# Super-soft symmetry energy encountering non-Newtonian gravity in neutron stars

###### Abstract

Considering the non-Newtonian gravity proposed in the grand unification theories, we show that the stability and observed global properties of neutron stars can not rule out the super-soft nuclear symmetry energies at supra-saturation densities. The degree of possible violation of the Inverse-Square-Law of gravity in neutron stars is estimated using an Equation of State (EOS) of neutron-rich nuclear matter consistent with the available terrestrial laboratory data.

###### pacs:

26.60.-c, 97.60.Jd, 14.70.pWThe density dependence of nuclear symmetry energy is an important ingredient for understanding many interesting phenomena in astrophysics, cosmology Sum94 (); Bom01 (); Lat04 (); Ste05a () and nuclear physics LiBA98 (); Dan02a (); Bar05 (); LCK08 (); Bro00 (). However, theoretical predictions on the especially at supra-saturation densities are currently rather diverse LCK08 (); Bro00 (); Kut94 (); Kub99 (); Szm06 (); Dip03 (). Unfortunately, there is no known first-principle guiding the high-density behavior of the . Presently, while many theories, see, e.g., refs. Ste05a (); Lee98 (); Hor01a (); Dip03 (); Che07 (); LiZH06 (), predict that the increases continuously at all densities, many other models, see, e.g., refs. Pan72 (); Fri81 (); Wir88a (); eft (); Kra06 (); Bro00 (); Cha97 (); Sto03 (); Che05b (); Dec80 (); Das03 (); Kho96 (); Bas07 (); MS (); Ban00 (); Ch09 (), predict that the first increases and then decreases above certain supra-saturation densities. The may even become negative at high densities Bom01 (); Wir88a (); LiBA98 (); LCK08 (); Bro00 (); Sto03 (); Kut94 (); Kub99 (); Szm06 (). This latter kind of symmetry energy functions are generally regarded as being soft. Some (e.g., the UV14+TNI in Wir88a () and group II in Sto03 ()) of them can describe very well all observed properties of neutron stars (NSs). However, the super-soft ones (e.g., the original Gogny-Hartree-Fock (GHF) prediction Das03 () and group III in Sto03 ()) that quickly drops to zero around three times the saturation density either can not keep the NSs stable or predict maximum NS masses significantly below depending on the EOS used for symmetric nuclear matter. Given the above theoretical situation, experimental indications on the high density are thus utmost important. Very interestingly, circumstantial evidence for a super-soft Xiao09 () was found very recently from analyzing the FOPI/GSI experimental data on the ratio in relativistic heavy-ion collisions Rei07 () within a transport model IBUU04 () using the MDI (Momentum-Dependent-Interaction) EOS Das03 (). While the symmetric part of the MDI EOS is consistent with the existing terrestrial nuclear laboratory dataDan02a (); LCK08 (), the total pressure of NS matter obtained using the super-soft (which is actually the original GHF prediction) preferred by the FOPI/GSI data can not keep neutron stars stable. Among possibly many important ramifications in astrophysics and cosmology, this finding posts immediately a serious scientific challenge: how can the NSs be stable with such kind of super-soft symmetry energies? In fact, this question has been raised repeatedly and the answer has been negative long before any experimental indication was available. In the literature, the super-soft symmetry energies were often regarded by some people as either “unpleasant”, see, e.g., Cha97 (), or “unphysical”, see, e.g., Glen (); Sto03 (); Stone05 (). These assertions, of course, are all based on the assumption that gravity is well understood. However, it is really remarkable that gravity, despite being the first to be discovered, is actually still considered by far the most poorly understood force Pea01 (); Hoy03 (); Ark98 (). In fact, in pursuit of unifying gravity with the three other fundamental forces, conventional understanding about gravity has to be modified due to either the geometrical effect of the extra space-time dimensions predicted by string theories and/or the exchange of the weakly interacting bosons newly proposed in the super-symmetric extension of the Standard Model, see, e.g., refs. Fis99 (); Adel03 () for reviews. Consequently, the Inverse-Square-Law (ISL) of gravity is expected to be violated. In stable neutron stars at equilibrium which is determined by the weak and electromagnetic interactions, the gravity has to be balanced by the strong interaction. Neutron stars are thus a natural testing ground of grand unification theories. In this Letter, we show that the super-soft preferred by the FOPI/GSI data can readily keep neutron stars stable if the non-Newtonian gravity is considered.

