# Neutron star tidal deformabilities constrained by chiral effective field theory

###### Abstract

We investigate neutron star tidal deformabilities from cold dense matter equations of state constrained in the low-energy regime (up to twice nuclear saturation density) by chiral effective field theory. We generate over 72,000 energy density functionals fitted simultaneously to the neutron matter equation of state and the properties of symmetric nuclear matter around saturation density. These functionals are then used to compute consistently the neutron star equation of state from the outer crust to the inner core, assuming a composition consisting of protons, neutrons, and electrons. Our results for the neutron star mass-radius relation and dimensionless tidal deformabilities are within the range of new constraints deduced from gravitational wave and electromagnetic observations of the GW170817 event. In particular we find that for a 1.4 solar-mass neutron star the tidal deformability lies in the range . Our predictions for the maximum neutron star mass are largely consistent with new upper bounds predicted from numerical relativity simulations, but we find that lower bounds on the neutron star tidal deformability may be very important for constraining theories of the dense matter equation of state. Finally, we investigate correlations between the density-dependent isospin asymmetry energy and the tidal deformability.

###### pacs:

21.30.-x, 21.65.Ef,[name=Yeunhwan, color=magenta]YH

## I Introduction

Gravitational wave and electromagnetic signals from binary neutron star mergers offer a unique probe for studying the properties of ultra-dense matter. The recent observation of gravitational wave event GW170817 Abbott et al. (2017a) and the associated electromagnetic counterpart Abbott et al. (2017b) suggest the source to be a merger of two neutron stars with combined mass that left behind a relatively long-lived hypermassive neutron star remnant. Measurements of the late inspiral gravitational waveform from GW170817 were sufficient to place a conservative upper limit of on the tidal deformability of a neutron star, competitive with bounds Steiner et al. (2015) deduced from current neutron star mass and radius measurements. Subsequent works Bauswein et al. (2017); Margalit and Metzger (2017); Shibata et al. (2017); Radice et al. (2018); Rezzolla et al. (2018); Ruiz et al. (2018); Annala et al. (2017) have inferred constraints on a broader set of bulk neutron star properties such as the maximum mass Bauswein et al. (2017); Margalit and Metzger (2017); Shibata et al. (2017); Rezzolla et al. (2018); Ruiz et al. (2018), radii Bauswein et al. (2017); Margalit and Metzger (2017); Annala et al. (2017), and tidal deformabilities Radice et al. (2018); Annala et al. (2017) from a combination of observational data and numerical relativity simulations. Ultimately, it will be equally important to infer complementary constraints on specific properties of the dense matter equation of state itself Krastev and Li (2018), such as the symmetry energy and its density dependence.

In the present work we study neutron star masses, radii, and tidal deformabilities from a large class of energy density functionals constrained by state of the art chiral effective field theory Weinberg (1979); Epelbaum et al. (2009); Machleidt and Entem (2011) predictions for the cold neutron matter equation of state and empirical information on the symmetric nuclear matter equation of state in the vicinity of nuclear saturation density. The work builds upon previous studies Lim and Holt (2017); Zhang et al. (2017) by the authors in which traditional and extended Skyrme effective interactions were fitted to constraints from chiral effective field theory and energies of doubly-magic nuclei. Chiral effective field theory has been used in the past to predict neutron star radii and masses Hebeler et al. (2010, 2013) and their impact on gravitational wave measurements Bauswein et al. (2012) by extending the neutron matter equation of state to higher densities using piecewise polytropes. For instance, a neutron star was found to have a radius in the range . The stiffest equations of state considered in Ref. Hebeler et al. (2010) generate neutron stars with a maximum mass up to nearly . Recent numerical relativity simulations Bauswein et al. (2017); Margalit and Metzger (2017); Shibata et al. (2017); Rezzolla et al. (2018); Ruiz et al. (2018) that place an upper bound on the maximum mass of a nonrotating spherical neutron star, , may therefore help to rule out possible equations of state generated from extrapolating chiral effective field theory results to higher densities and thereby better constrain our theories of dense nuclear matter.

