⚛️ general relativity

Rotating Fermion-Boson Stars in $R$ -squared Gravity

This paper constructs and analyzes static and rotating fermion-boson star models within $R$ -squared gravity, demonstrating that the modified gravitational framework enhances maximum supported masses and expands the range of stable equilibrium solutions while remaining consistent with current astrophysical and gravitational-wave constraints.

Original authors: Saeed Fakhry, Jorge Castelo Mourelle, Nicolas Sanchis-Gual, Daniela Doneva, Stoytcho Yazadjiev, José A. Font

Published 2026-07-07

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CC BY 4.0

Original authors: Saeed Fakhry, Jorge Castelo Mourelle, Nicolas Sanchis-Gual, Daniela Doneva, Stoytcho Yazadjiev, José A. Font

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic construction site. For decades, physicists have been building models of the densest, most extreme objects in the cosmos: neutron stars. Think of these stars as the "cosmic anvils"—matter so squeezed together that a single teaspoon would weigh a billion tons on Earth.

For a long time, scientists have used a specific set of blueprints called General Relativity (Einstein's theory of gravity) to design these anvils. But, just like any good engineer, physicists suspect the blueprints might be missing a few details, especially in the most extreme conditions.

This paper is about testing a new, slightly modified set of blueprints called R-squared Gravity. Here is a simple breakdown of what the researchers did and what they found, using everyday analogies.

1. The New Ingredient: A "Ghost" Force

In standard Einstein gravity, gravity is just the bending of space and time. In this new R-squared Gravity, there is an extra ingredient: a hidden "scalar field."

The Analogy: Imagine you are trying to stack heavy bricks (matter) to build a tower. In standard gravity, the bricks just sit there, and gravity pulls them down. In this new theory, it's as if there is an invisible, elastic cushion (the scalar field) between the bricks. This cushion pushes back slightly, changing how the tower holds together.
The Goal: The researchers wanted to see if this "elastic cushion" changes the shape and strength of neutron stars.

2. The Special Star: A "Two-Flavor" Dessert

Usually, neutron stars are made of one thing: super-dense atomic matter (fermions). But the universe might be full of dark matter, which we can't see but know is there.

The researchers built models of "Fermion-Boson Stars."

The Analogy: Think of these stars as a cosmic parfait.
- The bottom layer is the "fermion" part: the normal, heavy neutron star matter (the fruit and granola).
- The top layer is the "boson" part: a cloud of dark matter (the whipped cream).
The Twist: In this study, the "whipped cream" (dark matter) doesn't just sit on top; it mixes with the fruit, and the whole thing is spinning. The researchers asked: If we add this invisible dark matter cloud and spin the star, how does the new "elastic cushion" gravity affect the mix?

3. The Experiment: Spinning the Cosmic Top

The team used powerful computers to simulate these stars. They looked at two scenarios:

Static Stars: Not spinning (like a rock sitting on a table).
Rotating Stars: Spinning very fast (like a figure skater doing a pirouette).

They compared three versions of the universe:

Version A: Standard Einstein gravity (the old blueprints).
Version B: R-squared gravity with a "medium-strength" elastic cushion.
Version C: R-squared gravity with a "very strong" elastic cushion.

4. The Findings: Bigger, Stronger, and More Flexible

Here is what happened when they turned on the "elastic cushion" (the R-squared gravity):

The Stars Got Bigger: The new gravity allowed the stars to support more weight before collapsing. It's like the elastic cushion gave the star extra structural integrity.
- The Result: The maximum mass a star could hold increased significantly compared to standard gravity.
The Shape Changed: The "whipped cream" (dark matter) didn't just sit in the middle. Depending on how fast the star spun, the dark matter spread out into a doughnut shape (torus) around the center, or formed a halo around the star.
The "Sweet Spot" for Mystery Objects: One of the biggest puzzles in astronomy is a "gap" in the mass of cosmic objects. We see small black holes and heavy neutron stars, but there is a gap in the middle where we don't know what objects exist.
- The Discovery: In standard gravity, it's very hard to build a star heavy enough to fill this gap without it collapsing into a black hole. However, with the R-squared gravity (especially the strong version), the researchers found they could build stable stars that fit perfectly into this mysterious gap.

5. Why This Matters

The paper concludes that if our universe actually follows these "R-squared" rules (with the extra elastic cushion), then:

Neutron stars can be heavier and larger than we thought.
The mysterious objects found in the "black hole low-mass gap" (detected by gravitational waves) might actually be these special Fermion-Boson stars made of normal matter mixed with dark matter, rather than small black holes.

In Summary:
The researchers took a standard model of a neutron star, added a cloud of dark matter, spun it up, and tested it against a new theory of gravity that includes an extra "push." They found that this new gravity makes these stars stronger and more flexible, offering a potential explanation for some of the strangest, heaviest objects we see in the sky.

Technical Summary: Rotating Fermion-Boson Stars in R-squared Gravity

Problem Statement
Compact objects, particularly neutron stars (NSs), serve as critical laboratories for testing gravity in the strong-field regime. While General Relativity (GR) remains the standard, extensions such as $f(R)$ gravity introduce additional scalar degrees of freedom (scalarons) that may alter stellar structure. Concurrently, the potential accumulation of dark matter within NSs—modeled as a bosonic component interacting only gravitationally with fermionic matter—offers a mechanism to explain unusually massive compact objects or those falling within the "black-hole low-mass gap" (e.g., the secondary component of GW190814). Previous studies have addressed static mixed stars in modified gravity or rotating stars in GR separately. However, a self-consistent treatment combining modified gravity (specifically $R$ -squared gravity), uniform rotation, and a mixed fermion-boson sector has been lacking. This work addresses the gap by constructing static and uniformly rotating equilibrium models of fermion-boson stars within the $f(R) = R + aR^2$ framework to investigate how the scalar degree of freedom and rotation jointly influence the mass-radius relations and internal structure.

