Timothy Clifton
email: tclifton@astro.ox.ac.uk

Máximo Bañados
email: maxbanados@gmail.com

Constantinos Skordis
email: skordis@nottingham.ac.uk

The Parameterised Post-Newtonian Limit of Bimetric Theories of Gravity

Keywords: numbers: 04, 25, Nx, 04, 50, Kd

We consider the post-Newtonian limit of a general class of bimetric theories of gravity, in which both metrics are dynamical. The established parameterised post-Newtonian approach is followed as closely as possible, although new potentials are found that do not exist within the standard frame-work. It is found that these theories can evade solar system tests of post-Newtonian gravity remarkably well. We show that the perturbations about Minkowski space in these theories contain both massless and massive degrees of freedom, and that in general there are two different types of massive mode, each with a different mass. If both of these masses are sufficiently large then the predictions of the most general class of theories we consider are indistinguishable from those of general relativity, up to post-Newtonian order in a weak field, low velocity expansion. In the limit that the massive modes become massless, we find that these general theories do not exhibit a van Dam-Veltman-Zakharov-like discontinuity in their γ parameter, although there are discontinuities in other post-Newtonian parameters as the massless limit is approached. This smooth behaviour in γ is due to the discontinuities from each of the two different massive modes cancelling each other out. Such cancellations cannot occur in special cases with only one mass parameter, such as the Isham-Salam-Strathdee theory.

Introduction

There exists a long history in gravitational physics of considering theories that are generalisations of, or alternatives to, general relativity. These theories can take on a variety of different forms, and often involve the introduction of extra scalar [START_REF] Brans | Mach's principle and a relativistic theory of gravitation[END_REF], vector [START_REF] Will | Conservation laws and preferred frames in relativistic gravity. I. Preferred-frame theories and an extended PPN formalism[END_REF] or tensor fields [START_REF] Isham | F-dominance of gravity[END_REF] in the gravitational action. Here we consider the latter of these: Bimetric theories in which there are two dynamical rank-2 symmetric tensor fields, rather than the usual one. Our goal is to calculate the Parameterised Post-Newtonian (PPN) limit of a general class of these theories, so that their consequences for observational and experimental gravitational physics can be determined in a straightforward way.

The PPN formalism [START_REF] Nordtvedt | Equivalence principle for massive bodies. II. Theory[END_REF][START_REF] Will | Theoretical frameworks for testing relativistic gravity. II. Parameterized post-Newtonian hydrodynamics and the Nordtvedt effect[END_REF] is now the standard frame-work within which investigations of the phenomenology of relativistic gravitational physics are performed [START_REF] Will | Theory and experiment in gravitational physics[END_REF]. It has at its heart a simple and compelling rationale; that by observationally constraining terms in a general 'test metric', it is then possible to impose observational and experimental constraints on a variety of different metric based gravitational theories without having to go through the rigmarole of directly calculating their physical consequences every time. This allows the space-time geometry of the physical environments in question to be constrained in a theory independent way, as well as supplying a quick and direct route to determining the observational validity (or otherwise) of particular theories.

The problem of determining the PPN limit of bimetric theories of gravity has been approached in the past. The PPN limit of the bimetric theory of Rosen [START_REF] Rosen | A bi-metric theory of gravitation[END_REF] was found by Lee, Caves, Ni and Will [START_REF] Lee | Theoretical frameworks for testing relativistic gravity. V. Post-Newtonian limit of Rosen's theory[END_REF]. The PPN limits of Rastall's bimetric theory [START_REF] Rastall | A theory of gravity[END_REF][START_REF] Rastall | A note on a theory of gravity[END_REF], and Lightman and Lee's bimetric theory [START_REF] Lightman | New two metric theory of gravity with prior geometry[END_REF] are also known. These theories, however, contain only one dynamical rank-2 tensor each, the other being a priori specified as being Riemann flat. Such non-dynamical fields can cause serious problems for these theories, particularly with respect to the emission of gravitational waves from binary systems. For example, Rosen's theory predicts an unobserved increase in the rotational period of the Hulse-Taylor binary pulsar system PSR 1913+16 [START_REF] Will | Dipole gravitational radiation in Rosen's theory of gravity: Observable effects in the binary system PSR 1913+16[END_REF], and it can be argued that similar behaviour should be expected in all theories with 'priori geometry' [START_REF] Will | Theory and experiment in gravitational physics[END_REF]. The bigravity theories considered in the present article do not have any such non-dynamical fields, and are therefore not expected to fall foul of binary pulsar observations in the way discussed above. To the best of our knowledge, the case of determining the PPN limit of a general class of bimetric theories with two dynamical rank-2 tensor fields has yet to be investigated in detail. It is this subject that we intend to investigate here.

As motivation for this study we use the recent astronomical observations that have revealed that most of the matter in the Universe is in the form of unknown "dark components" (i.e. particles or fields that are not present in the standard model, and that do not interact electromagnetically). Despite ongoing efforts these fields have yet to be observed directly, and it seems natural to consider extensions of general relativity that may explain them. In this context, bimetric gravity has recently been put forward as an alternative to both dark matter and dark energy. For details of the way in which the extra degrees of freedom in these theories can be made to reproduce the observations usually attributed to the dark components of the Universe we refer the reader to [START_REF] Damour | Non-linear bigravity and cosmic acceleration[END_REF][START_REF] Bañados | Eddington-Born-Infeld action for dark matter and dark energy[END_REF][START_REF] Bañados | Eddington-Born-Infeld gravity and the large scale structure of the Universe[END_REF][START_REF] Bañados | A note on bigravity and dark matter[END_REF][START_REF] Skordis | Eddington-Born-Infeld theory and the dark sector[END_REF][START_REF] Milgrom | Bimetric MOND gravity[END_REF][START_REF] Clifton | FRW cosmology in Milgrom's bimetric theory of gravity[END_REF]. Here we will not concern ourselves with the cosmological implications of these theories directly, but will instead study their weak field limit. If theories of this type are to be considered observationally viable it is absolutely necessary that we properly understand their weak field limit, and how they are constrained by observations of post-Newtonian gravitational phenomena.

The plan of this paper is as follows. In Section 2 we introduce and discuss the bimetric theories of gravity that we will be investigating, giving their gravitational action and field equations. In Section 3 we briefly introduce the PPN formalism, the perturbative expansion it relies upon, and the way in which it is used to constrain gravitational theories. Section 4 contains a calculation of metric perturbations up to Newtonian accuracy. We then proceed in Sections 5 and 6 to derive the full post-Newtonian limit of the metric perturbations about both of the metrics involved in these theories. In Section 7 we transform into the standard post-Newtonian gauge, and calculate the form of the perturbations in the metric that couples to matter. Section 8 provides a discussion of the extent to which these theories can be subject to observational constraint, and in Section 9 we provide a brief discussion of our results.

Bimetric Theories of Gravity

One of the most straightforward ways to extend general relativity is to take the already existing concept of a dynamical rank-2 symmetric tensor field, and replace it with a multiplet of N g such fields. Multigravity theories can then be constructed by considering an extension of the Yang-Mills approach to gauge theories. That is, if we write our N g fields as g a µν , where a = 1, 2, ..., N g , then our gravitational action can be written

S[g a ] ∼ d 4 x Ng a=1 det(g a )R(g a ) -I int (g a ) , (1)
where I int is an interaction term that can depend on any or all of the metrics. Although it is impossible to construct a non-trivial I int term that preserves the original N gdimensional symmetries of the free theory [START_REF] Boulanger | No consistent cross-interactions for a collection of massless spin-2 fields[END_REF], the action above is manifestly invariant under diagonal diffeomorphisms acting on all metrics. The potential therefore breaks the symmetry group down to the diagonal subgroup of diffeomorphisms acting on all metrics with the same parameter. This approach to gravity has a long history, especially the N g = 2 incarnation known as 'bigravity'. It began in the early 1970s with the pioneering work of Isham, Salam and Strathdee [START_REF] Isham | F-dominance of gravity[END_REF], while today actions like (1) are mostly used as covariant actions for massive gravitons. A classification of different interaction terms, as well as a variety of different motivations for bigravity, are explored in [START_REF] Damour | Effective Lagrangians and universality classes of nonlinear bigravity[END_REF].

In order to make progress we must specialise the general action (1) somewhat. The theory we will consider for the rest of this paper is therefore the N g = 2 bigravity theory

S = 1 16πG d 4 x √ -g(R -2Λ 0 ) + √ -q(K -2λ) - 1 l 2 √ -q κ 0 J + κ 1 J 2 + κ 2 (q -1 ) µν (q -1 ) µν . ( 2
)
Here we have written the two metric fields as g µν and q µν . The Ricci scalars constructed from these metrics are written as R and K, respectively, and the two constants Λ 0 and λ are the bare cosmological constants on g µν and q µν . The three constants κ 0 , κ 1 and κ 2 parameterise the interactions between g µν and q µν . The inverse (q -1 ) µν is defined such that (q -1 ) µα q µβ = δ α β , and all raising or lowering of the indices of these tensors is done with g µν , so that e.g. (q -1 ) µν ≡ (q -1 ) αβ g αµ g βν . We have also defined J ≡ (q -1 ) µν g µν = (q -1 ) µ µ . The theory above is a generalization of the often considered Isham-Salam-Strathdee (ISS) action

S = 1 16πG d 4 x √ -g(R -2Λ 0 ) + σ -f ( K -2 λ) - σ l 2 -f I int , (3)
with the interaction term

I int = (g µν -f µν )(g αβ -f αβ )((f -1 ) µα (f -1 ) νβ -(f -1 ) µν (f -1 ) αβ ).
Here the scalar K is the Ricci scalar constructed from the metric f µν , and (f -1 ) µα f µβ = δ α β . The generalised theory (2) can be seen to reduce to the ISS action (3) under the metric rescaling q µν = σf µν , together with the identifications κ 0 = 6, κ 1 = -σ and κ 2 = σ, and the redefinition λ = λ/σ -6/(l 2 σ).

