1/2 to 2/3 around t = t mod .We show a typical profile of the function p(t) in Fig. 9.The parameters are chosen as σ * = 14, βref = 10 −2 and R mod rad = 10 −5 .As we can see from (3.33)As this plot shows, the value of σ * (i.e., C rad ) just shifts the profile to the time direction without changing its shape.

βref -dependence
The left plot in Fig. 11 shows the profile of p(t) for various values of βref .For βref ≲ 1, the profile of p(t) is almost independent of βref , and the initial value of p(t) for t ≪ t rad is 9/5, which is the value of the 6D radiation-dominated universe.For 2 < βref < 20, the value of p(t) at early times (t ≪ t rad ) decreases as βref increases.This can be understood from Fig. 8.A larger value of βref in this region indicates that the temperature is not high enough for the radiation to feel the compact space S 2 completely, and the 3D space expands less rapidly.For βref > 15, the radiation no longer feels the compact space, and the expansion rate of the 3D space is almost the same as the 4D radiation-dominated one.
The right plot in Fig. 11 shows that a small change of β corresponds to a large change of t in early times.This explains the plateau for t ≪ t rad in the left plot.
R mod rad -dependence Fig. 12 shows the R mod rad -dependence of p(t).Recall that R mod rad defined in (3.31) parame- terizes the ratio of the energy density for the moduli-oscillation to that for the radiation at t = t ref .
Since the latter energy density decreases faster than the former, the former eventually dominates the total energy density at late times, and p will approach to 2/3.The parameter R mod rad determines t mod , at which the moduli-oscillation contribution starts to dominate.For a smaller value of R mod rad , it takes more time to dominate the total energy density, and thus t mod becomes larger.When R mod rad > 1, on the other hand, the moduli oscillation will dominate the energy density before the universe behaves as the 4D radiation-dominated one.

Estimation of transition times
So far, we have worked in the 6D Planck unit.For phenomenological discussions, however, it is more convenient to translate the physical quantities in the 4D unit.First, let us restore the dependence of the 6D Planck mass M 6 .
The 4D Planck mass M 4 is defined after the extra dimensions are stabilized, and is related to the 6D Planck mass M 6 as where V 2 * ≡ 4π(e B * l 6 ) 2 (l 6 ≡ M −1 6 : 6D Planck length) is the volume of the compact space S 2 after the moduli stabilization.Thus, the quantities in (4.5) are expressed as 6) = 1.46 × 10 −18 e B * t (6) GeV −1 = 8.61 × 10 −42 e B * t (6) sec, where t (6) and β (6) are the values of the time and the inverse temperature measured by M 6 .Fig. 13 shows the transition times t rad and t mod defined in (4.4) as functions of the initial inverse temperature normalized by the KK mass scale βI ≡ m KK β I .The solid, dashed and dotted lines correspond to the case of σ * = 10 (m KK = 1.2 × 10 14 GeV), 13 (6.2 × 10 12 GeV) and 16 (3.1 × 10 11 GeV).From this plot, we see that t mod increases as the initial inverse temperature βI increases.This can be understood by noting that the induced moduli oscillation has a larger amplitude for high initial temperature.Namely, for a large value of βI (i.e., low initial temperature), the pressure p rad 2 is small and the induced moduli oscillation has a small amplitude, which leads to a small value of R mod rad .As shown in Fig. 12, this means that the moduli oscillation dominates the energy density at later time.In contrast, the transition from the 6D to 4D radiation-dominated eras occurs when the moduli-oscillation energy density is negligible.Therefore, t rad is almost independent of βI , as can be seen from the plot.Next, we see the dependence of the mass parameter m in the moduli potential (2.2).For a smaller value of m, the potential becomes shallower, and the moduli can move from the potential minimum (2.4) by the pressure p rad 2 more easily.Therefore, the amplitude of the moduli oscillation becomes larger, and the value of R mod rad increases.As a result, we have a smaller value of t mod .This behavior can be seen in Fig. 14, which shows the dependence of t mod on m.
From (4.1) and (3.23), the transition time t rad in the M 6 unit is approximated as Here, we have assumed that βmod ≫ βref and used that for x ≫ 1.Thus, t mod in the unit of second is Since we are considering the case that βref ≪ 1, we have from (3.9).Plugging these and (3.19) into (4.13),we can estimate the value of t mod .

Moduli decay
The moduli-oscillation-domination era will end by the decay of the moduli.After the lifetime of the moduli t dc , the moduli oscillation is converted into radiation.In this subsection, we will see this effect.
If we introduce the effect of the moduli decay, and (3.20) and (4.17) are modified as where Γ mod ≡ 1/t dc is the total decay rate of the moduli.Recall that where t c is given by (3.39).Now, we numerically evaluate p at each time.Denote the value of a quantity q at βi ≡ βref e ∆i , where ∆ ≪ 1 is a small positive constant, as q i .Then, we have the following recurrence relations.
At the last equality, we have used that

.22)
Using these quantities, the effective power p at t = t i is calculated as Fig. 15 shows the effective power p as a function of t in the unit of second.As expected, p rapidly decreases to the radiation-dominated value 1/2 at t = t dc .

