Tidal deformability of neutron stars with realistic equations of state and their gravitational wave signatures in binary inspiral
Abstract
The early part of the gravitational wave signal of binary neutron star inspirals can potentially yield robust information on the nuclear equation of state. The influence of a star’s internal structure on the waveform is characterized by a single parameter: the tidal deformability , which measures the star’s quadrupole deformation in response to the companion’s perturbing tidal field. We calculate for a wide range of equations of state and find that the value of spans an order of magnitude for the range of equation of state models considered.
An analysis of the feasibility of discriminating between neutron star equations of state with gravitational wave observations of the early part of the inspiral reveals that the measurement error in increases steeply with the total mass of the binary. Comparing the errors with the expected range of , we find that Advanced LIGO observations of binaries at a distance of 100 Mpc will probe only unusually stiff equations of state, while the proposed Einstein Telescope is likely to see a clean tidal signature.
pacs:
04.40.Dg, 26.60.Kp, 97.60.Jd, 95.85.SzI Introduction and summary
The observation of inspiraling binary neutron stars (NSs) with groundbased gravitationalwave detectors such as LIGO and Virgo may provide significantly more information about neutronstar structure, and the highly uncertain equation of state (EOS) of neutronstar matter, than is currently available. The available electromagnetic observations of neutron stars provide weak constraints from properties such as the star’s mass, spin, and gravitational redshift (see for example Lattimer and Prakash (2007); Read et al. (2009a)). Simultaneous measurements of both the mass and radius of a neutron star Özel (2006); Leahy et al. (2008); Özel et al. (2009); Güver et al. (2008); Leahy et al. (2009), on the other hand, have the potential to make significantly stronger constraints. These measurements, however, depend on detailed modeling of the radiation mechanisms at the neutronstar surface and absorption in the interstellar medium, and are subject to systematic uncertainties.
Another possibility for obtaining information about the neutron star EOS is from the inspiral of binary neutron stars due to gravitational radiation. The tidal distortion of neutron stars in a binary system links the EOS describing neutronstar matter to the gravitationalwave emission during the inspiral. Initial estimates showed that for LIGO, tidal effects change the phase evolution only at the end of inspiral, and that point particle postNewtonian waveforms can be used for templatebased detection Kochanek (1992); Bildsten and Cutler (1992); Lai and Wiseman (1996). With the projected sensitivities of latergeneration detectors, however, effects which can be neglected for the purpose of detection may become measurable in the strongest observed signals.
While EOS effects are largest during the late inspiral and merger of two neutron stars where numerical simulations must be used to predict the signal, Flanagan and Hinderer showed that a small but clean tidal signature arises in the inspiral below 400 Hz Flanagan and Hinderer (2008a). This signature amounts to a phase correction which can be described in terms of a single EOSdependent tidal deformability parameter , namely the ratio of each star’s induced quadrupole to the tidal field of its companion. The parameter depends on the EOS via both the NS radius and a dimensionless quantity , called the Love number Brooker and Olle (1955); Mora and Will (2004); Berti et al. (2008): .
The relativistic Love numbers of polytropic^{1}^{1}1Polytropes are often written in two forms. The first form is expressed as , where is the pressure, is the energy density, is a pressure constant, and is the polytropic index. The second form, is given by , where is the restmass density, defined as the baryon number density times the baryon rest mass. The first form was mainly used in the recent papers Hinderer (2008); Damour and Nagar (2009a); Binnington and Poisson (2009). However, the second form is more commonly used in the neutronstar literature and is more closely tied to the thermodynamics of a Fermi gas. We will use both forms as was done in Ref. Damour and Nagar (2009a). EOS were examined first by Flanagan and Hinderer Flanagan and Hinderer (2008a); Hinderer (2008) and later by others in more detail Damour and Nagar (2009a); Binnington and Poisson (2009). Flanagan and Hinderer also examined the measurability of the tidal deformability of polytropes and suggested that Advanced LIGO could start to place interesting constraints on for nearby events. However, they used incorrect values for , which overestimated by a factor of and were therefore overly optimistic about the potential measurability. In addition, polytropes are known to be a poor approximation to the neutron star equation of state, and there may be significant differences in the tidal deformability between polytropes and “realistic” EOS. In this paper, we calculate the deformability for realistic EOS, and show that a tidal signature is actually only marginally detectable with Advanced LIGO.
