Nuclear constraints on gravitational waves from deformed pulsars
The recent direct detection of gravitational waves (GWs) from binary black hole mergers [1, 2] opens up an entirely new non-electromagnetic window into the Universe making it possible to probe physics that has been hidden or dark to electromagnetic observations. In addition to cataclysmic events involving black holes, GWs can be triggered by physical processes and systems involving neutron stars. Properties of neutron stars are largely determined by the equation of state (EOS) of neutron-rich matter, which is the major ingredient in calculating the stellar structure and properties of related phenomena, such as gravitational wave emission from elliptically deformed pulsars and neutron star binaries. Although the EOS of neutron-rich matter is still rather uncertain mainly due to the poorly known density dependence of nuclear symmetry energy at high densities, significant progress has been made recently in constraining the symmetry energy using data from terrestrial nuclear laboratories. These constraints could provide useful information on the limits of GWs expected from neutron stars. Here after briefly reviewing our previous work on constraining gravitational radiation from elliptically deformed pulsars with terrestrial nuclear laboratory data in light of the recent gravitational wave detection, we estimate the maximum gravitational wave strain amplitude, using an optimistic value for the breaking strain of the neutron star crust, for 15 pulsars at distances 0.16 kpc to 0.91 kpc from Earth, and find it to be in the range of , depending on the details of the EOS used to compute the neutron star properties. Implications are discussed.
PACS: 97.60.Gb, 97.60.Jd, 04.30.-w, 26.60.Kp, 21.65.Mn
Keywords: pulsars, gravitational waves, dense matter
Gravitational waves are tiny disturbances in space-time predicted by General Relativity in 1916. A century after the fundamental predictions of Albert Einstein, the Laser Interferometer Gravitational-Wave Observatory (LIGO) has detected directly gravitational waves from black hole mergers [1, 2]. This detection has the potential to transform profoundly our understanding of the Universe, as gravitational wave astronomy opens up the possibility to probe physics that has been hidden to current electromagnetic observations . Because gravity interacts extremely weakly with matter, gravitational waves carry much cleaner and detailed picture of their sources as opposed to their electromagnetic counterparts .
Several types of gravitational waves and their corresponding sources have been discussed in the literature: (i) Inspiral gravitational waves are triggered during the latest stages of compact binary systems where the two objects finally collide and merge. These systems typically consist of two neutron stars, two black holes, as demonstrated by LIGO [1, 2], or a neutron star and a black hole whose orbit decayed to the point that the two masses are about to coalesce. This requires one of the original stars to be massive enough to undergo collapse to a compact object without destroying its companion, and without disrupting the bound orbit . (ii) Burst gravitational waves are generated in sudden cataclysmic events, such as supernova explosions or gamma-ray bursts . A spherically symmetric explosion cannot lead to emission of GWs in General Relativity. In order to produce GWs a supernova must exhibit some asymmetry. It is known that many pulsars formed in supernovae have large speeds relative to neighboring stars (high ”birth kicks”), and this suggests strongly that certain spernovae do exhibit considerable non-spherical motion . (iii) Stochastic gravitational waves could even have been produced during the very early Universe, well before any stars had been formed, merely as a consequence of the dynamics and expansion of the Cosmos . Very similar to the Cosmic Micro-wave Background (CMB), which is an electromagnetic leftover from the Big Bang, these GWs originate from many independent events amounting to a Cosmic Gravitational-wave Background (CGB). Such gravitational waves may carry critical information about the very early Universe, as they would have been stretched out as the Cosmos expanded. (iv) Continuous gravitational waves are generated by sources with steady and well-defined frequency. Examples of such sources are binary neutron star or black hole systems orbiting the common center of mass, or a (rapidly) spinning neutron star with some long-living axial asymmetry .
