Implications of the Neutron Star Merger GW170817 for CosmologicalScalar-Tensor Theories

Jeremy Sakstein1, ∗ and Bhuvnesh Jain1, †

1Department of Physics and Astronomy, Center for Particle Cosmology,University of Pennsylvania, 209 S. 33rd St., Philadelphia, PA 19104, USA

The LIGO/VIRGO collaboration has recently announced the detection of gravitationalwaves from a neutron star-neutron star merger (GW170817) and the simultaneous measure-ment of an optical counterpart (the gamma-ray burst GRB 170817A). The close arrival timeof the gravitational and electromagnetic waves limits the difference in speed of photons andgravitons to be less than about one part in 1015. This has three important implications forcosmological scalar-tensor gravity theories that are often touted as dark energy candidatesand alternatives to ΛCDM. First, for the most general scalar-tensor theories—beyond Horn-deski models—three of the five parameters appearing in the effective theory of dark energycan now be severely constrained on astrophysical scales; we present the results of combiningthe new gravity wave results with galaxy cluster observations. Second, the combination withthe lack of strong equivalence principle violations exhibited by the supermassive black holein M87, constrains the quartic galileon model to be cosmologically irrelevant. Finally, wederive a new bound on the disformal coupling to photons that implies that such couplingsare irrelevant for the cosmic evolution of the field.

The terms dark energy and modified gravityare closely connected at the most general level;all but the simplest alternatives to the ΛCDMmodel typically invoke some modification of gen-eral relativity (GR) (see [1–5] for reviews). Themost widely studied of these are scalar-tensortheories where a new scalar φ mediates an addi-tional gravitational interaction between matterthat is suppressed in the solar system by screen-ing mechanisms (see [6–9] for reviews) but thatbecomes relevant on cosmological scales. Thishas motivated an intense theoretical effort to-wards finding the most general scalar-tensor the-ory that is pathology free, and the modern ap-proach to dark energy model building can beepitomized by the class of models called beyondHorndeski (BH) [10, 11]. BH theories are acomplete and general framework for constructingdark energy/modified gravity models (includingcommonly studied paragons for modified grav-ity such as chameleons [12] and galileons [13]),many of which can accelerate without a cos-mological constant (self-accelerate). They aretherefore viewed as alternatives to the ΛCDMcosmological model and there is much effort fo-cused on how well upcoming cosmological sur-veys will constrain them [14].

BH theories make a striking prediction: thespeed of gravitational waves in the cosmological

background differs in general from the speed oflight [15–18]. Recently, the LIGO/VIRGO con-sortium has announced the observation of neu-tron star merger GW170817 [19], a neutron star-neutron star merger that has been localized tothe galaxy NGC 4993, about 40 Mpc from theMilky Way. The simultaneous observation of anoptical counterpart (the gamma-ray burst GRB170817A) by the Fermi gamma-ray telescope [20]and several optical telescopes [21] implies thatthe two speeds can differ by at most one part in1015, more specifically1 (c2T −c2)/c2 ≤ 6×10−15,where cT is the speed of gravitational waves andc is the speed of light (this limit comes from thetime lag between the LIGO and Fermi detec-tions). This has severe implications for cosmo-logical scalar-tensor theories that we delineate inthis letter.

Cosmologically, deviations for ΛCDM thatfall into the BH class of models can be pa-rameterized by five free functions of timeαM , αK , αB, αH , αT [16, 24]. These are typ-

1 Note that the sign is positive, i.e. the observation ofthe gravitational waves before the optical counterpartconstrains gravitons to move at a faster speed than pho-tons (cT > c). This therefore probes theories where thiseffect is predicted, unlike previous bounds that con-strain the difference in propagation speeds if cT < c[22, 23].

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ically referred to as the effective theory of darkenergy [25–27], and constraining both their val-ues and their cosmological time-dependence isone of the goals of upcoming dark energy mis-sions such as DESI, LSST, Euclid and WFIRST(see [14] for example). The first describes therunning of the Planck mass and the second thekinetic term for the scalar; we will not dis-cuss these here. The third, αB, describes thekinetic-mixing of the scalar and graviton andthe fourth, αH describes the so-called disfor-mal properties of the theory [28–32]. The fifth,αT = (c2T − c2)/c2 is none other than the frac-tional difference between the speed of gravitonsand photons. The observation of optical counter-parts therefore implies that this is now known:αT ≈ 0.