Of course, we also want to couple matter to these theories. We do this by including a Lagrangian density of the form L m = L m (ψ, ĝµν ), where the matter fields ψ are coupled to a linear combination of the metrics

ĝµν = mg µν + nq µν , (4)
where m and n are constants. After varying with respect to each of the metric fields this then results in the field equations

l 2 R µν -8πGl 2 m ĝ g T µν - 1 2 g µν T α α -α 0 g µν (5) = - q g (κ 0 + 2κ 1 J) (q -1 ) µν - 1 2 g µν J + 2κ 2 (q -1 ) µα (q -1 ) α ν - 1 2 (q -1 ) αβ (q -1 ) αβ g µν , and
l 2 K µν -8πGl 2 n ĝ q q µα q νβ T αβ - 1 2 q µν T αβ q αβ -αq µν (6) = - 1 2 κ 1 J 2 + κ 2 (q -1 ) αβ (q -1 ) αβ q µν + [κ 0 + 2κ 1 J] g µν + 2κ 2 (q -1 ) µν ,
where α 0 ≡ Λ 0 l 2 and α ≡ λl 2 , and where T µν is defined by

T µν = - 2 √ -ĝ δL m δĝ µν . (7)
Expressing the field equations in this form, without R and K, the perturbation equations take on their simplest form. Now, we should be aware that these theories contain within their spectrum a massive spin-2 field, and that such terms often have associated with them certain discontinuities and instabilities [START_REF] Blas | Global structure of bigravity solutions[END_REF][START_REF] Blas | Bigravity and Lorentz-violating Massive Gravity[END_REF][START_REF] Blas | Aspects of Infrared Modifications of Gravity[END_REF][START_REF] Berezhiani | Spontaneous Lorentz breaking and massive gravity[END_REF][START_REF] Berezhiani | Exact Spherically Symmetric Solutions in Massive Gravity[END_REF][START_REF] Berezhiani | Gravity Modification with Yukawa-type Potential: Dark Matter and Mirror Gravity[END_REF][START_REF] Blas | Lorentz Breaking Massive Gravity in Curved Space[END_REF]. These include the van Dam-Veltman-Zakharov (vDVZ) discontinuity [START_REF] Van Dam | Massive and massless Yang-Mills and gravitational fields[END_REF][START_REF] Zakharov | Linearized gravitation theory and the graviton mass[END_REF], and the Boulware-Deser (BD) instability [START_REF] Boulware | Can gravitation have a finite range?[END_REF]. Both of these are potentially very serious problems. The vDVZ discontinuity involves the zeromass limit of massive theories of gravity. Instead of recovering the solutions of general relativity in this limit, it can instead sometimes be found that space-time geometry approaches a limit which is inconsistent with observed gravitational phenomena. For example, the prediction for weak lensing by the Sun in the Pauli-Fierz theory [START_REF] Pauli | On relativistic wave equations for particles of arbitrary spin in an electromagnetic field[END_REF] yields results which disagree with general relativity by 25%. The BD instability involves the possible existence of negative energy states at the non-linear level, and a lack of boundedness from below. We refer the reader to [START_REF] Dubovsky | Phases of massive gravity[END_REF][START_REF] Rubakov | Infrared-modified gravities and massive gravitons[END_REF][START_REF] Chamseddine | Higgs for Graviton: Simple and Elegant Solution[END_REF] for recent discussions on massive (single metric) gravitons.

While deserving of careful study, these problems are not necessarily fatal for the bimetric theories we are considering. The vDVZ discontinuity and the BD instability were both identified in the context of bimetric theories in which only one of the metrics is dynamical (the other being a priori specified as Riemann flat). It remains to be seen to what extent these problems endure when both tensor fields are dynamical. Furthermore, in the case of the vDVZ discontinuity it is thought that in the zero-mass limit the graviton can become strongly-coupled, such that the usual perturbative analysis can break down on scales smaller than the Vainshtein radius, so that general relativity is recovered [START_REF] Vainshtein | To the problem of non-vanishing gravitation mass[END_REF][START_REF] Arkani-Hamed | Effective field theory for massive gravitons and gravity in theory space[END_REF]. This 'Vainshtein mechanism' can be demonstrated numerically [START_REF] Babichev | Recovering General Relativity from massive gravity[END_REF]. Also, in the theories we are considering the vDVZ discontinuity is not necessarily a problem as both massless and massive modes can be shown to exist when Minkowski space is perturbed. The mass parameters can therefore be assumed to be large, and there will still always exist long-ranged massless modes to carry the Newtonian gravitational force (as long as matter doesn't couple to the massive modes only). The relevant question in this case then becomes whether or not general relativity can be recovered in the limit M → ∞, rather than M → 0. This is primarily the question we will concern ourselves with here, although we will also pause in the final sections to consider the zero-mass limit of the theory.

The PPN Approach

Here we will briefly recap the essential elements of the PPN formalism, as required for the internal coherence of this article. The PPN formalism is a perturbative treatment, and requires a small parameter to expand in. An "order of smallness" is therefore defined by

U ∼ v 2 ∼ p ρ ∼ Π ∼ O(2) ( 8
)
where U is the Newtonian potential, v is the velocity of a fluid element, p is the fluid pressure, ρ is its rest-mass density and Π is the ratio of energy to rest-mass densities.

We also take time derivatives to add an extra order of smallness each:

∂ ∂t ∼ O(1). (9)
The PPN formalism then proceeds using an expansion in this order of smallness. For our theories, we will now perturb our two fundamental tensors as

g µν = η µν + h µν (10)
q µν = ηµν + µν , (11)
where η µν is the metric of Minkowski space, and where we take ηµν = X 2 0 η µν , where X 0 is a constant. This last equation expresses the fact that, in general, one set of coordinates cannot be used to correspond to the same proper separation between events in the geometries associated with each of the two metrics. To be fully general one could consider different constants in front of the different components of η µν . We will not do this here.

Let us note that Minkowski space is not always a solution to the field equations ( 5) and ( 6), due to constant terms in the action. For q µν = X 2 0 g µν these terms correspond to the 'cosmological constants' terms

ρ Λ = 1 8πG 2 α 0 + X 2 0 κ 0 + 8κ 1 + 2κ 2 (12)
ρ λ = 1 8πG 2 α + κ 0 X 2 0 , (13)
for g and q, respectively. These can, in general, be non-zero even in the absence of α and α 0 , resulting in a de Sitter background §. For vanishingly small ρ Λ and ρ λ , however, we can approximate this as Minkowski space. This corresponds to the conditions

α 0 → -X 2 0 κ 0 -8κ 1 -2κ 2 (14) α → - κ 0 X 2 0 , (15)
which we will use to eliminate α 0 and α in the perturbed equations that follow. Now consider the energy-momentum tensor, T µν . For a perfect fluid this is given by T µν = [ρ(1 + Π) + p] u µ u ν + pĝ µν , where ρ is the density of rest mass, p is pressure, Π is internal energy per unit rest mass, and u µ is the 4-velocity of matter. Neglecting O(1) contributions from h 0i and 0i , the time-like normalisation u µ u ν ĝµν = -1 then gives

u µ = 1 m + nX 2 0 1 + 1 2 v 2 + (mh 00 + n 00 ) 2(m + nX 2 0 ) + O(4); v i + O(3) , ( 16
) § An exception is the ISS theory, for which ρ Λ = α 0 /8πG 2 .
to the relevant order in perturbations the components of the energy-momentum tensor are then given by

T 00 = ρ (m + nX 2 0 ) 1 + Π + v 2 + (mh 00 + n 00 ) (m + nX 2 0 ) + O(6) (17)
T 0i = ρv i (m + nX 2 0 ) + O(5) (18)
T ij = (ρv i v j + pδ ij ) (m + nX 2 0 ) + O(6). (19)
The two expanded metrics ( 10) and ( 11) can now be substituted into the field equations ( 5) and ( 6), along with ( 17), ( 18) and ( 19). The field equations can then be solved for order by order in smallness of perturbations, and gauge transformations of the form

x µ → x µ + ξ µ (20)
can be used to transform it into the "standard post-Newtonian gauge", where the spatial part of the metric is diagonal, and where terms containing time derivatives are removed wherever possible. The metrics that result then allow us to determine the post-Newtonian parameters of these theories by comparing to the test metric

ĝ00 = -1 + 2GU -2βG 2 U 2 -2ξG 2 Φ W + (2γ + 2 + α 3 + ζ 1 -2ξ)GΦ 1 +2(1 + 3γ -2β + ζ 2 + ξ)G 2 Φ 2 + 2(1 + ζ 3 )GΦ 3 +2(3γ + 3ζ 4 -2ξ)GΦ 4 -(ζ 1 -2ξ)GA (21) ĝ0i = - 1 2 (3 + 4γ + α 1 -α 2 + ζ 1 -2ξ)GV i - 1 2 (1 + α 2 -ζ 1 + 2ξ)GW i ( 22
) ĝij = (1 + 2γGU )δ ij , (23)
where

β, γ, ξ, ζ 1 , ζ 2 , ζ 3 , ζ 4 , α 1
, α 2 and α 3 are the post-Newtonian parameters, U is the Newtonian gravitational potential that solves Poisson's equation, and Φ W , Φ 1 , Φ 2 , Φ 3 , Φ 4 , A, V i and W i are the post-Newtonian gravitational potentials given in [START_REF] Will | Theory and experiment in gravitational physics[END_REF]. The particular combination of parameters before each of the potentials in ( 21), ( 22) and ( 23) are given so that the parameters have particular physical significance, once gravitational phenomena have been computed. We have chosen to construct the test metric here in terms of the combined metric ĝµν , as this is the metric that couples to the matter fields, and hence is the one being constrained by observations of those fields.

In general relativity the PPN parameters in ( 21)-( 23) are given by β = γ = 1, with all other parameters equaling zero. The interpretation that is often given to these parameters is that γ is 'the spatial curvature per unit rest mass', β is 'the degree of nonlinearity in the law of gravity', the α i are due to 'preferred-frame effects', and ξ is due to 'preferred-location' effects. The parameters ζ i , as well as α 3 , are sometimes associated with the violation of conservation of momentum. These parameters are all constrained by observations, and for the present study the two constraints below are of particular interest [START_REF] Bertotti | A test of general relativity using radio links with the Cassini spacecraft[END_REF]:

γ -1 = (2.1 ± 2.3) × 10 -5 (24) |α 2 | 1.2 × 10 -7 . ( 25
)
The first of these comes from observations of the Shapiro time delay of radio signals from the Cassini space-craft as it passed behind the Sun [START_REF] Bertotti | A test of general relativity using radio links with the Cassini spacecraft[END_REF], and the second is derived from the observation that the Sun's spin axis is closely aligned with the angular momentum vector of the solar system [START_REF] Nordtvedt | Probing gravity to the second post-Newtonian order and to one part in 10 7 using the spin axis of the Sun[END_REF]. For further constraints the reader is referred to [START_REF] Will | Theory and experiment in gravitational physics[END_REF].

Finally, the equations of motion show that for time-like particles propagating along geodesics the Newtonian limit is given by ĝ00 (and hence g 00 and q 00 ) to O(2) and ĝ0i to O(1), with no other knowledge of the metric components beyond the background level being necessary . The post-Newtonian limit for time-like trajectories requires a knowledge of

ĝ00 to O(4) (26) ĝ0i to O(3) (27) ĝij to O(2), (28)
where Latin letters are used to denote spatial indices. To obtain the Newtonian limit of trajectories followed by null particles we only need to know the metric to background order. The post-Newtonian limit of null trajectories requires ĝ00 and ĝij both to O(2), as well as ĝ0i to O(1).

In order to proceed in calculating the form of the weak field metric ĝµν , to the orders of smallness specified above, it is useful to have expressions for various perturbed quantities involving the metric tensors, such as their determinants and inverses. These are given in Appendix A.

Newtonian Perturbations to O(1) and O(2)

4.1. The g 0i and q 0i terms, to O(1)

Unlike the case of gravitational theories with a single metric, for the theories being considered here we cannot automatically set both g 0i = 0 and q 0i = 0 to O(1) through gauge transformations. We must therefore calculate these terms explicitly.

