Investigating quark star properties through baryon number density $(n)$ within the framework of $f(Q)$ gravity

This paper constructs and analyzes a viable strange star model within $f(Q)$ modified gravity using an MIT bag model equation of state with a baryon number density-dependent bag constant, demonstrating that the framework successfully predicts the radii and stability of compact stars with masses up to $2.46~M\odot$, distinguishing between strange stars below $2.01~M\odot$ and di-quark stars at higher masses.

The Gist

Imagine the universe as a giant cosmic kitchen. For decades, physicists have been trying to understand the most extreme "dishes" in this kitchen: neutron stars and strange stars. These are the leftovers of massive stars that have collapsed under their own weight, packing more mass than our Sun into a sphere the size of a city.

This paper is like a new recipe book written by a team of physicists (Sourav Biswas and colleagues) who are trying to figure out exactly how these stellar "cakes" are baked, using a new set of cooking rules.

Here is the breakdown of their work in simple, everyday terms:

1. The Old Rules vs. The New Rules (Gravity)

For a long time, we used Einstein's General Relativity as the rulebook for gravity. It's like a classic cookbook that works perfectly for baking bread (planets and stars in our solar system). But when you try to bake a "cosmic soufflé" (a super-dense star), the old rules sometimes get messy or fail to explain what's happening.

The authors decided to try a new cooking method called f(Q) gravity.

• The Analogy: Imagine Einstein's gravity is like measuring ingredients by weight. The new method, f(Q), is like measuring them by how much they stretch or shrink the measuring cup itself. It's a different way of looking at the fabric of space and time. The authors found that this new method is mathematically simpler and easier to handle for these extreme stars.

2. The Secret Ingredient: The "Bag" of Quarks

Inside these strange stars, the matter is so squeezed that atoms break apart. Instead of whole atoms, you have a soup of tiny particles called quarks (the building blocks of protons and neutrons).

Physicists use a model called the MIT Bag Model to describe this soup.

• The Analogy: Think of the quarks as marbles trapped inside a rubber balloon (the "bag"). The rubber skin of the balloon pushes in, trying to squeeze the marbles, while the marbles push back.
• The Twist: In old models, scientists thought the rubber skin was always the same thickness (a constant "Bag Constant"). But this paper argues that the rubber skin changes depending on how many marbles are inside.
• The Wood-Saxon Connection: They used a formula called the "Wood-Saxon" profile (usually used for nuclear physics) to describe how the "bag" gets tighter or looser as the density of marbles changes. It's like saying the balloon gets stiffer the more you stuff it.

3. The Main Experiment: Testing the Recipe

The team built a mathematical model of a star using this new gravity rule (f(Q)) and the new "changing bag" recipe. They asked: "If we bake a star this way, does it hold together, or does it collapse?"

They checked three main things:

1. Does it make sense physically? (Is the pressure pushing out stronger than gravity pulling in?)
2. Is it stable? (If you poke it, does it wobble and fall apart, or does it bounce back?)
3. Does it match reality? (Do the stars we see in the sky look like the ones in their model?)

4. The Results: What They Found

The results were surprisingly good!

• The "Sweet Spot": They found that for stars with masses between 1.6 and 2.0 times the Sun's mass, the model predicts they are Strange Stars (made of three types of quarks: up, down, and strange).
• The Heavyweights: For stars heavier than 2.0 solar masses (up to 2.46), the model suggests they might be Di-quark stars (made of only two types of quarks).
• The Size Check: They calculated the size (radius) of these stars. When they compared their numbers to real observations of stars like 4U 1820-30, the numbers matched almost perfectly. It's like their recipe predicted the exact size of a cake that astronomers had already measured.

5. Why This Matters

Think of this paper as a stress test for a new theory of the universe.

• The Problem: We don't fully understand what happens inside a star when it's crushed to infinite density.
• The Solution: This paper shows that if we use the "f(Q)" gravity rules and a "smart bag" that changes with density, we can create a stable, realistic model of these stars.
• The Takeaway: It suggests that our current understanding of gravity might need a tweak (like adding a new spice to the recipe) to fully explain the most extreme objects in the universe.

Summary in One Sentence

This paper uses a new, simpler version of gravity and a "smart" model of quark matter to prove that we can successfully predict the size and stability of the universe's densest stars, matching real-world observations perfectly.

Technical Summary

1. Problem Statement

The nature of the Equation of State (EoS) for matter inside compact stars (neutron stars and strange stars) remains a significant unresolved issue in astrophysics, particularly at densities far exceeding nuclear saturation.

• Limitations of General Relativity (GR): While GR successfully describes many cosmic phenomena, it faces challenges regarding dark matter, dark energy, and inconsistencies in strong gravitational fields.
• Limitations of the Standard MIT Bag Model: The standard MIT bag model assumes a constant bag constant ($B$), representing the energy difference between perturbative and non-perturbative vacuums. However, experimental data (e.g., from CERN) suggests a phase transition from hadronic matter to Quark-Gluon Plasma (QGP) that depends on density. Treating $B$ as a constant fails to accurately model this transition and the resulting stability of strange quark matter (SQM).
• Objective: The authors aim to construct a physically viable model of compact stars (specifically strange stars) by:
  1. Replacing the constant bag parameter $B$ with a baryon number density-dependent parameter $B(n)$ using a Wood-Saxon profile.
  2. Investigating these stars within the framework of $f(Q)$ gravity (symmetric teleparallel gravity), which relies on non-metricity rather than curvature or torsion.

2. Methodology

A. Theoretical Framework: $f(Q)$ Gravity

The study utilizes the symmetric teleparallel equivalent of General Relativity (STEGR), where gravity is attributed to non-metricity ($Q$) rather than curvature.