An interesting property of BH theories wewill consider2 is that they satisfy solar systemtests of gravity perfectly using the Vainshteinscreening mechanism3, but they predict new andnovel deviations from GR inside astrophysicalbodies of the form [37–39]:

dΦ

dr= −GM(r)

r2− Υ1G

4

d2M(r)

dr2(1)

dΨ

dr= −GM(r)

r2+

5Υ2G

4r

dM(r)

dr, (2)

where

Υ1 =4α2

H

(1 + αT )(1 + αB)− αH − 1and (3)

Υ2 =4αH(αH − αB)

5[(1 + αT )(1 + αB)− αH − 1], (4)

and Φ and Ψ are the Newtonian potential andthe ij-component of the metric. This novel Vain-shtein breaking led to several suggestions forsmall-scale tests of Υi [38, 40–47], which couldbe used either as priors for cosmological searchesor as consistency checks. Note that Eqns. 1 and2 contain three unknown functions, so until nowthere was a degeneracy even allowing for the left

2 We are considering quartic theories with αH 6= 0(see [33–35] for discussions of extensions of these).This means we exclude theories that screen using thechameleon mechanism (and similar mechanisms).

3 Note that the Vainshtein mechanism does not screendeviations in the speed of photons and gravitons [36].

Dwarf Stars

Galaxy Clusters

-1.0 -0.5 0.0 0.5 1.0-2

-1

0

1

2

FIG. 1. The excluded regions in the αH–αB planenow that cT is known to be unity with very highprecision. The regions excluded by cluster tests anddwarf stars are labeled accordingly.

hand side to be constrained observationally (asdescribed below). With αT known to be neg-ligible these constraints translate uniquely intobounds on αB and αH at late times. We presentthese in Figure 1.

We have used two sets of constraints to ob-tain these bounds. The first comes from dwarfstars [40, 41]. There is a minimum mass forthe onset of hydrogen burning (MMHB) in stars(see [48] for a review); stars lighter than this arebrown dwarfs while heavier stars are red dwarfs.When Υ1 > 0, non-relativistic stars are typicallyless compact so that their cores are cooler andless dense, and therefore this MMHB is largerthan the GR value of 0.08M. Demanding thatthe lightest observed red dwarf is at least asheavy as the MMHB sets the bound Υ1 < 1.6.The second constraint come from galaxy clus-ters. The equivalence of Φ and Ψ in GR impliesthat the mass measured by weak lensing (lensingmass, sensitive to Φ + Ψ) and X-ray data (thesurface brightness measured in X-ray data is aprobe of the hydrostatic mass, sensitive to Φ)should agree. In BH, this equivalence is brokenand so any deviation (or lack thereof) constrainsthe parameters Υ1 and Υ2. Reference [44] hasperformed such a test using weak lensing data

3

from CFHTLenS and X-ray data from XMM-Newton, including measurement errors and sys-tematic uncertainty due to non-thermal pressure(see [49–51] for studies on the consistency of X-ray and lensing masses). They obtain the con-straints Υ1 = −0.11+0.93

−0.67 and Υ2 = −0.22+1.22−1.19.

(There is a stronger bound than the lower boundon Υ1, Υ1 > −2/3 that we include in Figure 1;one cannot form stable stars if this is violated[39].)

Figure 1 shows that a large region of the αB–αH plane is ruled out4. The allowed region canbe further constrained by data on galaxy clus-ters with forthcoming surveys. Modified gravitymodels that can explain dark energy typicallypredict αi ∼ O(1) so our results are severelyconstraining for these models. Stage-IV cos-mological surveys could constrain αi to levelsof O(10−1) but these make several assumptionsabout the evolution of these parameters and theamount of screening [14]. Our results are in-dependent of these assumptions and are com-pletely general.. Note that there are two otherparameters αM and αK that are completely un-constrained on small scales (although one maybe able to constrain αM using tests of the time-variation of Newton’s constant). It is also worthnoting that the line αH = 0 is completely un-constrained, as it should be given equations(3) and (4). This line corresponds to a largesubset of models known as Horndeski theories5

[56, 57]. These theories can still be constrainedusing gravitational waves, but one needs a sec-ond probe since Vainshtein screening works in-side objects for these theories. This probe comes

4 Note that the cluster constraints of [44] apply at z =0.33 and that the parameters αi can, in principle, betime-varying. Models where there is a strong variationfrom 0 to 1 between z = 0.33 and z = 0 can evade thecluster bounds but not the dwarf star bounds.

5 Historically, Horndeski theories are those which givemanifestly second-order equations of motion and aretherefore free of Ostrogradski instabilities. Later, thesewere extended to BH theories, which have higher-orderequations but propagate three degrees of freedom andare therefore also ghost-free. See [11, 52–55] for moredetails and other extensions of the Horndeski theory.We emphasize that the distinction is purely histori-cal, and that Horndeski theories should be considereda subset of the more general BH class.

in the form of strong equivalence principle (SEP)violations, which we describe next.

