Effects of spatial curvature and anisotropy on the asymptotic regimes in Einstein–Gauss–Bonnet gravity
Abstract
In this paper we address two important issues which could affect reaching the exponential and Kasner asymptotes in Einstein–Gauss–Bonnet cosmologies—spatial curvature and anisotropy in both three and extradimensional subspaces. In the first part of the paper we consider the cosmological evolution of spaces that are the product of two isotropic and spatially curved subspaces. It is demonstrated that the dynamics in \(D=2\) (the number of extra dimensions) and \(D \ge 3\) is different. It was already known that for the \(\Lambda \)term case there is a regime with “stabilization” of extra dimensions, where the expansion rate of the threedimensional subspace as well as the scale factor (the “size”) associated with extra dimensions reaches a constant value. This regime is achieved if the curvature of the extra dimensions is negative. We demonstrate that it takes place only if the number of extra dimensions is \(D \ge 3\). In the second part of the paper we study the influence of the initial anisotropy. Our study reveals that the transition from Gauss–Bonnet Kasner regime to anisotropic exponential expansion (with three expanding and contracting extra dimensions) is stable with respect to breaking the symmetry within both three and extradimensional subspaces. However, the details of the dynamics in \(D=2\) and \(D \ge 3\) are different. Combining the two described effects allows us to construct a scenario in \(D \ge 3\), where isotropization of outer and inner subspaces is reached dynamically from rather general anisotropic initial conditions.
1 Introduction
Extradimensional theories had been known [1] even prior to General Relativity (GR) [2], but relatively well known they have become after work by Kaluza and Klein [3, 4, 5]. Since then the extradimensional theories evolved a lot but the main motivation behind them remains the same—unification of interactions. Nowadays one of the promising candidate for a unified theory is M/string theory.
While considering extradimensional theories, regardless of the model, we need to explain where the additional dimensions are. Indeed, with our current level of experiments, we clearly sense three spatial dimensions and sense no presence of extra dimensions. The common explanation is that they are “compactified”, meaning that they are so small that we cannot detect them. Perhaps, the simplest class of such theories are the theories with “spontaneous compactification”. Exact solutions of this class have been known for a long time [13], but especially relevant for cosmology are those with a dynamical size of the extra dimensions (see [14, 15, 16, 17] for different models). Notable recent studies include [18], where dynamical compactification of the (5+1) Einstein–Gauss–Bonnet model was considered; [19, 20], where different metric Ansätze for scale factors corresponding to (3+1) and extradimensional parts were studied, and [21, 22, 23], where we investigated general (e.g., without any Ansatz) scale factors and curved manifolds. Also, apart from cosmology, the recent analysis has focused on properties of black holes in Gauss–Bonnet [24, 25, 26, 27, 28] and Lovelock [29, 30, 31, 32, 33] gravities, features of gravitational collapse in these theories [34, 35, 36], general features of sphericalsymmetric solutions [37], and many others.
When it comes to exact cosmological solutions, two most common Ansätze used for the scale factor are exponential and power law versions. Exponential solutions represent de Sitter asymptotic stages while powerlaw solutions represent Friedmannlike stages. Powerlaw solutions have been analyzed in [14, 38] and more recently in [39, 40, 41, 42, 43] so that by now there is an almost complete description of the solutions of this kind (see also [44] for comments regarding physical branches of the powerlaw solutions). One of the first considerations of the extradimensional exponential solutions was done by Ishihara [45]; later considerations included [46], as well as the models with both variable [47] and constant [48] volume; the general scheme for constructing solutions in EGB gravity was developed and generalized for general Lovelock gravity of any order and in any dimensions [49]. Also, the stability of the solutions was addressed in [50] (see also [51] for stability of general exponential solutions in EGB gravity), and it was demonstrated that only a handful of the solutions could be called “stable”, while most of them are either unstable or have neutral/marginal stability.
