# One-loop effective action for non-local modified Gauss-Bonnet
gravity

in de Sitter space

###### Abstract

We discuss the classical and quantum properties of non-local modified Gauss-Bonnet gravity in de Sitter space, using its equivalent representation via string-inspired local scalar-Gauss-Bonnet gravity with a scalar potential. A classical, multiply de Sitter universe solution is found where one of the de Sitter phases corresponds to the primordial inflationary epoch, while the other de Sitter space solution—the one with the smallest Hubble rate—describes the late-time acceleration of our universe. A Chameleon scenario for the theory under investigation is developed, and it is successfully used to show that the theory complies with gravitational tests. An explicit expression for the one-loop effective action for this non-local modified Gauss-Bonnet gravity in the de Sitter space is obtained. It is argued that this effective action might be an important step towards the solution of the cosmological constant problem.

## I Introduction

The discovery of the accelerated expansion of the late-time universe was the starting point of a wide spectrum of different theoretical constructions which aim is to provide a reasonable explanation of this acceleration. The simplest qualitative possibility for such construction is to consider a gravitational modification of General Relativity (GR), as compared with the introduction of extra, exotic dark components of the energy in an ordinary GR scheme. There are a number of different candidates which qualify for gravitational alternatives of dark energy (a general review of those theories can be found in review ). Recently, as a new key proposal for dark energy, non-local gravitational theories have been considered n1 ; n2 ; Capozziello:2008gu . It has been demonstrated that some versions of these non-local gravities, which depend on the curvature and its (inverse) derivatives only, are definitely able to pass the Solar System tests n1 ; n2 . A scalar-tensor representation for such theories has been developed too n2 , making explicit the connection with local gravities. Furthermore, the possibility to construct a unified description of the early-time inflation epoch with the late-time acceleration period becomes quite natural in such theories n2 (for related, string-inspired non-local theories with scalars as dark-energy models, see also n3 and the references therein).

In this paper we study classical and quantum aspects of the non-local Gauss-Bonnet gravities introduced by some of the present authors in Capozziello:2008gu , in the de Sitter space. Their massive and scalar-potential versions are proposed, and their relation with local string-inspired scalar-Gauss-Bonnet gravity will be investigated. Using the above equivalence, the corresponding de Sitter solution with a constant scalar field will be explicitly constructed. The Chameleon scenario for this theory will be investigated too, and it will be shown that the theory under consideration passes the local tests (Newton’s law, absence of instabilities). The one-loop effective action for the theory under discussion (again, using its equivalence with the local version) will be explicitly evaluated on the de Sitter space. The corresponding effective action is then explicitly obtained by using zeta-regularization, and it is used in the important discussion of the induced cosmological constant. Finally, some different version of massive non-local GB gravity, which may also be presented as a local multi-scalar-GB theory, is proposed. A classical, multiple de Sitter solution of this theory is found, where one of the de Sitter points can serve for the description of the inflationary epoch, while the other de Sitter solution, with much smaller value of the corresponding Hubble parameter, can be used for the description of the late-time acceleration period.

## Ii Non-local modified Gauss-Bonnet gravity as string-inspired scalar-Gauss-Bonnet theory

We consider the non-local Gauss-Bonnet (GB) model introduced in Capozziello:2008gu ,

(1) |

where is the matter action, the scalar curvature, the Gauss-Bonnet invariant, and the d’Alembertian operator in the metric , with determinant . Finally, is related to the Newton constant by . By introducing the scalar field , one can rewrite the action (1) in a local form, namely

(2) |

In fact, one of the field equations of the latter action gives . By substituting this expression into (2), one obtains (1). Note that the action (2) corresponds to string-inspired scalar GB gravity, which was proposed as a dark energy model in Ref. scalar-Gauss-Bonnet ; Ito:2009nk .

For the sake of generality, we add a potential to the Lagrangian density in (2), that is

(3) |

If we eliminate by using the field equation , we obtain a rather untractable non-local theory (here and in the following mean derivatives with respect to the argument). There is however a simple case, that is

(4) |

where a mass term is present, which, typically, may be thought of as a non-perturbative string correction. In case of (4), by eliminating the scalar field , we obtain

(5) |

Coming back to action (3), by assuming the metric to be the FRW one, with flat spatial section, and to depend on the cosmological time only, the equations of motion—neglecting for simplicity the contribution due to matter—read

(6) |

As usual, represents the Hubble parameter and the dot, as in , means derivative with respect to the cosmological time. If we further restrict to de Sitter space, that is, if we take and to be constant, we obtain

(7) |

Thus, we see that there is always a de Sitter solution if the potential satisfies the condition (7) for a specific value of .

