Limiting masses and radii of neutron stars and their implications

We combine the equation of state of dense matter up to twice nuclear saturation density $n_{\mathrm{sat}}$ obtained using chiral effective field theory $\left(\chi \mathrm{EFT}\right)$ and recent observations of neutron stars to gain insights about the high-density matter encountered in their cores. A key element in our study is the recent Bayesian analysis of correlated EFT truncation errors based on order-by-order calculations up to next-to-next-to-next-to-leading order in the $\chi \mathrm{EFT}$ expansion. We refine the bounds on the maximum mass imposed by causality at high densities and provide stringent limits on the maximum and minimum radii of $\sim 1.4{\mathrm{M}}_{\odot }$ and $\sim 2.0{\mathrm{M}}_{\odot }$ stars. Including $\chi \mathrm{EFT}$ predictions from $n_{\mathrm{sat}}$ to $2n_{\mathrm{sat}}$ reduces the permitted ranges of the radius of a $1.4{\mathrm{M}}_{\odot }$ star, $R_{1.4}$, by $\sim 3.5\text{km}$. If observations indicate $R_{1.4}<11.2\text{km}$, then our study implies that either the squared speed of sound $c_s^2>1/2$ for densities above $2n_{\mathrm{sat}}$ or that $\chi \mathrm{EFT}$ breaks down below $2n_{\mathrm{sat}}$. We also comment on the nature of the secondary compact object in GW190814 with mass $\simeq 2.6{\mathrm{M}}_{\odot }$ and discuss the implications of massive neutron stars $>2.1{\mathrm{M}}_{\odot }\left(2.6{\mathrm{M}}_{\odot }\right)$ in future radio and gravitational-wave searches. Some form of strongly interacting matter with $c_s^2>0.35\left(0.55\right)$ must be realized in the cores of such massive neutron stars. In the absence of phase transitions below $2n_{\mathrm{sat}}$, the small tidal deformability inferred from GW170817 lends support for the relatively small pressure predicted by $\chi \mathrm{EFT}$ for the baryon density $n_{\mathrm{B}}$ in the range $1\text{–}2n_{\mathrm{sat}}$. Together they imply that the rapid stiffening required to support a high maximum mass should occur only when $n_{\mathrm{B}}\gtrsim 1.5\text{–}1.8n_{\mathrm{sat}}$.

Figure 1

(a) Pressure of neutron-star matter (NSM) in the outer core as a function of the baryon number density at ${\mathrm{N}}^2\mathrm{LO}$ (orange-shaded band) and ${\mathrm{N}}^3\mathrm{LO}$ (blue-shaded band) in the chiral expansion; (b) differences of the PNM and NSM in the outer core with the same notation. Uncertainty bands depict $1\sigma$ confidence regions.

Figure 2

(a) $M\text{−}R$ diagram for NSM based on MBPT calculations shown in Fig. 1, including ${\mathrm{N}}^2\mathrm{LO}$-NSM (orange-shaded band) and ${\mathrm{N}}^3\mathrm{LO}$-NSM (blue-shaded band) in the outer core matched to a linear causal ($c_{s,\mathrm{match}}^2=1.0$) EOS at $n_{\mathrm{m}}=2.0n_{\mathrm{sat}}$, and the maximally compact EOS for self-bound stars with the same value of $M_{\max }$ (black solid). Horizontal lines indicate $M=1.4$, 2.0, $2.6{\mathrm{M}}_{\odot }$. The colored bands above $n_{\mathrm{m}}=2.0n_{\mathrm{sat}}$ represent upper bounds on the NS radius for a given mass, as the high-density matter is assumed maximally stiff without discontinuities in the overall EOS (see detailed discussions in Sec. 3a). (b) Similar to panel (a) but with a lower matching density, $n_{\mathrm{m}}=1.5n_{\mathrm{sat}}$.

Figure 3

Radius bounds obtained by combining $\mathrm{N}{}^3\mathrm{LO}\text{−}\chi \mathrm{EFT}$ predictions up to $n_{\mathrm{m}}=2.0n_{\mathrm{sat}}$ and maximum-mass information is shown. The orange bands show the upper bound on the NS radius, while the black and purple bands depict the lowers bounds corresponding to $M_{\max }=2.0{\mathrm{M}}_{\odot }$ and $M_{\max }=2.6{\mathrm{M}}_{\odot }$, respectively.

Figure 4

Similar to Fig. 3 but obtained using the polytropic extrapolation of the $\chi \mathrm{EFT}$ EOS up to $n_{\mathrm{m}}=3.0n_{\mathrm{sat}}$.

Figure 5

$M\text{−}R$ relations for NSM EOSs extrapolated to $n_{\mathrm{B}}\ge 2.0n_{\mathrm{sat}}$ beyond the $\chi \mathrm{EFT}$ calculations using the ZL parametrization [60]; the thin black line indicates where the NS central densities are $3.0n_{\mathrm{sat}}$. From left to right, the colored dotted curves represent $L=45\text{MeV}$ to $L=75\text{MeV}$ in increments of $5\text{MeV}$, and the black-solid (black-dashed) curves refer to $\chi \mathrm{EFT}\text{−}{\mathrm{N}}^3\mathrm{LO}$ (${\mathrm{N}}^2\mathrm{LO}$) with $\pm 1\sigma$ uncertainties. The $L=50\text{MeV}$ (red) and $L=60\text{MeV}$ (green) ZL EOSs are used in Fig. 11 because they best represent $\pm 1\sigma$ bounds.

