Dark Energy Stars in Finch-Skea Spacetime with a Schwarzschild-(Anti)-de Sitter Exterior

Abstract: In this work, the effects of the cosmological constant on dark energy stars in Finch-Skea spacetime are systematically investigated within the framework of Einstein gravity. A static and anisotropic stellar configuration composed of ordinary matter and dark energy is constructed by employing the complexity factor formalism to obtain the temporal metric potential. The interior solution is smoothly matched to the exterior Schwarzschild-(anti-)de Sitter spacetime through appropriate boundary conditions. The influence of positive and negative cosmological constants on the structural and stability properties of the compact star candidate Vela X-1 is investigated in detail. The results show that positive values of the cosmological constant produce larger and less compact configurations, whereas negative values lead to denser and more compact stars with stronger gravitational effects. The energy conditions are satisfied throughout the stellar interior for all considered cases. However, sufficiently large values of the cosmological constant introduce deviations from hydrostatic equilibrium and affect the causal and dynamical stability behavior of the model. Overall, the results demonstrate that the cosmological constant plays an important role in determining the compactness, equilibrium and stability behavior of dark energy stars in Finch-Skea spacetime.

• The paper presents a model of dark energy stars using a two-fluid approach in Finch–Skea spacetime, highlighting the impact of a cosmological constant on stellar structure.
• It employs the complexity factor formalism to derive a corresponding temporal metric potential and ensures a smooth match with a Schwarzschild–(anti-)de Sitter exterior.
• Numerical analysis on Vela X-1 shows that positive Λ increases star radii while negative Λ enhances compactness and redshift, with large |Λ| leading to instability and acausal behavior.

Dark Energy Stars in Finch–Skea Spacetime with Schwarzschild–(Anti-)de Sitter Exterior

Introduction and Context

This work presents an in-depth study of compact objects modeled as dark energy stars within Finch–Skea spacetime, incorporating the cosmological constant as a key parameter through a Schwarzschild–(anti-)de Sitter exterior (2605.30398). The modeling framework systematically combines ordinary matter and dark energy components, utilizing the complexity factor formalism to derive the temporal metric potential. The analysis explores both the structural and stability properties of such stars, with explicit focus on the impact of both positive and negative values of the cosmological constant ($\Lambda$). The canonical neutron star candidate Vela X-1 serves as the sample astrophysical object for all numerical computations.

Model Construction: Interior Solution and Complexity Factor Approach

The analysis begins with Einstein's field equations incorporating a cosmological constant. The internal metric is assumed to be static, spherically symmetric, and anisotropic. The radial metric potential is selected as $e^{\lambda(r)} = 1 + r^2/R_*^2$. The temporal metric potential $e^{\nu(r)}$ is then constructed using the complexity factor formalism, following the prescription for zero complexity (vanishing $Y_{TF}$), which imposes physically motivated structure and reduces the arbitrariness of the model.

The two-fluid model introduces both ordinary matter and a dark energy component, with the dark energy assigned a negative equation of state, $p_r^{de} = -\rho_{de}$. The two components are coupled via a parameter $\alpha=1$, implying a balanced contribution from both. Analytical, closed-form solutions are derived for all effective thermodynamic quantities, including $\rho_{eff}(r)$, $(p_r)_{eff}(r)$, $(p_t)_{eff}(r)$, with explicit dependence on $\Lambda$ and the model parameters.

Boundary Conditions and Matching to Exterior Spacetimes

The interior solution is smoothly matched at the surface to an exterior Schwarzschild–(anti-)de Sitter geometry. This ensures the continuity of $e^{\lambda(r)} = 1 + r^2/R_*^2$0, $e^{\lambda(r)} = 1 + r^2/R_*^2$1, and their derivatives on the boundary. The matching conditions yield explicit formulae for the unknown constants $e^{\lambda(r)} = 1 + r^2/R_*^2$2 and $e^{\lambda(r)} = 1 + r^2/R_*^2$3 in terms of $e^{\lambda(r)} = 1 + r^2/R_*^2$4, $e^{\lambda(r)} = 1 + r^2/R_*^2$5, and $e^{\lambda(r)} = 1 + r^2/R_*^2$6, and also provide an algebraic relation dictating the stellar radius for any given mass and cosmological constant.

Variations in $e^{\lambda(r)} = 1 + r^2/R_*^2$7 directly control the equilibrium configuration's size and structural properties: positive $e^{\lambda(r)} = 1 + r^2/R_*^2$8 yields larger radii, while negative $e^{\lambda(r)} = 1 + r^2/R_*^2$9 compresses the star.

Spacetime Structure and Matter Profile Analysis

The radial profiles of the spacetime metric potentials are demonstrated to be monotonic, regular, and positive throughout the interior.

Figure 1: Radial variation of the metric potentials $e^{\nu(r)}$0 and $e^{\nu(r)}$1 for different cosmological constants $e^{\nu(r)}$2.

All matter variables are well-behaved, with the density peaking at the core and decreasing monotonically outwards. The center exhibits finite values for all thermodynamic quantities, and radial pressure vanishes at the stellar surface, defining the physical boundary.

