# Effective One Body description of tidal effects in inspiralling compact binaries

###### Abstract

The late part of the gravitational wave signal of binary neutron star inspirals can in principle yield crucial information on the nuclear equation of state via its dependence on relativistic tidal parameters. In the hope of analytically describing the gravitational wave phasing during the late inspiral (essentially up to contact) we propose an extension of the effective one body (EOB) formalism which includes tidal effects. We compare the prediction of this tidal-EOB formalism to recently computed nonconformally flat quasi-equilibrium circular sequences of binary neutron star systems. Our analysis suggests the importance of higher-order (post-Newtonian) corrections to tidal effects, even beyond the first post-Newtonian order, and their tendency to significantly increase the “effective tidal polarizability” of neutron stars. We compare the EOB predictions to some recently advocated, nonresummed, post-Newtonian based (“Taylor-T4”) description of the phasing of inspiralling systems. This comparison shows the strong sensitivity of the late-inspiral phasing to the choice of the analytical model, but raises the hope that a sufficiently accurate numerical–relativity–“calibrated” EOB model might give us a reliable handle on the nuclear equation of state.

###### pacs:

04.25.Nx, 04.30.-w, 04.40.Dg, 95.30.Sf,## I Introduction

Some of the prime targets of the currently operating network of ground-based detectors of gravitational waves (GWs) are the signals emitted by inspiralling and coalescing compact binaries. Here, “compact binary” refers to a binary system made either of two black holes, a black hole and a neutron star, or two neutron stars. The GW signal emitted by binary black hole (BBH) systems has been the subject of intense theoretical studies, based either on analytical methods or on numerical ones. In particular, recent progress in the application of the effective one body (EOB) approach to BBH systems has led to a remarkable agreement between the (analytical) EOB predictions and the best current numerical relativity results Damour:2009kr ; Buonanno:2009qa (see also Yunes:2009ef ). By contrast, much less work has been devoted to the study of the GW signal emitted by compact binaries comprising neutron stars: either black-hole-neutron-star (BHNS) systems or binary neutron-star (BNS) ones. During the inspiral phase (before contact), these systems differ from the BBH ones by the presence of tidal interactions which affect both the dynamics of the inspiral and the emitted waveform. During the merger and coalescence phase, the presence of neutron stars drastically modifies the GW signal Baiotti:2009gk ; Giacomazzo:2009mp ; Baiotti:2008ra . The coalescence signal involves (especially in the BNS case) a lot of complicated physics and astrophysics, and is, probably, not amenable to the type of accurate analytical description which worked in the BBH case. Early works on this problem have tried to approximately relate some qualitative features of the merger GW signal linked, e.g., to “tidal disruption”, to analytically describable inputs Bildsten:1992my ; Kochanek:1992wk ; Vallisneri:1999nq .

Recently, Flanagan and Hinderer Flanagan:2007ix ; Hinderer:2007mb ; Hinderer:2009 have initiated the program of studying the quantitative influence of tidal effects Hinderer:2007mb ; Damour:2009vw ; Binnington:2009bb in inspiralling BNS systems. However, they only considered the early (lower frequency) portion of the GW inspiral signal, mainly because they were using a post-Newtonian based description of the binary dynamics whose validity is restricted to low enough frequencies. In particular, one of the results of the recent work of Hinderer et al. Hinderer:2009 is to show that the accumulated GW phase due to tidal interactions is, for most realistic NS models of mass smaller than the “uncertainty” in the PN-based description of GW phasing (see the central panel of their Fig. 4 where the thin-dashed and thin-dotted lines are two measures of the PN “uncertainty”. [These measures are larger than the inspiral tidal signal except for the extreme case where the radius of the NS is taken to be km].

