Constraining neutron star tidal Love numbers with gravitational wave detectors
Ground-based gravitational wave detectors may be able to constrain the nuclear equation of state using the early, low frequency portion of the signal of detected neutron star - neutron star inspirals. In this early adiabatic regime, the influence of a neutron star’s internal structure on the phase of the waveform depends only on a single parameter of the star related to its tidal Love number, namely the ratio of the induced quadrupole moment to the perturbing tidal gravitational field. We analyze the information obtainable from gravitational wave frequencies smaller than a cutoff frequency of , where corrections to the internal-structure signal are less than . For an inspiral of two non-spinning neutron stars at a distance of Megaparsecs, LIGO II detectors will be able to constrain to with confidence. Fully relativistic stellar models show that the corresponding constraint on radius for neutron stars would be for a polytrope with equation of state .
Background and motivation: Coalescing binary neutron stars are one of the most important sources for gravitational wave (GW) detectors Cutler:2002me . LIGO I observations have established upper limits on the event rate Abbott:2007xi , and at design sensitivity LIGO II is expected to detect inspirals at a rate of day 2004ApJ…601…L179 .
One of the key scientific goals of detecting neutron star (NS) binaries is to obtain information about the nuclear equation of state (EoS), which is at present fairly unconstrained in the relevant density range 2000ARNPS..50..481H . The conventional view has been that for most of the inspiral, finite-size effects have a negligible influence on the GW signal, and that only during the last several orbits and merger at GW frequencies Hz can the effect of the internal structure be seen.
There have been many investigations of how well the EoS can be constrained using these last several orbits and merger, including constraints from the GW energy spectrum 2002PhRvL..89w1102F , and, for black hole/NS inspirals, from the NS tidal disruption signal 2000PhRvL..84.3519V . Several numerical simulations have studied the dependence of the GW spectrum on the radius 2003PhR…376…41B . However, there are a number of difficulties associated with trying to extract equation of state information from this late time regime: (i) The highly complex behavior requires solving the full nonlinear equations of general relativity together with relativistic hydrodynamics. (ii) The signal depends on unknown quantities such as the spins of the stars. (iii) The signals from the hydrodynamic merger (at frequencies 1000 Hz) are outside of LIGO’s most sensitive band.
The purpose of this paper is to demonstrate the potential feasibility of instead obtaining EoS information from the early, low frequency part of the signal. Here, the influence of tidal effects is a small correction to the waveform’s phase, but it is very clean and depends only on one parameter of the NS – its Love number Mora:2003wt .
Tidal interactions in compact binaries: The influence of tidal interactions on the waveform’s phase has been studied previously using various approaches 1995MNRAS.275..301K ; 1993ApJ…406L..63L ; 1992ApJ…398..234K ; 1998PhRvD..58h4012T ; 2002PhRvD..65j4021P ; Mora:2003wt . We extend those studies by (i) computing the effect of the tidal interactions for fully relativistic neutron stars, i.e. to all orders in the strength of internal gravity in each star, (ii) computing the phase shift analytically without the assumption that the mode frequency is much larger that the orbital frequency, and (iii) performing a computation of how accurately the Love number can be measured.
The basic physical effect is the following: the fundamental f-modes of the star can be treated as forced, damped harmonic oscillators driven by the tidal field of the companion at frequencies below their resonant frequencies. Assuming circular orbits they obey equations of motion of the form Dong94
where is the mode amplitude, the mode frequency, a damping constant, is the mode azimuthal quantum number, is the orbital phase of the binary, and is a slowly varying amplitude. The orbital frequency and evolve on the radiation reaction timescale which is much longer than . In this limit the oscillator evolves adiabatically, always tracking the minimum of its time-dependent potential. The energy absorbed by the oscillator up to time is
The second term here describes a cumulative, dissipative effect which dominates over the first term for tidal interactions of main sequence stars. For NS-NS binaries, however, this term is unimportant due to the small viscosity 1992ApJ…398..234K , and the first, instantaneous term dominates.