A main purpose of the present study is to investigate as well the extent to which lower bounds on the tidal deformability Radice et al. (2018) can reduce the range of allowed neutron star equations of state. In our modeling the maximum neutron star mass naturally falls below about , but many of the equations of state produce neutron stars with small tidal deformabilities. A second motivation is to develop a framework for implementing future constraints on the nuclear equation of state from measurements of neutron star masses, radii, and tidal deformabilities together with theory constraints from chiral effective field theory.

The paper is organized as follows. In Section II we present the equations relating the tidal deformability to the nuclear equation of state and describe our method for implementing constraints from chiral effective field theory and the properties of symmetric nuclear matter. We present results for the neutron star mass vs. radius relation in Section III together with predictions for the tidal deformability as a function of mass. We then explore correlations between the tidal deformability and the slope of the isospin asymmetry energy at nuclear saturation density. We end with a summary and conclusions.

## Ii Tidal deformability from the neutron star equation of state

The gravitational wave signal from the late inspiral phase of binary neutron star coalescence is connected Hinderer (2008); Read et al. (2009) to the neutron star equation of state through the dimensionless tidal deformability , which can be determined from the Love number defined through

where is the neutron star compactness and is the solution at the neutron star surface to the first order differential equation

(2) | ||||

Here is the energy density and is the pressure obtained from the equation of state. In Eq. (2), is the metric function for a spherical star

(3) |

and

(4) | ||||

These equations are solved together with the Tolman-Oppenheimer-Volkoff equations for hydrostatic equilibrium to obtain the neutron star mass vs. radius relation and tidal deformabilities. The tidal deformability of neutron stars has been studied Postnikov et al. (2010); Hinderer et al. (2010); Read et al. (2013); Lackey and Wade (2015); Hotokezaka et al. (2016) using many equations of state, including polytropes, realistic nuclear interaction models, and even including the presence of quark matter. The gravitational event GW170817 has led to many new efforts Fattoyev et al. (2017); Krastev and Li (2018) to predict tidal deformabilities for inspiraling neutron stars.

We take as a starting point for the calculation of the neutron star equation of state a range of predictions from chiral effective field theory (EFT) for the energy per particle of neutron matter as a function of the nucleon number density Coraggio et al. (2013); Holt and Kaiser (2017); Holt et al. (2016). Three-body forces are included at next-to-next-to-leading order (N2LO) in the chiral expansion, and progress toward the consistent inclusion of N3LO three-body forces is being made Tews et al. (2013); Drischler et al. (2016). The chiral interactions considered in the present work have been used extensively in studies of nuclear dynamics and thermodynamics (for recent reviews, see Refs. Holt et al. (2013, 2016)). In order to assess the theoretical uncertainties, we vary the resolution scale MeV and the chiral order of the underlying nucleon-nucleon interaction Entem and Machleidt (2003); Coraggio et al. (2007, 2013, 2014); Sammarruca et al. (2015). Uncertainties associated with the order-by-order convergence of the ground state energy in many-body pertubation theory are much smaller (for details see Ref. Holt and Kaiser (2017)) and neglected in the present work. In the inset to Fig. 1 we show the resulting neutron matter equation of state up to , where fm, from the five chiral nuclear potentials considered.

In order to extrapolate the neutron matter equations of state to arbitrary proton fractions and higher densities, we construct a set of 72,849 energy density functionals of the form

(5) | ||||

where

(6) |

is the proton fraction, and . Keeping fewer terms in the expansion of the energy in powers of the density results in a poorer description of the density dependence of the neutron matter equations of state. We assume a quadratic dependence of the energy per particle on the isospin asymmetry, , as in Refs. Wiringa et al. (1988); Bombaci and Lombardo (1991); Wellenhofer et al. (2015); Papakonstantinou et al. (2018). Variational calculations performed by Lagaris and Pandharipande Lagaris and Pandharipande (1981) found that higher-order terms are negligible, even though a power series expansion in generically breaks down Wellenhofer et al. (2016).

Starting from the chiral EFT predictions up to , we set the upper and lower bounds and construct a set of 41 interpolating equations of state, evenly spaced in energy at each chosen density. While many more equations of state would fit within the allowed uncertainty band, our present method retains a proper correlation between the symmetry energy and slope of the symmetry energy .

