Abstract
The strong-field gravity in general relativity (GR) realized in neutron stars (NSs) renders the equation of state (EOS) of supradense neutron star matter to be essentially nonlinear and refines the upper bound for to be much smaller than the special relativity (SR) requirement with linear EOSs, where and are respectively the pressure and energy density of the system considered. Specifically, a tight bound is obtained by perturbatively anatomizing the intrinsic structures of the scaled Tolman–Oppenheimer–Volkoff (TOV) equations without using any input nuclear EOS. New insights gained from this novel analysis provide EOS-model-independent constraints on the properties (e.g., density profiles of the sound speed squared and trace anomaly ) of cold supradense matter in NS cores. Using the gravity-matter duality in theories describing NSs, we investigate the impact of gravity on supradense matter EOS in NSs. In particular, we show that the NS mass , radius , and compactness scale with certain combinations of its central pressure and energy density (encapsulating its central EOS). Thus, observational data on these properties of NSs can straightforwardly constrain NS central EOSs without relying on any specific nuclear EOS model.
1 Introduction
The speed of sound squared (SSS) (Landau and Lifshitz, 1987) quantifies the stiffness of the equation of state (EOS) expressed in terms of the relationship between the pressure and the energy density of the system considered. The principle of causality of special relativity (SR) requires the speed of sound of any signal to stay smaller than the speed of light , that is, . For a linear EOS of the form with being some constant, the condition is globally equivalent to . For such EOSs, the causality condition can be equivalently written as follows:
The indicated equivalence between and could be demonstrated as follows: If could be greater than somewhere, then the curve of may unavoidably across the line from below to above, indicating the slope at the crossing point is necessarily larger than 1, as illustrated in Figure 1. In the following, we use the above causality requirement on with linear EOSs as a reference in discussing properties of supradense matter in strong-field gravity.
FIGURE 1
The EOS of nuclear matter may be strongly nonlinear depending on both the internal interactions and the external environment/constraint of the system; this means that is necessary but not sufficient to ensure supradense matter in all NSs always stays causal. For example, the EOS of noninteracting degenerate fermions (e.g., electrons) can be written in the polytropic form (Shapiro and Teukolsky, 1983), where for non-relativistic and for extremely relativistic electrons; consequently, . Similarly, many years ago, Zel’dovich considered the EOS of an isolated ultra-dense system of baryons interacting through a vector field (Zel’dovich, 1961). In this case, ; here, is the baryon number density. Consequently, is obtained. The EOS of dense nuclear matter where nucleons interact through both the-meson and-meson in the Walecka model (Walecka, 1974) is an example of this type. In particular, the-field scales at asymptotically large density aswhile the-field scaleswith the scalar densityapproaching some constant for(Cai and Li, 2016); therefore, the vector field dominates at these densities. More generally, however, going beyond the vector field, the baryon density dependence of either or could be very complicated and nontrivial. The resulting EOS could also be significantly nonlinear. The EOS of supradense matter under the intense gravity of NSs could be forced to be nonlinear as the equilibrium state of NSs is determined by extremizing the total action of the matter–gravity system through Hamilton’s variational principle. It is well known that the strong-field gravity in general relativity (GR) is fundamentally nonlinear; the EOS of NS matter, especially in its core, is thus also expected to be nonlinear. Therefore, the causality condition may be appreciably different from , and it may also effectively render the upper bound for to be smaller than 1. Accurately determining an upper bound of (equivalently a lower bound of the dimensionless trace anomaly ) will thus help constrain properties of supradense matter in strong-field gravity.
The upper bound for is a fundamental quantity essentially encapsulating the strong-field properties of gravity in GR. Its accurate determination may help improve our understanding of the nature of gravity. The latter is presently the least known among the four fundamental forces despite being the first one discovered in nature (Hoyle, 2003). An upper bound on substantially different from 1 then vividly characterizes how GR affects the supradense matter existing in NSs. In some physical senses, this is similar to the effort in determining the Bertsch parameter. The latter was introduced as the ratio of the EOS of a unitary Fermi gas (UFG) over that of the free Fermi gas (FFG) (Giorgini et al., 2008); here,andare the energies per particle in the two systems considered. The EOS characterizes the strong interactions among fermions under the unitary condition. Extensive theoretical and experimental efforts have been made to constrain/fix the Bertsch parameter. Indeed, its accurate determination has already made a strong impact on understanding strongly interacting fermions (Giorgini et al., 2008; Bloch et al., 2008).