In Sec. II we describe how the Love number and tidal deformability can be calculated for tabulated EOS. We use the equations for developed in Hinderer (2008), which arise from a linear perturbation of the OppenheimerVolkoff (OV) equations of hydrostatic equilibrium. In Sec. III we then calculate and as a function of mass for several EOS commonly found in the literature. We find that, in contrast to the Love number, the tidal deformability has a wide range of values, spanning roughly an order of magnitude over the observed mass range of neutron stars in binary systems.
As discussed above, the direct practical importance of the stars’ tidal deformability for gravitational wave observations of NS binary inspirals is that it encodes the EOS influence on the waveform’s phase evolution during the early portion of the signal, where it is accurately modeled by postNewtonian (PN) methods. In this regime, the influence of tidal effects is only a small correction to the pointmass dynamics. However, when the signal is integrated against a theoretical waveform template over many cycles, even a small contribution to the phase evolution can be detected and could give information about the NS structure.
Following Flanagan and Hinderer (2008a), we calculate in Sec. IV the measurability of the tidal deformability for a wide range of equal and unequal mass binaries, covering the entire expected range of NS masses and EOS, and with proposed detector sensitivity curves for second and third generation detectors. We show that the measurability of is quite sensitive to the total mass of the system, with very lowmass neutron stars contributing significant phase corrections that are optimistically detectable in Advanced LIGO, while largermass neutron stars are more difficult to distinguish from the case of black holes Damour and Nagar (2009a); Binnington and Poisson (2009). In thirdgeneration detectors, however, the tenfold increase in sensitivity allows a finer discrimination between equations of state leading to potential measurability of a large portion of proposed EOSs over most of the expected neutron star mass range.
We conclude by briefly considering how the errors could be improved with a more careful analysis of the detectors and extension of the understanding of EOS effects to higher frequencies.
Finally, in the Appendix we compute the leading order EOSdependent corrections to our model of the tidal effect and derive explicit expressions for the resulting corrections to the waveform’s phase evolution, extending the analysis of Ref. Flanagan and Hinderer (2008a). Estimates of the size of the phase corrections show that the main source of error are post1 Newtonian corrections to the Newtonian tidal effect itself, which are approximately twice as large as other, EOSdependent corrections at a frequency of 450 Hz. Since these pointparticle corrections do not depend on unknown NS physics, they can easily be incorporated into the analysis. A derivation of the explicit postNewtonian correction terms is the subject of Ref. Flanagan and Vines (2009).
Conventions: We set .
Ii Calculation of the Love number and tidal deformability
As in Flanagan and Hinderer (2008a) and Hinderer (2008), we consider a static, spherically symmetric star, placed in a static external quadrupolar tidal field . To linear order, we define the tidal deformability relating the star’s induced quadrupole moment to the external tidal field,
(1) 
The coefficient is related to the dimensionless tidal Love number by
(2) 
The star’s quadrupole moment and the external tidal field are defined to be coefficients in an asymptotic expansion of the total metric at large distances from the star. This expansion includes, for the metric component in asymptotically Cartesian, masscentered coordinates, the standard gravitational potential , plus two leading order terms arising from the perturbation, one describing an external tidal field growing with and one describing the resulting tidal distortion decaying with :
where and and are both symmetric and traceless. The relative size of these multipole components of the perturbed spacetime gives the constant relating the quadrupole deformation to the external tidal field as in Eq. (1).
To compute the metric (LABEL:eq:metric), we use the method discussed in Hinderer (2008). We consider the problem of a linear static perturbation expanded in spherical harmonics following Thorne and Campolattaro (1967). Without loss of generality we can set the azimuthal number , as the tidal deformation will be axisymmetric around the line connecting the two stars which we take as the axis for the spherical harmonic decomposition. Since we will be interested in applications to the early stage of binary inspirals, we will also specialize to the leading order for tidal effects, .
Introducing a linear perturbation onto the spherically symmetric star results in a static (zerofrequency), evenparity perturbation of the metric, which in the ReggeWheeler gauge Regge and Wheeler (1957) can be simplified Hinderer (2008) to give
where is related to by . Here primes denote derivatives with respect to . The corresponding perturbations of the perfect fluid stressenergy tensor components are and , where is the energy density and the pressure. The function satisfies the differential equation
(5) 
Here is given by
(6) 
which for slow changes in matter configurations corresponds to .