Substituting the expressions derived above into the 0 -i components of the field equations ( 5) and ( 6), and eliminating α 0 and α using ( 14) and ( 15), gives

l 2 - 1 2 ∇ 2 h 0i + 1 2 h 0j,ji = κ 0 + 8κ 1 X 2 0 + 4κ 2 X 2 0 0i -X 2 0 h 0i (29)
l 2 - 1 2 ∇ 2 0i + 1 2 0j,ji = κ 0 + 8κ 1 X 2 0 + 4κ 2 X 2 0 X 2 0 h 0i -0i . ( 30
)
The left-hand side of these equations vanishes when acted upon with ∇•, due to the Bianchi identities. We must therefore have 0i,i = X 2 0 h 0i,i . The equations ( 29) and (30) can then be decoupled by defining two new variables:

h (m) 0i ≡ h 0i - 0i X 2 0 (31) h (0) 0i ≡ h 0i + 0i , (32)
to get

∇ 2 h (m) 0i = M 2 h (m) 0i (33)
∇ 2 h (0) 0i = h (0) 0j,ji , (34)
where we have defined

M 2 ≡ 2(1 + X 2 0 ) X 2 0 l 2 [X 2 0 κ 0 + 8κ 1 + 4κ 2 ]. ( 35
)
We can now set h (0) 0i = 0 by a gauge transformation; the h (m) 0j mode, however, is gauge invariant, and so cannot be made to vanish in this way. We now note that h

(m) 0i,i = 0, so h (m)
0i is the curl of some vector potential A i , with the solution

h (m) 0i = (∇ × A) i = (c 1 ) i e -M |x-x | |x -x | + (c 2 ) i e M |x-x | |x -x | . ( 36
) If M ∈ R then h (m)
0i is a hyperbolic function, and if M ∈ I then it is oscillatory. This solution does not depend on the matter content of the space-time, and exists in a vacuum where only g µν and q µν are present. It also has a preferred point in space, x , and h (m) 0i , being the curl of a vector, looks like a rotation field. Finally, we impose c 2 = 0 as a boundary condition, in order to maintain asymptotic flatness at infinity. This is one of the PPN assumptions, up to cosmological terms.

The g 00 and q 00 terms, to O(2)

We will now proceed to calculate g 00 and q 00 to O(2). To do this we will need both the 0-0 field equations to O(2), and the trace of the i-j equations to O(2). As before, in order to decouple the equations let us now define massive and massless combinations

h (m) µν ≡ h µν - µν X 2 0 ( 37
)
h (0) µν ≡ h µν + µν . ( 38
)
With the gauge choice h (0) 0i = 0, and by using h

(m)
0i,i = 0, we can then write the relevant massless combination of ( 5) and (6) as

∇ 2 h (0) 00 + 8πG(m + nX 2 0 ) 2 ρ = X 2 0 (1 + X 2 0 ) h (m)2 0i,j -h (m) 0i,j h (m) 0j,i + M 2 X 2 0 2(1 + X 2 0 ) h (m)2 0i , (39)
while the massive combination is given by

∇ 2 h (m) 00 -M 2 h (m) 00 + 8πG(m + nX 2 0 )(m -n)ρ = (1 -X 2 0 ) (1 + X 2 0 ) h (m) 0i,j h (m) 0j,i -h (m)2 0i,j + 2(2κ 1 + κ 2 )(1 + X 2 0 ) l 2 X 2 0 A + (X 4 0 κ 0 -8(1 -X 2 0 )κ 1 -2(1 -3X 2 0 )κ 2 ) l 2 X 2 0 h (m)2 0i . (40)
Here we have defined

A ≡ h (m) ii -h (m) 00 , (41)
which is given by the solution of

X 2 0 κ 0 + 12κ 1 + 6κ 2 ∇ 2 A -κ 0 M 2 X 2 0 A = - 8πGX 2 0 M 2 l 2 (m + nX 2 0 )(m -n) (1 + X 2 0 ) ρ + 4M 2 (8κ 1 + (3 -X 2 0 )κ 2 ) (1 + X 2 0 ) h (m)2 0i + (X 2 0 (1 -X 2 0 )κ 0 + 8(3 -X 2 0 )κ 1 + 8(1 -X 2 0 )κ 2 ) (1 + X 2 0 ) h (m)2 0i,j + (X 2 0 (1 -X 2 0 )κ 0 + 8(1 -X 2 0 )κ 1 + 8κ 2 ) (1 + X 2 0 ) h (m) 0i,j h (m) 0j,i . (42)
To find the equations above we have made use of the expression

∇ 2 h (m)2 0i = 2h (m)2 0i,j + 2M 2 h (m)2 0i , as well as the Bianchi identities to O(2): M 2 l 2 X 2 0 2(1 + X 2 0 ) h (m) ij,ij = [X 2 0 (1 -X 2 0 )κ 0 + 8(1 -X 2 0 )κ 1 + 2(3 -X 2 0 )κ 2 ] (1 + X 2 0 ) h (m) 0j,i h (m) 0i,j - [X 2 0 (1 -X 2 0 )κ 0 -8X 2 0 κ 1 + 2(1 -X 2 0 )κ 2 ] 2(1 + X 2 0 ) ∇ 2 h (m)2 0i + 1 2 (X 2 0 κ 0 + 4κ 1 + 2κ 2 )∇ 2 h (m) ii -h (m) 00 . ( 43
)
It is interesting to note that in the ISS case the factor multiplying the differential operator in (42) vanishes. The equation above then becomes an algebraic relation between A, ρ and h (m) 0i

(and its derivatives). It can also be seen from ( 42) that in the case κ 0 = 0 the Green's functions for A are those of Laplace's equation, rather than those of Helmholtz's equation. In this case A becomes long-ranged. In what follows we will therefore consider the ISS and κ 0 = 0 cases separately from the more general case, as they clearly have different behaviour.

We can now integrate Eq. ( 39) using the Green's function of Laplace's equation. The massless mode h (0) 00 is then given by

h (0) 00 = 2G(m + nX 2 0 ) 2 U - M 2 X 2 0 8π(1 + X 2 0 ) V h (m)2 0i - X 2 0 4π(1 + X 2 0 ) V h (m)2 0i,j -V h (m) 0i,j h (m) 0j,i (44)
where the Newtonian potential U is

U = ρ(x ) |x -x | d 3 x (45)
and we have defined the additional functional

V(φ) ≡ φ(x ) |x -x | d 3 x . ( 46
)
If M is large, we are then left with the long-ranged part of this metric component being given by

h (0) 00 =2G(m + nX 2 0 ) 2 U. (47)
Here, and throughout, the symbol = will be taken to mean 'equal up to exponentially suppressed terms'. Such suppression is guaranteed as long as M -1 is much smaller than the length scale over which observations are being made, and can be arranged by choice of the parameters κ 0 , κ 1 , κ 2 , l and X 0 .

Let us now consider the massive modes. These are different, depending on the theory in question.

ISS theory

For the ISS theory we can write the solution to ( 40) and (42) as

h (m) 00 = 8G(m + nσ)(m -n) 3 W M (ρ) + (11 -9σ) 24π(1 + σ) W M h (m)2 0i,j - (1 -3σ) 24π(1 + σ) W M h (m) 0i,j h (m) 0j,i + (1 -4σ) 6πl 2 W M h (m)2 0i , (48)
where we have defined the new functional

W c (φ) ≡ φ(x ) |x -x | e -c|x-x | d 3 x , (49)
and suppressed the exponentially increasing mode. For sufficiently large M, the longranged part of this massive mode is zero:

h (m) 00 =0. ( 50
)
κ 0 = 0 theory When κ 0 = 0 the solution to equations ( 40) and ( 42) is

h (m) 00 = 8G 3 (m + nX 2 0 )(m -n)W M (ρ) - A 4 + (1 -X 2 0 ) 4π(1 + X 2 0 ) W M h (m)2 0i,j -W M h (m) 0i,j h (m) 0j,i + (4(1 -X 2 0 )κ 1 + (1 -3X 2 0 )κ 2 ) 2πl 2 X 2 0 W M h (m)2 0i , (51)
where

A = 8G(m + nX 2 0 )(m -n) 3 U - ((3 -X 2 0 )κ 1 + (1 -X 2 0 )κ 2 ) 3π(1 + X 2 0 )(2κ 1 + κ 2 ) V h (m)2 0i,j - ((1 -X 2 0 )κ 1 + κ 2 ) 3π(1 + X 2 0 )(2κ 1 + κ 2 ) V h (m) 0i,j h (m) 0j,i - 4(8κ 1 + (3 -X 2 0 )κ 2 ) 3πl 2 X 2 0 V h (m)2 0i , (52)
and where we have used

3M 2 W M (A) = 12πA -32πG(m + nX 2 0 )(m -n)W M (ρ).
From the above it can be seen that even for large M the massive modes are not short-ranged:

h (m) 00 = - 2G 3 (m + nX 2 0 )(m -n)U. ( 53
)

Other theories

For all other theories the solution to equations ( 40) and ( 42) is

h (m) 00 = 8G 3 (m + nX 2 0 )(m -n)W M (ρ) - κ 0 (1 + X 2 0 ) 3l 2 N 2 A + (1 -X 2 0 ) 4π(1 + X 2 0 ) W M h (m)2 0i,j -W M h (m) 0i,j h (m) 0j,i - (X 4 0 κ 0 -8(1 -X 2 0 )κ 1 -2(1 -3X 2 0 )κ 2 ) 4πl 2 X 2 0 W M h (m)2 0i , (54)
where

A = 2GX 2 0 M 2 l 2 (m + nX 2 0 )(m -n) (X 2 0 κ 0 + 12κ 1 + 6κ 2 )(1 + X 2 0 ) W N (ρ) - (X 2 0 (1 -X 2 0 )κ 0 + 8(3 -X 2 0 )κ 1 + 8(1 -X 2 0 )κ 2 ) 4π(X 2 0 κ 0 + 12κ 1 + 6κ 2 )(1 + X 2 0 ) W N h (m)2 0i,j - (X 2 0 (1 -X 2 0 )κ 0 + 8(1 -X 2 0 )κ 1 + 8κ 2 ) 4π(1 + X 2 0 )(X 2 0 κ 0 + 12κ 1 + 6κ 2 ) W N h (m) 0i,j h (m) 0j,i - M 2 (8κ 1 + (3 -X 2 0 )κ 2 ) π(1 + X 2 0 )(X 2 0 κ 0 + 12κ 1 + 6κ 2 ) W N h (m)2 0i , (55)
and where we have defined

N 2 ≡ 2κ 0 (1 + X 2 0 )(X 2 0 κ 0 + 8κ 1 + 4κ 2 ) l 2 (X 2 0 κ 0 + 12κ 1 + 6κ 2 ) , (56)
and used 2κ

0 (1 + X 2 0 ) l 2 N 2 A = 4G(m+nX 2 0 )(m-n)W M (ρ)+ 3(2κ 1 + κ 2 )(1 + X 2 0 ) πl 2 X 2 0 W M (A).(57)
Now, if both M and N are large enough then there are no long-ranged terms in the massive mode:

h (m) 00 =0.
(58)

Post-Newtonian Perturbations to O(2) and O(3)

5.1. The g ij and q ij terms, to O(2)

Let us now consider the g ij and q ij modes to O(2), which we call 'post-Newtonian tensor modes', as they are tensors under transformations of the spatial coordinate system.

