Ghosts in classes of non-local gravity

We consider a class of non-local gravity theories where the Lagrangian is a function of powers of the inverse d'Alembertian operator acting on the Ricci scalar. We take an approach in which the non-local Lagrangian is made local by introducing auxiliary scalar fields, and study the degrees of freedom of the localized Lagrangian. We find that among the auxiliary scalar fields introduced, some of them are always ghost-like. That is, in the Einstein frame they develop a negative kinetic term. Because of this, except for a particular case already known in the literature, in general, it is not clear how to quantize these models and how to interpret this theory in the light of standard field theory.


I. INTRODUCTION
Among the theories introduced to describe the late-time acceleration of the universe, the modified-gravity paradigm has attracted much interest, because it explicitly states that the reason for the acceleration of the universe is due to a modified gravity law which is mostly felt at very large scales. The exploration of different ways of modifying gravity have started since the pioneeristic works in the so-called f (R) gravity. Many other theories have been proposed since then. Among others, let us mention a few of them here: the extension of f (R) theories to f (R, G) theories where G stands for the Gauss-Bonnet term, the DGP model motivated by the possible existence of spatial extra-dimensions, Galileon theories and general scalar-tensor theories of the Horndeski Lagrangian with second order differential equations. All these theories generalize the Einstein-Hilbert Lagrangian by introducing second order Lagrangians (or to Lagrangians which reduce to them, as in the f (R) case) for gravity and some extra scalar degrees of freedom. More recently a new class of modifications of gravity has been introduced, so-called non-local theories of gravity. The Lagrangian of these theories consists of terms which are non-local in the form f (· · · , −1 R, · · ·) [1]. These theories have attracted some attention both theoretically [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and phenomenologically [21][22][23][24][25][26], as a possible alternative to dark energy that renders the universe accelerated at late times.
How to deal with this kind of Lagrangian is a non-trivial topic. We will consider here the case of a general function studied recently in the literature [27] L = √ −g f (R, −1 R, · · · , −n R) , with n < +∞ , and we will try to understand its content. The meaning of such terms in the Lagrangian is obscure, and not very well understood. Some people take the point of view (see e.g. [4]), that, in order to make it sensible, the −1 operator, must be replaced by the operator −1 ret where the subscript "ret" means the retarded boundary condition. This point of view is non-standard in the conventional context of variational calculus where setting initial data is a defining constituent of a theory at level of the action. Further it is unconformable to the usual quantization procedure known for known theories based on the Lagrangian formalism.
In this paper, we take a different approach. Pursuing the standard picture of classical/quantum field theory, we interpret the non-local Lagrangian (1) as equivalent to another, local Lagrangian which can be derived by introducing auxiliary fields. The resulting Lagrangian can be studied with the usual tools of field theory. Namely we consider the Lagrangian, Having the new local Lagrangian (2), we can perform the usual study of the degrees of freedom in the theory. We then find that such a Lagrangian contains in general n ghost-like propagating degrees of freedom in any background. Special cases are also studied, such as the case ∂ 2 f /∂σ 2 = 0, separately. In all these subcases we find a finite number of ghost degrees of freedom except for the n = 1 case. These ghosts are unavoidable, in the sense that they cannot be gauged away. Therefore their presence would make these models, in general, unviable, unless one tunes the mass of these modes to values larger than the cut-off of the theory. This paper is organized as follows. In Sec. II we rewrite the general non-local Lagrangian in the form of a localized Lagrangian as given by Eq. (2) and analyze its physical degrees of freedom. In Sec. III, we focus on a special case where the Lagrangian is linear in the Ricci scalar, that is, the case ∂ 2 f /∂σ 2 = 0 in Eq. (2). Section IV is devoted to conclusion and discussions.

