Inhomogeneous compact extra dimensions and de Sitter cosmology

In the framework of multidimensional f(R) gravity, we study the metrics of compact extra dimensions assuming that our 4D space has the de Sitter metric. Manifolds described by such metrics could be formed at the inflationary and even higher energy scales. It is shown that in the presence of a scalar field, it is possible to obtain a variety of inhomogeneous metrics in the extra factor space M_2. Each of these metrics leads to a certain value of the 4D cosmological constant Lambda_4, and in particular, it is possible to obtain Lambda_4 =0, as is confirmed by numerically obtained solutions. A nontrivial scalar field distribution in the extra dimensions is an important feature of this family of metrics. It is shown that the extra space volume is strongly related to the Hubble parameter.

Stabilization of the extra space as a pure gravitational effect in f (R) and more general multidimensional theories with maximally symmetric extra spaces has been studied in [23][24][25], as well as their ability to describe both early and late inflationary expansion [26][27][28]. In [15], it was shown that an f (R) model with inhomogeneous extra space is compatible with 4D Minkowski or very weakly curved space-times.
The structure of this paper is as follows. In Section 2 we choose the metric and dimensionality of our manifold, the Lagrangian containing gravity with higher derivatives and a scalar field and derive the set of classical equations. In Section 3 we discuss the boundary conditions that are necessary in order to obtain a set of solutions, as well as their physical meaning. Also, a number of numerical solutions are presented. Conclusion are made in Section 4.

Basic equations
We will consider 6D metrics of the general form where u and ϕ are extra spatial coordinates, and ϕ ∈ [0, 2π) is assumed to be a polar angle, while the 4D metric tensor g µν may depend on both 4D coordinates (making it possible to consider, for example, cosmological or static models) and the "radial" fifth coordinate u. The extra factor space M 2 parametrized by (u, ϕ) is thus a surface of rotation, which can be compact if r(u) ≡ e β(u) tends to zero at two boundary values of u.
In such space-times, we consider a theory with the action where g 6 = det(g AB ), f (R) and V (φ) are some functions (to be chosen later) of the 6D scalar curvature R and the scalar field φ, respectively. Variation of (2.2) with respect to φ and g AB leads to the field equations where the stress-energy tensor of the scalar field φ = φ(y) reads Before writing the particular equations to be solved, it is helpful to present brief expressions for the Ricci tensor components R A B assuming a general diagonal metric in arbitrary dimensions, where b A (X) are arbitrary functions of x A , and η A = ±1. Then for the diagonal components of R A B we have where no summing is assumed over an underlined index, for an arbitrary function f (X), and g = | det(g M N )|. The off-diagonal components of the Ricci tensor are more conveniently written with lower indices, namely, In the present study, we consider a cosmological (de Sitter) metric in 4D space-time and the extra dimensions using the Gaussian u coordinate (length along the coordinate axis of x 4 = u), so that the metric (2.1) takes the form where H = const is the Hubble constant. Accordingly, in terms of (2.6) we now have Assuming φ = φ(u), Eq. (2.3) and noncoinciding equations from (2.4) may be written as where f R = df R /du, etc. Equations (2.15) and (2.12) give Another combination, (2.15)+(2.16)-(2.14)-f R · (2.12) leads to 3 Models with inhomogeneous extra space

Equations and boundary conditions
For our calculations, it is convenient to consider the Ricci scalar R(u) as an unknown function. in addition to r(u) and φ(u). As three independent equations for this system, we can take, for example, (2.13), (2.12) and a combination of (2.16) and (2.12): resolved with respect to the higher derivatives φ , r , R . Equation with lower-order derivatives plays the role of a restriction on the solutions of the coupled second order differential equations.
As boundary conditions, we use the requirement of u = 0 being a regular center on the (u, φ) surface and the corresponding requirements for φ and R: where all quantities with the index 0 are constants. These initial parameters are related by the condition following from Eq. (2.18), This means that for given f (R) the quantity R 0 is related to H and φ 0 , so that any two of these three parameters are free. We also have from (2.17) and (3.1) for u → 0 We will seek solutions for u > 0 in which the circular radius r → 0 at some u = u max , which provides compactness of the extra space parametrized by u and ϕ.
The total energy on the (u, ϕ) surface for a specific solution is it may be interpreted as the energy density of the scalar field stored in the extra dimensions. This energy density depends on the parameter φ 0 expressing the boundary scalar field value in M 2 .

