Preliminary study on the impact of stress-energy tensor compared to scalar field in Nonminimal Derivative model

Abstract: In this article, we report the results of comparing the effect of using trace of stress-energy tensor versus real-valued scalar field in Nonminimal Derivative Coupling gravitation model, respectively denoted as NMDC-T and NMDC-phi. We employ the model into an incompressible star and see the effect of both models NMDC-T and NMDC-phi on the compactness and mass-radius relation. We find that coupling parameters of NMDC-T is less sensitive than NMDC-phi.

• The paper demonstrates that NMDC models based on the stress-energy tensor avoid the complex scalar field issues seen in NMDC-phi.
• It employs numerical shooting and recursion methods with an incompressible equation of state to modify Tolman–Oppenheimer–Volkoff equations.
• Results indicate that negative coupling in NMDC-T can support higher neutron star masses, marking it as a viable alternative to NMDC-phi.

Impact of Stress-Energy Tensor Versus Scalar Field in Nonminimal Derivative Coupling Models for Incompressible Stars

Introduction

This paper, "Preliminary study on the impact of stress-energy tensor compared to scalar field in Nonminimal Derivative model" (2604.04617), rigorously investigates Nonminimal Derivative Coupling (NMDC) gravitational models where the nonminimal (derivative) coupling is expressed either through a real-valued scalar field (NMDC-phi) or via the trace of the stress-energy tensor (NMDC-T). Both models are embedded in the broader context of Horndeski and Fab Four theories, with the aim of addressing their influence on compact stars, specifically using the simplified case of an incompressible stellar equation of state (EoS).

Theoretical Formulation

NMDC-phi Model

The NMDC-phi framework is based on an action containing a coupling between the scalar field derivatives and the Einstein tensor, as delineated by

$$S = \int d^4x \sqrt{-g} [ \kappa(R - 2\Lambda) - (\zeta g_{ab} - \eta G_{ab}) \nabla^a \phi \nabla^b \phi ] + S_m.$$

Here, only the nonminimal coupling parameter $\eta$ is considered relevant for the current investigation (the minimal term $\zeta$ is set to zero). The field equations are derived following Horndeski-type procedures, incorporating static spherically symmetric metric ansatz and a time-plus-radial split for the scalar field, $\phi = Qt + F(r)$. Importantly, the scalar field current conservation equation, $J^a = 0$, provides a critical constraint, impacting the structural equations for the star. The resulting set of equations, including a modified Tolman-Oppenheimer-Volkoff (TOV) equation, is closed and amenable to direct numerical integration without small-coupling expansions.

The authors highlight a pathology previously observed for $\eta < 0$, where the requirement $F'(r)^2 > 0$ may be violated within a physically plausible compact star, leading to the introduction of complex scalar field solutions—a manifest breakdown of the model’s self-consistency.

NMDC-T Model

The NMDC-T scenario replaces the scalar field with the trace $T = g^{ab} T_{ab}$ of the matter stress-energy tensor, motivated by recent proposals in cosmology. The action is formulated as

$$S = \int d^4x \sqrt{-g} [ \kappa(R-2\Lambda) + \alpha G_{ab}T^{ab} + \beta G_{ab} \nabla^a T \nabla^b T ] + S_m,$$

with a focus on the nonminimal derivative term ($\beta$), setting $\eta$0 for this analysis. In the plausible case of perfect fluids ($\eta$1), $\eta$2 remains real, bypassing the complexification pathology noted in NMDC-phi.

However, the inclusion of $\eta$3 terms leads to higher-derivative terms ($\eta$4, $\eta$5) in the modified equations. To guarantee consistency with the GR TOV equation in the decoupling limit, the derivation uses a recursion expansion, truncating at first order in $\eta$6. This approximation is non-exact and thus cannot capture strong-coupling effects, in contrast to NMDC-phi, where such an expansion is unnecessary.

