Critical Combinations of Higher Order Terms in Einstein-Maxwell theory and Compactification

We discuss the role of a particular combination of higher derivative terms in higher dimensional theories, especially in the background of spontaneous compactification. We find that the special extension of Einstein-Maxwell theory with higher order terms admits interesting cosmological solutions.


I. INTRODUCTION
The gravitational theory with higher derivative terms [1, 2] is one of the most interesting subjects to study in physics for many years. The treatment of Einstein gravity with perturbative field quantization naturally leads to inclusion of such nonlinear terms in the spacetime curvatures in effective actions.
The generalization of Einstein gravity has been developed extensively by many authors.
The Lanczos-Lovelock gravity [3][4][5][6] 1 , which is dictated by the action with the dimensionally continued Euler forms, is defined in a generic dimensional spacetime and it describes a massless spin-two fluctuation despite of the existence of higher order terms in its action. In a few decades ago, it is discovered that the effective field theory arising from string theory involves similar terms at least the lowest-order correction to the Einstein-Hilbert term [8,9].
The other specific extension is found in the study on so-called critical gravity [10][11][12] (although the word "critical" originally implied the emergence of Log gravity [13][14][15]). In such a theory of gravity, the behavior of spin-two modes is described by the Lagrangian of special combination of higher order terms in the Ricci tensor and the scalar curvature. There is a ghost field but no spin-zero mode in the theory. In the three dimensional spacetime, the model governed by a similar Lagrangian expresses the new massive gravity [16,17].
In this paper, we consider incorporation of Maxwell fields to the (next-to) critical gravity. The critical gravity has been studied under the assumption of the maximally symmetric background spacetime. The higher derivative theory on the other type of background fields is worth studying. Inclusion of a non-vanishing flux field brings about the partial compactification with a maximally symmetric extra space and might provide an interesting scenario for a very early stage of the universe.
The assumption of the partially symmetric solution can yield another possibility. It is feasible to attach the other type of higher order terms to the Lagrangian which realizes the spontaneous compactification. Because of the symmetry, use of the Riemann tensor is possible in the additional terms. They are expressed by special combinations of Gauss-Bonnet terms and Horndeski's generalized Maxwell term [18]. This can induce a second order field equation displaying the cosmological evolution of the background geometry. 2 This paper is organized as follows. In Sec. II, we build the Lagrangian of our first model, which involves the quadratic terms of the Ricci tensor, the Ricci scalar and the twoform flux field, in a similar way revealed in the study of critical gravity. The spontaneous compactification in the first model is studied in Sec. III and cosmological solutions are examined in Sec. IV. In Sec. V, we present the second model containing the quadratic term of the Riemann tensor. We also investigate the cosmological solution qualitatively in this model. The last section is devoted to summary and prospects.
Throughout the present paper, we deal with our models at the classical level.

II. A CRITICAL MODIFICATION OF EINSTEIN-MAXWELL THEORY
In the work on critical gravity, it is found that the Lagrangian of the specific combination of curvatures yields a spin-two massless graviton and a ghost mode; there appears no scalar mode as in Einstein gravity. In this section, we propose a model of higher order extension of Maxwell-Einstein theory. We will follow the method for constructing the Lagrangian examined in our previous work [22,23].
First, we start with the action for the Einstein gravity and electromagnetism in D dimensions: where R and Λ denote the scalar curvature obtained from the metric g M N and the cosmo- Applying the variational principle to the action results in the following equations of motion: Next, we construct a Lagrangian of our model as follows: (2.6) Note that the solution of T M N = 0 is the solution for the equation of motion derived from L, provided that the field equation for the vector field is satisfied (though the stability is not necessarily guaranteed). 3 The Lagrangian L can then be written as Incidentally, we can pick up some 'critical' cases: If α = D 4(D−1) βΛ, the cosmological constant is zero in the D dimensional spacetime; If α = D−2 2(D−1) βΛ, the Einstein-Hilbert term (R) is absent from the D dimensional action; If α = D−4 2(D−1) βΛ, the pure Maxwell term (F 2 ) is absent from the D dimensional action.
We should note that the auxiliary field method [14,16,17,23], which can be used in critical higher order gravity, also leads to the additional terms, such as where S M N is an auxiliary symmetric tensor field and S ≡ S M N g M N . Varying the auxiliary field S M N , we obtain 9) or, solving that with respect to S M N , we get (2.10) Therefore, one can see the equivalence between the Lagrangians L and L ′ . We can derive the linearized field equation using the action I ′ = d D x L ′ . Now, we investigate the behavior of linear fluctuations around the background geometry, which satisfies T M N = 0. We decompose the metric as follows: (2.11) The indices are raised and lowered by the background metricḡ M N . Then the trace of the fluctuation is expressed as For conveniences, we write down the curvature tensors up to the linear order in h M N here [24]: where the nabla stands for the covariant derivative and the barred symbols show that they are constructed from the background metricḡ M N .

