Generalised Quadratic Curvature, Non-Local Infrared Modifications of Gravity and Newtonian Potentials

Generalised Quadratic Curvature, Non-Local Infrared Modifications of Gravity and Newtonian Potentials

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Metric theories of gravity are studied, beginning with a general action that is quadratic in curvature and allows arbitrary inverse powers of the d’Alembertian operator, resulting in infrared non-local extensions of general relativity. The field equations are derived in full generality and their consistency is checked by verifying the Bianchi identities. The weak-field limit is computed and a straightforward algorithm is presented to infer the post-Newtonian corrections directly from the action. This is then applied to various infrared gravity models including non-local dark energy and non-local massive gravity models. Generically, the Newtonian potentials are not identical and deviate from the behaviour at large distances. However, the former does not occur in a specific class of theories that does not introduce additional degrees of freedom in flat spacetime. A new non-local model within this class is proposed, defined by the exponential of the inverse d’Alembertian. This model exhibits novel features, such as the weakening of the gravity in the infrared, suggesting de-gravitation of the cosmological constant.

a]Aindriú Conroy, b]Tomi Koivisto, a]Anupam Mazumdar a]and Ali Teimouri


[a]Consortium for Fundamental Physics, Lancaster University, Lancaster, LA1 4YB, UK \affiliation[b]Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden

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1 Introduction

It has been known for some time that higher derivative theory of gravity can be renormalisable but only at the cost of unitarity [1]. There is some evidence that in an infinite-order higher derivative, i.e. non-local theory of gravity, one can avoid the issue of ghosts and other pathologies while recovering general relativity (GR) at low energies. The propagator for the most general metric theory was recently derived in [2], see also e.g. [3, 4, 5]. In non-locally improved theories, gravity becomes weak in the ultraviolet (UV), yielding non-singular black hole, gravitational wave and cosmological solutions. Many aspects of ghost-free and singularity-free gravity have been studied in the context of early Universe cosmology [6, 7, 8, 9, 2, 10, 11, 12].

However, non-local operators in the gravity sector may also play a role in the infrared (IR). In particular, they could filter out the contribution of the cosmological constant to the gravitating energy density, possibly providing the key to solving one of the most notorious problems in physics [13], see also [14, 15, 16, 17, 18]. Cosmological implications of non-local terms in the gravity action, such as , have been investigated for instance motivated by the possibility to render the Euclidean action finite [19]. More recently, many studies of cosmologies in infrared, non-locally modified gravity models have been undertaken, stimulated by the problems of dark matter and dark energy [20, 21]. In this paper we will focus on the infrared non-localities of gravity in a very generic manner. We consider metric covariant quadratic curvature theories of gravity, allowing arbitrary (including up to infinite order) inverse derivatives in the action. Typically the corrections are then suppressed by some IR scale, which we shall denote as everywhere in the following.

Deser and Woodard proposed non-local corrections to the gravity action in the form [22]. By such means one could perhaps address some of the fine-tuning problems of dark energy: the curvature scalar is negligible with respect to the radiation density early on, which might help to understand why the corrections become significant only during the matter dominated epoch, and on the other hand, being a dimensionless combination, modifications at the scale of dark energy might be generated without introduction of tiny mass scales into the theory [23]. The models have been studied extensively [24, 25, 16, 17, 26, 27, 28, 29], and in particular, cosmological perturbations have been analysed in Refs. [30, 31, 32, 33], with the conclusion that the current structure formation data clearly favours GR over the models when the background evolution of the latter is fixed to be identical to the CDM. In fact, this model is ruled out at the confidence level of several sigmas [33]. Furthermore, in more general models, involving tensorial non-local terms such as , an additional, potentially dangerous, growing mode appears in the cosmological perturbations [34, 35].

However, this does not render all non-local gravity dark energy models incompatible with cosmological constraints, as demonstrated by two interesting viable examples recently put forward by Maggiore et al. One model was defined by a term added to the Einstein field equations [36, 37, 38] and its viability was verified against a number of large scale structure data [39]. The other model was defined by adding a modification to the Lagrangian [40]. This introduces a single-parameter alternative to the CDM cosmology, whose perturbation evolution has also been shown to produce a matter power spectrum that matches well with current measurements [41]. The possibility that dark matter could be a manifestation of a non-local deviation from Einstein’s gravity has been investigated by several authors as well [42, 43, 44, 45, 46, 47, 48, 49].

