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We construct the higher order terms of curvatures in Lagrangians of the scale factor in the

It is well known that higher-derivative gravity has a scalar degree of freedom in general [

The dimensionally continued Euler densities have also been studied [

In recent years, it turned out that there is a special case where a scalar mode disappears in higher-derivative gravity. Originally, this fact was found in research of a three-dimensional theory of massive gravity [

In the present paper, we generalize the structure of the Lagrangian of critical gravity, to models with higher order terms in curvature tensors in higher dimensions. We show that such extensions can be attained by use of the Lovelock tensors. In order to offer a systematic way to construct the required higher order term, we take an explanatory approach by assuming the Robertson-Walker (RW) metric. In this approach, the absence of second derivatives of the scale factor from the Lagrangian with appropriate total derivatives is considered as a necessary condition for disappearing scalar modes.

It should be noted that the combination defined in

The RW geometry is known to be conformally flat [

In the present paper, we show that it is possible to continue the higher-curvature terms beyond the number of dimensions.

The outline of this paper is as follows. In Section

In this section, we construct the higher order term in curvatures by generalizing that of critical gravity. We will verify the absence of scalar modes in cosmological models with the terms in the next section. First, we introduce the dimensionally continued Euler density

Next, we consider the Lovelock tensor [

Here, we construct the new combination of the Lovelock tensor and the metric multiplied by the trace of the Lovelock tensor. That is,

Now, we find that the natural generalization of this is given by

In the next section, we confirm that this combination is suitable for an extension of critical gravity in higher dimensions, by utilizing the RW metric.

We consider the following RW metric in

First in this section, we examine the Lovelock tensors of

By the combinatorial property of the generalized Kronecker delta, we can find the Lovelock tensor for a general

Next, we calculate the generalized Schouten tensor

Now, we consider the combined term

The combination

Up to now, we become aware of a similarity to the Lovelock Lagrangian. It takes the form for the RW metric as follows:

Before closing this section, discussion on relation to the work of Meissner and Olechowski [

Later, we show that the restriction by the dimensions can be overcome. To exhibit the discussion on the subject, we examine the cosmological action in the present model again in the next section.

We consider the action for

Then, the action for the scale factor

Using the expansion

Now, we define the Lagrangian

The Hamiltonian constraint is regarded as the variation equation

It is known that the variation of the lapse function corresponds to the variation of

The generalized Lovelock tensor can be written as variation of the action, which is proportional to

In the similar manner, the corresponding generalized Schouten tensor is defined as in the previous section,

Again, we obtain the same functional form as those constructed from the Lovelock tensors in the previous section. More iterative operations yield the more higher order terms. For this time, however, there is no limitation on the relation between the number of dimensions

Unfortunately, only for

Another way to cross the dimensional limitation, which is inspired by the work of Meissner and Olechowski [

In the next section and after, we will turn our attention to apply the higher order Lagrangian obtained here to cosmological models.

In this section, we investigate the possible inflationary stage in evolution of the universe in the model with higher order terms discussed in the previous sections.

Let us consider the general cosmological action

As usual, the energy momentum tensor of matter is taken as

We here give a few models in four-dimensional spacetime with focus on the possibility of an inflationary growth of the scale factor. Furthermore, we take

If we can choose the coefficient

Now, we specify the function

If the energy density is sufficiently low, such as

Another model can be chosen, in which the correction is a monomial of the higher order term. That is,

Even though we show special toy models here, we find that the existence of higher order terms yields the inflationary growth of the scale factor with ordinary matter, and with no scalar mode and no redefinition of the metric.

We will apply the previous discussion on a conformally flat metric to solutions for domain walls (“thick branes”). The

Substituting the metric (

The field equations can now be derived in a usual manner. The differential equation for the scalar is

Define

Note that

Now, we take a BPS ansatz [

If the function

For example, we try to express the sine-Gordon equation. We take

In the present paper, we have attempted to show the possible quasilinear second-order theory of gravity in conformally flat spacetimes. Models with arbitrary higher order of curvatures have been obtained. As long as we adopt an isotropic and homogeneous cosmological setting, the energy conservation holds in the models because there is no scalar mode and no requirement of frame rescaling. In spite of them, inflationary expansion can be found in the models. We have also found that the domain wall solution in the present type of the higher order gravity can be obtained naturally with the potential having many minima.

Our work corresponds to the extension of the Lovelock higher-curvature gravity in arbitrarily higher order terms in higher dimensions. Our analysis, however, has been limited for the case with the conformally flat spacetime, and the explicit form of the action written in curvature tensors and derivatives has been still ambiguous. The stability of the solutions obtained here is problematic for anisotropic perturbations or tensor modes. To study it, we should classify the possible form of the Lagrangian when the Weyl tensor vanishes. It is known that the dimensionally continued Euler forms have the property of factorization in terms of those of lower orders when the spacetime is represented by the direct product of spaces [

Nevertheless, we emphasize that our arbitrarily high order gravity can be applied to many models in various contexts. The problem of initial singularity can be reconsidered by studying classical bouncing universes in our model. On the other hand, the Wheeler-DeWitt equation of the Lagrangian should lead to higher-derivative quantum cosmology. The equation must be difficult to treat with, but the study on it may shed new light on quantum gravity. The possible black hole solutions are interesting in both cases of asymptotically flat and asymptotically AdS spacetime. We will return to some of the problems in future.