If we want to find all possible regimes in EGB cosmology, we need to go beyond an exponential or powerlaw Ansatz and keep the scale factor generic. We are particularly interested in models that allow for dynamical compactification, so that we consider the spatial part as the warped product of threedimensional and extradimensional parts. In that case the threedimensional part is “our Universe” and we expect this part to expand, while the extradimensional part should be suppressed in size with respect to the threedimensional one. In [21] we demonstrated the existence of a regime when the curvature of the extra dimensions is negative and the Einstein–Gauss–Bonnet theory does not admit a maximally symmetric solution. In this case both the threedimensional Hubble parameter and the extradimensional scale factor asymptotically tend to constant values. In [22] we performed a detailed analysis of the cosmological dynamics in this model with generic couplings. Later in [23] we studied this model and demonstrated that, with an additional constraint on the couplings, Friedmann latetime dynamics in the threedimensional part could be restored.
The current paper is a natural continuation of our previous research on the properties of cosmological dynamics in EGB gravity. After a thorough investigation of spatially flat cases in [52, 53, 54], it is natural to consider spatially nonflat cases. Indeed, the spatial curvature affects inflation [64, 65], so that it could change asymptotic regimes in other highenergy stages of the Universe evolution, and we are considering one of them. We already investigated the cases with negative curvature of the extra dimensions in [21, 22, 23], but to complete our description it is necessary to consider all possible cases. We are going to consider all possible curvature combinations to see their influence on the dynamics—we know the regime for the case with both subspaces being spatially flat and will see the change in the dynamics with the curvatures being nonflat. This allows us to find all possible asymptotic regimes in the spatially nonflat case; together with the results for the flat case, it will complete this topic.
Another important issue we are going to consider is the anisotropy within subspaces. Indeed, the analysis in [52, 53, 54] is performed under the conjecture that both three and extradimensional subspaces are isotropic. The question is if the results are stable under small (or not very small) deviations of the isotropy of these subspaces. Finally, if we consider both effects, we could build a twostep scheme which allows us to qualitatively describe the dynamical compactification of an anisotropic curved spacetime.
The structure of the manuscript is as follows: first we write down the equations of motion for the case under consideration. Next, we study the effects of curvature—we add all possible curvature combinations to all known existing flat regimes and describe the changes in the dynamics. After that we draw conclusions separately for the vacuum and the \(\Lambda \)term regimes and describe their differences and generalities. After that we investigate the effects of anisotropy and find stability areas for different cases. Finally, we use both effects to build a twostep scheme which allow us to describe the dynamics of a wide class spatially curved models. In the end, we discuss the results obtained and draw the conclusions.
2 Equations of motion
3 Influence of curvature
In this section we investigate the impact of the spatial curvature on the cosmological regimes. As a “background” we use the results obtained in [52, 53, 54]—exact regimes for \(\gamma _{\left( 3 \right) }= \gamma _{\left( \mathbf {D}\right) }\equiv 0\) for both the vacuum and the \(\Lambda \)term cases. As we use them as a “background” solutions, it is worth to quickly describe them all. All solutions found for both vacuum and \(\Lambda \)term cases could be split into two groups—those with “standard” regimes as both past and future asymptotes and those with nonstandard singularity as one (or both) of the asymptotes. By the “standard” regimes we mean Kasner (generalized powerlaw) and exponential. In our study me encounter two different Kasner regimes: the “classical” GR Kasner regime (with \(\sum p_i = \sum p_i^2 = 1\) where \(p_i\) is the Kasner exponent from the definition of powerlaw behavior \(a_i (t) = t^{p_i}\)), which we denote \(K_1\) (as \(\sum p_i = 1) \) and it is lowenergy regime; and the GB Kasner regime (with \(\sum p_i = 3\)), which we denote \(K_3\) and it is highenergy regime. For a realistic cosmology we should have a highenergy regime as past asymptote and a lowenergy as future asymptote, but our investigation demonstrates that potentially both \(K_1\) and \(K_3\) could play a role as past and future asymptotes [52]. Also we should note that \(K_1\) exists only in the vacuum regime, while \(K_3\) as past asymptote we encounter in both the vacuum and the \(\Lambda \)term regimes (see [53] for details). The exponential regimes (where the scale factors depend upon time exponentially, so the Hubble parameters are constant) could be seen in both the vacuum and the \(\Lambda \)term regimes and there are two of them—the isotropic and the anisotropic ones. The former of them corresponds to the case where all the directions are isotropized and, since we work in the multidimensional case, it does not fit the observations. On the contrary, the latter of them have different Hubble parameters for three and extradimensional subspaces. For realistic compactification we demand expansion of the three and contraction of the extradimensional spaces. The exponential solutions are denoted \(E_\text {iso}\) for the isotropic and \(E_{3+D}\) for the anisotropic case, where D is the number of extra dimensions (so that, say, in \(D=2\) the anisotropic exponential solution is denoted \(E_{3+2}\)).