The instantaneous effect is somewhat larger than often estimated for several reasons: (i) The GWs from the time varying stellar quadrupole are phase coherent with the orbital GWs, and thus there is a contribution to the energy flux that is linear in the mode amplitude. This affects the rate of inspiral and gives a correction of the same order as the energy absorbed by the mode 1993ApJ…406L..63L . (ii) Some papers 1992ApJ…398..234K ; 1995MNRAS.275..301K ; 1998PhRvD..58h4012T compute the orbital phase error as a function of orbital radius . This is insufficient as one has to express it in the end as a function of the observable frequency, and there is a correction to the radius-frequency relation which comes in at the same order. (iii) The effect scales as the fifth power of neutron star radius , and most previous estimates took . Larger NS models with e.g. give an effect that is larger by a factor of .
Tidal Love number: Consider a static, spherically symmetric star of mass placed in a time-independent external quadrupolar tidal field The star will develop in response a quadrupole moment In the star’s local asymptotic rest frame (asymptotically mass centered Cartesian coordinates) at large the metric coefficient is given by (in units with ) Thorne:1998kt :
where this expansion defines the traceless tensors and To linear order, the induced quadrupole will be of the form
Here is a constant which we will call the tidal Love number (although that name is usually reserved for the dimensionless quantity ). The relation (4) between and defines the Love number for both Newtonian and relativistic stars. For a Newtonian star, is the Newtonian potential, and is related to the density perturbation by .
We have calculated the Love numbers for a variety of fully relativistic NS models with a polytropic pressure-density relation . Most realistic EoS’s resemble a polytrope with effective index in the range 2001ApJ…550..426L . The equilibrium stellar model is obtained by numerical integration of the Tolman-Oppenheimer-Volkhov equations. We calculate the linear static perturbations to the Schwarzschild spacetime following the method of 1967ApJ…149..591T . The perturbed Einstein equations can be combined into a second order differential equation for the perturbation to . Matching the exterior solution and its derivative to the asymptotic expansion (3) gives the Love number. For our results agree well with the Newtonian results of Refs. 1995MNRAS.275..301K ; 1955MNRAS.115..101B . Figure 1 shows the range of Love numbers for , corresponding to the surface redshift that has been measured for EXO0748-676 2002Natur.420…51C . Details of this computation will be published elsewhere.
Effect on gravitational wave signal: Consider a binary with masses , and Love numbers , . For simplicity, we compute only the excitation of star 1; the signals from the two stars simply add in the phase. Let , and be the frequency, the contribution to and the contribution to of modes of the star with and with radial nodes, so that and . Writing the relative displacement as , the action for the system is
Here and are the total and reduced masses, and is the tidal field. This action is valid to leading order in the orbital potential but to all orders in the internal potentials of the NSs, except that it neglects GW dissipation, because and are defined in the star’s local asymptotic rest frame TH .
Using the action (5), adding the leading order, Burke-Thorne GW dissipation terms, and defining the total quadrupole with , gives the equations of motion
By repeatedly differentiating and eliminating second order time derivative terms using the conservative parts of Eqs. (6), we can express in terms of , , and and obtain a second order set of equations; this casts Eqs. (6) into a numerically integrable form.
When GW damping is neglected, there exist equilibrium solutions with , for which is static in the rotating frame. Working to leading order in , we have , , , , where
and . Substituting these solutions back into the action (5), and into the quadrupole formula for the GW damping, provides an effective description of the orbital dynamics for quasicircular inspirals in the adiabatic limit. We obtain for the orbital radius, energy and energy time derivative
where , , , and . Using the formula for the phase of the Fourier transform of the GW signal at GW frequency Tichy now gives for the tidal phase correction
Here , is an arbitrary constant related to the initial time and phase of the waveform, and . In the limit assumed in most previous analyses 1995MNRAS.275..301K ; 1992ApJ…398..234K ; 1998PhRvD..58h4012T ; Mora:2003wt , we get
which depends on internal structure only through and . Here we have added the contribution from star 2. The phase (10) is formally of post-5-Newtonian (P5N) order, but it is larger than the point-particle P5N terms (which are currently unknown) by .
Accuracy of Model: We will analyze the information contained in the portion of the signal before . This frequency was chosen to be at least smaller than the frequency of the innermost stable circular orbit 1996PhRvD..54.3958L for a conservatively large polytropic NS model with , , and km. We now argue that in this frequency band, the simple model (10) of the phase correction is sufficiently accurate for our purposes.