There are fundamental physics issues regarding both strong-field gravity and supradense matter EOS and their couplings. What is gravity? Is a new theory of light and matter needed to explain what happens at very high energies and temperatures? These are among the eleven greatest unanswered physics questions for this century, as identified in 2003 by the National Research Council of the U.S. National Academies (National Research Council, 2003). Compact stars provide far more extreme conditions necessary to test possible answers to these questions than terrestrial laboratories. A gravity-matter duality exists in theories describing NS properties; see, for example, Psaltis (2008) and Shao (2019) for recent reviews. Neutron stars are natural testing grounds for our knowledge of these issues. Some of their observational properties may help break the gravity-matter duality; see, for example, DeDeo and Psaltis (2003), Wen et al. (2009), Lin et al. (2014), He et al. (2015), Yang et al. (2020). Naturally, these issues are intertwined, and one may gain new insights into the EOS of supradense matter by analyzing features of strong-field gravity or vice versa. The matter-gravity duality reflects the deep connection between the microscopic physics of supradense matter and the powerful gravity effects of NSs. They both must be fully understood to unravel mysteries associated with compact objects in the Universe. In this brief review, we summarize the main physics motivation, formalism, and results of our recent efforts to gain new insights into the EOS of supradense matter in NS cores by perturbatively dissecting the intrinsic structures of the Tolman–Oppenheimer–Volkoff (TOV) equations (Tolman, 1939; Oppenheimer and Volkoff, 1939) without using any input nuclear EOS. For more details, we refer the readers to our original publications in Cai et al. (2023b), (Cai et al., 2023a), Cai and Li (2024a), and (Cai and Li, 2024b).
The rest of this article is organized as follows: First, in Section 2, we make a few remarks about some existing constraints on the EOS of supradense NS matter. Section 3 introduces the scaled TOV equations from which one can execute an effective perturbative expansion; the central SSS is obtained in Section 4. We then infer an upper bound for the ratio of central pressure over central energy density for NSs at the maximum-mass configuration along the M-R curve. The generalization for the upper bound of is also studied in Section 4. In Section 5, we compare our prediction on the lower bound of with existing predictions in the literature. We summarize in Section 6 and give some perspectives for future studies. In the Appendix, we discuss an effective correction to obtained in Section 4.
2 Remarks on some existing constraints on supradense NS matter
Understanding the EOS of supradense matter has long been an important issue in both nuclear physics and astrophysics (Walecka, 1974; Chin, 1977; Freedman and McLerran, 1977; Baluni, 1978; Wiringa et al., 1988; Akmal et al., 1998; Migdal, 1978; Morley and Kislinger, 1979; Shuryak, 1980; Bailin and Love, 1984; Lattimer and Prakash, 2001; Danielewicz et al., 2002; Steiner et al., 2005; Lattimer and Prakash, 2007; Alford et al., 2008; Li et al., 2008; Watts et al., 2016; Özel and Freire, 2016; Oertel et al., 2017; Vidaña, 2018). In fact, it has been an outstanding driver at many research facilities in both fields. For example, finding the EOS of the densest visible matter existing in our Universe is an ultimate goal of astrophysics in the era of high-precision multimessenger astronomy (Sathyaprakash et al., 2019). However, despite much effort and progress made during the last few decades using various observational data and models, especially since the discovery of GW170817 (Abbott et al., 2017a; 2018), GW190425 (Abbott et al., 2020a), GW190814 (Abbott et al., 2020b) and the recent NASA’s NICER (Neutron Star Interior Composition Explorer) mass-radius measurements for PSR J0740 + 6,620 (Fonseca et al., 2021; Riley et al., 2021; Miller et al., 2021; Salmi et al., 2022; Dittmann et al., 2024; Salmi et al., 2024), PSR J0030 + 0451 (Riley et al., 2019; Miller et al., 2019; Vinciguerra et al., 2024), and PSR J0437-4715 (Choudhury et al., 2024; Reardon et al., 2024), knowledge about the core NS EOS remains ambiguous and quite elusive (see, for example, Bose et al., 2018; De et al., 2018; Fattoyev et al., 2018; Lim and Holt, 2018; Most et al., 2018; Radice et al., 2018; Tews et al., 2018; Zhang et al., 2018; Bauswein et al., 2019; 2020; Baym et al., 2019; McLerran and Reddy, 2019; Most et al., 2019; Annala et al., 2020; 2023; Sedrakian et al., 2020; Zhao and Lattimer, 2020; Weih et al., 2020; Xie and Li, 2019; 2020; 2021; Drischler et al., 2020; 2021a; Li et al., 2020; Bombaci et al., 2021; Al-Mamun et al., 2021; Nathanail et al., 2021; Raaijmakers et al., 2021; Altiparmak et al., 2022; Breschi et al., 2022; Komoltsev and Kurkela, 2022; Perego et al., 2022; Huang et al., 2022; Tan et al., 2022a; b; Brandes et al., 2023b; a; Gorda et al., 2023; Han et al., 2023; Jiang et al., 2023; Ofengeim et al., 2023; Mroczek et al., 2023; Raithel and Most, 2023; Somasundaram et al., 2023; Zhang and Li, 2020; 2021; 2023b; a; Pang et al., 2023; Fujimoto et al., 2024; Providência et al., 2024; Rutherford et al., 2024). See recent reviews for additional discussion (for example, Baym et al., 2018; Baiotti, 2019; Li et al., 2019; Orsaria et al., 2019; Blaschke et al., 2020; Capano et al., 2020; Chatziioannou, 2020; Burgio et al., 2021; Dexheimer et al., 2021; Drischler et al., 2021b; Lattimer, 2021; Li et al., 2021; Lovato et al., 2022; Sedrakian et al., 2023; Kumar et al., 2024; Sorensen et al., 2024; Tsang et al., 2024).