Pure gravity (φ = const)
Let us first consider the case φ = φ 0 , in which the scalar field is distributed uniformly in space and does not depend on time, and the equations can be solved analytically. In this case the scalar field potential is constant, Equation (2.18) in this case leads to which means that also R = R 0 = const, hence the 2D extra space is maximally symmetric for any given f (R), and from (2.12) it follows Now, the difference of equations (2.14)-(2.16) reduces to If we assume that f R (R 0 ) = 0, and also notice that and we also have Under our conditions at u = 0, the solution of (3.13) reads 15) and the metric has the form the extra space being a 2-sphere.
We have shown that with any choice of the initial function f (R) the only solution with the metric (2.9) for pure gravity (or with a constant scalar field) corresponds to a spherical extra space.
This result deserves attention at high energies where the Hubble parameter is large enough. A common starting point is to fix the properties of extra dimensions, their size in particular. These properties depend on the Lagrangian parameters, include the topology of extra space, but do not depend on our 4D metric. According to (3.14), (3.16), the state of affairs is different at least for the class of models containing all sorts of f (R). The extra space is inevitably maximally symmetric, and its radius is stiffly related to the Hubble constant, In particular, if we choose f (R) = aR 2 +R+(c−V ) (see Eq. (3.18) further on), Eq. (3.10) gives the following relation between the parameters: The possibility of complex roots in this expression shows that not any choice of the parameters leads to a valid solution, since obviously H 2 must be real. At the inflationary stage of the Universe evolution, r is close to 10 −27 cm, and it is about 10 −33 cm at the Planck scale. However, if we consider very small H, for example, corresponding to the present epoch, the Ricci scalar of extra dimensions will be close to zero, meaning their huge size, incompatible with observations. To avoid such a strong constraint, one can add matter fields (a scalar one in our case) or/and widen the Lagrangian by adding other invariants like the Ricci tensor squared, making it possible to obtain inhomogeneous extra dimensions. A detailed discussion on the basis of other extra space metrics can be found in [29,30].

Numerical solutions for φ = const
To obtain examples of numerical solutions of interest, let us choose the following functions in the action (2.2): The figures below present solutions for different values of the parameters.      A variation of the parameter values can lead to qualitatively different metrics in M 2 . For example, the curvature may change its sign, as is seen from Fig. 6.
Evidently, the classical equations written above are invalid at energy scales larger than the Planck scale. For 4D physics the corresponding length scale is about l 4 = 1/m 4 ∼ 10 −33 cm. This means that quantum fluctuations are strong, and any solution to the classical equations is invalid at length scales smaller than l 4 . In a D-dimensional world the analogous scale is l D = 1/m D = 1 by our convention. There are two consequences if we intend to work on the classical level: (i) the size of extra dimensions must be much larger than unity; (ii) any peculiarities with the coordinate interval δu l D = 1 are meaningless without thorough analysis of quantum effects. In particular, if a classical solution contains a singularity, as it happens in our solutions at u = u max , it is reasonable to suppose that such a singularity is smoothed by quantum effects and should not be taken seriously.

