Constraints on Born-Infeld gravity from the speed of gravitational waves after GW170817 and GRB 170817A

# Constraints on Born-Infeld gravity from the speed of gravitational waves after GW170817 and GRB 170817A

Soumya Jana111sjana@prl.res.in    Girish Kumar Chakravarty222girish20@prl.res.in    Subhendra Mohanty333mohanty@prl.res.in Theoretical Physics Division, Physical Research Laboratory, Ahmedabad 380009, India
###### Abstract

The observations of gravitational waves from the binary neutron star merger event GW170817 and the subsequent observation of its electromagnetic counterparts from the gamma-ray burst GRB 170817A provide us a significant opportunity to study theories of gravity beyond general relativity. An important outcome of these observations is that they constrain the difference between the speed of gravity and the speed of light to less than . Also, the time delay between the arrivals of gravitational waves at different detectors constrains the speed of gravity at the Earth to be in the range . We use these results to constrain a widely studied modified theory of gravity: Eddington-inspired Born-Infeld (EiBI) gravity. We show that, in EiBI theory, the speed of gravitational waves in matter deviates from . From the time delay in arrival of gravitational wave signals at Earth-based detectors, we obtain the bound on the theory parameter as . Similarly, from the time delay between the signals of GW170817 and GRB 170817A, in a background Friedmann-Robertson-Walker universe, we obtain . Although the bounds on are weak compared to other earlier bounds from the study of neutron stars, stellar evolution, primordial nucleosynthesis, etc., our bounds are from the direct observations and thus worth noting.

###### pacs:
04.20.-q, 04.20.Jb

## I Introduction

General relativity (GR) is extremely successful as a classical theory of gravity. Over the years, it has been scrutinized in vacuum or in the weak-field regime through several precision tests, and no significant deviation from GR has been found Will (2014). Still, there exist many unsolved puzzles in GR such as the problem of singularities (which is expected to be resolved by quantum gravity), understanding the dark matter and dark energy, etc. In order to address some of these problems, many researchers actively pursue modified gravity theories in the classical domain which deviate from GR inside matter distributions, or in the strong-field regime. One such modification is inspired by the well-known Born-Infeld electrodynamics Born and Infeld (1934) where, even at the classical level, it is possible to avoid the infinity in the electric field at the location of a point charge. Deser and Gibbons Deser and Gibbons (1998) first suggested a gravity theory in the metric formalism consisting of a similar determinantal structure as in the action of Born-Infeld electrodynamics. In fact, the determinantal form of the gravitational action is not a new concept; it existed earlier in Eddington’s reformulation of GR in de Sitter spacetime Eddington (1924). This is essentially an affine formalism where the affine connection is the basic variable instead of the metric; however, the coupling of matter to gravity remained a problem.

Later, the Palatini (metric-affine) formulation in Born-Infeld gravity was introduced by Vollick Vollick (2004). He worked on various related aspects and also introduced a nontrivial and somewhat artificial way of coupling matter in such a theory Vollick (2005, 2006). More recently, Bañados and Ferreira Bañados and Ferreira (2010) have come up with a formulation, popularly known as Eddington-inspired Born-Infeld (EiBI) gravity, where the matter coupling is different and simpler compared to Vollick’s proposal. For a recent review on Born-Infeld gravity, see Ref. Beltrán Jiménez et al. (2018) and for its cosmological, astrophysical, and other applications see Refs. Pani et al. (2011); Delsate and Steinhoff (2012); Pani and Sotiriou (2012); Scargill et al. (2012); Cho et al. (2012, 2013); Escamilla-Rivera et al. (2012); Yang et al. (2013); Wei et al. (2015); Jana and Kar (2013, 2015, 2016, 2017); Shaikh (2015); Sotani and Miyamoto (2014); Olmo et al. (2014); Bazeia et al. (2017); Casanellas et al. (2012); Avelino (2012a); Sham et al. (2012, 2013); Harko et al. (2013); Sotani (2014a, b, 2015); Wei et al. (2015); Sotani and Miyamoto (2014); Odintsov et al. (2014); Fernandes and Lahiri (2015); Delhom-Latorre et al. (2018) and references therein.

