Constraining the Speed of Sound inside Neutron Stars with Chiral Effective Field Theory Interactions and Observations

In this work, we use auxiliary-field diffusion Monte Carlo (AFDMC) to find the many-body ground state for a given nonrelativistic nuclear Hamiltonian (Carlson et al. 2015). In general, the nuclear Hamiltonian contains two-body (NN), three-body (3N), and higher many-body (AN) forces,

$$\begin{eqnarray}&&{ \mathcal H }={ \mathcal T }+{{ \mathcal V }}_{\mathrm{NN}}+{{ \mathcal V }}_{3{\rm{N}}}+{{ \mathcal V }}_{\mathrm{AN}},\end{eqnarray} \tag{ 2 }$$

which can be obtained from chiral effective field theory (EFT) at low-density (see, for instance, Epelbaum et al. 2009 and Machleidt & Entem 2011). Chiral EFT is a systematic framework for low-energy hadronic interactions, that naturally includes both two-body and many-body forces and allows for systematic uncertainty estimates. It has been successfully used to calculate nuclei and nuclear matter, see, for instance, Hebeler et al. (2015) and references therein.

In this paper, we extend the AFDMC calculations of PNM of Lynn et al. (2016) with recently developed local chiral N²LO interactions, including two- and three-body forces of Gezerlis et al. (2013, 2014) and Tews et al. (2016), to higher densities. We find that, despite the rapid increase of the error estimates, EFT-based interactions remain useful up to n = 0.32 fm⁻³ and our results for the energy per particle in neutron matter are shown in Figure 2. We plot the results for local chiral interactions at LO, NLO, and N²LO with three different 3N interactions defined in Lynn et al. (2016): 3N interactions with only the two-pion exchange (TPE-only), and 3N interactions containing the TPE plus shorter-range contact terms with two different spin-isospin operators (TPE + ${V}_{E,{\mathbb{1}}}$ and TPE + V_E,τ), see Lynn et al. (2016) for details. The uncertainty bands for the individual N²LO interactions are obtained as suggested by Epelbaum et al. (2015), i.e., the uncertainty ${\rm{\Delta }}{X}^{{{\rm{N}}}^{2}\mathrm{LO}}$ at order N²LO is given by

$$\begin{eqnarray}{\rm{\Delta }}{X}^{{{\rm{N}}}^{2}\mathrm{LO}} & = & \max ({Q}^{4}| {X}^{\mathrm{LO}}-{X}^{\mathrm{free}}| ,{Q}^{2}| {X}^{\mathrm{NLO}}\\ & & -{X}^{\mathrm{LO}}| ,Q| {X}^{{{\rm{N}}}^{2}\mathrm{LO}}-{X}^{\mathrm{NLO}}| ),\end{eqnarray} \tag{ 3 }$$

and similar for lower orders. The dimensionless expansion parameter Q is given by Q = max(p/Λ_B, m_π/Λ_B) with p being a typical momentum scale of the system, m_π being the pion mass, and Λ_B ≈ 500 MeV being the breakdown scale of chiral EFT. For PNM in Figure 2, we use the scale $p=\sqrt{3/5}{k}_{F};$ see Lynn et al. (2017). The results indicate that the EOS of PNM is well constrained up to saturation density, but the associated error grows rapidly with density. Nonetheless, the order-by-order convergence is still consistent with EFT expectations in the range n₀ – 2n₀. In addition, although these interactions are only fit to low-energy scattering data up to laboratory energies of 150 MeV, they also describe phase shifts at much higher energies within uncertainties; see, e.g., Epelbaum et al. (2015). For comparison, we also show results obtained using the phenomenological Argonne v8' NN interaction (AV8') and the Urbana IX 3N interactions (UIX) in Figure 2.

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We present the energy per particle and the pressure at n₀ and 2n₀ at LO, NLO, and N²LO in Table 1 to provide supporting arguments for the convergence of the chiral expansion in this density range. Uncertainty estimates in Table 1 are provided by assuming a typical momentum scale $p=\sqrt{3/5}{k}_{F}$ and, in addition, a more conservative choice p = k_F. At saturation density, we find a systematic order-by-order convergence for both energy and pressure: the results at different orders overlap within their uncertainties, which decrease order-by-order.

