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Gravitational-Wave Asteroseismology with Fundamental Modes from Compact Binary Inspirals ✉ ✉ ✉ Geraint Pratten 1,2 , Patricia Schmidt 1 & Tanja Hinderer3,4

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Gravitational-Wave Asteroseismology with Fundamental Modes from Compact Binary Inspirals ✉ ✉ ✉ Geraint Pratten 1,2 , Patricia Schmidt 1 & Tanja Hinderer3,4 ARTICLE https://doi.org/10.1038/s41467-020-15984-5 OPEN Gravitational-wave asteroseismology with fundamental modes from compact binary inspirals ✉ ✉ ✉ Geraint Pratten 1,2 , Patricia Schmidt 1 & Tanja Hinderer3,4 Gravitational waves (GWs) from binary neutron stars encode unique information about ultra- dense matter through characterisic signatures associated with a variety of phenomena including tidal effects during the inspiral. The main tidal signature depends predominantly on Λ 1234567890():,; the equation of state (EoS)-related tidal deformability parameter , but at late times is also characterised by the frequency of the star’s fundamental oscillation mode (f-mode). In General Relativity and for nuclear matter, Λ and the f-modes are related by universal relations which may not hold for alternative theories of gravity or exotic matter. Independently measuring Λ and the f-mode frequency enables tests of gravity and the nature of compact binaries. Here we present directly measured constraints on the f-mode frequencies of the companions of GW170817. We also show that future GW detector networks will measure f- mode frequencies to within tens of Hz, enabling precision GW asteroseismology with binary inspiral signals alone. 1 School of Physics and Astronomy and Institute for Gravitational Wave Astronomy, University of Birmingham, Edgbaston, Birmingham B15 9TT, United Kingdom. 2 Universitat de les Illes Balears, Crta. Valldemossa km 7.5, E-07122 Palma, Spain. 3 GRAPPA, Anton Pannekoek Institute for Astronomy and Institute of High-Energy Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands. 4 Delta Institute for Theoretical ✉ Physics, Science Park 904, 1090 GL Amsterdam, The Netherlands. email: [email protected]; [email protected]; [email protected] NATURE COMMUNICATIONS | (2020) 11:2553 | https://doi.org/10.1038/s41467-020-15984-5 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-15984-5 fi he rst detection of gravitational waves (GWs) from the Table 1 Marginalised 90% lower bound or 90% credible 1 Tbinary neutron star (NS) inspiral GW170817 provided a interval of the quadrupolar f-mode frequencies in distinctive avenue for probing ultra-dense matter and GW170817. fundamental interactions in extreme regimes, at the frontier of nuclear physics and astrophysics. This observation yielded the f f first constraints on the yet unknown equation of state (EoS) of NS Tidal phase model 2,1 [kHz] 2,2 [kHz] + f matter from the imprint of tidal interactions in the GW signal. (a) 6.5PN ad. 2 dyn. 1.47 1.57 + f Tidal signatures in GWs arise from the response of a matter (b) 7.5PN ad. 2 dyn. 1.43 1.59 object to the spacetime curvature sourced by its binary compa- (c) Combined 1.45 1.58 (d) 6.5PN ad. + f + f dyn. 1.40 1.49 nion. They crucially depend on the EoS and are predominantly 2 3 (e) 7.5PN ad. + f2 + f3 dyn. 1.37 1.47 characterised by the tidal deformability parameters Λℓ2, where ℓ = (f) Combined 1.39 1.48 2, 3 denotes the quadrupole and octupole respectively. As the (g) 6.5PN ad. + URs 1.36–2.83 1.42–3.08 binary evolves towards merger, additional dynamical tidal effects (h) 7.5PN ad. + URs 1.37–2.90 1.43–3.16 become important when variations in the tidal fields occurring on (i) Combined 1.38–2.86 1.43–3.12 ’ the orbital timescale approach a resonance with the star s internal (j) 6.5PN ad. + f2 dyn. + URs 1.42–2.88 1.48–3.17 oscillation modes, giving rise to new characteristic signatures in (k) 7.5PN ad. + f2 + f3 dyn. + URs 1.44–2.92 1.50–3.18 the GWs associated with specific modes. Among these modes, the (l) Combined 1.43–2.90 1.48–3.18 fundamental (fℓ-)modes have the strongest tidal coupling and We list results for all considered combinations of adiabatic and dynamical tide contributions lead to a f-mode frequency-dependent amplification of tidal sig- in the waveform model for both companions of GW170817 assuming a uniform prior on Ω2,A ∈ [0, 0.5]. For results which assume URs, we also provide the upper bounds. Fig. 1b natures in the GW signal that starts to accumulate long before the f 3–5 displays cases (f), (i) and (l). Cases where 3 is included also include the 7PN adiabatic octupolar resonance . In General Relativity (GR), and for a range of effects52. The results in this table demonstrate that the data disfavours hyper-excited dynamical proposed nuclear EoSs for NSs, the fℓ-mode frequencies are tides independent of PN systematics. empirically found to be related to Λℓ through approximate uni- versal relations (URs)6,7. However, not imposing such URs opens up the possibility of using GW observations to perform para- phase developed in ref. 5 (see Methods, Waveform Model). We meterised tests of GR, to understand properties of matter at perform coherent Bayesian inference using the nested sampling supranuclear densities such as