Direct detection of gravitational waves can measure the time variation of the Planck mass

Luca Amendola,1 Ignacy Sawicki,2 Martin Kunz,3 and Ippocratis D. Saltas2 1Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany 2CEICO, Institute of Physics of the Czech Academy of Sciences, Na Slovance 2, 182 21 Praha 8, Czechia 3Départment de Physique Théorique and Center for Astroparticle Physics, Université de Genève, Quai E. Ansermet 24, CH-1211 Genève 4, Switzerland The recent discovery of a γ-ray counterpart to a event has put extremely stringent constraints on the speed of gravitational waves at the present epoch. In turn, these constraints place strong theoretical pressure on potential modifications of gravity, essentially allowing only a conformally-coupled scalar to be active in the present Universe. In this paper, we show that direct detection of gravitational waves from optically identified sources can also measure or constrain the strength of the conformal coupling in scalar–tensor models through the time variation of the Planck mass. As a first rough estimate, we find that the LISA satellite can measure the dimensionless time variation of the Planck mass (the so-called parameter αM ) at around 1.5 with an error of about 0.03 to 0.13, depending on the assumptions concerning future observations. Stronger constraints can be achieved once reliable distance indicators at z > 2 are developed, or with GW detectors that extend the capabilities of LISA, like the proposed Observer. We emphasize that, just like the constraints on the gravitational speed, the bound on αM is independent of the cosmological model.

INTRODUCTION evolution and GWs [10] 1. A similar structure exists for general vector-tensor theories: Einstein-Aether [11] and generalized Proca [12, 13]. The recent almost simultaneous observation of gravi- The complete effect of gravity modification at the level tational waves (GWs) and γ-rays from bina- of linear perturbations can be described by a small set ries [1] has opened a new era of multi-messenger astron- of functions of time alone [14–16]. For example, Ref. [17] omy. One of the most interesting aspects of this event has makes the choice, which we will adopt in this paper, to been the first-ever precise measurement of the GW speed parametrize the general Horndeski theory with four func- c , confirming what (GR) predicted, T tions αM , αT , αB, αK , which in turn depend on the Gi namely that the present value of cT equals the speed of functions appearing in the Lagrangian. From those, only light c to an astonishing precision, |c /c−1| 10−15 [2]. T . the running of the Planck mass αM and the excess in the Modified gravity theories that couple extra scalar or tensors’ speed αT , vector degrees of freedom to curvature, change the prop- 2 −1 d ln M∗ agation of gravitational waves in one of two ways: they αM = H , (1) allow in general for a propagation speed c different from dt T 2 αT ≡ c − 1, (2) c, and make the effective Planck mass M∗ time and/or T position dependent. At the same time, the metric sourced express the non-minimal interaction with gravity. In by massive bodies is modified by the so-called gravita- turn, the non-minimal interaction between the scalar and tional slip, see e.g for earlier and more recent discussions curvature can be separated into a conformal part (i.e., [3–6]. In this context, we use as a working definition of the sector that can be absorbed into a conformal rescal- modified gravity the one we chose in the above works, ing of the metric which redefines the Planck mass), and a that is, any theory which modifies the propagation of GWs. For example, scalar-tensor theories with second-order 1 equations of motion find their most general formulation Whenever ∂G4(φ, X)/∂X or G5 6= 0, the speed of GWs is af- fected, otherwise, this sector reduces to the standard, confor- arXiv:1712.08623v2 [astro-ph.CO] 13 Dec 2018 in the so-called Horndeski Lagrangian [7–9]. The four mal coupling to curvature. We should also add here that, the free functions that enter this Lagrangian, often denoted so-called Kinetic Gravity Braiding term described by the free as G2,G3,G4,G5, are functions of the scalar field, and are function G3(φ, X) does not modify the propagation of GWs not associated with two sharply separate sectors: the first one does it produce gravitational slip and hence does not fall into (G ,G ) only affects the scalar-field evolution, while the our chosen definition