2 Alternative Theories of Gravity

In this section, we discuss the many possible alternative theories that have been studied so far in the context of gravitational-wave tests. We begin with a description of the theoretically desirable properties that such theories must have. We then proceed with a review of the theories so far explored as far as gravitational waves are concerned. We will leave out the description of many theories in this chapter, especially those which currently lack a gravitational-wave analysis. We will conclude with a brief description of unexplored theories as possible avenues for future research.

2.1 Desirable theoretical properties

The space of possible theories is infinite, and thus, one is tempted to reduce it by considering a subspace that satisfies a certain number of properties. Although the number and details of such properties depend on the theorist’s taste, there is at least one fundamental property that all scientists would agree on:

  1. Precision Tests. The theory must produce predictions that pass all solar system, binary pulsar, cosmological and experimental tests that have been carried out so far.

This requirement can be further divided into the following:

    1. General Relativity Limit. There must exist some limit, continuous or discontinuous, such as the weak-field one, in which the predictions of the theory are consistent with those of GR within experimental precision.
    2. Existence of Known Solutions [426*]. The theory must admit solutions that correspond to observed phenomena, including but not limited to (nearly) flat spacetime, (nearly) Newtonian stars, and cosmological solutions.
    3. Stability of Solutions [426*]. The special solutions described in property (1.b) must be stable to small perturbations on timescales smaller than the age of the universe. For example, perturbations to (nearly) Newtonian stars, such as impact by asteroids, should not render such solutions unstable.

Of course, these properties are not all necessarily independent, as the existence of a weak-field limit usually also implies the existence of known solutions. On the other hand, the mere existence of solutions does not necessarily imply that these are stable.

In addition to these fundamental requirements, one might also wish to require that any new modified gravity theory possesses certain theoretical properties. These properties will vary depending on the theorist, but the two most common ones are listed below:

  1. Well-motivated from Fundamental Physics. There must be some fundamental theory or principle from which the modified theory (effective or not) derives. This fundamental theory would solve some fundamental problem in physics, such as late-time acceleration or the incompatibility between quantum mechanics and GR.
  2. Well-posed Initial Value Formulation [426]. A wide class of freely specifiable initial data must exist, such that there is a uniquely determined solution to the modified field equations that depends continuously on this data.

The second property goes without saying at some level, as one expects modified-gravity–theory constructions to be motivated from some (perhaps yet incomplete) quantum-gravitational description of nature. As for the third property, the continuity requirement is necessary because otherwise the theory would lose predictive power, given that initial conditions can only be measured to a finite accuracy. Moreover, small changes in the initial data should not lead to solutions outside the causal future of the data; that is, causality must be preserved. Section 2.2 expands on this well-posedness property further.

One might be concerned that Property (2) automatically implies that any predicted deviation to astrophysical observables will be too small to be detectable. This argument usually goes as follows. Any quantum gravitational correction to the action will “naturally” introduce at least one new scale, and this, by dimensional analysis, must be the Planck scale. Since this scale is usually assumed to be larger than 1 TeV in natural units (or 10−35 m in geometric units), gravitational-wave observations will never be able to observe quantum-gravitational modifications (see, e.g., [155*] for a similar argument). Although this might be true, in our view such arguments can be extremely dangerous, since they induce a certain theoretical bias in the search for new phenomena. For example, let us consider the supernova observations of the late-time expansion of the universe that led to the discovery of the cosmological constant. The above argument certainly fails for the cosmological constant, which on dimensional arguments is over 100 orders of magnitude too small. If the supernova teams had respected this argument, they would not have searched for a cosmological constant in their data. Today, we try to explain our way out of the failure of such dimensional arguments by claiming that there must be some exquisite cancellation that renders the cosmological constant small; but this, of course, came only after the constant had been measured. One is not trying to argue here that cancellations of this type are common and that quantum gravitational modifications are necessarily expected in gravitational-wave observations. Rather, we are arguing that one should remain agnostic about what is expected and what is not, and allow oneself to be surprised without suppressing the potential for new discoveries that will accompany the new era of gravitational-wave astrophysics.

One last property that we wish to consider for the purposes of this review is:

  1. Strong Field Inconsistency. The theory must lead to observable deviations from GR in the strong-field regime.

Many modified gravity models have been proposed that pose infrared or cosmological modifications to GR, aimed at explaining certain astrophysical or cosmological observables, like the late expansion of the universe. Such modified models usually reduce to GR in the strong-field regime, for example via a Vainshtein-like mechanism [413*, 140*, 45*] in a static spherically-symmetric context. Extending this mechanism to highly-dynamical strong-field scenarios has not been fully worked out yet [137*, 138*]. Gravitational-wave tests of GR, however, are concerned with modified theories that predict deviations in the strong-field, precisely where cosmological modified models do not. Clearly, Property (4) is not necessary for a theory to be a valid description of nature. This is because a theory might be identical to GR in the weak and strong fields, yet different at the Planck scale, where it would be unified with quantum mechanics. However, Property (4) is a desirable feature if one is to test this theory with gravitational wave observations.

2.2 Well-posedness and effective theories

Property (3) not only requires the existence of an initial value formulation, but also that it be well posed, which is not necessarily guaranteed. For example, the Cauchy–Kowalewski theorem states that a system of n partial differential equations for n unknown functions ϕi of the form ϕi,tt = Fi(xμ;ϕj,μ; ϕj,ti;ϕj,ik), with Fi analytic functions has an initial value formulation (see, e.g., [425*]). However, this theorem does not guarantee continuity or the causal conditions described above. For this, one has to rely on more general energy arguments, for example constructing a suitable energy measure that obeys the dominant energy condition and using it to show well-posedness (see, e.g., [225, 425*]). One can show that second-order, hyperbolic partial differential equations, i.e., equations of the form

μ μ ∇ ∇ μϕ + A ∇ μϕ + Bϕ + C = 0, (5 )
where Aμ is an arbitrary vector field and (B, C) are smooth functions, have a well-posed initial value formulation. Moreover, the Leray theorem proves that any quasilinear, diagonal, second-order hyperbolic system also has a well-posed initial value formulation [425*].

Proving the well-posedness of an initial-value formulation for systems of higher-than-second-order, partial differential equations is much more difficult. In fact, to our knowledge, no general theorems exist of the type described above that apply to third, fourth or higher-order, partial, non-linear and coupled differential equations. Usually, one resorts to the Ostrogradski theorem [337*] to rule out (or at the very least cast serious doubt on) theories that lead to such higher-order field equations. Ostrogradski’s theorem states that Lagrangians that contain terms with higher-than-first-time derivatives possess a linear instability in the Hamiltonian (see, e.g., [443*] for a nice review).2 As an example, consider the Lagrangian density

m m ω2 gm ℒ = --˙q2 − ----q2 − ---2¨q2, (6 ) 2 2 2 ω
whose equations of motion,
2 -g-.... ¨q + ω q = − ω2 q , (7 )
obviously contain higher derivatives. The exact solution to this differential equation is
q = A1 cos k1t + B1 sink1t + A2 cosk2t + B2 sin k2t, (8 )
where (Ai,Bi) are constants and 2 2 √ ------- k1,2∕ ω = (1 ∓ 1 − 4g )∕(2g). The on-shell Hamiltonian is then
m-∘ -------2( 2 2) m- ∘ -------2 ( 2 2) H = 2 1 − 4gk 1 A 1 + B1 − 2 1 − 4gk2 A2 + B 2 , (9 )
from which it is clear that mode 1 carries positive energy, while mode 2 carries negative energy and forces the Hamiltonian to be unbounded from below. The latter implies that dynamical degrees of freedom can reach arbitrarily negative energy states. If interactions are present, then an “empty” state would instantaneously decay into a collection of positive and negative energy particles, which cannot describe the universe we live in [443].