From now on we will not consider further the contribution of the h (m) 0i ∼ O(1) terms to the metric perturbations. If they do not have an effect at the Newtonian level, then it seems unlikely they will be required at higher orders. Now, if we apply the gauge condition

h (0)µ i,µ - 1 2 h (0)µ µ,i = 0 ( 59
)
then the massless i-j equation, to O(2), becomes

∇ 2 h (0) ij = -8πG(m + nX 2 0 ) 2 δ ij ρ, (60)
which has the solution

h (0) ij = 2G(m + nX 2 0 ) 2 δ ij U. (61)
The massive modes, however, are once again dependent on the theory that is being considered. Now, the i-j equations, to O(2), are

∇ 2 h (m) ij -M 2 h (m) ij + 8πG(m + nX 2 0 )(m -n)δ ij ρ = - 2(2κ 1 + κ 2 ) (X 2 0 κ 0 + 8κ 1 + 4κ 2 ) A ,ij - 2(2κ 1 + κ 2 )(1 + X 2 0 ) l 2 X 2 0 δ ij A, (62)
where we have used the Bianchi identities to O(2), Equation (43), and where A is again given by the solution of Equation (42).

ISS theory

In the ISS theory the solution to (42) and ( 62) is given by

h (m) ij = - 4G 3M 2 (m+nσ)(m-n)W M (ρ ,ij )+ 4G 3 (m+nσ)(m-n)δ ij W M (ρ).( 63
)
The term involving W M (ρ ,ij ) is not in keeping with the usual PPN philosophy of avoiding terms with derivatives of ρ involved. Normally, one could perform a gauge transformation to remove such terms. Here, the massive modes are gauge invariant, but in subsequent sections we will find it possible to remove these terms from the combination of metrics that couples to the matter fields. For large M we have

h (m) ij =0. ( 64
)
κ 0 = 0 theory When κ 0 = 0 the solution to (42) and ( 62) is

h (m) ij = 2G 3 (m + nX 2 0 )(m -n)δ ij U + 4G 3 (m + nX 2 0 )(m -n)δ ij W M (ρ) + G 3π (m + nX 2 0 )(m -n)W M (U ,ij ). ( 65
)
There is again an unsightly term containing derivatives of its integrand, which we will gauge transform away later on when considering the combined metric ĝij . The longrange component of h

(m) ij is now given by h (m) ij = 2G 3 (m + nX 2 0 )(m -n)δ ij U. ( 66
)

Other theories

For all other theories the solutions to (42) and ( 62) are given by

h (m) ij = 4G 3 (m + nX 2 0 )(m -n)δ ij W M (ρ) + 2G 3 (m + nX 2 0 )(m -n)δ ij W N (ρ) + (2κ 1 + κ 2 ) 2π(X 2 0 κ 0 + 8κ 1 + 4κ 2 ) W M (A ,ij ), ( 67
)
where A is given by ( 55). Once again it can be seen that in the general case there are two different massive modes, with masses M and N . Once again, there is also a term containing derivatives. If both M and N are large enough then all of the terms in this massive mode are suppressed:

h (m) ij =0. ( 68
)

The g 0i and q 0i terms, to O(3)

Now consider the g 0i and q 0i modes to O(3), which we call 'post-Newtonian vector modes', due to their properties under spatial coordinate transformations. If we again apply the gauge condition (59), together with the new condition

h (0)µ 0,µ - 1 2 h (0)µ µ,0 = - 1 2 h (0) 00,0 , (69)
then we can write the massless combination of the field equations ( 5) and ( 6) as

∇ 2 h (0) 0i = G 2 (m + nX 2 0 ) 2 ∇ 2 (V i -W i ) + 16πG(m + nX 2 0 ) 2 ρv i , (70)
where we have used U ,0i = 1 2 ∇ 2 (W i -V i ), as well as the previously found solution for h (0) 00 . It can then be seen that equation (70) has the solution

h (0) 0i = - 7G 2 (m + nX 2 0 ) 2 V i - G 2 (m + nX 2 0 ) 2 W i , (71)
where V i and W i are the usual vector post-Newtonian potentials, as defined in [START_REF] Will | Theory and experiment in gravitational physics[END_REF].

If we now use the O(2) Bianchi identities, (43), as well as the Bianchi identities to

O(3), 2(X 2 0 κ 0 +8κ 1 +4κ 2 )h (m) 0i,i = (X 2 0 κ 0 +4κ 1 +2κ 2 )h (m) ii,0 +(X 2 0 κ 0 +12κ 1 +6κ 2 )h (m)
00,0 ,(72) then we can write the relevant combination of field equations from ( 5) and ( 6) as

∇ 2 h (m) 0i -M 2 h (m) 0i = - 2(2κ 1 + κ 2 ) (X 2 0 κ 0 + 8κ 1 + 4κ 2 ) A ,0i +16πG(m+nX 2 0 )(m-n)ρv i .( 73
)
As above, the form of the massive modes in ĝ0i depend on the theory being considered.

ISS theory

The solution to Equation (73) in the ISS theory is given by

h (m) 0i = - Gl 2 (m + nσ)(m -n) 3(1 + σ) W M (ρ ,0i )-4G(m+nσ)(m-n)W M (ρv i ).( 74
)
As with the O(2) components, there is still only one mass, M. If M is large enough then

h (m) 0i =0. (75)
κ 0 = 0 theory When κ 0 = 0 the solution to Equation ( 73) is given by

h (m) 0i = Gl 2 X 2 0 (m + nX 2 0 )(m -n) 6(1 + X 2 0 )(2κ 1 + κ 2 ) U ,0i -4G(m + nX 2 0 )(m -n)W M (ρv i ) - Gl 2 X 2 0 (m + nX 2 0 )(m -n) 6(1 + X 2 0 )(2κ 1 + κ 2 ) W M (ρ ,0i ). ( 76
)
Possible long-ranged forces are now given by

h (m) 0i = Gl 2 X 2 0 (m + nX 2 0 )(m -n) 6(1 + X 2 0 )(2κ 1 + κ 2 ) U ,0i . (77)

Other theories

For all other theories we have that the solution to (73) is given by

h (m) 0i = 2κ 0 (1 + X 2 0 ) 3l 2 N 2 M 2 A ,0i - 4G(m + nX 2 0 )(m -n) 3M 2 W M (ρ ,0i ) -4G(m + nX 2 0 )(m -n)W M (ρv i ). ( 78
)
To O(3) we have no long-ranged terms in h (79)

Post-Newtonian Perturbations to O(4)

We will now consider the g 00 and q 00 component of the metric to O(4). Using known solutions and identities we can write the massless combination of the field equations ( 5) and ( 6) as

∇ 2 h (0) 00 = -8πG(m + nX 2 0 ) 2 ρ 1 + Π + 2v 2 + 3 p ρ + 8πGX 2 0 (m + nX 2 0 )(m -n)ρh (m) 00 (1 + X 2 0 ) - h (0)2 00,i (1 + X 2 0 ) - X 2 0 (h (m)2 00,i -h (m) 00,ij h (m) ij ) (1 + X 2 0 ) - 2(2κ 1 + κ 2 ) M 2 l 2 h (m) 00,i A ,i - X 2 0 κ 0 2l 2 h (m)2 00 - 1 4l 2 (X 2 0 κ 0 + 12κ 1 + 6κ 2 )A 2 + M 2 X 2 0 4(1 + X 2 0 ) h (m)2 ij . ( 80
)
This equation can be integrated to give

h (0) 00 = 2(m + nX 2 0 ) 2 G U - (m + nX 2 0 ) 2 (1 + X 2 0 ) GU 2 + 2Φ 1 + 2 (m + nX 2 0 ) 2 (1 + X 2 0 ) GΦ 2 + Φ 3 + 3Φ 4 - 2GX 2 0 (m + nX 2 0 )(m -n) (1 + X 2 0 ) V(ρh (m) 00 ) + X 2 0 κ 0 8πl 2 V(h (m)2 00 ) + X 2 0 V(h (m)2 00,i ) -V(h (m) 00,ij h (m) ij ) 4π(1 + X 2 0 ) + (2κ 1 + κ 2 ) 2πM 2 l 2 V(h (m) 00,i A ,i ) + (X 2 0 κ 0 + 12κ 1 + 6κ 2 ) 16πl 2 V(A 2 ) - M 2 X 2 0 16π(1 + X 2 0 ) V(h (m)2 ij ), ( 81
)
where the h

(m) 00 , h (m) ij
and A to O(2) are taken to be given by the expressions found in previous sections. The long-ranged modes in h (0) 00 are then given, for our various different theories by the following:

ISS theory h (0) 00 =2(m + nσ) 2 G U - (m + nσ) 2 (1 + σ) GU 2 + 2Φ 1 + 2 (m + nσ) 2 (1 + σ) GΦ 2 + Φ 3 + 3Φ 4 + 4πσG 2 l 2 (m + nσ) 2 (m -n) 2 3(1 + σ) 2 V(ρ 2 ). ( 82
)
κ 0 = 0 theory h (0) 00 =2(m + nX 2 0 ) 2 G U - (m + nX 2 0 ) 2 (1 + X 2 0 ) GU 2 + 2Φ 1 + 2 (m + nX 2 0 ) 2 (1 + X 2 0 ) GΦ 2 + Φ 3 + 3Φ 4 + 8X 2 0 G 2 (m + nX 2 0 ) 2 (m -n) 2 3(1 + X 2 0 ) Φ 2 + M 2 X 2 0 G 2 (m + nX 2 0 ) 2 (m -n) 2 4π(1 + X 2 0 ) V(U 2 ) - 2X 2 0 G 2 (m + nX 2 0 ) 2 (m -n) 2 9(1 + X 2 0 )M 2 |∇U | 2 - 4X 2 0 G 2 (m + nX 2 0 ) 2 (m -n) 2 3M 2 (1 + X 2 0 ) V(ρ ,i U ,i ) + 4X 2 0 G 2 (m + nX 2 0 ) 2 (m -n) 2 3M 2 (1 + X 2 0 ) V(ρ ,i W M (ρ) ,i ). ( 83
)
Other theories

h (0) 00 =2(m + nX 2 0 ) 2 G U - (m + nX 2 0 ) 2 (1 + X 2 0 ) GU 2 + 2Φ 1 + 2 (m + nX 2 0 ) 2 (1 + X 2 0 ) GΦ 2 + Φ 3 + 3Φ 4 . ( 84
)
In deriving these expressions we have used

|∇U | 2 = 1 2 ∇ 2 U 2 -∇ 2 Φ 2 ,

as well as

A ,i B ,i = 1 2 ∇ 2 (AB) + 2πσA + 2πρB - 1 2 (M 2 + N 2 )AB (85) A ,ij B ,ij = 1 2 ∇ 2 (A ,i B ,i ) + 2πσ ,i A ,i + 2πρ ,i B ,i - 1 4 (M 2 + N 2 )∇ 2 (AB) + 1 4 (M 2 + N 2 ) 2 AB -π(M 2 + N 2 )(Aσ + Bρ) (86)
where A = W M (ρ) and B = W N (σ). Further manipulation has also been done by noting that, for example, V(ρ ,i W M (ρ) ,i ) =4πV(ρ 2 ). These manipulations are performed so that the potentials are written without derivatives acting on the matter fields, as is usual in the PPN formalism.

