Neutron stars with primary scalar hair

Abstract: We investigate static and spherically symmetric neutron star solutions endowed with primary scalar hair in a subfamily of Degenerate-Higher-Order-Scalar-Tensor (DHOST) theories of gravity. By solving the modified Tolman-Oppenheimer-Volkoff (TOV) equations, we construct equilibrium configurations for polytropic and realistic equations of state and analyse the impact of the scalar hair on the stellar structure. We examine the resulting metric and scalar field profiles as well as the mass-radius relation, showing deviations from the predictions of General Relativity (GR). Positive scalar charges lead to more compact stars than in GR and, above a critical threshold, to singularities. Observations could therefore put stringent constraints on the parameters characterising the beyond-GR effects in these theories and their potential scalar hair.

• The paper extends primary scalar hair concepts from black holes to neutron stars in DHOST gravity via modified TOV equations.
• It quantifies the interplay of the scalar charge, length scale, and EOS, revealing non-monotonic pressure and density profiles.
• Parameter space analysis shows critical bounds where excess scalar hair induces singular or negative ADM mass configurations.

Neutron Stars with Primary Scalar Hair in DHOST Gravity

DHOST Framework and Primary Scalar Hair

This paper presents a detailed study of neutron stars with primary scalar hair in a sub-class of Degenerate Higher-Order Scalar-Tensor (DHOST) gravity, where the Lagrangian exhibits shift and parity symmetry. These models propagate a single scalar degree of freedom via carefully engineered degeneracy relations that eliminate Ostrogradsky ghosts, encompassing Beyond Horndeski as a special case. Crucially, static spherically symmetric (SSS) solutions can feature nontrivial primary scalar hair—an independent integration constant associated with a time-dependent scalar profile in the presence of a global shift symmetry.

This work builds directly on previous black hole solutions exhibiting primary scalar hair in this DHOST sector, extending their construction to compact stars. The focus is on a DHOST model with an explicit kinetic term dependent on a power of the gradient squared $$X = -\frac{1}{2} \partial_\mu \phi \partial^\mu \phi$$, where—for analytic and phenomenological clarity—the primary example considered is the $p=2$ case.

Modified Stellar Structure Equations

The authors derive the modified Tolman-Oppenheimer-Volkoff (TOV) system governing static equilibrium configurations. The effect of the scalar hair (quantified by a star-dependent parameter $\xi_2$) and a theory-defined length scale $\lambda$ is encoded as a radially varying effective energy-momentum tensor resembling a perfect fluid with $P_{\rm eff} = -\rho_{\rm eff}$. For each EOS and central density, the stellar structure is thus determined by $\xi_2$, $\lambda$, and the chosen EOS.

A nontrivial result is that, unlike in GR or many Horndeski theories, the scalar profile inside the star couples non-universally to matter, leading to strong backreaction effects even when matter couplings are purely metric. The scalar field and metric components can exhibit unconventional radial behavior—pressure and density may rise outward in the core, and the $A$ metric component may display non-monotonicity.

Structure and Parameter Space Constraints

Parameter Bounds

Systematic numerical integration reveals strong upper limits in the ($\xi_2, \rho_c$) parameter plane: for each $\lambda$, increasing $p=2$0 or the central density produces a singularity (divergence in the equations) beyond a critical curve. The criticality arises from the kinetic structure of the modified TOV equations—most notably from factors involving the effective sound speed and the nontrivial scalar backreaction on the pressure gradient, which can reverse sign and give rise to pathological configurations. The domain of allowed solutions is thus sharply delimited.

Figure 1: Upper bounds in the parameter space $p=2$1, defined by the maximal central density $p=2$2 allowed for each value of scalar charge and length scale.

Radial Profiles

The radial profiles of the metric functions, matter pressure, energy density, and the effective energy density are extensively studied for both polytropic and realistic (SLy, BSk21, BSk22) EOSs. Key findings include:

• For large positive $p=2$3 or small $p=2$4, pressure can peak non-centrally and decrease only near the surface—a major qualitative departure from GR.
• Metric components $p=2$5, $p=2$6 can both deviate substantially from GR—interpreted as enhanced or suppressed central compactness.
• The effective energy density $p=2$7 is generically positive in the core and can become negative in the outer layers, leading to modifications in observable mass profiles and the local versus ADM mass.

  Figure 2: Radial profiles of $p=2$8, $p=2$9, and $\xi_2$0 for varying $\xi_2$1, with $\xi_2$2 and a polytropic EoS; deviations from GR become pronounced as $\xi_2$3 increases.