• Action: The gravitational action is defined with a linear function $f(Q) = \alpha_0 + \alpha_1 Q$, where $\alpha_0$ and $\alpha_1$ are model parameters.
• Field Equations: The authors derive the modified Einstein Field Equations (EFE) for an anisotropic perfect fluid distribution.
• Metric Ansatz: To solve the system of differential equations, the authors adopt the Finch-Skea metric potential for the radial component:
  $$e^{2\lambda} = 1 + kr^2$$
  where $k$ is a constant determined by boundary conditions.

B. Equation of State (EoS) and Bag Parameter

• MIT Bag Model EoS: The radial pressure is defined as $p_r = \frac{1}{3}(\rho - 4B)$.
• Density-Dependent Bag Constant: Instead of a constant $B$, the authors introduce a Wood-Saxon-like parameterization dependent on baryon number density ($n$):
  $$B(n) = B_{as} + (B_0 - B_{as})e^{-\beta_n (n/n_0)^2}$$
  • $B_0 = 200$ MeV/fm$^3$ (at zero density).
  • $B_{as} = 38$ MeV/fm$^3$ (at asymptotic density).
  • Parameters are tuned based on Relativistic Mean Field (RMF) theory and CERN transition data.
• Stability Criterion: The stability of the quark matter is evaluated by calculating the energy per baryon ($E_B$).
  • Stable SQM requires $E_B < 930.4$ MeV (mass of $^{56}$Fe).
  • Metastable states exist for $930.4 < E_B < 939$ MeV.
  • Unstable hadronic matter exists for $E_B > 939$ MeV.

C. Model Construction and Boundary Conditions

• Interior Solution: The field equations are solved to obtain expressions for energy density ($\rho$), radial pressure ($p_r$), tangential pressure ($p_t$), and anisotropy ($\Delta = p_t - p_r$).
• Exterior Solution: The interior solution is matched to the Schwarzschild exterior solution at the stellar surface ($r=R$) where $p_r(R)=0$. This matching determines the constant $k$.
• Physical Analysis: The model is tested against:
  • Causality conditions ($v_s^2 \leq 1$).
  • Energy conditions (NEC, WEC, SEC, DEC).
  • Stability criteria: Generalized TOV equation, Herrera's cracking condition, and the adiabatic index ($\Gamma$).

3. Key Contributions

1. Novel EoS in $f(Q)$ Gravity: The paper successfully integrates a density-dependent bag parameter $B(n)$ into the $f(Q)$ gravity framework, moving beyond the limitations of the constant bag model.
2. Classification of Compact Stars: The study provides a theoretical basis for distinguishing between Strange Stars (SS) and Di-quark stars based on mass and bag parameter ranges:
   • Strange Stars (SS): Mass range $1.59 - 2.01 M_\odot$ (Bag parameter $B \approx 57.55 - 91.55$ MeV/fm$^3$).
   • Di-quark Stars: Mass range $2.01 - 2.46 M_\odot$ (Bag parameter $B < 57.55$ MeV/fm$^3$, implying only up and down quarks).
3. Analytical Solutions: The authors derived exact analytical solutions for the metric potentials and physical variables using the Finch-Skea ansatz within linear $f(Q)$ gravity.

4. Results

• Mass-Radius Relationship:
  • The model predicts a maximum mass of $2.46 M_\odot$ with a corresponding radius of $13.42$ km for the highest baryon density ($n = 1.11$ fm$^{-3}$).
  • As the baryon number density ($n$) increases, the bag constant ($B$) decreases, leading to an increase in the maximum supported mass.
• Physical Viability:
  • Energy Density & Pressure: Both energy density ($\rho$) and pressures ($p_r, p_t$) are positive, finite at the center, and monotonically decrease towards the surface.
  • Anisotropy: The anisotropy factor ($\Delta$) is positive ($p_t > p_r$), indicating a repulsive force that aids in stabilizing the star against gravitational collapse.
  • Causality: The sound speeds ($v_r^2$ and $v_t^2$) remain within the causal limit ($0 < v^2 \leq 1$) throughout the star.
  • Energy Conditions: All standard energy conditions (NEC, WEC, SEC, DEC) are satisfied.
• Stability Analysis:
  • TOV Equation: The gravitational, hydrostatic, and anisotropic forces are in equilibrium ($F_g + F_h + F_a = 0$).
  • Cracking Condition: The condition $|v_r^2 - v_t^2| \leq 1$ is satisfied, ensuring stability against local perturbations.
  • Adiabatic Index: The adiabatic index $\Gamma$ remains above the critical value ($\Gamma > 4/3$), confirming dynamical stability.
• Observational Comparison:
  • The model successfully predicts the radii of known compact stars (e.g., 4U 1820-30, Her X-1, LMC X-4) with high accuracy, matching recent observational data within error margins.

5. Significance

This research offers a significant advancement in the modeling of compact objects by addressing two critical gaps:

1. Realistic Microphysics: By making the bag constant density-dependent, the model better reflects the phase transition physics observed in high-energy experiments, providing a more realistic description of the interior of strange stars.
2. Modified Gravity Context: It demonstrates that $f(Q)$ gravity, with its simpler second-order field equations and focus on non-metricity, is a viable alternative to GR for describing compact stars. The model not only supports the existence of massive strange stars but also predicts a transition to di-quark stars at higher masses, offering new theoretical insights into the composition of the densest matter in the universe.

The study concludes that the proposed model is physically consistent, stable, and capable of explaining the observed properties of compact stars, thereby validating the utility of $f(Q)$ gravity and density-dependent bag models in astrophysical research.