The deviation from the ISL of gravity can be characterized effectively by adding a Yukawa term to the normal gravitational potential Fujii71 (); Long03 (), i.e.,

(1) |

where is a dimensionless strength parameter, is the length scale and is the universal gravitational constant. Alternatively, the Yukawa term can also be considered as due to the putative “fifth force” Fis99 (); Adel03 (); Fujii71 () coexisting with gravity or a non-universal gravitational “constant” Fis99 (); Uzan03 () of . In the scalar/vector boson exchange picture, and (in natural units). The , and are the boson-baryon coupling constant, the boson and baryon mass, respectively. To reduce gravity from the ISL, the exchange of a vector boson is necessary. It is worth noting that a neutral spin-1 vector -boson has been a favorite candidate. It is very weakly coupled to baryons Kri09 (), can mediate the interactions among Dark Matter (DM) candidates Fayet (); Boe04 () and has been used to explain the 511 keV -ray observation from the galactic bulge Jean03 (); Boehm04a (); Zhu07 ().

According to Fujii Fuj2 (), the Yukawa term is simply part of the matter system in general relativity. Consequently, the Einstein equation remains the same and only the EOS is modified. Within the mean-field approximation, the extra energy density due to the Yukawa term is Long03 (); Kri09 ()

(2) |

where is the normalization volume, is the baryon number density and . The corresponding addition to the pressure is then Assuming a constant boson mass independent of the density, one obtains For the purposes of the present study, it is sufficient to consider neutron stars as simply consist of neutrons (n), protons (p) and electrons (e). Including the Yukawa term the total pressure inside neutron stars is . For the inner and outer crusts we use for the EOS of Carriere et al. Hor03 () and that of Baym et al. BPS (), respectively. They are smoothly connected to the EOS in the core Xu09 (). For the latter we use . The value of the isospin asymmetry at equilibrium is determined by the chemical equilibrium condition and the charge neutrality requirement . The and obtained consistently within the modified GHF approximation are Das03 (); Xu09 (), respectively,

(3) | |||||

where is the Fermi momentum for symmetric nuclear matter at density . The coefficients and . The values of the parameters are , MeV, MeV, MeV and (Das03, ). The resulting symmetric EOS contribution to the pressure is consistent with that extracted from studying kaon production and nuclear collective flow in relativistic heavy-ion collisions using hadronic transport models assuming no hadron to Quark-Gluon Plasma phase transition up to about Dan02a (); LCK08 (). The parameter in Eq. 3 was introduced to vary the density dependence of the without changing any property of symmetric nuclear matter and the value of MeV Das03 (). Shown in the inset of Fig.1 are two typical denoted as MDIx1 and MDIx0 obtained by using and , respectively. While the MDIx0 increases continuously, the MDIx1 becomes negative above . Only the MDIx1 can reproduce the FOPI/GSI pion production data within the transport model analysis Xiao09 (). It is seen that the corresponding MDIx1 pressure decreases with increasing density as shown with the lowest curve in Fig. 1. However, the Yukawa term makes the pressure grow continuously with increasing density with a value of higher than about 10 GeV.