The second large group are the regimes which have a nonstandard singularity as either of the asymptotes or even both of them. The nonstandard singularity is the situation which arises in nonlinear theories and in our particular case it corresponds to the point of the evolution where \(\dot{H}\) (the derivative of the Hubble parameter) diverges at the final H; we denote it as nS. This kind of singularity is “weak” by Tipler’s classification [67] and is type II in the classification by Kitaura and Wheeler [68, 69]. Our previous research reveals that the nonstandard singularity is a widespread phenomenon in EGB cosmology, for instance, in the \((4+1)\)dimensional BianchiI vacuum case all the trajectories have nS as either past or future asymptote [41]. Since a nonstandard singularity means the beginning or end of the dynamical evolution, either higher or lower values of H are not reached and so the entire evolution from high to low energies cannot be restored; for this reason we disregard the trajectories with nS in the present paper.
Thus, the viable (or realistic) regimes are limited to \(K_3 \rightarrow K_1\) and \(K_3 \rightarrow E_{3+D}\) for the vacuum case and \(K_3 \rightarrow E_{3+D}\) for the \(\Lambda \)term; we further investigate these regimes in the presence of curvature.
3.1 Vacuum \(K_3 \rightarrow K_1\) transition with curvature
First we want to investigate the influence of the curvature on the vacuum Kasner transition—the transition from Gauss–Bonnet Kasner regime \(K_3\) to the standard GR Kasner \(K_1\). We add curvature to either and both three and extradimensional manifolds and see the changes in the regimes. We label the cases as \((\gamma _3, \gamma _D)\) where \(\gamma _3\) is the spatial curvature of the threedimensional manifold and \(\gamma _D\)—of the extradimensional. Therefore, for the (0, 0) case—the flat case—we have \(K_3 \rightarrow K_1\), as reported in [52]. Now if we introduce nonzero curvature, both (1, 0) and \((1, 0)\) do not change the regime and there remains \(K_3\rightarrow K_1\). So we can conclude that \(\gamma _3\) alone does not affect the dynamics. On the contrary, \(\gamma _D\) does—(0, 1) has the transition changes to \(K_3 \rightarrow nS\), while \((0, 1)\) changes the transition to \(K_3 \rightarrow K_D\). This \(K_D\) is a new but nonviable regime with \(p_3 \rightarrow 0\) and \(p_D \rightarrow 1\)—a regime with constantsize three dimensions and expanding as powerlaw extra dimensions, which makes the behavior in the expanding subspace Milnelike, caused by the negative curvature. Therefore, the curvature of the extra dimensions alone makes future asymptotes nonviable. Let us also note that for \(D=2\) the original \(K_3 \rightarrow K_1\) regime exists for both \(\alpha > 0\) and \(\alpha < 0\), while for \(D \ge 3\) it exists solely for \(\alpha < 0\). It appears that the \(\alpha < 0\) branch of the \(D=2\) Kasner transition does not have \(K_3^D\)—instead, it has nS, so that \(K_3^D\) exists only for \(\alpha > 0\) in \(D=2\) and \(\alpha < 0\) in \(D \ge 3\). If we include both curvatures, the situation changes as follows: for (1, 1) we have \(K_3 \rightarrow nS\); for \((1, 1)\) it is \(K_3 \rightarrow K_3^D\); for \((1, 1)\) it is \(K_3 \rightarrow nS\) and finally for \((1, 1)\) it is \(K_3 \rightarrow K_3^D\).
The described regimes require some explanations. First of all, as we reported in [52], viable regimes have \(p_a > 0\) and \(p_D < 0\)—indeed, we want expanding threedimensional space and contracting extra dimensions to achieve compactification. Then it is clear why \(\gamma _3\) alone does not change anything—with expanding scale factor, the effect of curvature vanishes. But most interesting is the effect of \(\gamma _D = 1\)—indeed, negative curvature not just stops the contraction of the extra dimensions but starts their expansion, which changes the entire dynamics drastically. Now the extradimensional scale factor “dominates” and the threedimensional one goes to a constant.