By contrast, our aim in this work will be to propose a way of describing the binary dynamics (including tidal effects) whose validity does not have the limitations of PN-based descriptions and therefore is not apriori limited to the low frequency part, but extends to significantly higher frequencies. This might be crucial to increase the detectability of the GW signal and thereby have a handle on the nuclear equation of state (EOS). Indeed, our proposal consists in extending the EOB method by incorporating tidal effects in it. Our hope is that such a tidally-extended EOB framework will be able to describe with sufficient approximation not only the early inspiral phase, but also the late inspiral up to the moment (that we shall consistently determine within our scheme) of “contact”. We think that the present EOB description of tidal effects is likely to be more accurate than any of the possible “post-Newtonian-based” descriptions involving supplementary tidal terms (such as Flanagan:2007ix or Hinderer:2009 ). This should be especially true in the BHNS systems which, in the limiting case , are known to be well described by the EOB approach (and rather badly described by post-Newtonian-based approaches). We will give some evidence of the validity of the EOB description of close neutron star systems by comparing our analytical predictions to recently calculated quasi-equilibrium neutron star (NS) sequences of circular orbits Uryu:2009ye (see also Uryu:2005vv ).

## Ii Effective-action description of tidal effects in two-body systems

### ii.1 General formalism

The general relativistic tidal properties of neutron stars have been recently studied in Refs. Hinderer:2007mb ; Damour:2009vw ; Binnington:2009bb ; Hinderer:2009 . As emphasized in Damour:2009vw , there are (at least) three different types of tidal responses of a neutron star to an external tidal solicitation, which are measured by three different tidal coefficients: (i) a gravito-electric-type coefficient measuring the -order mass multipolar moment induced in a star by an external -order gravito-electric tidal field ; (ii) a gravito-magnetic-type coefficient measuring the spin multipole moment induced in a star by an external -order gravito-magnetic tidal field ; and (iii) a dimensionless “shape” Love number measuring the distorsion of the shape of the surface of a star by an external -order gravito-electric tidal field. It was found in Damour:2009vw ; Binnington:2009bb that all those coefficients have a strong sensitivity to the value of the star’s “compactness” (where we denote by the velocity of light, to be distinguished from the compactness ). This means, in particular, that the numerical values of the tidal coefficients of NS’s should not be evaluated by using Newtonian estimates. Indeed, the dimensionless version of , traditionally denoted as (“second Love number”) and defined as

(1) |

where denotes the areal radius of the NS, is typically three times smaller than its Newtonian counterpart (computed from the same equation of state). A similar, though less drastic, “quenching” also occurs for the “first Love number” . In particular, though Newtonian ’s are larger than (and equal to , see Eq. (81) of Damour:2009vw ), the typical relativistic values of are smaller than . This will play a useful role in our analysis below of the moment where the tidal distortion of the NS becomes too large for continuing to use an analytical approach.

It was shown in Damour_cras80 ; Damour1983 that the motion and radiation of two black holes can be described, up to the fifth post-Newtonian (5PN) approximation, by an effective action of the form

(2) |

where

(3) |

is a “skeletonized” description of black holes, as “point masses”. To give meaning to the addition of point-mass sources to the nonlinear Einstein equations, one needs to use a covariant regularization method. Refs. Damour_cras80 ; Damour1983 mainly used Riesz’ analytic regularization, but it was already mentioned at the time that one could equivalently use dimensional regularization. The efficiency and consistency of the latter method was shown by the calculations of the dynamics, and radiation, of BBH systems at the 3PN level Damour:2001bu ; Blanchet:2003gy ; Blanchet:2004ek . Let us also recall that the limitation to the 5PN level in Ref. Damour1983 is precisely linked to the possible appearance of ambiguities in BBH dynamics appearing at the level where tidal effects start entering the picture. Indeed, it is well-known in effective field theory that finite-size effects correspond to augmenting the point-mass action 2 by non-minimal (worldline) couplings involving higher-order derivatives of the field [see Damour:1995kt ; Goldberger:2004jt and Appendix A of Ref. Damour:1998jk ]. More precisely, the two tidal effects parametrized by and correspond to augmenting the leading point-particle effective action, (2), (3), by the following nonminimal worldline couplings

