Strong constraints on cosmological gravity from GW170817 and GRB 170817A. T. Baker,1 E. Bellini,1 P. G. Ferreira,1 M. Lagos,2 J. Noller,3 and I. Sawicki4 1University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK 2Kavli Institue for Cosmological Physics, The University of Chicago, Chicago, IL 60637, USA 3Institute for Theoretical Studies, ETH Zurich, Clausiusstrasse 47, 8092 Zurich, Switzerland 4CEICO, Fyzikální ústav Akademie vˇed CR,ˇ Na Slovance 2, 182 21 Praha 8, Czechia (Dated: October 18, 2017) The detection of an electromagnetic counterpart (GRB 170817A) to the gravitational wave signal (GW170817) from the merger of two neutron stars opens a completely new arena for testing theories of gravity. We show that this measurement allows us to place stringent constraints on general scalar-tensor and vector- tensor theories, while allowing us to place an independent bound on the graviton mass in bimetric theories of gravity. These constraints severely reduce the viable range of cosmological models that have been proposed as alternatives to general relativistic cosmology. Introduction: The advanced Laser Interferometer Gravita- cosmic rays [16]. Up to now, the only upper bound on the tional Observatory (aLIGO) and the VIRGO interferometer, propagation speed of GWs comes from measuring the travel have recently announced the detection of gravitational waves time between the two detectors of aLIGO, and is αT < 0.42 (GW170817) from the merger of a neutron star (NS) bi- [17, 18]. nary located near NGC 4993 [1]. A gamma ray burst (GRB In the regime we are considering (a gravitational wave 170817A), occurring within 1.7 seconds, and in the vicinity propagatingin effectively empty space, other than the medium of GW170817, was observed by the Fermi Gamma-ray Burst provided by the new degree of freedom) the linearized action Monitor, and the Anti-Coincidence Shield for the Spectrom- (1) is sufficient. It is conceivable (but unlikely) that there may eter for the International Gamma-Ray Astrophysics Labora- be some exotic behavior close the GW sources, in regions of tory [2, 3]. There is strong evidence that this event is an elec- strong gravity (for example, as occurs with the screening of tromagnetic counterpart to the NS-NS merger [4, 5]. Com- scalar forces) that leads to non-linear corrections. Such ef- paring the travel time of light and gravitational waves (here- fects could alter GW production, but will have no bearing on after GW), we can place stringent constraints on cosmological the gravitational wave propagation during the bulk of its travel gravity, and cosmology more generally [6–14]. time. Also, though Eq. (1) is valid for a wide range of gravi- We will assume that constraints on Lorentz violation in the tational theories, it does not encompass bimetric theories. electromagnetic sector are sufficiently strong that the speed Constraint on tensor speed excess: We consider the geomet- of light is c 1. In vacuum, Lorentz symmetry implies that = ric optics limit of Eq. (1) so that cT is indeed the speed of all massless waves propagate at the speed of light. How- gravitational waves. ever, when a medium is present, Lorentz symmetry is sponta- Let ts be the time of emission for both the gravitational neously violated and propagation speeds can differ. Alterna- waves and photons; there can be some delay between the two tive theories of gravity, directly coupling extra degrees of free- emission peaks (for a NS-NS merger this will be a few sec- dom to curvature, provide such a medium when the new de- onds, of the order of the GRB duration), but this would only gree of freedom takes a configuration that defines a preferred serve to tighten the bound we derive below. Let tT be the direction (such as the time direction in cosmology). The ac- merger time identified in the gravitational wave train, and tc tion for linearized gravitational waves in such a medium takes be the measured peak brightness time in the optical signal. To the form: good accuracy, then, the transit time of the GW and photon 1 3 2 ˙2 2 2 signals are cT (tT ts)= ds and (tc ts)= ds, where ds 40 Sh = d xdtM hA cT (∇hA) . (1) − − ≃ 2 Z Mpc is the distance to the source. We then have (tc tT )/ds = ∗ − − arXiv:1710.06394v1 [astro-ph.CO] 17 Oct 2017 1 1/c . Taylor expanding this gives α 2∆t/d , where ∆t We have decomposed the metric as g = η + h — with T T s αβ αβ αβ is− the measured difference in arrival times≃ between the GW η the Minkowski metric — by choosing locally inertial αβ merger peak and the peak optical brightness. An arrival delay coordinates with time chosen to be the direction defined by of ∆t 1.7 seconds implies that the medium. We have expanded hαβ in polarization states, ≃ 15 εA, with amplitudes hA, where A = ,+. M is the effective αT . 