Exhibiting stable model of dark energy compact star with Tolman-VI solution under complexity free system

Abstract Several recent developments have highlighted the significance of the vanishing complexity factor formalism in understanding the structure and evolution of stellar relativistic compact objects. This formalism, introduced through a novel definition proposed by Herrera (Phys. Rev. D 97:044010, 2018), offers valuable insights into the dynamics of such systems. In this manuscript, we explored a class of realistic solutions to the static and spherically symmetric field equations characterized by two fluid distributions: ordinary stellar matter and dark energy, within the framework of this formalism. Utilizing the well-known Tolman- VI solution as the seed ansatz for the metric coefficient $$g_{rr},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mi>rr</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> we employed the complexity-free format to derive an analytic solution for the other metric coefficient, $$g_{tt}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mi>tt</mml:mi> </mml:mrow> </mml:msub> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Subsequently, we obtained the solutions of gravitational field equations for our proposed spacetime model by incorporating the linear dark energy equation of state. These results were applied to the astrophysical compact star candidate LMC X -4, with $$M =1.04 M_\odot $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1.04</mml:mn> <mml:msub> <mml:mi>M</mml:mi> <mml:mo>⊙</mml:mo> </mml:msub> </mml:mrow> </mml:math> and $$R =8.4~\text {km}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>=</mml:mo> <mml:mn>8.4</mml:mn> <mml:mspace/> <mml:mtext>km</mml:mtext> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> The potential viability and credibility of the proposed dark star solutions were thoroughly analyzed by examining key constraints, including the regularity of metric functions, physical adequacy through matter variables, state parameter behavior, energy conditions, stability tests (such as pressure anisotropy and hydrostatic equilibrium), the speed of sound, and the mass–radius relation for this compact star candidate. Notably, the estimated values of the dark energy coupling factor, presented in Table 1, highlight the exotic nature of the fluid distribution and effectively quantify the contribution of dark energy to the structure and evolution of an ultra-relativistic dark compact star. These findings strongly support our model solutions and demonstrate improvements over previously reported results in Rej et al. (Chin J Phys 87:608, 2024).