Table 1.  Energy per Particle and Pressure in PNM at n₀ and 2n₀ for Local Chiral Hamiltonians

		Free	Pheno.	LO	NLO	N2LO (TPE-only)	N2LO (+${V}_{E,{\mathbb{1}}}$)	N2LO (+VE,τ)
E/A	n0	35.1	19.1	15.5 ± 5.2 (8.6)	14.3 ± 2.7 (5.7)	15.6 ± 1.4 (3.8)	17.3 ± 1.5 (3.8)	13.5 ± 1.4 (3.8)
	2n0	55.7	49.9	20.9 ± 14.6 (24.3)	27.0 ± 9.4 (20.3)	27.2 ± 6.1 (16.9)	36.9 ± 6.4 (16.9)	14.3 ± 8.2 (16.9)
P	n0	3.7	3.3	1.3 ± 0.7 (1.1)	1.6 ± 0.4 (0.8)	1.8 ± 0.2 (0.5)	2.4 ± 0.4 (0.6)	1.1 ± 0.3 (0.5)
	2n0	11.9	25.8	3.1 ± 3.7 (6.1)	9.8 ± 4.4 (5.6)	7.8 ± 2.8 (4.7)	15.1 ± 3.4 (4.7)	−2.6 ± 8.1 (10.4)

Note. Energy per particle and pressure in PNM at n₀ and 2n₀ for local Chiral Hamiltonians at LO, NLO, and N²LO with three different short-range operators. The uncertainties are obtained using Equation (3) with two different expansion parameters p/Λ_B with $p=\sqrt{3/5}{k}_{F}$ and p = k_F in parenthesis. We also give the values for the free Fermi gas and the phenomenological AV8'+UIX interaction.

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From Figure 2 and Table 1, we see that the shorter-range 3N interactions significantly influence the convergence, but are still within conservative uncertainty estimates. These shorter-range 3N forces appear at N²LO in the chiral power counting and contribute to systems containing triples of both neutrons and protons with S = 1/2 and T = 1/2, where S and T are the total spin and isospin, respectively. They are typically fit to few-body observables, such as the ${}^{4}\mathrm{He}$ binding energy and neutron-alpha scattering (see Lynn et al. 2017 for a detailed discussion). In PNM, where all triples have T = 3/2, these 3N forces usually vanish and it can be shown that at N²LO only the long-range 3N TPE interaction contributes (see, for instance, Hebeler & Schwenk 2010), while shorter-range contributions to PNM only appear at higher-order in the chiral expansion. In our case, however, contributions to PNM from short-range N²LO 3N interactions arise as artifacts from using local regulators. Local forces in coordinate space are essential for their incorporation in AFDMC but introduce regulator artifacts; see Huth et al. (2017) for a detailed discussion in the NN sector. In the case of three-nucleon forces, these regulator artifacts appear in form of shorter-range 3N interactions mixed into triples with S = 3/2 or T = 3/2, e.g., triples containing three neutrons. We include these additional contributions, which we denote as N²LO TPE+${V}_{E,{\mathbb{1}}}$ and N²LO TPE+V_E,τ, to provide even more conservative uncertainty estimates.

It is interesting to analyze the predictions for the energy per particle and the pressure of PNM at 2n₀. From Table 1 and comparing the LO, NLO, and N²LO TPE-only predictions, the pattern of convergence for the energy per particle is quite reasonable, with clear signs of an order-by-order improvement. For the pressure, the NLO contribution is larger than expected from naive power counting arguments. A plausible resolution of this discrepancy could be important contributions due to the Δ isobar, which are only included at N²LO in Δ-less chiral EFT. We note, however, that the more conservative error estimates in brackets assuage the tension somewhat. The improvement in predictions for the pressure at 2n₀ from NLO to N²LO (TPE-only), on the other hand, is consistent with EFT expectations. The situation for the TPE+${V}_{E,{\mathbb{1}}}$ and TPE+V_E,τ interactions is less satisfactory. For the TPE+${V}_{E,{\mathbb{1}}}$ interaction, the predicted pressure at N²LO has some overlap with the error band at NLO, while for the TPE+V_E,τ interaction the overlap is negligible. It is particularly troubling that the TPE+V_E,τ interaction predicts negative pressures at 2n₀.

As argued earlier, these shorter-range contributions are an artifact of using local regulators. The large attractive contribution of the TPE+V_E,τ interaction is unphysical and should be discarded for the following reasons: (i) the full N²LO 3N contributions in momentum-space calculations by Hebeler & Schwenk (2010), Tews et al. (2013), Hagen et al. (2014), Sammarruca et al. (2015), Wlazowski et al. (2014), and Drischler et al. (2016) are all found to be repulsive in PNM; (ii) the inclusion of N³LO 3N contributions does not change the repulsive nature of 3N forces in PNM, see Krüger et al. (2013) and Drischler et al. (2016); and (iii) local regulators lead to less repulsion from the 3N TPE compared to typically used nonlocal regulators (Tews et al. 2016; Dyhdalo et al. 2016). For these reasons, we will ignore the N²LO TPE+V_E,τ interaction in the following but include the TPE+${V}_{E,{\mathbb{1}}}$ interaction to investigate more repulsive short-range 3N forces.