The scheme above has one interesting feature—as we described, \(\gamma _{\left( \mathbf {D}\right) }< 0\) gives rise to regime with \(p_3 \rightarrow 0\) and \(p_D \rightarrow 1\)—but in \(D=3\) this gives us a“would be” viable regime—indeed, if both subspaces are threedimensional, as long as one is expanding and another is not, we could just call the expanding one “our Universe” and we have stabilized “extra dimensions”. So that in \(D=3\) there exists a regime with stabilized extra dimensions and a powerlaw expanding threedimensional “our Universe”. However, the viability of this regime needs more checks, and we leave this question to further study.
Thus the negative curvature of extra dimensions gives rise to new and an interesting regime—\(K_D\) with expanding extra dimensions and constantsized threedimensional subspace. It is not presented in the spatially flat vacuum case, but it is also nonviable (except for \(D=3\)), so that it does not improve the chances for successful compactification. The only viable case is \(K_3 \rightarrow K_1\), which remains unchanged for \(\gamma _D = 0\).
3.2 Vacuum \(K_3 \rightarrow E_{3+D}\) transition with curvature
Now let us examine the effect of curvature on another viable vacuum regime—the transition from GB Kasner \(K_3\) to the anisotropic exponential solution \(E_{3+D}\). Similar to the previously considered cases, for an anisotropic exponential solution to be considered as “viable”, we require the expansion rate of the threedimensional subspace to be positive and for the extra dimensions to be negative. Let us see what happens if we add a nonzero spatial curvature.
Similar to the previous case, the curvature of the threedimensional subspace \(\gamma _3\) alone does not change the dynamics—(1, 0) and \((1, 0)\) both have the \(K_3 \rightarrow E_{3+D}\) regime. But unlike the previous case, the curvature of the extra dimensions \(\gamma _D\) alone makes the future asymptotes singular—a powerlawtype finitetime future singularity in the case of \(\gamma _D = +1\) and nonstandard singularity in the case of \(\gamma _D = 1\). The same situation remains in the cases that both subspaces have curvature—as long as \(\gamma _D \ne 0\), the future asymptote is singular—either powerlaw or nonstandard, depending on the sign of the curvature.
Therefore, similar to the previous case, the only viable regime is unchanged, \(K_3 \rightarrow E_{3+D}\), which occurs if \(\gamma _D = 0\). But unlike the previous case, this one does not give us interesting nonsingular regimes.
3.3 \(\Lambda \)term \(K_3 \rightarrow E_{3+D}\) transition with curvature
Finally, let us describe the effect of curvature on the only viable \(\Lambda \)term regime—the \(K_3 \rightarrow E_{3+D}\) transition described in [53, 54]. The condition for viability is the same as in the described above cases—expansion of the threedimensional subspace and contraction of the extra dimensions. Our investigation suggests that the cases with \(D=2\) and \(D\ge 3\) are different; let us first describe the \(D=2\) case. According to [53, 54], there are three domains for the \(\Lambda \)term case where the \(K_3 \rightarrow E_{3+D}\) transition take place.
