Maximum mass limit of strange stars in quadratic curvature-matter coupled gravity

This paper investigates the maximum mass limit of strange stars within quadratic curvature-matter coupled gravity using the MIT bag model, demonstrating that the theory allows for masses up to 3.11 solar masses and suggesting the lighter companion of the GW190814 event could plausibly be a strange star.

The Gist

Imagine the universe is filled with cosmic "heavyweights." Among the most extreme of these are Neutron Stars and Strange Stars. These are the dead cores of massive stars, crushed so tightly that a teaspoon of their material would weigh as much as a mountain. For decades, physicists have been trying to figure out exactly how heavy these stars can get before they collapse into black holes.

This paper is like a new set of instructions for building a stronger, heavier version of these cosmic giants. Here is the breakdown in simple terms:

1. The Problem: The "Weight Limit" Puzzle

In our current understanding of physics (General Relativity, or Einstein's theory), there is a strict speed limit and a strict weight limit for these stars. If a star gets too heavy (about 2 to 2.5 times the mass of our Sun), it should collapse into a black hole.

However, astronomers recently spotted a mysterious object in a gravitational wave event called GW190814. This object was about 2.6 times the mass of the Sun. It was too heavy to be a normal neutron star, but too light to be a typical black hole. It was stuck in a "mass gap." Scientists were scratching their heads: How does a star this heavy exist without collapsing?

2. The Solution: A New "Gravity Glue"

The authors of this paper propose a new theory of gravity. Think of Einstein's gravity as a trampoline. If you put a bowling ball on it, the fabric curves. That's how gravity works.

But in this new theory, the authors suggest the trampoline fabric has a special, stretchy coating (called "quadratic curvature") and is also glued to the bowling ball itself (called "non-minimal matter coupling").

• The Quadratic Curvature ($\alpha R^2$): Imagine the trampoline isn't just flexible; it has a hidden springiness that kicks in when the weight gets really heavy. This extra "spring" helps support more weight without the fabric tearing (collapsing).
• The Matter Coupling ($\beta T$): Imagine the bowling ball and the trampoline are sticky. The more the ball presses down, the more the trampoline pushes back, not just because of the curve, but because they are chemically bonded. This extra push helps hold the star up.

3. The "Strange Star" Recipe

To test this, the scientists used a model for Strange Stars.

• Normal Neutron Stars are like a giant ball of neutrons (like a giant atomic nucleus).
• Strange Stars are like a giant ball of "quark soup." Inside, the neutrons have broken apart into their smaller ingredients (quarks). The authors used a recipe called the MIT Bag Model, which is like a virtual plastic bag holding this soup. The bag has a specific pressure that keeps the soup from exploding.

4. The Experiment: Cranking the Dials

The scientists took their new gravity theory and the "quark soup" recipe and cranked the knobs on two settings:

• Knob A ($\alpha$): How strong is the "springy" trampoline coating?
• Knob B ($\beta$): How sticky is the glue between the star and the trampoline?

They ran thousands of simulations to see how heavy the star could get before it collapsed.

5. The Big Discovery

The results were exciting:

• Old Physics (Einstein): The limit was around 2.0 solar masses.
• New Physics (This Paper): By adjusting the knobs, they found they could support a strange star up to 3.11 solar masses!

This is a massive jump. It means that the mysterious 2.6-solar-mass object from GW190814 could actually be a Strange Star held up by this new type of gravity, rather than a black hole.

6. Does it Match Reality?

The authors checked their numbers against real-world data from telescopes and gravitational wave detectors. They found that their new theory predicts star sizes (radii) that match what we actually see in the sky.

• The Analogy: It's like a tailor making a new suit. If the suit fits the customer perfectly (matches the observations), the tailor's pattern is likely good. Their "pattern" (theory) fits the "customer" (the universe) very well.

Summary

This paper suggests that the universe might be a bit more flexible than we thought. By adding a little bit of "extra spring" and "sticky glue" to the laws of gravity, we can explain how some of the heaviest, strangest stars in the universe manage to stay standing without turning into black holes. It offers a fresh, plausible explanation for some of the most puzzling cosmic objects we've ever found.

Technical Summary

1. Problem Statement

The internal composition and maximum mass limits of compact stars, particularly Neutron Stars (NS) and Strange Stars (SS), remain a significant puzzle in astrophysics. Recent observational data, such as the massive pulsar PSR J0740+6620 ($\approx 2.14 M_\odot$) and the secondary component of the gravitational wave event GW190814 ($2.59^{+0.08}_{-0.09} M_\odot$), challenge the standard General Relativity (GR) framework.

• The Conflict: In standard GR, using conventional Equations of State (EoS) like the MIT bag model, the maximum mass of a strange star is typically capped around $2.0 M_\odot$. Explaining objects with masses $>2.5 M_\odot$ (like the GW190814 companion) often requires invoking extremely stiff EoSs or exotic matter distributions that may be physically unrealistic.
• The Gap: There is a need for a theoretical framework that can accommodate these high-mass compact objects without violating physical stability conditions, potentially by modifying the theory of gravity itself rather than just the matter properties.

2. Methodology

The authors investigate the structure of strange stars within the framework of quadratic curvature-matter coupled gravity, specifically using the modified action:
$$f(\tilde{R}, T) = R + \alpha R^2 + 2\beta T$$
Where:

• $R$ is the Ricci scalar.
• $T$ is the trace of the energy-momentum tensor.
• $\alpha$ is the parameter for quadratic curvature correction (higher-order gravity).
• $\beta$ is the coupling parameter for non-minimal matter-geometry interaction.