The EiBI theory reduces to GR in vacuum but differs in the presence of matter. Therefore most stringent tests of this theory come from neutron stars and stellar evolution Pani et al. (2011); Casanellas et al. (2012). However, the recent direct detection of gravitational wave (GW) signals from the binary black hole and neutron star mergers Abbott et al. (2016, 2017a) not only provide direct confirmation of one of the major predictions of GR, but also give a platform to probe the gravity deeper into the strong-field regime. The time delay between the observations of gravitational wave signals from the binary neutron star merger GW170817 and the observation of its associated electromagnetic counterparts from the gamma-ray burst GRB-170817A constrain the difference between the speed of gravity and the speed of light to less than ; more specifically,  Abbott et al. (2017b). Also, the observations of GWs at several Earth-based detectors constrain the speed of gravity between  Cornish et al. (2017). These observations have been used to constrain the theories of gravity beyond GR Lombriser and Taylor (2016); Lombriser and Lima (2017); Bettoni et al. (2017); Shoemaker and Murase (2017); Baker et al. (2017); Sakstein and Jain (2017); Creminelli and Vernizzi (2017); Ezquiaga and Zumalacárregui (2017); Nojiri and Odintsov (2018). In this paper, we point out that another theory where the speed of gravitational waves deviates from the speed of light is EiBI gravity. Note that, like GR, also in EiBI gravity, the graviton is massless and there are only two polarization modes. Although, gravitational waves propagate in vacuum in exactly the same manner as in GR, there should be some differences due to matter distributions.

Various upper bounds on the theory parameter (, illustrated in the next section) of EiBI gravity exist in the literature from astrophysical and cosmological observations. For example, a strong constraint on the theory with and m comes from the existence of self-gravitating compact objects like neutron stars Pani et al. (2011). Stellar equilibrium and the evolution of the Sun lead to m Casanellas et al. (2012). Assuming , the constraint m was obtained from the conditions for primordial nucleosynthesis Avelino (2012a). Requiring that the electromagnetic force dominates over the gravitational force at the subatomic scale leads to the very strong constraint m Avelino (2012b). Note that we express in dimensions of [length], i.e., in units of . In most of the literature, the units used are kgms with the assumption of and, therefore, one should divide the numbers by for the conversion. However, most of these bounds are somewhat indirect. In this article, we obtain constraints on from the bounds on the speed of gravitational waves as mentioned above. In Sec. II, we obtain the gravitational-wave propagation equation in the background of Earth’s gravitational field, and using this we put a constraint on from the time delay in the arrival of gravitational-wave signals at widely separated earth based detectors. In Section III, we put constraint on from the time delay between the gravitational wave signal from the recently detected neutron star merger event GW170817 and the electromagnetic signal from the associated -ray burst event GRB 170817A. Finally, we conclude and summarize our results in Sec. IV.

## Ii Speed of gravity through the Earth in Born-Infeld gravity

First, we briefly recall EiBI gravity. The central feature here is the existence of a physical metric which couples to matter, and another auxiliary metric which is not used for matter couplings. One needs to solve for both metrics through the field equations. The action for the theory developed in Ref. Bañados and Ferreira (2010) is given as

 SBI(g,Γ,Ψ) = c38πGκ∫d4x[√−|gμν+κRμν(Γ)| (1) −λ√−g]+SM(g,Ψ)

where , with being the cosmological constant. As mentioned earlier, is the constant parameter of the theory with dimensions of [length] and, for sufficiently small , the action reduces to the known Einstein-Hilbert action. Variation with respect to (assuming symmetric and ) gives

 qμν=gμν+κRμν(Γ), (2)

where is called the auxiliary metric which satisfies the compatibility condition with respect to , which gives

 Γρμν=12qρα(qαμ,ν+qνα,μ−qμν,α). (3)

Variation with respect to gives the field equation

 √−qqμν=λ√−ggμν−8πGc4κ√−gTμν, (4)

where the components are in the coordinate frame.