for modified gravity. Nonetheless, it does 2 3 still modify non-trivially the constraint structure of gravity com- second in principle couples non-minimally (i.e. beyond pared to GR owing to the kinetic mixing between the scalar and the standard gravitational coupling) the scalar field to gravity and thus changes the effective Newton’s constant for per- turbations. gravity (G4,G5), the latter affecting both the scalar’s 2 non-conformal part (the sector which changes the speed In contrast, we shall emphasise that the method we of GWs). An important consequence is that the scalar propose here is independent of the underlying cosmolog- field is only minimally coupled to gravity if, and only if, ical model and of the precise model of modified gravity. αM = αT = 0. In this case, gravity is no longer mod- Another advantage with respect to CMB or BBN con- ified and GWs propagate as in standard GR. A similar straints is that one can in principle map the evolution of parametrization can be adopted for other theories, such αM in an extended redshift range from today to z ≈ 8. as beyond Horndeski [18, 19] and vector-tensor models, with the same functions αM , αT describing fully the non- minimal interaction with gravity [20]. GW PROPAGATION In view of the above parametrisation, the LIGO event [1], then, tells us that αT = 0 with great precision. This We consider a flat Friedmann-Robertson-Walker implies that any non-conformal coupling between the (FRW) spacetime with scale factor a and conformal Hub- scalar and curvature vanishes at the present epoch [21– ble function H. As it has been shown in [4], in such 24]. (see also [25, 26]). While this constraint is enough to a cosmological background the GW amplitude h in any forbid any sort of non-minimal coupling in vector-tensor modified gravity theory which does not give a theories [24, 27], for scalar-tensor theories a conformal mass, obeys the equation non-minimal coupling is still allowed. It is also possi- ¨ ˙ 2 2 h + (2 + αM )Hh + cT k h = 0, (3) ble to use this event to constrain αM , but the bound is extremely weak [28]. where the dot stands for a derivative with respect to In this paper, we show that GWs from sources with conformal time t, cT is the speed of GWs. The quan- identifiable redshift can also measure and constrain the tity αM , already defined in the Introduction, expresses evolution of the effective Planck mass, i.e. constrain the the time variation of the time-dependent effective Planck 2 second modified-gravity parameter, αM strongly, and mass M∗ (see [17]). M∗ is defined as the normaliza- with this, the remaining conformal coupling for scalar- tion of the kinetic term for the metric fluctuations h in tensor theories. In this way, GWs can constrain or rule the action for perturbations. For example, in the sim- out the entire modified-gravity sector of both vector- ple case of a Brans-Dicke gravity with parameter ω, one tensor and scalar-tensor models. finds αM = 1/(1 + ω). In more general models of mod- One generically expects that if there is such a confor- ified gravity, the GW equation is also of this form, see mal coupling of gravity, the model must feature screen- Refs [4, 33] for details. ing so that precision tests of gravity do not already rule The GW event reported in Ref. [1] has shown that it out. This screening mechanism would act as to sup- cT = 1 with extreme precision, at least for the present Universe. Here we would like to investigate the observ- press the Solar-System value of αM , which is essentially the rate of change per Hubble time of the gravitational able effects of αM on the GW signal, remembering that, constant, compared to that in the wider cosmology. The fixing αM , αT , as already mentioned, amounts to com- pletely fixing the non-minimal scalar-tensor interaction. present and local value of |αM | can indeed be constrained to be less than 0.01÷0.03 in the laboratory and in the Let us define the field v ≡ M∗ah. This quantity obeys Solar System (see for instance a recent summary of re- the equation of motion sults and a positive detection in [29]). A cosmological v¨ + k2v − µ2v = 0, (4) constraint from Big-Bang nucleosynthesis (BBN) is also with tachyonic mass µ of order H, and given by 4µ2 ≡ a stringent one, |GBBN/G0 − 1| 0.2 [30]. As will be . 