Modified gravitational wave propagation and the binary neutron star mass function Andreas Finke,1 Stefano Foffa,1 Francesco Iacovelli,1 Michele Maggiore,1 and Michele Mancarella1 1D´epartement de Physique Th´eoriqueand Center for Astroparticle Physics, Universit´ede Gen`eve,24 quai Ansermet, CH{1211 Gen`eve4, Switzerland Modified gravitational wave (GW) propagation is a generic phenomenon in modified gravity. It affects the reconstruction of the redshift of coalescing binaries from the luminosity distance measured by GW detectors, and therefore the reconstruction of the actual masses of the component compact stars from the observed (`detector-frame') masses. We show that, thanks to the narrowness of the mass distribution of binary neutron stars, this effect can provide a clear signature of modified gravity, particularly for the redshifts explored by third generation GW detectors such as Einstein Telescope and Cosmic Explorer. In recent years, modified GW propagation has come by eq. (1), in theories with extra dimensions, where it is to attention as one of the most promising ways of test- rather due to the loss of gravitons to the bulk [17, 18]. ing deviations from General Relativity (GR) on cosmo- The modified friction term in eq. (1) changes the evo- logical scales. The effect is encoded in the propagation lution of the GW amplitude in the propagation across equation of gravitational waves (GWs) across cosmologi- cosmological distances. Since, in GR, the amplitude of a cal distances which, in modified gravity theories, can take coalescing binary is proportional to 1=dL, where dL is the the form [1{8]1 luminosity distance, this introduces a bias in the lumi- nosity distance inferred from GW observations. In par- ~00 ~0 2 2~ hA + 2 [1 δ(η)]hA + c k hA = 0 : (1) ticular, if δ(η) < 0, the damping term is stronger and, H − after propagation from the source to the detector, the The difference with respect to GR is given by a non- GW has a smaller amplitude. If interpreted within GR, vanishing function δ(η). Several other modifications with it would therefore appear to come from a distance larger respect to GR are possible in the GW sector. The than its actual distance (and vice versa for δ(η) > 0). It most immediate options are a deviation of the speed of is then useful to introduce a distinction between the stan- GWs from the speed of light, or a graviton mass. Both dard luminosity distance, that, in this context, is called would rather modify the c2k2 term in eq. (1), but are the `electromagnetic luminosity distance' and denoted by em now very significantly constrained: a deviation of the dL , and the luminosity distance extracted from the ob- speed of GWs from the speed of light is excluded at servation of the GWs of a compact binary coalescence, −15 the level cgw c =c < (10 ) by the observation of that is called the `GW luminosity distance' [5] and de- j − j O gw GW170817 [9], while limits on the graviton mass are noted by dL . The two quantities are related by [5,7] in the range (10−32 10−22) eV, depending on the Z z 0 probes used [10O]; several− other modifications, in general gw em dz 0 dL (z) = dL (z) exp 0 δ(z ) ; (2) related to rather specific classes of modified gravity the- − 0 1 + z ories, have been tested or proposed, such as extra polar- where δ(z) δ[η(z)]. A useful parametrization of this ef- izations [11], Lorentz-violating dispersion relations [12], fect, which≡ catches the redshift dependence predicted by parity-violating effects [13], or scale dependent modifica- almost all explicit models in terms of just two parameters tions of the speed of GWs [14]. (Ξ ; n), is obtained writing [7], Modified GW propagation, in the form described by 0 gw eq. (1), was first found in some explicit scalar-tensor theo- dL (z) 1 Ξ0 em = Ξ0 + − n ; (3) arXiv:2108.04065v1 [gr-qc] 9 Aug 2021 ries of the Horndeski class [1{4] and, in [5,7], in non-local dL (z) (1 + z) infrared modifications of gravity (i.e. in theories where gw em the underlying classical action is still GR, but non-local which interpolates between d =dL = 1 as z 0 and L ! terms, relevant in the infrared, are assumed to be gener- an asymptotic value Ξ0 at large z, with a power-law be- ated by non-perturbative effects in the quantum effective havior in a = 1=(1 + z) fixed by n. GR is recovered when action [15]; see [16] for recent review). However, it has Ξ0 = 1 (for all n). The study of explicit modified gravity been understood