Trace of the energy-momentum tensor and macroscopic properties of neutron stars

A generic feature of scalar extensions of general relativity is the coupling of the scalar degrees of freedom to the trace $T$ of the energy-momentum tensor of matter fields. Interesting phenomenology arises when the trace becomes positive—when pressure exceeds one third of the energy density—a condition that may be satisfied in the core of neutron stars. In this work, we study how the positiveness of the trace of the energy-momentum tensor correlates with macroscopic properties of neutron stars. We first show that the compactness for which $T=0$ at the stellar center is approximately equation-of-state independent, and given by $C=0.262_{-0.017}^{+0.011}$ (90% confidence interval). Next, we exploit Bayesian inference to derive a probability distribution function for the value of $T$ at the stellar center given a putative measurement of the compactness of a neutron star. This investigation is a necessary step in order to use present and future observations of neutron star properties to constrain scalar-tensor theories based on effects that depend on the sign of $T$.

Figure 1

Mass-radius curves for 136 piecewise-polytropic causal EoS (cf. Sec. 2a). Configurations for which $T<0$ in the entire stellar interior are displayed in orange, while those for which $T>0$ in some region inside the star are depicted in blue. We highlight in cyan the configurations for which $T=0$ at the stellar center, which marks the transition between the previous regimes. The red line corresponds to $M=(c^2/G)CR$, with $C=0.262$, as discussed in the main text.

Figure 2

Probability distribution functions for $T_c$, the central value of the trace of the energy-momentum tensor, corresponding to six values of a compactness measurement, ranging from $C=0.200$ (leftmost curve) to $C=0.325$ (rightmost curve). The assumed uncertainty in the measurement is fixed to ${\sigma }_C=0.03$ in all cases. Shades give an estimate of the error in the marginalization procedure (cf. Sec. 4).

Figure 3

$T=3p-\epsilon$ as a function of the areal radial coordinate for the most massive star allowed by four realistic EoS. We adopt the piecewise-polytropic approximation to the APR4, H4, SLy, and MPA1 models, as given in Table III of Ref. [22].

Figure 4

Histogram of the compactness of a stable and causal star with $T_c=0$, computed for 20 000 EoS.

Figure 5

Histogram of the maximum speed of sound inside a star with $T_c=0$, computed for 15 000 sampled EoS. Vertical lines highlight the values $c_{s,\max }=c/\sqrt{3}$ and $c_{s,\max }=c$.

Figure 6

Central value of the trace of the energy-momentum tensor $T_c$ as a function of the stellar compactness $C$. Curves are drawn for 100 causal EoS, and they terminate at the maximum value of $C$ for a stable configuration according to each EoS. It should be noted that while $T_c$ is a single-valued function of $C$ for each EoS, $C$ can be a multivalued function of $T_c$.

Figure 7

Probability distribution functions for $T_c$, for two values of a compactness measurement, $C=0.225$ and $C=0.325$. For the dashed curves, a maximum-mass cutoff of $2.3M_{\odot }$ was included in the EoS prior, while no maximum mass cutoff was imposed for the solid curves. In all cases, ${\sigma }_C=0.03$.

Figure 8

Probability distribution functions for $T_c$, for two values of the measured compactness, $C=0.275$ and $C=0.325$, and ${\sigma }_C=0.03$ (solid curves) or ${\sigma }_C=0.01$ (dashed curves). In the upper panel the PDF is shown for an EoS prior that includes no maximum mass cutoff, while in the lower panel a maximum mass cutoff of $2.3M_{\odot }$ is considered.

Figure 9

Probability distribution function for $T_c$, for a measured stellar compactness of $C=0.275$, with ${\sigma }_C=0.03$, and either no concurrent mass measurement or mass measurements of $1.8M_{\odot }$ or $2.1M_{\odot }$, with ${\sigma }_M=0.05M_{\odot }$. Shades represent the error in the marginalization procedure.