Numerical Approach

Both models are implemented for stars with incompressible EoS ($\eta$7). Dimensionless variables are enforced by suitable rescalings, matching the physical dimensions of the coupling parameters ($\eta$8). For NMDC-phi, a shooting method is required to satisfy the junction condition $\eta$9 at the star's surface, iteratively adjusting both the central metric function $\zeta$0 and the “frequency” parameter $\zeta$1. In the case of NMDC-T, the absence of time dependence obviates the need to iterate over $\zeta$2.

Results

Trends in Mass-Radius and Compactness Relations

The quantitative impact on solutions is probed via variations in $\zeta$3 (NMDC-phi) and $\zeta$4 (NMDC-T):

• Positive Coupling: For both models, taking positive values of the respective coupling parameters decreases the compactness ($\zeta$5) and the stellar mass for a fixed central density.
• Negative Coupling: For NMDC-T, negative $\zeta$6 increases mass—potentially allowing explanations of high observed neutron star masses—without encountering pathologies. In contrast, for NMDC-phi, negative $\zeta$7 generically yields regions of complex scalar field values, making such solutions physically inadmissible.

A critical numerical result is that NMDC-T requires coupling parameters roughly two orders of magnitude larger than NMDC-phi to produce comparable deviations in mass or compactness, exposing either a suppression due to the linearized treatment or an inherent reduced sensitivity in the NMDC-T coupling.

Both models, under the incompressible EoS scenario, yield noticeable but bounded modifications to the mass-radius (MR) relation. The shift in the MR relation is more pronounced for NMDC-phi, while the central pressure versus compactness behavior is comparable across models.

Pathologies and Physical Admissibility

The most salient claim is the exclusion of NMDC-phi (with $\zeta$8) for realistic neutron star scenarios: the emergence of complex-valued fields for physically reasonable parameters precludes its utility. Conversely, NMDC-T can accommodate large neutron star masses for negative $\zeta$9 without introducing unphysical solutions, at least within the scope of linearized corrections.

Implications and Outlook

Theoretical Relevance

The distinction in sensitivity and pathology between NMDC-phi and NMDC-T underlines the importance of the coupling structure in modified gravity models for astrophysical compact objects. The inability to obtain closed nonlinear equations for NMDC-T at the nonlinear level, as opposed to the exact closure in NMDC-phi, limits strong-field phenomenological applications of the former unless higher-order expansions or resummation strategies are developed.

Furthermore, the absence of pathologies in NMDC-T for negative $\phi = Qt + F(r)$0 strengthens its standing as a candidate for explaining the high-mass tail ($\phi = Qt + F(r)$1) of the neutron star population, as required by observations such as GW170817 (2604.04617).

Numerical and Physical Generality

Due to the simplifying assumption of an incompressible EoS, correction terms involving $\phi = Qt + F(r)$2 and $\phi = Qt + F(r)$3 are absent. More physically realistic, microphysics-informed EoS will introduce additional structure into the NMDC-T correction terms, possibly enhancing or mitigating deviations from GR and altering the order-of-magnitude sensitivity to $\phi = Qt + F(r)$4.

Future Directions

Addressing the nonlinearity and strong-coupling behavior of NMDC-T will require going beyond the first-order expansion in $\phi = Qt + F(r)$5 and dealing rigorously with higher derivatives in the field equations. Additionally, exploring more physical EoS and incorporating detailed microphysics is essential for robust phenomenological constraints.

Conclusion

This comparative analysis demonstrates that nonminimal derivative coupling built from the trace of the stress-energy tensor (NMDC-T) circumvents the internal inconsistencies afflicting the scalar NMDC-phi model for physically motivated parameter regimes, and is capable of supporting higher mass neutron stars for $\phi = Qt + F(r)$6, provided the linear expansion remains a good approximation. The study strongly suggests that NMDC-T is more viable for compact star modeling from both a physical and phenomenological standpoint, with further work needed to probe its nonlinearity and confront it with realistic microphysical inputs.