Methodology
The authors construct equilibrium solutions using a numerical framework that extends the RNS code (based on the Komatsu-Eriguchi-Hachisu self-consistent field method) to include $f(R)$ gravity and a mixed matter sector.

Theoretical Framework:
- Gravity: The study adopts the $R$ -squared model, $f(R) = R + aR^2$ , where $a$ is a positive parameter controlling the deviation from GR. This theory is dynamically equivalent to a scalar-tensor theory with a massive scalar field. The equations are formulated in the Einstein frame via a conformal transformation, introducing a scalar field $\phi$ with a specific potential $V(\phi)$ .
- Matter Sector: The system consists of two components:
  - Fermionic Sector: Modeled as a perfect fluid with a zero-temperature equation of state (EOS) using the tabulated AkmalPR EOS.
  - Bosonic Sector: Represented by a self-interacting complex scalar field with a potential $V_s(|\Phi|^2) = \mu_b^2|\Phi|^2 + \frac{1}{2}\lambda|\Phi|^4$ .
- Approximation: The bosonic sector is treated using the RPV (Rigidly Rotating Potential/High-Coupling) approximation. In the strong self-interaction regime ( $\lambda$ large), the bosonic field is approximated as an effective perfect fluid with compact support. The exponentially decaying "tail" of the field is neglected, and the field is set to zero outside a specific radius determined by the field parameters. This allows the bosonic contribution to be treated as a source term similar to the fermionic fluid, simplifying the numerical implementation.
Numerical Implementation:
- The field equations for the metric potentials and the scalar field are solved iteratively using Green's functions.
- The bosonic field ansatz for rotating configurations includes a harmonic time and azimuthal dependence: $\Phi(t, r, \theta, \psi) = \phi(r, \theta)e^{i(\eta\psi - wt)}$ .
- The fermionic fluid is assumed to rotate rigidly, while the bosonic sector exhibits an effective differential rotation determined by field parameters, though the RPV approximation renders this profile closer to rigid rotation.
- Global quantities (mass $M$ , angular momentum $J$ ) are computed via Komar integrals in the Einstein frame, which coincide with Jordan-frame quantities at infinity due to the vanishing scalar field.

Key Contributions and Results
The paper presents equilibrium sequences for static and uniformly rotating fermion-boson stars for three gravitational models: GR ( $a=0$ ), and two $R$ -squared cases ( $a=10$ and $a=10^4$ ).

Structural Modifications:
- Scalar Field Influence: The presence of the scalar degree of freedom systematically modifies the spatial distribution of both matter sectors. Compared to GR, $R$ -squared gravity leads to a suppression of the bosonic field amplitude (particularly in the inner regions) and a redistribution of fermionic energy density toward more spatially extended profiles.
- Mass Redistribution: For a fixed total mass, the modified gravity models require lower central densities. The scalar interaction effectively reduces the localized matter concentration needed to support a given mass, leading to larger stellar radii and a "softer" effective gravitational coupling in the core.
- Rotation Effects: Rotating configurations exhibit a toroidal bosonic structure (vanishing on the rotation axis). The dimensionless internal frequency $\bar{w}$ controls whether the bosonic component forms a core-like structure or a halo. Increasing $a$ enlarges the parameter space accessible to compact, core-like mixed configurations, even at lower internal frequencies where such solutions would not exist in GR.
Mass-Radius Relations and Maximum Masses:
- Enhanced Maximum Mass: The $R$ $R$ -squared term increases the maximum supported mass for both static and Keplerian (mass-shedding) limits relative to GR.
  - For $a=10$ , the static maximum mass increases by $\sim 2.9\%$ and the Keplerian limit by $\sim 6.2\%$ .
  - For $a=10^4$ , the static maximum mass increases by $\sim 7.5\%$ and the Keplerian limit by $\sim 13.9\%$ .
- Keplerian vs. Static: The relative increase in maximum mass is consistently larger at the Keplerian limit than in the static limit.
Observational Constraints:
- The models are tested against current astrophysical constraints, including NICER measurements of PSR J0030+0451 and PSR J0740+6620, GW170817 tidal deformability, and the mass of the secondary component in GW190814 ( $\sim 2.5\text{--}2.67 M_\odot$ ).
- GW190814 Interpretation: In GR, reproducing the GW190814 secondary mass with the AkmalPR EOS requires near-Keplerian rotation, which inflates the radius beyond constraints from PSR J0740+6620. In contrast, the $R$ -squared models (particularly $a=10^4$ ) can accommodate this mass even in near-static sequences without violating radius constraints. This suggests that mixed fermion-boson stars in $R$ -squared gravity offer a viable explanation for objects in the low-mass black-hole gap without invoking extreme nuclear stiffness or rapid rotation.

Significance and Claims
The paper claims that fermion-boson stars in $R$ -squared gravity provide a promising framework for investigating the combined effects of dark bosonic matter, rotation, and strong-field gravity modifications. The primary significance lies in demonstrating that the scalar degree of freedom in $f(R)$ gravity can enlarge the domain of admissible equilibrium solutions and increase maximum supported masses relative to GR.

The authors modestly conclude that while their models remain compatible with current observational constraints, the specific ability of $R$ -squared gravity to support the GW190814 secondary mass in a non-rotating or slowly rotating state distinguishes it from standard GR models with conventional nuclear matter. They emphasize that this result is specific to the chosen EOS and approximation (RPV) and that the framework offers a testable alternative for compact objects in the black-hole low-mass gap. The work does not claim to definitively prove the existence of such stars but establishes them as a physically consistent and observationally viable candidate class within modified gravity theories.