Extensive theoretical investigations about the EOS of supradense NS matter have been conducted, and many interesting predictions have been made. For example, the realization of approximate conformal symmetry of quark matter at extremely high densities with the nuclear saturation density implies the corresponding EOS approaches that of an ultra-relativistic Fermi gas (URFG) from below, namely (Bjorken, 1983; Kurkela et al., 2010):
For the URFG, . Therefore, is at least upper bounded to be below 1/3 at these densities; equivalently, a lower bound on the dimensionless trace anomaly emerges:
This prompts the question of whether the bound holds globally for dense matter or if some other bound(s) on may exist. In this sense, massive NSs like PSR J1614-2230 (Demorest et al., 2010; Arzoumanian et al., 2018), PSR J0348 + 0432 (Antoniadis et al., 2013), PSR J0740 + 6,620 (Fonseca et al., 2021; Riley et al., 2021; Miller et al., 2021; Salmi et al., 2022; Dittmann et al., 2024; Salmi et al., 2024), and PSR J2215 + 5135 (Sullivan and Romani, 2024) provide an ideal testing bed for exploring such quantity. A sizable arises for NSs but not for ordinary stars or low-density nuclear matter (Cai and Li, 2024a). For example, considering stars such as white dwarfs (WDs), one has and ; thus, . The could be even smaller for main-sequence stars like the Sun. Specifically, the pressure and energy density in the solar core are approximately and , respectively, and therefore . These stars are Newtonian in the sense that GR effects are almost absent. Similarly, for NS matter around nuclear saturation density, the pressure is estimated to be. Its isospin-dependent part iswith(Li et al., 2018; 2021) being the slope parameter of nuclear symmetry energyat,is the isospin asymmetry of the system (), andis the pressure of symmetric nuclear matter (SNM) at. The energy density atis similarly estimated aswiththe nucleon static mass,the binding energy atfor SNM, and(Li, 2017), leading to.
Based on the dimensional analysis and the definition of sound speed, we may write out the SSS generally as (we use the units in which )where is dimensionless. For low-density matter, such as matter in ordinary stars and WDs or the nuclear matter around saturation density , the ratio is also small (as estimated in the last paragraph), indicating that could be expanded around as , where (to guarantee the stability condition ). Keeping the first leading-order term enables us to obtain , so has a similar value of if , and the EOS does not take a linear form (except for ). Moreover, the causality principle requires . The at from chiral effective field calculations (Essick et al., 2021) confirms our order-of-magnitude estimate on . If the next-leading-order term is small and positive, then the upper bound for becomes , which is even reduced compared with . The exact form of should be worked out/analyzed by the general-relativistic structure equations for NSs (Tolman, 1939; Oppenheimer and Volkoff, 1939). By doing that, we demonstrated earlier that is upper bounded as near the centers of stable NSs (Cai et al., 2023b; Cai et al., 2023a; Cai and Li, 2024a; Cai and Li, 2024b). The corresponding trace anomaly in NS cores is thus bounded to be above 0.04. In the next sections, we first show the main steps leading to these conclusions and then discuss their ramifications compared with existing predictions on in the literature.