As a second—non trivial—example we consider the potential

(9) |

with positive constants and . The model has a de Sitter solution, where

(10) |

In Capozziello:2008gu it has been shown that a classical de Sitter solution exists in the absence of the potential, too, but in such case the background field is a time-dependent function. Extending this formulation one can construct de Sitter solution with time-dependent scalars also in the presence of the potential. However, such solutions will not be discussed here, due to fact that we are primarily interesting in the quantum properties of non-local modified GB gravity on the de Sitter background with constant scalars.

## Iii Chameleon scenario in non-local modified Gauss-Bonnet gravity

Many dark energy models generically include propagating scalar modes, which might render a large correction to Newton’s law, if the scalar field couples with usual matter. In order to avoid this problem, a scenario called the Chameleon mechanism has been proposed Chame ; Mota:2003tc . In this scenario, the mass of the scalar mode becomes large due to the coupling with matter or the scalar curvature in the Solar System and/or on the Earth. Since the range of the force mediated by the scalar field is given by the Compton length, if the mass is large enough–and therefore the Compton length is short enough–the correction to Newton’s law becomes very small and it cannot be observed. The Chameleon mechanism has been used to obtain realistic models of -gravity Hu:2007nk ; acc ; acc1 . In this section, we are going to consider the variant of the Chameleon mechanism which is generated by the coupling of the scalar field with the GB invariant in the action. A mechanism of this kind has been proposed in Ito:2009nk . As we have shown, the non-local GB theory can be rewritten as a model with a scalar field coupled with the Gauss-Bonnet invariant. Then, the Chameleon mechanism could work in a theory of this class, which we will investigate in the present section.

For the action (3) the mass of the field is given by

(11) |

Here is the background value of in a local region, like on the Earth or the entire Solar System. In the relevant region in which we are investigating the possible corrections to Newton’s law, the curvature and the background scalar field could be almost constant. Then the equation given by the variation of the action (3) with respect to takes the following form:

(12) |

Here, is the background value of . For example, on the Earth, we find that .

For the model (4) the mass (11) is given by

(13) |

Since and , by using (7) we find

(14) |

In order that the Compton length could be , that is, , we find that , which is very small but, anyway, the Chameleon mechanism could perfectly work.

For the model (9) Eq. (12) has the form:

(15) |

and the mass (13) is given by

(16) |

Then, by using (10), we find

(17) |

Thus, when the Compton length could be .

For a different example, we can now consider the following model,

(18) |

Here and are positive constants, and is a constant. As long as , is a positive and smooth function of . In this case, Eqs. (6) have the following form

(19) |

By eliminating in the two above equations (19), we obtain

(20) |

which can be solved as follows:

(21) |

Since , as long as , we find . Then, is surely positive and therefore there are two de Sitter solutions where the Hubble rate is given by

(22) |

The smaller one could be identified with the present acceleratedly expanding universe and the larger one, with the inflation epoch in the early universe. We should note that, since there is no singularity in for , the two solutions are connected smoothly and, therefore, the transition from the inflationary era to the Dark Energy universe is possible, in principle. On Earth or at the Solar System scale, Eq. (12) acquires the following form:

(23) |

which could be solved with respect to . The mass (13) is given by

(24) |

We can conveniently choose the parameters so that becomes large enough in order not to give any measurable correction to Newton’s law.

We have thus shown that the Chameleon mechanism can actually work even in the new situation when we deal with a non-local GB theory which is equivalent to a local scalar-Einstein-GB theory, where the scalar field couples with the Gauss-Bonnet invariant. There, even though the non-local GB theory contains a scalar mode, this scalar mode does not provide any observable correction to the Newtonian law and, thus, a theory of this kind could emerge as a perfectly viable theory. Moreover, as we were able to see, the possibility to unify early-time inflation with late-time acceleration becomes quite natural in this context, what is an added bonus worth mentioning. Note also that the understanding of the equivalence principle in modified gravity thomas may be somehow different from its standard formulation.

## Iv One-loop effective action in the non-local Gauss-Bonnet gravity on de Sitter space

Here we discuss the one-loop quantization (for a review see Ref. buch ) of the classical models we will be dealing with here, on a maximally symmetric space in the Euclidean approach. One-loop contributions can be important, especially during the inflationary phase. In any case, as it was shown in Cognola:2005sg , their analysis also provides an alternative method to study the stability with respect to non-homogeneous perturbations around de Sitter solutions in modified gravitational models, in agreement with fara ; monica .