It is worth noting that the pressure predicted by the phenomenological AV8'+UIX interaction at 2n₀ is significantly larger than the N²LO (TPE-only) prediction. Even the inclusion of additional repulsion through the N²LO TPE+${V}_{E{\mathbb{1}}}$ interaction does not alleviate this discrepancy. With some reserve, we propose that our calculations with the N²LO TPE+${V}_{E,{\mathbb{1}}}$ interaction provide an upper limit to the pressure. Calculations with local chiral interactions suggest that P (2n₀) < 20 MeV fm⁻³ in PNM, see Table 1.

In the following, we use the PNM results from the chiral N²LO TPE-only and TPE+ ${V}_{E,{\mathbb{1}}}$ interactions up to a transition density n_tr, which we vary in the range of 1–2n₀. Although the uncertainties are sizable at 2n₀, we shall find that PNM calculations still provide useful constraints. The phenomenological AV8'+UIX interaction will also be analyzed in the same density interval for comparison. Even though the expansion parameter  ≃k_F/Λ_B only increases by about 25% when the density is increased from n₀ to 2n₀, we have chosen 2n₀ as an upper limit to the validity of nuclear Hamiltonians for the following reasons. First, the previous discussion of uncertainties and the order-by-order convergence of the energy per particle and pressure in neutron matter has shown that while the convergence for the energies is consistent with EFT expectations, the situation is less satisfactory for the pressure at 2n₀. Second, the accuracy of chiral nuclear interactions in describing typical momenta in nuclei and nucleon–nucleon scattering data decreases with increasing density. Third, at higher densities, additional degrees of freedom, e.g., hyperonic dof, might appear (e.g., Ambartsumyan & Saakyan 1960; Glendenning 1982; Lonardoni et al. 2015; Gal et al. 2016). Fourth, at densities above 2n₀, typical momenta in neutron matter are comparable to the cutoff scales employed in the calculation, which further increases the size of regulator artifacts. Based on these reasons, we believe that 2n₀ is a reasonable upper bound for calculations with the local chiral Hamiltonians that we employ here.

Matter in NS is in β—equilibrium, and at the relevant densities a small fraction of protons will be present. The proton fraction, denoted by x, increases with density but remains small and x ≲ 10% even at 2n₀. Although the proton contribution to the EOS can be expected to be small compared to the intrinsic uncertainty associated with the nuclear Hamiltonian discussed earlier, we shall extend the PNM results to finite proton fraction. To achieve this, we use the parameterization introduced by Hebeler et al. (2013), given by

$$\begin{eqnarray}\displaystyle \frac{{E}^{\mathrm{nuc}}}{A}(n,x) & = & {T}_{0}\left[\displaystyle \frac{3}{5}({x}^{\tfrac{5}{3}}+{(1-x)}^{\tfrac{5}{3}}){\left(\displaystyle \frac{2n}{{n}_{0}}\right)}^{\tfrac{2}{3}}\right.\\ & & \,-[(2\alpha -4{\alpha }_{L})x(1-x)+{\alpha }_{L}]\cdot \displaystyle \frac{n}{{n}_{0}}\\ & & \,\left.+[(2\eta -4{\eta }_{L})x(1-x)+{\eta }_{L}]\cdot {\left(\displaystyle \frac{n}{{n}_{0}}\right)}^{\gamma }\right],\end{eqnarray} \tag{ 4 }$$

where ${T}_{0}={(3{\pi }^{2}{n}_{0}/2)}^{2/3}{{\hslash }}^{2}/(2{m}_{N})$ is the Fermi energy of noninteracting symmetric nuclear matter at saturation density, x = n_p/n is the proton fraction (n_p is the proton density), and α, α_L, η, η_L, and γ are parameters that are fit to the neutron-matter results and the saturation point of symmetric nuclear matter, $\tfrac{{E}^{\mathrm{nuc}}}{A}({n}_{0},0.5)=-16\,\mathrm{MeV}$, and P^nuc(n₀, 0.5) = 0. The saturation point determines α and η, while the parameters α_L, η_L, and γ are determined by the PNM results. This parameterization provides a faithful reproduction of the PNM results obtained using AFDMC for densities up to 2n₀, and has also been shown to provide a good representation of results for asymmetric nuclear matter obtained in many-body perturbation theory (Drischler et al. 2014).