Key Steps in the Methodology:

1. Field Equations Derivation: The authors derived the modified gravitational field equations from the action using the metric formalism. They explicitly showed that the non-minimal coupling ($2\beta T$) leads to a non-conservation of the energy-momentum tensor ($\nabla_\mu T^{\mu\nu} \neq 0$), introducing an extra force term dependent on the gradient of $T$.
2. TOV Equations: For a static, spherically symmetric spacetime with an isotropic perfect fluid, they derived the modified Tolman-Oppenheimer-Volkoff (TOV) equations. These equations govern the hydrostatic equilibrium, incorporating terms from both the $R^2$ correction and the $\beta T$ coupling.
3. Equation of State (EoS): The study employs the MIT bag model EoS for strange quark matter:
   $$p = \frac{1}{3}(\rho - 4B_g)$$
   where $p$ is pressure, $\rho$ is energy density, and $B_g$ is the bag constant. Two values were tested:
   • Lower bound: $B_g = 57.55 \, \text{MeV/fm}^3$ (ensuring stability against nuclear matter).
   • Upper bound: $B_g = 95.11 \, \text{MeV/fm}^3$ (ensuring stability against iron nuclei).
4. Numerical Solution: The modified TOV equations were solved numerically with boundary conditions $m(0)=0$ and central density $\rho_c$. The authors systematically varied parameters $\alpha$ and $\beta$ to map the Mass-Radius ($M-R$) relationships.
5. Stability Analysis: The models were validated using:
   • Adiabatic Index ($\Gamma$): Ensuring $\Gamma > 4/3$ throughout the star.
   • Harrison-Zel'dovich-Novikov Criterion: Checking that $dM/d\rho_c > 0$ for stable configurations.

3. Key Contributions

• Novel Framework: The paper presents a unified framework combining quadratic curvature corrections ($R^2$) and non-minimal matter coupling ($T$) specifically applied to strange stars. This extends previous works that treated these effects separately.
• Recovery of GR: The formalism explicitly demonstrates that in the limit $\alpha \to 0$ and $\beta \to 0$, the standard GR results are recovered, validating the model's consistency.
• Parameter Space Exploration: The study provides a comprehensive mapping of how $\alpha$ (curvature strength) and $\beta$ (matter coupling strength) influence stellar structure, identifying viable ranges where physical solutions exist.
• GW190814 Interpretation: The work offers a concrete theoretical explanation for the "mass gap" object in GW190814, suggesting it could be a strange star supported by modified gravity effects rather than a black hole or an impossible neutron star.

4. Key Results

The numerical analysis yielded several critical findings regarding the maximum mass ($M_{max}$) and radius ($R$) of strange stars:

• Exceeding GR Limits: The maximum mass limit of strange stars in this framework can significantly exceed the GR counterpart ($\approx 2.0 M_\odot$).
  • For $B_g = 57.55 \, \text{MeV/fm}^3$, $M_{max}$ reaches $2.42 M_\odot$ (with $\beta=0, \alpha=52$).
  • For $B_g = 95.11 \, \text{MeV/fm}^3$, $M_{max}$ reaches $3.11 M_\odot$ (with $\beta=0, \alpha=78.2$).
• Impact of Parameters:
  • Negative $\beta$ ($\beta < 0$): Corresponds to a weakly coupled system. Increasing $\alpha$ in this regime stiffens the EoS, leading to higher maximum masses and larger radii.
  • Positive $\beta$ ($\beta > 0$): Corresponds to strong gravity-matter coupling. This tends to soften the EoS, reducing the maximum mass and radius as $\alpha$ increases.
  • Curvature ($\alpha$): Generally, higher-order curvature corrections ($R^2$) allow for more massive configurations by providing additional support against gravitational collapse, provided $\beta$ is not too large and positive.
• Observational Consistency: The predicted mass-radius relationships for various combinations of $\alpha$ and $\beta$ align well with recent observational data from:
  • X-ray binaries (e.g., HER X-1, EXO 1745-248).
  • Gravitational wave events (GW170817 and GW190814).
  • Specifically, the model predicts a radius of $\approx 10.41$ km for the GW190814 secondary component ($2.59 M_\odot$), fitting the "strange star" hypothesis.
• Stability: All derived configurations satisfy the adiabatic index condition ($\Gamma > 1.33$) and the stability criterion ($dM/d\rho_c > 0$), confirming that these high-mass solutions are physically stable.

5. Significance

This research is significant for several reasons:

1. Resolving the Mass Gap: It provides a viable alternative to the black hole interpretation of the GW190814 secondary component. By showing that modified gravity can support strange stars up to $3.11 M_\odot$, it bridges the gap between canonical neutron stars and low-mass black holes.
2. Testing Gravity in Strong Fields: The study demonstrates that compact stars serve as unique laboratories for testing deviations from General Relativity. The results suggest that quadratic curvature terms and matter-geometry couplings are non-negligible in supranuclear density environments.
3. Multi-Messenger Astronomy: The paper outlines specific observational signatures (tidal deformability, post-merger gravitational wave frequencies, cooling curves, and moment of inertia) that can be used to constrain the parameters $\alpha$ and $\beta$ using data from LIGO/Virgo/KAGRA, NICER, and XMM-Newton.
4. Theoretical Robustness: By recovering GR in the appropriate limits and satisfying stability conditions, the model offers a robust, minimal extension to Einstein's gravity that is consistent with both cosmological constraints (via the Starobinsky $R^2$ term) and local astrophysical observations.

In conclusion, the paper establishes that quadratic curvature-matter coupled gravity is a promising framework for explaining the existence of ultra-massive compact objects, suggesting that the lighter companion of GW190814 could plausibly be a strange star stabilized by these modified gravity effects.