We have: i) \(\alpha > 0\), \(\Lambda > 0\), \(\alpha \Lambda \le \zeta _0\) with \(\zeta _0 = 1/2\) for \(D=2, 3\) and \(\zeta _0 = (3D^2  7D + 6)/(4D(D1))\), ii) entire \(\alpha > 0\), \(\Lambda < 0\) domain and iii) \(\alpha < 0\), \(\Lambda > 0\), \(\alpha \Lambda \le 3/2\). Formally i) and ii) supplement each other to form a single domain \(\alpha > 0\), \(\alpha \Lambda \le \zeta _0\), but in i) there also exist isotropic exponential solutions, which, as we will see, affects the dynamics, so we consider these two domains separately. So for the domain of i), we have a regime unchanged if \(\gamma _{\left( \mathbf {D}\right) }= 0\), isotropization (\(K_3 \rightarrow E_\text {iso}\)) if \(\gamma _{\left( \mathbf {D}\right) }< 0\) and a nonstandard singularity nS if \(\gamma _{\left( \mathbf {D}\right) }> 0\). In the domain of ii), we again have unchanged \(K_3 \rightarrow E_{3+2}\) if \(\gamma _{\left( \mathbf {D}\right) }= 0\) and nS in all other (i.e. \(\gamma _{\left( \mathbf {D}\right) }\ne 0\)) cases. Already here we can see the difference between i) and ii) domains. Finally, the domain of iii) has the same dynamics as ii). Therefore, in the domain where isotropic and anisotropic exponential solutions coexist, we have slightly richer dynamics, but neither of the regimes are viable; the only viable regime is unchanged \(K_3 \rightarrow E_{3+2}\) and it takes place if \(\gamma _{\left( \mathbf {D}\right) }= 0\). Now if we consider the general \(D\ge 3\) case, the resulting regimes are as follows: now the of i) and ii) have the same structure—opposite to the \(D=2\) case, the structure is as follows: the only viable regime is unchanged \(K_3 \rightarrow E_{3+D}\), which exists if \(\gamma _{\left( \mathbf {D}\right) }= 0\); if \(\gamma _{\left( \mathbf {D}\right) }\ne 0\), we always have nS. The domain of iii) has the structure: unchanged \(K_3 \rightarrow E_{3+D}\) if \(\gamma _{\left( \mathbf {D}\right) }= 0\), the “stabilization” (or “geometric frustration” regime [21, 22]) if \(\gamma _{\left( \mathbf {D}\right) }< 0\) and nS if \(\gamma _{\left( \mathbf {D}\right) }> 0\). This “stabilization” regime is the regime which naturally appears in the “geometric frustration” case and is described in [21, 22]. In this regime the Hubble parameter, associated with threedimensional subspace reaches a constant value while the Hubble parameter associated with the extra dimensions reaches zero (and so the corresponding scale factor—the “size” of the extra dimensions—reaches a constant value; the size of he extra dimensions “stabilizes”).
Thus, in this last case—the \(\Lambda \)term \(K_3 \rightarrow E_{3+D}\) transition—the “original” regime remains unchanged for \(\gamma _{\left( \mathbf {D}\right) }= 0\). For nonzero curvature of extra dimensions, if it is positive, the future asymptote is singular, if it is negative, and \(D \ge 3\), in the future we could have the regime with stabilization of extra dimensions, otherwise it is also singular.
We remind a reader that the geometric frustration proposal suggests that the dynamical compactification with stabilization of extra dimensions occurs only for those coupling constants in EGB gravity for which maximally symmetric solutions are absent. In turn, the absence of the maximally symmetric solutions means the absence of the isotropic exponential solutions, so that with negative curvature of the extra dimensions, isotropic and anisotropic exponential solutions cannot “coexist”, which means that, for any set of couplings and parameters, only one of them could exist. The validity of this proposal has been checked numerically in [53, 54] for a larger number of extra dimensions, and now we see that it is valid also for the \(D=3\) case.
It is not the same in the flat case—for instance, for \(\alpha > 0\), \(\Lambda > 0\) [53, 54] we have both \(K_3 \rightarrow E_\text {iso}\) and \(K_3 \rightarrow E_{3+D}\) on different branches. If we turn on the negative curvature \(\gamma _{\left( \mathbf {D}\right) }< 0\), the former of them remains, while the latter turns to \(K_3 \rightarrow nS\), a nonstandard singularity in \(D=2\), or to the stabilization regime in \(D > 2\). This way we can see that \(D=2\) is somehow pathological—in the presence of curvature, there are no realistic regimes in \(D=2\) but they do exist in \(D \ge 3\).
Finally, we made the same analysis starting from the exponential regime instead of the GB Kasner regime with the same number of expanding and contracting dimensions. The final fate of all trajectories appears to be the same. We will use this remark later in Sect. 5.
3.4 Summary
All three considered cases have the original regimes unchanged as long as \(\gamma _{\left( \mathbf {D}\right) }= 0\). This means that the curvature of the threedimensional world alone cannot change the future asymptote. For nonzero curvature of the extra dimensions, the situation is different in all three cases: in the vacuum \(K_3 \rightarrow E_{3+D}\) case all trajectories with \(\gamma _{\left( \mathbf {D}\right) }\ne 0\) are singular; in vacuum \(K_3 \rightarrow K_1\) we have a new regime but it is nonviable; finally, in \(\Lambda \)term \(K_3 \rightarrow E_{3+D}\) case if \(\gamma _{\left( \mathbf {D}\right) }> 0\) the future asymptote is singular, while for \(\gamma _{\left( \mathbf {D}\right) }< 0\) there could be a viable regime with stabilization of the extra dimensions, but this regime occurs only when an isotropic exponential solution cannot exist and we have \(D \ge 3\).