Using the parameterization in Equation (4), we follow Tews (2017) to construct a consistent crust model up to the crust-core transition density, n_cc ≈ n₀/2. For densities between n_cc and the chosen transition density n_tr, we extend the PNM results to β equilibrium. From this procedure, the neutron-star equation of state P() and the speed of sound ${c}_{S}^{2}(n)$ are determined. In Figure 3, we show the speed of sound in NS matter up to two times nuclear saturation density for the chiral N²LO TPE-only (green band), N²LO TPE+${V}_{E,{\mathbb{1}}}$ (red band), and AV8'+UIX (black line) interactions.

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In this scenario ${c}_{S}^{2}\lt 1/3$ at all densities, and we extend the speed of sound to densities above those described by the nuclear EOS using a simple three-parameter curve,

$$\begin{eqnarray}&&{c}_{S}^{2}=\displaystyle \frac{1}{3}-{c}_{1}\exp \left[-\displaystyle \frac{{(n-{c}_{2})}^{2}}{{n}_{\mathrm{BL}}^{2}}\right],\end{eqnarray} \tag{ 5 }$$

where two of the parameters, c₁ and c₂, are fit to the speed of sound and its derivative at n_tr. The remaining parameter, n_BL, controls the width of the curve and presents a density interval in which the speed of sound changes considerably. The smaller this parameter, the stronger the change of the speed of sound with density. By varying this remaining parameter, we can easily control the slope of the speed of sound after the transition density, see Figure 4. The choice of an exponential function may seem ad hoc at this stage, but will be well motivated shortly when we find that to obtain 2 M_⊙ NSs, c_S must increase rapidly over a narrow interval n_BL ≲ n₀.

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From Equation (5), we can construct the EOS using the following procedure. Starting at n_tr, where (n_tr), p(n_tr), and '(n_tr) are known, we take successive small steps Δn in density,

$$\begin{eqnarray}&&{n}_{i+1}={n}_{i}+{\rm{\Delta }}n\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\epsilon }_{i+1}={\epsilon }_{i}+{\rm{\Delta }}\epsilon ={\epsilon }_{i}+{\rm{\Delta }}n\cdot \left(\displaystyle \frac{{\epsilon }_{i}+{p}_{i}}{{n}_{i}}\right)\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{p}_{i+1}={p}_{i}+{c}_{S}^{2}({n}_{i})\cdot {\rm{\Delta }}\epsilon ,\end{eqnarray} \tag{ 8 }$$

to iteratively obtain the high-density EOS, where the index i = 0 defines the transition density n_tr. Note, in the second line we have used the thermodynamic relation $p=n\partial \epsilon /\partial n-\epsilon$ valid at zero temperature. We then use the resulting equation of state to solve the Tolman–Oppenheimer–Volkoff (TOV) equations, and to obtain the mass–radius relation for NS.

In Figure 5, we show the maximum NS mass as a function of the baseline width n_BL of Equation (5) (describing the slope) for two transition densities, n_tr,1 = 1.1n₀ = 0.18 fm⁻³ and n_tr,2 = 2n₀ = 0.32 fm⁻³, and for the nuclear interactions of Figure 3. We compare these to the current NS maximum-mass constraint of 2.01 ± 0.04 M_⊙ (Antoniadis et al. 2013). Depending on the transition density and the nuclear Hamiltonian, we observe the tension between current nuclear and astrophysical constraints and the ${c}_{S}^{2}\lt 1/3$ bound set by the conformal limit, first noted in Bedaque & Steiner (2015; see also Moustakidis et al. 2017). At the lower transition density, there is some freedom to find an EOS that satisfies both the conformal bound and the maximum-mass constraint. However, the inferred value of n_BL is small, indicating that c_S must increase much more rapidly in the density interval n₀ – 2n₀ than would be compatible with EFT predictions. For n_tr = 2n₀, chiral EFT, even with the repulsion beyond the TPE-only interaction, is unable to support the 2.01 ± 0.04 M_⊙ NS if the conformal bound is observed. The stiffest EOS predicted by the phenomenological Hamiltonian AV8'+UIX (black line) barely satisfies both bounds in that case.

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In the left panel of Figure 6, we show the evolution of c²_S for the largest values of n_BL compatible with a two-solar-mass NS for several transition densities and for the softest and stiffest nuclear interaction under investigation (without uncertainty bands for clarity). In the right panel, we show this maximal width as a function of n_tr. We find that for the chiral N²LO TPE-only interaction there is no curve that satisfies both the speed-of-sound and the NS-mass bounds for transition densities larger than ≈0.25 fm⁻³. For the phenomenological nuclear Hamiltonian, the situation is only slightly different. We find that for low transition densities a moderate slope of the speed of sound is sufficient to reproduce a two-solar-mass NS. With increasing transition density, again, the speed of sound has to increase on a smaller density interval to fulfill the mass constraint. Above transition densities of ≈0.3 fm⁻³, we find strong tension between both bounds.