  

  Figure 3: Radial profiles of $\xi_2$4, $\xi_2$5, and $\xi_2$6 under variation of $\xi_2$7, fixing $\xi_2$8 and $\xi_2$9; decreasing $\lambda$0 sharpens non-monotonic effects in pressure and density.

Nontrivial Scalar Gradient Dynamics

In the SLy EOS case, for certain parameter choices and high central density, the scalar kinetic density $\lambda$1 can flip sign in the interior (implying a change from timelike to spacelike scalar profile over finite radius), only to revert in the outer region. This is permitted at the level of the background solution, although the associated stability and physical acceptability remain uncertain. The system remains mathematically regular as $\lambda$2 crosses zero.

Figure 4: Radial profile of $\lambda$3 showing regions where the scalar density $\lambda$4 changes sign, for select SLy models.

Mass-Radius Relations and Negative-Energy Configurations

The mass-radius (M–R) relation is mapped for each EOS and parameter set, exhibiting clear systematic shifts under variation of the theory-fixed ($\lambda$5) and star-dependent ($\lambda$6) parameters:

• At fixed $\lambda$7, increasing $\lambda$8 enhances mass for given radius; for sufficiently large $\lambda$9, the ADM mass scales additively as $P_{\rm eff} = -\rho_{\rm eff}$0 in the small-$P_{\rm eff} = -\rho_{\rm eff}$1 limit.
• The $P_{\rm eff} = -\rho_{\rm eff}$2 curve terminates suddenly at the leftmost point, reflecting the onset of singular configurations at maximal central density.

Figure 5: Mass-radius relation $P_{\rm eff} = -\rho_{\rm eff}$3 for polytropic EOS; left: effect of $P_{\rm eff} = -\rho_{\rm eff}$4; right: effect of $P_{\rm eff} = -\rho_{\rm eff}$5.

Figure 6: Mass-radius relation $P_{\rm eff} = -\rho_{\rm eff}$6 for the SLy EOS; dependence on both $P_{\rm eff} = -\rho_{\rm eff}$7 and $P_{\rm eff} = -\rho_{\rm eff}$8 is evident, especially at large $P_{\rm eff} = -\rho_{\rm eff}$9.

Moreover, negative scalar charge ($\xi_2$0) yields striking and potentially problematic regimes:

• The effective energy density is negative in the core, positive near the surface.
• For large negative $\xi_2$1, the ADM mass can become negative, especially when $\xi_2$2 is large—an outcome disallowed in standard GR.
• Mass and radius both increase with central density at high compression, leading to physically questionable sequences.

  Figure 7: Mass-radius relation $\xi_2$3 for the SLy EOS with negative scalar charge, revealing negative ADM mass regimes at large $\xi_2$4 and illustrating the departure from GR behavior.

Implications and Perspectives

This research demonstrates that neutron stars in shift-symmetric Beyond Horndeski DHOST gravity can circumvent classic no-hair theorems and support configurations with primary scalar hair, parameterized independently of mass or equation of state. Observational signatures—such as systematic deviations in M–R relations, maximum mass constraints, and local versus ADM mass shifts—offer concrete targets for gravitational wave and X-ray observations. The presence of singularities at high compactness and the possibility of non-monotonic matter profiles serve as strong theoretical constraints on the allowed DHOST parameter space.

Potentially observable effects include:

• Modifications to the maximum allowable neutron star mass, impacting interpretation of LIGO/Virgo events in the mass gap regime.
• Differences between locally measured mass and ADM mass at large distances, arising from the scalar field profile and its scale $\xi_2$5.
• Non-monotonic pressure and density distributions and associated effects on stability and potential instabilities observable via oscillation modes or in the GW inspiral regime.

The possibility that $\xi_2$6 may become spacelike inside the star, together with the occurrence of negative mass configurations for negative scalar charge, raises nontrivial interpretational and stability challenges. The paper suggests that future work should prioritize a full perturbative stability analysis (including both radial and nonradial modes), as well as explore further the observational constraints from current and future multimessenger astrophysics.

Conclusion

By constructing neutron star equilibrium solutions with primary scalar hair in DHOST gravity, the paper provides a comprehensive model that generalizes and extends scalar-tensor predictions beyond GR. The existence of strong upper parameter-space bounds, pronounced modification of mass-radius relations, and the appearance of new instabilities have direct implications for both the theoretical landscape of modified gravity models and their possible astrophysical realization or exclusion. This study highlights the importance of incorporating both theoretical self-consistency and observational signatures in the program of testing gravity with compact objects (2607.01937).