To conclude, it seems that the only important player in this case is the curvature of extra dimensions. And it is clear why is it so—from the requirements of viability we demand that the threedimensional subspace should expand, while the extra dimensions should contract. The expansion of the three dimensions cannot be stopped by \(\gamma _{\left( 3 \right) }> 0\) nor by \(\gamma _{\left( 3 \right) }< 0\), which is why \(\gamma _{\left( 3 \right) }\) does not influence the dynamics. On the other hand, extra dimensions are contracting, so both signs of extradimensional curvature affect it—a positive sign usually leads to a singularity (standard or not), while a negative sign could turn it to expansion (which is what we see in the \(K_D\) regime).
4 Influence of anisotropy
We start with vacuum regimes; according to [52], in the vacuum \(D=2\) case at high enough \(H_0\) (initial value for the Hubble parameter, associated with threedimensional subspace), there are four combinations (two of them, \(h_1\) and \(h_2\)) and \(\alpha \lessgtr 0\). The first of the cases, \(\alpha > 0\) and \(h_1\), gives the \(K_3 \rightarrow K_1\) transition. If we break the symmetries in both spaces, the stability of the regime is broken as well—in Fig. 1a we present the analysis of this case. There we present the regime depending on the initial conditions—we seek the regime change around the exact solution \(H_1 = H_2 = H_3 = 2.0\), and \(H_4 = H_5 = h_0\) is being found from the constraint equation (7); we fix \(H_3 = 2.0\) and \(H_4 = h_0\) and change \(H_1\) and \(H_2\) and find \(H_5\) from the constraint equation. The exact solution in question (\(H_1 = H_2 = H_3 = 2.0\), \(H_4 = H_5 = h_0\)) is depicted as a circle. The shaded area corresponds to the \(K_3\rightarrow K_1\) regime, while the area which surrounds it corresponds to \(K_3 \rightarrow nS\). One can see that the stability region is quite small and any substantial deviation from the exact solution causes a nonstandard singularity. The second case, \(\alpha > 0\) and \(h_2\), has the \(K_3 \rightarrow E_{3+2}\) regime. With broken symmetry the regime is conserved much better than the previous one—in Fig. 1b we present the analysis of this case. One can see that not just the area of the regime stability covers very large initial conditions, but this area is also unbounded. The typical evolution of such a transition is illustrated in Fig. 2a. The next case, \(\alpha < 0\) and \(h_1\), has the \(K_3 \rightarrow K_1\) transition, just like the first one, and their stability is similar. Finally, the last case, \(\alpha < 0\) and \(h_2\), governs the \(K_3 \rightarrow E_\text {iso}\) transition. If we break the symmetry for this case, the resulting stability area is quite similar to that of \(K_3 \rightarrow K_1\).
To summarize the results for the vacuum case, only \(K_3 \rightarrow E_{3+2}\)—the transition from GB Kasner to anisotropic exponential solution—is stable. All other regimes—transitions to isotropic exponential solution and to GR Kasner—have much smaller stability areas and could be called “metastable”. Formally, the basin of attraction of \(K_1\) and isotropic expansion is nonzero and they are stable within it, but on the other hand its area is much smaller than that of \(E_{3+2}\); so that comparing the two we decided to call \(K_3 \rightarrow E_{3+2}\) “stable”, while \(K_3 \rightarrow K_1\) and \(K_3 \rightarrow E_\text {iso}\) are called “metastable”.
Now let us consider the \(\Lambda \)term case. According to [53], in the presence of the \(\Lambda \)term the variety of the regimes is a bit different from the vacuum case. Again, there are two branches (\(h_1\) and \(h_2\)) and now in addition to variation in \(\alpha \) there is variation in \(\Lambda \) and in their product \(\alpha \Lambda \).