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To summarize, since AFDMC predictions for the EOS using EFT interactions are very reliable, and order-by-order convergence at these densities was observed, we argue that our results suggest that either QCD violates the conformal bound, or a chiral EFT-based description breaks down at densities below 2n₀ due to new physical effects.

We shall now allow the speed of sound to exceed the conformal limit of $1/\sqrt{3}$ at the densities encountered in the NS core and only require that c_S < 1. We model its evolution with density in a minimalist approach constrained by knowledge of its behavior at low and high density by adding a skewed Gaussian to the form defined in Equation (5),

$$\begin{eqnarray}{c}_{S}^{2} & = & \displaystyle \frac{1}{3}-{c}_{1}\exp \left[-\displaystyle \frac{{(n-{c}_{2})}^{2}}{{n}_{\mathrm{BL}}^{2}}\right]\\ & & +{h}_{{\rm{P}}}\exp \left[-\displaystyle \frac{{(n-{n}_{{\rm{P}}})}^{2}}{{w}_{{\rm{P}}}^{2}}\right]\left(1+\mathrm{erf}\left[{s}_{{\rm{P}}}\displaystyle \frac{(n-{n}_{{\rm{P}}})}{{w}_{{\rm{P}}}}\right]\right),\end{eqnarray} \tag{ 9 }$$

where the peak is described by its height h_P, position n_P, width w_P, and the shape or skewness parameter s_P. The coefficients c₁ and c₂ are again adjusted to the microscopic calculations up to the transition density n_tr to make sure that the speed of sound and its derivative are continuous. By varying the remaining parameters, we can model many possible curves for the speed of sound. In Figure 7, we show four such parameterizations where we also indicate the maximal central density reached for each EOS. As one can see, this approach is very versatile and can produce very different shapes for the speed of sound, some of which resemble the result by Paeng & Rho (2016).

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The chosen form in Equation (9) allows c_S to be a nonmonotonic function of density. The peak at intermediate density can be interpreted as describing a crossover transition that may be realized inside NSs. However, the high-density behavior, where c_S² ≈ 1/3, is realized at densities well beyond those accessed in NSs. While this functional form is rather simple and has only five parameters, it is comparable in complexity to a polytropic extension with three polytropic segments, and can be easily extended. We have also tested different functional forms for the speed of sound and found our results to be robust: the average radius for a typical 1.4 M_⊙ NS varied only on the percent level when different functional forms were chosen.

We sample values for the baseline width and the four peak parameters that determine c_S to construct the high-density EOS. We use the full EOS and solve the TOV equations to obtain NS mass–radius curves and the NS maximum mass; when this is found to be greater than the 2.01 M_⊙ mass constraint, we accept the parameter set, and reject it otherwise. We sample the five parameters from uniform distributions within the following ranges: n_BL between 0.01 and 3.0 fm⁻³, h_P between 0.0 and 0.9, w_P between 0.1 and 5.0 fm⁻³, n_P between (n_tr + 0.08) – 5.0 fm⁻³, and s_P between (−50) − 50, and enforce that $0\leqslant {c}_{S}^{2}\leqslant 1$.

For the chiral interactions, we additionally sample from the uncertainty bands for c_S by randomly choosing a factor f^err between −1 and 1, that linearly interpolates between the lower and upper bounds of the uncertainty band,

$$\begin{eqnarray}&&{c}_{S}(n)={c}_{S}^{{{\rm{N}}}^{2}\mathrm{LO}}(n)+{f}^{\mathrm{err}}{\rm{\Delta }}{c}_{S}^{\mathrm{EKM}}(n),\end{eqnarray} \tag{ 10 }$$

where ${c}_{S}^{{{\rm{N}}}^{2}\mathrm{LO}}(n)$ is the chiral result at N²LO and ${\rm{\Delta }}{c}_{S}^{\mathrm{EKM}}(n)$ is its uncertainty.

As stated before, we analyze the results for two different transition densities and generate a few thousand accepted parameter sets for each transition density. We show histograms for the resulting speed of sound, the mass–radius relation, and the EOS in Figures 8 and 9 for both transition densities. For the mass–radius histograms, we also show the average radius for each mass as well as 68% confidence intervals.

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We find that the speed of sound increases rapidly in a small density range above n_tr. This increase is more drastic for softer nuclear interactions. For stiffer interactions, c_S increases slowly and peaks at higher densities. In all cases, for a large fraction of parameterizations, the speed of sound increases to values around c_S ≈ 0.9. For the smaller transition density, there exist parameterizations that observe the conformal limit at all densities, while for the higher transition density all parameterizations violate this bound, consistent with our previous findings.