3 Analyzing scaled TOV equations, mass/radius scalings, and central SSS
The TOV equations describe the radial evolution of pressure and mass of an NS under static hydrodynamic equilibrium conditions (Tolman, 1939; Oppenheimer and Volkoff, 1939). In particular, we have (adopting )Here, the mass , pressure , and energy density are functions of the distance from NS center. The central energy density is a specific and important quantity, which straightforwardly connects the central pressure via the EOS . Using , we can construct a mass scale and a length scale :respectively. Here, the second relations follow with . Using and , we can rewrite the TOV equations in the following dimensionless form (Cai et al., 2023b; Cai et al., 2023a; Cai and Li, 2024a; Cai and Li, 2024b),where , , and . The general smallness of
together with the smallness of
near NS centers enable us to develop effective/controllable expansion of a relevant quantity over and as Cai et al. (2023b), (Cai et al., 2023a), Cai and Li (2024a), (Cai and Li, 2024b):Here, is the quantity at the center. Because both GR and its Newtonian counterpart with small and are nonlinear, the TOV equations are also nonlinear. One often solves the more involved nonlinear TOV equations by adopting numerical algorithms via a selected and an input-dense matter EOS (Cai and Li, 2016; Li et al., 2022) as well as the termination condition:which defines the NS radius . The NS mass is given as
Starting from the scaled TOV Equation 7, we can show that both and are even under the transformation , while is odd (Cai and Li, 2024a). Therefore, we can write the general expansions for , and near :the expansion for follows directly from that for . As a direct consequence, we find that ; that is, there would be no odd terms in in the expansion of over . The relationships between and are determined by the scaled TOV Equation 7; and the results are (Cai et al., 2023b)etc., and all the odd terms of and are 0. The coefficient can be expressed in terms of via the SSS because
Evaluating it at gives , or inversely, . Because and , we find ; that is, the energy density is a monotonically decreasing function of near .
According to the definition of NS radius given in Equation 11, we obtain from the truncated equation that , and therefore, the radius (Cai et al., 2023b):
Similarly, the NS mass scales as Cai et al. (2023b).
Consequently, the NS compactness scales as Cai and Li (2024b).
For small (Newtonian limit), . Relation (22) implies that is the source and also a measure of NS compactness (Cai and Li, 2024b). The correlation betweenand is studied and fitted numerically in the form ofusing various EOS models (Saes and Mendes, 2022). Such fitting schemes eventually become effective as enough parameters,, are used.However, the real correlation between and is somehow lost. In particular, our correlation tells thatwithand.
The maximum-mass configuration (or the TOV configuration) along the NS M-R curve is a special point. Consider a typical NS M-R curve near the TOV configuration from right to left, the radius (mass ) eventually decreases (increases), the compactness correspondingly increases and reaches its maximum value at the TOV configuration. When going to the left along the M-R curve even further, the stars become unstable and may collapse into black holes (BHs). The NS at the TOV configuration is denser than its surroundings, and the cores of such NSs contain the densest stable visible matter in the Universe. The TOV configuration is indicated on a typical M-R sequence in Figure 2. Mathematically, the TOV configuration is described as
FIGURE 2
Using the NS mass scaling of Equation 21, we obtain
Inversely, we obtain the expression for the central SSS (Cai et al., 2023a; Cai and Li, 2024a),where
We see that the SSS is in the form of Equation 4. For NSs at the TOV configuration, we have
because now, . Using the of Equation 27 for NSs at the TOV configuration, we can calculate the derivative of NS radius with respect to around the TOV point, that is, Cai et al. (2023b).That is, as increases, the radius decreases (self-gravitating property), as expected. On the other hand, for stable NSs along the M-R curve with a nonzero , we have ; this means if is approximately 2, the dependence of the radius on would be weak.
For verification, the scaling - (panel (a)) of Equation 20 and the scaling - (panel (b)) of Equation 21 are shown in Figure 3 by using 87 phenomenological and 17 extra microscopic NS EOSs with and/or without considering hadron-quark phase transitions and hyperons by solving the original TOV equations numerically. See Cai et al. (2023b) for more details on these EOS samples. The observed strong linear correlations demonstrate vividly that the - and - scalings are nearly universal. While it is presently unclear where the mass threshold for massive NSs to collapse into BHs is located, the TOV configuration is the closest to it theoretically. It is also well known that certain properties of BHs are universal and only depend on quantities like mass, charge, and angular momentum. One thus expects the NS mass and radius scalings near the TOV configuration to be more EOS-independent than those for light NSs. It is also interesting to notice that EOSs allowing phase transitions and/or hyperon formations consistently predict the same scalings.
FIGURE 3
By performing linear fits of the results obtained from the EOS samples, the quantitative scaling relations are (Cai et al., 2023b; Cai et al., 2023a; Cai and Li, 2024a)
with their Pearson’s coefficients approximately 0.958 and 0.986, respectively. Here, and are measured in . In addition, the standard errors for the radius and mass fittings are approximately 0.031 and 0.003 for these EOS samples. In Figure 3, the condition used is necessary to mitigate influences of uncertainties in modeling the crust EOS (Baym et al., 1971; Iida and Sato, 1997; Xu et al., 2009) for low-mass NSs. For the heavier NSs studied here, it is reassuring to see that although the above 104 EOSs predicted quite different crust properties, they all fall closely around the same scaling lines consistently, especially for the - relation.