II. GENERAL NON-LOCAL GRAVITY ACTION
Let us consider the general action, where f is a general function of −k R (k = 0, 1, 2, · · · , max(n, m)), where n and m are positive integers, i.e. 1 ≤ (m, n) < ∞, and the function f 1 is chosen by the condition that it satisfies Thus n is the largest integer for which this inequality holds. Note that the choice of f 1 is not unique, given the function f , but this ambiguity does not affect our discussion below. As already mentioned in the Introduction, on allowing ourselves to interpret the action (1) as a model which can be redefined in terms of a local action (without e.g. assuming the d'Alambertian operators restricted on particular or prior-given boundary conditions, which would result in considering different theories), we can rewrite the action as or On taking the equations of motion for the fields σ, and λ i (i = 1, · · · , n), we find for m ≤ n, and the additional equations, for m > n. Therefore provided that ∂ 2 f 1 /∂σ 2 = 0, we obtain for m ≤ n, and additionally for m > n. We regard the original non-local Lagrangian (3) as equivalent to the new one, (5) or (6). The importance of the new action, (5) or (6), is that it is now clear how many degrees of freedom are present, and their scalar nature. In fact, we can rewrite them as and Let us make a field redefinition as which can be solved for U n provided which is guaranteed by definition, as given by Eq. (4). Notice that Eq. (22), or, in our approach, its equivalent form (4), excludes General Relativity in this class of theories. Therefore the set of theories considered here, are those ones for which it is possible to solve Eq. (21) in terms of the field U n . In fact, the field U n becomes a function of the other n + 2 fields as In this section, we also assume ∂ 2 f 1 /∂σ 2 = 0. The particular case ∂ 2 f 1 /∂σ 2 = 0 will be discussed separately in the next section. We find, on differentiating the constraint (21), that Recalling that U n is a function of the other n + 2 fields, we may rewrite the above as This constraint has solution for where we have replaced f 1 by f for notational simplicity, which is allowed because ∂f 1 /∂σ = ∂f /∂σ by definition.
Using the above result, the action is further rewritten as or On performing the following conformal transformation where we can see that the new action, for m ≤ n, becomes, up to a total derivative, where j = 1, · · · , n − 1, and On the other hand, for m > n, we have where j = 1, · · · , n − 1, k = n + 1, · · · , m and In the following, we will consider the two cases, m ≤ n and m > n, separately.
A. Case m ≤ n Let us make a further field redefinition by constant rescaling as where we have included the rescaling of U n although it is not an independent field. Notice also that this rescaling is not necessary for the function f 2 as this quantity only enters in the definition of the potential, i.e. it does not affect the kinetic term of any of the fields. Thus in total we have 2n + 1 fields that we have named q l where l = 1, · · · , 2n + 1.
Then the action takes the following form: where (k, l) = 1, · · · , 2n + 1, and the kinetic-term metric G kl is a field-dependent symmetric matrix G kl whose only non-zero elements are Let us analyze the kinetic matrix G. In order to examine whether the fields q have positive kinetic terms, we need to study whether G is positive-definite or not. For this purpose, we notice that it enters in the Lagrangian in the form of L ∋ v T · G · v where v is a (2n + 1)-dimensional vector. Hence it suffices to look for a linear transformation of v in the form v = A · w which diagonalizes the matrix G. Such a transformation is found as Then the new kinetic matrixG = A T · G · A becomes diagonal with the elements given bỹ As clear from the above, for this theory, we conclude that there always exist n ghosts independently of the sign of q 1 .

B. Case m > n
In this case we perform the field redefinition, where again we have rescaled the dependent field U n as well. Thus we have 2m + 1 fields in total, and the action takes the form, where (k.l) = 1, · · · , 2m + 1. The only non-zero elements of the kinetic-term metric G kl are where j = 1, · · · , n − 1 and r = n + 1, · · · , m. Once again the kinetic matrix enters the Lagrangian in the form v T · G · v, and whether it is positive definite or not can be examined by diagonalizing the matrix by a transformation of the form v = A · w. We find that the transformation, diagonalizes the matrix G. The new kinetic matrixG = A T · G · A becomes diagonal with the elements given bỹ Therefore, similar to the previous case, there always exist m ghosts independently of the sign of q 1 .
To summarize, for the Lagrangian of the form (3), there always exist max(n, m) ghosts provided f 1 is nonlineaer in the Ricci scalar R.