The g (m) 00

and q (m) 00

terms, to O(4)

The equations involved in finding g (m) 00 and q (m) 00 to O(4) are very lengthy, and we therefore choose to present them in Appendix B. Here we only state the form of the equations, and give their long-ranged components when M and N are large. We find the equation for h

(m) 00 looks like ∇ 2 h (m) 00 -M 2 h (m) 00 = .... (87)
The Green's function for this equation is therefore that of Helmholtz's equation, with the same mass term, M, as at O(2). The right-hand side of this equation has terms containing A to O(4). This quantity is given in Appendix B by the solution to an equation of the form

(X 2 0 κ 0 + 12κ 1 + 6κ 2 )∇ 2 A -M 2 X 2 0 κ 0 A = .... (88)
The Green's function for this equation can also be seen to be the same as the corresponding O(2) case. The long-ranged components of the massive modes of h 00 to O(4) are given below.

ISS theory

h (m) 00 = π 2 l 4 G 2 18(1 + σ) 4 (m + nσ) 2 (m -n) 2 ρ 2 . ( 89
)
κ 0 = 0 theory
In this case the massive modes h (m) 00

and h

(m) ij
to O(2) both have long-ranged components, and so the full expressions do not simplify greatly when the exponentially suppressed modes are neglected. The reader is therefore referred to equations ( 135) and (139) in Appendix B, where h (m) 00 to O( 4) is given explicitly.

Other theories

h (m) 00 =0. ( 90
)
To find these expressions we have made use of ( 85) and ( 86), as well as the following relations:

W M (W N (X)) = 4π (M 2 -N 2 ) (W N (X) -W M (X)) (91) W M (V(X)) = 4π M 2 V(X). (92)

Perturbations Coupled to Matter Fields

We have so far calculated the form of the massless and massive combinations of the perturbations to g µν and q µν . We now want to know what these tell us about the perturbations to the metric ĝµν . Using previous relations we can write ĝµν as ĝµν = ηµν + ĥµν (93)

where ĥµν = (m + nX 2 0 ) (1 + X 2 0 ) h (0) µν + X 2 0 (m -n) (1 + X 2 0 ) h (m) µν ( 94
)
and ηµν = η µν + ηµν . It is now straightforward to write down the components of ĥµν .

To proceed in doing this it is useful to define two new constants:

G N ≡ G(m + nX 2 0 ) 3 (1 + X 2 0 ) (95) G N ≡ G(m + nX 2 0 ) 3(1 + X 2 0 ) 3(m + nX 2 0 ) 2 -X 2 0 (m -n) 2 . ( 96
)
The reason for doing this is that although the constant G appears in the gravitational action (2) in the same way as it does in general relativity, it is not necessarily the value of Newton's constant that is determined by Cavendish type experiments. For the κ 0 = 0 theories this value of Newton's constant is given by G N (when M is large), while for the ISS theory and all other theories it is given by G N (when M is large for the ISS theory, and when both M and N are large for all other theories).

It is also useful at this point to start using infinitesimal gauge transformations to put our results in the 'standard post-Newtonian gauge'. In the preceding section we used the gauge specified by conditions (59) and (69). This has been a convenient choice, and has allowed integration of the field equations to post-Newtonian accuracy. It has, however, resulted in the metrics in question having off-diagonal components in the spatial part of the metric, as well as non-zero O(1) terms in the 0-i components of the metric. These can be removed by making an infinitesimal coordinate transformation of the form x µ → x µ + ξ µ . The metric that couples to matter, ĝµν , is then transformed in such a way that ¶

ĥµν → ĥµν -ξ µ;ν -ξ ν;µ + O(ξ 2 ), (97)
where covariant derivatives are with respect to ĝµν , and where indices have been lowered with ĝµν . By making coordinate transformations such that ξ 0 ∼ O(1) or O(3) and ξ i ∼ O(2) the metric perturbations then transform as

ĥij → ĥij -2ξ (i,j) (98) ĥ0i → ĥ0i -ξ 0,i -ξ i,0 . (99)
Transformations of this kind will be used in what follows to remove all contributions from the O(1) massive modes in ĥ0i , as well as to diagonalise ĥij , and to remove the potentials found above that contain derivatives on the functions in their integrands. As in the previous section, many of the expressions involved here are quite lengthy. We will therefore once again present the full equation in Appendix C, and quote here only the terms that are long-ranged when M and N are large (after the gauge transformations discussed above have been performed + ). To O(4) in ĥ00 , O(3) in ĥ0i and O(2) in ĥij we find the results below.

ISS theory

ĥ00 =2G N U - G N (m + nσ) U 2 + 2Φ 1 + 2G N (m + nσ) Φ 2 + Φ 3 + 3Φ 4 + 4πσG 2 N l 2 (m -n) 2 3(1 + σ)(m + nσ) 3 V(ρ 2 ) + π 2 l 4 G 2 N σ(m -n) 3 18(1 + σ) 3 (m + nσ) ρ 2 (100) ĥ0i = - 7G N 2 V i - G N 2 W i (101) ĥij =2G N δ ij U. ( 102
)
¶ This is true as long the matter does not couple to the massive combination of metrics only, i.e. as long as m = -nX 2 0 . + These transformations are given explicitly in Appendix C.

κ 0 = 0 theory ĥ00 =2G N U + O(4) (103) ĥ0i = - 7G N 2 3(m + nX 2 0 ) 2 (3(m + nX 2 0 ) 2 -X 2 0 (m -n) 2 ) V i - G N 2 3(m + nX 2 0 ) 2 (3(m + nX 2 0 ) 2 -X 2 0 (m -n) 2 ) W i (104) ĥij =2G N (3(m + nX 2 0 ) 2 + X 2 0 (m -n) 2 ) (3(m + nX 2 0 ) 2 -X 2 0 (m -n) 2 ) δ ij U. (105)
Other theories

ĥ00 =2G N U - G N (m + nX 2 0 ) U 2 + 2Φ 1 + 2G N (m + nX 2 0 ) Φ 2 + Φ 3 + 3Φ 4 (106) ĥ0i = - 7G N 2 V i - G N 2 W i (107) ĥij =2G N δ ij U. (108)
In the κ 0 = 0 theory we have not written the O(4) part of ĥ00 explicitly, because, as mentioned before, it contains a large number of long-ranged potentials. Appendices B and C contain enough information for the reader to calculate these terms explicitly, if they are required.

Post-Newtonian Parameters, and Observational Constraints

It is convenient at this point to choose units so that m + nX 2 0 = 1. The unperturbed metric ηµν then takes its usual form. Using the results of the previous section we can now obtain the PPN parameters of the theories we are considering, and determine the extent to which they can be observationally constrained.

ISS theory

It can be seen by comparing (100), ( 101) and ( 102) with ( 21), ( 22) and ( 23) that, when M is large, all PPN parameters take the same value as in General Relativity. We do, however, have two new potentials in the ĥ00 term at O(4). These are

+ 4πσG 2 N l 2 (1 -n(1 + σ)) 2 3(1 + σ) V(ρ 2 ) and + π 2 l 4 G 2 N σ(1 -n(1 + σ)) 3 18(1 + σ) 3 ρ 2 . ( 109
)
No such terms appear in the usual PPN prescription. The second of these is particularly unfamiliar as it involves ρ directly, and not ρ integrated over any volume. Such a term, however, can be expected to be small in comparison to V(ρ 2 ) so long as l d, where d is the length scale over which the observational phenomena are being measured. Outside of any matter distribution, of course, this term will vanish entirely.

In the absence of any observational constraints on these new potentials, the only constraint that can currently be placed on this particular theory is therefore Md 1, or equivalently

4(1 + σ) l 2 d 2 . ( 110
)
No other constraints are available from standard PPN formalism.

κ 0 = 0 theory

In the κ 0 = 0 theory it can be seen by comparing (103), ( 104) and ( 105) with ( 21), ( 22) and ( 23) that, among other deviations from general relativity, we have

γ = (3 + X 2 0 (1 -n(1 -X 2 0 )) 2 ) (3 -X 2 0 (1 -n(1 -X 2 0 )) 2 ) . ( 111
)
As in the ISS theory, the κ 0 = 0 theory must also obey Md 1 in order for the massive modes to be exponentially suppressed. For this theory, this corresponds to

8(1 + X 2 0 ) X 2 0 (2κ 1 + κ 2 ) l 2 d 2 . ( 112
)
Unlike the ISS theory, however, we can also use the PPN constraint on γ from observations of the Cassini spacecraft to constrain the theory. To be compatible with observations γ must satisfy equation [START_REF] Blas | Aspects of Infrared Modifications of Gravity[END_REF]. We then have the constraint

X 2 0 (1 -n(1 -X 2 0 )) 2 = (3.1 ± 3.3) × 10 -5 . (113)
While tight, it is worth noting that this constraint can always be evaded by an appropriate choice of X 0 . Unless X 0 is known from other considerations (cosmology, for example) it is not possible to constrain the theory with this information only. It is also worth noting that this is a constraint on n only, and not the parameters determining the interactions in the theory (i.e. κ 1 and κ 2 ). Further constraints are also available on this theory from the ĝ00 metric component at O(4).

Other theories

Comparing (106), ( 107) and ( 108) with ( 21), ( 22) and ( 23) shows, as long as both Md 1 and N d 1, that all of the PPN parameters in this general class of theories are the same as in general relativity. The only constraints that need to be satisfied by these theories are therefore

2(1 + X 2 0 ) X 2 0 (X 2 0 κ 0 + 8κ 1 + 4κ 2 ) l 2 d 2 (114) and 2κ 0 (1 + X 2 0 ) (X 2 0 κ 0 + 8κ 1 + 4κ 2 ) (X 2 0 κ 0 + 12κ 1 + 6κ 2 ) l 2 d 2 . ( 115
)
If these conditions are met then all the post-Newtonian predictions of these theories are indistinguishable from those of general relativity. This shows the importance of the κ 0 term in the interaction Lagrangian.

Discussion

We have investigated here the weak field limit of a general class of bimetric theories, generated from the gravitational action (2), and coupled to matter via a linear combination of the two metrics involved. We have calculated the post-Newtonian limit of these theories, up to O(4) in g 00 , O(3) in g 0i and O(2) in g ij , for a perfect fluid matter content distributed without any symmetries. We find that, to all orders considered, there is a natural decomposition of the perturbations to the two metrics into modes we have called 'massive' and 'massless'. These modes are defined through the combinations

h (m) µν = h µν - µν X 2 0 and h (0) µν = h µν + µν , (116)
where h µν and µν denote the perturbations to the two metrics involved, and X 0 is a constant associated with the unperturbed background space-times. We find that, in general, the O(1) contributions to the massive modes h (m) 0i are nonzero, and cannot be made to vanish by a coordinate transformation. These quantities satisfy a Helmholtz equation with mass M (as given by Eq. ( 35)), and while the 0-i component of the combined metric that couples to matter can always be set to zero by a gauge transformation (as long as the matter does not couple exclusively to the massive modes), these O(1) terms can in principle make a non-negligible contribution to the other components of the metric. We therefore include the effects of these terms while calculating the 0-0 component of the combined metric to O(2), but neglect them to higher orders, as we see no reason to expect them to contribute substantially to post-Newtonian accuracy if they are negligible at the Newtonian level.