4 Gravitational upper bound on , its generalizations, and the impact on supradense NS matter EOS
Based on Equation 27 and the principle of causality of SR, we obtain immediately (Cai et al., 2023b)
Although the causality condition requires apparently , the supradense nature of core NS matter indicated by the nonlinear dependence of on essentially renders it to be much smaller.
A small was, in fact, indicated earlier in the literature (Koranda et al., 1997; Saes and Mendes, 2022). For example, in Koranda et al. (1997), the minimum-period EOS of the form for and for was adopted; here, is a free parameter of the model. Such an EOS is simplified and unrealistic in the following senses: (1) both the parameter and the central energy density are unrealistically large for a 1.442 NS (Koranda et al., 1997); the consequent ratio in this model is ; (2) the central SSS of 1 of such model is inconsistent with Equation 27. Actually, only with or can one make this EOS model consistent with Equation 27. That is, the parameter space for is limited; however, a vanishing pressure up to is fundamentally unsatisfactory. Therefore, is only qualitatively meaningful.
The bound (
31) is obtained under the specific condition that it gives the upper limit for
at the center of NSs at TOV configurations. In order to bound a general
, we need to take three generalizations of
obtained from
Equation 31by asking (
Cai et al., 2023a).
(a) How does behave at a finite for the maximum-mass configuration ?
(b) How does the limit modify when considering stable NSs on the M-R curve away from the TOV configuration?
(c) By combining (a) and (b), how does behave for stable NSs at finite distances away from their centers?
For the first question, because the pressure and are both decreasing functions of , that is, and (notice and ), we obtain by taking their ratio:
Generally, , the small- expansions of of Equation 27 and of Equation 16 are used in the last step. This means that not only and decrease for finite but also does their ratio . Therefore, for NSs at the TOV configuration of the M-R curves, we have . Considering the second question and for stable NSs on the M-R curve, one has (of Equation 26), and Equation 25 induces an even smaller upper bound for than 0.374. Furthermore, for the last question (c), the inequality (32) still holds and is slightly modified for small as
which implies that for also decreases with .
Combining the above three aspects, we find
Nevertheless, the validity of this conclusion is limited to small due to the perturbative nature of the expansions of and . Whether could exceed such upper limit at even larger distances away from the centers depends on the joint analysis of and , for example, by including more higher-order contributions of the expansions (Cai et al., 2023a). The upper bound (at least near the NS centers) is an intrinsic property of the TOV equations, which embody the strong-field aspects of gravity in GR, especially the strong self-gravitating nature. In this sense, there is no guarantee a priori that this bound is consistent with all microscopic nuclear EOSs (either relativistic or non-relativistic). This is mainly because the latter were conventionally constructed without considering the strong-field ingredients of gravity. The robustness of such an upper bound for can be checked only by observable astrophysical quantities/processes involving strong-field aspects of gravity such as NS M-R data, NS-NS mergers, and/or NS-BH mergers (Baumgarte and Shapiro, 2010; Shibata, 2015; Baiotti and Rezzolla, 2017; Kyutoku et al., 2021). As mentioned earlier, in the NS matter-gravity inseparable system, the total action determines the matter state and the NS structure. Thus, to our best knowledge, there is no physics requirement that the EOS of supradense matter created in vacuum from high-energy heavy-ion collisions or other laboratory experiments where effects of gravity can be neglected must be the same as EOSs in NSs, as the nuclear matter in the two situations is in very different environments. Nevertheless, the ramifications of the above findings and logical arguments should be further investigated.
Next, we consider the Newtonian limit where and are small. We can neglect in the coefficient ; consequently, is obtained (Chandrasekhar, 2010). In such case, we shall obtain from Equation 27:and the principle of causality requires . The latter can be applied to nuclear matter created in laboratory experiments where the effects of gravity can be neglected. Turning on gravity in NSs, we see that the nonlinearity of Newtonian gravity has already reduced the upper bound for from 1 obtained by requiring in SR via a linear EOS of the form to 3/4; the even stronger nonlinearity of the gravity in GR reduces it further. These effects of gravity on are illustrated in Figure 4. It is seen that the strong-field gravity in GR brings a relative reduction on the upper bound for by approximately 100%. Though the or in Newtonian gravity is generally smaller, the upper bound for or is, however, larger than its GR counterpart. The index , being greater than 1 in both Newtonian gravity and in GR, implies that the central EOS in NSs once considering the gravity effect could not be linear or conformal.
