Probing geometrically perturbed strange stars with minimal decoupling using millisecond pulsar timing observations

Abstract: We construct a gravitationally decoupled anisotropic strange star model using the minimal geometric deformation approach with a MIT bag equation of state and an additional source sector controlled by a deformation parameter $β$ and a radial perturbation scale $Ψ$ through $g(r)=\sin(Ψr<sup>{2})$. The resulting Einstein system is consistently split into seed and $θ$-sectors and matched to an exterior Schwarzschild geometry. The model is constrained by high-mass pulsars: PSR J0740+6620 $(2.08\pm0.07\,M_\odot)$, PSR J1810+1744 $(2.13\pm0.04\,M_\odot)$, PSR J1959+2048 $(2.18\pm0.09\,M_\odot)$, and PSR J2215+5135 $(2.28<sup>{+0.10}<em>{-0.09}\,M</em>\odot)$. It reproduces these objects with predicted radii $R \approx 11.3$--$12.9$ km. The maximum mass reaches $M_{\max} \approx 2.28\,M_\odot$ for $β= 3\times 10<sup>{-3}$ and $Ψ\approx 0.03\,\text{km}<sup>{-2}$, while for $β= 10<sup>{-3}$ the configuration yields $M_{\max} \approx 2.12\,M_\odot$ with $R \approx 12.2$ km. The central density lies in $ρ_c \approx (2.4$--$3.1)\times 10<sup>{-4}\,\text{km}<sup>{-2}$, decreasing smoothly to $$ρ_s \approx 2.0\times 10<sup>{-4}\,\text{km}<sup>{-2}$$. The anisotropy increases from zero at the center to $Δ\approx (0.25$--$0.45)\times 10<sup>{-4}\,\text{km}<sup>{-2}$ near the surface, generating additional outward support that enhances compactness by $\sim 15\%$. The compactness parameter spans $C \approx 0.17$--$0.22$, safely below the Buchdahl limit, while the surface redshift reaches $z_s \approx 0.25$--$0.38$. The condition $dM/dρ_c &gt; 0$ is satisfied throughout, confirming dynamical stability. Overall, $β$ enhances the maximum mass by up to $\sim 15\%$, while $Ψ$ introduces controlled oscillatory structure without violating observational constraints, producing stable ultra-compact stars consistent with current pulsar data.

• The paper demonstrates that minimal geometric deformation via harmonic perturbations successfully models strange stars consistent with millisecond pulsar observations.
• The methodology splits the Einstein field equations into a seed GR fluid and an auxiliary sector, applying the MIT bag model EOS and a sinusoidal perturbation ansatz.
• The work predicts up to a 15% mass increase and accurate radii ranges while ensuring stability, causality, and compatibility with high-mass pulsar data.

Geometrically Perturbed Strange Stars with Minimal Decoupling: Analysis via Millisecond Pulsar Timing

Introduction and Motivation

This work addresses the structure and stability of strange stars (SSs) by deploying a perturbatively extended framework in general relativity: the Minimal Geometric Deformation (MGD) approach. The central aim is to reconcile analytically tractable models of self-bound quark compact objects with the stringent empirical constraints provided by massive millisecond pulsar timing (notably, the radii of high-mass pulsars such as PSR J0740+6620, PSR J1810+1744, PSR J1959+2048, and PSR J2215+5135). Recognizing the difficulty of direct integration within nonlinear Einstein field equations for realistic equations of state (EOS), the paper adopts a gravitational decoupling scheme—splitting the system into a "seed" GR fluid and an auxiliary source sector, parametrized by a geometric deformation $\beta$ and a radial perturbation frequency $\Psi$.

Theoretical Framework and Methodology

The stellar matter is modeled by the MIT bag model EOS under the massless, non-interacting quark assumption:

$P_r = \frac{1}{3}(\rho - 4\mathcal{B}_g),$

where $\mathcal{B}_g$ is the bag constant.

