Possible existence of viable models of bi-gravity with detectable graviton oscillations by gravitational wave detectors

We discuss graviton oscillations based on the ghost free bi-gravity theory. We point out that this theory possesses a natural cosmological background solution which is very close to the case of general relativity. Furthermore, interesting parameter range of the graviton mass, which can be explored by the observations of gravitational waves, is not at all excluded by the constraint from the solar system tests. Therefore the graviton oscillation with possible inverse chirp signal would be an interesting scientific target of KAGRA, adv LIGO, adv Virgo and GEO.

Introduction: Many works have been done for the detection possibility of modified propagation of gravitational waves due to finite graviton mass [1,2]. However, adding mass to graviton was thought to be theoretically problematic due to the so-called Boulware-Deser (BD) ghost [3].
Recently, Hassan and Rosen proposed the first example of ghost-free bi-gravity models [4], based on the fully nonlinear massive gravity theory in which the Boulware-Deser ghost is removed by construction [5][6][7]. We consider two metrics expressed by ds 2 = g µν dx µ dx ν , ds 2 =g µν dx µ dx ν .
We introduce a ghost free action S = d 4 xL with where M 2 G = 1/(8πG N ); G N is the gravitational constant; Y µ ν = √ g µαg αν ; R andR are the Ricci scalars with respect to g µν andg µν , respectively; g andg are the determinants of g µν andg µν , respectively; κ is a constant which expresses the ratio between the two gravitational constants forg µν and g µν ; c n (n = 0, . . . , 4) are dimensionless constants, and L m is the Lagrangian of the matter which interacts only with g µν . By expressing the trace of Y n as [Y n ] = tr(Y n ) = Y α0 α1 Y α1 α2 · · · Y αn−1 α0 , we can write V n 's as The variation of the action with respect to g µν andg µν yields the field equations as where T µν is the energy momentum tensor of the ordinary matter, whereas B µν andB µν come from the variations of the mass term. B µν andB µν , as well as T µν , satisfy conservation laws, which are explicitly given by where ∇ and∇ are the covariant derivative operators with respect to g µν andg µν , respectively. The cosmological background: The background cosmology of this theory has been widely studied in Refs. [8][9][10], but here our focus is on a particularly healthy branch. We assume that the two metrics can be written as where a,ã, andc are functions of the time coordinate t. The Friedmann equation for the physical metric reads where we have introduced the Hubble parameter H ≡ a/a 2 , the matter energy density ρ m including the dark energy, and the energy density due to the mass term where Γ(ξ) ≡ c 1 ξ + 4c 2 ξ 2 + 6c 3 ξ 3 . This equation can be solved by imposing Γ(ξ) = 0 orcaH − (ȧ/ã) = 0, which implies the existence of two branches. In the following we will discuss the physical branch, defined by the latter condition, since the other branch is pathological 1 . Combining this condition with Eqs. (2) and (3), we obtain an algebraic equation for ξ If m 2 ρ m /M 2 G , the r.h.s of Eq. (4) should be very small. Denoting a value of ξ at which the right hand side vanishes by ξ c , we focus on a cosmological background solution for which ξ asymptotes to ξ c for ρ m → 0. As we can absorb the constant part of ρ V (ξ) into the cosmological constant in ρ m , we also assume that ρ V (ξ c ) = 0.
For this type of solution, we can expand ξ around ξ c at low energies. Keeping only the linear order in ξ − ξ c , Eq. (4) becomes where Γ c ≡ Γ(ξ c ). Substituting this relation into Eq. (2), we recover the usual Friedmann equation as with the effective gravitational constant given bỹ On using the definition of ξ, the relationcaH =ȧ/ã impliesξ = (c − 1)aHξ. Substituting the differentiation of Eq. (4) into this relation, we obtaiñ at low energies, where P m is the matter pressure density. The above relation implies that the light cone of the hidden metric automatically gets closer to the physical one as the matter energy density is diluted. Propagation of the gravitational waves: We now discuss the propagation of gravitational waves. We introduce tensor-type perturbations as , with tr(ε + ε + ) = 1 = tr(ε × ε × ), and tr(ε + ε × ) = 0. The gravitational waves propagate at the speed of light for the physical sector, whereas at the speedc ≈ 1 + O(H 2 /m 2 ) for the hidden sector. However, the physical and hidden gravitons, because of the coupling through the mass term, will oscillate from one to the other. Keeping only the leading effect of the deviation ofc from unity, and neglecting the cosmic expansion effects, we write the propagation equations as [10] where we have omitted the +/× index. For this set of equations, we write down the dispersion relation, assumingc − 1 1 but the magnitude of is moderate, where we have defined Then, for a given gravitational wave frequency f , two eigen wave numbers are given by and the corresponding eigen functions h 1 and h 2 are related to h andh as with the mixing angle We find that µ is the graviton mass of the second mode in the Minkowski limit (x → 0). When we consider the propagation over a distance D, the phase shifts, due to the modified dispersion relation for their respective modes, are given by Notice that this factor is symmetric under the replacement x → 1/x. In the limit x → 0, the first mode becomes massless. Although this mode also has nontrivial dispersion relation, its magnitude of modification tends to be suppressed. The factor µD √c − 1 = 3(1 + κξ 2 c )Ω 0 HD becomes O(1) only after propagating over a cosmological distance unless κξ 2 c is extremely large, where Ω 0 is the energy fraction of the dust matter at the present epoch. On the other hand, the remaining factor takes the maximum value 2 −1/2 − (2 + 2κξ 2 ) −1/2 at x = 1, which is also at most O(1). In contrast to the first mode, the phase shift of the second mode can be significantly large when x is small or large. Here we plot δΦ 1,2 in Fig. 1 for κξ 2 c = 0.2, 1 and 100. Gravitational potential around a star in the Minkowski limit: In the above, we find that, unless κξ 2 c is extremely large, a relatively small value of λ µ together with the excitation of the second mode is required for an observable magnitude of the phase shifts due to the non-trivial dispersion relation. Here we show that in the present bi-gravity models even with such a small value of λ µ we can easily evade the solar system constraint from the precision measurement of gravity.