FIGURE 4
We emphasize that all of the analyses above based on SR and GR are general from analyzing perturbatively analytical solutions of the scaled TOV equations without using any specific nuclear EOS. Because the TOV equations are the results of a hydrodynamical equilibrium of NS matter in the environment of a strong-field gravity from extremizing the total action of the matter-gravity system, features revealed above from SR and GR inherent in the TOV equations must be matched by the nuclear EOS. This requirement can then put strong constraints on the latter. In particular, the upper bound for as of Equation 31 enables us to limit the density dependence of nuclear EOS relevant for NS modeling.
In the following, we provide an example illustrating how the strong-field gravity can restrict the behavior of superdense matter in NSs. For simplicity, we assume that the energy per baryon takes the following form:where the first term is the kinetic energy of an FFG of neutrons in NSs with MeV being its known value at , and the second term is the contribution from interactions described with the parameters and . The pressure and the energy density are obtained from and , respectively. The ratio and the SSS could be obtained correspondingly. After denoting the reduced density , where and , as (e.g., for realistic NSs), the following constraining equation for is obtained:
Thus, effectively restricts the index characterizing the stiffness of nuclear EOS. There are two solutions of Equation 37, with one being greater than 1 (denoted as ) and the other smaller than 1 (denoted as ). They can be explicitly written aswhere
The expression for the coefficient iswhich depends on through . As a numerical example, using , , and leads to and or and (this second solution is unphysical because is necessarily required to make at NS densities). If one artificially takes , then the two solutions (38) approach
Now, neither solution is physical because for while for from above, according to Equation 40. The general -dependence of and of Equation 38 is shown in Figure 5. It is seen that only as does the EOS approach a linear form (so and ) at large densities (magenta line), which is consistent with our general analyses and expectation.
FIGURE 5
Because the parameterization (36) is over-simplified, more density-dependent terms should be included for general cases; that is, . We may then obtain two related equations from and as (for either or ):
These constraints for and should be taken appropriately into account when writing an effective NS EOS based on density expansions. For example, when extending Equation 36 to be under two conditions for pure neutron matter with and , using together with , we may obtain and (as well as and ), respectively. This example quantitatively shows that the gravitational bound naturally leads to a constraint on the nuclear EOS and the underlying interactions in NSs.
5 Gravitational lower bound on trace anomaly in supradense NS matter
After the above general demonstration on the gravitational upper limit for near NS centers given by (31) or (34), we equivalently obtain a lower limit on the dimensionless trace anomaly as
It is very interesting to notice that such a GR bound on is very close to the one predicted by perturbative QCD (pQCD) at extremely high densities owning to the realization of approximate conformal symmetry of quark matter (Bjorken, 1983; Fujimoto et al., 2022), as shown in Figure 6 using certain NS modelings. A possible negative in NSs was first pointed out by Fujimoto et al. (2022). Since then, several studies have been made on this issue. In the following, we summarize the main findings of these studies by others and compare them with what we found above when possible.
FIGURE 6
The analysis in Ecker and Rezzolla (2022) using an agnostic EOS showed that is very close to 0 for and may be slightly negative for even more massive NSs (e.g., for ); the radial dependence of is shown in the upper panel of Figure 7 from which one finds the for NS at the TOV configuration is much deeper than that in a canonical NS. Moreover, incorporating the pQCD effects was found to effectively increase the inference on . An updated analysis of Ecker and Rezzolla (2022) was given in Musolino et al. (2024), where or was obtained under the constraint without or with considering the pQCD effects; see the lower panel of Figure 7 for the PDFs. Similarly, if was required, these two limits become and (Musolino et al., 2024), respectively. In Takátsy et al. (2023), the central minimum value of is found to be about 0.04 using the NICER data together with the tidal deformability from GW170817, and a value of was inferred considering additionally the second component of GW190814 as an NS with mass approximately (Abbott R. et al., 2020) using two hadronic EOS models (Takátsy et al., 2023); see the upper panel of Figure 8. By incorporating the constraints from AT2017gfo (Abbott et al., 2017b), it was found (Pang et al., 2024) that the minimum of is very close to 0 (approximately to 0.05), as shown in the lower panel of Figure 8. Using similar low-density nuclear constraints as well as astrophysical data, including the black widow pulsar PSR J0952-0607 (Romani et al., 2022), Brandes et al. (2023a) predicted taken at . Another analysis within the Bayesian framework considering the state-of-the-art theoretical calculations showed that (Annala et al., 2023) (where is assumed). Furthermore, by considering the slope and curvature of energy per particle in NSs, Marczenko et al. (2024) showed that is lower bounded for to be approximately . In addition, Cao and Chen (2023) found that the should be roughly larger than about in self-bound quark stars while that in a normal NS is generally greater than zero.