A non-singular, monotonically decreasing energy density profile is imposed:

$$\rho(r) = \rho_0 \left[ 1 - \left(1 - \frac{\rho_s}{\rho_0}\right)\frac{r^2}{r_s^2}\right],$$

with $\rho_0$ and $\rho_s$ as the central and surface densities.

MGD is implemented by deforming only the radial metric component through an additive function $\beta g(r)$. Specifically, the perturbation ansatz:

$g(r) = \sin(\Psi r^2)$

models small-amplitude, spatially oscillatory deformations representing minimal external disturbances (e.g., weak accretion events or transient gravitational waves). The choice of this harmonic form, regular at the center, is justified on grounds of analytic tractability and physical relevance for quadrupolar even-parity perturbations.

Separation of the Einstein field equations is performed into a seed sector (solved exactly for the MIT bag EOS profile) and a secondary sector sourced by the extra energy-momentum tensor $\theta_{ij}$. The star–exterior matching (at $\Psi$0) uses Israel–Darmois conditions.

Matter Structure: Energy Density and Pressure Profiles

The resulting model ensures physically reasonable stellar structure: non-singular, positive-definite central energy density and pressures, monotonically decreasing toward the surface, and vanishing radial pressure at the boundary (see below).

Figure 1: Energy density $\Psi$1 as a function of radius for various $\Psi$2 (left) and $\Psi$3 (right); showing monotonic decrease from the center outward consistent across parameter choices.

Figure 2: Radial and transverse pressures for varying $\Psi$4 (left) and $\Psi$5 (right); both components decrease radially, upholding hydrostatic equilibrium and the MIT bag EOS's predictions.

Anisotropy and the Role of Geometric Deformation

Decoupling introduces anisotropic stresses—quantified by $\Psi$6—with clear consequences for stability and maximum mass.

Figure 3: Pressure anisotropy profiles showing that increased $\Psi$7 or $\Psi$8 enhance the outward-directed stress, supporting higher compactness and mass.

The anisotropy vanishes at the center and grows smoothly outward, reaching $\Psi$9 near the surface for parameters matching observed massive pulsars.

Mass-Radius Relations and Astrophysical Constraints

The $P_r = \frac{1}{3}(\rho - 4\mathcal{B}_g),$0-$P_r = \frac{1}{3}(\rho - 4\mathcal{B}_g),$1 relation, computed for a range of deformation/perturbation parameters, robustly accommodates all considered high-mass millisecond pulsars. Notably, $P_r = \frac{1}{3}(\rho - 4\mathcal{B}_g),$2 provides a $P_r = \frac{1}{3}(\rho - 4\mathcal{B}_g),$3 increase of maximum mass over the unperturbed case, directly counteracting constraints from the so-called "mass gap" between the most massive neutron stars and low-mass black holes.

Figure 4: Mass–radius trajectories for increasing $P_r = \frac{1}{3}(\rho - 4\mathcal{B}_g),$4 (left) and $P_r = \frac{1}{3}(\rho - 4\mathcal{B}_g),$5 (right); higher $P_r = \frac{1}{3}(\rho - 4\mathcal{B}_g),$6 broadens the allowed mass range up to $P_r = \frac{1}{3}(\rho - 4\mathcal{B}_g),$7 while $P_r = \frac{1}{3}(\rho - 4\mathcal{B}_g),$8 controls fine oscillatory structure and stability.

Figure 5: Explicit dependence of $P_r = \frac{1}{3}(\rho - 4\mathcal{B}_g),$9 and $\mathcal{B}_g$0 on $\mathcal{B}_g$1 and $\mathcal{B}_g$2, confirming that deformation increases both mass and radius up to observational thresholds before damping sets in.

Radii for the highest-mass stars are predicted in the $\mathcal{B}_g$3–$\mathcal{B}_g$4 km range, agreeing with NICER and X-ray modeling within systematic uncertainties.