(4) | |||||

Here^{1}^{1}1We use here the notation of Damour:1990pi , notably for multi-indices .
and are the
gravito-electric and gravito-magnetic “external” tidal gradients evaluated along the worldline of the
considered star (labelled by ), in the local frames (attached to body ) defined in Damour:1990pi .
If needed, they can be reexpressed in terms of covariant derivatives of the Riemann (or Weyl)
tensor. For instance, using Eq. (3.40) of Damour:1990pi , the leading,
quadrupolar terms in Eq. (4) read

(5) | |||||

where , , with , and being the four-velocity along the considered worldline. As explained in Appendix A of Ref. Damour:1998jk , one can, modulo some suitable “field redefinitions” that do not affect the leading result, indifferently use the Weyl tensor or the Riemann tensor in evaluating the and entering Eq. (5). being the dual of the Weyl tensor

The effective-action terms (4), (5) can be used to compute the various observable
effects linked to the relativistic tidal coefficients and ^{2}^{2}2More precisely,
Eq. (4) describes only the effects that are linear in tidal
deformations (and which preserve parity). If one wished to also
consider nonlinear tidal effects one should augment the quadratic-only terms (5) by
higher-order nonminimal worldline couplings which are cubic, quartic, etc in and
its gradients. The coefficients of such terms would then parametrize some nonlinear tidal effects,
which have not been considered in the linear treatments of Refs. Damour:2009vw ; Binnington:2009bb ..
In particular, they imply both: (i) additional terms in the dynamics of the considered binary system, and
(ii) additional terms in the gravitational radiation emitted by the considered binary system.
Both types of additional terms can, in principle, be evaluated with any needed relativistic accuracy
from Eq. (4), i.e. computed either in a “post-Minkowskian” (PM) expansion in powers of
, or (after a further re-expansion in powers of ), in a “post-Newtonian” (PN)
expansion in powers of . Let us remark in passing that the PM expansion can be
conveniently expressed in terms of Feynman-like diagrams, as was explicitly discussed (for
tensor-scalar gravity) at the 2PN level in Damour:1995kt .

Here we shall use the extra terms (4), (5) as a way to add to the description of binary black hole systems the effects linked to the replacement of one or two of the black holes by a neutron star. From this point of view, we shall conventionally consider that the tidal coefficients of a black hole vanish: Damour:2009vw ; Binnington:2009bb . However, as emphasized in Damour:2009vw , more work is needed to clarify whether this is exact, i.e. whether the description of BBH’s by an effective action requires or not the presence of additional couplings of the type of Eqs. (4), (5), as “counter terms” to absorb dimensional regularization poles (such poles are indeed linked to the possible ambiguities expected to arise at 5PN in the point-mass dynamics; see the discussion in Sec. 5 of Damour1983 ; see also Sec. 7 of Damour:2009sm ) . We leave to future work a clarification of this subtle issue.

### ii.2 Leading-Order tidal effects in the two-body interaction Lagrangian

Let us first consider the dynamical effects, implied by (4) i.e. the tidal contribution to the “Fokker” Lagrangian describing the dynamics of two compact bodies after having integrated out the gravitational field, say

(6) |

Here, denotes the (time-symmetric) interaction Lagrangian following from the point-mass action (2) (say after a suitable redefinition of position variables to eliminate higher derivatives). It is currently known at the 3PN level. The supplementary term in Eq. (6) is of the symbolic form (keeping only powers of and )

(7) | |||||

Let us start by discussing the leading order contributions associated to each tidal coefficient or . The leading term in the contribution linked to is simply obtained from (4) by inserting the leading-order value of , i.e.

