11.3 Relativistic stars in metric f (R) gravity

The results presented above are valid for BD theory including metric f (R) gravity with the coupling √ -- Q = − 1∕ 6. While the analysis was carried out in the Einstein frame, thin-shell solutions were numerically found in the Jordan frame of metric f (R) gravity for the models (4.83View Equation) and (4.84View Equation[43Jump To The Next Citation Point600Jump To The Next Citation Point42Jump To The Next Citation Point]. In these models the field ∘ ---- ϕ = 3∕2 lnF in the region of high density (R ≫ Rc) is very close to the curvature singularity at ϕ = 0. Originally it was claimed in [266Jump To The Next Citation Point349Jump To The Next Citation Point] that relativistic stars are absent because of the presence of this accessible singularity. However, as we have discussed in Section 11.2, the crucial point for obtaining thin-shell solutions is not the existence of the curvature singularity but the choice of appropriate boundary conditions around the center of the star. For the correct choice of boundary conditions the field does not reach the singularity and thin-shell field profiles can be instead realized. In the Starobinsky’s model (4.84View Equation), static configurations of a constant density star have been found for the gravitational potential Φc smaller than 0.345 [600Jump To The Next Citation Point].
View Image

Figure 11: The profile of the field ∘ ---- ϕ = 3∕2 ln F (in units of Mpl) versus the radius &tidle;r (denoted as r in the figure, in units of −1∕2 Mpl ρcenter) for the model (4.84View Equation) with n = 1, R∞ ∕Rc = 3.6, and v0 = 10− 4 (shown as a solid line). The dashed line corresponds to the value ϕmin for the minimum of the effective potential. (Inset) The enlarged figure in the region 0 < &tidle;r < 2.5. From [43Jump To The Next Citation Point].

For the star with an equation of state &tidle;ρM < 3P&tidle;M, the effective potential of the field ϕ (in the presence of a matter coupling) does not have an extremum, see Eq. (11.7View Equation). In those cases the analytic results in Section 11.2 are no longer valid. For the equation of state ρ&tidle;M < 3 &tidle;PM there is a tachyonic instability that tends to prevent the existence of a static star configuration [42Jump To The Next Citation Point]. For realistic neutron stars, however, the equation of state proposed in the literature satisfies the condition &tidle; &tidle;ρM > 3PM throughout the star.

Babichev and Langlois [43Jump To The Next Citation Point42Jump To The Next Citation Point] chose a polytropic equation of state for the energy density ρM and the pressure PM in the Jordan frame:

( 2) 2 ρ (n ) = m n + K n-- , P (n ) = Km n--, (11.39 ) M B n0 M Bn0
where −27 mB = 1.66 × 10 kg, n0 = 0.1 fm−1, and K = 0.1. Solving the continuity equation ∇ μT μν = 0 coupled with Einstein equations, [43Jump To The Next Citation Point42Jump To The Next Citation Point] showed that 3P&tidle;M can remain smaller than &tidle;ρM for realistic neutron stars. Note that the energy density is a decreasing function with respect to the distance from the center of star. Even for such a varying energy density, static star configurations have been shown to exist [43Jump To The Next Citation Point42Jump To The Next Citation Point].

The ratio between the central density ρcenter and the cosmological density at infinity is parameterized by the quantity v0 = M 2plRc∕ρcenter. Realistic values of v0 are extremely small and it is a challenging to perform precise numerical simulations in such cases. We also note that the field mass mA in the relativistic star is very much larger than its cosmological mass and hence a very high accuracy is required for solving the field equation numerically [600Jump To The Next Citation Point581]. The authors in [43Jump To The Next Citation Point600Jump To The Next Citation Point42Jump To The Next Citation Point] carried out numerical simulations for the values of v0 of the order of − 3 −4 10 –10. Figure 11View Image illustrates an example of the thin-shell field profile for the polytropic equation of state (11.39View Equation) in the model (4.84View Equation) with n = 1 and v0 = 10− 4 [43Jump To The Next Citation Point]. In the regime 0 < &tidle;r < 1.5 the field is nearly frozen around the extremum of the effective potential, but it starts to evolve toward its asymptotic value ϕ = ϕ B for &tidle;r > 1.5.

Although the above analysis is based on the f (R) models (4.83View Equation) and (4.84View Equation) having a curvature singularity at ϕ = 0, such a singularity can be cured by adding the R2 term [350]. The presence of the R2 term has an advantage of realizing inflation in the early universe. However, the f (R) models (4.83View Equation) and (4.84View Equation) plus the 2 R term cannot relate the epoch of two accelerations smoothly [37Jump To The Next Citation Point]. An example of viable models that can allow a smooth transition without a curvature singularity is [37Jump To The Next Citation Point]

[ ] cosh(R∕𝜖 − b) R2 Rc f(R ) = (1 − c)R + c𝜖ln -------------- + ----2, 𝜖 ≡ --------------, (11.40 ) cosh b 6M b + ln(2 coshb)
where b, c (0 < c < 1∕2), Rc, and M are constants. In [42Jump To The Next Citation Point] a static field profile was numerically obtained even for the model (11.40View Equation).

Although we have focused on the stellar configuration with Φc ≲ 0.3, there are also works of finding static or rotating black hole solutions in f (R) gravity [193Jump To The Next Citation Point497]. Cruz-Dombriz et al. [193] derived static and spherically symmetric solutions by imposing that the curvature is constant. They also used a perturbative approach around the Einstein–Hilbert action and found that only solutions of the Schwarzschild–Anti de Sitter type are present up to second order in perturbations. The existence of general black hole solutions in f (R) gravity certainly deserves for further detailed study. It will be also of interest to study the transition from neutron stars to a strong-scalar-field state in f (R) gravity [464]. While such an analysis was carried out for a massless field in scalar-tensor theory, we need to take into account the field mass in the region of high density for realistic models of f (R) gravity.

Pun et al. [498] studied physical properties of matter forming an accretion disk in the spherically symmetric metric in f (R) models and found that specific signatures of modified gravity can appear in the electromagnetic spectrum. In [92] the virial theorem for galaxy clustering in metric f (R) gravity was derived by using the collisionless Boltzmann equation. In [398] the construction of traversable wormhole geometries was discussed in metric f (R) gravity. It was found that the choice of specific shape functions and several equations of state gives rise to some exact solutions for f (R).


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