Now we derive the equation for the propagation of gravitational waves in the background of Earth’s gravity. For the background field equations, we work in the Newtonian limit or, more precisely, the nonrelativistic limit of EiBI theory Bañados and Ferreira (2010). We consider a time-independent metric,

 ds2=−(1+2Φc2)c2dt2+(1−2Φc2)δijdxidxj, (5)

coupled to the energy-momentum tensor , in the comoving frame. Here is the matter density and is the pressure. In the nonrelativistic limit and is the Newtonian potential. Also, here, the cosmological constant is irrelevant. We additionally assume a time-independent auxiliary metric,

 ds2q=−(1+2Φqc2)c2dt2+(1−2Φqc2)δijdxidxj. (6)

From the (temporal indices) field equations [Eqs. (4) and (2)], we have

 Φ=Φq+2πGκρ, (7) Φ=Φq+κ2∇2Φq, (8)

where we keep only linear terms in , and . In Eq. (8), we used . From Eqs. (7) and (8) we get

 ∇2Φq=4πGρ. (9)

Taking the Laplacian of both sides of Eq. (7) we get

 ∇2Φ=∇2Φq+2πGκ∇2ρ, (10)

and using Eq. (9) in Eq. (10), we finally get the modified Poisson equation

 ∇2Φ=4πGρ+2πGκ∇2ρ. (11)

Using this modified Poisson equation, it was shown in Ref. Pani et al. (2011) that this theory supports a stable pressureless neutron star. They it was also shown that nonrelativistic dust collapse does not lead to a singularity for , which is a completely different result from that in Newtonian gravity. The second term in the modified Poisson equation may play a role of repulsive role and may be important in a highly dense region of matter. However, for a nearly constant matter density (such as the Earth), this effect is negligible and we get back Newtonian gravity. But we will see below that, even for the Earth, there will be a nonzero contribution in the equation of gravitational-wave propagation.

Now, in the presence of gravitational waves, the perturbed line elements will take the following forms:

 ds2=−(1+2Φc2)c2dt2+(1−2Φc2)(δij+hij)dxidxj, ds2q=−(1+2Φqc2)c2dt2+(1−2Φqc2)(δij+γij)dxidxj,

where and are transverse and traceless, i.e., and . In our computations, we keep the terms in first order of and as well as and . However, we keep the terms like “” to capture the effect of Earth’s gravity. The following identities are used to construct the perturbed field equations:

 √−g ≈ c(1−2Φc2)+O(h2), √−q ≈ c(1−2Φqc2)+O(γ2), gij ≈ (1+2Φc2)(δij−hij), qij ≈ (1+2Φqc2)(δij−γij).

The , (spatial indices) field equation from variation [Eq. (4)] becomes

 (1−2Φqc2)(1+2Φqc2)(δij−γij) =(1−2Φc2)(1+2Φc2)(δij−hij),

which simplifies greatly to

 γij=hij. (12)

The exactly same result is seen in the case of gravitational waves in a Friedmann-Robertson-Walker (FRW) background Escamilla-Rivera et al. (2012); Beltrán Jiménez et al. (2017).

Using Eq. (3), we compute the perturbed :

 R00(q) = ∇2Φq−Φq,i,jγij, Rij(q) = 12c2(1−4Φqc2)¨γij+1c2∇2Φq(δij+γij) −12∇2γij−1c2Φq,k,lγklδij.

From the , perturbed field equation [Eq. (2)], we get . Using this relation in the , equation, we get

 (1−2Φqc2)(δij+γij) = (1−2Φc2)(δij+hij)

which [after using Eqs. (8) and (12)] simplifies to

 ¨hij−c2(1+4Φqc2)∇2hij=0. (13)

The two polarization modes and of the radial component of the gravitational-wave amplitude satisfy the following one-dimensional wave equation:

 ¨A+c2(1+4Φqc2)k2A=0. (14)

We have used , where is the wave number. Let us take a plan wave solution of (14)

 A(t,ω)=~A(ω)ei(kr−ωt). (15)

Inserting this into the wave equation (14) gives the dispersion relation as

 −ω2+c2(1+4Φqc2)k2=0. (16)

Thus the speed of gravitational waves in the background of Earth’s gravitational field becomes

 vgw≡dωdk = c(1+4Φqc2)1/2 (17) ≈ c(1+2Φc2−4πGκρc2),

where we used Eq. (7). Note that in GR the gravitational waves propagate with the speed of light, which in this case is . At the surface of Earth, .