The method of calculating the tidal perturbation for a general equation of state table is similar to the method of calculating moment of inertia in the slow rotation approximation Hartle and Thorne (1969). The specific implementation we use follows the moment of inertia calculation in Appendix A of Read et al. (2009a), via an augmentation of the OV system of equations^{2}^{2}2Here we present the equations in terms of the radial coordinate ; the extension to the enthalpy variable used in Read et al. (2009a) is straightforward.:
(7)  
(8)  
(9)  
(10) 
The secondorder differential equation for is separated into a firstorder system of ODEs in terms of the usual OV quantities ^{3}^{3}3We use for the mass enclosed within radius instead of to avoid confusion with the total mass of the star, which we will label ., , and , as well as the additional functions , , and the equation of state function (recall ):
(11)  
These are combined with Eqs. (7)–(10), and the augmented system is solved simultaneously. The system is integrated outward starting just outside the center using the expansions and as . The constant determines how much the star is deformed and can be chosen arbitrarily as it cancels in the expression for the Love number. The ODE for outside the star, where , has a general solution in terms of associated Legendre functions at large , and at large . The boundary conditions that determine the unique choice of this solution follow from matching the interior and exterior solutions and their first derivatives at the boundary of the star, where . By comparison with Eq. (LABEL:eq:metric), the coefficients of the external solution can then be identified with the axisymmetric tidal field and quadrupole moment via , and as was done in Hinderer (2008). Here, and are the magnitudes of the spherical harmonic coefficients of the tidal tensor and quadrupole moment respectively.
Defining the quantity
(13) 
for the internal solution, the Love number is
where is the compactness of the star.
For stars with a nonzero density at the surface (for example strange quark matter or an incompressible polytrope), the term in Eq. (5) blows up at the surface and is no longer continuous across the surface. Following the discussion in Damour and Nagar (2009b) for an polytrope, this discontinuity leads to an extra term in the expression above for :
(15) 
where is the density just inside the surface.
Iii Love numbers and tidal deformabilities for candidate EOS
Differences between candidate EOS can have a significant effect on the tidal interactions of neutron stars. In this paper we consider a sample of EOS from Refs. Lattimer (2001); Read et al. (2009a) with a variety of generation methods and particle species. The sample is chosen to include EOS with the largest range of behaviors for , and rather than to fairly represent the different generation methods. We also restrict ourselves to stars with a maximum mass greater than 1.5 , which is conservatively low given recent neutronstar mass observations Ransom et al. (2005); Freire et al. (2008a); Champion et al. (2008); Verbiest et al. (2008); Freire et al. (2008b). We consider 7 EOS with just normal matter (SLY Douchin and Haensel (2001), AP1 and AP3 Akmal et al. (1998), FPS Friedman and Pandharipande (1981), MPA1 Muther et al. (1987), MS1 and MS2 Müller and Serot (1996)), 8 EOS that also incorporate some combination of hyperons, pion condensates, and quarks (PS Pandharipande and Smith (1975), BGN1H1 Balberg and Gal (1997), GNH3 Glendenning (1985), H1 and H4 Lackey et al. (2006), PCL2 Prakash et al. (1995), ALF1 and ALF2 Alford et al. (2005)), and 3 selfbound strange quark matter EOS (SQM13 Prakash et al. (1995)). A brief description of these EOS and their properties can be found in Lattimer (2001); Read et al. (2009a).
The generic behavior of the Love number is shown in the top panel of Fig. 1 as a function of compactness for different types of EOS. The two types of polytropes, energy and restmass density polytropes, are shown in gray. They coincide in the limit where as the star’s density goes to zero, and in the limit where and are both constant. This can be seen from the first law of thermodynamics,
(16) 
which relates to .
The sequences labeled “Normal” correspond to the 15 EOS with a standard nuclear matter crust, and the 3 sequences labeled “SQM” correspond to the crustless EOS SQM13 where the pressure is zero below a few times nuclear density. Within these two classes, there is little variation in behavior, so we do not explicitly label each candidate EOS.