(8) |

where denotes the companion of body in the considered binary system (), and the distance between the two bodies. In addition , with , denotes the differentiation with respect to that appear after taking the limit where the field point tends to on the worldline of body . Using

(9) |

where , , and where the hat denotes a symmetric trace-free (STF) projection, and the fact that (see, e.g., Eq. (A25) of BD86 )

(10) |

one easily finds that the leading Lagrangian contribution proportional to reads

(11) | |||||

Here we have used (1) to replace in terms of the dimensionless Love number , and of the areal radius of the NS. Note that, in a BNS system, one has to add two different contributions: . By contrast, in a BHNS system one has only if denotes the NS.

Let us also evaluate the leading “magnetic-type” contribution, i.e. the term in (6). It is obtained by inserting in (4) the “Newtonian”-level value of the gravito-magnetic quadrupolar field exterted by body on body . This is given by Eq. (6.27a) of Damour:1991yw , namely

(12) |

where is the relative velocity between and . A straightforward calculation then yields

(13) |

Note that the leading quadrupolar gravito-magnetic contribution (13) is smaller than the corresponding quadrupolar gravito-electric contribution

(14) |

by a factor

(15) |

In terms of the corresponding dimensionless Love numbers (defined in Damour:2009vw ) and , the prefactor is equal to the dimensionless ratio . However, it was found in Damour:2009vw ; Binnington:2009bb that the magnetic Love number was much smaller than . Typically, for a -polytrope and a compactness , one has , while , so that . In other words, the leading gravito-magnetic interaction (13) is equivalent (say for circular orbits) to a 1PN fractional correction factor, , modifying the leading gravito-electric contribution (14), with . As we shall discuss below, the 1PN correction to (14), implied by (4), involves coefficients of order unity. We will therefore, in the following, neglect the contribution (13) which represents only a small fractional modification to the 1PN correction to (14). On the other hand, we shall retain some of the higher-degree gravito-electric contributions. Indeed, though, for instance, formally corresponds to a 2PN correction to , its coefficient is much larger than that corresponding to an order-unity 2PN correction to Eq. (14) [see Table 1 below].

Summarizing: the leading-order tidal contributions to the two-body interaction Lagrangian are (from Eq. (11))

(16) |

where denotes the dimensionless Love number of a NS Hinderer:2007mb ; Damour:2009vw ; Binnington:2009bb . Note that the plus sign in Eq. (16) expresses the fact that the tidal interactions are attractive.

### ii.3 Structure of subleading (post-Newtonian) dynamical tidal effects

Leaving to future work DEF09 a detailed computation of higher-order relativistic tidal effects, let us indicate their general structure. Here, we shall neglect the effects which are nonlinear in the worldline couplings of Eq. (4) (e.g. effects ) for two reasons. On the one hand, such effects are numerically quite small, even for close neutron stars (as we shall check below). On the other hand, a fully consistent discussion of such effects requires that one considers a more general version of nonminimal worldline couplings, involving terms which are cubic (or more nonlinear) in the curvature tensor and its covariant derivatives. Indeed, it is easily seen that a nonminimal coupling which is cubic in contributes to the dynamics at the same level that a 1PN correction to the coupling quadratic in .

In the ”quadratic-in-curvature” approximation of Eq. (4) the part of the tidal interaction which is proportional to will have the symbolic structure

(17) |

where we only indicate the dependence on and , leaving out all the coefficients (symbolically replaced by 1), which depend on positions and velocities. The presence of an overall factor comes from the fact that in Eq. (4) (which denotes the regularized value of some gradient of the curvature tensor as the field point tends to on the worldline of ) is proportional to , so that it is vanishing when , i.e. in the limit of a one-body system. [We are considering here a two-body system; in the more general case of an -body system we would have .] In a diagrammatic language (see e.g. Damour:1995kt ) the higher-order terms on the right hand side (r.h.s.) of Eq. (II.3) correspond to diagrams where, besides having the basic (quadratic in ) vertex on the worldline being connected by two gravity propagators to two “sources” on the worldline, we also have some further gravity propagators connecting one of the worldlines either to one of the worldline vertices, or to some intermediate “field” vertex. Note that the information about the 1PN corrections to both gravito-electric () and gravito-magnetic () multipolar interactions (of any degree ) is contained in the work of Damour, Soffel and Xu Damour:1991yw ; Damour:1992qi ; Damour:1993zn . We shall discuss below the effect of the subleading (post-Newtonian) terms in (II.3) on the EOB description of the dynamics of tidally interacting binary systems.

