[Paper Review] Study of Anisotropic Compact Stars in Starobinsky Model

M. Zubair, G. Abbas|arXiv (Cornell University)|Dec 3, 2014

Cosmology and Gravitation Theories3 citations

TL;DR

This paper investigates anisotropic compact stars in the Starobinsky $f(R) = R + \lambda R^2$ model of modified gravity, using the Krori-Barua metric to model interior spacetime and matching it to the Schwarzschild exterior solution. The study confirms that these stars satisfy energy conditions, maintain regular matter profiles, exhibit stable anisotropic pressure, and support masses consistent with observed X-ray binaries such as 4U1820-30, Her X-1, and SAXJ1808-3658, with surface redshifts up to $Z_s = 0.845$, indicating enhanced stability and viability in $f(R)$ gravity.

ABSTRACT

The aim of this paper is to study the formation of anisotropic compact stars in modified $f(R)$ theory of gravity, which is the generalization of the Einstein's gravity. To this end, we have used the solution of Krori and Barua to the anisotropic distribution of matter in $f(R)$ gravity. Further, we have matched the interior solution with the exterior solution to determine the constants of Krori and Barua solution. Finally the constant have been determined by using the data of compact compact stars like 4$U1820-30, Her X-1, SAX J 1808-3658$. Using the evaluated form of the solutions, we have discussed the regularity of matter components at the center as well as on the boundary, energy conditions, anisotropy, stability analysis and mass-radius relation of the compact stars 4$U1820-30$, $Her X-1$, $SAX J 1808-3658.$

Motivation & Objective

• To investigate the existence and stability of anisotropic compact stars in the $f(R) = R + \lambda R^2$ modified gravity model.
• To determine whether $f(R)$ gravity can support realistic compact star configurations with physical matter profiles and energy conditions.
• To constrain the model parameters using observational data from known compact stars like 4U1820-30, Her X-1, and SAXJ1808-3658.
• To analyze the role of anisotropy and curvature corrections in enabling more massive stellar configurations.

Proposed method

• Adopting the Krori and Barua metric ansatz for spherically symmetric, static spacetime in $f(R)$ gravity.
• Deriving the field equations for anisotropic matter sources in the $f(R) = R + \lambda R^2$ model, including curvature-induced stress-energy contributions.
• Matching the interior Krori-Barua solution to the exterior Schwarzschild metric at the star's boundary to fix integration constants.
• Using observed masses and radii of compact stars (4U1820-30, Her X-1, SAXJ1808-3658) to numerically determine the model parameters $A$, $B$, $C$, and $\lambda$.
• Evaluating energy conditions, anisotropy factor $\Delta = p_t - p_r$, sound speed profiles, and surface redshift to assess physical viability.
• Performing stability analysis via the relativistic adiabatic index and checking for causality ($v^2_{sr}, v^2_{st} < 1$) and regularity at center and boundary.

Experimental results

Research questions

• RQ1 Can the $f(R) = R + \lambda R^2$ model support stable, anisotropic compact star solutions with regular matter profiles and finite central densities?
• RQ2 What are the constraints on the model parameter $\lambda$ and metric constants derived from observational data of known compact stars?
• RQ3 How does the inclusion of $R^2$ curvature correction affect the anisotropy, energy conditions, and maximum surface redshift compared to general relativity?
• RQ4 Does the modified gravity framework allow for higher mass configurations due to repulsive anisotropic forces?
• RQ5 Are the sound speeds within the star causal and stable under the $f(R)$ gravity model?

Key findings

• The energy conditions (null, weak, dominant) are satisfied throughout the stellar interior for all considered compact stars, indicating physically viable matter distributions.
• The energy density and radial/transverse pressures remain finite and positive everywhere, with maximum values at the center, confirming singularity-free solutions.
• The anisotropy factor $\Delta = p_t - p_r > 0$ for all stars, indicating a repulsive force that supports more massive configurations.
• The radial and transverse sound speeds satisfy $0 < v^2_{sr}, v^2_{st} < 1$ and $v^2_{sr} > v^2_{st}$, with $|v^2_{st} - v^2_{sr}| \leq 1$, confirming dynamical stability.
• The surface redshift reaches a maximum of $Z_s = 0.845$ for Her X-1 (radius 7 km), significantly lower than the GR upper bound of $Z_s \leq 2$, indicating reduced gravitational redshift in $f(R)$ gravity.
• The equation of state parameters satisfy $0 < \omega_r(r), \omega_t(r) < 1$, confirming the presence of ordinary matter with a non-negligible $f(R)$ gravity correction.