We first determine the unknown coefficients in Eq. (6) related to pure neutron matter by fitting the energy per baryon () up to . For symmetric nuclear matter, we start with empirical constraints from Ref. Dutra et al. (2012):

(7) | ||||

where

(8) |

evaluated at . We then obtain from the analysis of 205 Skyrme force models the Gaussian probability distribution functions

(9) |

for the four nuclear matter properties in Eq. (7). In Table 1 we show the resulting mean values and standard deviations for the quantities and .

Nuc. Prop. | unit | ||
---|---|---|---|

In constructing the symmetric nuclear matter equation of state, we sample uniformly {} from the bounds in Eq. (7), but in the final analysis of neutron star masses and radii we assign them Gaussian weights according to Table 1. From the values of and we can fix the remaining unknown coefficients in Eq. (6). Fig. 1 shows the resulting uncertainty bands on the nuclear and neutron matter equations of state up to .

Once the energy density functional in Eq. (5) is fixed, we construct the full neutron star equation of state from outer crust to inner core. Certain combinations of the neutron matter and nuclear matter equations of state lead to unphysical behavior at very high densities. We remove from the complete set those equations of state for which pure neutron matter has a lower energy per particle than symmetric nuclear matter at some density present in the neutron star. We also ensure that the speed of sound remains subluminal for all densities present in the neutron star. This reduces the initially large number of equations of state down to 72,849 which we use for subsequent statistical analysis. Compared with previous calculations for the tidal deformability , we construct a crust equation of state using the liquid drop model technique for each nuclear force model. This is a unified approach that allows the inclusion of nuclear pasta phases and is necessary for the consistent treatment of the neutron star equation of state. Additional details can be found in Ref. Lim and Holt (2017).

() | (km) | (km) | (km) | (km) | (km) |
---|---|---|---|---|---|

1.0 | 11.58 | 11.89 | 12.27 | 12.60 | 12.89 |

1.1 | 11.62 | 11.92 | 12.29 | 12.60 | 12.89 |

1.2 | 11.66 | 11.95 | 12.30 | 12.60 | 12.88 |

1.3 | 11.66 | 11.95 | 12.29 | 12.58 | 12.87 |

1.4 | 11.65 | 11.94 | 12.27 | 12.56 | 12.84 |

1.5 | 11.62 | 11.90 | 12.24 | 12.52 | 12.80 |

1.6 | 11.56 | 11.84 | 12.18 | 12.46 | 12.74 |

1.7 | 11.46 | 11.76 | 12.09 | 12.38 | 12.66 |

1.8 | 11.32 | 11.64 | 11.98 | 12.26 | 12.55 |

1.9 | 11.11 | 11.46 | 11.81 | 12.11 | 12.40 |

2.0 | 10.75 | 11.18 | 11.55 | 11.88 | 12.18 |

## Iii Results

In Fig. 2 we show the mass vs. radius distribution that results from a statistical analysis of the nearly 73,000 equations of state with Gaussian weight obtained from the construction described in Section II. We have shown for comparison in the enclosed “dashed” region the mass vs. radius constraints obtained by analyzing X-ray burst data from Steiner et al. Steiner et al. (2010). We observe that for a neutron star, the radius lies within the range and the distribution peaks just under km. We find a relatively small probability for obtaining a maximum neutron star mass larger than due to our removal of equations of state with superluminar speeds of sound. Our results are therefore already consistent with most of the recent numerical relativity simulations Bauswein et al. (2017); Margalit and Metzger (2017); Shibata et al. (2017); Rezzolla et al. (2018); Ruiz et al. (2018) that have imposed upper bounds on the neutron star maximum mass around . In Table 2 we show the 68% and 95% confidence intervals for the neutron star radius as a function of mass for the range . Generically the range of allowed radii decreases with increasing mass.