For the mass–radius relation, we find a rather broad radius distribution at lower transition densities that narrows with increasing transition density. This highlights the fact that PNM calculations at densities ∼2n₀ provide valuable information despite sizable uncertainties. We highlight this fact in Figure 10, where we show the radius of a typical 1.4 M_⊙ NS as a function of n_tr for the chiral EFT interactions. At n_tr,1, we find a radius range of 9.4–14.0 km (10.0–14.1 km) with a 68% confidence interval of 12.0 ± 1.0 km (12.3 ± 0.9 km) for the TPE-only (TPE+${V}_{E,{\mathbb{1}}}$) interaction. This range reduces to 9.4–11.8 km (10.2–12.3 km) with a 68% confidence interval of 10.7 ± 0.5 km (${11.5}_{-0.4}^{+0.3}$ km) for n_tr,2.

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For the phenomenological interaction, the mass–radius relation is much narrower than for the chiral interactions because the EOS is much stiffer and uncertainties associated with the interaction are unknown. For a typical NS, we find a radius range of 11.4–14.3 km with a 68% confidence interval of ${12.7}_{-0.6}^{+0.7}$ for n_tr,1 and a very narrow range of 12.8–12.9 km for n_tr,2.

In all histograms, we compare our findings to the corresponding envelopes of Hebeler et al. (2013) for a polytropic expansion with three polytropes and find very good agreement for all interaction models. This suggests that our extension is general enough to capture similar effects as the polytropic extension. Our results are also consistent with other radius constraints using EOSs obtained with the AFDMC method (Steiner & Gandolfi 2012; Steiner et al. 2015).

In Table 2, we show the maximum masses and the maximal central densities for all interactions and both transition densities. The upper limit for the maximum mass strongly depends on the chosen transition density but not on the interaction. For the smaller transition density, the highest achievable maximum mass is ≈3.6 M_⊙, independent of the interaction. For the higher transition density, the highest achievable maximum mass is ≈2.8 M_⊙ for all interactions. The maximal central density reached inside the NS, n_c,max, that is, the central density in the maximum-mass NS, ranges between 2.7 and 8.9n₀ for the lower transition density, and 4.4–9.3n₀ for the larger transition density.

Table 2.  Maximal NS Mass and Maximal Central Densities

ntr (fm−3)	Interaction	Mmax (M⊙)	nc,max (n0)
	TPE-only	2.01–3.63	2.8–8.6
0.18	TPE+${V}_{{\rm{E}},{\mathbb{1}}}$	2.01–3.66	2.7–8.9
	AV8'+UIX	2.01–3.57	2.9–8.4
	TPE-only	2.01–2.79	4.6–8.7
0.32	TPE+${V}_{{\rm{E}},{\mathbb{1}}}$	2.01–2.81	4.5–9.3
	AV8'+UIX	2.01–2.84	4.4–8.9

Note. Maximal NS mass and maximal central densities for all interactions and both transition densities.

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Kurkela et al. (2014), Fraga et al. (2016), and Annala et al. (2018) discussed the possibility of additionally constraining the EOS with information from pQCD at very high densities. Their calculation constrains the pressure at  ≈ 15000 MeV fm⁻³ to lie in the range of p ≈ 2600–4000 MeV fm⁻³. However, we find that the highest pressure inside any stable NS p_max ≈ 1570 MeV fm⁻³ is significantly smaller. Since we cannot infer the EOS at densities above the maximum central density from NS observations, it is always possible to find curves for the speed of sound with $0\leqslant {c}_{S}^{2}\leqslant 1$ that connect our models at the maximal central density with the pQCD constraint.

In the following, we want to discuss correlations between n_c,max and M_max, and ${c}_{S,\max }^{2}$ and M_max. We show histograms for these correlations in Figure 11 for the chiral interactions and n_tr,2, but our observations do not change significantly for the other interactions or transition densities.

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The correlation between n_c,max and M_max indicates, as expected, that the central densities of the heaviest NSs decrease with maximum mass. The same is true for the uncertainties of the central density, which also decrease with maximum mass. Furthermore, these central densities are well below the upper limit established by Lattimer & Prakash (2005).

For the correlation between M_max and ${c}_{S,\max }^{2}$, we find that the highest masses can only be reached for the stiffest EOS, i.e., EOS with highest possible ${c}_{S,\max }^{2}=1$. Furthermore, for every maximum mass there is a minimum for ${c}_{S,\max }^{2}$ that is needed to support this mass. It follows that a maximum mass observation will give a lower bound to the maximum speed of sound. For the current maximum mass constraint, M_max = 2 M_⊙, ${c}_{S,\max }^{2}\geqslant 0.4$ for n_tr,2, which is in excellent agreement with the findings of Alsing et al. (2018). An observation of a 2.4 M_⊙ NS would require ${c}_{S,\max }^{2}\geqslant 0.6$.