To Newtonian order, we find that in general the relevant metric perturbations have massless modes (that are always long-ranged), as well as massive modes with two different masses: M and N (where N is given by Eq. ( 56)). We find two special cases in which the second mass, N , does not appear. These are the ISS theory, as specified in Section 2, and theories with κ 0 = 0, in which the interaction term √ -q(q -1 ) µν g µν is missing from the action. In the ISS theory this occurs because the ∇h (m) 00 = ... equation can be decoupled from the h (m) ij terms without having to solve any other differential equations. In all other cases this is not possible, and a second equation of the form ∇ 2 A -N 2 A = ... must be solved, before one can integrate the h (m) 00 equation. In the second anomalous case, when κ 0 → 0 it is found that the mass parameter N → 0. Theories of this kind therefore have extra massless modes, beyond the usual Newtonian term that occurs from integrating the ∇h (0) 00 = ... equation. We then proceed to calculate the metric perturbations to post-Newtonian accuracy. These terms all involve the same two masses, M and N , that are required to Newtonian order (except for the ISS and κ 0 = 0 cases, which only require M). By expressing these perturbations in the standard post-Newtonian gauge we find that the predictions of these theories are, in general, very similar to those of general relativity. In particular, if M and N are both large enough then they can evade all preferred frame experiments, and are indistinguishable from general relativity by any experiments that constrain the PPN parameters β and γ (including the stringent tests available from Shapiro time-delay observations [START_REF] Bertotti | A test of general relativity using radio links with the Cassini spacecraft[END_REF], and lunar laser ranging measurements [START_REF] Williams | Progress in lunar laser ranging tests of relativistic gravity[END_REF]). The ISS theory also does a remarkably good job of satisfying weak field tests of gravity when M is large, with PPN parameters that are again the same as in general relativity. In this case, however, we do find a couple of anomalous terms appearing at O(4) in the 0-0 component of the metric that couples to matter. These are given in Eq. ( 109), and could potentially be used to observationally distinguish between general relativity and ISS theory. The κ 0 = 0 theory does not fare quite so well, due to its extra long-ranged modes. In this case observations of γ can be used to place constraints on a particular combination of the constant X 0 , and the mixing angle between the two fundamental spin-2 fields that couples to matter, even when M is large.

As well as the large mass limit, one can also consider the small mass limit. In this case a viable Newtonian term can again appear in the 0-0 component of the metric, and we must then consider the problem of the vDVZ discontinuity that was discussed in Section 1. To do this we note that as M → 0 and N → 0 we get W M (ρ) → U * and W N (ρ) → U † , respectively (we have used the superscripts * and † here, even though U * = U † = U , so that we can keep track of where the potential has come from). Neglecting contributions from h (m) 0i

at O(1), the full expressions in Appendix C then give, for the most general theories,

ĥ00 = 2G N U + 8G N X 2 0 (1 -n(1 + X 2 0 )) 2 U * 3 - 2G N X 2 0 (1 -n(1 + X 2 0 )) 2 U † 3 (117) ĥij = 2G N U + 4G N X 2 0 (1 -n(1 + X 2 0 )) 2 U * 3 + 2G N X 2 0 (1 -n(1 + X 2 0 )) 2 U † 3 . (118)
The vDVZ discontinuity can be seen to be manifest in the U * terms of these two equations. If these were the only terms in these expressions then we would indeed get the familiar γ = 1/2 that is anticipated from massive gravity. Here, however, we have two additional terms in both ĥ00 and ĥij -one from the terms that are long-ranged even when M and N are large, and another from the W N (ρ) terms. The cumulative effect of all three of these contributions added together shows that there is no vDVZ discontinuity in the γ parameter of the most general class of these theories, as in this case we end up with ĥij = δ ij ĥ00 , and hence γ = 1. The same is true in the special case in which κ 0 = 0, as should be expected, as these modes already correspond to the N → 0 limit of the more general theories. The ISS theory, however, does not have γ = 1 as M → 0. Instead we find

γ → 3 + 2(1 -n(1 + σ)) 2 σ 3 + 4(1 -n(1 + σ)) 2 σ . ( 119
)
This is essentially because the W N (ρ) terms that cancel the discontinuity from the W M (ρ) terms are absent in the ISS theory.

It should be noted, however, that although the general relativistic value of γ is recovered in the most general theories as M and N → 0, this does not mean that all PPN parameters will be. For example, for the parameters α 1 and α 2 that parameterise deviations from general relativity in the 0-i components of the field equations, we find that while α 1 → 0 (as in general relativity), we also have

α 2 → - (1 -n(1 + X 2 0 )) 2 X 2 0 (2 -n(1 + X 2 0 )) 2 X 2 0 . (120)
In terms of this expression, equation ( 25) then gives the constraint

X 2 0 (1 -n(1 + X 2 0 )) 2 1.2 × 10 -7 . (121)
This is the same combination of quantities that was constrained in the large M limit of the κ 0 = 0 theory using the parameter γ, but the constraint here is from an entirely different gravitational effect, and is even tighter. On should also note that in the limit

M → 0 the O(1) terms in h (m)
0i are no longer short-ranged. These terms are, however, set by initial conditions, rather than by the matter content of the space-time. One may then be able to limit their magnitude by applying suitable observational constraints, but this will place constraints on the initial conditions only, and not on the gravitational theory itself.

Finally, we note that when the matter fields are coupled to the massless combination of metrics only (so that m = n) then many of the constraints found above are automatically satisfied. These observations can therefore be though of as constraining the amount of massive mode that is allowed to be mixed with the massless modes.

The expressions above should be understood to have h and A given to O(2) by the lower order solutions found in previous sections. It now remains to find A to O(4). To do this we use the field equations ( 5) and ( 6), as well as known identities and lower order solutions, to find

(X 2 0 κ 0 + 12κ 1 + 6κ 2 )∇ 2 A -M 2 X 2 0 κ 0 A = - 3M 2 l 2 X 2 0 2(1 + X 2 0 ) h (m) 00,00 - 4πGM 2 l 2 X 2 0 (1 + X 2 0 ) 3mn(1 -X 2 0 ) + (m -n) 2 X 2 0 (1 + X 2 0 ) ρh (m) 00 +6(2κ 1 + κ 2 )A ,00 + 4πGM 2 l 2 X 2 0 3(1 + X 2 0 ) 2mn(1 + X 2 0 ) - (m -n) 2 X 2 0 (1 + X 2 0 ) ρh (m) ii - 8πGM 2 l 2 X 2 0 (m + nX 2 0 )(m -n) (1 + X 2 0 ) ρ 1 + Π -3 p ρ + 2h
(0) 00 The solutions to this equation are given below.

(1 + X 2 0 ) + M 2 (2κ 1 + κ 2 ) (1 + X 2 0 ) 2h (
- 2(X 2 0 κ 0 + 8κ 1 + 2(3 + X 2 0 )κ 2 ) (1 + X 2 0 ) h (m)2 ij,j + 2h (m) ij h (m) jk,ki + h (m) ij,k h (m) jk,i - M 2 l 2 X 2 0 (1 -X 2 0 ) 4(1 + X 2 0 ) 2
In the above we have made use of the Bianchi identities to O(4), which are given by: (X 2 0 κ 0 + 8κ

  M and N are both sufficiently large:h (m) 0i =0.

  1 + 4κ 2 )h

								(m)
								ij,ij
	=	1 2	(X 2 0 κ 0 + 4κ 1 + 2κ 2 )∇ 2 A +	(X 2 0 κ 0 + 4κ 1 + 4κ 2 ) 2	A ,i h	(m) ij,j + Ah (m) ij,ij
		+	(X 2 0 κ 0 + 2(1 + X 2 0 )κ 2 ) 2(1 + X 2 0 )		h	(m) ij,k h	(m) ij,k + h	(m) ij ∇ 2 h	(m) ij
		-	(X 2 0 κ 0 + 2κ 2 ) 4	A i h	(m) jj,i + A∇ 2 h	(m) ii -h	(m) jj,i h	(m) 00,i -h	(m) ii ∇ 2 h	(m) 00
		-	(X 2 0 κ 0 + 4κ 1 + 2κ 2 ) 2(1 + X 2 0 )	h	(0) 00,i h	(m) jj,i + h	(0) 00 ∇ 2 h	(m) ii + h	(0) 00,i h	(m) 00,i + h	(0) 00 ∇ 2 h	(m) 00
		+	(X 2 0 κ 0 + 4(1 -X 2 0 )κ 1 + 4κ 2 ) 2(1 + X 2 0 )	h	(m) ij,i h	(m) kk,j + h	(m) ij h	(m) kk,ij
		-	(X 2 0 κ 0 + 6κ 1 + 3κ 2 ) (1 + X 2 0 )	h	(m) 00,i h	(0) 00,i + h	(m) 00 ∇ 2 h	(0) 00
		+	(2κ 1 + κ 2 ) (1 + X 2 0 )	h	(m) jj,i h	(0) 00,i + h	(m) ii ∇ 2 h	(0) 00 -(X 2 0 κ 0 + 8κ 1 + 4κ 2 )h	(m) 0i,0i
		+	(X 2 0 (1 -5X 2 0 )κ 0 -32X 2 0 κ 1 + 2(1 -7X 2 0 )κ 2 ) 4(1 + X 2 0 )	h	(m)2 00,i + h	(m) 00 ∇ 2 h	(m) 00
		-	(X 2 0 κ 0 + 8κ 1 + 2(3 + X 2 0 )κ 2 ) (1 + X 2 0 )	h	(m) ij,i h	(m) jk,k + 2h	(m) ij h	(m) jk,ki + h	(m) ij,k h	(m) jk,i
		+	(X 2 0 κ 0 + 8κ 1 + 4κ 2 ) (1 + X 2 0 )	h	(0) 00 h	(m) ij,ij -h	(m) ij h	(0) 00,ij
		-	((1 + 2X 2 0 )(X 2 0 κ 0 + 4κ 2 ) + 4(1 + 3X 2 0 )κ 1 ) 2(1 + X 2 0 )	h	(m) ij,i h	(m) 00,j + h	(m) ij h	(m) 00,ij . (141)

The O(1) terms in ĝ0i are usually omitted from the beginning in theories with a single metric, as they can be removed by a suitable gauge choice. In what follows we will find that the situation is somewhat more complicated in multigravity.

Acknowledgements

TC acknowledges the support of Jesus College, Oxford and the BIPAC. MB would like to thank the Oxford Astrophysics department for hospitality during a sabbatical year. MB was partially supported by the JS Guggenheim Memorial Foundation, Fondecyt (Chile) Grant # 1100282 and Alma-Conicyt Grant # 31080001.

‡ JS Guggenheim Memorial Foundation

Appendix A: Perturbed Metric Quantities

The perturbed form of the metric that we will be using is specified in [START_REF] Rastall | A note on a theory of gravity[END_REF] and [START_REF] Lightman | New two metric theory of gravity with prior geometry[END_REF]. In order to solve [START_REF] Will | Theoretical frameworks for testing relativistic gravity. II. Parameterized post-Newtonian hydrodynamics and the Nordtvedt effect[END_REF] and [START_REF] Will | Theory and experiment in gravitational physics[END_REF], however, we also need to know the perturbed Ricci tensor to O(3) in its off-diagonal components, and O(4) in all of its diagonal components * , as well as the relevant expressions for the perturbed metric determinants, and inverse metrics. The perturbed Ricci tensor components take on their usual lengthy form, and we will not reproduce them explicitly here.