() | |||||
---|---|---|---|---|---|

1.0 | 2712.86 | 2981.06 | 3270.50 | 3522.92 | 3769.92 |

1.1 | 1584.90 | 1751.70 | 1930.08 | 2086.54 | 2238.42 |

1.2 | 950.27 | 1056.60 | 1169.93 | 1269.87 | 1365.85 |

1.3 | 579.43 | 649.43 | 723.35 | 788.82 | 851.16 |

1.4 | 357.51 | 404.35 | 453.39 | 497.07 | 538.43 |

1.5 | 221.40 | 253.36 | 286.42 | 316.05 | 343.85 |

1.6 | 136.66 | 158.71 | 181.23 | 201.53 | 220.43 |

1.7 | 83.19 | 98.56 | 114.01 | 128.05 | 141.05 |

1.8 | 49.11 | 59.93 | 70.57 | 80.32 | 89.29 |

1.9 | 27.20 | 34.91 | 42.24 | 49.05 | 55.27 |

2.0 | 13.02 | 18.52 | 23.59 | 28.36 | 32.70 |

In Fig. 3 we show the dimensionless tidal deformability and associated statistical uncertainties as a function of the neutron star mass for both full neutron star structure (upper right) and neglecting the contribution from the neutron star crust (lower left). We see that increases rapidly as the mass of the neutron star decreases. In fact, we find that the relation between and the neutron star mass follows the simple relation:

(10) |

For our results, we find . This formula gives about a 20% error on the tidal deformability of a neutron star and about a 10% error for that of a neutron star.

For a neutron star, we find a range of tidal deformabilities . Given the smaller range of allowed radii for more massive stars, the spread in the tidal deformability is also naturally much tighter than for lower-mass neutron stars. We see that only neutron stars with have tidal deformabilities larger than , and therefore nearly all of the present equations of state are consistent with the upper bound of from GW170817 and x-ray observations. Potentially much more restrictive to our present theories of the dense matter equation of state would be lower bounds on the tidal deformability, such as the recent suggestion Radice et al. (2018) that is needed for a binary neutron star merger remnant to avoid an immediate or short-timescale collapse to a black hole, where

(11) |

In Fig. 4 we plot against the chirp mass . Note that the chirp mass for equal mass companions is smaller by a factor of . For neutron star merger GW170817 the chirp mass was found to be , and our 95% confidence interval leads to a value of , which lies just above the lower bound predicted in Ref. Radice et al. (2018). In Table 3 we also present uncertainty bounds at the 68% and 95% confidence levels for the dimensionless tidal deformability for neutron stars with masses ranging from .

Combining our predictions for neutron star radii and tidal deformabilities from the constrained energy density functionals constructed in this work, we plot in Fig. 5 the two-dimensional probability contour for and assuming a neutron star. Larger values of the neutron star compactness (smaller radii for a given ) naturally lead to smaller tidal deformations, since in this case the neutron star will be less affected by gradients of the gravitational field.

Neutron star radii are strongly correlated with the slope of the symmetry energy at nuclear saturation density, and therefore we anticipate a similar correlation between and the tidal deformability. In Fig. 6 we plot the two-dimensional correlation between and for a neutron star. We extract a linear relation from the statistical analysis of our results:

(12) |

where and MeV fit the data with correlation coefficient . Our result does not give a very strong correlation between and because EFT neutron matter calculations give a relatively small uncertainty in at nuclear saturation density, and consequently is highly localized at the most probable zone. Since has a strong correlation with neutron skin thicknesses, , in medium-mass and heavy nuclei, it is also possible to derive an empirical relation between and . Roca-Maza et. al. Roca-Maza et al. (2011) obtained an empirical formula between the neutron skin of Pb and that implies a neutron skin thickness of when we apply Eq. (12) and the gravitational wave constraint . The lower bound on the tidal deformation tell us that .

During the preparation of the manuscript, we became aware of a very similar study Most et al. (2018) that reaches some of the same conclusions as our work. In particular, the authors of Ref. Most et al. (2018) employ a different set of chiral nuclear potential models to construct the neutron matter equation of state, which they extrapolate to higher densities using piecewise polytropes. In contrast to our equations of state, those in Ref. Most et al. (2018) are strongly constrained by new upper bounds on the maximum neutron star mass. Both analyses, however, point to the importance of lower bounds on the tidal deformability for placing limitations on equation of state modeling. In comparison to Ref. Most et al. (2018), our predictions for the mass vs. radius relation are similar but we find radii that are systematically lower by about km but with a similar uncertainty band of km for neutron stars.

