⚛️ general relativity

Neutron stars in $f(Q) = Q +ξQ^2$ gravity

This paper investigates the structure of neutron stars within the $f(Q) = Q + \xi Q^2$ gravity framework by deriving modified Tolman-Oppenheimer-Volkoff equations, numerically solving them with realistic equations of state to determine mass-radius relations and maximum masses, and analyzing the behavior of the nonmetricity scalar to compare results with observational constraints.

Original authors: J. C. N. de Araujo, H. G. M. Fortes

Published 2026-07-07

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CC BY 4.0

Original authors: J. C. N. de Araujo, H. G. M. Fortes

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, flexible trampoline. For nearly a century, our best description of how this trampoline works has been Albert Einstein's theory of General Relativity. In Einstein's view, massive objects like stars create "dents" in the trampoline (curvature), and other objects roll toward those dents because of gravity.

However, scientists have noticed some things in the universe—like why the cosmos is speeding up its expansion—that Einstein's trampoline doesn't quite explain. This has led researchers to ask: "What if the trampoline works slightly differently?"

This paper explores one specific alternative idea called $f(Q)$ gravity.

The Three Ways to Describe Gravity

The authors explain that there are actually three different "languages" we can use to describe gravity, all of which agree with Einstein's theory under normal circumstances:

Curvature (Einstein's way): Gravity is a bend in space.
Torsion (Twist): Gravity is a twist in space.
Non-metricity (The focus of this paper): Gravity is a change in the "ruler" we use to measure distances.

Think of Non-metricity like a magical ruler that changes its length depending on where you are. In the standard Einstein model, your ruler stays the same size everywhere. In this new model, the ruler itself stretches or shrinks near massive objects. This paper investigates what happens when we use a ruler that changes size, specifically using a formula that adds a "squared" term to the stretching effect.

The Experiment: Neutron Stars as Cosmic Laboratories

To test if this new "stretching ruler" theory makes sense, the authors looked at neutron stars. These are the dead, super-dense cores of exploded stars. They are so heavy that a teaspoon of their material would weigh a billion tons on Earth. They are the ultimate stress test for gravity theories because the gravity inside them is incredibly strong.

The researchers used four different "recipes" (called Equations of State) to describe how the matter inside these stars behaves. Think of these recipes as different types of dough:

Soft dough (FPS, SLy): Easier to squish.
Stiff dough (ENG, MPA1): Harder to squish, holds its shape better.

They then ran computer simulations to see how big and heavy these stars could get under the new gravity rules compared to Einstein's rules.

The Main Findings: The "Knob" of Gravity

The new theory has a special "knob" called $\xi$ (xi). By turning this knob, the scientists could change how the "stretching ruler" behaves.

Turning the knob Negative ( $\xi < 0$ ): This makes gravity act slightly weaker inside the star. Because the inward pull is a bit less intense, the star can support more weight before collapsing. The result? The stars can become heavier and larger than Einstein's theory predicts.
Turning the knob Positive ( $\xi > 0$ ): This makes gravity act stronger inside. The star gets squeezed tighter, meaning it can support less weight before collapsing. The result? The stars become lighter and smaller.

The Sweet Spot: The authors found that if the knob is turned to a negative value, these neutron stars could potentially be much more massive than we thought possible. This is exciting because astronomers have recently spotted some incredibly heavy neutron stars (like the mysterious object in the GW190814 event) that are hard to explain with Einstein's original theory. This new model offers a way to explain how they could exist.

What Happens Inside the Star?

The paper also looked at the "stretching ruler" (the non-metricity scalar) inside the star:

At the very center: The ruler is normal (no stretching).
In the middle layers: The ruler stretches the most. This is where the new gravity effects are strongest.
Outside the star: Once you leave the star, the ruler returns to normal. The "stretching" disappears, and the gravity outside the star looks exactly like Einstein's gravity.

This is a crucial detail: the new theory doesn't break the universe; it only changes the rules inside the most extreme objects, and then seamlessly hands the baton back to Einstein's rules once you step outside.

Summary

In simple terms, this paper suggests that if gravity involves a "stretching ruler" that changes length based on a specific mathematical tweak (a negative value for the knob), then neutron stars can be much heavier than we previously thought. This provides a potential explanation for the existence of some of the heaviest, most mysterious objects in the universe, without breaking the laws of physics we use for everything else.

Technical Summary: Neutron Stars in $f(Q) = Q + \xi Q^2$ Gravity

Problem Statement
While General Relativity (GR) has been successful across various scales, high-precision cosmological data and the existence of extremely massive neutron stars (e.g., PSR J1614-2230, PSR J0740+6620) motivate the investigation of modified gravity theories. Specifically, there is a need to determine if alternative geometric frameworks can accommodate stellar configurations with masses or compactness exceeding the limits predicted by GR (such as the Buchdahl limit). This work addresses the structure of neutron stars within the framework of $f(Q)$ gravity, a theory based on the symmetric teleparallel geometry where gravity is encoded in the nonmetricity scalar $Q$ rather than curvature or torsion. Previous studies in this framework utilized polytropic equations of state (EOS); this paper extends that analysis to realistic microphysical models to provide a more astrophysically consistent description of dense matter.