Figure 6

Similar to Figs. 3 and 4, but displaying the minimum and maximum radii of $1.4{\mathrm{M}}_{\odot }$ (a) and $2.0{\mathrm{M}}_{\odot }$ (b) stars as a function of the matching density $n_{\mathrm{m}}=1.0-3.0n_{\mathrm{sat}}$. Additionally, $R_{\min }$ contours and uncertainty bands for the case $M_{\max }=2.3{\mathrm{M}}_{\odot }$ are shown.

Figure 7

$M_{\max }$ contours on the $(c_{s,\mathrm{match}}^2,n_{\mathrm{m}})$ plane, obtained when $\mathrm{\Delta }{\varepsilon }_{\mathrm{m}}=0$. For each value of $M_{\max }$, the central solid curve shows results with the central value of $\chi \mathrm{EFT}\text{−}{\mathrm{N}}^3\mathrm{LO}$; dashed lines indicate $\pm 1\sigma$ bounds. Extensions to $n_{\mathrm{m}}>2.0n_{\mathrm{sat}}$ for the central and $+\sigma$ bound are also shown as solid curves. The gray-shaded region is excluded by the binary tidal deformability constraint ${\tilde{\mathrm{\Lambda }}}_{1.186}\le 720$ from GW170817 at the 90% credibility level [6] if ${\mathrm{N}}^3\mathrm{LO}$-cen is assumed; the dot-dashed lines refer to constraints with the ${\mathrm{N}}^3\mathrm{LO}$ $\pm 1\sigma$ boundaries. The GW170817 bounds will be shifted downwards if there is a first-order transition at such low densities.

Figure 8

The maximum (orange) and minimum (black) bounds on $R_{1.4}$ and $R_{2.0}$ assuming $c_s^2\le 1/3$ above $n_{\mathrm{B}}=n_{\mathrm{m}}$; $\chi \mathrm{EFT}\text{−}{\mathrm{N}}^3\mathrm{LO}$ uncertainties are indicated (darker bands: $\pm 1\sigma$; lighter bands: $\pm 2\sigma$). The bands merge and terminate at critical matching densities above which $M_{\max }<2.0{\mathrm{M}}_{\odot }$.

Figure 9

Similar to Fig. 8, but assuming $c_s^2\le 1/2$ above $n_{\mathrm{B}}=n_{\mathrm{m}}$.

Figure 10

The left panel shows bounds on the tidal deformability $\mathrm{\Lambda }$ obtained using $\chi \mathrm{EFT}$ ${\mathrm{N}}^3\mathrm{LO}$ EOS up to $n_{\mathrm{m}}=2.0n_{\mathrm{sat}}$, and the right panel extends the low-density EOS to $n_{\mathrm{m}}=3.0n_{\mathrm{sat}}$ using the polytropic extrapolation. As in Fig. 3, the orange bands show the upper bound, while the lower bounds corresponding to $M_{\max }=2.0{\mathrm{M}}_{\odot }$ and $M_{\max }=2.6{\mathrm{M}}_{\odot }$ are shown by the black and purple bands, respectively. The vertical solid line depicts the constraint inferred from GW170817, $70\le {\mathrm{\Lambda }}_{1.4}\le 580$. When $n_{\mathrm{m}}=3.0n_{\mathrm{sat}}$, $M_{\max }<2.6{\mathrm{M}}_{\odot }$.

Figure 11

(a) The solid lines show contours of $n_{\mathrm{m}}$ in the $M_{\max }\text{−}{c}_{s,\mathrm{match}}^2$ plane, and the dashed lines bracket ${\mathrm{N}}^3\mathrm{LO}$ $\pm 1\sigma$ uncertainties. The upper horizontal line indicates $M_{\max }=2.6{\mathrm{M}}_{\odot }$; see also examples later in Fig. 14. The gray-shaded region is excluded by the binary tidal deformability constraint ${\tilde{\mathrm{\Lambda }}}_{1.186}\le 720$ from GW170817 at the 90% credibility level [6] if ${\mathrm{N}}^3\mathrm{LO}$-cen is assumed; the dot-dashed lines refer to constraints with the ${\mathrm{N}}^3\mathrm{LO}$ $\pm 1\sigma$ boundaries. The thin dotted line indicates a lower upper bound with ${\mathrm{N}}^3\mathrm{LO}$-cen and ${\tilde{\mathrm{\Lambda }}}_{1.186}\le 600$. (b) Same as panel (a), except that contours of $c_{s,\mathrm{match}}^2$ are displayed in the $M_{\max }\text{−}{n}_{\mathrm{m}}$ plane; the upper-right gray-shaded region is excluded by causality. For $n_{\mathrm{m}}\in [2.0,3.0]n_{\mathrm{sat}}$, extrapolations from $\chi \mathrm{EFT}$ using ZL models with $L=50\text{MeV}$ and $L=60\text{MeV}$ are applied (see Fig. 5).