## Iv Conclusion

In this work we have computed neutron star masses, radii, and tidal deformabilities from equations of state fitted to chiral effective field theory predictions for pure neutron matter together with statistical modeling of the symmetric nuclear matter equation of state around saturation density. We have found that neutron star crusts have an important impact on the tidal deformability, and our models provide a consistent description of the outer crust, inner crust, and core of a neutron star in beta equilibrium. Since our uncertainties regarding the properties of dense nuclear matter grow rapidly above , tightening the upper and lower bounds on the tidal deformability with future binary neutron star merger observations will be invaluable for further constraining the nuclear equation of state, and the present work provides the framework for such a program.

###### Acknowledgements.

We thank James M. Lattimer for useful discussions. Work supported by the National Science Foundation under Grant No. PHY1652199. Portions of this research were conducted with the advanced computing resources provided by Texas A&M High Performance Research Computing.## References

- Abbott et al. (2017a) B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 119, 161101 (2017a).
- Abbott et al. (2017b) B. P. Abbott et al., Astrophys. J. Lett. 848, L12 (2017b).
- Steiner et al. (2015) A. W. Steiner, S. Gandolfi, F. J. Fattoyev, and W. G. Newton, Phys. Rev. C 91, 015804 (2015).
- Bauswein et al. (2017) A. Bauswein, O. Just, H.-T. Janka, and N. Stergioulas, Astrophys. J. Lett. 850, L34 (2017).
- Margalit and Metzger (2017) B. Margalit and B. D. Metzger, Astrophys. J. Lett. 850, L19 (2017).
- Shibata et al. (2017) M. Shibata, S. Fujibayashi, K. Hotokezaka, K. Kiuchi, K. Kyutoku, Y. Sekiguchi, and M. Tanaka, Phys. Rev. D 96, 123012 (2017).
- Radice et al. (2018) D. Radice, A. Perego, F. Zappa, and S. Bernuzzi, Astrophys. J. Lett. 852, L29 (2018).
- Rezzolla et al. (2018) L. Rezzolla, E. R. Most, and L. R. Weih, Astrophys. J. Lett. 852, L25 (2018).
- Ruiz et al. (2018) M. Ruiz, S. L. Shapiro, and A. Tsokaros, Phys. Rev. D 97, 021501 (2018).
- Annala et al. (2017) E. Annala, T. Gorda, A. Kurkela, and A. Vuorinen, arXiv:1711.02644 (2017).
- Krastev and Li (2018) P. G. Krastev and B.-A. Li, arXiv:1801.04620 (2018).
- Weinberg (1979) S. Weinberg, Physica A 96, 327 (1979).
- Epelbaum et al. (2009) E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Rev. Mod. Phys. 81, 1773 (2009).
- Machleidt and Entem (2011) R. Machleidt and D. R. Entem, Phys. Rept. 503, 1 (2011).
- Lim and Holt (2017) Y. Lim and J. W. Holt, Phys. Rev. C 95, 065805 (2017).
- Zhang et al. (2017) Z. Zhang, Y. Lim, J. W. Holt, and C.-M. Ko, Phys. Lett. B 777, 73 (2017).
- Hebeler et al. (2010) K. Hebeler, J. M. Lattimer, C. J. Pethick, and A. Schwenk, Phys. Rev. Lett. 105, 161102 (2010).
- Hebeler et al. (2013) K. Hebeler, J. M. Lattimer, C. J. Pethick, and A. Schwenk, Astrophys. J. 773, 11 (2013).
- Bauswein et al. (2012) A. Bauswein, H.-T. Janka, K. Hebeler, and A. Schwenk, Phys. Rev. D 86, 063001 (2012).