Shown in Fig. 2 is the mass-radius relation of static neutron stars obtained from solving the Tolman-Oppenheimer-Volkoff (TOV) equation using the MDIx1 and various values for the . The result obtained using the MDIx0 without including the Yukawa term is included as a reference LiBA06 (). The causality Lat04 () and rotational constraint Haensel09 () are also shown. The Keplerian (mass-shedding) frequency is approximately Haensel09 () So far, the fastest pulsar observed is the J1748-2446ad spinning at 716 Hz Hessels06 (). Taking 716 Hz as the Keplerian frequency, the M-R relation is restricted to the left side of the rotational limit. The latter restricts the value of to less than 150 GeV. It is seen that to produce a neutron star with a maximum mass above , the has to be higher than about 50 GeV. More specifically, with the MDIx1 and the GeV, or equivalently , neutron stars can have a maximum mass between 1.4 and 2.5 and a corresponding radius between 12 and 18 km.

For canonical neutron stars of 1.4 , the radius is quite sensitive to the value used. Thus, besides the accurate measurement of neutron star radii, additional measurements related to the mass distribution, such as the moment of inertia, will be very useful in setting astrophysical constraints on the and . According to Lattimer and Schutz Lattimer05 (), at the slow rotation limit the moment of inertia can be well approximated as

Shown in Fig. 3 is the as a function of M. For , the MDIx0 without the Yukawa contribution gives an no more than Worley08 (). However, significantly larger values are obtained with the MDIx1 and the Yukawa contribution. The discovery of the double-pulsar system PSR J0737-3039 provides a great opportunity to determine accurately the moment of inertia of the star Lyne04 (); Lattimer07 (). Our results shown here add to the importance of measuring the moment of inertia accurately.

To constrain the values of and has been a longstanding goal of many terrestrial experiments and astrophysical observations as limits on them may provide useful guidance for developing grand unification theories, see, e.g., refs. Fis99 (); Bordag01 (); Adel03 (); Fujii71 (); Gib81 (); Ser05 (); Decca05 (); Ade07 (); Kap07 (); Kamyshkov08 (). These studies have estimated various upper limits on the . In the range of and m, there is a clear trend of increased strength at shorter length . What we have constrained is the value of or equivalently the from the pressure necessary to support both static neutron stars and the fastest pulsars. While we expect that the range parameter has to be much larger (smaller) than the radii of finite nuclei (neutron stars), we can not set separate constraints on the values of and . Compared to other efforts to constrain the and , our study here is unique in that the estimated minimum value of is a lower limit satisfying all known constraints from both terrestrial nuclear experiments and observations of global properties of neutron stars. Moreover, very interestingly, our estimated range of overlaps well with the upper limits estimated from analyzing the neutron-proton and neutron-lead scattering data in the range of m Kamyshkov08 (); BARB75 (); POKO06 (); NESV08 ().

In summary, neutron stars are a natural testing ground of grand unification theories of fundamental forces. Considering the possible violation of the ISL of gravity, the stability and observed properties of NSs can not rule out super-soft symmetry energies at supra-saturation densities. Given the uncertainties and model dependence involved in extracting information about the EOS and symmetry energy from heavy-ion reactions, it is very important to test the possible super-soft symmetry energy at supra-saturation densities using several observables simultaneously from independent experiments analyzed using different models. If confirmed, it may point towards a violation of the ISL in neutron stars.

We would like to thank M. I. Krivoruchenko for useful communications and G. C. Yong, C. Xu and J. Xu for helpful discussions. The work is supported in part by the National Natural Science Foundation of China under Grant No. 10647116, 10710172, 10575119, 10675082 and 10975097, the Young Teachers’ Training Program from China Scholarship Council under Grant No. 2007109651, the MOE of China under project NCET-05-0392, Shanghai Rising-Star Program under Grant No.06QA14024, the SRF for ROCS, SEM of China, and the National Basic Research Program of China (973 Program) under Contract No.2007CB815004, the US National Science Foundation under Grants No. PHY0652548 and No. PHY0757839, the Research Corporation under Award No. 7123 and the Texas Coordinating Board of Higher Education Grant No.003565-0004-2007.

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