However, the Ostrogradski theorem [337] can be evaded if the Lagrangian in Eq. (6*) describes an effective theory, i.e., a theory that is a truncation of a more general or complete theory. Let us reconsider the particular example above, assuming now that the coupling constant g is an effective theory parameter and Eq. (6*) is only valid to linear order in g. One approach is to search for perturbative solutions of the form qpert = x0 + gx1 + ..., which leads to the system of differential equations

2 1-..x.. ¨xn + ω xn = − ω2 n−1, (10 )
with x = 0 −1. Solving this set of n differential equations and resumming, one finds
qpert = A1 cosk1t + B1 sin k1t. (11 )
Notice that qpert contains only the positive (well-behaved) energy solution of Eq. (8*), i.e., perturbation theory acts to retain only the well-behaved, stable solution of the full theory in the g → 0 limit. One can also think of the perturbative theory as the full theory with additional constraints, i.e., the removal of unstable modes, which is why such an analysis is sometimes called perturbative constraints [117, 118, 466*].

Another way to approach effective field theories that lead to equations of motion with higher-order derivatives is to apply the method of order reduction. In this method, one substitutes the low-order derivatives of the field equations into the high-order derivative part, thus rendering the resulting new theory usually well posed. One can think of this as a series resummation, where one changes the non-linear behavior of a function by adding uncontrolled, higher-order terms. Let us provide an explicit example by reconsidering the theory in Eq. (6*). To lowest order in g, the equation of motion is that of a simple harmonic oscillator,

2 ¨q + ω q = 𝒪 (g), (12 )
which is obviously well posed. One can then order-reduce the full equation of motion, Eq. (7*), by substituting Eq. (12*) into the right-hand side of Eq. (7*). Doing so, one obtains the order-reduced equation of motion
¨q + ω2q = g ¨q + 𝒪 (g2), (13 )
which now clearly has no high-order derivatives and is well posed, provided g ≪ 1. The solution to this order-reduced differential equation is qpert once more, but with k1 linearized in g ≪ 1. Therefore, the solutions obtained with a perturbative decomposition and with the order-reduced equation of motion are the same to linear order in g. Of course, since an effective field theory is only defined to a certain order in its perturbative parameter, both treatments are equally valid, with the unstable mode effectively removed in both cases.

However, such a perturbative analysis can say nothing about the well-posedness of the full theory from which the effective theory derives, or of the effective theory if treated as an exact one (i.e., not as a perturbative expansion). In fact, a well-posed full theory may have both stable and unstable solutions. The arguments presented above only discuss the stability of solutions in an effective theory, and thus, they are self-consistent only within their perturbative scheme. A full theory may have non-perturbative instabilities, but these can only be studied once one has a full (non-truncated in g) theory, from which Eq. (6*) derives as a truncated expansion. Lacking a full quantum theory of nature, quantum gravitational models are usually studied in a truncated low-energy expansion, where the leading-order piece is GR and higher-order pieces are multiplied by a small coupling constant. One can perturbatively explore the well-behaved sector of the truncated theory about solutions to the leading-order theory. However, such an analysis is incapable of answering questions about well-posedness or non-linear stability of the full theory.

2.3 Explored theories

In this subsection we briefly describe the theories that have so far been studied in some depth as far as gravitational waves are concerned. In particular, we focus only on those theories that have been sufficiently studied so that predictions of the expected gravitational waveforms (the observables of gravitational-wave detectors) have been obtained for at least a typical source, such as the quasi-circular inspiral of a compact binary.

2.3.1 Scalar-tensor theories

Scalar-tensor theories in the Einstein frame [82, 129*, 166, 165, 181, 197] are defined by the action (where we will restore Newton’s gravitational constant G in this section)

∫ (E) 1 4 √--- μν 2 S ST = ------ d x − g [R − 2g (∂ μφ)(∂νφ ) − V(φ )] + Smat[ψmat,A (φ)gμν], (14 ) 16πG
where φ is a scalar field, A (φ ) is a coupling function, V (φ) is a potential function, ψmat represents matter degrees of freedom and G is Newton’s constant in the Einstein frame. For more details on this theory, we refer the interested reader to the reviews [438*, 435*]. Of course, one can consider more complicated scalar-tensor theories, for example by including multiple scalar fields, but we will ignore such generalizations here.

The Einstein frame is not the frame where the metric governs clocks and rods, and thus, it is convenient to recast the theory in the Jordan frame through the conformal transformation &tidle;gμν = A2(φ )g μν:

1 ∫ ∘ ---[ ω (ϕ) ] S(SJT)= ------ d4x − &tidle;g ϕ &tidle;R − -----&tidle;gμν (∂μϕ)(∂νϕ ) − ϕ2V + Smat [ψmat, &tidle;gμν], (15 ) 16πG ϕ
where &tidle;gμν is the physical metric, the new scalar field ϕ is defined via ϕ ≡ A− 2, the coupling field is −2 ω (ϕ) ≡ (α − 3)∕2 and α ≡ A,φ∕A. When cast in the Jordan frame, it is clear that scalar-tensor theories are metric theories (see [438*] for a definition), since the matter sector depends only on matter degrees of freedom and the physical metric (without a direct coupling of the scalar field). When the coupling ω(ϕ) = ωBD is constant, then Eq. (15*) reduces to the massless version of Jordan–Fierz–Brans–Dicke theory [82].

The modified field equations in the Einstein frame are



[ ] (φ) 1 1 ,δ 1 Tμν = --- φ,μφ,ν − --gμνφ,δφ − --gμνV(φ ) (17 ) 4π 2 4
is a stress-energy tensor for the scalar field. The matter stress–energy tensor is not constructed from the Einstein-frame metric alone, but by the combination A (φ)2gμν. In the Jordan frame and neglecting the potential, the modified field equations are [435*]

where T mat is the trace of the matter stress-energy tensor Tmat μν constructed from the physical metric &tidle;g μν. The form of the modified field equations in Jordan frame suggest that in the weak-field limit one may consider scalar-tensor theories as modifying Newton’s gravitational constant via G → G(ϕ ) = G ∕ϕ.

Using the decompositions of Eqs. (3*)-(4*), the field equations of massless Jordan–Fierz–Brans–Dicke theory can be linearized in the Jordan frame to find (see, e.g., [441*])

μν μν □ η𝜃 = − 16πτ , □ ηψ = − 16πS, (19 )
where □η is the D’Alembertian operator of flat spacetime, we have defined a new metric perturbation
1 ψ 𝜃μν = h μν − --ημνh − ---ημν, (20 ) 2 ϕ0
i.e., the metric perturbation in the Einstein frame, with h the trace of the metric perturbation and

with cubic remainders in either the metric perturbation or the scalar perturbation. The quantity mat ∂T ∕∂ϕ arises in an effective point-particle theory, where the matter action is a functional of both the Jordan-frame metric and the scalar field. The quantity μν t is a function of quadratic or higher order in 𝜃 μν or ψ. These equations can now be solved given a particular physical system, as done for quasi-circular binaries in [441*, 374, 336]. Given the above evolution equations, Jordan–Fierz–Brans–Dicke theory possesses a scalar (spin-0) mode, in addition to the two transverse-traceless (spin-2) modes of GR, i.e., Jordan–Fierz–Brans–Dicke theory is of Type N3 in the E (2) classification [161*, 438*].