Bound on from the speed of gravity in the Earth: Cornish et al. Cornish et al. (2017) obtained upper and lower bounds on the speed of gravitational wave propagation from the time delay between gravitational-wave signals (reported by the LIGO Scientific and Virgo Collaborations) arriving at widely separated Earth-based detectors. Their bounds are given by . Although this bound is very crude and will improve with more detections and more detectors joining the worldwide network, there may be a signature of modified gravity where the speed of gravitational waves is different from the speed of light. This feature indeedp resent in EiBI theory, and we can use the bounds on to put a bound on . Assuming , we get , where we have used the matter density of Earth in Eq. (17).

## Iii Speed of gravity in the background of FRW Universe

The time delay between the recently detected gravitational-wave signal GW170817 and the associated -ray burst GRB 170817A Abbott et al. (2017a) also gives bounds on Abbott et al. (2017b). The source was localized at a luminosity distance of 40 Mpc. Since the signals were propagating over an intergalactic distance, we consider the gravitational wave propagating in a background FRW spacetime in cosmology. Thus, we solve the EiBI field equations in the FRW background for the tensor perturbations and obtain the gravitational-wave propagation equation (in cosmic time) as Escamilla-Rivera et al. (2012); Beltrán Jiménez et al. (2017)

 ¨hij+(3H+˙αα)˙hij−c2β2∇2a2hij=0, (18)

where

 α=λ+8πGκρc2,        β2 = ∣∣ ∣ ∣∣λ−8πGκpc4λ+8πGκρc2∣∣ ∣ ∣∣,

and and are the energy density and pressure of the matter in the Universe, is the Hubble parameter, and is the scale factor in the FRW spacetime. From the coefficient of in Eq. (18), we see that in addition to the cosmological damping proportional to , there is also an extra damping factor due to EiBI gravity proportional to . On subhorizon propagation distance scales this term is small.

We go to Fourier space and redefine the perturbations as . The wave equation (18) in conformal time , defined via , can be written as

 μ′′ij+[−a′′a−α′′2α−a′α′aα−α′24α2+c2β2k2]μij=0, (19)

where prime denotes a derivative with respect to the conformal time  . On subhorizon scales we can safely ignore all of the terms inside the square brackets compared to the mode scale , and the above equation reduces to

 μ′′ij+c2β2k2μij=0. (20)

The two polarization modes and of the radial component of the gravitational wave amplitude satisfy the following one dimensional wave equation at large distances from the source

 A′′+c2β2k2A=0. (21)

Let the solution of Eq. (21) be

 A(τ,ω)=~A(ω)eikr−∫iωdτ. (22)

Inserting this into the wave equation (21) gives the dispersion relation as

 −ω2+c2β2k2=0, (23)

and therefore the speed of gravitational waves can be given as

 vgw≡dωdk=cβ=c∣∣ ∣ ∣∣λ−8πGκpc4λ+8πGκρc2∣∣ ∣ ∣∣1/2. (24)

The difference between the speed of gravity and the speed of light in vacuum becomes

 vgw−cc=∣∣ ∣ ∣∣λ−8πGκpc4λ+8πGκρc2∣∣ ∣ ∣∣1/2−1. (25)

Bound on from the speed of gravity in a FRW Universe: The observed time delay of s between the gravity wave from the neutron star merger event and the light from the subsequent -ray burst constrain the difference between the speed of gravity and the speed of light to be between and . Inserting , the present energy density of the Universe to be at the critical density , the cosmological constant , and pressure into (25), we obtain .

## Iv Conclusions

In this article we demonstrated how the signature of some of the modified theories of gravity may be imprinted in the speed of gravitational waves. In particular, we put constraints on EiBI gravity from the bounds on the speed of gravity from the recent direct detection of GW signals from binary black hole and neutron star mergers.

We derived the gravitational-wave propagation equation in the background of Earth’s interior and used it to put a bound on . From the bound on the speed of gravity derived from the time delay of GW signals at different Earth-based detectors, we obtained . Similarly, from the time delay in the signals from GW170817 and GRB 170817A events, in the background of a FRW Universe, we obtained the bound . These bounds are weaker than other bounds from neutron stars, stellar evolution, nucleosynthesis, etc. However, they constitute direct constraints on from observations. Also, we note that there is no dispersion of GW in matter in EiBI gravity. Future observations of the speed of GWs and their dispersion will put tighter constraints on theories of gravity beyond GR like EiBI theory.

## Acknowledgments

S.J. acknowledges Sayan Kar for useful discussions.

## References

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