1 10− . (3) Planck mass, which in media provided× by alternative∗ gravity | | × theories can differ from the standard MP. cT is the speed of Comparing this to current cosmological constraints (where gravitational waves; we will find it convenient to parametrize σα 1 [19]) or forecast cosmological constraints (where T ∼ this as [15], σα 0.1[20]), this constraint is remarkable. For all intents T ∼ 2 and purposes, we will hereafter consider αT 0 and attempt cT = 1 + αT . (2) to understand its consequences for cosmology.≃ In principle, αT could adopt either positive or negative values. Implications for scalar-tensor theories: We begin by con- However, negative values (cT < c) are constrained to αT > sidering scalar-tensor theories. The Horndeski action is 10 15 by a lack of observed gravi-Cerenkovˇ radiation from the most general scalar-tensor theory with second-order − − 2 equations of motion [21, 22], and is given by S = the scalar fluctuations is equal to that of light. On the other 4 5 L L L d x√ g ∑i=2 i[φ,gµν ]+ M[gµν , ] , where M is hand, the strength of the fifth force, f,φ , is allowed to be simi- theR minimally− coupled matter action.··· The scalar field la- lar to gravity. grangian is built of four terms: two minimally coupled to In Class (ii), the scalar evolves quickly, X H2M2, and ∼ ∗ gravity, L2 = K and L3 = G3φ and two terms explicitly non-canonical kinetic terms in G2 and G3 play a signifi- − involving the Ricci curvature, R, and the Einstein tensor, Gµν : cant role: they can give rise to self-acceleration, significantly changing the equation of state and the sound speed. Con- 2 µ ν L4 = G4R + G4,X (φ) ∇µ ∇ν φ∇ ∇ φ , straints on the evolution of the Planck mass [31, 32] restrict − µ ν 1 3 µ ν the strength of coupling to gravity f,φ to be small, since the L5 = G5Gµν ∇ ∇ φ G5 X (∇φ) 3∇ ∇ φ∇µ ∇ν φφ − 6 , − scalar runs during the entire history of the universe in these ν α µ models. We reiterate that perturbative control of quantum cor- +2∇ ∇µ φ∇ ∇ν φ∇ ∇α φ . (4) rections in the fast-moving models depends on shift symme- ν Here K and Gi are functions only of φ and X ∇ φ∇ν φ/2, try, which would disallow any dependence on φ in the action, and subscript commas denote derivatives. On≡− a cosmological specifically the conformal coupling f (φ). background, Horndeski models give [23] Horndeski theory is not the most general scalar-tensor the- 2 ory propagating one single extra degree of freedom. New M αT 2X 2G4,X 2G5,φ φ¨ φ˙H G5,X (5) ∗ ≡ − − − terms can be added to construct the “beyond” Horndeski La- grangian [33, 34] at the price of third derivatives in equations where M2 2 G 2XG + XG φ˙HXG . 4 4,X 5,φ 5,X of motion and new constraints to remove any extra degrees of One way∗ ≡ of satisfying− α 0 is through− a delicate cancel- T freedom naively implied by them. This extension is described lation between G , G and∼G . If G = 0, this cancel- 4,X 5,φ 5,X 5,X by two new free functions, G˜ φ X and G˜ φ X correct- lation is trivial, since it implies that the theory is minimally 4 ( , ) 5 ( , ) ing L and L (see [33] for the complete expressions) and coupled. Any non-trivial cancellation would not only have 4 5 modifying Eq. (5) to to be time-dependent, but also sensitive to the matter con- tent of the universe due to the dependence on H and φ¨. Thus 2 αTM = 4X G4,X G˜ 4,X G5,φ 2φ¨XG5,X even a small change in e.g. the dark matter density, or devia- ∗ − − − ˙ ˜ tions from isotropy and homogeneity, would severely violate + 2φHX G5,X G5,X , (6) − it. Furthermore, any such a cancellation would be accidental, 2 with no symmetry to protect it. Some shift symmetric Horn- where M = 2G4 4X G4,X G˜ 4,X + 2XG5,φ ∗ − − − deski actions (i.e. not dependent on φ) are, to some degree, 2φ˙HX G5 X G˜ 5 X . , − , stable to radiative corrections. In flat spacetime, for K,Gi It is clear from Eq. (6) that one option is to set all the linear in X (galileons [24]), there exists an exact quantum terms contributing to αT to zero, as in the Horndeski case. non-renormalization theorem [25–27] — there are no correc- An intriguing alternative is to choose G5,X = G˜5,X = 0 and tions to these operators. The corrections remain under con- G˜ 4 X = G4 X G5 φ , which indeed leads to αT = 0 but also al- , , − , trol when the galilean symmetry is weakly broken [28], as lows for M = MP and αH = 0, where αH is the additional it must be in curved spacetime.