Non-local gravity poses many theoretical and technical issues. The initial value problem has been considered [50, 51] and practical methods of solving non-local systems of differential equations, such as the diffusion equation approach, have been developed [52, 53, 54, 55]. In the non-local framework, the graviton can be given a mass without introducing an additional metric [56, 57]. One could thus speculate that non-local gravity may describe massive ghost-free gravity, once the additional metric has been integrated out along the lines of [58]. Indeed it has been argued that the infrared non-local gravity models proposed to date can only be taken as phenomenological effective theories [37, 36, 59]. Two techniques have been employed to generate causal and conserved field equations: either by varying an invariant non-local effective action and then enforcing causality by the ad hoc replacement of any advanced Green’s function with its retarded counterpart, or by introducing causal non-locality into a general ansatz for the field equations and then enforcing conservation. These approaches are implemented in the two examples of dark energy models mentioned above, respectively.

In this paper we adopt the first approach: our starting point is a general non-local action. The most general linear equations were analysed in [2] and [60], where the authors have presented the most general non-linear field equations for non-local gravity up to quadratic order in the curvature, with the aim to understand the UV properties of gravity. Here, we proceed by extending the analysis into the IR. We begin, in Section 2 by describing the quadratic action and deriving the field equations in full generality. Due to the complicated nature of these calculations, it is useful to perform a consistency check by verifying that they satisfy the Bianchi identities. This non-trivial calculation is outlined briefly. In Section 3 we consider the weak-field limit of these theories and present an algorithm to compute the Newtonian potentials. These can be very useful in determining the observational viability of a theory at the level of astrophysics and classical tests of gravity within the Solar system. In the following section, we apply our formalism to specific models, by way of three examples, before proposing a new model featuring the exponential of the inverse d’Alembertian operator. We give some concluding remarks in Section 5 and some technical details have been confined to the appendices, along with a scalar presentation of a restricted class of models (Appendix A).

2 General Quadratic Action and Field Equations

In a pioneering paper, Schmidt considered the field equations in quadratic-curvature gravity theories of arbitrarily high derivative order [61], then restricting to modifications of GR in terms of the Ricci scalar. Only recently, the full nonlinear analysis was generalised to arbitrary curvature terms [60], motivated by the progress made with such theories in Ref. [2] where it was shown that gravity in the UV can be made asymptotically-free without violating basic principles of physics such as unitarity and general covariance. Here we extend the action into the IR, and describe it as follows


with , where is a constant to ensure correct dimensionality and is some infrared mass scale. To derive the field equations from this action, we need to first understand the properties of the inverse d’Alembertian operator under variations of the metric.

2.1 Variation of the Inverse D’Alembertian

We compute the equations of motion of (2) by straightforwardly taking the variation of the action. We note that most of the terms, can be found by adhering to the prescription given in [60]. However, one particular brand of term requires more attention, namely the -type terms, which we shall discuss briefly below.

Following the prescription of [34], [62], in order to preserve causality and the conservation of the energy-momentum tensor, we consider only solutions with vanishing homogenous solution, namely where

Further details of this can be found in Appendix B. Applying the product rule, we find

Using the defintion of the function (2) and substituting (2.1) into the above equation, we find the general form of to be

Finally, from [60] , we know how acts upon the curvature scalar, Ricci tensor and Weyl tensor and following the prescription therein, we can read off the equations of motion. Details of these variational terms are given in Appendix C.

2.2 Equations of Motion

The field equations are:

where we have defined the following tensors

2.3 Bianchi Identity Test

The stress-energy tensor of any minimally coupled diffeomorphism invariant gravitational action must be conserved,

Furthermore, it should be noted that the Bianchi identities should hold for each ’part’ of the action (2), with the first ’part’ comprised of the Einstein-Hilbert action and the following three accounting for the , and sections, as each of these sections are independent of each other. Clearly, the Einstein-Hilbert action satisfies the Bianchi identity as the Einstein tensor satisfies .

Let us begin with the piece

Expanding the tensors given in (2.2), we may write the equation of motion for (2.3) as follows:

We then take the covariant derivative and cancel like terms

Next we use

to find

and substitute to obtain

All remaining terms will cancel by noting that

and thus the Bianchi identities are satisfied. A similar method may be used to test for the Bianchi identities of the entire action using the general formula

Further details are given in Appendix D for the and pieces of the action.

3 Weak-Field Limit

In order to make a step towards understanding the physical implications of the theories analysed in Section 4 and to make contact with observations, let us consider the weak-field limit of the general field equations.