possible phase transitions from algorithm implemented in LALINFERENCE17,18, as summarised hadronic to quark matter8, and to test for the existence of exotic in Methods, Parameter Estimation. We adopt the narrow spin compact objects such as boson stars or gravastars9. priors of16, limiting the dimensionless neutron star spins to be In the past, prospects for f-modes asteroseismology with GW ≤0.05, which is consistent with the observed population of binary observations were focused on the post-merger signal10–12. With neutron stars that will merge within a Hubble time16. For the f- the observation of the inspiral of GW170817, recent work has mode frequencies, we adopt an agnostic uniform prior. The substituted the constraints on the tidal deformability derived choice of priors is further discussed in Methods, Choice of Priors from GW170817 into empirical URs to indirectly make predic- and Parameterisation. We further demonstrate the robustness of tions for bounds on other NS parameters, such as the moment of our results under systematic uncertainties from incomplete inertia or f-mode frequencies. In ref. 13 bounds on f-modes were knowledge of PN corrections to adiabatic tidal effects by per- obtained by extrapolating a point estimate of the tidal deform- forming the analysis using different combinations of adiabatic ability for GW170817 assuming URs. and dynamical tide contributions, as summarised in Table 1. The Here we show direct constraints on f-modes from the GW data results presented in the text below will be taken from datasets (f) alone, without the assumption of URs, by using a model for the and (l) unless stated otherwise. GW signal that explicitly depends both on the tidal deformability Figure 1 summarises our findings for the larger-mass (labelled = and the f-mode frequency5. In this UR-independent analysis, we here as object A 1 with mass m1) companion of GW170817; we find that the data from GW170817 disfavours anomalously small obtain similar results for the lower-mass (m2) companion values of fℓ. Our analysis enables tests of GR and exotic compact resulting in the same conclusions as illustrated in Supplementary objects that would not be possible using point estimates and Fig. 1. Figure 1a shows the joint posterior probability distribution = ω π empirical URs. Our work thus demonstrates the concrete example function (PDF) for the quadrupolar f-mode frequency f2 /(2 ) Λ of f-mode astereoseismology with GWs from binary inspirals, and corresponding tidal deformability 2. We recall that in the opening up this highly anticipated method for future detailed subscripts {2, 1} the first label denotes the multipolar index ℓ, probes of compact object interiors. the second specifies the object label A, with A = 1 taken to be the larger mass companion here. The black solid curve indicates the 90% credible region (CR) of our analysis with independent Results parameters, i.e., without imposing the UR. We find that the data f-mode constraints from GW170817. In order to constrain disfavours large tidal deformabilities and small f-mode frequen- fundamental oscillation mode frequencies in GW170817, we re- cies. Complementary to the joint two-dimensional posterior, 14,15 analyse the publicly available strain data treating the tidal Fig. 1b shows the marginalised one-dimensional PDF of f2.Itis Λ deformabilities ℓ,A and the dimensionless angular f-mode fre- evident that without imposing the UR for NSs on the dynamical 3 quencies Ωℓ,A ≡ GmAωℓ,A/c of the A-th component object as tides terms (green curve), we can rule out anomalously small independent parameters without imposing the aforementioned values of the f-mode frequency and place a lower bound on f2. empirical URs or any requirement that the two compact objects This can be understood from the scaling behaviour of the fℓ- À25 obey the same EoSs. We then repeat the analysis imposing URs dependent phase contribution which is proportional to Λ‘f ‘ , that relate the f-mode frequency and the tidal deformability, implying that small values of fℓ result in hyper-excited dynamical Ω = ∑ ξi ξ ¼ ðΛ Þ fi ℓ,A iai , where ln ‘;A and ai are numerical coef - tides that are inconsistent with the data. This is reflected in the cients6. For our analyses, we use the efficient post-Newtonian shape of the posterior (green curve), where we see a dramatic (PN) model of the frequency-domain waveform ‘TaylorF2” that drop in posterior support as f → 0. Conversely, increasing the includes adiabatic tidal effects (see references in ref. 16) aug- fundamental frequency leads to a suppression of dynamical tidal mented with the dynamical f-mode tidal contribution to the effects and the waveform becomes indistinguishable from the 2 NATURE COMMUNICATIONS | (2020) 11:2553 | https://doi.org/10.1038/s41467-020-15984-5 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-15984-5 ARTICLE a 3500 b NS APR4 Combined : (f) NS MPA1 Combined + UR : (l) 3000 NS H4 0.04 BS (1.4, 1197) Adiabatic + UR : (i) BS (1.24, 1047) Uniform prior 2500 BS (1.06, 449) 0.03 2000 ) 2,1 f [Hz] ( p 2,1 f 1500 0.02 1000 0.01 500 0 0.00 0 200 400 600 800 1000 1200 1400 0 1000 2000 3000 4000 5000 Λ 2,1 f2,1 [Hz] Fig.
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