We begin by assuming that the compactness of a NS has been observed. It was claimed by Hambaryan et al. (2017) that the compactness of the NS RX J0720.4-3125 can be inferred to be (M/M_⊙)/(R/km) = 0.105 ± 0.002. Margueron et al. (2018) used this information to constrain the corresponding NS mass to be 1.33 ± 0.04 M_⊙ with a radius of 12.7 ± 0.3 km.

Investigating the effect of such a compactness observation, we find a rather broad range for possible neutron-star masses and radii, ranging from M = 1.00–1.38 M_⊙ and R = 9.8–12.8 km. The chiral interactions favor smaller NSs, with the weight of the distribution around the point (11 km, 1.1 M_⊙) for the TPE-only and (12 km, 1.2 M_⊙) for the TPE+${V}_{E,{\mathbb{1}}}$ interaction. The AV8'+UIX interaction leads to a prediction of M = 1.32–1.38 M_⊙ with a radius of 12.8 km, in good agreement with the prediction of Margueron et al. (2018).

The observation of the compactness alone is naturally not sufficient to further constrain the EOS. However, by observing the compactness of a star with known mass, which is equivalent to a radius measurement, additional constraints can be found. We discuss this in the next section.

In the following, we investigate how possible radius measurements by the NICER mission will impact our findings for the EOS and the mass–radius curve and help to constrain the microscopic equation of state. The NICER mission, which was launched in 2017 July, is expected to measure the compactness (and, thus, radius) of at least three NSs with known masses with an accuracy of 5%–10%. Among these are (Arzoumanian et al. 2014) the NSs PSR J1023+0038 with M = 1.71 ± 0.16 M_⊙ (Deller et al. 2012), PSR J0437-4715 with M = 1.44 ± 0.07 M_⊙ (Reardon et al. 2016), and a proposal for using NICER to measure PSR J1614-2230 with a mass of ≈2.0 M_⊙ (Miller 2016).

We show histograms for the radius of a typical 1.4 M_⊙ NS, R_1.4, versus the pressure at two times the nuclear saturation density, p(2n₀) = p₂, and the pressure at four times the nuclear saturation density, p(4n₀) = p₄ in Figure 13, for all interactions and n_tr,2. All interactions predict different p₂, as expected, but overlapping distributions for p₄. The chiral interactions predict similar radii around 10–12 km, and the AV8'+UIX interaction predicts a higher radius of ${R}_{1.4}\approx 12.8$ km, which is not consistent with the chiral models; see also Figure 9.

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We now assume that the radius of such a 1.4 M_⊙ NS is observed with an accuracy of 10%. We will focus on two extreme cases: R_obs = (10 ± 1.0) km, which is the lower bound on the suggested radius range of Ozel & Freire (2016), and R_obs = (13 ± 1.3) km, which is on the upper end of the currently accepted radius range. We show the corresponding histograms for the mass–radius relation and the speed of sound in Figure 14.

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The observation of a small 10 km NS would eliminate a sizable part of the stiffer parameterizations with higher p₂ and p₄ (see also Figure 13) and would allow us to obtain additional constraints on the microscopic EOS at n_tr,2: It would (i) exclude the stiffest chiral interactions; (ii) rule out the AV8'+UIX interaction; (iii) reduce the allowed maximum mass to ≈2.5 M_⊙; and (iv) suggest that the speed of sound changes less drastically above 2n₀ and peaks at densities of ∼4–5n₀ with ${c}_{S}^{2}\approx 0.8\pm 0.1{c}^{2}$.

The observation of a large 13 km NS, instead, would exclude the softest interactions. For the TPE-only interaction, for example, only a very small fraction of parameterizations would survive, and a major fraction of the uncertainty band for that interaction could be ruled out. Also, in this case, the speed of sound has to increase quickly above 2n₀.

Clearly, observations of more extreme radii and/or with smaller uncertainties would present even stronger constraints. For example, an observation of an 13 km NS with 5% uncertainty would rule out the TPE-only interaction. The prospect that NICER may achieve this precision soon with better understood systematic errors is exciting for nuclear physics and we eagerly await its results.

As we have shown in the previous section, the observation of a single NS radius might prove useful to constrain nuclear interactions, if the observed radius is either on the lower or on the upper side of the currently accepted range of 12 ± 2 km for typical 1.4 M_⊙ NSs. We now investigate to which extent the observation of two NS radii with 5% uncertainty for stars with known masses can be used to constrain both the EOS and nuclear interactions.

