Modified Gravitational Wave Propagation and the Binary Neutron

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 1Département de Physique Théoriqueand Center for Astroparticle Physics, Universitéde Genève,24 quai Ansermet, CH–1211 Genève4, 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 luminosity 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- | − | 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. frame masses that would be inferred in GR, mGR,i, by Since the effect of modified GW propagation increases with redshift (at least until the ratio in eq. (3) saturates m(det),i 1 + zGR gw em mtrue,i = = mGR,i , (5) to its large z limit dL /dL Ξ0), third generation (3G) 1 + ztrue 1 + ztrue ground based GW detectors' such the Einstein Telescope (ET) [28, 29] and Cosmic Explorer (CE) [30], or the space where mGR,i = m(det),i/(1 + zGR). The same multi- interferometer LISA [31], are particularly well suited to plicative bias factor will appear in any other combi- study it, and several forecasts have been made on the nation (with dimensions of mass) of the source-frame accuracy that future observations can reach on Ξ0, using masses of the component stars, such as the total source- different techniques [8, 32–38]. frame mass mtot = m1 + m2, or the source-frame chirp 3/5 1/5 The aim of this paper is to discuss another technique mass Mc = (m1m2) /mtot . Fig.2 shows the ratio for bounding, or observing, modified GW propagation, MGR/Mtrue, as a function of ztrue, for any such mass based on the use of the BNS mass function. The idea was scale. We see that, at redshifts accessible to ET and CE, proposed in [39, 40] in the context of the determination and for values of Ξ0 consistent with current limits, the 3 effect can be very large. For instance, setting Ξ0 = 1.8, Ξ0 = 0.5 for a NS with mtrue = 1.35M at ztrue = 1 (that, with Ξ0 = 1.8 this value of Ξ0, corresponds to zGR 1.45), the mass ' Ξ0 = 5.5 incorrectly inferred from GR would be mGR 1.10M ; ' Ξ = 1 at ztrue = 2 (zGR 3.10) for the same system in GR one 0 ' would infer mGR 0.99M ; and, for a BNS with the ' 5 same mass at ztrue = 5 (zGR 8.20), which could still ' be accessible to CE, one would find mGR 0.88M . Furthermore, exactly the same factor affects' the two 4 component stars (which is not the case in general for astrophysical effects), so a BNS with (1.35 + 1.35)M 3 would appear as a (1.10 + 1.10)M system for ztrue = 1, zGR as a (0.99 + 0.99)M system for ztrue = 2, and as a 2 (0.88 + 0.88)M system at ztrue = 5. Compared to the narrowness of the neutron star (NS) mass distribution, this is a huge effect. The total mass of the BNSs found 1 with electromagnetic observations can be described by a Gaussian distribution with mean 2.66M and standard 0 deviation 0.13M [42], or by a flat distribution between a minimum and a maximum mass, with a similar width 0 1 2 3 4 5 GR (somewhat broader limits are obtained from an analysis mtot (M ) using only the NSs in BNS or in BH-NS systems de- tected by GW observations [43], although this sample, of six NSs, is very limited). FIG. 3. The evolution in redshift of the distribution of the BNS (source-frame) total mass, inferred using GR, for differ- These estimates show that even a single BNS at large z ent values of Ξ0. We assume a Gaussian distribution. would have a significant constraining power on Ξ0. Still, if one would find just a single system that, interpreted within GR, corresponds to, say, a (1.0 + 1.0)M binary not change with redshift (black dotted line). In the pres- at zGR 3.1, as in one of the examples above, one would remain' in doubt on whether this is a binary made of ence of modified GW propagation, the masses wrongly exotic compact objects, such as primordial black holes, inferred using GR are narrowly distributed around com- or a signal of modified GW propagation. The power of pletely different mean values. the method, however, is that the same effect will affect Actually, it is convenient to use the chirp mass, rather all BNS system, by a factor that depends only on z. If than the total mass, because, in GW observations, the Nature is described by a modified gravity theory with chirp mass is measured much more accurately than the a large deviation from GR such as, say, Ξ0 = 1.8, as individual component masses or the total mass. The in the examples above, at large redshifts ET and CE corresponding mass distribution can again be fitted by will not find a single BNS whose component masses, in- a Gaussian distribution or by a flat distribution be- terpreting the data within GR, will be near the typical tween minimum and maximum values, where, for BNS mass of 1.35M . When interpreted within GR, all BNS that merge within a Hubble time, as reported in Ta- z ble 1 of [42], one could take Mc,min 1.10M and with true = 1 would appear to have component masses ' . M z Mc,max 1.25M . The corresponding apparent evolu- around 1 10 ; all BNS at true = 2 would appear to ' have masses around 0.99M , and so on. The detection tion in redshift is shown in Fig.4. rate of BNS at ET and CE will be impressive, of order The above discussion is still idealized, because it ne- of 7 104 events per year already for a single detector glected the errors on the measurements. The relative ac- × such as ET [32, 44, 45] and, among these, within a GR curacy on the detector-frame chirp mass c = (1+z)Mc M interpretation, there would not be a single ‘normal’ neu- is of order ∆ c/ c 1/ c, where c is the num- tron star at large z, but rather a plethora of objects with ber of inspiralM cyclesM of∼ the signalN in theN detector band- puzzling masses. The situation is illustrated in Fig.3, width (see, e.g., eq. (7.187) of [46]). For a lower cut- GR where mtot denotes the total (source-frame) mass of the off of the detector near 3 Hz, as in the design of ET, 5 BNS inferred in GR, for different values of Ξ0. Here we and the chirp mass of a BNS, we have c 10 (us- have assumed that the distribution of the source-frame ing eq. (4.23) of [46]). The error on theN detector-frame' total mass of the binary is a Gaussian, with mean 2.66M chirp mass is therefore negligible. More important is and standard deviation 0.13M , as in ref. [42]. In the ab- the error on the redshift due to the observational error gw sence of astrophysical evolutionary effects (for which, cur- on dL , which affects the reconstruction of the source- rently, there is no observational information, but which frame chirp mass. From Mc = c/(1 + z) [where we M are not expected by any means to give effects compara- set (Mc = Mc,true, z = ztrue) but the same computation ble to those shown in the figure), the distribution would holds for (Mc = Mc,GR, z = zGR)], and the fact that the 4