The bottom panel of Fig. 1 shows for the realistic EOS, which is more astrophysically relevant because mass, not compactness, is the measurable quantity during binary inspiral. Unlike the quantity , depends on the constant for polytropes, so polytropic EOS are not shown. There is more variation in for fixed mass than for fixed compactness.
The behavior of these curves can be understood as follows: The Love number measures how easily the bulk of the matter in a star is deformed. If most of the star’s mass is concentrated at the center (centrally condensed), the tidal deformation will be smaller. For polytropes, matter with a higher polytropic index is softer and more compressible, so these polytropes are more centrally condensed. As a result, decreases as increases. The limiting case represents a uniform density star and has the largest Love number possible. The Love number also decreases with increasing compactness, and from Eq. (LABEL:eq:k2) it can be seen that vanishes at the compactness of a black hole () regardless of the EOS dependent quantity Damour and Nagar (2009a); Binnington and Poisson (2009).
Normal matter EOS behave approximately as polytropes for large compactness. However, for smaller compactness, the softer crust becomes a greater fraction of the star, so the star is more centrally condensed and smaller. For strange quark matter, the EOS is extremely stiff near the minimum density, and the star behaves approximately as an polytrope for small compactness. As the central density and compactness increase, the softer part of the EOS has a larger effect, and the star becomes more centrally condensed.
The parameter that is directly measurable by gravitational wave observations of a binary neutron star inspiral is proportional to the tidal deformability , which is shown for each candidate EOS in Fig. 2. The values of for the candidate EOS show a much wider range of behaviors than for because is proportional to , and the candidate EOS produce a wide range of radii (9.4–15.5 km for a 1.4 star for normal EOS and 8.9–10.9 km for the SQM EOS). See Table 1.
EOS  (km)  g cm s  

SLY  11.74  0.176  0.0763  1.70 
AP1  9.36  0.221  0.0512  0.368 
AP3  12.09  0.171  0.0858  2.22 
FPS  10.85  0.191  0.0663  1.00 
MPA1  12.47  0.166  0.0924  2.79 
MS1  14.92  0.139  0.110  8.15 
MS2  13.71  0.151  0.0883  4.28 
PS  15.47  0.134  0.104  9.19 
BGN1H1  12.90  0.160  0.0868  3.10 
GNH3  14.20  0.146  0.0867  5.01 
H1  12.86  0.161  0.0738  2.59 
H4  13.76  0.150  0.104  5.13 
PCL2  11.76  0.176  0.0577  1.30 
ALF1  9.90  0.209  0.0541  0.513 
ALF2  13.19  0.157  0.107  4.28 
SQM1  8.86  0.233  0.098  0.536 
SQM2  10.03  0.206  0.136  1.38 
SQM3  10.87  0.190  0.166  2.52 
For normal matter, becomes large for stars near the minimum mass configuration at roughly because they have a large radius. For masses in the expected mass range for binary inspirals, there are several differences between EOS with only matter and those with condensates. EOS with condensates have, on average, a larger , primarily because they have, on average, larger radii. The quark hybrid EOS ALF1 with a small radius (9.9 km for a star) and the nuclear matter only EOSs MS1 and MS2 with large radii (14.9 km and 14.5 km, respectively, at ) are exceptions to this trend.
For strange quark matter stars, there is no minimum mass, so the radius (and therefore ) approaches zero as the mass approaches zero. At larger masses, the tidal deformability of SQM stars remains smaller than most normal matter stars because, despite having large Love numbers, the radii of SQM stars are typically smaller.
Error estimates for an equalmass binary inspiral at 100 Mpc are also shown in Fig. 2 for both Advanced LIGO and the Einstein Telescope. They will be discussed in the next section.
Iv Measuring effects on gravitational radiation
We wish to calculate the contribution from realistic tidal effects to the phase evolution and resulting gravitational wave spectrum of an inspiraling neutron star binary. In the secular limit, where the orbital period is much shorter than the gravitational radiation reaction timescale, we consider the tidal contribution to the energy and energy flux for a quasicircular inspiral using the formalism developed by Flanagan and Hinderer Flanagan and Hinderer (2008a), which adds the following leadingorder terms to the postNewtonian pointparticle corrections (PNPP corr.):
(17)  
(18)  
Here and are the tidal deformabilities of stars 1 and 2, respectively. is the total mass, is the dimensionless reduced mass, and is the postNewtonian dimensionless parameter given by , where is the orbital angular frequency. One can then use
(19) 
to estimate the evolution of the quadrupole gravitational wave phase via .
