Hybrid metric-Palatini stars

We consider the internal structure and the physical properties of specific classes of neutron, quark and Bose-Einstein condensate stars in the recently proposed hybrid metric-Palatini gravity theory, which is a combination of the metric and Palatini $f(R)$ formalisms. It turns out that the theory is very successful in accounting for the observed phenomenology, since it unifies local constraints at the Solar System level and the late-time cosmic acceleration, even if the scalar field is very light. In this paper, we derive the equilibrium equations for a spherically symmetric configuration (mass continuity and Tolman-Oppenheimer-Volkoff) in the framework of the scalar-tensor representation of the hybrid metric-Palatini theory, and we investigate their solutions numerically for different equations of state of neutron and quark matter, by adopting for the scalar field potential a Higgs-type form. It turns out that the scalar-tensor definition of the potential can be represented as an Clairaut differential equation, and provides an explicit form for $f(\mathcal{R})$ given by $f(\mathcal{R})\sim \mathcal{R}+{\mathrm{\Lambda }}_{\mathrm{eff}}$, where ${\mathrm{\Lambda }}_{\mathrm{eff}}$ is an effective cosmological constant. Furthermore, stellar models, described by the stiff fluid, radiation-like, bag model and the Bose-Einstein condensate equations of state are explicitly constructed in both general relativity and hybrid metric-Palatini gravity, thus allowing an in-depth comparison between the predictions of these two gravitational theories. As a general result it turns out that for all the considered equations of state, hybrid gravity stars are more massive than their general relativistic counterparts. Furthermore, two classes of stellar models corresponding to two particular choices of the functional form of the scalar field (constant value, and logarithmic form, respectively) are also investigated. Interestingly enough, in the case of a constant scalar field the equation of state of the matter takes the form of the bag model equation of state describing quark matter. As a possible astrophysical application of the obtained results, we suggest that stellar mass black holes, with masses in the range of 3.8 and $6M_{\odot }$, respectively, could be in fact hybrid metric-Palatini gravity neutron or quark stars.

Figure 1

Mass-radius relation for stiff fluid stars in hybrid metric-Palatini gravity theory, for $\mu ={10}^{-5}{\mathrm{cm}}^{-1}$, $\xi =8.5\times {10}^{-10}{\mathrm{cm}}^{-2}$, ${\mathrm{\Phi }}^{\prime }(0)=-1.8\times {10}^{-16}$ to $-5.7\times {10}^{-16}{\mathrm{cm}}^{-1}$ for the central densities considered, and for different values of $\mathrm{\Phi }(0)$: $\mathrm{\Phi }\equiv 0$ (standard general relativistic limit) (solid curve), $\mathrm{\Phi }(0)=0.14$ (dotted curve), $\mathrm{\Phi }(0)=0.22$ (short dashed curve), $\mathrm{\Phi }(0)=0.28$ (dashed curve), and $\mathrm{\Phi }(0)=0.30$ (long dashed curve).

Figure 2

Variation of the dimensionless scalar field $\mathrm{\Phi }$ (left) and of the Higgs type potential $u$ (right) for a stiff fluid star in the hybrid metric-Palatini gravity theory for $\mathrm{\Phi }(0)=0.73$, and for different values of the potential parameters ${\mu }_0$ and ${\xi }_0$ and of the central values of the derivatives of the scalar field: ${\mu }_0=1.05$, ${\xi }_0=1.15$, ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-1.32\times {10}^{-4}$ (solid curve), ${\mu }_0=2.05$, ${\xi }_0=1.25$, ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-3\times {10}^{-4}$ (dotted curve), ${\mu }_0=3.05$, ${\xi }_0=1.35$, ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-5.79\times {10}^{-4}$ (short dashed curve), ${\mu }_0=4.05$, ${\xi }_0=1.45$, ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-9.68\times {10}^{-4}$ (dashed curve), and ${\mu }_0=5.05$, ${\xi }_0=1.55$, ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-1.47\times {10}^{-3}$ (long dashed curve). The initial conditions for the central density and dimensionless mass used to numerically integrate the hybrid metric-Palatini gravity structure equations are $\theta (0)=1$ and $M_{\mathrm{eff}}(0)=0$, respectively.