We assume that radii of a 1.4 M_⊙ neutron star, R_1.4, and of a 2.0 M_⊙ neutron star, R_2.0, have been observed, and present the pressure at twice saturation density, p₂, in Table 3 and the pressure at four times saturation density, p₄, in Table 4 for all interactions and assuming n_tr,2.

Table 3.  Pressure at Two Times Saturation Density for Hypothetical Radius Measurements

p2 (MeV fm−3)
				R1.4 (km)
			10% ± 5%	11% ± 5%	12% ± 5%	13% ± 5%
		TPE-only	6.5–10.7
	9% ± 5%	TPE+${V}_{{\rm{E}},{\mathbb{1}}}$
		AV8'+UIX
		TPE-only	5.9–10.9	5.9–10.9
	10% ± 5%	TPE+${V}_{{\rm{E}},{\mathbb{1}}}$	12.8–13.6	12.8–18.1	14.7–18.9
		AV8'+UIX
		TPE-only	5.9–10.4	5.9–10.9	9.8–10.8
R2.0 [km]	11% ± 5%	TPE+${V}_{{\rm{E}},{\mathbb{1}}}$		12.8–18.4	12.8–18.9
		AV8'+UIX				26.0
		TPE-only		5.9–10.9	6.1–10.9
	12% ± 5%	TPE+${V}_{{\rm{E}},{\mathbb{1}}}$		12.8–16.8	12.8–18.8
		AV8'+UIX				26.0
		TPE-only			10.8–10.9
	13% ± 5%	TPE+${V}_{{\rm{E}},{\mathbb{1}}}$			13.1–18.8
		AV8'+UIX				26.0

Note. Pressure at two times saturation density, p₂, for all interactions and n_tr,2 = 0.32 fm⁻³, for hypothetical simultaneous measurements of the radii of a 1.4 M_⊙ and a 2.0 M_⊙ NS, R_1.4 and R_2.0, respectively.

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Table 4.  Pressure at Four Times Saturation Density for Hypothetical Radius Measurements

p4 (MeV fm−3)
				R1.4 (km)
			10% ± 5%	11% ± 5%	12% ± 5%	13% ± 5%
		TPE-only	52–92
	9% ± 5%	TPE+${V}_{{\rm{E}},{\mathbb{1}}}$
		AV8'+UIX
		TPE-only	69–182	113–159
	10% ± 5%	TPE+${V}_{{\rm{E}},{\mathbb{1}}}$	72–112	84–152	107–141
		AV8'+UIX
		TPE-only	151–242	145–310	168–192
R2.0 [km]	11% ± 5%	TPE+${V}_{{\rm{E}},{\mathbb{1}}}$		122–265	117–252
		AV8'+UIX				147–152
		TPE-only		164–428	178–428
	12% ± 5%	TPE+${V}_{{\rm{E}},{\mathbb{1}}}$		182–306	163–442
		AV8'+UIX				148–285
		TPE-only			425–426
	13% ± 5%	TPE+${V}_{{\rm{E}},{\mathbb{1}}}$			230–446
		AV8'+UIX				171–461

Note. Pressure at four times saturation density, p₄, for all interactions and n_tr,2 = 0.32 fm⁻³, for hypothetical simultaneous measurements of the radii of a 1.4 M_⊙ and a 2.0 M_⊙ NS, R_1.4 and R_2.0, respectively.

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For n_tr,2, the pressure p₂ is set by the nuclear input EOS and independent of the extension. As we have shown in Section 3.2, different interactions are compatible with different ranges for R_1.4. The chiral interactions lead to R_1.4 ≈ 10–12 km and the AV8'+UIX interaction is only compatible with R_1.4 = 13 km ± 5%. If, for instance, R_1.4 = 10 km ± 5% was observed, the AV8'+UIX interaction would be ruled out.

An additional observation of R_2.0 would permit further constraints. For instance, both chiral interactions permit R_1.4 = 12 km, but only the TPE+${V}_{{\rm{E}},{\mathbb{1}}}$ interaction could simultaneously lead to R_2.0 = 10 km. Also, the second radius observation might prove useful to constrain p₄ and the high-density equation of state. For a single radius measurement, the predicted ranges for p₄ are very large. For example, for R_1.4 = 12 km, the predicted p₄ = 107–446 MeV fm⁻³. A second radius measurement would allow us to clearly reduce this range in most cases. If, for example, R_2.0 = 11 km, the range for p₄ would reduce to p₄ = 117–252 MeV fm⁻³.

The observation of two NS radii could be very useful to constrain both low- and high-density EOS, and will hopefully be made available by the NICER mission.