Each equation of state gives in this approximation a known phase contribution as a function of and , or as a function of the total mass and the mass ratio , via and for that EOS. Although we calculated for individual neutron stars, the universality of the neutron star core equation of state allows us to predict the tidal phase contribution for a given binary system from each EOS. Following Flanagan and Hinderer (2008a), we discuss the constraint on the weighted average
(20) 
which reduces to in the equal mass case. The contribution to from the tidal deformation, which adds linearly to the known PP phase evolution, is
(21) 
The weighted average is plotted as a function of chirp mass in Fig. 3 for three of the EOS and for three values of : equal mass (), large but plausible mass ratio Bulik et al. (2004) (), and extremely large mass ratio ().
We can determine the significance of the tidal effect on gravitational waveforms in a given frequency range by considering the resulting change in phase accumulated as a function of frequency. In the case of templatebased searches, for example, a drift in phase of half a cycle leads to destructive interference between the signal and template, halting the accumulation of signal to noise ratio. The phase contributions to binary neutron stars of various masses from a range of realistic tidal deformabilities are plotted in Fig. 4.
The postNewtonian formalism itself is sensitive to highorder corrections at the frequencies at which the tidal effect becomes significant; as reference, we show in Fig. 4 the phase difference between the 3.0PN and 3.5PN expansions, as well as that from varying the form of the postNewtonian Taylor expansion from T4 to T1.^{4}^{4}4For an explanation of the differences between T4 and T1, see Damour et al. (2001); Boyle et al. (2007). An accurate knowledge of the underlying pointparticle dynamics will be important to resolve the effects of tidal deformation on the gravitational wave phase evolution at these frequencies.
The halfcycle or more contribution to the gravitational wave phase at relatively low frequencies suggests that this effect could be measurable. Flanagan and Hinderer Flanagan and Hinderer (2008a) first calculated the measurability for frequencies below 400 Hz, where the approximations leading to the tidal phase correction are welljustified. We extend the same computation of measurability to a range of masses and mass ratios. We take noise curves from the projected NSNS optimized Advanced LIGO configuration Shoemaker (2009), as well as a proposed noise spectrum of the Einstein Telescope Hild et al. (2008). These noise curves are representative of the anticipated sensitivities of the two detectors. Our results do not change significantly for alternate configurations which have similar sensitivities in the frequency range of interest.
We also extend the computation to a slightly higher cutoff frequency. As estimated in the Appendix, our calculation should still be fairly robust at 450 Hz, as the contributions to the phase evolution from various higher order effects are of the leading order tidal contribution. The uncertainty in the phase contribution from a given EOS is therefore significantly smaller than the order of magnitude range of phase contributions over the full set of realistic EOS.
The rms uncertainty in the measurement of is computed using the standard Fisher matrix formalism Poisson and Will (1995). Assuming a strong signal and Gaussian detector noise, the signal parameters have probability distribution , where is the difference between the parameters and their bestfit values and is the Fisher information matrix. The parentheses denote the inner product defined in Poisson and Will (1995). The rms measurement error in is given by a diagonal element of the inverse Fisher, or covariance, matrix: .
Using the stationary phase approximation and neglecting postNewtonian corrections to the amplitude, the Fourier transform of the waveform for spinning point masses is given by , where the pointmass contribution to the phase is given to 3.5 postNewtonian order in Ref. Blanchet (2006). The tidal term
(22) 
obtained from Eq. (28) adds linearly to this, yielding a phase model with 7 parameters (), where and are spin parameters. We incorporate the maximum spin constraint for the NSs by assuming a Gaussian prior for and as in Poisson and Will (1995). The uncertainties computed will depend on the choice of pointparticle phase evolution, but we assume this to be exactly the 3.5PN form for the current analysis.
The rms measurement uncertainty of , along with the uncertainties in chirp mass and dimensionless reduced mass , are given in Table 2 and plotted in Figs. 2 and 3, from a singledetector observation of a binary at 100 Mpc distance with amplitude averaged over inclinations and sky positions. If the bestfit is zero, this represents a 1 upper bound on the physical . A signal with bestfit would allow a measurement rather than a constraint of , with 1 uncertainty of .
