&Bullet; physics 14, 66

The combination of gravitational wave and X-ray observations of neutron stars offers new insights into the structure of these stars as well as a new confirmation of Einstein’s theory of gravity.

After black holes, neutron stars are some of the most compact objects in the universe. Their enormous densities create strong gravitational fields and a large curvature in the surrounding space-time. We want to test gravity under these conditions, but we don’t know how matter behaves at such extreme densities. In particular, we do not know the equation of state of nuclear matter, which in its simplest form relates the density of matter to pressure. One way to get around this limitation is to use universal relationships that connect different neutron star properties independently of the equation of state [1, 2] . Hector Silva from the Max Planck Institute for Gravitational Physics in Germany and his colleagues at the University of Illinois in Urbana-Champaign used such relationships to derive the moment of inertia, quadrupole moment and surface eccentricity of an isolated neutron star from recent X-ray observations [3] . By combining these mass distribution estimates with gravitational wave observations of merging neutron stars, they demonstrated an effective way to constrain the strong-field gravity regime. In particular, this “multimessenger” analysis of neutron star data offers new limits for the violation of parity symmetry in the gravitational interaction.

Observations of neutron stars can help us to answer a longstanding question: Is Einstein’s general theory of relativity the correct theory of gravity in the regime of strong fields? Many theoretical considerations in basic physics that relate to the quantization of gravity or the unification of all fundamental interactions involve modifications of the general theory of relativity. For example, some alternative theories predict that gravitational interactions, such as the weak interactions, would exhibit parity asymmetry (or “handedness”). While researchers have tested Einstein’s theory with remarkable accuracy for weak gravitational fields, investigation of the regime of large space-time curvature was largely inaccessible until the recent advances in gravitational wave observation. Neutron stars, observed by both gravitational waves and electromagnetic waves, are unique test beds for Einstein’s theory in this great curvature regime.

The density in the neutron star core reaches several times

$1{0}^{15}\phantom{\rule{2.22198pt}{0ex}}{\text{g / cm}}^{3rd}$

While terrestrial nuclear experiments can only determine the behavior of matter with good accuracy up to approximately

$1{0}^{11}\phantom{\rule{2.22198pt}{0ex}}{\text{g / cm}}^{3rd}$

. Theorists try to bridge the large gap between these two densities, but their models suffer from great uncertainties. One of the main goals of neutron star observation is to limit these uncertainties. A recent attempt in this direction is the Neicron Star Interior Composition Explorer (NICER), which can deduce the star mass and the radius of a neutron star from X-ray observations of hot spots on the star’s surface (see Dimensioning of the Most Massive Neutron Star). Another powerful channel for astrophysical observations are gravitational waves. In 2017, the LIGO and Virgo collaborations observed a gravitational wave signal named GW170817 that came from a pair of merging neutron stars (see Viewpoint: Neutron Star Fusion Seen and Heard). From the data, the researchers determined the tidal love numbers of the neutron stars, which – roughly speaking – characterize the dimensional stiffness of the body under tidal forces (see Synopsis: Rising Tides on Black Holes).

One way of extracting more information from the various neutron star observations is through the so-called universal relationships, which have become popular in recent years [4–6] . These relationships are constructed by calculating neutron star properties with a variety of different equations of state of nuclear matter. Although these equations of state can lead to large differences in neutron star structure, they show consistency in their predictions of various relationships, such as one linking moment of inertia to tidal love number. Since the derived relationships have the “luxury” of being almost independent of the model of nuclear matter, they can be used to analyze astrophysical observations of neutron stars [1, 2] . Silva and co-workers built on such universal relationships and combined the compactness of the neutron star (the ratio of the neutron star’s mass to its radius) with several parameters that, roughly speaking, are related to the star’s mass distribution. The team used compactness values ​​derived from two independent analyzes of the NICER observation data from the isolated pulsar PSR J0030 + 0451 [7, 8] . By combining the results for neutron star compactness with the corresponding universal relationships, they succeeded in inferring the moment of inertia (I), Quadrupole moment (Q.) and the surface eccentricity of PSR J0030 + 0451, which represents the first time these parameters have been determined for an isolated neutron star. They also showed that the results were weakly dependent on the data analysis approach used.