Methodology
The authors employ the covariant formulation of $f(Q)$ gravity to derive the modified Tolman–Oppenheimer–Volkoff (TOV) equations. This approach is chosen to avoid the limitations of the coincident gauge, which can reduce the theory to linear dependence on $Q$ (effectively reproducing GR). The specific model investigated is $f(Q) = Q + \xi Q^2$ , where $\xi$ is a real constant parameter.

The study utilizes four realistic equations of state (EOS) representing increasing stiffness: FPS, SLy, ENG, and MPA1. The numerical procedure involves:

System of Equations: Solving a coupled system of differential equations for the metric functions $A(r)$ and $B(r)$ , and the pressure $P(r)$ , derived from the field equations and the conservation of energy-momentum.
Boundary Conditions: Imposing regularity at the center ( $r=0$ ) with $A'(0)=0$ , $B(0)=0$ , and $P(0)=P_c$ . The stellar radius $R$ is defined where $P(R)=0$ .
Mass Definitions: Calculating the gravitational mass $M_S$ (derived from the exterior Schwarzschild metric matching) and the total rest mass $M_0$ . The condition $M_S < M_0$ is used as a criterion for physical consistency.
Parameter Variation: Numerical integration is performed for $\xi$ values of $-2, -1, 0, 1, 2$ (where $\xi=0$ corresponds to GR). The parameter $\xi$ is given in units of the square of the Sun's gravitational radius ( $r_{gs}^2 \approx 2.95 \text{ km}^2$ ).
Analysis: The authors generate mass-radius ( $M_S$ vs. $R$ ) and mass-central density ( $M_S$ vs. $\rho_c$ ) sequences and analyze the radial profiles of the nonmetricity scalar $Q(r)$ and metric functions.

Key Contributions and Results

Modified TOV Equations: The paper derives the specific set of differential equations required to model neutron stars in the $f(Q) = Q + \xi Q^2$ framework using realistic EOSs, extending previous work based on polytropic models.
Impact of $\xi$ on Maximum Mass: The results demonstrate a clear dependence of stellar structure on the sign and magnitude of $\xi$ $ξ$ :
- Negative $\xi$ : Shifts mass-radius curves toward higher masses, allowing for maximum masses significantly larger than those predicted by GR. This effect is interpreted as a reduction in the effective strength of gravity inside the star.
- Positive $\xi$ : Reduces the maximum mass supported by the star, increasing the effective gravitational strength.
- Stiffness Dependence: The deviations from GR become more pronounced as the EOS stiffness increases (from FPS to MPA1). The quadratic correction in $Q$ has a more significant impact on stiffer matter.
Nonmetricity Scalar Behavior: The nonmetricity scalar $Q(r)$ $Q (r)$ is found to be non-positive ( $Q \leq 0$ $Q \leq 0$ ) throughout the stellar interior and vanishes exactly at the surface ( $r \geq R$ $r \geq R$ ).
- $Q(r)$ exhibits a "potential well" profile: it is zero at the center, reaches a minimum (maximum magnitude) in an intermediate region, and increases monotonically to zero at the surface.
- Unlike the polytropic case where the minimum occurs near the center, realistic EOSs shift the minimum of $Q(r)$ to larger radii, indicating that the region of maximal deviation from GR depends sensitively on the internal matter distribution.
Vacuum Solutions: The study confirms that vacuum solutions in this $f(Q)$ model reduce to the Schwarzschild metric of GR, ensuring that deviations are confined to the high-density interior of the star.
Physical Consistency: The analysis identifies regions where the condition $M_S < M_0$ is violated (specifically for certain positive $\xi$ values and high central densities), suggesting these configurations may be unstable or unphysical.

Significance and Claims
The paper claims that the quadratic correction in $f(Q)$ gravity provides a viable framework for describing neutron stars that allows for significant deviations from General Relativity in the strong-field regime. The authors assert that:

Negative values of the parameter $\xi$ effectively "stiffen" the equation of state, enabling the support of more massive neutron stars, which could offer a theoretical explanation for observed massive pulsars or high-mass compact objects in gravitational wave events (like GW190814) that challenge standard GR limits.
Compact stars serve as excellent laboratories for testing modified gravity, as the effects of the quadratic term become relevant only at high densities and naturally vanish in vacuum.
The use of realistic EOSs confirms that the qualitative trends observed in simplified polytropic models persist under more astrophysically consistent conditions, reinforcing the potential of $f(Q)$ gravity to address open problems in compact object physics.

The authors remain modest regarding specific experimental proposals, noting that while the model accommodates massive configurations, the physical consistency of these solutions (specifically the $M_S < M_0$ condition) must be carefully evaluated for each parameter set.