Ground-based gravitational wave observatories, such as LIGO and VIRGO, will also allow studies of a large population of neutron stars. Additionally, in space eLISA(Evolved Laser Interferometric Space Antenna), a planned space mission by the European Space Agency expected to launch in the early 2020s, will provide an unprecedented instrument for GWs search and detection . The eLISA will survey the low-frequency gravitational-wave bandwidth and is expected to detect a wide variety of events and systems in Space.
Rotating neutron stars are considered to be one of the main prospects for sources of continuous GWs potentially detectable by the LIGO  and VIRGO (e.g. Ref. ) laser interferometric observatories. According to General Relativity, a perfectly symmetric rotating object self-bound by gravity does not generate GWs. To emit GWs over an extended time period, a pulsar needs to exhibit a long-living axial asymmetry, e.g., a ”mountain” on its surface . In the literature, several different mechanisms causing such asymmetries have been discussed:
Anisotropic stress built up during the crystallization period of the neutron star crust may be able to support long term asymmetries, such as static ”mountains” on the neutron star surface .
In addition, because of its violent formation in supernova the rotational axis of a neutron star may not necessarily be aligned with its principal moment of inertia axis, which results in a neutron star precession . Because of this, even if the pulsar remains symmetric with respect to its rotational axis, it generates GWs [12, 13].
Also, neutron stars have extremely strong magnetic fields, which could create magnetic pressure and, in turn, deform the pulsar, if the magnetic and rotational axes do not coincide .
These mechanisms generally cause a triaxial pulsar configuration. GWs are characterized by a very small dimensionless strain amplitude, . The magnitude of the strain amplitude depends on how much the pulsar is distorted from axial symmetry which depends on details of the EOS of neutron-rich matter. The EOS of nuclear matter under extreme pressure, density and/or isospin asymmetry is still largely uncertain and rather theoretically controversial. A major source of these uncertainties is the rather poorly known density dependence of the nuclear symmetry energy, which encodes the information about the energy associated with the neutron-proton asymmetry in neutron-rich matter, see, e.g. [15, 16, 17, 18, 19, 20]. Fortunately, both nuclear structures and reactions provide useful means to probe the in terrestrial nuclear laboratories . For example, several observables in heavy-ion reactions are known to be sensitive to the from sub-saturation to supra-saturation densities, see, e.g., Refs. [22, 23, 24, 25]. For a comprehensive recent review, see articles in Ref. . Thanks to the hard work of many people in both nuclear physics and astrophysical communities, considerable progress has been achieved in recent years in constraining the symmetry energy especially around and below the nuclear matter saturation density using data from both terrestrial experiments and astrophysical observations, see, e.g., Refs. [26, 27, 28, 29, 30]. In this chapter, we first briefly review our earlier work on constraining the GWs expected from elliptically deformed pulsars  in light of the recent direct gravitational wave detections [1, 2]. Applying several nucleonic EOSs constrained by laboratory nuclear experiments, we then estimate the GW strain amplitude for fifteen pulsars for selected neutron star configurations. Our focus is on illustrating effects of nuclear symmetry energy on the GW strain amplitude from deformed pulsars. Recent reviews on effects of the density dependence of nuclear symmetry energy on GWs from various neutron star oscillations , binary neutron star mergers , the pure general relativistic w-mode  as well as the EOS-gravity degeneracy  can be found in the referred references.
