Exploring the stability of ultra-compact anisotropic dark energy stars with maximum allowable mass in $$R+\chi (R^{2}+\eta R^{3})$$ gravity

Abstract In recent years, the modeling of compact astrophysical objects ( COs ) has garnered significant attention from various research groups, particularly in efforts to determine their stable structures. This interest has been further amplified by the incorporation of dark energy as an additional source within the relativistic interior geometries of these stellar objects. In this paper, we present new structural properties of anisotropic, static, and spherically symmetric compact stars, characterized by a two-fluid distribution comprising ordinary baryonic matter and dark energy, within the framework of the gravity theory f ( R ). We derive a novel class of exact analytical solutions to the modified field equations by employing the well-established Tolman–Buchdahl solutions as seed ansatz for the $$g_{tt}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mi>tt</mml:mi> </mml:mrow> </mml:msub> </mml:math> and $$g_{rr}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mi>rr</mml:mi> </mml:mrow> </mml:msub> </mml:math> metric potentials, in conjunction with a linear dark energy equation of state. The unknown parameters involved in these seed solutions, along with the dark energy coupling factor $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> , are determined by a smooth matching of the interior and exterior regions in the hypersurface of the boundary. To analyze the physical viability of our model, we apply it to the compact star PSR $$J1614-2230$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>J</mml:mi> <mml:mn>1614</mml:mn> <mml:mo>-</mml:mo> <mml:mn>2230</mml:mn> </mml:mrow> </mml:math> , using the widely studied and cosmologically consistent $$f(R)=R+\chi \Big (R^{2}+\eta R^{3}\Big )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> <mml:mi>χ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>η</mml:mi> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> gravity model. The obtained space-time geometry is assessed on the basis of several physical constraints, including the regularity of metric components, the viability of matter variables, the validity of state parameters and energy conditions, and various stability factors. Additionally, we examine the mass-radius profile, compactness, and surface redshift to ensure the model’s physical acceptability. Notably, our analysis reveals that the maximum allowable mass and compactness of the proposed dark energy star model exceed observational data, providing strong implications within this modified gravity framework. This suggests that such a model has the potential to surpass conventional observational predictions. In conclusion, our results confirm that the proposed solutions are physically viable and realistic, effectively mimicking a stable ultra-compact dark-energy star. These findings offer new insights into the relativistic stellar framework, highlighting the intricate interplay between extended gravity theories and a two-fluid distribution.