Differentiating between sharp and smoother phase transitions in neutron stars

The internal composition of neutron stars is still an open issue in astrophysics. Their innermost regions are impervious to light propagation and gravitational waves mostly carry global aspects of stars, meaning that only indirect inferences of their interiors could be obtained. Here we assume a hypothetical future scenario in which an equation of state softening due to a phase transition is identified and estimate the observational accuracy to differentiate a sharp phase transition from a smoother one (which we take to be associated with a mixed phase/state due to the unknown value of the surface tension of dense matter) in a region of a hybrid star by means of some electromagnetic and gravitational wave observables. We show that different transition constructions lead to similar sequences of stellar configurations due to their shared thermodynamic properties. In the most optimistic case—a strong quark-hadron density jump phase transition—radius observations require fractional uncertainties smaller than 1%–2% to differentiate mixed states from sharp phase transitions. For tidal deformabilities, relative uncertainties should be smaller than 5%–10%. However, for masses around the onset of stable quark cores, relative tidal deformability differences associated with strong sharp phase transitions and mixed states connecting the two pure phases could be much larger (up to around 20%–30%). All the above suggests that 2.5- and third-generation gravitational wave detectors and near-term electromagnetic missions may be able to start assessing some particular aspects of phase transitions in neutron stars. In addition, it points to some limitations on the equation of state recovery using typical neutron star observables and the impact of systematic uncertainties on modelings of the equation of state of hybrid stars. Finally, we briefly discuss other observables that may also be relevant for the probe of mixed states in stars.

Figure 1

Examples of polytropic EOSs and resulting sequences of NS parameters (solutions of TOV equations): sharp phase transition (solid blue curves), and three mixed state realizations (dash-dotted orange curves with $n_1=0.375{\mathrm{fm}}^{-3}$, dashed green curves with $n_1=0.4{\mathrm{fm}}^{-3}$ and dotted red curves with $n_1=0.425{\mathrm{fm}}^{-3}$). The sharp phase transition EOS parameters are ${\gamma }_1=3.5$, ${\gamma }_2=6$, density jump (in terms of the baryon density $n$) between $n_0=0.475{\mathrm{fm}}^{-3}$ and $n_{02}=0.76{\mathrm{fm}}^{-3}$. The leftmost panel contains the mass-radius $M(R)$ sequences (the inset plot presents a closeup of the region around the maximum mass), the middle panel is the chemical potential-pressure $\mu (P)$ relation, the upper right panel is the pressure-density $P(\rho )$ relation, whereas the lower right one is the mass-tidal deformability $M(\mathrm{\Lambda })$ relation. Green and red dots mark the beginning and the end of the mixed-state region in the case of the $n_1=0.4{\mathrm{fm}}^{-3}$ EOS; correspondingly, stellar configurations in the other panels have central parameters equal to the beginning (green dot) and the end (red dot) of the mixed state. The inset in the $\mu (P)$ plot shows the definition of ${\mathrm{\Delta }}_p$—marked by an arrow—on the example of the $n_1=0.4{\mathrm{fm}}^{-3}$ EOS, marked by the green dashed line. $P({\mu }_0)$ is denoted by $P_0$. Note that the for the $M(R)$ sequences the mixed state curves are below the sharp one in the vicinity of the phase transition point, but $M_{\max }$ is larger for the mixed state EOSs. For the $n_1=0.4{\mathrm{fm}}^{-3}$ EOS, the mixed state and the sharp transition EOSs have the same mass and radius parameters at $M\approx 1.68M_{\odot }$ and $\approx 10.92\mathrm{km}$, marked by a magenta cross.

Figure 2

Mass differences of hybrid stars with mixed states and sharp phase transitions for the maximum masses for several polytropic equations of state as a function of ${\mathrm{\Delta }}_p$.

Figure 3

Same as Fig. 2 but now taking into account the pressure difference between the bottom and the top of the mixed state. The fit in the plot has the form $y=a_p{x}^p$, with $a_p=2.911\times {10}^{-2}$ and $p=1.5317$.

Figure 4

Maximum mass dependence on the chemical potentials at the borders of the mixed state. The power-law fit of Fig. 2 gives $a_p=2.573\times {10}^{-4}$ and $p=1.7316$.

Figure 5

Radius differences associated with the maximum masses for stars with mixed states and sharp phase transitions for several polytropic equations of state as a function of ${\mathrm{\Delta }}_p$. The parameters for the power-law fit ($y=a_p{x}^p$) are $a_p=0.2557$ and $p=1.930$.

Figure 6

Masses of the shells containing mixed states (within pressures $P_1$ and $P_3$) subtracted by masses encompassed within the same pressures in stars with sharp phase transitions, normalized by the reference (“ref”) masses (either 1.4 or 1.8 solar masses). For $1.4M_{\odot }$ stars, the fit ($y=a_p{x}^p$) is such that $a_p=1.506\times {10}^{-2}$ and $p=1.697$. For $1.8M_{\odot }$, it follows that $a_p=5.020\times {10}^{-3}$ and $p=1.818$.