We start with the non-local GB-gravity related to the generalized model in (3). For the sake of simplicity, in the present section we will neglect the matter action , since it is irrelevant for our aims, and we shall use units in which the speed of light and the Newton constant . Our analysis is going to be very general: we will consider a non-minimal interacting term between the scalar and gravity of the form , with being an arbitrary function. In this way, the model is described by the (Euclidean) action

(25) |

When is a constant, this action reduces to Einstein’s gravity minimally coupled to the scalar field.

In accordance with the background field method, now we consider the small fluctuations of the fields around the de Sitter manifold , of the kind

(26) |

This is a classical solution if the potential satisfies the conditions

(27) |

which are the analog of (7) written in terms of the scalar curvature . We see that, if ,then the Minkowski solution emerges.

For the arbitrary solutions of the field equations, we set

(28) |

and perform a Taylor expansion of the action around the de Sitter manifold, up to second order in the small perturbations . Before we proceed with the expansion, it is convenient to write the action in order to take into account the fact that is a topological invariant. Then we observe that the non minimal interacting term between gravity and the scalar field, around the background solution, can be written in the form

(29) | |||||

Here is the value of the Gauss-Bonnet invariant evaluated on the de Sitter background. We note that the first term on the right hand side of the latter equation does not give contributions to the classical field equations and can be dropped out, while the second, proportional to , modifies the scalar potential. Then, for our aim it is convenient to write the classical action (25) in the final form

(30) |

where we have introduced the effective potential

(31) |

One can check that action (30) is equivalent to the original one (2) when .

We are now ready to perform the expansion. In the following we will use the compact notation , , and so on. After a straightforward computation, along the same lines as for one-loop -gravity Cognola:2005de , one obtains

(32) |

where represents the quadratic contribution in the fluctuation fields . Disregarding total derivatives, this reads

(33) | |||||

Here and represent the covariant derivative and the Laplace operator, respectively, in the unperturbed metric . We have also carried out the standard expansion of the tensor field in irreducible components frad ; buch , that is

(34) |

where is the scalar component, while and are the vector and tensor components, respectively, with the properties

(35) |

As is well known, invariance under diffeomorphisms renders the operator in the sector non-invertible. One needs a gauge fixing term and a corresponding ghost compensating term. We consider the class of gauge conditions

parameterized by the real parameter . As gauge fixing, we choose the quite general term buch

(36) |

where the term proportional to is the one normally used in Einstein’s gravity. The corresponding ghost Lagrangian reads buch

(37) |

where and are the ghost and anti-ghost vector fields, respectively, while is the variation of the gauge condition due to an infinitesimal gauge transformation of the field. It reads

(38) |

Neglecting total derivatives one has

(39) |

where we have set

(40) |

In terms of irreducible components, one finally obtains

(41) | |||||

(42) | |||||

where ghost irreducible components are defined by

(43) |

In order to compute the one-loop contributions to the effective action we have to consider the path integral for the bilinear part

(44) |

of the total Lagrangian, and take into account the Jacobian due to the change of variables with respect to the original ones. In this way, the Euclidean one-loop partition function reads frad ; buch

(45) | |||||

where and are the Jacobians coming from the change of variables in the ghost and tensor sectors, respectively. They read buch

(46) |

Finally, the determinant of the operator , acting on vectors assumes, in our gauge, the form

(47) |

while it is trivial in the simplest case . By () we indicate the Laplacians acting on scalar, vector, and transverse tensor fields, respectively.

Now, a straightforward computation, disregarding zero gravity modes and the multiplicative anomaly as well (Elizalde:1997nd )), leads to the expression of the one-loop contribution to the Euclidean partition function. It is a quite involved expression, which depends on the gauge parameters, and for this reason we will only write it explicitly in the Landau gauge corresponding to the choice . On-shell, is independent of the gauge and it is compatible with a similar expression obtained in frad for Einstein’s theory with a cosmological constant , but in presence of a scalar field. In fact, we get

(48) |

The latter term is due to the scalar field, but it also depends on the coupling with the Gauss-Bonnet invariant. It has to be noted that the on-shell partition function is obtained by imposing conditions (27) and . The latter condition in the expression of the gauge-dependent one-loop partition function is equivalent to dropping all terms proportional to .

