Post-Tolman-Oppenheimer-Volkoff formalism for relativistic stars

Besides their astrophysical interest, compact stars also provide an arena for understanding the properties of theories of gravity that differ from Einstein’s general relativity. Numerous studies have shown that different modified theories of gravity can modify the bulk properties (such as mass and radius) of neutron stars for given assumptions on the microphysics. What is not usually stressed though is the strong degeneracy in the predictions of these theories for the stellar mass and radius. Motivated by this observation, in this paper we take an alternative route and construct a stellar structure formalism which, without adhering to any particular theory of gravity, describes in a simple parametrized form the departure from compact stars in general relativity. This “post-Tolman-Oppenheimer-Volkoff (TOV)” formalism for spherical static stars is inspired by the well-known parametrized post-Newtonian theory, extended to second post-Newtonian order by adding suitable correction terms to the fully relativistic TOV equations. We show how neutron star properties are modified within our formalism, paying special attention to the effect of each correction term. We also show that the formalism is equivalent to general relativity with an “effective” (gravity-modified) equation of state.

Figure 1

The gravity-theory degeneracy problem. The mass-radius relations in different modified theories of gravity for EOS APR [19]. Masses are measured in solar masses, and radii in kilometers. The theory parameters used for this plot are $\alpha =20M_{\odot }^2$ and ${\beta }^2=1$ (Einstein-dilaton-Gauss-Bonnet [22]), $c_{14}=0.3$ (Einstein-aether [24]), $\beta =-4.5$ (scalar-tensor theory [20]), and $\kappa =\pm 0.005$ (Eddington-inspired-Born-Infeld gravity [26]). Even if the high-density EOS were known, it would be hard to distinguish the effects of competing theories of gravity on the bulk properties of neutron stars.

Figure 2

2PN-order post-TOV corrections on the mass-radius curves. We show the modification induced by different families of post-TOV terms on the general relativistic mass-radius curve, assuming the APR EOS. Left to right and top to bottom, the different panels show the effect of the pressure terms, proportional to ${\pi }_i$ ($i=1,\dots ,4$), and of the mass terms, proportional to ${\mu }_i$ ($i=1,\dots ,5$).

Figure 3

The radial profile of the ${\mathrm{\Lambda }}_2$ function. We exhibit the behavior of $\overline{\rho }{\mathrm{\Lambda }}_2$, where $\overline{\rho }=\rho /{\rho }_c$ and ${\mathrm{\Lambda }}_2$ is given in Eq. (47), for a stellar model using the APR EOS, with ${\epsilon }_{\mathrm{c}}/c^2=0.86\times {10}^{15}\mathrm{g}/{\mathrm{cm}}^3$, $M=1.51M_{\odot }$, and $R=12.3\mathrm{km}$. The curves are labeled according to the respective values of $(\alpha ,\beta ,\theta )$. From the top row to the bottom row the index $\theta$ takes on the values (0,1,2,3), respectively. Despite the multitude of possible dimensionally correct 2PN terms, their self-similarity—which is clear when we compare terms along the bottom-left to top-right diagonals in this “grid” of plots—allows us to group them into a relatively small number of families (see text for details). The contributions plotted in the three panels at the bottom right of the grid (marked as “Excluded”) would lead to divergences in the hydrostatic equilibrium equations, and therefore they can be discarded as unphysical.

Figure 4

The family-representative 2PN terms. Here we show the selected representative terms from each of the families depicted in Fig. 3, as listed in Eq. (54), for three different EOSs: FPS (left panel), APR (middle panel) and an $n=0.6$ polytrope (right panel). Each term illustrates the qualitative behavior of each family of possible 2PN contributions to the structure equations. The high degree of invariance of the ${\mathrm{\Lambda }}_2$ profiles with respect to the EOS is evident in this figure. The GR background stellar models utilized in the figure have the following bulk properties: ${\epsilon }_{\mathrm{c}}=0.861\times {10}^{15}\mathrm{g}/{\mathrm{cm}}^3$ ($\lambda \equiv p_{\mathrm{c}}/{\epsilon }_{\mathrm{c}}=0.165$), $M=1.51M_{\odot }$ and $R=12.3\mathrm{km}$ (left panel); ${\epsilon }_{\mathrm{c}}=1.450\times {10}^{15}\mathrm{g}/{\mathrm{cm}}^3$ ($\lambda =0.198$), $M=1.50M_{\odot }$ and $R=10.7\mathrm{km}$ (center panel); $\lambda =0.165$, $M=1.50M_{\odot }$ and $R=11.75\mathrm{km}$ (right panel).