Let us now discuss whether scalar-tensor theories satisfy the properties discussed in Section 2.1. Massless Jordan–Fierz–Brans–Dicke theory agrees with all known experimental tests provided 4 ωBD > 4 × 10, a bound imposed by the tracking of the Cassini spacecraft through observations of the Shapiro time delay [73*]. Massive Jordan–Fierz–Brans–Dicke theory has been recently constrained to ωBD > 4 × 104 and ms < 2.5 × 10−20 eV, with ms the mass of the scalar field [348, 20*]. Of course, these bounds are not independent, as when ms → 0 one recovers the standard massless constraint, while when m → ∞ s, ω BD cannot be bounded as the scalar becomes non-dynamical. Observations of the Nordtvedt effect with Lunar Laser Ranging observations, as well as observations of the orbital period derivative of white-dwarf/neutron-star binaries, yield similar constraints [131*, 132*, 20*, 177*]. Neglecting any homogeneous, cosmological solutions to the scalar-field evolution equation, it is clear that in the limit ω → ∞ one recovers GR, i.e., scalar-tensor theories have a continuous limit to Einstein’s theory, but see [164*] for caveats for certain spacetimes. Moreover, [375, 278, 425] have verified that scalar-tensor theories with minimal or non-minimal coupling in the Jordan frame can be cast in a strongly-hyperbolic form, and thus, they possess a well-posed initial-value formulation. Therefore, scalar-tensor theories possess both Properties (1) and (3).

Scalar-tensor theories also possess Property (2), since they can be derived from the low-energy limit of certain string theories. The integration of string quantum fluctuations leads to a higher-dimensional string theoretical action that reduces locally to a field theory similar to a scalar-tensor one [189, 176], the mapping being ϕ = e−2ψ, with ψ one of the string moduli fields [133, 134]. Moreover, scalar-tensor theories can be mapped to f(R ) theories, where one replaces the Ricci scalar by some functional of R. In particular, one can show that f (R) theories are equivalent to Brans–Dicke theory with ωBD = 0, via the mapping ϕ = df (R)∕dR and V (ϕ) = Rdf (R )∕dR − f (R) [104, 396]. For a recent review on this topic, see [135].

Black holes and stars continue to exist in scalar-tensor theories. Stellar configurations are modified from their GR profile [441*, 131*, 214, 215, 410, 132*, 394, 139, 393, 235*], while black holes are not, provided one neglects homogeneous, cosmological solutions to the scalar field evolution equation. Indeed, Hawking [224*, 159*, 222*, 98*, 244*, 363*] has proven that Brans–Dicke black holes that are stationary and the endpoint of gravitational collapse are identical to those of GR. This proof has recently been extended to a general class of scalar-tensor models [398*]. That is, stationary black holes radiate any excess “hair”, i.e., additional degrees of freedom, after gravitational collapse, a result sometimes referred to as the no-hair theorem for black holes in scalar-tensor theories. This result has recently been extended even further to allow for quasi-stationary scenarios in generic scalar-tensor theories through the study of extreme–mass-ratio inspirals [465*] (small black hole in orbit around a much larger one), post-Newtonian comparable-mass inspirals [315*] and numerical simulations of comparable-mass black-hole mergers [230*, 67*].

Damour and Esposito-Farèse [129*, 130*] proposed a different type of scalar-tensor theory, one that can be defined by the action in Eq. (15*) but with the conformal factor αφ+βφ2∕2 A (φ) = e or the coupling function ω(ϕ ) = − 3∕2 − 2 πG ∕(β log ϕ), where α and β are constants. When β = 0 one recovers standard Brans–Dicke theory. When β ≲ − 4, non-perturbative effects that develop if the gravitational energy is large enough can force neutron stars to spontaneously acquire a non-trivial scalar field profile, to spontaneously scalarize. Through this process, a neutron-star binary that initially had no scalar hair in its early inspiral would acquire it before merger, when the binding energy exceeded some threshold [51*]. Binary pulsar observations have constrained this theory in the (α, β) space; very roughly speaking β > − 4 and α < 10−2 [131, 132, 177]

As for Property (4), scalar tensor theories are not built with the aim of introducing strong-field corrections to GR.3 Instead, they naturally lead to modifications of Einstein’s theory in the weak-field, modifications that dominate in scenarios with sufficiently weak gravitational interactions. Although this might seem strange, it is natural if one considers, for example, one of the key modifications introduced by scalar-tensor theories: the emission of dipolar gravitational radiation. Such dipolar emission dominates over the general relativistic quadrupolar emission for systems in the weak to intermediate field regime, such as in binary pulsars or in the very early inspiral of compact binaries. Therefore, one would expect scalar-tensor theories to be best constrained by experiments or observations of weakly-gravitating systems, as it has recently been explicitly shown in [465*].

2.3.2 Massive graviton theories and Lorentz violation

Massive graviton theories are those in which the gravitational interaction is propagated by a massive gauge boson, i.e., a graviton with mass mg ⁄= 0 or Compton wavelength λg ≡ h∕(mgc ) < ∞. Einstein’s theory predicts massless gravitons and thus gravitational propagation at light speed, but if this were not the case, then a certain delay would develop between electromagnetic and gravitational signals emitted simultaneously at the source. Fierz and Pauli [169*] were the first to write down an action for a free massive graviton, and ever since then, much work has gone into the construction of such models. For a detailed review, see, e.g., [232].

Gravitational theories with massive gravitons are somewhat well-motivated from a fundamental physics perspective, and thus, one can say they possess Property (2). Indeed, in loop quantum cosmology [42, 77], the cosmological extension to loop quantum gravity, the graviton dispersion relation acquires holonomy corrections during loop quantization that endow the graviton with a mass [78*] mg = Δ −1∕2γ−1(ρ∕ρc), with γ the Barbero–Immirzi parameter, Δ the area operator, and ρ and ρc the total and critical energy density respectively. In string-theory–inspired effective theories, such as Dvali’s compact, extra-dimensional theory [157], such massive modes also arise.

Massive graviton modes also occur in many other modified gravity models. In Rosen’s bimetric theory [365*], for example, photons and gravitons follow null geodesics of different metrics [438*, 435*]. In Visser’s massive graviton theory [424*], the graviton is given a mass at the level of the action through an effective perturbative description of gravity, at the cost of introducing a non-dynamical background metric, i.e., a prior geometry. A recent re-incarnation of this model goes by the name of bigravity, where again two metric tensors are introduced [349*, 346*, 219*, 220*]. In Bekenstein’s Tensor-Vector-Scalar (TeVeS) theory [54], the existence of a scalar and a vector field lead to subluminal gravitational-wave propagation.

Massive graviton theories have a theoretical issue, the van Dam–Veltman–Zakharov (vDVZ) discontinuity [418, 475], which is associated with Property 1.a, i.e., a GR limit. The problem is that certain predictions of massive graviton theories do not reduce to those of GR in the mg → 0 limit. This can be understood qualitatively by studying how the 5 spin states of the graviton behave in this limit. Two of them become the two GR helicity states of the massless graviton. Another two become helicity states of a massless vector that decouples from the tensor perturbations in the mg → 0 limit. However, the last state, the scalar mode, retains a finite coupling to the trace of the stress-energy tensor in this limit. Therefore, massive graviton theories in the mg → 0 limit do not reduce to GR, since the scalar mode does not decouple.

However, the vDVZ discontinuity can be evaded, for example, by carefully including non-linearities. Vainshtein [413, 269, 140, 45] showed that around any spherically-symmetric source of mass M, there exists a certain radius r < rV ≡ (rSλ4g)1∕5, with rS the Schwarzschild radius, where linear theory cannot be trusted. Since r → ∞ V as m → 0 g, this implies that there is no radius at which the linear approximation (and thus vDVZ discontinuity) can be trusted. Of course, to determine then whether massive graviton theories have a continuous limit to GR, one must include non-linear corrections to the action (see also an argument by [34]), which are more difficult to uniquely predict from fundamental theory. Recently, there has been much activity in the development of new, non-linear massive gravity theories [60*, 136*, 211, 61, 137, 138].