The black circle in Fig. 1c corresponds to the exact \(E_{3+3}\) solution and one can see that the initial conditions are aligned along \(H_i^{(0)} \sim H_j^{(0)}\). The same could be seen from the \(D=2\) case as well (see Fig. 1b). The reason for it is quite clear—indeed, with appropriate \(H_i^{(0)} = H_j^{(0)}\) the exact \(E_{3+D}\) solution is achieved explicitly, so that it is natural for the initial conditions to tend to this relation.
To conclude, we see that all \(K_3 \rightarrow E_{3+D}\) regimes in the \(\Lambda \)term case are stable with respect to breaking the symmetry of both subspaces. On the other hand, another nonsingular regime, \(K_3 \rightarrow E_\text {iso}\), is stable only for (\(\alpha > 0\), \(\Lambda > 0\)) and (\(\alpha < 0\), \(\Lambda < 0\)). Finally, \(K_3 \rightarrow K_1\) in vacuum is also stable, but its basin of attraction is quite small and any substantial deviation from the exact solution destroys it.
5 Twostep scheme for general spatially curved case

the evolution starts from a rather general anisotropic initial conditions,

the evolution ends in a state with three isotropic big expanding dimensions and stabilized isotropic extra dimensions.
In higher dimensions the situation is worsening, on the one hand—as it is seen from Fig. 1c, in \(D \ge 3\), there is more than one stable anisotropic exponential solution, so that starting from the vicinity of exact \(E_{3+3}\) solution we could end up in \(E_{4+2}\) solution, which does not have realistic compactification. However, on the other hand, initial conditions with two expanding and four contracting dimensions can end up in \(E_{3+3}\).
Suppose also that a negative spatial curvature is small enough at the beginning and starts to be important only after this transition to an exponential solution (which is established in the present paper only for a flat Universe) had already occurred. This condition allows us to glue the second part of the scenario, which requires negative spatial curvature of the inner space. We have seen in Sect. 3 that the exponential solution turns to the solution with stabilized extra dimensions in this case. As a result of these two stages a Universe starting from an initially anisotropic state with both outer expanding threedimensional space and contracting inner space evolves naturally to the final stage with isotropic three big dimensions and isotropic and stabilized inner dimensions. The only additional condition for this scenario to be realized (in addition to starting from the appropriate zone in the initial conditions space) is that spatial curvature should become dynamically important only after the transition to an exponential solution occurs. As we mentioned in Sect. 3, this part (and so the entire scheme as well) works only for \(D \ge 3\).
6 Discussions and conclusions
Prior to this paper, we completed a study of the most simple (but the most important as well) cases. The spatial part of these cases is the product of three and extradimensional subspaces which are spatially flat and isotropic [52, 53, 54]. Thus, the obvious next step is consideration of these subspaces being nonflat and anisotropic, and that is what we have done in the present paper. Nonflatness is addressed by assuming that both subspaces have constant curvature, while anisotropy is addressed by breaking the symmetry between the spatial directions. The results of the curvature study suggest that the only viable regimes are those from the flat case with the requirement \(\gamma _{\left( \mathbf {D}\right) }= 0\). Additionally, in the \(\Lambda \)term case there is a “geometric frustration” regime, described in [21, 22] and further investigated in [23] with the requirement \(\gamma _{\left( \mathbf {D}\right) }< 0\).
Our study reveals that there is a difference between the cases with \(\gamma _{\left( \mathbf {D}\right) }= 0\) and \(\gamma _{\left( \mathbf {D}\right) }< 0\): the former of them have only exponential solutions and the isotropic and anisotropic solutions coexist; the latter have the regime with stabilization of the extra dimensions (instead of a “pure” anisotropic exponential regime) and isotropic exponential regimes cannot coexist with regimes of stabilization—this difference was not noted before. The curvature effects also differ in cases with different D—in \(D=2\) there is no stabilization of the extra dimensions, while in \(D \ge 3\) there is.
In \(D=3\) and \(\gamma _{\left( \mathbf {D}\right) }< 0\) there is also an interesting regime in the vacuum case—the regime with stabilization of one and powerlaw expansion of the other threedimensional subspaces; the viability of this regime for some compactification scenario needs further investigations.