Figure 7

Thickness differences of the mixed states of stars with respect to sharp phase transitions for given reference masses (and for completeness the same central pressure $P_c$ for a sharp EOS and its associated mixed EOSs), normalized by the radius of stars with sharp phase transitions ($R_{\mathrm{ref}}^{\text{sharp}}$). For $1.4M_{\odot }$ stars, the fit $y=a_p{x}^p$ implies that $a_p=6.509\times {10}^{-7}$ and $p=2.812$. For $1.8M_{\odot }$, it follows that $a_p=4.185\times {10}^{-7}$ and $p=2.693$. When the central pressure of a star with a sharp phase transition and another one with a mixed state is the same (their masses are not the same; we choose $P_c\mathrm{s}$ in stars with mixed-state EOSs so their masses are $1.8M_{\odot }$), we have that $a_p=1.622\times {10}^{-4}$ and $p=1.326$.

Figure 8

Relative radius differences for $1.4M_{\odot }$ stars with mixed states, sharp phase transitions and even purely hadronic (“had”)/one phase. Here, 2% is a representative level of accuracy of near-future electromagnetic/GW missions. The largest differences are connected with the comparison of one-phase stars with those with mixed states. The smallest differences [$\lesssim (1–2)\%$], though, come from stars with mixed states and those with sharp phase transitions and quark cores.

Figure 9

Relative thicknesses and masses of mixed states in hybrid stars with $1.8M_{\odot }$. Straight lines ($y=a_{1}x$) fit well the data; for the relative thickness size of the mixed state, $a_1=0.1189$; for its relative mass, $a_1=0.1969$. The size and mass of the mixed state is non-negligible when compared to the corresponding aspects of the star in general, and it increases with the increase of ${\gamma }_m$.

Figure 10

Relative tidal deformabilities of $1.8M_{\odot }$ hybrid stars (with mixed states and sharp phase transitions and quark cores) as a function of the chemical potential difference at the beginning and the end of the mixed state. The threshold of around 5% precision is reached for chemical potential differences larger than approximately 250 MeV and ${\gamma }_m\gtrsim 1.5$.

Figure 11

Relative tidal deformabilities of $1.4M_{\odot }$ stars relaxing the constraint that they only have quark cores and central pressures larger than the ones fully encompassing the mixed state: here they can also be purely hadronic (“had”) or have mixed states at their centers. The tidal deformability normalization has been chosen to be those of $1.4M_{\odot }$ stars either presenting sharp phase transitions or being purely hadronic, which depends on the phase transition masses due to their EOSs. The largest differences are obtained when one compares an one-phase star with another one having a mixed state (for a same mass), for large values of ${\gamma }_m$. Fractional differences even exceeding 10% could happen for several EOSs and ${\gamma }_m\mathrm{s}$. When stars only with sharp phase transitions and mixed states are compared, maximum tidal deformations are of a few percent ($\lesssim 2\%–3\%$).

Figure 12

Mass-radius relations around the appearance of a quark phase for stars with and without the mixed state for $\eta =0.71$ (same sharp-phase-transition EOS of Fig. 1). The dot-dashed cyan curve is an enlargement of the correspondent curve in Fig. 1 for a polytropic construction of the mixed state ($n_1=0.4{\mathrm{fm}}^{-3}$ EOS) around the appearance of the quark phase. The darker (lighter) box corresponds to masses and radii with (representative) 1% (2%) fractional uncertainties, centered around the critical point for the Maxwell construction ($M=1.63M_{\odot }$, $R=11.84\mathrm{km}$).

Figure 13

Mass-tidal deformability relations around the appearance of a quark phase for stars with and without the mixed state for $\eta =0.71$. The dot-dashed cyan curve is also an enlargement of the correspondent plot in Fig. 1 for the $n_1=0.4{\mathrm{fm}}^{-3}$ EOS. The darker (lighter) box corresponds to tidal deformability with (representative) 5% (10%) fractional uncertainties. The mass uncertainties are the same of Fig. 12. Around the onset of stable quark phases ($M=1.63M_{\odot }$ and $\mathrm{\Lambda }=120$), relative tidal deformability differences associated with strong sharp phase transitions and mixed states could be up to around 25%.

Figure 14

Same as Fig. 12 but for $\eta =0.391$ (weak phase transition). We centered uncertainty boxes at $M=1.31M_{\odot }$, $R=12.75\mathrm{km}$ (inflection point of the ${\mathrm{\Delta }}_p=2\%$ curve).

Figure 15

Same as Fig. 13 but for weak phase transitions. The uncertainty boxes are centered at the same mass of Fig. 14 and its associated tidal deformability. Maximum fractional tidal differences here are much smaller than in the strong phase transition case.