We obtain the following approximate formula for the rms measurement uncertainty , which is accurate to better than for the range of masses and cutoff frequencies :
(23) 
where g cm s for a single Advanced LIGO detector and g cm s for a single Einstein Telescope detector.
Our results show that the measurability of tidal effects decreases steeply with the total mass of the binary. Estimates of the measurement uncertainty for an equalmass binary inspiral in a single detector with projected sensitivities of Advanced LIGO and the Einstein Telescope, at a volumeaveraged distance of 100 Mpc and using only the portion of the signal between Hz, are shown in Fig. 2, together with the values of predicted by various EOS models. Measurability is less sensitive to mass ratio, as seen in Fig. 3. Comparing the magnitude of the resulting upper bounds on with the expected range for realistic EOS, we find that the predicted are greatest and the measurement uncertainty is smallest for neutron stars at the low end of the expected mass range for NSNS inspirals of () Stairs (2004).
In a single Advanced LIGO detector, only extremely stiff EOS could be constrained with a typical 100 Mpc observation. However, a rare nearby event could allow more interesting constraints, as the uncertainty scales as the distance to the source. Rate estimates for detection of binary neutron stars are often given in terms of a minimum signaltonoise ; a recent estimate O’Shaughnessy et al. (2009) is between 2 and 64 binary neutron star detections per year for a single Advanced LIGO interferometer with a volume averaged range of 187 Mpc. The rate of binaries with a volume averaged distance smaller than 100 Mpc translates to roughly of this total detection rate, but over multiple years of observation a rare event could give measurements of with uncertainties smaller than the values in Table 2 (e.g. with half the tabled uncertainty at the total NSNS rate).
Using information from a network of detectors with the same sensitivity decreases the measurement uncertainty by approximately a factor of Cutler and Flanagan (1994), giving more reason for optimism. However, we should also note that, in some ways, our estimates of uncertainty are already too optimistic. First, only represents a confidence in the measurement; a error bar would give a more reasonable confidence. In addition, our Fisher matrix estimates are likely to somewhat underestimate the measurement uncertainty in real nonGaussian noise.
In contrast to Advanced LIGO, an Einstein Telescope detector with currently projected noise would be sensitive to tidal effects for typical binaries, using only the signal below 450 Hz at 100 Mpc. The tidal signal in this regime would provide a clean signature of the neutron star core equation of state. However, an accurate understanding of the underlying pointparticle phase evolution is still important to confidently distinguish EOS effects.
Advanced LIGO
M 
g cm s  

2.0  1.0  0.00028  0.073  8.4  27 
2.8  1.0  0.00037  0.055  19.3  35 
3.4  1.0  0.00046  0.047  31.3  41 
2.0  0.7  0.00026  0.058  8.2  26 
2.8  0.7  0.00027  0.058  18.9  35 
3.4  0.7  0.00028  0.055  30.5  41 
2.8  0.5  0.00037  0.06  17.8  33 
Einstein Telescope
M  g cm s  

2.0  1.0  0.000015  0.0058  0.70  354 
2.8  1.0  0.000021  0.0043  1.60  469 
3.4  1.0  0.000025  0.0038  2.58  552 
2.0  0.7  0.000015  0.0058  0.68  349 
2.8  0.7  0.000021  0.0045  1.56  462 
3.4  0.7  0.000025  0.0038  2.52  543 
2.8  0.5  0.000020  0.0048  1.46  442 
Expected measurement uncertainty will decrease if we can extend the calculation later into the inspiral. From Eq. (23), at 500 Hz is approximately 79% of its value at 450 Hz. The dominant source of error in the tidal phasing at these frequencies are postNewtonian effects which scale as and do not depend on any additional EOS parameters. These terms are computed in Ref. Flanagan and Vines (2009), and when they are incorporated into the analysis, the resulting phase evolution model can be used at slightly higher frequencies. These terms also add to the strength of the tidal signature.
Higherorder tidal effects and nonlinear hydrodynamic couplings, which depend on unknown NS microphysics, are smaller than postNewtonian effects by factors of and , so they become important later in the inspiral, where the adiabatic approximation that the mode frequency is large compared to the orbital frequency also breaks down. At this point we can no longer measure only , but an EOS dependent combination of effects including higher multipoles, nonlinearity, and tidal resonances.