## Iii Incorporating dynamical tidal effects in the Effective One-Body (EOB) formalism

### iii.1 General proposal

The EOB formalism Buonanno:1998gg ; Buonanno:2000ef ; Damour:2001tu replaces the two-body interaction Lagrangian (or Hamiltonian) by a Hamiltonian, of a specific form, which depends only on the relative position and momentum of the binary system, say . For a non spinning BBH system, it has been shown that its dynamics, up to the 3PN level, can be described by the following EOB Hamiltonian (in polar coordinates, within the plane of the motion):

(18) |

where

(19) |

Here is the total mass, is the symmetric mass ratio and . In addition we are using rescaled dimensionless (effective) variables, notably and , and is canonically conjugated to a “tortoise” modification of Damour:2009ic .

A remarkable feature of the EOB formalism is that the complicated, original 3PN Hamiltonian (which contains many corrections to the basic Newtonian Hamiltonian ) can be replaced by the simple structure (18), (19) whose two crucial ingredients are: (i) a “double square-root” structure , and (ii) the “condensation” of most of the nonlinear relativistic gravitational interactions in one function of the (EOB) radial variable: the basic “radial potential” . In addition, the structure of the function is quite simple. At the 3PN level it is simply equal to

(20) |

where , and . It was recently found Damour:2009kr that an excellent description of the dynamics of BBH systems is obtained by: (i) augmenting the presently computed terms in the PN expansion (20) by additional 4PN and 5PN terms, and by (ii) Padé-resumming the corresponding 5PN “Taylor” expansion of the function. In other words, BBH (or “point mass”) dynamics is well described by a function of the form

(21) |

where denotes an Padé approximant. It was found in Ref. Damour:2009kr that a good agreement between EOB and numerical relativity binary black hole waveforms is obtained in an extended “banana-like” region in the plane approximately extending between the points and . In this work we shall select the values , which lie within this good region.

Our proposal for incorporating dynamical tidal effects in the EOB formalism consists in preserving the simple general structure (18), (19) of the EOB Hamiltonian, but to modify the BBH radial potential (21) (which corresponds to the point-mass action (2)) by augmenting it by some “tidal contribution”. In other words the proposal is to use Eqs. (18), (19) with

(22) |

### iii.2 Incorporating leading order (LO) dynamical tidal interactions

Let us show that, at the leading order (LO), one can use a tidal contribution of the form

(23) |

with some dimensionless coefficient .

Indeed, if we keep only the Newtonian approximation of the full EOB Hamiltonian (18), (19) (using with being 1PN small as ) one finds (with )

(24) |

which exhibits the role of , and remembering that there is a sign reversal between the interaction energy and the interaction Lagrangian, we see that the terms (16) can be converted in a contribution to the potential of the form (23), if the coefficients take the values as being the interaction energy. Decomposing

(25) | |||||

In the second form, we have introduced the compactness parameters of the stars: . It is interesting to note that the dimensionless tidal parameters that enter the EOB dynamics are (when ) the ratios , rather than the Love numbers . Let us also note that the velocity of light formally appears in the numerator of . This is related to the fact that, contrary to the coefficients of the successive powers of that enter the BBH EOB potential which are (roughly speaking) pure numbers of order unity, the coefficients entering the tidal contribution will tend to be much larger than unity (and to increase with ). For instance, we shall typically find that . This numerical difference makes it consistent to add to (which is known for sure only up to terms, i.e. the 3PN level) additional terms that would formally correspond to 5PN 7PN contributions if their coefficients were “of order unity” (at least in the parametric sense).