In the low energy limit, it would be natural to assume the hierarchy, k 2 µ 2 H 2 . Since the limit H → 0 is smooth, the H-dependent terms in the action appear as a positive power in H. Since such terms will not give any dominant contribution under the assumption of the above hierarchy, we set H = 0 from the beginning here.
Let us now consider static spherical symmetric perturbations for both metrics induced by non-relativistic matter energy density ρ m , which is coupled only to the physical metric. We can write the respective perturbed metrics as ds 2 = −e u−v dt 2 + e u+v (dr 2 + r 2 dΩ 2 ), c eũ −ṽ dt 2 + ξ 2 c eũ +ṽ (dr 2 +r 2 dΩ 2 ), without loss of generality. Herer is related to r bỹ r = e R (r)r, and R(r) is another perturbation variable. We adopted the parametrization such that u vanishes in the case of general relativity. Now we write down the equations of motion and eliminate the variables on the hidden metric side,ũ,ṽ and R. However, doing this is not so straightforward. In order to simplify the manipulation, we truncate the perturbation equations at second order and also neglect higher order terms in µ appropriately.
When we compute the terms second order in perturbation, we notice that there are terms enhanced by the factor 1/µ 2 . If we scrutinize these terms, some of them contain the factor Our assumption here is that the energy scale of the bigravity theory itself is relatively high but the graviton mass µ is suppressed by a certain mechanism. Under this assumption, we pick up only the terms enhanced by the factor C/µ 2 from the second order terms in the equations of motion. Then, after a little calculation, we obtain where ∂ i is the differentiation with respect to the coordinates r(sin θ cos φ, sin θ sin φ, cos θ) and ≡ ∂ i ∂ i , the standard three dimensional Laplacian operator, and we have definedC ≡ C(1 + κξ 2 c )/(κξ 2 c ). At this level, the expressions were recast into the form that does not assume spherical symmetry, where there is no ambiguity.
Although we have truncated the equations at second order for simplicity, the higher order terms will not be suppressed once the second order terms become important. Nevertheless, such higher order terms will not change the following discussion as to the order of magnitude estimate on the correction to the Newton's law.
First we focus on Eq. (7). Notice that non-vanishing u is the origin of the vDVZ discontinuity [12]. This equation tells us that the Vainshtein radius [13], within which the second term dominates the first term on the left hand side in Eq. (7), is given by where r g is the gravitational radius of the star. From the above estimate, we find that the Vainshtein radius can be made arbitrarily large even with a large graviton mass, if C is sufficiently large. Thus, the solar system can be easily contained within the Vainshtein radius, where the second or even higher order terms on the left hand side of Eq. (7) dominate. Then, we have u ≤ O [κξ 2 c /(1 + κξ 2 c )] r g r/Cλ 2 µ . Even if we require u to be smaller than 10 −9 in the solar system r ≈ 10 13 cm, λ 2 µ can be left arbitrarily small depending on the value of C [14].
Once u is suppressed on the scale of the solar system, Eq. (8) tells us that the equation for v does not largely deviate from the one in the Newtonian case: v = −M −2 G ρ m , and the gravitational constant is not different from the cosmological one. In Eq. (8) the terms second order in u are eliminated with the aid of Eq. (7) to make the effective gravitational source for v to be manifest. Solving the equation, we find that the correction to v is at most of O(u). Notice that the mass term for v in Eq. (8) is absent. Therefore, v does not suffer from the Yukawa-type correction.