Now, we explicitly write the off-shell partition function in Landau’s gauge. It reads

(49) | |||||

The quantities , , , which depend in general on , are the roots of the third-order algebraic equation

(50) |

where

(51) | |||||

The one-loop effective action can now be evaluated by making use of zeta-function regularization, and actually computing the zeta-functions related to the differential-elliptic Laplace-like operators eli94 . Using the same notations as in Ref. Cognola:2005de , we have (see the Appendix)

(52) | |||||

where

(53) |

(54) |

and , , .

The effective equation for the induced cosmological constant can be obtained by varying the effective action with respect to frad . We would now like to study the role of the Gauss-Bonnet term. For the sake of simplicity, we choose a scalar potential satisfying the conditions , . This means that, in the absence of the term, namely when , the potential is constant and the action becomes the Einsteinian one with a cosmological constant , while, when , there is a coupling of the form .

We consider separately the two cases and . In the first case we get , while and so they do not give a contribution. Thus, we get

(55) |

Here we have introduced the dimensionless variable , so the effective equation for the induced cosmological constant can be obtained by taking the derivative of the effective action with respect to , since

(56) |

where is the classical action on the background, which reads

(57) |

(58) | |||||

For the first case, at lower order in , from (56) we obtain the equation

(59) | |||||

In the second case we also have vector and tensor contributions, as in the previous one but, in addition, we get three scalar contributions related with the roots of (50). Using (52) and (56), at lowest order in we obtain

(60) | |||||

In Figs. (1) and (2) we have plotted the right hand sides of Eqs. (59) and (60), respectively, as functions of , for two different sets of values of the parameters. Notice that the zeros of these functions correspond, in each case, to a zero value of the induced cosmological constant. Aside from the cases here explicitly depicted, it can be easily seen that in our effective models the possibilities to get an induced cosmological constant which is exactly zero are reasonably high, since for values of the parameters lying in wide regions of the domain of expected values, either one or two roots of the equations exist, yielding the value of the induced cosmological constant exactly zero. As is well known frad , there is a possible resolution of the the induced -term problem coming from the contribution of higher-loop terms. This is explained in frad precisely in the example of an effective action in pure Einstein gravity for a de Sitter background: one can get a very small effective term irrespective of the tree level cosmological constant. But an even better possibility is to start from a zero tree level cosmological term, since quantum corrections will respect this property. We see that in our model there are good chances to realize this latter situation. One remark is in order. We got the one-loop effective action for non-local GB gravity using its classical equivalence with local scalar-GB gravity and working in terms of such local theory. It is quite well-known that such classical equivalence may be broken already at one-loop level. However, the equivalence is restored on-shell, i.e. using the one-loop corrected equations of motion. That is precisely the situation in which the induced cosmological constant has been here discussed.

Now, we come back to the original action (2) with and , which is equivalent to the non-local action (1). Also for this case there is a de Sitter solution , but is not a constant. This means that the term in (29) cannot be dropped out and, thus, it gives a contribution to . In principle, it is possible to take such contribution into account, but technically this is quite complicated, since it contains a lot of independent terms which mix scalar with vector and tensor components, so that the partition function is given in terms of the determinant of an involved matrix of differential operators.

When the original model has, however, a Minkowskian solution. This means that, using (52) with , we can compute the cosmological constant induced by quantum fluctuations around the Minkowski background, and this may be interpreted as the spontaneous creation of a de Sitter universe starting from a flat one, which is a quite interesting feature.

## V Other classical non-local GB models and their de Sitter solutions

Another model giving rise to an interesting non-local action is the following

(61) |

where is a dimensional constant and is an adequate function of . By introducing three scalar fields, , , and , one can rewrite the action (61) under the following form

(62) |

We may further add a potential to the action

(63) |

In the FRW universe, this action leads to the following equations

(64) |

However, after adding the potential , it is difficult to get the corresponding non-local action explicitly.

When , assuming that , , , and , with constant , , , and , the equations in (V) reduce to the algebraic ones

(65) |

We can solve Eqs. (65) with respect to , , and as follows: , , , and we find

(66) |

Here .

For example, if we choose

(67) |

(66) gives

(68) |

which has a trivial solution corresponding to the flat background and

(69) |

which corresponds to the de Sitter universe.

As another example, one can choose

(70) |

Here , , and are positive constants. In order that can be real, we restrict the value of as . Then Eq. (66) yields