The entire class of Horndeski theories is vast,and one typically focuses on specific models thatencapsulate the relevant physics in order to pro-vide a concrete realization of their cosmologi-cal consequences. The quintessential paradigmis the covariant quartic galileon [58] with La-grangian

L√g

=K(X) +G4(X)R

+G4, X

((φ)2 −∇µ∇νφ∇µ∇νφ

), (5)

where X = −(∂φ)2/2, K(X) = X, and

G4(X) =Mpl

2

2+ 2c0

φ

Mpl+ 2

c4Λ64

X2. (6)

(We have chosen the notation to match thatcommonly used in the literature.) The free pa-rameters c0 (often called α or β elsewhere in theliterature) and c4 parameterize the strength ofthe coupling to matter and the strength of thenew interaction respectively. The speed of grav-itational waves in this theory is given by [59]∣∣∣∣c2T − c2c2

∣∣∣∣ =

∣∣∣∣ 4c4x2

1− 3c4x2

∣∣∣∣ < 6× 10−15 (7)

imposing the LIGO/VIRGO-Fermi bound. Theparameter x = φ/(HMpl) where φ is the time-derivative of the scalar (using cosmic time) en-codes information about the cosmology of thegalileon. Indeed, one has [60]

Ωφ = c0x2 +

x2

6+

15c4x2

2. (8)

where Ωφ is the density parameter for galileons.The speed of gravitational waves therefore con-strains a combination of c4 and the cosmology ofthe galileon.

On smaller scales, reference [61] has recentlyobtained new bounds on the parameters c0 andc4 using the lack of SEP violations predicted inthese theories [62, 63]. Black holes in galileontheories have no scalar hair and therefore donot couple to external fields. Non-relativisticbaryons do couple to galileon fields, and there-fore black holes and baryons fall at different rates

4

in external gravitational fields, signifying a vio-lation of the SEP. The acceleration of a satel-lite galaxy infalling towards the center of a clus-ter would have a sub-dominant galileon compo-nent not felt by its central supermassive blackhole (SMBH). This would cause the SMBH tolag behind the rest of the galaxy and becomeoffset from the center by an observable amount(O(kpc)) given by the distance where the miss-ing galileon component is balanced by the restor-ing force from the baryons left at the center. Us-ing the techniques of [64] applied to the galaxyM87 (located in the Virgo cluster), [61] were ableto place strong constraints on c0

6 and c4.Taken together, the SMBH and

LIGO/VRIGO-Fermi constraints allow oneto constrain the cosmological contribution ofthe quartic galileon to the Universe’s energybudget using equation (8). In Figure 2 we showthe corresponding constraints in the c4–Ωφ

plane for representative values of c0 ∼ O(1)7.The LIGO/VIRGO-Fermi bounds constrainlarge values of c4, which correspond to largedifferences in the speed of photons and gravitonsas well as strong screening; whereas SMBHconstrains small c4, where galaxy clusters areless screened and the speed of photons andgravitons are similar. The speed of gravitonstherefore rules out larger values of Ωφ at largec4 while SMBH constraints rule out lower valuesat low c4. The two constraints are thereforecomplementary, and the combination rules outall galileon cosmologies except those whereΩφ 1. In particular, the combination ofSMBH and LIGO/VIRGO-Fermi constraintsrules out regions where the galileon is more im-

6 When applied to cosmological galileons, the effectivecoupling to matter is a combination of c0 and the cos-mological parameters [65, 66], and so it is this combi-nation that is constrained. We account for this in thepresent work.

7 Order-unity matter couplings are considered natural inscalar-tensor theories. One can write c0φ/Mpl = φ/Mwith M = Mpl/c0 being the relevant mass-scale forthe interaction. c0 1 ⇒ M Mpl and the theoryhas a trans-Planckian mass-scale whereas c0 1 ⇒M Mpl so that there is a low cut-off for the effectivefield theory (at least naively, the Vainshtein mechanismalters the cut-off for the theory and the quantum prop-erties of galileons are still uncertain [67–71]).

portant than radiation in the late-time Universe.Clearly, the galileon can have little to nothingto say about dark energy. One could potentiallyavoid these harsh restrictions by adding otherterms such as a cubic galileon (which itself isheavily constrained by a prediction of a toolarge integrated Sachs-Wolfe effect [72, 73]) ora quintic galileon (which typically destabilizesVainshtein screening [65, 66]). We will notdo so here because, ultimately, one is simplyadding more parameters to the theory, in whichcase there are bound to be tunings that cancircumvent constraints.