Figure 2: Radial profile of the effective energy density $e^{\nu(r)}$3, showing the central concentration and outward decay for various $e^{\nu(r)}$4.

Anisotropy, quantified as $e^{\nu(r)}$5, is induced exclusively by the cosmological constant. The central anisotropy is nonzero for $e^{\nu(r)}$6, vanishing only in the pure Schwarzschild case.

Macroscopic Quantities: Mass, Compactness, Redshift

The effective mass function increases monotonically with radius, attaining its maximum at the surface and serving as a diagnostic for comparing different $e^{\nu(r)}$7 scenarios.

Figure 3: Effective mass function $e^{\nu(r)}$8 for several choices of $e^{\nu(r)}$9.

The stellar compactness and surface redshift respond distinctly to $Y_{TF}$0:

• Positive $Y_{TF}$1: Decreased compactness and redshift, associated with larger, less dense stars.
• Negative $Y_{TF}$2: Increased compactness and redshift, resulting in denser, more compact configurations.

These findings are robust across parameter variations and are consistent with prior treatments of the cosmological constant's gravitational effects in compact objects.

Energy Conditions and Physical Admissibility

All relevant energy conditions—Weak, Null, Dominant, and Strong—are rigorously verified across the interior for each tested value of $Y_{TF}$3. This ensures that the model maintains physical plausibility in all explored scenarios.

Figure 4: Verification of the weak energy condition ($Y_{TF}$4) as a function of radius for all $Y_{TF}$5 considered.

Hydrostatic Equilibrium, Causality, and Stability Diagnostics

Hydrostatic equilibrium is analyzed via the generalized Tolman–Oppenheimer–Volkoff (TOV) equation, including gravity, pressure, and anisotropic terms. For small magnitudes of $Y_{TF}$6, these forces remain balanced, but substantial positive or negative $Y_{TF}$7 introduces clear deviations, particularly near the core.

Figure 5: Radial distribution of the net total force including gravity, pressure, and anisotropy, showing deviations from equilibrium at large $Y_{TF}$8.

Causality is checked by computing the squared sound speeds in radial and tangential directions. Explicit violations occur for large positive $Y_{TF}$9 in internal regions, while negative $p_r^{de} = -\rho_{de}$0 scenarios demonstrate improved causal behavior, with violations confined more centrally.

Stability is further probed via Herrera's cracking criterion, which is always satisfied in these models, and by examining the adiabatic index. For large negative $p_r^{de} = -\rho_{de}$1, the adiabatic indices fall below the Chandrasekhar limit near the stellar core, indicating a trend toward dynamical instability.

Implications, Contradictions, and Future Directions

Key numerical findings include:

• For Vela X-1 ($p_r^{de} = -\rho_{de}$2), $p_r^{de} = -\rho_{de}$3 increases from 9.82 km ($p_r^{de} = -\rho_{de}$4) to 11.45 km ($p_r^{de} = -\rho_{de}$5).
• Compactness and redshift are maximized for negative $p_r^{de} = -\rho_{de}$6 and minimized for positive $p_r^{de} = -\rho_{de}$7.
• All energy conditions remain satisfied for the entire parameter space explored.

A particularly notable claim, supported by detailed structural analysis, is that "the cosmological constant introduces a central anisotropy of purely geometric origin, not present in standard models without $p_r^{de} = -\rho_{de}$8." Another substantive result is the demonstration that larger values of $p_r^{de} = -\rho_{de}$9—especially positive—can lead to unbalanced forces and acausal propagation speeds, thus rendering such configurations physically unviable, despite mathematical solvability.

The study supports the assertion that the cosmological constant modulates equilibrium, compactness, and surface properties of compact stars, and can play a non-negligible 'local' astrophysical role beyond its usual cosmological setting.

Practical implications include new constraints on the allowable values of the effective local cosmological constant within compact objects, with potentially observable consequences for mass–radius relations and redshift signatures. Theoretically, the results suggest that extensions of the dark energy star paradigm must carefully consider both causality and dynamical stability diagnostics when introducing strong vacuum terms.

Future developments may involve:

• Extending beyond static, spherically symmetric metrics to include rotation or strong magnetic fields;
• Considering variable $\alpha=1$0 profiles or further couplings to modified gravity theories for higher astrophysical fidelity;
• Integrating observational constraints from neutron star and quark star measurements, including those from X-ray timing and gravitational wave data, to further bound $\alpha=1$1 in realistic compact objects.

Conclusion

This study establishes that the cosmological constant exerts significant, and in some regimes, destabilizing influence on the structure and stability of dark energy stars constructed within the Finch–Skea metric. While all models retain regularity and physical admissibility for small $\alpha=1$2, large values—especially positive—can trigger acausal and non-equilibrium behavior, tightly constraining the parameter space for viable stars in this framework. The research enhances understanding of the interplay between dark energy components, interior geometry, and global spacetime structure in relativistic astrophysics, and opens pathways for testing such effects in compact star observations and in extended gravitational theories.