2. Strain amplitude of gravitational waves from deformed pulsars
For consistency, we first recall the formalism applied to compute the gravitational wave strain amplitude following closely the discussion of Ref. . A rotating neutron star generates GWs if it has some long-living axial asymmetry. As already discussed in the introduction, there are several mechanisms that could lead to stellar deformations, and in turn gravitational wave emission. Generally, such processes result in triaxial neutron star configuration, which in the quadrupole approximation, would generate GWs at twice the rotational frequency of the star . These waves are characterized by a strain amplitude at the Earth’s vicinity given by 
In the above equation is the neutron star rotational frequency, is the stellar principal moment of inertia, is its equatorial ellipticity, and is the distance to Earth. The ellipticity is related to the maximum quadrupole moment of the star via 
where for slowly rotating (and static) neutron stars can be written as 
In this expression is the breaking strain of the neutron star crust which is rather uncertain at present time. Although earlier studies estimated the value of the breaking strain to be in the range , more recent investigations using molecular dynamics suggested that the breaking strain could be as large as . Using a rather conservative value of in a previous work  we reported the nuclear constraints on the gravitational wave signals to be expected from several pulsars close to Earth. Here, to estimate the maximum value of , we take and revisit some key issues. From Eqs. (1) and (2) it is clear that does not depend on the moment of inertia , and the total dependence upon the underlying EOS is carried by the quadrupole moment . Thus Eq. (1) can be rewritten as
In a previous work  we have computed the moment of inertia of both static and rotating neutron stars. In the case of slowly rotating neutron stars, in Ref. , it has been shown that the following empirical relation
to compute the neutron star mass, , and radius, . We then proceed to calculate the moment of inertia and the quadrupole moment using Eq. (5) and Eq. (3) respectively, and the neutron star ellipticity, , via Eq. (2). For rotational frequencies up to , global properties of rotating neutron stars stay approximately constant . Therefore for slowly rotating stars, if one knows the pulsar’s rotational frequency and its distance from the Earth, the above formalism can be applied to estimate the GW strain amplitude. Then these estimates can be compared with the current upper limits for the sensitivity of the laser interferometric observatories, such as LIGO and VIRGO .
3. Model EOSs constrained by terrestrial nuclear experiments
For completeness, below we summarize relevant EOS details following closely Ref.. We assume a simple model of stellar matter of nucleons and light leptons (electrons and muons) in beta-equilibrium. For many astrophysical studies, it is more convenient to express the EOS in terms of the pressure as a function of density and isospin asymmetry where and are the densities of neutrons and protons. In Fig. 2. we show the pressure as a function of density for two extreme cases: symmetric (upper panel) and pure neutron matter (lower panel). We pay particular attention to the EOS calculated with the MDI [43, 44] (momentum-dependent) interaction because its symmetry energy has been constrained in the subsaturation density region by the available nuclear laboratory data . Here we emphasize that the EOS of symmetric nuclear matter with the MDI interaction is constrained by the available data on collective flow in relativistic heavy-ion collisions . The single-particle potential corresponding to the MDI EOS is given by
where () for neutrons (protons), , , , , ,, , and are parameters discussed in Ref. . The parameter is introduced in Eq. (8) to account for the largely uncertain density behavior of the nuclear symmetry energy as predicted by various models of the nuclear interaction and/or many-body approaches. As illustrated in Fig. 3., the different values of result in a wide range of possible behaviors for the density dependence of the nuclear symmetry energy. It was demonstrated that only EOSs with in the range between -1 and 0 have symmetry energies consistent with the isospin-diffusion laboratory data  and the fiducial value for the neutron-skin of . We therefore consider only these two limiting cases in calculating the boundaries of the possible neutron star configurations.
The first column identifies the equation of state. The remaining columns exhibit the following quantities at the nuclear saturation density: saturation (baryon) density; energy-per-particle; compression modulus; nucleon effective mass; symmetry energy. Taken from Ref. .
For modeling neutron stars, at baryon densities below approximately the EOSs used in this work are supplemented by a crustal EOS, which is more applicable at lower densities. For the inner crust we apply the EOS by Pethick et al.  and for the outer crust the one by Haensel and Pichon . At higher densities we assume a continuous functional for the equations of state applied here. (For details of the extrapolation procedure of the DBHF+Bonn B EOS see Ref. .) The MDI EOS has been used in various calculations of neutron star properties and related phenomena. For example, it has been used to constrain the neutron star radius  with an estimated range consistent with the observational data . It has been also applied to constrain possible time variations of the gravitational constant  with the help of the gravitochemical heating approach developed by Jofre et al. . In addition, we applied the MDI EOS to limit a number of other global, transport and thermal properties of both static and rapidly rotating neutron stars [39, 58, 59, 60].