Figure 5

The family-representative terms in the structure equations. This figure illustrates the behavior of each of the family-representative 2PN terms [see Eq. (54)] multiplied by the Newtonian prefactors in the post-TOV equations. The stellar parameters are identical to the ones used in Fig. 4. Left panel: The combination $(\rho m/r^2){\mathrm{\Lambda }}_2(\alpha ,\beta ,\theta )$ appearing in the pressure equation. Right panel: The combination $\rho r^2{\mathrm{\Lambda }}_2(\alpha ,\beta ,\theta )$ appearing in the mass equation. The top panels correspond to EOS APR; the bottom panels correspond to a relativistic polytrope with polytropic index $n=0.6$. The divergence at the origin of the F5 term justifies its exclusion from the pressure equation.

Figure 6

Self-similarity in mass-radius curves—I. Numerical integrations show that 2PN terms belonging to the same family result in self-similar deviations from GR in the mass-radius relation. This figure illustrates this remarkable property for pressure terms (top row) and mass terms (bottom row) belonging to families F2, F3, and F4 (from left to right). In each panel, the solid line corresponds to GR, the long-dashed line corresponds to a positive-sign correction due to the chosen term in each family, and the short-dashed line corresponds to a negative-sign correction due to the chosen term in each family. The various symbols show that nearly identical corrections can be produced using different terms belonging to the same family, as long as we appropriately rescale their post-TOV coefficients.

Figure 7

Self-similarity in mass-radius curves—II. Same as in Fig. 6, but for the F5 family, which only admits post-TOV corrections with ${\mu }_5<0$ (see text).

Figure 8

Fractional deviations induced by the post-TOV parameters on the stellar mass and radius. Here we illustrate the fractional changes caused by the post-TOV parameters in neutron star masses and radii. For a fixed central energy density and EOS APR, we plot the relative deviations from GR in mass and radius that result from varying the post-TOV parameters within the range indicated in the legends. Top row: ${\epsilon }_{\mathrm{c}}/c^2=8.61\times {10}^{14}\mathrm{g}/{\mathrm{cm}}^3$, $M_{\mathrm{GR}}=1.51M_{\odot }$, and $R_{\mathrm{GR}}=12.3\mathrm{km}$. Bottom row: ${\epsilon }_{\mathrm{c}}/c^2=1.20\times {10}^{15}\mathrm{g}/{\mathrm{cm}}^3$, $M_{\mathrm{GR}}=2.04M_{\odot }$, and $R_{\mathrm{GR}}=11.9\mathrm{km}$. Left panels: Effect of the post-TOV terms that enter in the pressure equation. Right panels: Effect of the post-TOV terms that enter in the mass equation. The circles represent contours of fixed relative deviation from GR.

Figure 9

Directions of the post-TOV induced deviations. This schematic diagram shows which sign of individual post-TOV parameters produces smaller or larger masses/radii with respect to GR, cf. Fig. 8.

Figure 10

Deviations induced by the post-TOV parameters on the stellar compactness. Here we consider the influence of the post-TOV parameters on the compactness $\mathcal{C}=M/R$ of neutron stars. Deviations from GR are calculated assuming the same APR EOS models as in the top and bottom rows of Fig. 8. Left panels: Effect of the post-TOV terms appearing in the pressure equation. Right panels: Effect of the post-TOV terms appearing in the mass equation.

Figure 11

Integral PN potentials and 1PN terms. The radial profiles of the integral potentials $U$, $E/m_{\mathrm{N}}$, and $\mathrm{\Omega }/m_{\mathrm{N}}$ are well fitted by linear functions of the nonintegral potentials $p/\rho$ and $r^{3}p/m_{\mathrm{N}}$. In all plots, the radial coordinate is normalized to the stellar radius $R$.