Figure 12

(a) Scaling relations between $M_{\max }$ and $n_{\max }$; (b) scaling relations between $M_{\max }$ and $R_{M_{\max }}$. Both relations, shown as dot-dashed lines, follow from the maximally compact EOS (see Appendix pp1). The black dashed curves correspond to the presence of a low-density nuclear mantle (crust $+$ ${\mathrm{N}}^3\mathrm{LO}$ EOS) for $n_{\mathrm{B}}\le n_{\mathrm{m}}$, with fixed sound speeds $c_{s,\mathrm{match}}^2=0.33$ and $c_{s,\mathrm{match}}^2=1.0$ for $n_{\mathrm{B}}>n_{\mathrm{m}}$. The gray-shaded region is excluded by GW170817 (${\tilde{\mathrm{\Lambda }}}_{1.186}\le 720$ and ${\mathrm{N}}^3\mathrm{LO}$-cen). The solid colored curves show contours of $n_{\mathrm{m}}=1.0,1.5,2.0n_{\mathrm{sat}}$ for ${\mathrm{N}}^3\mathrm{LO}$-cen; dashed colored curves show $\pm 1\sigma$ uncertainties. For EOSs that accommodate $M_{\max }\ge 2.6{\mathrm{M}}_{\odot }$, the permitted ranges of $n_{\max }$ and $R_{M_{\max }}$ are severely restricted.

Figure 13

Radius differences $\mathrm{\Delta }R=R_{2.0}-R_{1.4}$ using the ZL extrapolations with $L=50\text{MeV}$ and $L=60\text{MeV}$ joined continuously to linear EOSs at $n_{\mathrm{m}}$ between $2.0n_{\mathrm{sat}}$ and $3.0n_{\mathrm{sat}}$.

Figure 14

(a) $M\text{−}R$ diagram for matched linear EOSs that give rise to $M_{\max }=2.6{\mathrm{M}}_{\odot }$ with ${\mathrm{N}}^3\mathrm{LO}$-NSM ($\pm 1\sigma$) applied for low densities $\le 2.0n_{\mathrm{sat}}$. Corresponding values of $n_{\mathrm{m}}$ and $c_{s,\mathrm{match}}^2$ are indicated [see also Fig. 11]. (b) Sound speed profiles $c_s^2\left(n_{\mathrm{B}}\right)$ for ${\mathrm{N}}^3\mathrm{LO}$-NSM only (black-solid for the central value and black-dashed for $\pm 1\sigma$ uncertainties) and matched linear EOSs with different values of $c_{s,\mathrm{match}}^2$ associated with $M_{\max }=2.6{\mathrm{M}}_{\odot }$ in panel (a) (colored horizontal lines). The open triangles mark the central densities of the maximum-mass stars $M_{\max }=2.6{\mathrm{M}}_{\odot }$.

Figure 15

$\tilde{\mathrm{\Lambda }}\text{−}\mathcal{M}$ and $\mathrm{\Lambda }\text{−}M$ relations confronted with constraints from GW170817 [6, 7] (vertical lines with arrows), with fixed $M_{\max }=2.6{\mathrm{M}}_{\odot }$ as an example. Parameters for matched EOSs are the same as in Fig. 14, except for the special case with $n_{\mathrm{m}}=1.674n_{\mathrm{sat}}$ (with $c_{s,\mathrm{match}}^2=0.5643$), which refers to the minimum matching density that survives $\tilde{\mathrm{\Lambda }}(\mathcal{M}=1.186{\mathrm{M}}_{\odot })\le 720$.

Figure 16

The mass as a function of the radius for the EOS Eq. (A9) with $P_0=0$, $P=0$ for $\varepsilon <{\varepsilon }_0$, and various values for $c_s^2$ with fixed $M_{\max }=2.6{\mathrm{M}}_{\odot }$, are shown as five black curves (see legend). These curves correspond to the minimum possible radius $R_{\min }\left(M\right)$, for different maximum values of the sound speed. The four red curves correspond to ${\varepsilon }_0={\varepsilon }_{\mathrm{sat}}$, either $c_s^2=1$ (and $M_{\max }\simeq 4.09{\mathrm{M}}_{\odot }$) or $c_s^2=1/3$ (and $M_{\max }\simeq 2.48{\mathrm{M}}_{\odot }$) for $P>P_0$, and either $P_0=0$ (self-bound) or $P_0=0.02{\varepsilon }_0\simeq 3\text{MeV}{\text{fm}}^{-3}$ and a normal crust EOS for $P<P_0$ [maximum possible radii $R_{\max }\left(M\right)$]; the configuration where ${\varepsilon }_c={\varepsilon }_{\mathrm{sat}}$ is indicated by a diamond.

Figure 17

Correlations among $P\left(n_{\mathrm{B}}\right)$, $R_{1.4}$, $R_{2.0}$, and $M_{\max }$ for six EOS parametrizations (see text for details). “Average” refers to the mean of all models. Blue histograms show the summed distributions of the central densities of the relevant stars.