From and the definition of the Christoffel symbols and the Riemann tensor, one can find the weak-field limit of the Riemann tensor, Ricci tensor and curvature scalar,

as well as the Weyl tensor which is somewhat lengthier and is given in appendix E.

In the weak-field limit, we may discount terms of order and higher. With this in mind, the equation of motion (2.2) reduces significantly,

into which we can then substitute the above values for the Riemann tensor, Ricci tensor and curvature scalar (3) to obtain

Here we have set for convenience. We can rewrite this as

where we have defined


and have recovered the same constraints as in the UV 111We note that the forms of these constraints differ to those of Ref. [2]. This is due to different conventions, namely, in [2], the authors take the signature to be "mostly negative", where as in Ref. [60], we take the signature to be “mostly positive" with . Secondly, the presence of the Weyl tensor rather than the Riemann tensor in the action has an effect on the terms. Having said this, when these convention changes are taken into account, we find that the above constraints are the same as those in [2] and [60] with the exception that we are now considering in the IR as opposed to in the UV. [2, 60]:


These equalities we found by explicit evaluation of the respective terms, can be understood as a consequence of the Bianchi identities. In the linearised limit, , and it suffices to take the partial derivative of (3) as


This divergence should vanish identically, and when the coefficients of each independent term is zero due to (2), it does. It is this classical conservation structure of the theory that also sets the coefficients of the effective stress energy terms and identical (denoted here) in the first place.

We then close this section with a brief remark concerning massive gravity. The Fierz-Pauli term would have the form . We can now indeed recover such a term, without resorting to Lorenz violation or additional metrics, with an action specified by an arbitrary by setting and . However, there will inevitably then also appear additional terms in the stress energy tensor, due to (2), and thus the linearised theory doesn’t quite coincide with the pure Fierz-Pauli theory.

3.1 Newtonian Potentials

We wish to compute the Newtonian potentials. In order to do so, we consider the weak field (i.e. ) static (i.e. ) limit. The trace and the -component of the field equation (3) are


where we have assumed negligible pressures for , so that and . We then impose the spherically symmetric metric

and note

so that the pair of equations (4,5) becomes


Solving for , we can then, upon performing a Fourier transform and restoring the , express the Newtonian potential as the integral

where222However, we should point out a subtlety that though there is no ambiguity in the case of derivative operators, but we have identically that , in the case of inverse derivative operators implies a choice of boundary conditions for the operator . These boundary conditions should be understood as specification of the operator and thus a property of the theory itself rather than free parameters for each solution. The boundary conditions adopted here amount to setting the homogeneous solution of the flat-space Green functions to zero, as seems most reasonable in this case. -A constant associated to the homogeneous modes was tuned in the screening mechanism of [16, 17] to cancel the cosmological constant in cosmological background. It is unclear if such a prescription for the operator would be viable in other backgrounds. , and is the mass of the test particle. Similarly we get for


Thus we have now a complete algorithm to determine the Newtonian potentials of an arbitrary metric theory of gravity: given any form of local or non-local action, one may readily expand it up to quadratic order in the curvature, read off the functions and and obtain the post-Newtonian potentials by performing the above two integrals (3.1, 3.1).

We immediately see that, generally speaking, these will differ from each other, We describe their ratio using the Eddington parameter , which is defined as


and is constrained by the Cassini tracking experiment to have the following upper bound [63] and therefore the discrepancy can provide useful constraints on generic non-local models. However, we also notice that in the class of theories with , i.e. , the Newtonian potentials will be identically the same, thus , but the potential can still deviate from the behaviour at large distances. This is in complete accordance with the results of [2, 5], where the class of theories was found to introduce no new degrees of freedom, since the additional scalar contribution in the propagator disappears at the limit , while the function still modulates the usual graviton propagator. In the case of non-analytical (inverse powers of the d’Alembertian) functions, however, the propagator may be an ill-defined object, the intuition is retained here that the special class of theories is devoid of an extra scalar and thus features only one independent Newtonian potential in the weak field limit. To recapitulate, we have two classes of theories:

  • , i.e. . In this case, we have an extra scalar degree of freedom and the two gravitational potentials are not independent, i.e. .

  • , i.e. . In this case, there are no additional modes and thus i.e. .

In the following section, we will consider explicit examples from both classes of theories.

4 Examples

Armed with the machinery to study generic metric theories, we illustrate its power by applying it to several non-local models found in the literature, before proposing a new model featuring the exponential of the inverse d’Alembertian operator.