The results of the anisotropy study reveal that the \(K_3 \rightarrow E_{3+D}\) regime is always stable with respect to breaking the isotropy in both subspaces, meaning that within some vicinity of the exact \(K_3 \rightarrow E_{3+D}\) transition, all initial conditions still lead to this regime (see Fig. 2a). Although the area of the basin of attraction for this regime depends on the number of extra dimensions D, in \(D=2\) it is quite vast (see Fig. 1b) and there are no other anisotropic exponential solutions, in \(D=3\) (and higher number of extra dimensions) it seems smaller^{2} and there are initial conditions in the vicinity of \(E_{3+D}\) which leads to other exponential solutions. In our particular example, \(D=3\), presented in Fig. 1c, some of the initial conditions from the vicinity of \(E_{3+3}\) end up in \(E_{4+2}\) instead. We expect that in a higher number of extra dimensions the situation for \(E_{3+D}\) would be more complicated and requires a special analysis.
Another viable regime, \(K_3 \rightarrow K_1\) from the vacuum case, as well as other nonviable regimes, are “metastable”—formally they are stable, but their basin of attraction is much smaller compared to that of \(E_{3+D}\) (see Fig. 1a).
Our study clearly demonstrates that the dynamics of the nonflat cosmologies could be different from flat cases and even some new regimes could emerge. In this paper we covered only the simplest case with constantcurvature subspaces leaving the most complicated cases aside—we are going to investigate some of them deeper in the papers to follow.
Now with both effects—the spatial curvature and anisotropy within both subspaces—being described, let us combine them. In the totally anisotropic case, as we demonstrated, a wide area of the initial conditions leads to anisotropic exponential solution (for the values of couplings and parameters when isotropic exponential solutions do not exist). Therefore, if we start from some vicinity of the exact exponential solution, and if the initial scale factors are large enough for the curvature effects to be small, we shall reach the anisotropic exponential solution with expanding three and contracting extra dimensions. After that the curvature effects in the expanding subspace are nullified, while in the contracting dimensions they are not. If it is the vacuum case, as we have shown earlier, as long as \(\gamma _{\left( \mathbf {D}\right) }\ne 0\) we encountered a nonstandard singularity, so that the vacuum case is pathological in this scenario. In the \(\Lambda \)term case, as we reported earlier, for \(\gamma _{\left( \mathbf {D}\right) }= 0\) we recover the same exponential regime, for \(\gamma _{\left( \mathbf {D}\right) }> 0\) the behavior is singular and only for \(\gamma _{\left( \mathbf {D}\right) }< 0\) we obtain the “geometric frustration” scenario [21, 22] with stabilization of the extra dimensions.
We can see that the proposed twostep scheme works only for the \(\Lambda \)term case and only if \(\gamma _{\left( \mathbf {D}\right) }< 0\)—in all other cases it either provides trivial regimes, or it leads to singular behavior. Also, there is a minor problem with the number of extra dimensions—as we noted, the first stage of this scheme—reaching the exponential asymptote from initial anisotropy—is best achieved in \(D=2\) and the probability of reaching \(E_{3+D}\) could decrease with the growth of D. On the other hand, the second stage—when the negative curvature changes the contracting exponential solution for the extra dimensions into stabilization—is not present in \(D=2\) and only manifest itself in \(D \ge 3\). Thus, the described twostage scheme works only in \(D \ge 3\) and in this case the initial conditions for the first stage are already not so wide, though a finetuning of the initial conditions is not needed.
This finalize our paper: the analysis presented suggests that more indepth investigations of both curvature and anisotropy effects are required—we have investigated and described the most simple but still very important cases—constantcurvature and flat anisotropic (BianchiItype) geometries; in the papers to follow we are going to consider more complicated topologies.
Footnotes
 1.
We consider the Ansatz for spacetime in the form of a warped product \(M_4\times b(t)M_D\), where \(M_4\) is a Friedmann–Robertson–Walker manifold with scale factor a(t), whereas \(M_D\) is a Ddimensional Euclidean compact and constantcurvature manifold with scale factor b(t).
 2.
To quantitatively address this question we need to introduce an appropriate measure and since the area is unbounded, this is not an easy task. Also, the answer will depends on the chosen measure, so we skip the quantitative analysis.
Notes
Acknowledgements
The work of A.T. is supported by RFBR grant 170201008 and by the Russian Government Program of Competitive Growth of Kazan Federal University. The authors are grateful to Alex Giacomini (ICFMUACh, Valdivia, Chile) for discussions.
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