2 2 ˙ shown in the following, the completely independent test (2+αM ) H +2(2+αM )H+2α ˙ M H. So, provided that the we propose here can reach similar or even better sensi- wavelength of the GW is subhorizon, k  H, v evolves 2 tivity. according to the standard wave equation, v¨+k v = 0, i.e. subhorizon GWs in the Jordan frame evolve according to The idea of using GWs to test αM and αT was put for- ward for the first time in [31], where it was shown that B- i(kx−ωt) h = hae , haaM∗ = const, (5) modes created by primordial GWs in the polarized Cos- mic Microwave Background (CMB) sky can in principle where ha is the wave’s amplitude. This result implies constrain both quantities. The Planck’s CMB analysis that ha is sensitive only to the ratio of the effective Planck [32] produced, for some classes of functional parametriza- mass and scale factors at emission and observation (see also the discussion of a phase shift in Ref. [33]). tion of αM (t), errors around 0.05 at 95% confidence level In GR, the GW amplitude can be related to the lumi- for the present value of αM . These errors, however, de- pend on the assumption of a standard cosmological model nosity distance dL of the source from the observer – the and, in particular, of a ΛCDM background. Therefore, potential evolution of M∗ is the only modification here, these are tests of structure formation for particular mod- so that   ified gravity models, rather than direct tests of generic M∗,em ha = × hs , (6) modifications of gravity. M∗,obs 3 where hs is the standard amplitude expression that, for would be yet another test of theories beyond GR in this merging binaries, can be approximated as (see e.g. equa- context, however this does not come free of subtleties. tion (4.189) of [34]) Nonetheless, in gravity modifed at low curvatures, used as a model for late-time cosmology, the coupling of the  5/3  2/3 4 GMc πfGW fifth-force scalar is more suppressed the more compact hs = 2 , (7) dL c c the object, with black holes completely decoupled [36]. We thus expect no change in the waveform for black with Mc the so-called and fGW the GW fre- holes and only at most small corrections when dealing quency measured by the observer. with neutron-star mergers. If gravity were in addition The observable signal in the two polarizations h+, h× modified at high curvatures, an accurate prediction of is finally obtained by multiplying h by sinusoidal oscilla- the waveform would require a good accounting of the tions and by the factors cos i (for the × polarization) and various astrophysical factors such as the environment of 2 the (1+cos i)/2 (for the + polarization) that depend on the merger, the respective equations of state etc., which the inclination i of the binary orbit with respect to the could potentially be degenerate with modified gravity ef- line of sight. fects (see e.g [37] for a detailed discussion). In this regard, As a concrete example, in the rest of this paper we modified gravity effects could also affect the predictions assume for simplicity that αM is constant in the region for quantities such as e.g. the chirp mass. 2 of observability (i.e. for z ≤ 2 roughly). Then we have On the other hand, the merging stars are sensitive to that, the local value of M∗, which is a function of the scalar-

αM field configuration at the site of the merger. This is M∗ ∼ a 2 , (8) not only determined by the cosmological value, but is affected by the profile of the mass in the galaxy and its and surrounding environment. Screening mechanisms such as − αM chameleon or Vainshtein, make this effect of mass distri- ha = (1 + z) 2 × hs. (9) bution on the scalar large, rather than a small perturba- In Fig.1, we show a solution to (3) for a ΛCDM back- tion. When the emitting galaxies are screened, the local ground and an illustrative choice of constant αM , com- M∗ will still drift together with cosmology, although with paring it to the evolution of the amplitude as given by a smaller amplitude and there will now be a scatter of (5). Assuming the constancy of αM all the way to recom- values from different galaxies located at the same red- bination is excluded by e.g. the Planck analysis [32] at shift. This would result in a scatter in the effective GW least when some extra assumptions about the model are luminosity distance, reducing the sensitivity. This issue taken, but it provides a clear illustration of the physics. deserves a more thorough investigation, and the develop- Equations (9) and (7) allow us to define an effective ment of numerical simulations of mergers within modified GW “luminosity distance” for the constant αM case [33] gravity.