An alternative to the covariant quar-tic galileon is the beyond Horndeski quarticgalileon8. This model gives an identical cos-mology to the model studied above and has anidentical expression for c2T [74]. Therefore, thismodel is also tightly constrained. Going be-yond this, a large portion of Horndeski and BHmodels are now excluded as dark energy can-didates, as are several more complicated theo-ries such as vector-tensor and degenerate higher-order scalar-tensor theories [75–77]. We empha-size that models such as Einstein-dilaton-Gauss-Bonnet that do not lead an accelerating universe,of interest for example for deviations detectablevia black hole tests [78], are still allowed.

Finally, we consider one additional quantitythat can be bounded by the LIGO/VIRGO-Fermi observation: the disformal coupling tophotons. Disformal couplings refer to derivativecouplings of the scalar to a matter species i viathe metric

g(i)µν = gµν +∂µφ∂νφ

M4i

. (9)

If Mi =M then the field couples to all speciesuniversally and there is no violation of the equiv-alence principle. But there is no a priori reasonfor one to expect this to be the case and, inparticular, reference [79] have investigated theeffects of having different disformal couplings tophotons and baryons. In the simplest case wherethere is no disformal coupling to matter9, the

8 With Lagrangian L/√−g = Mpl

2R/2 + X +2c4X

2/Λ64[(φ)2 −∇µ∇νφ∇µ∇νφ].

9 More complicated theories where the cosmic accelera-

5

SpeedofGravitons

SMBHs

-10 -8 -6 -4 -2 0-10

-8

-6

-4

-2

0

FIG. 2. Constraints in the c4–Ωφ plane coming fromthe near equivalence of the speed of gravitons andphotons (orange) and the lack of an offset supermas-sive black hole in M87 (red). The shaded regionscorrespond to c0 = 1, and we indicate the extent ofthe graviton speed constraint using the solid orangeline. Also shown using the dashed and dotted linesare the equivalent regions for c0 = 3 from the LIGO-Fermi and SMBH bounds respectively.

speed of photons is given by

c2γc2

= 1− φ2

M4γ

, (10)

and so the LIGO/VIRGO-Fermi bound impliesthat φ2/M4

γ<∼ 6 × 10−15. Cosmologically-

relevant scalars typically have φ ∼ H0Mpl [81],which implies that Mγ

>∼ 10 MeV. This isstronger than the constraint inferred from theabsence of any vacuum Cerenkov radiation ob-served at LEP [23] and is comparable with thebound from constraints on energy-loss by thePrimakov process in the Sun. It is supersededby the same constraints coming from horizontalbranch stars, which give Mγ

>∼ 100 MeV [82],although our bound is free from the degenera-cies of stellar physics (such as metallicity). Atthis level, the disformal coupling to photons can

tion arises from disformal couplings [80] are also con-strained as photons travel slower than gravitons inthese theories.

have no significant effect on the cosmic evolutionof the scalar [32].

To summarize, in this letter we have high-lighted three important consequences of the ob-servation of gravitational waves and an opti-cal counterpart from the binary neutron starmerger GW170817 [19–21] for cosmologicalscalar-tensor theories. The close arrival time(less than a minute) constrains the speed ofgravitons and photons to differ by at most onepart in 10−15. For beyond Horndeski theo-ries, a very general framework for constructingpathology-free dark energy models, one of thefive functions that describes the cosmology ofthese theories (αT = c2T /c

2 − 1) is now knownto be negligible. Furthermore, this implies thattwo of the other functions, αB and αH , canbe constrained using astrophysical tests. Wehave presented these constraints here for the firsttime. Second, combining the LIGO/VIRGO-Fermi bound with separate bounds coming fromthe lack of any strong equivalence principle vi-olations by the central supermassive black holein M87, we have shown that the covariant quar-tic galileon, a common paradigm for modifiedgravity as dark energy, must be cosmologicallyirrelevant. Finally, we have constrained the dis-formal coupling to photons, and shown that thiscan play no significant role in the cosmologicalevolution of scalar fields.

Acknowledgements: We are grateful fordiscussions with Tessa Baker, Jeremy Heyl,Justin Khoury, Kazuya Koyama, Mark Trod-den, and Filippo Vernizzi. BJ is supported inpart by the US Department of Energy grant de-sc0007901. JS is supported by funds provided tothe Center for Particle Cosmology by the Uni-versity of Pennsylvania.

∗ Email: sakstein@physics.upenn.edu† Email: bjain@physics.upenn.edu

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