Assuming the background fields satisfiesT
2 Λḡ M N = 0, we can use δ(2S P Q T P Q − S P Q S P Q + S 2 ) = 2S P Q δT P Q +(higher order terms in small fluctuations from the background) , (2. 16) to obtain the field equation at the linearized order: The linearized equation obtained from (2.9) is The important observation is that, using (2.9), the field equation (2.17) becomes In the D dimensional Minkowski vacuum, this equation is just the Fierz-Pauli equation [25] for the spin-two wave with mass-squared m 2 = α β . Therefore we recognize that S M N corresponds to the massive ghost field. The postulate of no tachyonic ghost requires α β > 0 in the present model.
The linear fluctuation from the background fields can be argued by using above equations.
In the next section, we consider the solution for spontaneous compactification of spacetime as the background.

III. SPONTANEOUS COMPACTIFICATION IN THE CRITICALLY MODIFIED HIGHER ORDER EINSTEIN-MAXWELL THEORY
In this section, we study the partial compactification of space with a non-trivial flux in our model. The extra dimensions are considered to be compactified into a sufficiently small size.
A maximally symmetric solution in our model is compactification of the form where M D−2 is the (D − 2) dimensional Minkowski spacetime and S 2 is a two dimensional sphere, as in the work of Randjbar-Daemi, Salam and Strathdee (RSS) [26] where D = 6 is considered. We consider the solution for flat D − 2 dimensional spacetime and study its classical stability in this section, though other maximally symmetric spacetimes are also interesting. Our notation is that the suffices µ, ν, . . . run over 0, 1, 2, D − 3 and the suffices m, n, . . . are used for the extra dimensions D − 2 and D − 1.
As stated in the previous section, the solution of T M N = 0 and J M = 0 is a solution in our model. A suitable ansatz for the flux field is where ε mn is the antisymmetric symbol and the constant q is the strength of the two-form flux. 4 This monopole flux yields the relation Then, the equation T M N = 0 can be decomposed to for the spacetime of a direct product. As for the flat D − 2 dimensional spacetime, where R µν = 0, we must take a fine-tuning of parameters as and therefore, the solution for the background fields reads Substituting this compactified background solution into Eq. (2.20), we obtain To show complete particle spectrum for the ghost field is rather complicated, but we can see that the lowest mass-squared of the ghost is α/β for symmetric tensor fields (which do not couples to the Ricci curvature of the extra space) and vector bosons of the Kaluza-Klein origin (which come from the zero modes of the Lichnerowicz operator on S 2 , −∇ 2 +R mn ).
It is also remarkable that the other graviton and vector field fluctuations is absent in the linearized equation (3.7).
Thus, taking the analysis by RSS for other fluctuation modes [26] into account, in the classical and linearized analysis, we find that the RSS background solution M D−2 ⊗ S 2 is stable if α/β > 0, because there is no growing mode. 5 Before closing this section, we note that the effective D −2 dimensional Newton constant, which can be found by the coefficient of the Ricci scalar of the D − 2 dimensional spacetime in the action, is not affected by the higher order terms in our model and is independent of β. It might be related to the absence of the graviton and the ghost modes at the linearized level mentioned above.

IV. DE SITTER SOLUTIONS AND COSMOLOGY IN THE MODEL
In this section, we perform the analysis on the stability of the compactification and the cosmological solution in our model. We consider an effective potential V for the radius of extra space S 2 [27] in order to study them.
We can express the Riemann tensor of the S 2 as where the first equation is the fine-tuning condition between the parameters.
The effective potential V as a function of b, the scale of the extra space, is obtained from replacing the background fields by functions of b in −I with and √ detḡ mn ∝ b 2 . The effective potential for the RSS model is calculated from −I 0 and is given as 4) and then, the effective potential for our model of which Lagrangian is described by (2.7) is with y ≡ Λb 2 . A minimum of the potential is found at y = 1 and attains V (1) = 0. The stability of the radius of the extra sphere is determined by the sign of the second derivative of the potential: If D > 3 and α, β > 0, the scale b = 1/ √ Λ is found to be stable. We also find that it is impossible to create another potential minimum satisfying V = 0 by any tuning of β/α in the model.
At a large value of b, V can be indefinitely negative provided that D 4(D−1) βΛ − α > 0. In this case, the D dimensional cosmological constant becomes negative. This may not indicate instability of compactification against a large fluctuation, because the higher derivative terms cause the coupling between the compactification scale and the first and second derivatives of an expanding scale factor of the rest of space. Therefore, we investigate the cosmological equation including the dynamical scale factor and the compactification scale. The metric is usually assumed as where b(t) denotes the radius of the compactified sphere S 2 and dΩ 2 2 is the line element an the unit sphere. For the maximally symmetric (D − 3) dimensional space, the line element is denoted dΩ 2 D−3 here, and its Riemann tensor is normalized asR ij kl = k(δ i k δ j l − δ i l δ j k ) with k = 0, ±1, where i, j, . . . = 1, 2, . . . , D − 3. a(t) is the scale factor of D − 3 dimensional homogeneous and isotropic space.
The Riemann curvature is computed with the metric as where the dot (˙) denotes the derivative with respect to the cosmic time t.
Now we can obtain the effective action of the scale factor a(t) and the compactification scale b(t) for the RSS model as where the over-all constant which comes from the volume of space has been omitted. The action has been integrated by parts after the symbol '∼'.
The effective action for our model can be written, by adopting the auxiliary fields, as where we set S 0 0 = s 0 , S i j = s a δ i j and S m n = s b δ m n . Here, the equations of motion are omitted, but they can be obtained from the above action. 6 We seek for the de Sitter (⊗S 2 ) solution, where k = 0, a(t) = e H 0 t with constant H 0 , and b = b 0 =constant, are taken as an ansatz. In this case, all the equations of motion become 6 The Euler equations can be obtained from the action using the command EulerEquations in the add-on Substituting these auxiliary relations to the other equations, we obtain the following two equations at last: and another is These values for solutions are the same as those of the RSS model [29], as expected (if D = 6). In both cases, the values for s 0 , s a and s b are equal to zero.
Because the simultaneous equations (4.12,4.13) are the fourth order algebraic equations, there are other two sets of solutions 7 at most. One can find, for sufficiently small value for β, only two original solutions (4.14,4.15) are real solutions. For D = 6, only two real solutions (4.14,4.15) exist for βΛ/α <≈ 2.8. As a model of the universe, the multiple de Sitter phases would have some significance in cosmology.
We now study the behavior of the small fluctuation around the above exact solutions. In the neighborhood of (i), we set and ∆s a = s a , and so on. Then, the linearized equations of motion read The oscillatory behavior of ∆b as well as ∆s a is enough interesting. The variables ∆s 0 , ∆s a and ∆s b oscillate with a different frequency from that of ∆b, as seen from (4.18,4.19). 8 Thus, the tuning of initial conditions realizes enough time for accelerating expansion and a sufficient inflation can be achieved, though we cannot tell the choice of initial state in the present classical analysis of the model.
Since the analysis of the perturbation from (ii) can be considered and results in finding exponentially growing modes as found in [29], we omit the indication of equations here.
Some fine tuning gives an enough expansion time for a sufficient inflation as the de Sitter phase also in this case.