However, information in the late inspiral could also constrain the underlying neutronstar EOS. Read et al. Read et al. (2009b) estimated potential measurability of EOS effects in the last few orbits of binary inspiral, where the gravitational wave frequency is above 500 Hz, using full numerical simulations. The EOS used for the simulation was systematically varied by shifting the pressure in the core while keeping the crust fixed. The resulting models were parameterized, either by a fiducial pressure or by the radius of the isolated NS model, and measurability in Advanced LIGO was estimated. Such numerical simulations include all the higher order EOS effects described above, but the tidal deformability parameter should remain the dominant source of EOSdependent modification of the phase evolution. We therefore expect it to be a better choice for a single parameter to characterize EOS effects on the late inspiral.
The numerically simulated models of Read et al. (2009b) can be reparameterized by the of the 1.35 neutron stars considered^{5}^{5}5The piecewise polytrope EOS {2H, H, HB, B, 2B} have of {0.588, 1.343, 1.964, 2.828, 10.842} g cm s, respectively.. The uncertainty of measurement for the new parameter can be estimated from Tables IIV of Read et al. (2009b). In the broadband Advanced LIGO configuration of Table IV, it is between 0.3 and 4 g cm s for an optimally oriented 100 Mpc binary, or between 0.7 and 9 g cm s averaged over sky position and orientation. However, in the NSNS optimized LIGO configuration of Table III, which is most similar to the Advanced LIGO configuration considered in this paper, the expected measurement uncertainty is more than several times for all models. These estimates should be considered orderofmagnitude, as numerical simulation errors are significant, and the discrete sampling of a parameter space allows only a coarse measurability estimate which neglects parameter correlations. In contrast to the perturbative/postNewtonian estimate of EOS effects calculated in this paper, EOS information in the signal before the start of numerical simulations is neglected. The estimate is complementary to the measurability below 450 Hz estimated in this paper.
V Conclusion
We have calculated the relativistic Love number and resulting tidal deformability for a wide range of realistic EOS in addition to polytropes. These EOS have tidal deformabilities that differ by up to an order of magnitude in the mass range relevant for binary neutron stars. However, the estimated uncertainty for a binary neutron star inspiral at 100 Mpc using the Advanced LIGO sensitivity below 450 Hz is greater than the largest values of except for very lowmass binaries. The uncertainty for the Einstein Telescope, on the other hand, is approximately an order of magnitude smaller than for Advanced LIGO, and a measurement of will rule out a significant fraction of the EOS.
Advanced LIGO can place a constraint on the space of possible EOS by obtaining a confidence upper limit of . The tables in Sec. IV can also be scaled as follows: For a network of detectors the uncertainty scales roughly as , and for a closer signal we have .
Acknowledgements.
We thank S. Hughes, E. Flanagan, and J. Friedman for helpful suggestions and J. Creighton for carefully reading the manuscript. The work was supported in part by NSF Grant PHY0503366, and by the Deutsche Forschungsgemeinschaft SFB/TR7. TH gratefully acknowledges support from the Sherman Fairchild postdoctoral fellowship, and BL also thanks the Wisconsin Space Grant Consortium fellowship program for support. RNL was supported by NSF Grant PHY0449884 and the NASA Postdoctoral Program, administered by Oak Ridge Associated Universities through a contract with NASA. *Appendix A Accuracy of the phasing model
To assess the accuracy of the simple phase evolution model, we compute the corrections to the tidal phase perturbation due to several EOSdependent effects: the leading order finite modefrequency terms, higher order tidal effects, and nonlinear hydrodynamic couplings. For simplicity, we will only derive the phase corrections for one star with internal degrees of freedom coupled to a point mass. The terms for the other star simply add. For such a binary system, the Lagrangian can then be written as
(24)  
Here, the star’s static mass quadrupole parameterizes the modes of the star, which can be treated as harmonic oscillators that are driven below their resonant frequency by the companion’s tidal field. The tensor parameterizes the star’s mass octupole degrees of freedom, and and are the and tidal tensors respectively, which are given by and in Newtonian gravity. The deformability constant is defined by . The quantities and are the and mode frequencies, and is a coupling constant for the leading order nonlinear hydrodynamic interactions. In general, one would need to sum over the contributions from all the modes, but other modes contribute negligibly in the regime of interest for the above model (see Flanagan and Hinderer (2008b)). PostNewtonian effects on the Lagrangian for the binary are derived in Ref. Flanagan and Vines (2009) and can simply be added to those derived here.