We consider six types of corrections to (10). For each correction, we estimate its numerical value at the frequency Hz for a binary of two identical , , stars: (i) Corrections due to modes with which are excited by higher order tidal tensors . The correction to , computed using the above methods in the low frequency limit, is smaller than the contribution by a factor of , where , are apsidal constants. For Newtonian polytropes we have , Mora:2003wt , and the ratio is . (ii) To assess the accuracy of the limit underlying (10) we simplify the model (5) by taking
so that is independent of . This simplification does not affect (10) and increases the size of the finite frequency corrections in (Constraining neutron star tidal Love numbers with gravitational wave detectors) since 111Buoyancy forces and associated -modes for which have a negligible influence on the waveform’s phaseDong94 .. This will yield an upper bound on the size of the corrections. (Also the modes contribute typically less than of the Love number 1995MNRAS.275..301K .) Figure 2 shows the phase correction computed numerically from Eqs. (6), and the approximations (Constraining neutron star tidal Love numbers with gravitational wave detectors) and (10) in the limit (11). We see that the adiabatic approximation (Constraining neutron star tidal Love numbers with gravitational wave detectors) is extremely accurate, to better than , and so the dominant error is the difference between (Constraining neutron star tidal Love numbers with gravitational wave detectors) and (10). The fractional correction to (10) is , where , neglecting unobservable terms of the form . This ratio is for Hz and for Hz as is the case for -mode frequencies for most NS models 2002PhRvD..65j4021P . (iii) We have linearized in ; the corresponding fractional corrections scale as at 400 Hz. (iv) The leading nonlinear hydrodynamic corrections can be computed by adding a term to the Lagrangian (5), where is a constant. This corrects the phase shift (10) by a factor , where we have used the models of Ref. Dong1 to estimate . (v) Fractional corrections to the tidal signal due to spin scale as , where is the spin frequency and the maximum allowed spin frequency. These can be neglected as Hz for most models and is expected to be much smaller than this. (vi) Post-1-Newtonian corrections to the tidal signal (10) will be of order . However these corrections will depend only on when , and can easily be computed and included in the data analysis method we suggest here.
Thus, systematic errors in the measured value of due to errors in the model should be , which is small compared to the current uncertainty in (see Fig. 1).
Measuring the Love Number: The binary’s parameters are extracted from the noisy GW signal by integrating the waveform against theoretical inspiral templates , where are the parameters of the binary. The best-fit parameters are those that maximize the overlap integral. The probability distribution for the signal parameters for strong signals and Gaussian detector noise is 1994PhRvD..49.2658C , where , is the Fisher information matrix, and the inner product is defined by Eq. (2.4) of Ref. 1994PhRvD..49.2658C . The rms statistical measurement error in is then .
Using the stationary phase approximation and neglecting corrections to the amplitude, the Fourier transform of the waveform for spinning point masses is given by . Here the phase is
where , and are spin parameters, and is Euler’s constant lrr . The tidal term (10) adds linearly to this, yielding a phase model with 7 parameters (), where is a weighted average of and . We incorporate the maximum spin constraint for the NSs by assuming a Gaussian prior for and as in Ref. 1994PhRvD..49.2658C .
Figure 1 [bottom panel] shows the confidence upper limit we obtain for LIGO II (horizontal line) for two nonspinning NSs at a distance of Mpc (signal-to-noise of 95 in the frequency range Hz) with cutoff frequency , as well as the corresponding values of for relativistic polytropes with (dashed curve) and (solid line). The corresponding constraint on radius assuming identical stars would be for polytropes. Current NS models span the range .
Our phasing model (12) is the most accurate available model, containing terms up to post-3.5-Newtonian (P3.5N) order. We have experimented with using lower order phase models (P2N, P2.5N, P3N), and we find that the resulting upper bound on varies by factors of order . Thus there is some associated systematic uncertainty in our result. To be conservative, we have adopted the most pessimistic (largest) upper bound on , which is that obtained from the P3.5N waveform.
In conclusion, even if the internal structure signal is too small to be seen, the analysis method suggested here could start to give interesting constraints on NS internal structure for nearby events.
This research was supported in part by NSF grants PHY-0140209 and PHY-0457200. We thank an anonymous referee for helpful comments and suggestions.
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