Lacking a particular action for massive graviton theories that modifies the strong-field regime and is free of non-linear and radiatively-induced ghosts, it is difficult to ascertain many of its properties, but this does not prevent us from considering certain phenomenological effects. If the graviton is truly massive, whatever the action may be, two main modifications to Einstein’s theory will be introduced:

  • Modification to Newton’s laws;
  • Modification to gravitational wave propagation.

Modifications of class (i) correspond to the replacement of the Newtonian potential by a Yukawa type potential (in the non-radiative, near-zone of any body of mass M): V = (M ∕r) → (M ∕r)exp (− r ∕λg), where r is the distance to the body [437*]. Tests of such a Yukawa interaction have been proposed through observations of bound clusters, tidal interactions between galaxies [200] and weak gravitational lensing [106], but such tests are model dependent.

Modifications of class (ii) are in the form of a non-zero graviton mass that induces a modified gravitational-wave dispersion relation. Such a modification to the dispersion relation was originally parameterized via [437*]

v2 m2 c4 -2g= 1 − --g2-, (23 ) c E
where vg and mg are the speed and mass of the graviton, while E is its energy, usually associated to its frequency via the quantum mechanical relation E = hf. This modified dispersion relation is inspired by special relativity, a more general version of which, inspired by quantum gravitational theories, is [316*]
2 vg- α c2 = 1 − λ , (24 )
where α is now a parameter that depends on the theory and λ represents deviations from light-speed propagation. For example, in Rosen’s bimetric theory [365], the graviton does not travel at the speed of light, but at some other speed partially determined by the prior geometry. In metric theories of gravity, 2 4 2 λ = Am gc ∕E, where A is some amplitude that depends on the metric theory (see discussion in [316*]). Either modification to the dispersion relation has the net effect of slowing gravitons down, such that there is a difference in the time of arrival of photons and gravitons. Moreover, such an energy-dependent dispersion relation would also affect the accumulated gravitational-wave phase observed by gravitational-wave detectors, as we discuss in Section 5. Given these modifications to the dispersion relation, one would expect the generation of gravitational waves to also be greatly affected in such theories, but again, lacking a particular healthy action to consider, this topic remains today mostly unexplored.

From the structure of the above phenomenological modifications, it is clear that GR can be recovered in the mg → 0 limit, avoiding the vDVZ issue altogether by construction. Such phenomenological modifications have been constrained by several types of experiments and observations. Using the modification to Newton’s third law and precise observations of the motion of the inner planets of the solar system together with Kepler’s third law, [437*] found a bound of 12 λg > 2.8 × 10 km. Such a constraint is purely static, as it does not sample the radiative sector of the theory. Dynamical constraints, however, do exist: through observations of the decay of the orbital period of binary pulsars, [174*] found a bound of λg > 1.6 × 1010 km;4 by investigating the stability of Schwarzschild and Kerr black holes, [88*] placed the constraint 13 λg > 2.4 × 10 km in Fierz–Pauli theory [169*]. New constraints that use gravitational waves have been proposed, including measuring a difference in time of arrival of electromagnetic and gravitational waves [126*, 266], as well as direct observation of gravitational waves emitted by binary pulsars (see Section 5).

Although massive gravity theories unavoidably lead to a modification to the graviton dispersion relation, the converse is not necessarily true. A modification of the dispersion relation is usually accompanied by a modification to either the Lorentz group or its action in real or momentum space. Such Lorentz-violating effects are commonly found in quantum gravitational theories, including loop quantum gravity [78] and string theory [107, 403], as well as other effective models [58, 59]. In doubly-special relativity [26, 300, 27, 28], the graviton dispersion relation is modified at high energies by modifying the law of transformation of inertial observers. Modified graviton dispersion relations have also been shown to arise in generic extra-dimensional models [381], in Hořava–Lifshitz theory [233, 234*, 412, 76] and in theories with non-commutative geometries [186, 187, 188]. None of these theories necessarily requires a massive graviton, but rather the modification to the dispersion relation is introduced due to Lorentz-violating effects.

One might be concerned that the mass of the graviton and subsequent modifications to the graviton dispersion relation should be suppressed by the Planck scale. However, Collins, et al. [111, 110] have suggested that Lorentz violations in perturbative quantum field theories could be dramatically enhanced when one regularizes and renormalizes them. This is because terms that vanish upon renormalization due to Lorentz invariance do not vanish in Lorentz-violating theories, thus leading to an enhancement [185]. Whether such an enhancement is truly present cannot currently be ascertained.

2.3.3 Modified quadratic gravity

Modified quadratic gravity is a family of models first discussed in the context of black holes and gravitational waves in [473*, 447*]. The 4-dimensional action is given by


The quantity ∗R μνδσ = (1 ∕2)𝜖δσαβR μναβ is the dual to the Riemann tensor. The quantity ℒmat is the external matter Lagrangian, while fi(⋅) are functionals of the field 𝜗, with (αi,β) coupling constants and κ = (16πG )−1. Clearly, the two terms second to last in Eq. (25) represent a canonical kinetic energy term and a potential. At this stage, one might be tempted to set β = 1 or the αi = 1 via a rescaling of the scalar field functional, but we shall not do so here.

The action in Eq. (25) is well-motivated from fundamental theories, as it contains all possible quadratic, algebraic curvature scalars with running (i.e., non-constant) couplings. The only restriction here is that all quadratic terms are assumed to couple to the same field, which need not be the case. For example, in string theory some terms might couple to the dilaton (a scalar field), while others couple to the axion (a pseudo scalar field). Nevertheless, one can recover well-known and motivated modified gravity theories in simple cases. For example, dynamical Chern–Simons modified gravity [17*] is recovered when α4 = − αCS∕4 and all other αi = 0. Einstein-Dilaton-Gauss–Bonnet gravity [343*] is obtained when α4 = 0 and (α ,α ,α ) = (1,− 4,1)α 1 2 3 EDGB.5 Both theories unavoidably arise as low-energy expansions of heterotic string theory [203*, 204*, 12*, 89*]. As such, modified quadratic gravity theories should be treated as a class of effective field theories. Moreover, dynamical Chern–Simons gravity also arises in loop quantum gravity [43, 366] when the Barbero–Immirzi parameter is promoted to a field in the presence of fermions [41*, 16, 406*, 311*, 192*].

One should make a clean and clear distinction between the theory defined by the action of Eq. (25) and that of f (R ) theories. The latter are defined as functionals of the Ricci scalar only, while Eq. (25) contains terms proportional to the Ricci tensor and Riemann tensor squared. One could think of the subclass of f(R ) theories with f(R ) = R2 as the limit of modified quadratic gravity with only α1 ⁄= 0 and f1(𝜗) = 1. In that very special case, one can map quadratic gravity theories and f(R ) gravity to a scalar-tensor theory. Another important distinction is that f (R) theories are usually treated as exact, while the action presented above is to be interpreted as an effective theory [89*] truncated to quadratic order in the curvature in a low-energy expansion of a more fundamental theory. This implies that there are cubic, quartic, etc. terms in the Riemann tensor that are not included in Eq. (25) and that presumably depend on higher powers of αi. Thus, when studying such an effective theory one should also order-reduce the field equations and treat all quantities that depend on αi perturbatively, the small-coupling approximation. One can show that such an order reduction removes any additional polarization modes in propagating metric perturbations [390*, 400*] that naturally arise in f (R ) theories. In analogy to the treatment of the Ostrogradski instability in Section 2.1, one would also expect that order-reduction would lead to a theory with a well-posed initial-value formulation.