The perturbed metric determinants can be calculated using the equation det(g) = e tr(ln(g)) . To O(2) in perturbations this gives

and, if we ignoring O(1) contributions to h 0i and 0i , then to higher order we have

In order to compute the interaction terms in the field equations [START_REF] Will | Theoretical frameworks for testing relativistic gravity. II. Parameterized post-Newtonian hydrodynamics and the Nordtvedt effect[END_REF] and [START_REF] Will | Theory and experiment in gravitational physics[END_REF] we also need to know the inverse metric fluctuations (q -1 ) µν = (η -1 ) µν + (δq -1 ) µν . These can be found from the definition (q -1 ) µα q αν = δ µ ν , which gives (δq -1 ) µν = -(η -1 ) µα (η -1 ) νβ αβ + (η -1 ) µα (η -1 ) νδ (η -1 ) βγ αβ γδ + O(h 3 ).(127) The components of (δq -1 ) µν are then given by (δq -1 ) 00 = -00

Ignoring O(1) contributions to h 0i and 0i , the higher order components of (δq -1 ) µν are (δq -1 ) 00 = -00

(132) * The i-j components are usually only required to O(2), but here we also need higher order terms in order to calculate the O(4) part of the 0-0 components.

Appendix B: The g (m) 00

and q (m) 00

Equations, to O(4)

Using this expression, together with other known identities and lower order solutions, the field equations ( 5) and ( 6) give

The solutions to this equation are given below.

ISS theory

Other theories

) .

(136)

κ 0 = 0 theory

Other theories

Appendix C: The ĥµν Equations, and Gauge Transformations

As long as m = -nX 2 0 we can always use a coordinate transformation to remove all contributions from the O(1) massive modes to ĥ0i , so that ĥ0i → 0. We will therefore consider here only the higher order contributions to ĥµν .

Newtonian perturbations to O(2)

To O(2) in ĥ00 we find the results below.

ISS theory

.

Other theories

Post-Newtonian perturbations to O(2) and O(3)

To O(2) in ĥij and O(3) in ĥ0i we find the results below.

ISS theory

The arrows in the expressions indicate the infinitesimal coordinate transformations

where the arrows indicate the coordinate transformations

(2U ,0 -3W M (ρ) ,0 ). ( 156)

where the arrows indicate the coordinate transformations

We will not show the full expressions for ĥ00 to O(4) here as they are quite lengthy, and we do not feel that writing them out explicitly will add sufficient extra insight to justify their inclusion. The interested reader can calculate these quantities straightforwardly using the relevant expressions in Appendix B, and the coordinate transformations given above.


A L Dubov

J Teisseire

E Barthel

Elastic Instability and Contact Angles on Hydrophobic Surfaces with Periodic Textures

Keywords: 68, 08, Bc -Wetting 68, 03, Cd -Surface Tension 68, 35, Np -Adhesion

Surface textures are used to impart advanced wetting properties to surfaces. However understanding the surface response in relation to the nature of the texture is still a challenge. Here we have measured advancing and receding contact angles on model hydrophobic surfaces with cylindrical pillars as a function of the pillar spacing. We show that the dependances of both advancing and receding contact angles upon spacing are well accounted for by a simple model of the instability of the triple line, following the line elasticity theory by Joanny and de Gennes (J. Chem. Phys. 81 (1984) 552). This result demonstrates the prominent role of the triple line elasticity in determining the wetting properties of textured surfaces.

Introduction. -Controlling the work of adhesion of a liquid to a solid w is a necessity in most areas of wetting and fluidics. This work of adhesion, which is the energy necessary to separate the liquid from the solid, is more directly measured through the contact angle θ 0 . Indeed if the surface tension of the liquid is γ, the work of adhesion is given by (Young-Dupré)

w = γ(1 + cos θ 0 ) (1)
In the specific case of low adhesion, θ 0 is large, in the 150 • to 180 • range. This state of repellency has been considered in great detail because it leads to easy fluid detachment and low friction [1,2]. Because of the unavoidable polarisability of solid and liquid, low works of adhesion can be achieved only by partly lifting the liquid from the solid through surface roughness. This is the so-called Cassie state. The general understanding on the impact of roughness is summarized in the Cassie equation which expresses the work of adhesion of a rough surface as a function of the work of adhesion of a flat surface made out of the same material. We introduce the solid fraction φ ≤ 1, which is the ratio of the effectively wetted area to the apparent wetted area. The Cassie equation is the surface average of the work of adhesion and since areas not touching the surface do not contribute to the work of adhesion, it stands as

1 + cos θ = φ(1 + cos θ 0 ) (2)
These notions have been known for quite a long time, especially for stochastically rough surfaces [3], but the advent of microfabrication techniques has made it possible to generate a large variety of surfaces with well controlled, usually periodic, roughness. A host of recent experiments have explored the relation between roughness and work of adhesion [4][5][6][7][8][9]. Beyond the static contact angle, interest has also focused on contact angle hysteresis because it is directly related to a number of properties such as the detachment force of a drop, for example. More specifically very rough hydrophobic surfaces usually exhibit low hysteresis and under such circumstances droplets require low drag force for removal. Recent experiments have demonstrated strong impact of structure morphology on the contact angle hysteresis of hydrophobic surfaces [4,6,7]. However, quantitative understanding is far from complete. The Cassie equation is based on an equilibrium average of the work of adhesion over area. But if energy differences between neighboring configurations are larger than the thermal energy, the capacity of the Cassie equation to account for intrinsically irreversible processes should be called into doubts [10].

More advanced theories have been developed, where some sensitivity to local triple line conformations is introduced to inject irreversibility in the Cassie equation. Usually the validity of some rule of mixture for the wetting properties along the triple line is assumed [5,8]. One possibility is a macroscopic approach which extrapolates some local configuration energies from the measured contact angles [11]. This method does at least qualitatively account for the observed increase of the receding contact angle with decreasing pillar density. Another method focusses on local differentials of the energy landscape and this differential area (DA) model [8] has also proved successful in describing the receding contact angles for various types of textures.

However, generally speaking, the advancing contact angle remains a problem. It has consistently been shown experimentally that the advancing contact angle for microfabricated textured surfaces is nearly constant as the pillar spacing increases [4,6], although the value of the constant differs between experiments (160 • to 175 • ). Within the modified Cassie framework [8,11], the advancing contact angle is either predicted to be 180 • (because the liquid progresses on air), which is too large, or predicted to change with solid fraction as per the Cassie equation, which is not the case. In summary these advanced theories fail at providing a consistent picture of the wetting properties of surfaces decorated with such sharply defined textures.

A different path has been shown by Joanny and de Gennes [12]. They proposed a theory for the hysteresis of hydrophilic surfaces with random distributions of work of adhesion. Irreversibility is connected to the local elastic instabilities of the triple line jumping on and off these pinning sites. Application of the concept to the calculation of receding contact angles (or effectively hysteresis) on hydrophobic pillar arrays has proved successful [9].

In this paper, we investigate the issue of the pinning of the triple line on hydrophobic surfaces textured with pillars in more detail. We have synthetized hydrophobic surfaces with model textures and measured advancing and receding contact angles in the Cassie state as a function of pillar spacing. The evolutions of advancing and receding contact angles with solid fraction (or more specifically pillar spacing), for constant feature size, are found to be consistent with previous experimental results. We show that a simple elastic line pinning model provides a consistent description of both advancing and receding contact angles. The asymmetry between these two cases proceeds from the nature of the boundary conditions when the deformation of the triple line becomes unstable. The results provide a simple demonstration of the prominent role of the elasticity of the triple line in the wetting properties of these surfaces.

Experiments. -Superhydrophobic surfaces were fabricated via silica sol-gel nanoimprint lithography (SSG-NIL) [13]. With SSG-NIL we obtain silica structures with the simple and versatile conditions of sol-gel chemistry.

The patterns (Fig. 1) are rigid, transparent and defectfree. In a typical experiment, 4.5 g of methyltriethoxysilane (MTES) is mixed with 1.2 g of HCl (pH=2) and stirred for 3 hours. The sol is spin coated on a smooth glass substrate. The mold is made of polydimethylsiloxane, which easily replicates masters obtained by photolithography on silicon wafers. It is applied onto the spin-coated gel film with a 50 N load at 50 • C for 30 min followed by 90 • C for another 30 min. As-prepared samples were oxidized by UV-ozone treatment, hydrophobized with 1,1,2,2-tetrahydroperfluorodecyltrichlorosilane in an evacuated desiccator during 3 hours and carefully rinsed with isopropanol.

The geometry of the patterned samples was characterized by scanning electron microscopy and interference profilometry and proved to match the geometry of the masters. The patterns are cylindrical pillar arrays on square lattices. The size of the pillars is fixed (diameter d=10 µm, aspect ratio 1) and the lattice parameter d + l ranges from 2d to 6d, varying the pillar density

n = (d + l) -2 (3) from 2.9×10 8 to 2.5×10 9 m -2 .
The contact angle measurements were performed with water in the Cassie state. We used the volume oscillation method on a DSA100 (Krüss, Germany). During the experiment the volume was slowly oscillated in the range 7 ± 5µL for the samples with d + l = 2d to 5d and in the range 5 ± 2µL for the surface with d + l = 6d. For large pillar spacings the Cassie state is quite unstable and the experiments must be conducted with care.

The results are shown in Fig. 2 where normalized works of adhesion w/γ for advancing and receding triple lines are plotted as a function of pillar spacing. Overall the results are similar to other data in the literature [4,7,9]. The advancing contact angle is constant with an average value θ adv = 161 • i.e. a very small normalized work of adhesion (around 0.05), while the receding contact angle increases from 135 • to 152 • (and the work of adhesion decreases) when the normalized spacing l/d increases from 1 to 5.

Model. -General ideas on pinning emerged in the mid 80s when theoretical predictions were developed for the deformation of elastic lines propagating through heterogeneous energy landscapes. In the field of fracture, Rice has studied the morphology of crack fronts evolving in a bulk with heterogeneous toughness [14]. In the field of wetting, Joanny and de Gennes have calculated the morphology of a triple line propagating in a landscape of heterogeneous work of adhesion [12]. The deformation of the line in the plane of propagation results in a long range elastic field (for a crack) or corrugation of the liquid surface (for a triple line). As a result there emerges an energy term which is quadratic in the deformation. This term is viewed as a line elasticity. In fact the relations between energy heterogeneities (toughness or work of adhesion) in the plane and line deformation are identical for cracks and for wetting.

The front shape can be calculated for a given pattern of energy. An extreme case appears when the energy gradient in the direction of propagation is stronger than the line stiffness, in which case local instabilities are to be expected. Series of instabilities may be triggered cooperatively resulting in avalanches. This complex response of the front shape in this so called strong pinning case leads to highly non trivial behaviour and has attracted a lot of attention in the field of statistical physics [15,16].