Hence, one can conclude that the correction to the Newtonian potential is at most O κξ 2 c r g r/Cλ 2 µ . Namely, we can avoid the constraint from the test in the solar system, keeping the graviton mass sufficiently large. In the above we assumed that C is large. On the other hand, in constructing the cosmological background we have used the linear approximation for the deviation from the conformal equivalence between the two metrics, i.e.c−1 1. If we further expand the background equations in terms ofc−1, we find terms enhanced by the factor C at second order. However, as long as Cλ 2 µ < H −2 , is satisfied, we can verify that the formulae for the background metric remain approximately valid. In the early universe, where H is larger, the non-linear terms become necessarily important. However, the terms second order in ξ − ξ c do not alter the effective Newton constant for the homogeneous background cosmology.
The equations forũ andṽ can be obtained similarly as Once u is suppressed, i.e. if the Vainshtein mechanism is at work, we findṽ ≈ v, which implies that metric perturbations on both sides are equally excited by the matter fields.
Graviton oscillations and inverse chirp signal: Here we begin with discussing the generation of gravitational waves. We found that the metric excitations are almost conformal within the Vainshtein radius of a star. If we consider the junction between the nearzone metric perturbation with the far-zone metric described as gravitational waves, both h andh are excited exactly as in the case of general relativity. This implies that both eigen modes h 1 and h 2 are excited unless One may suspect that the linear approximation to the gravitational wave perturbation equations (6) is not valid within the Vainshtein radius. However, the effective energy momentum tensor coming from the variation of the mass term, which gives corrections to the case of general relativity, is largely enhanced only for the terms purely composed of u (or equivalentlyũ), which behave as clouds around localized matter sources. Namely, it just contributes as the source of gravitational waves but does not change the wave propagation. The other corrections are suppressed as long as the amplitude of the deviation of the metric from the case of general relativity remains small.
Next, we analyze the gravitational waveform from inspirals of NS-NS binaries at a distance. For the current bi-gravity model, our detector signal becomes a linear combination of two components, whose relative amplitudes are determined by the mixing angle θ g . For simplicity, we here neglect the time dependence of θ g as well as all the cosmological effects. Using the stationary phase approximation and flux conservation, the observed signal is given in Fourier space as where the amplitude A(f ) (after angular average), B 1,2 and the phase function Φ(f, g) (truncated at 1.5PN order) are given by respectively. Here we plot B 1,2 in Fig. 2 for κξ 2 c = 0.2, 1 and 100.
For x 1, the excitation of the second mode h 2 is suppressed. Furthermore, δΦ 1 is suppressed in this regime. Therefore, the propagation of gravitational waves is similar to the case of general relativity. For x 1, both h 1 and h 2 are equally excited. However, since the gravitational wave detector can detect the perturbation of the physical metric only, we can observe only h 1 . Therefore, the frequencies at which both modes can be observed are limited to x ≈ 1. This is the meaning of Fig. 2.
When both modes are observable, graviton oscillations due to the interference between two modes can be detected beyond the distance scale where δΦ 2 − δΦ 1 becomes O(1). The difference of the phases δφ 1 − δφ 2 is minimum at x = 1 as shown in Fig. 1, which is evaluated as δφ 1 − δφ 2 | x=1 = √ 6Ω 0 HD. Therefore, one may think that the effect is really small as long as D H −1 . However, the average density of the universe is much lower than the average density in galaxies, where binaries are embedded. Therefore gravitational waves experience much lower value of x, typically x ≈ 10 −8 , during the propagation. Roughly speaking, δΦ 2 − δΦ 1 ≈ 3(1 + κξ 2 c )Ω 0 /2xHD for x 1. Hence, the effect can be largely enhanced.
Once d(δΦ 2 − δΦ 1 )/df becomes sufficiently large, the arrival times of two modes are different. Then, we may observe two chirp signals. Using the stationary phase approximation, the relation between the arrival time of the wave and the frequency is determined by t = d(Φ + δΦ i )/d(2πf ). As an illustrative purpose, in Fig. 3 we show the shifts of the arrival time compared with the case of general relativity for κξ 2 c = 100, D = 300Mpc H = 67.3km s −1 Mpc −1 , Ω 0 = 0.315 and µ = (0.001pc) −1 , for which x ≈ 4 × 10 −8 at 100Hz. One can see that the relation between the arrival time and the frequency is reversed for the second mode, i.e. inverse chirp signal may occur.
Since a large graviton mass can be consistent with the solar system test in our current model 2 , there is a possibility that we may detect the graviton oscillations or even the inverse chirp signal by the next generation gravitational wave detectors. In the present model, measurable effects are expected only when κξ 2 c is large. This requirement may cause some conflict with observations but at first analysis there seems to be no severe constraint. We think this model gives the first existence proof of models in which a measurable deviation from general relativity in the gravitational wave propagation can be expected.