Finally, to illustrate the typical numerical values of the EOB tidal parameters we give in Table 1 the values of for three paradigmatic systems, one equal-mass BNS and two BHNS of mass ratios and . The neutron star model is described with a “realistic”EOS SLy (with a piece-wise polytropic representation, see below) and has the following characteristics: mass , compactness , radius km. Note that the main dependence on the equation of state (EOS) in (say for the equal-mass BNS case) comes from . Therefore, if one were considering a NS of different radius (because of the use of a different EOS) with the same mass, would be approximately given by

Model | ||||
---|---|---|---|---|

BNS | 1 | 73.0426 | 165.2966 | 509.6131 |

BHNS | 4 | 1.4959 | 0.5416 | 0.2672 |

BHNS | 10 | 0.0726 | 0.0054 | 0.0005 |

One sees in Table 1 that the dimensionless tidal parameter is a strongly decreasing function of the mass ratio. This is analytically understood by looking at Eq. (25). If the label refers to a black hole (so that ), denoting , we have where

(26) |

Here denotes as above the compactness of the NS. Therefore, as soon as the mass ratio is significantly larger than one, we see that contains a small factor that suppresses the tidal contribution. As a consequence, GW-observable tidal effects will be strongly suppressed in realistic BHNS systems. Note, however, that it might be quite useful to compare numerical relativity simulations of “artificial” BHNS systems of mass ratio to their EOB description to probe the analytical understanding of the late inspiral and plunge phase. In particular, we note that, as a function of , vanishes both when and and reaches a maximum value when . Moreover the maximum value of is larger than the value of for a corresponding equal-mass BNS system by a factor . We suggest that the numerical study of such astrophysically irrelevant BHNS systems (with ) can be quite useful for improving our understanding of tidal interactions in strongly-interacting (near contact) regimes.

### iii.3 Parametrizing higher-order dynamical tidal corrections

Above we discussed the leading order (LO) contribution of tidal interactions to the EOB “radial potential” . We also discussed the structure of sub-leading (post-Newtonian) contributions to tidal interactions, Eq. (II.3). Comparing the structure (II.3) to the part of the EOB action linear in , which is proportional to the product of by reduced mass , we see that the general structure of the tidal contributions to the potential is

(27) |

where we invoked dimensional analysis to insert appropriate powers of the (EOB) radial separation . [Contrary to the action (II.3) which also depends on velocities (and higher-derivatives), the EOB radial potential depends only on the radius .]

In other words, if we separate, for each multipolar order, the and contributions to ,

(28) |

we can write

(29) |

where

(30) |

is the part of , Eq. (23), which is linear in , or , i.e.

(31) |

Similarly, one will have

(32) |

The coefficient represents the next to leading order (NLO) fractional correction to the leading order (i.e. a 1PN fractional correction), while represents the next-to-next to leading order (NNLO) correction (i.e. a 2PN fractional correction), etc. These coefficients are not pure numbers, but rather function of the two dimensionless mass ratios

(33) | ||||

(34) |

The coefficients entering Eq. (32) are obtained from those entering (29) by the interchange of and , i.e. . The symbolic structure (III.3) would naively suggest that is a linear combination of and and that is a combination of , and . However, as the reformulation of (II.3) in terms of an EOB potential (III.3) involves a “contact transformation” that depends on the symmetric mass ratio (see Ref. Buonanno:1998gg ), the mass-ratio dependence of might be more complicated. Note that, by using the identity , one can, e.g., express in terms of only. [Then will be the same function of than of .] Note also that, if one wishes, one can, for each value of factorize the total LO terms , and write

(35) |

where

(36) |

with

(37) |

Using Eqs. (4.27) and (4.29) of Damour:1992qi , or Eq. (3.33) of Damour:1993zn , together with effective action techniques, a recent calculation DEF09 gave the following result for the 1PN coefficient of multipolar order , , namely

(38) |

More work is needed to determine the higher degree and/or higher order coefficients , and thereby the coefficients entering Eq. (37). Below, we shall focus on the equal-mass case where the coefficients become pure numbers.