error on the source-frame chirp mass, induced by the ob- Ξ0 = 0.5 gw < servational error on dL , is below 6% up to z 9 for a Ξ0 = 1.8 network ET+CE+CE (where ET contributes to∼ BNS de- Ξ0 = 5.5 tections only up to z 3). Similarly, we find that it is Ξ0 = 1 below (5 6)% up to z'< 3 for ET alone. This is smaller than the− intrinsic relative∼ width of the BNS mass distri- 5 bution, ∆m/m 0.1 obtained from electromagnetic observations of BNS,∼ and therefore also of the correspond- 4 ing distribution of chirp masses. Therefore, the accuracy of the method appears to be limited mostly by the in- 3 trinsic width of the BNS mass distribution, rather than by observational errors on the reconstruction of the red- zGR shift. The error due to lensing from large scale structures 2 along the line of sight must also be taken into account. On linear scales, inhomogeneities induce a relative error 1 ∆dL/dL < 1% for all redshifts z < 5 [47] (see also Fig. 12 of [29]).∼ See [48] for a modeling of the effect of non-linear 0 scales. A full Bayesian analysis on mock data for different networks of 3G detectors, including observational errors 0.0 0.5 1.0 1.5 2.0 2.5 and selection effects, necessary to reliably quantify the GR Mc (M ) accuracy that can be obtained on Ξ0, will be presented in a separate work. Finally, another important signature of modified GW FIG. 4. The evolution in redshift of the distribution of the propagation will be given by how the BNS population BNS source-frame chirp mass, inferred using GR, for different is distributed in redshift (i.e., the absolute normaliza- values of Ξ0. We assume a flat distribution between Mc,min ' tion of the distributions, that in Fig.3 and4 have been 1.10M and Mc,max ' 1.25M . normalized to unity). Even if our prior information of the BNS merger rate is not as stringent as on the BNS mass function, still we expect that the rate will be de-

Ξ =0.5 0.06 0 scribed by a Madau-Dickinson form [49, 50] R(z) = Ξ0 =1.8 αz 1+z αz +βz Ξ0 =5.5 R0C0(1 + z) /[1 + ( ) ], where the normaliza- 0.05 1+zp Ξ0 = 1 tion constant C0(zp, αz, βz) ensures that R(0) = R0, and c 0.04 z is the peak of the star formation rate, which is known /M p c 0.03

M to be in the range zp (2 3). In a modiﬁed gravity ∆ ' − 0.02 theory, the diﬀerence between zGR and ztrue will lead to a

0.01 bias in the reconstruction of R(z). For instance we have

0.00 seen that, if Nature is described by a modified gravity 0 1 2 3 4 5 6 7 8 9 theory with our reference value Ξ0 = 1.8, and we rather z use GR to interpret the data, a BNS with ztrue = 2 would be wrongly interpreted as having a redshift zGR 3.10, FIG. 5. The relative error on the the source-frame chirp mass ' and ztrue = 3 corresponds to zGR 4.79. The peak of M d gw c due to the observational error on L , for different values the BNS merger distribution would' then appear to be at of Ξ0, for a network ET+CE+CE. redshifts larger than the peak of the star formation rate, leading to another puzzling result of the GR interpretation (that, for Ξ0 > 1, could not be explained in terms of error on c is negligible, it follows that M delay between formation and merger, since in this case gw gw one would find that the peak of the merger rate took ∆Mc dL ∆dL = gw gw . (6) place before the peak of the star formation rate). A joint Mc (1 + z)∂d /∂z d L L Bayesian inference on the BNS mass function and on the The expression in brackets goes from zero at z = 0 BNS rate parameters would therefore further strengthen to one at large z, with only mild dependence on Ξ0. the power of the method. gw gw For ∆dL /dL as a function of redshift we use the fit- In conclusion, for BNS at the large redshifts that will ting formulas provided in [32] [eq. (2.13) for ET, and be probed by third-generation detectors such as Einstein eq. (2.20) for a network ET+CE+CE], obtained from a Telescope and Cosmic Explorer, modified GW propaga- mock source catalog of BNS detections, averaging over tion has striking effects on the reconstruction of the BNS detector orientation, source inclination, and BNS mass mass function and of the BNS merger rate, that can pro- distribution. The results for ∆Mc/Mc at ET+CE+CE vide a clear and unambiguous signature of modifications is shown in Fig.5. We see that, on average, the relative of General Relativity on cosmological scales. 5

Acknowledgments. The work of the authors is sup- the SwissMap National Center for Competence in Re- ported by the Swiss National Science Foundation and by search.

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