Reduction to 4 dimensions and low energies
The study made above reveals that static inhomogeneous extra dimensions could exist. For given f (R) and V (φ), their shape and energy density also depend on the initial (random)l value φ 0 . Let us briefly discuss the observational manifestations of such solutions. As was shown in [16], there exist such extra-dimensional metrics that lead to the 4D cosmological constant Λ 4 arbitrarily close to zero. This effect is a result of interference between the gravitational and scalar field parts of the Lagrangian. The result obtained in [15] is based on approximate equations. In this section, we use the exact set of equations derived from the metric (2.9) and the action (2.2).
The quantity Λ 4 can be found by integrating out the internal coordinates in the action (2.2). We know that the Hubble parameter is at present almost zero as compared to the possible extra-dimensional scales. Hence let us put H = 0. In this case R 0 and φ 0 are related by (3.7), and Λ 4 (φ 0 ) is a function of the unique argument φ 0 . It remains to find this function and its zero points.
Let us consider static solutions found above and use the smallness of H as compared to the extra space Ricci scalar R 2 = −2r /r, see (2.12). After the decomposition Comparing this expression with the standard form of the 4D action we get the observed Planck mass in the units m D = 1, and The scalar field density stored in the extra space is neutralized by the gravitational term f (R 2 ), so that the cosmological constant (3.23) is small for the specific solution to Eqs. (3.1). Such a solution certainly exists due to the continuity of the set of solutions. The quantity φ 0 was used as an additional parameter, see (3.6), to find a specific distribution φ(u). We see that the set of the scalar field distributions is parametrized by the boundary value φ 0 . The same can be said about its energy-momentum tensor T AB (φ 0 ). Our analysis of the classical equations indicates that in the absence of matter fields only a maximally symmetric (spherical) metric in M 2 is possible. This analytic result shows that the Ricci scalar of the extra space is unambiguously related to the Hubble parameter, and hence the extra-dimensional radius is slowly varying with time at the inflationary stage, and a similar picture might be expected for the present epoch. However, at present this radius has to be unacceptably large, this shortcoming being cured by invoking a scalar field, which makes its role very important.
At high energy scales, quantum fluctuations perturb both the metric and the scalar field. It is widely assumed that our manifold was born at sub-Planckian energies, so that the scalar field randomly varies at those times. Let us estimate the conditions at which the fluctuations cannot disturb the extra space metric. Fluctuation are able to produce Kaluza-Klein excitations if the extra-dimensional scale l e is larger then the fluctuation wavelength 1/k, where k is magnitude of the its wave vector. For relativistic matter, the energy density ρ ∼ k 4 while the Hubble parameter is H ∼ ρ/m 2 4 . These estimates constrain the extradimensional scale as l e 1/ √ Hm 4 . This inequality allows us to impose a restriction on the extra space metric which is much stronger than those obtainable in collider experiments. Indeed, the cosmological constant and the gravitational constant do not vary within the present horizon. This means that fluctuations should be damped at the inflationary stage where H = H I 10 13 GeV, so that the scale l e of the extra dimensions should be smaller than 1/ √ H I m 4 . We conclude that the averaged size of the compact extra dimensions should be smaller than ∼ 10 −28 cm. This limit confirms those obtained in [17], where it was shown that the slow roll motion of the inflaton is forbidden if l e > H −1 . Recall that the collider limit is l e < 10 −18 cm, which is 10 orders of magnitude weaker than our prediction.
In conclusion, we would like to mention one more application of the idea of inhomogeneous extra dimensions. Consider, instead of (2.9), the 6D metric ds 2 = e 2γ(u) η µν dx µ dx ν − du 2 − r(u) 2 dϕ 2 , (4.1) where η µν is the 4D Minkowski metric, and the metric in M 2 is the same as in (2.9), but in terms of the metric (2.6) we now have b µ = γ(u), b 4 = 0, b 5 = ln r(u). Using the expressions (2.7) and (2.8), it is then straightforward to derive the explicit form of field equations for the unknowns γ(u), r(u) and φ(u). A tentative study has shown that there exist solutions with u-dependence of the circular radius r somewhat similar to that shown in Fig. 1 under boundary conditions similar to (3.5), (3.6), even if the scalar field is absent. If the size of M 2 is large enough and with proper dependences r(u) and φ(u), the solutions can admit interpretations in terms of the brane world concept, somewhat similar to [31,32]. Unlike solutions with the metric (2.9), mostly intended for the early Universe, those with (4.1) are able to describe the present-day universe with very small 4D curvature, and a study of their possible properties and applications is in progress.