Here we shall explore three possible proposals for including higher-order PN corrections in tidal effects. The first proposal consists in truncating Eq. (36) at 1PN order in a straightforward “Taylor” way, i.e. to consider a PN correcting factor to the EOB radial potential of the form

(39) |

The second proposal consists in considering a PN correcting factor which has a “Padé-resummed” structure, i.e.

(40) |

Our third proposal consists in considering a PN correcting factor which would result from having a “shift” between the EOB radial coordinate and the radial coordinate appearing most naturally in a Newtonian-like tidal interaction ().

(41) |

We use here a different notation for the 1PN coefficient, , as a reminder that, for instance, when , the parametrization (41) corresponds to a 1PN coefficient in the parametrization (39) given by

(42) |

## Iv Comparing EOB to numerical relativity results on ”waveless” circular binaries

The aim of this section is to compare stationary quasi-circular configurations of neutron star binaries computed, on the one hand, in the analytical framework outlined above and, on the other hand, in the numerical framework recently implemented by Ury et al. Uryu:2009ye (see also Uryu:2005vv ). The quantity from both frameworks that we shall compare is the binding energy as a function of the orbital frequency .

### iv.1 Tidally interacting BNS circular configurations in the EOB framework

#### iv.1.1 BNS binding energy in the EOB framework

As an application of the formalism discussed so far, we consider in this section binaries in exactly circular orbits, in absence of radiative effects (these will be discussed in the following section).

As the EOB formalism is based on a Hamiltonian description of the conservative dynamics, the stable circular orbits correspond to minima, with respect to , of the radial potential . Minimizing is equivalent to minimizing the corresponding effective Hamiltonian , or, its square, i.e.

(43) | |||||

Here, we have used the short-hand notation and . Minimizing (43) with respect to (or, equivalently, ), for a given (scaled) total angular momentum , yields the following equation

(44) |

where the prime denotes a -derivative. This leads to the following parametric representation of the squared angular momentum:

(45) |

where we use the letter to denote the value of along the sequence of circular orbits. Inserting this -parametric representation of in Eq. (19) defines the -parametric representation of the effective Hamiltonian . We can then obtain (at least numerically) as a function of by eliminating between and the corresponding -parametric representation of the frequency parameter obtained by the angular Hamilton equation of motion in the circular case

(46) |

where denotes the real EOB Hamiltonian

(47) |

In this situation, the binding energy of the system is simply given by

(48) |

where denotes, as above, the total mass of the system, and where one must eliminate between Eq. (46) and Eq. (48) to express the r.h.s. in terms of . Note that the function depends also on the choice of the following parameters: , and . Here we shall focus on the equal-mass case, and consider the dependence of only on and restrict the parametrization of 1PN tidal effects to the consideration of a single 1PN tidal parameter that is taken to be the same for the three values of that we consider. In addition, we will incorporate 1PN corrections to tidal effects in the three aforementioned functional forms, Eq. (39)-(41) and contrast their performances.

#### iv.1.2 BNS binding energy in the PN framework

We also want to constrast the performance of the EOB approach (which represents a resummation of the dynamics of the binary system) with the “standard” nonresummed PN-based description of the binding energy of tidally interacting BNS, as used for instance in Ref. Mora:2003wt . The PN-expanded binding energy is written in the form

(49) |

where

(50) |

is the 3PN accurate post-Newtonian binding energy of two point-masses as function of the orbital frequency parameter Damour:1999cr ; Damour:2001bu . The expression of the tidal contribution can be obtained for all values of the multipolar index by noting the following. Any (perturbative) power-law radial contribution to the interaction Hamiltonian of the form