LOVELOCK-HORNDESKI-NONLINEAR MAXWELL THEORY
In the solution for the spontaneous compactification in previous sections, one has found the following relation in the background fields: Similarly to the consideration in the previous sections, we come to an idea that the Minkowski compactification of the RSS model is attained in a higher derivative theory if the relation (5.1) is reflected in the higher order correction term.
We then propose an additional higher order term where the generalized Kronecker's delta is defined by  [30][31][32][33][34][35], the case with the RSS-type compactification with the Gauss-Bonnet term has been studied in a few papers including [36]. 9 As previously, we characterize the cosmological background as: and Then, the Lagrangian L 2 reduces to and we obtain the following action for a and b: Again, we omit exposition of equations of motion here.
We first explore the de Sitter phase, or the maximally symmetric product spacetime. To find this, we set k = 0, a(t) = e H 0 t and b(t) = b 0 , where H 0 and b 0 are constants. We then 9 The compactification with the Horndeski Lagrangian has been considered by one of the present authors [21]. The small arrows in the figures indicate the flow in the phase space. One can see that b = 1/ √ Λ is an attractor and the area of attracting region increases with the value of γ/α. 10 We conclude that an appropriate choice of the initial condition, which is in the flow passing through near by the branch point given by the de Sitter solution of (5.9,5.10), might cause a long expansion time. We here take a unit such that Λ = 1. We show (a) for γ/α = 0, (b) for γ/α = 1, (c) for γ/α = 2, respectively.

VI. SUMMARY AND PROSPECTS
In this paper, we have considered special cases for extension of Einstein-Maxwell theory with higher order terms. The models proposed in the present paper have the same solution for spontaneous compactification M D−2 ⊗S 2 utilizing the internal flux as the RSS model. The stability of the compactified vacuum have been confirmed at the classical level. Nevertheless, the models reveal the different cosmological behaviors because of the additional degrees of freedom or the existence of the additional de Sitter phases. We should investigate evolutional equations by using numerical calculations in future work.
Our toy models seem to be quite special, but the analysis of the models will exhibit some typical behaviors including inflation, and we can regard general higher order corrections as a modification from the critical models.
Recently, the black holes in higher derivative gravity attract renewed interest [38,39].
We consider that our model 1 is appropriate to examine magnetized black holes of a novel type.
The subjects we should consider are, for instance: the coupling to matter fields; compactifications on (S 2 ) N [40] in our models; the inclusion of a p-form flux field such as the Freund-Rubin compactification [41]; the treatment of our models in quantum cosmology [42][43][44]; the initial singularity problems [45] in the models; the possible supersymmetrization of the models; the field content of the model in 3 + 2 dimensions.
The generalization of our model 1 to more higher order theory is considered straightforwardly. If we introduce an auxiliary field S N M , the additional term can be replaced by which will be studied extensively elsewhere.
Finally, we must research the relation to other theories. We suppose that the structure of the model 1 owes to the matrix ∆ M N P Q , which appears also in the spin-two propagator.
It would be very interesting if our models were the effective theory of another theory.