- Hinderer (2008) T. Hinderer, Astrophys. J. 677, 1216 (2008).
- Read et al. (2009) J. S. Read, C. Markakis, M. Shibata, K. b. o. Uryū, J. D. E. Creighton, and J. L. Friedman, Phys. Rev. D 79, 124033 (2009).
- Postnikov et al. (2010) S. Postnikov, M. Prakash, and J. M. Lattimer, Phys. Rev. D 82, 024016 (2010).
- Hinderer et al. (2010) T. Hinderer, B. D. Lackey, R. N. Lang, and J. S. Read, Phys. Rev. D 81, 123016 (2010).
- Read et al. (2013) J. S. Read, L. Baiotti, J. D. E. Creighton, J. L. Friedman, B. Giacomazzo, K. Kyutoku, C. Markakis, L. Rezzolla, M. Shibata, and K. Taniguchi, Phys. Rev. D 88, 044042 (2013).
- Lackey and Wade (2015) B. D. Lackey and L. Wade, Phys. Rev. D 91, 043002 (2015).
- Hotokezaka et al. (2016) K. Hotokezaka, K. Kyutoku, Y.-i. Sekiguchi, and M. Shibata, Phys. Rev. D 93, 064082 (2016).
- Fattoyev et al. (2017) F. J. Fattoyev, J. Piekarewicz, and C. J. Horowitz, (2017), arXiv:1711.06615 [nucl-th] .
- Coraggio et al. (2013) L. Coraggio, J. W. Holt, N. Itaco, R. Machleidt, and F. Sammarruca, Phys. Rev. C 87, 014322 (2013).
- Holt and Kaiser (2017) J. W. Holt and N. Kaiser, Phys. Rev. C 95, 034326 (2017).
- Holt et al. (2016) J. W. Holt, M. Rho, and W. Weise, Phys. Rept. 621, 2 (2016).
- Tews et al. (2013) I. Tews, T. Krüger, K. Hebeler, and A. Schwenk, Phys. Rev. Lett. 110, 032504 (2013).
- Drischler et al. (2016) C. Drischler, A. Carbone, K. Hebeler, and A. Schwenk, Physical Review C 94, 054307 (2016).
- Holt et al. (2013) J. W. Holt, N. Kaiser, and W. Weise, Prog. Part. Nucl. Phys. 73, 35 (2013).
- Entem and Machleidt (2003) D. R. Entem and R. Machleidt, Phys. Rev. C 68, 041001 (2003).
- Coraggio et al. (2007) L. Coraggio, A. Covello, A. Gargano, N. Itaco, D. R. Entem, T. T. S. Kuo, and R. Machleidt, Phys. Rev. C 75, 024311 (2007).
- Coraggio et al. (2014) L. Coraggio, J. W. Holt, N. Itaco, R. Machleidt, L. E. Marcucci, and F. Sammarruca, Phys. Rev. C 89, 044321 (2014).
- Sammarruca et al. (2015) F. Sammarruca, L. Coraggio, J. W. Holt, N. Itaco, R. Machleidt, and L. E. Marcucci, Phys. Rev. C 91, 054311 (2015).
- Wiringa et al. (1988) R. B. Wiringa, V. Fiks, and A. Fabrocini, Phys. Rev. C 38, 1010 (1988).
- Bombaci and Lombardo (1991) I. Bombaci and U. Lombardo, Phys. Rev. C 44, 1892 (1991).
- Wellenhofer et al. (2015) C. Wellenhofer, J. W. Holt, and N. Kaiser, Phys. Rev. C 92, 015801 (2015).
- Papakonstantinou et al. (2018) P. Papakonstantinou, T.-S. Park, Y. Lim, and C. H. Hyun, Phys. Rev. C 97, 014312 (2018).
- Lagaris and Pandharipande (1981) I. Lagaris and V. Pandharipande, Nuclear Physics A 369, 470 (1981).
- Wellenhofer et al. (2016) C. Wellenhofer, J. W. Holt, and N. Kaiser, Phys. Rev. C 93, 055802 (2016).
- Dutra et al. (2012) M. Dutra, O. Lourenço, J. S. Sá Martins, A. Delfino, J. R. Stone, and P. D. Stevenson, Phys. Rev. C 85, 035201 (2012).
- Steiner et al. (2010) A. W. Steiner, J. M. Lattimer, and E. F. Brown, The Astrophysical Journal 722, 33 (2010).
- Roca-Maza et al. (2011) X. Roca-Maza, M. Centelles, X. Viñas, and M. Warda, Phys. Rev. Lett. 106, 252501 (2011).
- Most et al. (2018) E. R. Most, L. R. Weih, L. Rezzolla, and J. Schaffner-Bielich, arXiv:1803.00549 (2018).