(51) |

is easily shown to contribute a corresponding term

(52) |

where it should be noted that the sign of the tidal contribution flips between the Hamiltonian and the binding energy expressed as a function of the orbital frequency ( denoting the Newtonian value of corresponding to a given circular orbit of frequency ). As a result, we have the leading order contribution to the PN-tidal contribution

(53) |

We shall also explore the effect of correcting by a fractional 1PN contribution, i.e. to employ a PN tidal contribution of the form

(54) |

where the (approximate) link with the previously defined is

(55) |

Here the numerical coefficient arises as a consequence of the factor in the result above (considered for and ).

### iv.2 BNS circular configurations in numerical relativity

Model | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

2H | 13.847 | 1.35692 | 3 | 3 | 3 | 0.13097 | 1.3507 | 15.229 | 0.1342 | 0.0407 | 0.0168 |

HB | 14.151 | 1.35692 | 3 | 3 | 3 | 0.17181 | 1.3507 | 11.608 | 0.0946 | 0.0260 | 0.0097 |

2B | 14.334 | 1.35692 | 3 | 3 | 3 | 0.20500 | 1.3505 | 9.728 | 0.0686 | 0.0174 | 0.0059 |

SLy | 14.165 | 1.35692 | 3.005 | 2.988 | 2.851 | 0.17385 | 1.3499 | 11.466 | 0.0928 | 0.0254 | 0.0095 |

FPS | 14.220 | 1.35692 | 2.985 | 2.863 | 2.600 | 0.18631 | 1.3511 | 10.709 | 0.0805 | 0.0214 | 0.0077 |

BGN1H1 | 14.110 | 1.35692 | 3.258 | 1.472 | 2.464 | 0.15792 | 1.3490 | 12.614 | 0.1059 | 0.0307 | 0.0120 |

#### iv.2.1 Numerical framework of Ury et al.

In a recent paper, Ury et al. Uryu:2009ye constructed BNS systems in quasi-circular orbits by solving numerically the full set of Einstein’s equations. The important advance of this work with respect to previous analyses is the fact that Einstein equations are solved for all metric components, including the nonconformally flat part of the spatial metric. This goes beyond the common conformally flat approximation that is usually employed for the spatial geometry. The conformally flat approximation introduced systematic errors which enter the PN expansion already at the 2PN level [see the detailed calculation in the Appendix B of Ref. Damour:2000we ]. Consistently with this analytical argument, it was found in Ref. Uryu:2009ye that the difference between conformally flat and nonconformally flat calculations is so large that it can mask the effect of tidal interaction for close systems. See, in this respect, the location of the conformally flat (IWM) binding energy curves in the two upper panels of Fig. 3 in Ref. Uryu:2009ye . Below we shall however emphasize that the nonconformally flat calculations of Uryu:2009ye still introduce significant systematic errors which enter the PN expansion at the 3PN level.

Since the new nonconformally flat results of Ury et al. represent a definitive improvement with respect to previous calculations, it is appealing to see to what extent these new results agree with existing analytical descriptions. We extracted from Ref. Uryu:2009ye the six models which present the highest computational accuracy. These models were obtained by using EOS labelled 2H,HB,2B, SLy, FPS and BGN1H1. These labels refer to piecewise polytropic EOS. Note that in the case of SLy, FPS and BGN1H1 the corresponding piecewise polytropic EOS were proposed in Ref. Read:2008iy as approximations to original tabulated EOS. In the case of FPS and SLy, this implies that the tidal coefficients that we have computed for this work differ (by ) from the ones that we had previously computed in Ref. Damour:2009vw that used the original tabulated EOS. For example, in the case of a neutron star model described by the SLy EOS and having a compactness (which corresponds to a mass of ), we obtain a dimensionless Love number (which is consistent with the first line of Table I of Ref. Hinderer:2009 ) if we use the tabulated EOS, while we obtain