C. Case n = 0
In this case we want to discuss here the model described by the Lagrangian (6) where n = 0, that is the action can be written as and we will assume which can be described as a non-local term correction to an f 2 (R) gravity theory. The term f 2 alone is known not to introduce any ghosts, provided that ∂f 1 /∂R > 0, i.e. ∂f 1 /∂σ > 0. We can rewrite the action in the following form and define which, because of Eq. (79), can be inverted for the field σ, as σ = σ(Φ, λ 1 ), so that the action becomes On performing the conformal transformation we find where the potential V is defined as Also in this case, we can perform the following field redefinition, so that the action becomes or, the equivalent form, where (k.l) = 1, · · · , 2m + 1. The only non-zero elements of the symmetric kinetic-term metric G kl are where j = 1, · · · , m. On making the following final field redefinition then the new kinetic matrix becomes diagonal with elementŝ Therefore, for this theory, independently of the sign of q 1 , there will always exist m ghosts, independently of the sign of q 1 .

III. LINEAR-R NON-LOCAL GRAVITY THEORY
Let us consider a subcase of the theory where the Lagrangian is linear in R, which corresponds to the case ∂ 2 f 1 /∂σ 2 = 0 in the action (3) studied in the previous section. Namely we consider where we suppose n ≥ 2. For completeness, the special case n = 1, which has been already studied in the literature, will be discussed separately. We rewrite the action, along the same lines as the previous section, as for m ≤ n, and for m > n. These can be cast into the form, and respectively. Let us make the field redefinition as and let us use this equation to express U n in terms of the other fields as U n = U n (Φ − λ 1 , U j ) ; j = 1, · · · , n − 1 .
In this case the derivative of Eq. (98) gives or This implies Therefore, the actions for m ≤ n and m > n can be rewritten, respectively, as and Let us now perform a conformal transformation to the Einstein frame, as in the previous section. For m ≤ n we find where For m > n we find where Following the same procedure used in the previous section, we can rewrite the action as Let us make the field redefinition, and use this equation to express U 1 in terms of the other two fields as The derivative of Eq. (147) gives This implies Therefore, the action (146) can be rewritten as which may be transformed to the Einstein frame as We can still perform the field redefinition, Together with U 1 = u 1 /M Pl , we then find where u 1 is a function of a linear combination q 1 − q 2 , u 1 = u 1 (q 1 − q 2 ) . The action now takes the form, where i, j = 1, 2, and This matrix is positive definite when Only when this condition is satisfied, the theory can be made free from ghost [28].

IV. CONCLUSION
We considered a class of non-local gravity where the Lagrangian is a general function of −k R (k = 1, 2, · · · , n) where R is the Ricci scalar, and studied its formally equivalent local Lagrangian by introducing auxiliary fields. Taking the viewpoint that the physical degrees of freedom in thus localized theory properly represent those in the original non-local theory, we examined the kinetic term of the localized Lagrangian to see whether there are ghosts or not.
We found that, for a theory which contains a nonlinear function of R, there always exist n ghost fields for n ≥ 1, while for a theory linear in R, there always exist n − 1 ghost fields for n ≥ 2. The case of n = 1 with linear R has been already studied and it is known that one may or may not make the theory ghost-free depending on the choice of the parameters.
Thus except for the special case, this class of non-local gravity always suffers from the presence of a ghost. This result makes these theories problematic to be used as effective theories to describe the evolution of the universe at all times. The only possible way-out seems to be the case when the masses of these ghosts are individually tuned to be larger than the cut-off of the theory, e.g. larger than the Planck mass (in some other contexts, the cut-off mass can be lowered, but a careful choice of the functions f 1 and/or f 2 is needed in order to achieve large masses).
Of course, if we abandon our localization approach used to discuss the degrees of freedom, this ghost proliferation may be avoided. For example, one could regard the operator −1 in the Lagrangian as −1 ret . One would need a new formalism to deal with such a Lagrangian. In particular, it is not clear at all how to quantum the theory in this case. After all, a ghost field is fatally dangerous in quantum theory. Thus avoiding ghosts by using −1 ret in the Lagrangian may simply mean avoiding quantization.
Our result leads to discussion on the fundamentals of theories of non-local gravity with −k R (k = 1, 2, · · · , n), about how it is possible to make sense of it in terms of understanding the physical degrees of freedom. We hope this discussion will stimulate other studies on conditions for healthy extensions of a field theory and the quantization procedure for such non-standard Lagrangians.