This family of theories is usually simplified by making the assumption that coupling functions fi(⋅) admit a Taylor expansion: fi(𝜗) = fi(0) + f′i(0 )𝜗 + 𝒪(𝜗2) for small 𝜗, where fi(0) and f′i(0) are constants and 𝜗 is assumed to vanish at asymptotic spatial infinity. Reabsorbing fi(0) into the coupling constants (0) α i ≡ αifi(0) and ′ fi(0) into the constants (1) ′ α i ≡ αifi(0), Eq. (25) becomes S = SGR + S0 + S1 with


Here, SGR is the Einstein–Hilbert plus matter action, while S0 and S1 are corrections. The former is decoupled from 𝜗, where the omitted term proportional to (0) α4 does not affect the classical field equations since it is topological, i.e., it can be rewritten as the total 4-divergence of some 4-current. Similarly, if the α(0) i were chosen to reconstruct the Gauss–Bonnet invariant, (0) (0) (0) (α 1 ,α2 ,α 3 ) = (1,− 4,1)αGB, then this combination would also be topological and not affect the classical field equations. On the other hand, S1 is a modification to GR with a direct (non-minimal) coupling to 𝜗, such that as the field goes to zero, the modified theory reduces to GR.

Another restriction one usually makes to simplify modified gravity theories is to neglect the (0) αi terms and only consider the S1 modification, the restricted modified quadratic gravity. The α (0i) terms represent corrections that are non-dynamical. The term proportional to (0) α1 resembles a certain class of f(R ) theories. As such, it can be mapped to a scalar-tensor theory with a complicated potential, which has been heavily constrained by torsion-balance Eöt-Wash experiments to (0) −8 2 α 1 < 2 × 10 m [237, 259*, 62]. Moreover, these theories have a fixed coupling constant that does not run with energy or scale. In restricted modified gravity, the scalar field is effectively forcing the running of the coupling.

Then, let us concentrate on restricted modified quadratic gravity and drop the superscript in α(1) i. The modified field equations are


where we have defined


The 𝜗 stress-energy tensor is

[ 1 ( )] T(μ𝜗ν)= β (∇ μ𝜗 )(∇ ν𝜗 ) −--gμν ∇ δ𝜗∇ δ𝜗 − 2V (𝜗) . (29 ) 2
The field equations for the scalar field are

Notice that unlike traditional scalar-tensor theories, the scalar field is here sourced by the geometry and not by the matter distribution. This directly implies that black holes in such theories are likely to be hairy.

From the structure of the above equations, it should be clear that the dynamics of 𝜗 guarantee that the modified field equations are covariantly conserved exactly. That is, one can easily verify that the covariant divergence of Eq. (27) identically vanishes upon imposition of Eq. (30). Such a result had to be so, as the action is diffeomorphism invariant. If one neglected the kinetic and potential energies of 𝜗 in the action, as was originally done in [245*], the theory would possess preferred-frame effects and would not be covariantly conserved. Moreover, such a theory requires an additional constraint, i.e., the right-hand side of (30) would have to vanish, which is an unphysical consequence of treating 𝜗 as prior structure [470*, 207*].

One last simplification that is usually made when studying modified quadratic gravity theories is to ignore the potential V (𝜗), i.e., set V (𝜗) = 0. This potential can in principle be non-zero, for example if one wishes to endow 𝜗 with a mass or if one wishes to introduce a cosine driving term, like that for axions in field and string theory. However, reasons exist to restrict the functional form of such a potential. First, a mass for 𝜗 will modify the evolution of any gravitational degree of freedom only if this mass is comparable to the inverse length scale of the problem under consideration (such as a binary system). This could be possible if there is an incredibly large number of fields with different masses in the theory, such as perhaps in the string axiverse picture [40, 268, 303]. However, in that picture the moduli fields are endowed with a mass due to shift-symmetry breaking by non-perturbative effects; such masses are not expected to be comparable to the inverse length scale of binary systems. Second, no mass term may appear in a theory with a shift symmetry, i.e., invariance under the transformation 𝜗 → 𝜗 + const. Such symmetries are common in four-dimensional, low-energy, effective string theories [79, 204*, 203, 92, 89], such as dynamical Chern–Simons and Einstein-Dilaton-Gauss–Bonnet theory. Similar considerations apply to other more complicated potentials, such as a cosine term.

Given these field equations, one can linearize them about Minkowski space to find evolution equations for the perturbation in the small-coupling approximation. Doing so, one finds [447*]


where we have order-reduced the theory where possible and used the harmonic gauge condition (which is preserved in this class of theories [390*, 400*]). The corresponding equation for the metric perturbation is rather lengthy and can be found in Eqs. (17) – (24) in [447*]. Since these theories are to be considered effective, working always to leading order in αi, one can show that they are perturbatively of type N2 in the E (2) classification [161*], i.e., in the far zone, the only propagating modes that survive are the two transverse-traceless (spin-2) metric perturbations [390*]. However, in the strong-field region it is possible that additional modes are excited, although they decay rapidly as they propagate to future null infinity.

Lastly, let us discuss what is known about whether modified quadratic gravities satisfy the requirements discussed in Section 2.1. As it should be clear from the action itself, this modified gravity theory satisfies the fundamental requirement, i.e., passing all precision tests, provided the couplings αi are sufficiently small. This is because such theories have a continuous limit to GR as α → 0 i.6 Dynamical Chern–Simons gravity is constrained only weakly at the moment, 1∕4 8 ξ4 < 10 km, where ξ4 ≡ α24∕(βκ), only through observations of Lense–Thirring precession in the solar system [19*]. The Einstein-Dilaton-Gauss–Bonnet gravity coupling constant ξ3 ≡ α23∕(βκ), on the other hand, has been constrained by several experiments: solar system observations of the Shapiro time delay with the Cassini spacecraft placed the bound 1∕4 7 ξ3 < 1.3 × 10 km [73*, 29]; the requirement that neutron stars still exist in this theory placed the constraint 1∕4 ξ3 ≲ 26 km [342*], with the details depending somewhat on the central density of the neutron star; observations of the rate of change of the orbital period in the low-mass X-ray binary A0620–00 [358, 255*] has led to the constraint 1∕4 ξ3 < 1.9 km [445].

However, not all sub-properties of the fundamental requirement are known to be satisfied. One can show that certain members of modified quadratic gravity possess known solutions and these are stable, at least in the small-coupling approximation. For example, in dynamical Chern–Simons gravity, spherically-symmetric vacuum solutions are given by the Schwarzschild metric with constant 𝜗 to all orders in αi [245*, 470*]. Such a solution is stable to small perturbations [319, 190*], as also are non-spinning black holes and branes in anti de Sitter space [144]. On the other hand, spinning solutions continue to be elusive, with approximate solutions in the slow-rotation/small-coupling limit known both for black holes [466*, 272*, 345*, 455*] and stars [19*, 342*]; nothing is currently known about the stability of these spinning solutions. In Einstein-Dilaton-Gauss–Bonnet theory even spherically-symmetric solutions are modified [473*, 345*] and these are stable to axial perturbations [343*].

The study of modified quadratic gravity theories as effective theories is valid provided one is sufficiently far from its cut-off scale, i.e., the scale beyond which higher-order curvature terms cannot be neglected anymore. One can estimate the magnitude of this scale by studying the size of loop corrections to the quadratic curvature terms in the action due to n-point interactions [455*]. Simple counting requires that the number of scalar and graviton propagators, Ps and Pg, satisfy the following relation in terms of the number of vertices V:

V V Ps = --, Pg = (n − 1)--. (32 ) 2 2
Thus, loop corrections are suppressed by factors of αVi M p(2l−n)VΛnV, with Mpl the Planck mass and Λ the energy scale introduced by dimensional arguments. The cut-off scale above which the theory cannot be treated as an effective one can be approximated as the value of Λ at which the suppression factor becomes equal to unity:
Λ ≡ M 1−2∕nα1∕n, (33 ) c pl i
This cut-off scale automatically places a constraint on the magnitude of αi above which higher-curvature corrections must be included. Setting the largest value of Λc to be equal to 𝒪 (10μm), thus saturating bounds from table-top experiments [259*], and solving for αi, we find
1∕2 α i < 𝒪 (108 km ). (34 )
Current solar system bounds on αi already require the coupling constant to be smaller than 8 10 km, thus justifying the treatment of these theories as effective models.