Figure 3

Mass-radius relation for radiation fluid stars in hybrid metric-Palatini gravity theory, for $\mu ={10}^{-5}{\mathrm{cm}}^{-1}$, $\xi =8.5\times {10}^{-10}{\mathrm{cm}}^{-2}$ and for different values of $\mathrm{\Phi }(0)$ and ${\mathrm{\Phi }}^{\prime }(0)$: $\mathrm{\Phi }\equiv 0$ ${\mathrm{\Phi }}^{\prime }\equiv 0$ (standard general relativistic limit) (solid curve), $\mathrm{\Phi }(0)=0.10$, ${\mathrm{\Phi }}^{\prime }(0)=-2.71\times {10}^{-17}{\mathrm{cm}}^{-1}$ (dotted curve), $\mathrm{\Phi }(0)=0.15$, ${\mathrm{\Phi }}^{\prime }(0)=-1.01\times {10}^{-16}{\mathrm{cm}}^{-1}$ (short dashed curve), $\mathrm{\Phi }(0)=0.20$, ${\mathrm{\Phi }}^{\prime }(0)=-2.0\times {10}^{-16}{\mathrm{cm}}^{-1}$ (dashed curve), and $\mathrm{\Phi }(0)=0.25$, ${\mathrm{\Phi }}^{\prime }(0)=-2.8\times {10}^{-16}{\mathrm{cm}}^{-1}$ (long dashed curve).

Figure 4

Variation of the dimensionless scalar field $\mathrm{\Phi }$ (left) and of the Higgs type potential $u$ (right) for a radiation fluid star in the hybrid metric-Palatini gravity theory for different values of the potential parameters $\mu$ and $\xi$: $\mu =1.05$, $\xi =1.15$ (solid curve), $\mu =2.05$, $\xi =1.25$ (dotted curve), $\mu =3.05$, $\xi =1.35$ (short dashed curve), $\mu =4.05$, $\xi =1.45$ (dashed curve), and $\mu =5.05$, $\xi =1.55$ (long dashed curve). The initial conditions used to numerically integrate the hybrid metric-Palatini gravity structure equations are $\theta (0)=1$, $M_{\mathrm{eff}}(0)=0$, $\mathrm{\Phi }(0)=0.43$, while central values of the derivative of the scalar field, corresponding to the different values of the potential parameters ${\mu }_0$ and ${\xi }_0$ are ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-1.73\times {10}^{-5}$ (solid curve), ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-7.93\times {10}^{-5}$ (dotted curve), ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-1.81\times {10}^{-4}$ (short dashed curve), ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-3.24\times {10}^{-4}$ (dashed curve), and ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-5.07\times {10}^{-4}$ (long dashed curve).

Figure 5

Mass-radius relation for quark stars in hybrid metric-Palatini gravity theory, for $\mu ={10}^{-5}{\mathrm{cm}}^{-1}$, $\xi =8.5\times {10}^{-10}{\mathrm{cm}}^{-2}$, and for different values of $\mathrm{\Phi }(0)$ and ${\mathrm{\Phi }}^{\prime }(0)$: $\mathrm{\Phi }\equiv 0$, ${\mathrm{\Phi }}^{\prime }\equiv 0$ (standard general relativistic limit) (solid curve), $\mathrm{\Phi }(0)=0.13$, ${\mathrm{\Phi }}^{\prime }(0)=-1.4\times {10}^{-16}{\mathrm{cm}}^{-1}$ (dotted curve), $\mathrm{\Phi }(0)=0.16$, ${\mathrm{\Phi }}^{\prime }(0)=-2.0\times {10}^{-16}{\mathrm{cm}}^{-1}$ (short dashed curve), $\mathrm{\Phi }(0)=0.19$, ${\mathrm{\Phi }}^{\prime }(0)=-2.5\times {10}^{-16}{\mathrm{cm}}^{-1}$ (dashed curve), and $\mathrm{\Phi }(0)=0.22$, ${\mathrm{\Phi }}^{\prime }(0)=-2.9\times {10}^{-16}{\mathrm{cm}}^{-1}$ (long dashed curve).

Figure 6

Variation of the dimensionless scalar field $\mathrm{\Phi }$ (left) and of the Higgs type potential $u$ (right) for a quark star in the hybrid metric-Palatini gravity theory for different values of the potential parameters $\mu$ and $\xi$: $\mu =1.05$, $\xi =1.15$ (solid curve), $\mu =2.05$, $\xi =1.25$ (dotted curve), $\mu =3.05$, $\xi =1.35$ (short dashed curve), $\mu =4.05$, $\xi =1.45$ (dashed curve), and $\mu =5.05$, $\xi =1.55$ (long dashed curve). The initial conditions used to numerically integrate the hybrid metric-Palatini gravity structure equations are $\theta (0)=1$, $M_{\mathrm{eff}}(0)=0$, $\mathrm{\Phi }(0)=0.43$, while the central values of the derivatives of the scalar field, corresponding to different values of the potential parameters ${\mu }_0$ and ${\xi }_0$ are ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-1.26\times {10}^{-5}$ (solid curve), ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-7.47\times {10}^{-5}$ (dotted curve), ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-1.77\times {10}^{-4}$ (short dashed curve), ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-3.19\times {10}^{-4}$ (dashed curve), and ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-5.026\times {10}^{-4}$ (long dashed curve). For the bag constant $b$, we have adopted the value $b=0.047$.