that the phenomenon is completely gen- models shows that Ξ0 can be significantly different from eral and appears in all best studied modified gravity the- 1. In particular, in non-local gravity it can be as large ories [8]. It also appears, in a different form not described as 1:80 [16, 19], corresponding to a 80% deviation from GR, despite the fact that this model complies with ex- isting observational bounds, that force deviations from GR and from ΛCDM in the background evolution and 1 ~ in the scalar perturbation sector to be at most of a few We use standard notation: hA(η; k) is the Fourier transform of the GW perturbation, h0 = @h=@η where η is conformal time, percent [20, 21]. Thus, the newly opened window of GWs a(η) is the scale factor, H = a0=a, and A = +; × labels the two could give us the best opportunities for testing modified polarizations. gravity and dark energy. 2 Ξ0 = 0.5 Ξ0 = 0.5 17.5 1.6 Ξ0 = 1.8 Ξ0 = 1.8 Ξ0 = 5.5 Ξ0 = 5.5 15.0 1.4 Ξ0 = 1 12.5 1.2 true 10.0 1.0 /M true z GR 7.5 0.8 M 5.0 0.6 0.4 2.5 0.2 0.0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 zGR ztrue FIG. 1. The redshift ztrue of a source, as a function of the FIG. 2. The ratio of the mass MGR inferred in GR over the value zGR that would be incorrectly inferred using GR if Na- true mass Mtrue, as a function of the redshift ztrue. Here M ture is described by a modified gravity theory with Ξ0 =6 1, represent any mass scale made with the (source-frame) masses for different values of Ξ0 [we set for definiteness n = 1:9 in of the component stars, such as individual component masses, eq. (3), which is a value suggested by non-local gravity.] total mass, or chirp mass. Contrary to quantities such as the speed of GWs or of H0 within ΛCDM. We will see that it can become par- the graviton mass, the limits on the parameter Ξ0 (the ticularly powerful when applied to modified GW propa- main parameter that describes modified GW propaga- gation at 3G detectors, thanks to the fact that modified tion; the power n in eq. (3) only determines the precise GW propagation increases with distance, and ET and form of the interpolation between the asymptotic val- CE can detect BNS up to large redshifts, z 2 3 for ' − ues) are still quite broad. Using the binary neutron star ET, and even z 10 for CE [41]. ∼ (BNS) GW170817, with the redshift determined from the The starting point of our analysis is the fact that GW electromagnetic counterpart, only gives bounds of order detectors measure the GW luminosity distance of the gw Ξ0 < 14 (68% c.l., as all the other limits that we will give source, dL , which is different from the actual electro- em below)∼ [7] (see also [4, 22]). This is because the redshift magnetic luminosity distance dL if, in Nature, Ξ0 = 1. gw em 6 of GW170817 is very small, z 0:01, and d (z)=d (z) The redshift zGR of the source that would be inferred ' L L gw goes to one as z 0, for all Ξ0. A more significant limit, from the measured dL assuming GR and ΛCDM (with ! a given value of H0 and ΩM , that we keep for definiteness +3:2 Ξ0 = 2:1−1:2 ; (4) the same in GR and in the modified gravity theory under consideration), would therefore differ from the true value has been obtained in [23], using binary black hole (BBH) ztrue. The effect is shown in Fig.1, for a range of values coalescences without electromagnetic counterpart (`dark of Ξ0 consistent with eq. (4). We see that the effect can sirens') from the O1, O2 and O3a runs of the LIGO/Virgo become very significant at large redshifts. Collaboration (LVC) and correlating them with the In turn, this affects the reconstruction of the actual GLADE galaxy catalog [24]. An even more stringent masses mi (i = 1; 2) of the component stars (`source- bound is obtained under the tentative identification of frame' masses, as they are called in this context), from the flare ZTF19abanrhr as the electromagnetic counter- the `detector-frame' masses m(det);i (1 + z)mi, that part of the BBH coalescence GW190521, in which case are the quantities directly obtained≡ from the GW ob- +0:9 one gets Ξ0 = 1:8−0:6 [23] (see also [25]). However, this servations. If Nature is described by a modified gravity identification currently is not secure. A comparable limit theory with Ξ0 = 1, the true values of the source-frame 6 have been obtained in [26] using the BBH mass function, masses, mtrue;i, are related to the values of the source- following an idea originally proposed in [27] to infer H0.