As for the other requirements discussed in Section 2.1, it is clear that modified quadratic gravity is well-motivated from fundamental theory, but it is not clear at all whether it has a well-posed initial-value formulation. From an effective point of view, a perturbative treatment in α i naturally leads to stable solutions and a well-posed initial-value problem, but this is probably not the case when it is treated as an exact theory. In fact, if one were to treat such a theory as exact (to all orders in αi), then the evolution system would likely not be hyperbolic, as higher-than-second time derivatives now drive the evolution. Although no proof exists, it is likely that such an exact theory is not well-posed as an initial-value problem. Notice, however, that this says nothing about the fundamental theories that modified quadratic gravity derives from. This is because even if the truncated theory were ill posed, higher-order corrections that are neglected in the truncated version could restore well-posedness.

As for the last requirement (that the theory modifies the strong field), modified quadratic theories are ideal in this respect. This is because they introduce corrections to the action that depend on higher powers of the curvature. In the strong-field, such higher powers could potentially become non-negligible relative to the Einstein–Hilbert action. Moreover, since the curvature scales inversely with the mass of the objects under consideration, one expects the largest deviations in systems with small total mass, such as stellar-mass black-hole mergers. On the other hand, deviations from GR should be small for small compact objects spiraling into a supermassive black hole, since here the spacetime curvature is dominated by the large object, and thus it is small, as discussed in [390*].

2.3.4 Variable G theories and large extra dimensions

Variable G theories are defined as those where Newton’s gravitational constant is promoted to a spacetime function. Such a modification breaks the principle of equivalence (see [438*]) because the laws of physics now become local position dependent. In turn, this implies that experimental results now depend on the spacetime position of the laboratory frame at the time of the experiment.

Many known alternative theories that violate the principle of equivalence, and in particular, the strong equivalence principle, predict a varying gravitational constant. A classic example is scalar-tensor theory [435], which, as explained in Section 2.3.1, modifies the gravitational sector of the action by multiplying the Ricci scalar by a scalar field (in the Jordan frame). In such theories, one can effectively think of the scalar as promoting the coupling between gravity and matter to a field-dependent quantity G → G (ϕ), thus violating local position invariance when ϕ varies. Another example are bimetric theories, such as that of Lightman–Lee [293], where the gravitational constant becomes time-dependent even in the absence of matter, due to possibly time-dependent cosmological evolution of the prior geometry. A final example are higher-dimensional, brane-world scenarios, where enhanced Hawking radiation inexorably leads to a time-varying effective 4D gravitational constant [141], whose rate of change depends on the curvature radius of extra dimensions [255*].

One can also construct f(R )-type actions that introduce variability to Newton’s constant. For example, consider the f (R ) model [180*]

∫ [ ( ) ] 4 √ --- R S = d x − gκR 1 + α0ln --- + Smat, (35 ) R0
where κ = (16πG )−1, α0 is a coupling constant and R0 is a curvature scale. This action is motivated by certain renormalization group flow arguments [180*]. The field equations are

where we have defined the new constant

[ α0 ( R ) ] ¯κ := κ 1 + ---ln --- . (37 ) κ R0
Clearly, the new coupling constant ¯κ depends on the curvature scale involved in the problem, and thus, on the geometry, forcing G to run to zero in the ultraviolet limit.

An important point to address is whether variable G theories can lead to modifications to a vacuum spacetime, such as a black-hole–binary inspiral. In Einstein’s theory, G appears as the coupling constant between geometry, encoded by the Einstein tensor G μν, and matter, encoded by the stress energy tensor mat T μν. When considering vacuum spacetimes, mat Tμν = 0 and one might naively conclude that a variable G would not introduce any modification to such spacetimes. In fact, this is the case in scalar-tensor theories (without homogeneous, cosmological solutions to the scalar field equation), where the no-hair theorem establishes that black-hole solutions are not modified. On the other hand, scalar-tensor theories with a cosmological, homogeneous scalar field solution can violate the no-hair theorem, endowing black holes with time-dependent hair, which in turn would introduce variability into G even in vacuum spacetimes [246*, 236, 67*].

In general, Newton’s constant plays a much more fundamental role than merely a coupling constant: it defines the relationship between energy and length. For example, for the vacuum Schwarzschild solution, G establishes the relationship between the radius R of the black hole and the rest-mass energy E of the spacetime via R = 2GE ∕c4. Similarly, in a black-hole–binary spacetime, each black hole introduces an energy scale into the problem that is quantified by a specification of Newton’s constant. Therefore, one can treat variable G modifications as induced by some effective theory that modifies the mapping between the curvature scale and the energy scale of the problem, as is done for example in theories with extra dimensions.

An explicit example of this idea is realized in braneworld models. Superstring theory suggests that physics should be described by 4 large dimensions, plus another 6 that are compactified and very small [354, 355*]. The size of these extra dimensions is greatly constrained by particle theory experiments. However, braneworld models, where a certain higher-dimensional membrane is embedded in a higher-dimensional bulk spacetime, can evade this constraint as only gravitons can interact with the bulk. The ADD model [32, 33] is a particular example of such a braneworld, where the bulk is flat and compact and the brane is tensionless with ordinary fields localized on it. Since gravitational-wave experiments have not yet constrained deviations from Einstein’s theory in the strong field, the size of these extra dimensions is constrained to micrometer scales only by table-top experiments [259, 7*].

What is relevant to gravitational-wave experiments is that in many of these braneworld models black holes may not remain static [163, 405]. The argument goes roughly as follows: a five-dimensional black hole is dual to a four-dimensional one with conformal fields on it by the ADS/CFT conjecture [301, 9], but since the latter must evolve via Hawking radiation, the black hole must be losing mass. The Hawking mass loss rate is here enhanced by the large number of degrees of freedom in the conformal field theory, leading to an effective modification to Newton’s laws and to the emission of gravitational radiation. Effectively, one can think of the black-hole mass loss as due to the black hole being stretched away from the brane into the bulk due to a universal acceleration, that essentially reduces the size of the brane-localized black hole. For black-hole binaries, one can then draw an analogy between this induced time dependence in the black-hole mass and a variable G theory, where Newton’s constant decays due to the presence of black holes. Of course, this is only analogy, since large extra dimensions would not predict a time-evolving mass in neutron-star binaries.

Recently, however, Figueras et al. [170, 172, 171] numerically found stable solutions that do not require a radiation component. If such solutions were the ones realized in nature as a result of gravitational collapse on the brane, then the black hole mass would be time independent, up to quantum correction due to Hawking evaporation, a negligible effect for realistic astrophysical systems. Unfortunately, we currently lack numerical simulations of the dynamics of gravitational collapse in such scenarios.

Many experiments have been carried out to measure possible deviations from a constant G value, and they can broadly be classified into two groups: (a) those that search for the present or nearly present rate of variation (at redshifts close to zero); (b) those that search for secular variations over long time periods (at very large redshifts). Examples of experiments or observations of the first class include planetary radar ranging [350], surface temperature observations of low-redshift millisecond pulsars [249, 362], lunar ranging observations [442] and pulsar timing observations [260, 143], the latter two being the most stringent. Examples of experiments of the second class include the evolution of the sun [208] and Big-Bang Nucleosynthesis (BBN) calculations [119, 47], again with the latter being more stringent. For either class, the strongest constraints are about ˙ −13 − 1 G∕G ≲ 10 yr, varying somewhat from experiment to experiment.