Figure 7

Mass-radius relation for Bose-Einstein condensate stars in hybrid metric-Palatini gravity theory, for $\mu ={10}^{-5}{\mathrm{cm}}^{-1}$, $\xi =8.5\times {10}^{-10}{\mathrm{cm}}^{-2}$, ${\mathrm{\Phi }}^{\prime }(0)=-1.8\times {10}^{-16}$ to $-5.7\times {10}^{-16}{\mathrm{cm}}^{-1}$ for the range of the considered central densities, and for different values of $\mathrm{\Phi }(0)$: $\mathrm{\Phi }\equiv 0$ (standard general relativistic limit) (solid curve), $\mathrm{\Phi }(0)=0.05$ (dotted curve), $\mathrm{\Phi }(0)=0.08$ (short dashed curve), $\mathrm{\Phi }(0)=0.11$ (dashed curve), and $\mathrm{\Phi }(0)=0.14$ (long dashed curve).

Figure 8

Variation of the dimensionless scalar field $\mathrm{\Phi }$ (left) and of the Higgs type potential $u$ (right) for a Bose-Einstein condensate star in the hybrid metric-Palatini gravity theory for different values of the potential parameters $\mu$ and $\xi$: $\mu =1.05$, $\xi =1.15$ (solid curve), $\mu =2.05$, $\xi =1.25$ (dotted curve), $\mu =3.05$, $\xi =1.35$ (short dashed curve), $\mu =4.05$, $\xi =1.45$ (dashed curve), and $\mu =5.05$, $\xi =1.55$ (long dashed curve). The initial conditions used to numerically integrate the hybrid metric-Palatini gravity structure equations are $\theta (0)=1$, $M_{\mathrm{eff}}(0)=0$, $\mathrm{\Phi }(0)=0.27$, while the central values of the derivative of the potential, corresponding to different values of the potential parameters ${\mu }_0$ and ${\xi }_0$ are ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=1.02\times {10}^{-5}$ (solid curve), ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-1.46\times {10}^{-5}$ (dotted curve), ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-5.55\times {10}^{-5}$ (short dashed curve), ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-1.12\times {10}^{-4}$ (dashed curve), and ${(d\mathrm{\Phi }/d\eta )|}_{\eta =0}=-1.85\times {10}^{-4}$ (long dashed curve). For the coefficient $k$ in the polytropic equation of state, we have adopted the value $k=0.1$.

Figure 9

Mass-radius relation for hybrid metric-Palatini gravity stars for a constant scalar field, for $\mu ={10}^{-5}{\mathrm{cm}}^{-1}$, $\xi =8.5\times {10}^{-10}{\mathrm{cm}}^{-2}$, and for different values of $\mathrm{\Phi }$: $\mathrm{\Phi }\equiv 0$ (standard general relativistic quark star model) (solid curve), $\mathrm{\Phi }=0.35$ (dotted curve), $\mathrm{\Phi }=0.4$ (short dashed curve), $\mathrm{\Phi }=0.45$ (dashed curve), and $\mathrm{\Phi }=0.50$ (long dashed curve).

Figure 10

Mass-radius relation for hybrid metric-Palatini gravity stars for the scalar field $\mathrm{\Phi }(r)=\ln [1+{(Ar+B)}^4]$, for $A=100{\mathrm{cm}}^{-1}$, $U(0)={10}^{-3}{\mathrm{cm}}^{-2}$, and for different values of $\mathrm{\Phi }(0)$: $\mathrm{\Phi }(0)=2.0$ (dotted curve), $\mathrm{\Phi }(0)=2.1$ (short dashed curve), $\mathrm{\Phi }(0)=2.25$ (dashed curve), and $\mathrm{\Phi }=2.3$ (long dashed curve). The solid curve represents the standard general relativistic radiation fluid star model.

Figure 11

Variation of the scalar field potential $u$ as a function of $\eta$ (left) and of the scalar field potential $u$ as a function of $\mathrm{\Phi }$ (right) for a hybrid metric-Palatini gravity star with $\mathrm{\Phi }=\ln [1+{(\alpha \eta +\beta )}^4]$, for different values of the parameters $\alpha$ and $\beta$: $\alpha =0.03$, $\beta =0.15$ (solid curve), $\alpha =0.05$, $\beta =0.15$ (dotted curve), $\alpha =0.07$, $\beta =0.15$ (short dashed curve), $\alpha =0.09$, $\beta =0.10$ (dashed curve), and $\alpha =0.11$, $\beta =0.05$ (long dashed curve). The dot-dashed curve represents the solution of the structure equations for the radiation fluid star in standard GR. The initial conditions used to numerically integrate the hybrid metric-Palatini gravity structure equations are $\theta (0)=1$, $M_{\mathrm{eff}}(0)=0$ and $u(0)=0.15$,