αM dGW = (1 + z) 2 dL. (10) OBSERVING αM Since the chirp mass, the inclination angle i, and the fre- quency f can be measured independently of each other GW Given the result of the previous section, we now discuss from the GW signal (see e.g. [35]), GW experiments can how it can be applied to measure the time variation of measure directly the distance d . In the next Section GW the Planck mass through the parameter α . Taking the we discuss possible future observations of d and the M GW log of Eq. (10) we have that, constraints they can impose on αM . An important issue concerns the actual GW waveform 2 (log d − log d ) = α . (11) at the moment of emission. If gravity were modified, it log(1 + z) GW L M is possible that the waveform would show qualitative dif- ferences compared to the corresponding one within GR Then, assuming all variables to be Gaussian-distributed as a result of radiation into a new degree of freedom (e.g. and statistically uncorrelated, the error on αM can be the fifth-force scalar), or a different deformation of the estimated as merging stars as a result of extra forces. Indeed, this s  2 2  2 σGW σL σαM = 2 + 2 , (12) log(1 + z) dGW dL 2 The assumption α = const is just for convenience; as Eq. M 2 2 (6) shows, the GW amplitude depends on the ratio of the Planck where σGW, σL are the variances of dGW and dL, respec- mass at emission and observation independently of the functional tively. We neglect the error on z because it is likely to form of αM . be well below the errors on the other quantities. 4

LIGO will be likely to obtain optical counterparts only at very low . For instance, the event reported in 2.00 ΑM=0 [1] occurred at z ≈ 0.01. It is then clear from Eq. (12) 1.00 that the error on α is ∼ 200 times the combined relative ΑM=0.5 M 0.50 error of the distances and therefore very weak (see Ref. 3 [28]). We therefore focus on future prospects with the 0.20 LISA satellite.4 h 0.10 LISA will measure dGW up to very high redshifts. In particular, Ref. [38] has shown that massive black-hole 0.05 binaries (MBHB) can provide in 5 years’ operation 30- 50 identifiable optical counterparts distributed between 0.02 z = 1 and z = 8, 5-15 of which within z ≤ 2. From Ref. [38] (see e.g. their Fig. 1), an error of 5% on -4 -3 -2 -1 0 ln a dGW(z ≈ 1.5) seems feasible. More stringent limits, down to 1% or better, are quoted in [35]. FIG. 1. Numerical evolution of the GW amplitude |h| for il- The luminosity distance dL might be measurable with lustrative choice of initial condition, parameter αM = 0, 0.5 supernovae Ia and baryon acoustic oscillations (BAO) up and k/H0 = 100 as a function of loge a. The dashed lines to say z = 2. Although dL could be measured in the fu- represent the analytical result for the amplitude a−1−αM /2. ture to higher redshifts, for instance with real-time cos- The phase shift is mainly driven by the different value of µ in mology [39], we restrict here ourselves to measurements Eq. (4) and therefore a different horizon time, but is not rele- around z ≈ 1.5. With N supernovae Ia at this redshift, vant for GW wavelengths sourced by mergers from inspirals. each with magnitude error ∆m, one obtains a relative error on dL roughly equal to at the level of 0.1%. Below z = 0.1, the log(1 + z) de- ∆dL log 10 ∆m nominator in Eq. (12) weakens the constraints below the ≈ √ . (13) dL 5 N threshold of interest. In conclusion, we have discussed that GWs from op- Ref. [40] analyzed recently 15 supernovae Ia at z > 1, tically identified sources can probe not only the propa- nine of which within 1.5 < z < 2.3, with ∆m ≈ 0.2. gation speed of GWs (through the effective parameter Taking indicatively 10 supernovae Ia at z ≈ 1.5, we find αT ), but also a possible non-minimal, conformal cou- a relative error of 3% for dL(z ≈ 1.5), which is indeed pling between the scalar sector and curvature through close to the values reported in [40]. BAO measurements the time variation of the Planck mass, the latter being with SKA and Euclid should reach 1% accuracy at this parametrised by the parameter αM . This result holds redshift (see e.g. [41], Fig. 3). without any prior assumption about the particular cos- Taking now the conservative estimates of 5% and 3% mological model, and to a precision comparable or (with for dGW , dL, respectively, we obtain the BBO) superior to current tests. Acknowledgements. During the final stages of prepa- σ ≈ 0.13. (14) αM ration of this manuscript, ref. [43] appeared with sim- ilar conclusions to this work. We thank Michele This improves to 0.03 with the optimistic estimates (1% Maggiore and Filippo Vernizzi for useful discussions. for both distances). The work of L.A. is supported by the DFG through These estimates can be significantly improved in two TRR33 “The Dark Universe”. M.K. acknowledges ways, at higher and at lower redshifts. At redshifts higher funding by the Swiss National Science Foundation. than 2, several GWs will be detected by LISA, but at the I.S. and I.D.S. are supported by European Struc- moment we lack reliable distance indicators for d . At L tural and Investment Funds and Czech Ministry of redshifts between 0.1 and 1, conversely, we have very Education, Youth and Sports (Project CoGraDS – good distance indicators, but a dearth of strong GW CZ.02.1.01/0.0/0.0/15_003/0000437). sources (in particular MBHB) detectable by LISA. The proposed Big Bang Observer (BBO) [42] would be able to improve in this range, reaching a sensitivity on dGW

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