FIGURE 7
FIGURE 8
A very recent study classified the EOSs by using the local and/or global derivative of the resulting mass-radius sequences (Ferreira and Providência, 2024). Limiting the sign of to positive on the M-R curve for NS masses between about and , it was found that (Ferreira and Providência, 2024). On the other hand, if is required for all NS masses, then is found; see the upper left panel of Figure 9. Our understanding of this behavior is as follows: A negative slope along the whole M-R curve with (Ferreira and Providência, 2024) implies the radius of NS at the TOV configuration is relatively smaller than the one with a positive on a certain M-R segment, as indicated in the upper right panel of Figure 9. Thus, the NS compactness in the former case is relatively larger, which induces a larger via Equation 22 and, correspondingly, a smaller (Cai and Li, 2024b). The smaller radius also implies that the NS is much denser, so the maximum baryon density is correspondingly larger (Ferreira and Providência, 2024). The dense matter trace anomaly in twin stars satisfying relevant static and dynamic stability conditions was recently studied (Jiménez et al., 2024). The was found to be deeply bounded roughly as (Jiménez et al., 2024), as shown in the bottom panel of Figure 9. A deep negative implies a large or , so the compactness is correspondingly large according to Relation (22). We notice that the radii obtained in Jiménez et al. (2024) for certain NS masses (e.g., approximately ) may be small compared with the observational data, for example, PSR J0740 + 6,620 (Riley et al., 2021).
FIGURE 9
The above constraints on the lower limit of (realized in NSs) are summarized in the upper panel of Figure 10. Clearly, assuming all results are equally reliable within their individual errors indicated, there is a strong indication that the lower bound of is negative in NSs. Moreover, except for the prediction of Jiménez et al. (2024), the lower bounds of from various analyses are very close to the pQCD or GR limit . It is interesting to note that the and have no inner relation, to our best knowledge currently. However, we speculate that the matter-gravity duality in massive NSs mentioned earlier may be at work here. Certainly, this speculation deserves further study.
FIGURE 10
How relevant are the GR or pQCD limits for understanding the trace anomaly in NSs? The and its energy density dependence are crucial for studying the in NSs (Fujimoto et al., 2022). For instance, one can explore whether there would be a peaked structure in the density/radius profile of in NSs. Sketched in the lower panel of Figure 10 (Cai et al., 2023a) are two imagined functions versus the reduced energy density ; here, , around which the low-energy nuclear theories constrain the quite well. We notice that these two functions are educated guesses, certainly with biases. In fact, it has been pointed out that applying a particular EOS in extracting from observational data may influence the conclusion (Musolino et al., 2024). In the literature, there have been different imaginations/predictions/speculations on how the at finite energy density may vary and finally reach its pQCD limit of at very large energy densities (Fujimoto et al., 2022; Kurkela et al., 2010) or equivalently . The latter is far larger than the energy density reachable in the most massive NSs reported so far based on our present knowledge. The pQCD limit on is thus possibly relevant (Zhou, 2024) but not fundamental for explaining the inferred from NS observational data based on various microscopic and/or phenomenological models. On the other hand, we also have no confirmation in any way that the causality limit is reached in any NS. The magenta curve is based on the assumption that the causality limit under GR is reached in the most massive NSs observed so far. Based on most model calculations, in the cores of these NSs, the is roughly around 48. However, if the matter-gravity in massive NSs is indeed at work, we have no reason to expect that the GR limit is reached at an energy density lower than the one where the pQCD is applicable.
Keeping a positive attitude in our exploration of a completely uncharted area, we make a few more comments below on how the trace anomaly may reach the pQCD limit. As a negative is unlikely to be observed in ordinary NSs, the evolution of is probably more like the green curve in the lower panel of Figure 10. An (unconventional) exception may come from light but very compact NSs; for example, a NS at the TOV configuration with radius approximately 9.3 km has its because together with should be obtained via the mass and radius scalings of (30) and (29), and so . On the other hand, massive and compact NSs (masses ) are most relevant to observing a negative (as indicated by the magenta curves) and how it evolves to the pQCD bound, thus revealing more about properties of supradense matter (Cai et al., 2023a). Interestingly, both the green and magenta curves for the pattern are closely connected with the density dependence of the SSS using the trace anomaly decomposition of (Fujimoto et al., 2022) (we do not discuss these interesting topics in the current review). Unfortunately, the region with is largely inaccessible in NSs due to their self-gravitating nature.