Summary
We investigated the cosmological expansion of the 3D space in a model with two compact extra dimensions by solving the 6D evolution equations.We assumed that the whole 5D space is filled with the radiation and the moduli have already been stabilized at the initial time.In contrast to the conventional 4D effective theory analysis, the 6D evolution equations involve the pressure for the compact extra dimensions p rad 2 .When the temperature of the universe is higher than the compactification scale m KK , the pressure p rad 2 affects the moduli dynamics.In our previous work [8], we found that p rad 2 pushes out the moduli from the potential minimum, and induces the moduli oscillation.If the moduli lifetime is long enough, the oscillation will eventually dominate the energy density at late times.In that case, the 3D space expands as e A ∝ t 2/3 .If the temperature of the universe is higher than m KK , the radiation feels the whole 5D space, and the 3D space expands as e A ∝ t 5/9 .When the temperature goes down below m KK , the radiation ceases to feel the extra dimensions, and the expansion rate slows down as e A ∝ t 1/2 , which is the expansion law of the 4D radiation-dominated universe.In order to pursue these changes of the expansion rate, we define the effective power p in such a way that the 3D scale factor behaves as e A ∝ t p for each era.The nontrivial expansion of the 3D space is parameterized by two transition times t rad and t mod .The effective power p changes from 5/9 to 1/2 around t = t rad , and from 1/2 to 2/3 around t = t rad .
In our previous works [7,8], we evaluated the 3D scale factor by numerical computation.However, it is not easy to see how the transition times t rad and t mod depend on the model parameters and the initial temperature in such a numerical approach.Besides, we cannot pursue the whole history of the universe in this method due to the limitation of the computational power.In this paper, we derive analytic expressions for the 3D scale factor, the inverse temperature and the background moduli values by solving the 6D evolution equations under appropriate approximations, and provide analytic expressions for the transition times as functions of the model parameters m, σ * and the initial (inverse) temperature β I .The expressions we obtained enable us to pursue the cosmological evolution until much later times.
The first transition time t rad is determined solely by σ * (or m KK ), and is almost independent of the initial temperature.The second transition time t mod , on the other hand, depends on both the moduli potential scale m and the temperature.This is because t mod is determined by the oscillation amplitude induced by p rad 2 .The amplitude becomes larger for a shallower potential (i.e., a smaller value of m or a larger value of σ * ) or for higher initial temperature (i.e., a smaller value of β I ), and then the moduli oscillation dominates the energy density earlier (a smaller value of t mod ).
As shown in Ref. [7], the modulus B continues to increase for σ * ≳ 16, and the observed 4D universe cannot be obtained.Therefore, there is an upper bound for the stabilized value of the S 2 radius in our model.It is intriguing to study whether this is common to other models with extra dimensions or not.
For more realistic discussions, we need to extend our setup by including the inflaton sector.Our initial conditions in (2.11) with the radiation-domination assumption should be realized by the reheating process after the inflation.In such setups, the effective power p will enter the expression of the e-folding number for the 3D scale factor.
We will discuss these issues in separate papers.

Figure 5 .
Figure 5.The profile of F(x).The dotted line represents ln x + 0.833.

Figure 8 .
Figure 8.The profile of the function in (4.1).

Figure 9 .
Figure 9.The effective power p as a function of t.The parameters are chosen as σ * = 14, βref = 10 −2 and R mod rad = 10 −5 .The left and right vertical dashed lines denote t = t rad and t = t mod , respectively.

Figure 13 .Figure 14 .
Figure13.The transition times t mod and t rad defined in (4.4) as functions of βI ≡ m KK β I .The unit of the vertical axis is seconds.The solid, dashed and dotted lines correspond to the case of σ * = 10, 13 and 16, respectively.The upper (lower) line represents t mod (t rad ).The mass parameter m is chosen as m = 0.01.
Using this expression, the effective power p defined in(3.22) is expressed as and (3.43), the function p(t) depends on the parameters only through t ref , C rad , βref and R mod rad .Among them, we choose t ref much smaller than the second term of t( β) in(3.33), and thus its dependence can be neglected.Let us see the dependences on the other three parameters individually.
C rad -dependence From (3.23), C rad is determined only by σ * (or m KK ), and it only affects the overall time scale if t ref is negligible (see (3.33)).Fig. 10 shows the profile of p(t) for different values of σ * .The solid, dashed and dotted lines correspond σ * = 10, 13 and 16, respectively.