Lacking a particularly compelling action to describe variable G theories, one is usually left with a phenomenological model of how such a modification to Einstein’s theory would impact gravitational waves. Given that the part of the waveform that detectors are most sensitive to is the gravitational wave phase, one can model the effect of variable G theories by studying how the rate of change of its frequency would be modified. Assuming a Taylor expansion for Newton’s constant one can derive the modification to the evolution equation for the gravitational wave frequency, given whichever physical scenario one is considering. Solving such an evolution equation then leads to a modification in the accumulated gravitational-wave phase observed at detectors on Earth. In Section 5 we will provide an explicit example of this for a compact binary system.

Let us discuss whether such theories satisfy the criteria defined in Section 2.1. The fundamental property can be satisfied if the rate of change of Newton’s constant is small enough, as variable G theories usually have a continuous limit to GR (as all derivatives of G go to zero). Whether variable G theories are well-motivated from fundamental physics (Property 2) depends somewhat on the particular effective model or action that one considers. But in general, Property 2 is usually satisfied, considering that such variability naturally arises in theories with extra dimensions, and the latter are also natural in all string theories. However, variable G theories usually fail at introducing modifications in the strong-field region. Usually, such variability is parameterized as a Taylor expansion about some initial point with constant coefficients. That is, the variability of G is not usually constructed so as to become stronger closer to merger.

The well-posed property and the sub-properties of the fundamental property depend somewhat on the particular effective theory used to describe varying G modifications. In the f(R ) case, one can impose restrictions on the functional form f(⋅) such that no ghosts (′ f > 0) or instabilities (f′′ > 0) arise [180]. This, of course, does not guarantee that this (or any other such) theory is well posed. A much more detailed analysis would be required to prove well-posedness of the class of theories that lead to a variable Newton’s constant, but such is currently lacking.

2.3.5 Non-commutative geometry

Non-commutative geometry is a gravitational theory that generalizes the continuum Riemannian manifold of Einstein’s theory with the product of it with a tiny, discrete, finite non-commutative space, composed of only two points. Although the non-commutative space has zero spacetime dimension, as the product manifold remains four dimensional, its internal dimensions are 6 to account for Weyl and chiral fermions. This space is discrete to avoid the infinite tower of massive particles that would otherwise be generated, as in string theory. Through this construction, one can recover the standard model of elementary particles, while accounting for all (elementary particle) experimental data to date. Of course, the simple non-commutative space described above is expected to be replaced by a more complex model at Planckian energies. Thus, one is expected to treat such non-commutative geometry models as effective theories. Essentially nothing is currently known about the full non-commutative theory of which the theories described in this section are an effective low-energy limit.

Before proceeding with an action-principle description of non-commutative geometry theories, we must distinguish between the spectral geometry approach championed by Connes [114], and Moyal-type non-commutative geometries [389, 206, 322]. In the former, the manifold is promoted to a non-commutative object through the product of a Riemann manifold with a non-commutative space. In the latter, instead, a non-trivial set of commutation relations is imposed between operators corresponding to position. These two theories are in principle unrelated. In this review, we will concentrate only on the former, as it is the only type of non-commutative GR extension that has been studied in the context of gravitational-wave theory.

The effective action for spectral non-commutative geometry theories (henceforth, non-commutative geometries for short) is


where H is related to the Higgs field, C μνδσ is the Weyl tensor, (α ,τ ,ξ ) 0 0 0 are couplings constants and we have defined the quantity

R ∗R ∗ := 1-𝜖μνρσ𝜖αβγδR μναβR ρσγδ. (39 ) 4
Notice that this term integrates to the Euler characteristic, and since τ0 is a constant, it is topological and does not affect the classical field equations. The last term of Eq. (38) is usually ignored as H is assumed to be relevant only in the early universe. Finally, the second term can be rewritten in terms of the Riemann and Ricci tensors as
C Cμνδσ = 1-R2 − 2R R μν + R Rμνδσ. (40 ) μνδσ 3 μν μνδσ
Notice that this corresponds to the modified quadratic gravity action of Eq. (26) with all α (1) = 0 i and (0) (0) (0) (α 1 ,α2 ,α 3 ) = (1∕3,− 2,1), which is not the Gauss–Bonnet invariant. Notice also that this model is not usually studied in modified quadratic gravity theory, as one usually concentrates on the terms that have an explicit scalar field coupling.

The field equations of this theory can be read directly from Eq. (27), but we repeat them here for completeness:


One could in principle rewrite this in terms of the Riemann and Ricci tensors, but the expressions become quite complicated, as calculated explicitly in Eqs. (2) and (3) of [473*]. Due to the absence of a dynamical degree of freedom coupling to the modifications to the Einstein–Hilbert action, this theory is not covariantly conserved in vacuum. By this we mean that the covariant divergence of Eq. (41) does not vanish in vacuum, thus violating the weak-equivalence principle and leading to additional equations that might over-constrain the system. In the presence of matter, the equations of motion will not be given by the vanishing of the covariant divergence of the matter stress-energy alone, but now there will be additional geometric terms.

Given these field equations, one can linearize them about a flat background to find the evolution equations for the metric perturbation [326*, 325*]

(1 − β−2□ η)□ ηhμν = − 16πT mat, (42 ) μν
where the term proportional to β2 = (− 32 πα0)−1 acts like a mass term. Here, one has imposed the transverse-traceless gauge (a refinement of Lorenz gauge), which can be shown to exist [326*, 325*]. Clearly, even though the full non-linear equations are not covariantly conserved, its linearized version is, as one can easily show that the divergence of the left-hand side of Eq. (42*) vanishes. Because of these features, if one works perturbatively in −1 β, then such a theory will only possess the two usual transverse-traceless (spin-2) polarization modes, i.e., it is perturbatively of type N2 in the E (2) classification [161].

Let us now discuss whether such a theory satisfies the properties discussed in Section 2.1. Non-commutative geometry theories clearly possess the fundamental property, as one can always take α0 → 0 (or equivalently β −2 → 0) to recover GR. Therefore, there must exist a sufficiently small α0 such that all precision tests carried out to date are satisfied. As for the existence and stability of known solutions, [326*, 325*] have shown that Minkowski spacetime is stable only for α0 < 0, as otherwise a tachyonic term appears in the evolution of the metric perturbation, as can be seen from Eq. (42*). This then automatically implies that β must be real.

Current constraints on Weyl terms of this form come mostly from solar system experiments. Ni [328*] recently studied an action of the form of Eq. (38) minimally coupled to matter in light of solar system experiments. He calculated the relativistic Shapiro time-delay and light deflection about a massive body in this theory and found that observations of the Cassini satellite place constraints on |α0 |1∕2 < 5.7 km [328]. This is currently the strongest bound we are aware of on α0.

Many solutions of GR are preserved in non-commutative geometries. Regarding black holes, all solutions that are Ricci flat (vacuum solutions of the Einstein equations) are also solutions to Eq. (41). This is because by the second Bianchi identity, one can show that

κλ κ ∇ R μλνκ = ∇ νRμκ − □R μν, (43 )
and the right-hand side vanishes in vacuum, forcing the entire left-hand side of Eq. (41) to vanish. However, this is not so for neutron stars where the equations of motion are likely to be modified, unless they are static [324]. Moreover, as of now there has been no stability analysis of black-hole or stellar solutions and no study of whether the theory is well posed as an initial-value problem, even as an effective theory. Thus, except for the fundamental property, it is not clear that non-commutative geometries satisfy any of the other criteria listed in Section 2.1.