We will be interested in finding an effective description of the dynamics of the system for quasicircular inspirals in the adiabatic limit, where the radiation reaction timescale is long compared to the orbital timescale. From equilibrium solutions to the EulerLagrange equations derived from this Lagrangian, the following radiusfrequency relation is obtained:
(25) 
The equilibrium energy, obtained by reversing the signs of the potential energy terms in the Lagrangian, is given by:
(26) 
The correction to the energy flux , where is the total quadrupole moment, is
(27)  
Using the formula in the stationary phase approximation and integrating twice leads to the final expression for the tidal phase correction:
(28)  
We will analyze the information contained in the portion of the signal at frequencies . This is slightly higher than previously considered, and we now argue that in this frequency band, the simple model of the phase correction is still sufficiently accurate for our purposes. We will evaluate all of the corrections for the case of equal masses . An estimate of the fractional errors for the case of and km is given in parentheses.

Post1Newtonian corrections ().
These corrections give rise to terms that add to those in Eq. (28). The explicit form of these terms is computed in Ref. Flanagan and Vines (2009) and they depend on the NS physics only via the same parameter as the Newtonian tidal terms, so they can easily be incorporated into the data analysis method. Preliminary estimates indicate that for equal masses, these post1 Newtonian effects will increase the tidal signal. 
Adiabatic approximations ( ).
The approximation that the radiation reaction time is much longer than the orbital time is extremely accurate, to better than ; see Fig. 2 of Ref. Flanagan and Hinderer (2008a), which compares the phase error obtained from numerically integrating the equations of motion supplemented with the leading order gravitational wave dissipation terms to that obtained analytically using the adiabatic approximation.
The accuracy of the approximation can be estimated from the fractional correction to (28), which is , where and . For typical NS models the mode frequency is Kokkotas and Schmidt (1999)(29) so that the fractional correction is for Hz and for a conservatively low mode frequency of Hz.

Nonlinear hydrodynamic corrections ().
The leading nonlinear hydrodynamic corrections are characterized by the coupling coefficient in the action. The size of this parameter can be estimated by comparing the Newtonian to the coupling constants in Lai’s ellipsoidal models (e.g. Table 1 of Lai et al. (1993)) to be . The nonlinear selfcoupling term in Eq. (28) is smaller than the leading term by a factor . 
Spin corrections ().
Fractional corrections to the tidal signal due to spin scale as(30) where is the maximum rotational frequency the star can have before breakup, which for most NS models is . The observed NSNS binaries which will merge within a Hubble time have spin periods of ms, and near the coalescence they will have slowed down due to e.g. magnetic braking, with final spin periods of ms. The fractional corrections to the tidal signal due to the spin are then .
If the stars have spin, there will also be a spininduced correction to the phase. As discussed above, the slowrotation limit is likely to be the relevant regime for our purposes, and using similar methods as for the tidal corrections leads to a phase correction which scales as , where is the rotational Love number, which for Newtonian stars is the same as the tidal Love number and the spin frequency. The scaling of the spin term as shows that only at large separation do spin effects dominate over tidal effects, which scale as .

Nonlinear response to the tidal field ().
We have linearized in . Including terms gives a fractional correction in Eq. (28) of . 
Viscous dissipation (negligible).
There have been several analytical and numerical studies of the effect of viscosity during the early part of the inspiral, e.g. Kochanek (1992); Bildsten and Cutler (1992). They found that viscous dissipation is negligible during the early inspiral if the volumeaveraged shear viscosity is(31) The expected microscopic viscosity of NSs is Cutler and Lindblom (1987)
(32) which is orders of magnitude too small to lead to any significant effect. A variety of other likely sources of viscosity, e. g. the breaking or crumpling of the crust, are also insignificant Bildsten and Cutler (1992); Kochanek (1992) in the regime of interest to us.
Thus, systematic errors in the measured value of due to errors in the model should be , which is small compared to the current uncertainty of an order of magnitude in .
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