Stability Criteria and Dynamical Response

Adiabatic Index and Hydrostatic Stability

The adiabatic index $\mathcal{B}_g$5,

$\mathcal{B}_g$6

emerges as a central indicator of dynamical (radial) stability. Values remain in the $\mathcal{B}_g$7–$\mathcal{B}_g$8 interval; near the core, $\mathcal{B}_g$9 approaches the relativistic limit ($$\rho(r) = \rho_0 \left[ 1 - \left(1 - \frac{\rho_s}{\rho_0}\right)\frac{r^2}{r_s^2}\right],$$0), but remains globally above threshold, ensuring stability against radial oscillations.

Figure 6: Radial profiles of the adiabatic index for different $$\rho(r) = \rho_0 \left[ 1 - \left(1 - \frac{\rho_s}{\rho_0}\right)\frac{r^2}{r_s^2}\right],$$1 and $$\rho(r) = \rho_0 \left[ 1 - \left(1 - \frac{\rho_s}{\rho_0}\right)\frac{r^2}{r_s^2}\right],$$2. Perturbations soften the core slightly but do not induce instability within tested ranges.

Causal Sound Speeds

Radial and tangential sound speeds,

$$\rho(r) = \rho_0 \left[ 1 - \left(1 - \frac{\rho_s}{\rho_0}\right)\frac{r^2}{r_s^2}\right],$$3

remain subluminal ($$\rho(r) = \rho_0 \left[ 1 - \left(1 - \frac{\rho_s}{\rho_0}\right)\frac{r^2}{r_s^2}\right],$$4, $$\rho(r) = \rho_0 \left[ 1 - \left(1 - \frac{\rho_s}{\rho_0}\right)\frac{r^2}{r_s^2}\right],$$5), even at maximal deformation (before violation of causality and breakdown of the perturbative regime).

Figure 7: Sound speeds as functions of radius; $$\rho(r) = \rho_0 \left[ 1 - \left(1 - \frac{\rho_s}{\rho_0}\right)\frac{r^2}{r_s^2}\right],$$6 and $$\rho(r) = \rho_0 \left[ 1 - \left(1 - \frac{\rho_s}{\rho_0}\right)\frac{r^2}{r_s^2}\right],$$7 are always below the unit line, and thus respect causality for physically reasonable $$\rho(r) = \rho_0 \left[ 1 - \left(1 - \frac{\rho_s}{\rho_0}\right)\frac{r^2}{r_s^2}\right],$$8.

Harrison–Zel'dovich–Novikov Stability

The mass–central density relation $$\rho(r) = \rho_0 \left[ 1 - \left(1 - \frac{\rho_s}{\rho_0}\right)\frac{r^2}{r_s^2}\right],$$9 is maintained across the parameter space (mass increases monotonically with central density), confirming dynamical stability for all relevant configurations.

Figure 8: $\rho_0$0 versus $\rho_0$1 for varying $\rho_0$2 and $\rho_0$3; derivative positive throughout, no sign of dynamical instability.

Implications and Prospects

This model provides a parameter-dependent tuning of the maximum mass, compactness, and tidal response of strange stars under minimal geometric deformation—yielding analytic predictions compatible with current and future high-precision $\rho_0$4-$\rho_0$5 pulsar measurements. The formalism successfully reproduces observed masses and radii of the heaviest known neutron stars without exotic stiffening of the EOS or ad hoc assumptions about quark matter interactions.

The presence of spatially oscillatory perturbations, as controlled by $\rho_0$6, opens avenues for incorporating tidal deformability constraints from gravitational-wave events and provides a tractable method for analytic studies of tidal Love numbers in the ultra-relativistic regime.

Conclusion

The integration of minimal geometric deformation into compact star modeling, coupled with an explicit harmonic perturbation ansatz, yields a robust analytic platform for explaining the observed properties of massive neutron stars under the hypothesis of self-bound quark matter. The model remains physically viable—respecting causality, stability, and empirical $\rho_0$7-$\rho_0$8 constraints—and offers clear predictions for how small geometric or topological perturbations can influence compact object structure and stability. This approach is poised for extension to dynamical scenarios, nonradial modes, and multimessenger tests as observational data continue to advance.