Using the results for PSR J0030 + 0451, the researchers also estimated the moment of inertia of PSR J0737-3039A, the larger partner of the only known double pulsar. Ongoing pulsar timing observations of this object will provide an independent estimate of its moment of inertia over the next several years [9, 10] . The possibility of determining the same amount twice through completely different observations enables a further test of the nuclear matter at extreme densities and under strong gravity.

Most recently, Silva and co-workers performed a multimessenger test of general relativity using the moment of inertia from PSR J0030 + 0451 and the tidal deformability obtained from GW170817 (Fig. 1). For their test, the researchers used another relationship independent of the equation of state – part of the famous one I-Love-Q.trio [4] - this connects the normalized moment of inertia of the neutron star and the tidal love number. This universal relationship applies not only to general relativity, but also to various alternative theories of gravity (albeit with differences in the exact polynomial dependence that connects the normalized moment of inertia and the tidal love number). By independently inferring the moment of inertia (from the NICER data) and the tidal deformability (from the LIGO / Virgo data), they confirmed that Einstein’s theory was in agreement with the observations and showed that only a small amount of deviation was allowed . This result helped them greatly reduce the parity violation in the gravitational interaction and improve the previous limits by 7 orders of magnitude.

On the one hand, the research by Silva and co-workers is a culmination of efforts to use universal relationships to limit strong gravity with the help of astrophysical multi-messenger observations. On the other hand, it opens a new door to better understand nuclear matter at extreme densities and to test the laws of gravity. We can assume that this knowledge will soon increase as the observations progress rapidly. We live in an exciting time when high-precision instruments are finally available in both the electromagnetic and gravitational wave channels that allow us to explore fundamental physical concepts that were considered purely theoretical just a few decades ago. Stay tuned – multimessenger astrophysics is about to bring us new surprises that can challenge our understanding of nature.

## References

1. K. Yagi and N. Yunes, “Approximate Universal Relationships for Neutron Stars and Quark Stars” Phys. Rep.681, 1 (2017).
2. DD Doneva and G. Pappas, “Universal Relationships and Alternative Theories of Gravity”, Astrophys. Space Sci. Libr.457737 (2018).
3. HO Silva et al., “Astrophysical and theoretical physical implications from multimessenger neutron star observations” Phys. Rev. Lett.126181101 (2021).
4. K. Yagi and N. Yunes, “I-Love-Q: Unexpected Universal Relationships for Neutron Stars and Quark Stars”. science341365 (2013).
5. M. Bauböck et al., “Relationships between neutron star parameters in the Hartle-Thorne approximation” Astrophys. YY77768 (2013).
6. C. Breu and L. Rezzolla, “Maximum Mass, Moment of Inertia, and Compactness of Relativistic Stars”. Mon. Not. Roy. Astron. Soc.459646 (2016).
7. TE Riley et al., “A NICER View of PSR J0030 + 0451: Millisecond Pulsar Parameter Estimation” Astrophys. J. Lett887L21 (2019).
8. MC Miller et al., “PSR J0030 + 0451 Mass and radius from NICER data and effects on the properties of neutron star matter” Astrophys. J. Lett.887L24 (2019).
9. JM Lattimer and BF Schutz, “Restriction of the equation of state with measurements of the moment of inertia”, Astrophys. YY629979 (2005).
10. M. Kramer and N. Wex, “The Double Pulsar System: A Unique Laboratory for Gravity” Class. Quant. Grav.26073001 (2009).

Daniela Doneva is currently the head of the Emmy Noether Junior Research Group at the University of Tübingen. She received her Ph.D. at the University of Sofia, Bulgaria, under the supervision of Stoytcho Yazadjiev. Daniela then switched to a postdoc position in Tübingen, where she received a Humboldt grant, followed by a habilitation grant from Margarete von Wrangell in combination with an elite program grant from the Baden-Württemberg Foundation. Her research focuses on studying the structure and dynamics of neutron stars and black holes in Einstein’s theory of gravity and its generalizations.

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