6 Summary and future perspectives
In summary, perturbative analyses of the scaled TOV equations reveal interesting new insights into properties of supradense matter in NS cores without using any input nuclear EOS. In specific, the ratio of pressure over energy density (the corresponding trace anomaly ) in NS cores is bounded to be below 0.374 (above 0.04) by the causality condition under GR independent of the nuclear EOS. Moreover, we demonstrate that the NS mass , radius , and compactness strongly correlate with , and with , respectively; therefore observational data on and as well as on via red-shift measurements can directly constrain the central EOS in a model-independent manner. In addition to the topics we have already investigated (Cai et al., 2023b; Cai et al., 2023a; Cai and Li, 2024a; Cai and Li, 2024b), there are interesting issues to be further explored in this direction. Particularly, we notice:
1. The upper limit for near NS cores is obtained by truncating the perturbative expansion of and to low orders in reduced radius . While the results are quite consistent with existing constraints from state-of-the-art simulations/inferences, refinement by including even higher-order terms would be important for studying the radius profile of or in NSs. In the Appendix, we estimate such an effective correction.
2. Ironically, the upper bound from GR is very close to that from pQCD at extremely high densities (Bjorken, 1983; Kurkela et al., 2010; Fujimoto et al., 2022). While we speculated that the well-known matter-gravity duality in massive NSs may be at work, it is currently unclear whether there is a fundamental connection between them. Efforts to understand their relationships may provide useful hints for developing a unified theory for strong-field gravity and elementary particles in supradense matter.
Statements
Author contributions
B-JC: writing–original draft and writing–review and editing. B-AL: writing–original draft and writing–review and editing.
Funding
The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This work was supported in part by the U.S. Department of Energy, Office of Science, under Award Number DE-SC0013702 and the CUSTIPEN (China-U.S. Theory Institute for Physics with Exotic Nuclei) under the U.S. Department of Energy Grant No. DE-SC0009971.
Acknowledgments
We would like to thank James Lattimer and Zhen Zhang for their helpful discussions.
Conflict of interest
Author B-JC was employed by Shadow Creator, Inc.
The remaining author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declare that no generative AI was used in the creation of this manuscript.
Publisher’s note
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AppendixEstimate of an effective correction to
In this appendix, we estimate an effective correction to given in Equation 27 for NSs at the TOV configuration (Cai et al., 2023a). When writing down in Equation 21, we adopt , which only includes the first term in the systematic expansion (Equation 14); necessarily, we may include higher-order terms from Equation 14 in . As an effective correction, we now include from Equation 14 to the NS mass, which modifies Equation 21 aswhere is given by Equation 20 through , the coefficient is defined in Equation 21, and the general relation is used to write . The factor “” is actually the averaged reduced energy density by including the -term in of Equation 13, namely, with
Moreover, the in Equation A1 is now not given by Equation 27 but should include corrections due to including the -term in . Generally, we write it as:where is a coefficient to be determined. In addition, we have using the of Equation A3; taking with given by Equation A1 gives the expression for (which is quite complicated). We then expanding the latter over to order to giveMatching the two expressions (Equations A3, A4) at order gives . After that, we determine via , which is close to and consistent with 0.374 obtained in the main text; and similarly, . The magnitude of the correction “” in is smaller than 5% while the corresponding correction on is smaller than 2%. In addition, the NS mass now scales as
In order to obtain the corrections to more self-consistently and improve the accuracy of , one may include more terms in the expansion of over of Equation 14 (i.e., -term, -term and -term, etc.), the expansion of over of Equation 15 (i.e., -term, -term, -term, etc.), and in the mean while introduce corrections “” in as we did in Equation A3. Then, determine the coefficients , , and , etc. The procedure eventually becomes involved as more terms are included.
Summary
Keywords
equation of state, supradense matter, neutron star, Tolman–Oppenheimer–Volkoff equations, principle of causality, special relativity, speed of sound, generality relativity
Citation
Cai B-J and Li B-A (2024) New insights into supradense matter from dissecting scaled stellar structure equations. Front. Astron. Space Sci. 11:1502888. doi: 10.3389/fspas.2024.1502888
Received
27 September 2024
Accepted
28 October 2024
Published
11 December 2024
Volume
11 - 2024
Edited by
Armen Sedrakian, University of Wrocław, Poland
Reviewed by
Carlos Frajuca, Federal University of Rio Grande, Brazil
Jiajie Li, Southwest University, China
Updates
Copyright
© 2024 Cai and Li.
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*Correspondence: Bao-Jun Cai, bjcai87@gmail.com; Bao-An Li, bao-an.li@tamuc.edu
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