4.1 The Model

The non-local model proposed by Deser and Woodard [22] is defined by the action


As mentioned in the introduction, this model has been ruled out as an alternative to dark energy due to its impact on the structure formation [33]. It is, however, instructive to consider it first as perhaps the simplest example of a non-local modification of gravity in the infrared. Indeed, the parameter turns out to be simply a constant up to the first order in the post-Newtonian expansion. For this order we only need the quadratic term


We may read off from the action that and as well as the functions and , upon referring back to (1). Solving the integral in (3.1,7) for these values of and , we obtain


Newton’s constant is thus shifted - an occurrence which can be nullified with a redefinition. To first order, we obtain . This agrees with an earlier result given in [30] and derived using a scalar field formulation of the theory (see appendix A for such a treatment of theories). One can then test the validity of non-local models by substituting the prescribed value of . For example, [32] gives as follows

from which we deduce that

which is well within the Cassini bound. Conversely, for models of the type

as in [64], with and which is of order unity, we find that the best-fit values of the model are in disagreement with our constraints. In fact, must be less than to be contained within the limits.

4.2 Non-Local Massive Gravity

As explained in section 3, in the context of non-local theories, the graviton may acquire mass without the introduction of an external reference metric [56, 57]. The IR part of an action that has been proposed for this purpose reads

As we will see shortly, we can now interpret the IR scale as the mass of the graviton. From the action, we read off

so that


We observe that this model belongs to the class where there are no additional degrees of freedom. To solve for the gravitational potential (3.1), we need to integrate , that is

where represents the mass of the graviton. We observe that there are poles at on the upper half plane and on the lower half plane. We will take each pole separately and use the general contour integral formula

Pole is on the upper half plane so we only consider the portion of which encircles the pole, so that

which we consider of the form

with and , so that

Similarly for the pole

where, in this case we are moving into the lower half plane so that


Solving the integral (3.1), we find

We thus obtain precisely the expected Yukawa-type correction for massive gravity. We plot the solution (4.2) in Figure 2.

4.3 The Model and Generalisations

A variation of the previous model (4.2) where the tensorial piece is omitted was studied in Refs. [40, 41], where the instability arising from tensorial non-localities is avoided [34, 35]. However, as we have learned, an additional scalar mode will appear in flat space. The action to consider is333Here we chose the opposite sign for the -term from [40]. In the case of a scalar field at least, that sign choice would correspond to tachyonic mass-squared, which, however, is the choice that has been shown to lead to interesting cosmology [40, 41]. The Newtonian limit has been calculated for that case in Ref. [38], expectedly with different results from what we obtain here. Setting , one obtains oscillating -type corrections instead of the exponential we find here and in (4.2).


Thus, and . We proceed analogously to the previous two cases, further details can be found in Appendix F. As before, the integrals in (3.1, 7) can be completed by calculating their residuals, resulting in the following:


Thus, the gravitational potentials differ from each other and display the usual behaviour at distances , as expected.

Here, one may use the Cassini bound to put limits on the mass of the graviton in order to verify if it lies on the dark energy scale. We remind the reader that in order to be a suitable dark energy candidate, the graviton must be of the order of the present Hubble parameter . Taking

we then expand the exponential terms to first order, whilst setting or 1 billion kilometres as in the original Cassini experiment [63], from which we find the upper bound on the mass of the graviton in a theory of this type to be

Subsequently, with a small enough mass, this particular model of massive gravity is within the permissible limits of being a suitable candidate for dark energy.

The asymptotic value of the Eddington parameter coincides with that predicted for Solar System measurements in fourth order (i.e. local) metric theories [65] (for a unified analysis covering also e.g. the Palatini and non-minimally coupled theories, see [66]). The Newtonian potentials behave contrary to those in models: near to the source and at large distances . This can readily be seen by plugging a constant into our expressions, for instance :


It is nontrivial that non-local models of the type (12) exhibit the opposite behaviour with respect to the fourth order local models. Were this not the case, however, the former would of course be immediately ruled out.

We have checked that by considering higher powers of the inverse box operator, for example a model, one obtains qualitatively similar behaviour with exponential modification terms, further modulated by oscillatory functions. We illustrate this in figure 1, where we plot the for both the and models.

Figure 1: The Eddington parameter as a function of the distance from the source in models. The thick black line is for the model (12), the thin blue dotted line is for the local gravity and the dashed red line for a model. For higher powers of , the behaviour is qualitatively the same.

4.4 Degravitation with and