2.3.6 Gravitational parity violation

Parity, the symmetry transformation that flips the sign of the spatial triad, has been found to be broken in the standard model of elementary interactions. Only the combination of charge conjugation, parity transformation and time inversion (CPT) still remains a true symmetry of the standard model. Experimentally, it is curious that the weak interaction exhibits maximal parity violation, while other fundamental forces seem to not exhibit any. Theoretically, parity violation unavoidably arises in the standard model [55, 8, 21], as there exist one-loop chiral anomalies that give rise to parity-violating terms coupled to lepton number [428]. In certain sectors of string theory, such as in heterotic and Type I superstring theories, parity violation terms are also generated through the Green–Schwarz gauge anomaly-canceling mechanism [204, 355, 12]. Finally, in loop quantum gravity [41], the scalarization of the Barbero–Immirzi parameter coupled to fermions leads to an effective action that contains parity-violating terms [406, 90, 311, 192]. Even without a particular theoretical model, one can show that effective field theories of inflation generically contain non-vanishing, second-order, parity-violating curvature corrections to the Einstein–Hilbert action [429]. Alternatively, phenomenological parity-violating extensions of GR have been proposed through a scalarization of the fundamental constants of nature [115].

One is then naturally led to ask whether the gravitational interaction is parity invariant in the strong field. A violation of parity invariance would occur if the Einstein–Hilbert action were modified through a term that involved a Levi-Civita tensor and parity invariant tensors or scalars. Let us try to construct such terms with only single powers of the Riemann tensor and a single scalar field 𝜗:


Option (ia) and (ib) vanish by the Bianchi identities. Options (ic) and (id) include the commutator of covariant derivatives, which can be rewritten in terms of a Riemann tensor, and thus it leads to terms that are at least quadratic in the Riemann tensor. Therefore, no scalar can be constructed that includes contractions with the Levi-Civita tensor from a single Riemann curvature tensor and a single field. One can try to construct a scalar from the Ricci tensor


but again (iia) vanishes by the symmetries of the Ricci tensor, while (iib) involves the commutator of covariant derivatives, which introduces another power of the curvature tensor. Obviously, the only term one can write with the Ricci scalar would lead to a double commutator of covariant derivatives, leading to extra factors of the curvature tensor.

One is then forced to consider either theories with two mutually-independent fields or theories with quadratic curvature tensors. Of the latter, the only combination that can be constructed and that does not vanish by the Bianchi identities is the Pontryagin density, i.e., R∗R, and therefore, the action [245*, 17*]

∫ √ ---( α ) S = d4x − g κR + -𝜗R ∗R , (45 ) 4
is the most general, quadratic action with a single scalar field that violates parity invariance, where we have rescaled the α prefactor to follow historical conventions. This action defines non-dynamical Chern–Simons modified gravity, initially proposed by Jackiw and Pi [245*, 17*]. Notice that this is the same as the term proportional to α4 in the quadratic gravity action of Eq. (26), except that here 𝜗 is prior geometry, i.e., it does not possess self-consistent dynamics or an evolution equation. Such a term violates parity invariance because the Pontryagin density is a pseudo-scalar, while 𝜗 is assumed to be a scalar.

The field equations for this theory are7

α (1) 1 mat Gμν + ---𝒦 μν = ---Tμν , (46 ) 4κ 2κ
which is simply Eq. (27) with (α1,α2, α3) set to zero and no stress-energy for 𝜗. Clearly, these field equations are not covariantly conserved in vacuum, i.e., taking the covariant divergence one finds the constraint
αR ∗R = 0. (47 )
This constraint restricts the space of allowed solutions, for example disallowing the Kerr metric [207*]. Therefore, it might seem that the evolution equations for the metric are now overconstrained, given that the field equations provide 10 differential conditions for the 10 independent components of the metric tensor, while the constraint adds one additional, independent differential condition. Moreover, unless the Pontryagin constraint, Eq. (47*), is satisfied, matter fields will not evolve according to μ mat ∇ Tμν = 0, thus violating the equivalence principle.

From the field equations, we can derive an evolution equation for the metric perturbation when linearizing about a flat background, namely

α ( ) 2 □ηh μν + -- 𝜗,γ𝜖(μγδχ□ ηhν)δ,χ − 𝜗,γζ𝜖(μγδχh|δζ|,ν)χ + 𝜗,γζ𝜖(μγδχh ν)δ,χζ = − κ κ
in a transverse-traceless gauge, which can be shown to exist in this theory [11*, 460*]. The constraint of Eq. (47*) is identically satisfied to second order in the metric perturbation. However, without further information about 𝜗 one cannot proceed any further, except for a few general observations. As is clear from Eq. (48*), the evolution equation for the metric perturbation can contain third time derivatives, which generically will lead to instabilities. In fact, as shown in [13] the general solution to these equations will contain exponentially growing and decaying modes. However, the theory defined by Eq. (45*) is an effective theory, and thus, there can be higher-order operators not included in this action that may stabilize the solution. Regardless, when studying this theory order-reduction is necessary if one is to consider it an effective model.

Let us now discuss the properties of such an effective theory. Because of the structure of the modification to the field equations, one can always choose a sufficiently small value for α such that all solar system tests are satisfied. In fact, one can see from the equations in this section that in the limit α → 0, one recovers GR. Non-dynamical Chern–Simons gravity leads to modifications to the non-radiative (near-zone) metric in the gravitomagnetic sector, leading to corrections to Lense–Thirring precession [14, 15]. This fact has been used to constrain the theory through observations of the orbital motion of the LAGEOS satellites [388*] to (α∕κ )𝜗 ˙< 2 × 104 km, or equivalently (κ∕α ) ˙𝜗−1 ≳ 10 −14 eV. However, much better constraints can be placed through observations of the binary pulsar [472, 18]: ˙ (α4∕ κ)𝜗 < 0.8 km.

Some of the sub-properties of the fundamental requirement are satisfied in non-dynamical Chern–Simons gravity. On the one hand, all spherically-symmetric metrics that are solutions to the Einstein equations are also solutions in this theory for a “canonical” scalar field (𝜃 ∝ t[207]. On the other hand, axisymmetric solutions to the Einstein equations are generically not solutions in this theory. Moreover, although spherically-symmetric solutions are preserved, perturbations of such spacetimes that are solutions to the Einstein equations are not generically solutions to the modified theory [470*]. What is perhaps worse, the evolution of perturbations to non-spinning black holes have been found to be generically overconstrained [470]. This is a consequence of the lack of scalar field dynamics in the modified theory, which, via Eq. (47*), tends to overconstrain it. Such a conclusion also suggests that this theory does not posses a well-posed initial-value problem.

One can argue that non-dynamical Chern–Simons gravity is well-motivated from fundamental theories [17*], except that in the latter, the scalar field is always dynamical, instead of having to be prescribed a priori. Thus, perhaps the strongest motivation for such a model is as a phenomenological proxy to test whether the gravitational interaction remains parity invariant in the strong field, a test that is uniquely suited to this modified model.

2.4 Currently unexplored theories in the gravitational-wave sector

The list of theories we have described here is by no means exhaustive. In fact, there are many fascinating theories that we have chosen to leave out because they have not yet been analyzed in the gravitational wave context in detail. Examples of these include the following:

  • Einstein-Aether Theory [247] and Hořava–Lifshitz Theory [234];
  • Standard Model Extension [109];
  • Eddington-inspired Born–Infeld gravity [48];
  • New Massive Gravity [60, 136] and Bi-Gravity Theories [349, 346, 219, 220].

We will update this review with a description of these theories, once a detailed gravitational-wave study for compact binaries or supernovae sources is carried out and the predictions for the gravitational waveform observables are made for any physical system plausibly detectable by current or near future gravitational-wave experiments.

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