Experimental demonstrations of the phenomena however are scarce. For fractures, the applicability of the Rice equation has been shown by direct crack front monitoring in semi transparent interfaces with simple patterns [17]. Avalanche statistics have been investigated in model heterogeneous PMMA interfaces [18]. In the field of wetting, front shapes were demonstrated to agree with the Joannyde Gennes theory and their dynamics shown to derive from viscous dissipation [19]. Statistical experiments were developed to analyze cooperative line motion for wetting surfaces [20].

To analyse the wetting properties of our surfaces (Fig. 1) we follow [12]. Compared to the homogeneous surface, the wetting energy is affected by the elastic energy barrier resulting from the presence of the pinning sites. In this scaling approach, we will assume square defects with width d as in [12]. We also denote y the position of the deformed liquid front, y 0 its undeformed position, b the position of the edge of the defect (Fig. 3).

For a receding triple line, the wetting energy per defect is

E w = -wd(y -b) (4)
and the elastic energy is Here k is the elastic stiffness of the triple line characterizing the deformation energy of the liquid surface when the triple line is distorted. We assume that the line elasticity follows the same functional form as in the wetting case [12] so that k = κ ln 1 + l d (6) This assumption sounds reasonable: in the presence of an heterogeneity, the liquid surface deformation close to the triple line is expected to be similar for the hydrophilic and the hydrophobic cases, although the exact boundary conditions differ. Since this elastic response is ultimately due to the liquid surface tension, the constant κ is expected to be a fraction of the liquid surface tension γ. Indeed in the hydrophilic case we have [12]

E el = 1 2 k(y -y 0 ) 2 (5
κ = 1 2 πγθ s 2 (7)
where θ s is the contact angle of the defect-free, wetting, surface. In the present case, the exact expression for k is unknown. Adding up Eqs. 4 and 5, we find that unstable equilibrium occurs when k(y -y 0 ) = wd (8) At rupture the elastic energy stored in the deformation of the front is found equal to

E el = 1 2 (wd) 2 k (9)
This is a rupture energy for a constant force equal to wd, a value equal to the 90 • peel force of an adhesive strip of width d and adhesion w [21] for example. To propagate the triple line, this elastic energy barrier has to be overcome so that the effective work of adhesion for a receding contact is

γ(1 + cos θ rec ) = 1 2 n (wd) 2 k ( 10
)
where n is the pillar density. Following Joanny-de Gennes, this equation is obtained by summation over independent pinning sites, a process which is expected to be valid for low densities. However the result differs from the Joannyde Gennes theory because the distribution here is periodic while they assumed a random distribution to account for stochastic surfaces.

For the advancing contact angle, line must be deformed up to the next pillar. The critical front deformation y c -y 0 for instability is expected to be of the order of one pillar spacing (Fig. 4). The necessary elastic energy is evaluated as

E el = 1 2 k(y c -y 0 ) 2 (11)
In contrast to the receding triple line, the elastic energy stored at instability is calculated for a constant displacement (or fixed grip). Therefore we write that the critical front deformation y c -y 0 = λl where λ depends on geometrical and physical parameters, but should be of the order of 1. As a first approximation, we assume here that λ is a constant. Finally for an advancing triple line the work of adhesion is

γ(1 + cos θ adv ) = 1 2 nk(λl) 2 (12)
Normalizing the line stiffness by the surface tension

k = k γ ( 13
)
where k is given by Eq. 6, we can calculate the work of adhesion for a receding triple line by substituting Eq. 3 into Eq. 10. Normalizing the work of adhesion by γ we obtain

1 + cos θ rec = 1 2 (w/γ) 2 k(1 + l/d) 2 (14)
Similarly for an advancing triple line, by substituting Eq. 3 into Eq. 12 we obtain

1 + cos θ adv = 1 2 k (λl/d) 2 (1 + l/d) 2 (15)
For our surfaces d = 10 µm and θ 0 = 110 • which sets w/γ 0.65 according to Eq. 1. Following Eqs. 14 and 15, the receding and advancing contact angles for variable l are plotted on Fig. 2 with

κ ≡ κ γ = 0.11 (16)
λ = 1.5 (17)
A good agreement with the data is obtained for both receding and advancing contact angles. We also compare the results with the so-called Cassie equation (Eq. 2 ), which predicts an effective work of adhesion proportional to density n and reads

1 + cos θ = π 4 w/γ (1 + l/d) 2 (18)
From the shapes of the curves (Fig. 2) it is clear that the Cassie equation per se can account for the dependence of neither the receding nor the advancing contact angle, even assuming an effective local contact angle θ ef f to stand for an effective work of adhesion w ef f /γ. Turning to the DA model, we calculate the differential surface texture parameter φ d according to [8] and substitute this value for the solid fraction φ in the Cassie equation (Eq. 2). For circular pillars on a square array, and for a receding triple line, we find

φ d,rec = 1 1 + l d = 4 π φ (19)
which provides a good fit to the data (Fig. 2). For an advancing triple line however, the DA model predicts φ d,adv = 0, which does not account for the experimental results (Fig. 2).

In the pinning model, for a receding triple line (Eq. 14), the ln(1 + l/d) factor originating from the elastic stiffness k (Eqs. 6 and 13) significantly mitigates the dependence of the work of adhesion with density n (Eq. 3), as noted earlier [9]. Moreover for an advancing triple line (Eq. 15), the inverse variation of the elastic pinning energy f (l/d) = (l/d) 2 / ln(1 + l/d) very nearly compensates for the variations of the density n, resulting in an extremely flat response for a large range of pillar spacing. As a result the product n × f (l/d) is virtually constant over the interval 1 < l/d < 8 beyond which it starts to decrease more noticeably. In fact, over the range 1 < l/d < 8 the product function n × f (l/d) rises from about 0.36 for l/d = 1 to a maximum of no more than 0.41 for l/d 2.51 only to slowly decrease back to about 0.36 at l/d = 8. As a result the work of adhesion for an advancing triple line, although finite, is very nearly constant for almost all practical purposes.

There is a strong contrast between the pinning and the DA models for advancing contact angles. It is worth noting that in the DA model the deformation of the triple line holds an ambiguous status: deformation is accepted but energy is minimized at fixed deformation. As a result the DA model invariably predicts an advancing contact angle equal to 180 • for non connected features. In contrast, the measurements consistently return values around 165 • . This discrepancy is often ascribed to vibrations or other experimental imperfections [8]. In this paper we have emphasized that the role of line elasticity may actually be predominant in this regime. Indeed an interesting parallel may be drawn with peeling. When peeling an adhesive tape the direct work of the tension σ is G d = hσ(1 + cos θ) where θ is the supplement to the peel angle α (i.e. θ + α = π) and h is the thickness of the tape. At equilibrium G d = w where w is the adhesion energy of the tape; however as the peel angle α goes to zero G d goes to zero as well and this equation can no longer hold. Kendall has shown that in this case it is the (lineic) elastic energy stored in the tape G el = hσ 2 /2E which equals the work of adhesion [21]. For wetting, when the contact angle θ goes to 180 • , the direct work of the surface tension (Eq. 1) goes to zero. The pinning theory suggests that in this case, it is the contact line elasticity which takes over to mediate between work of adhesion and contact angle, in a manner similar to peeling.

The scaling model based on the elasticity of the pinned triple line is quite powerful at accounting for the overall dependence of both advancing and receding contact angles simultaneously. In this respect note also that the values of the two free parameters are fully consistent with the assumptions of the model: the stiffness constant κ is indeed a fraction the surface energy while the displacement parameter λ is of the order of 1. For an advancing triple line, the elastic energy of the line when it jumps to the next pinning site is controlled by the pillar spacing. As a result the advancing contact angle is nearly constant because the increasing elastic energy approximately cancels the decreasing number density. This quasi-constant is not zero (i.e. the contact angle is not 180 • ) and some energy is indeed expended in elastic deformation. In contrast, the receding contact angle is primarily controlled by the pillar size because during the instability the triple line is peeled off the top of the pillar. Hence the receding contact angle decreases roughly as the density, with a correction due to the weaker logarithmic factor which stems from triple line stiffness. For low enough densities, receding and advancing contact angles tend to converge to the same low adhesion value, resulting in low hysteresis.

Several refinements to the model can be considered. The elastic response of the triple line could be modeled more precisely and the instability conditions for both advancing and receding cases analyzed quantitatively. In particular the front shape and pillar geometry could then be taken into account accurately. This more in depth analysis can only be carried out using exact numerical solutions provided by classical fluid [22] or liquid models [23]. Then the direct contribution of the Cassie-like surface energy terms could be taken into account. Such calculations would improve the potential of the method for the design of structures with tailored wetting properties, but it is not expected that these more quantitative insights would radically alter the picture of the motion of a triple line on hydrophobic textured surfaces propounded here.

Conclusion. -Model, periodic, hydrophobic surfaces exhibit a well identified evolution of advancing and receding contact angles with solid area. A simple interpretation of the data can consistently be derived from a pinning theory which duly takes into account the elastic energy of deformation of the contact line. For an advancing triple line, the deformation occurs at a fixed displacement, and the line deformation is of the order of the pillar spacing, resulting in a nearly constant (and small) work of adhesion. In contrast line deformation proceeds at a fixed force for a receding triple line (this is similar to a peel force), giving a pronounced decrease of the work of adhesion as the pillar spacing increases. These opposite types of boundary conditions result in a strong contrast in energy barriers for advancing and receding triple lines and consistently account for the triple line response at this macroscopic level of description.

Fig. 1 :

 1 Fig. 1: SEM view of a surface textured by silica sol-gel nanoimprint lithography. Here the pillar spacing l equals 10 µm. Various surfaces were generated where the pillar geometry is kept constant and the spacings vary from 10 to 50 µm.

Fig. 2 :

 2 Fig. 2: Measured advancing (right triangle) and receding (left triangle) normalized work of adhesion w/γ = 1+cos θ as a funcof pillar spacing. Error bars are mostly smaller than symbol size. A typical Cassie model (small dash) cannot account for the results. The DA model (see text; dash-dot) provides a good prediction for the receding triple line but concludes that the work of adhesion is zero for an advancing triple line.In contrast, the pinning model (dash) simultaneously accounts for advancing and receding contact angles.

Fig. 3 :

 3 Fig. 3: Schematics of pinning for a receding triple line following Joanny and de Gennes [12]. The geometrical parameters of the instability are defect width d, defect position b, nominal triple line position y0 and triple line deflection y -y0. The triple line deflection at instability is determined by the competition between the peel force of the liquid from the pillar surface wd and the triple line stiffness k.

Fig. 4 :

 4 Fig.4: Schematics of pinning for an advancing triple line following Joanny and de Gennes[12]. The deflection y c -y 0 at instability is controlled by the spacing between defects in the direction of motion of the front. No attempt has been made at giving a realistic rendering of the actual deformation of the liquid surface near the front.

We thank Elie Raphaël for interesting discussions on triple line deformation and instabilities in the context of the present experiments. We are grateful to Antoine Chateauminois for help with the elaboration of the textured surfaces. This work was supported by the National Research Agency (ANR) within the framework of the DY-NALO project (project n • NT09_499845).