Evolution of Realistic Neutron star in the framework of f (Q) gravity

This paper investigates realistic neutron star models within the framework of $f(Q)$ gravity using the Krori-Barua metric and a linear $f(Q)$ function, demonstrating that four specific pulsars satisfy Buchdahl's limit and exhibit statistical consistency with observational data through Chi-Square analysis.

The Gist

Imagine the universe as a giant, cosmic construction site. For decades, physicists have been using a specific set of blueprints called General Relativity (Einstein's theory) to explain how gravity works. It's a great set of blueprints, but lately, scientists have been wondering if there are other, slightly different ways to draw the lines that might explain the universe's expansion even better.

This paper is like a team of architects (Samprity Das and Surajit Chattopadhyay) testing out a new, slightly modified blueprint called f(Q) gravity. Instead of just looking at how space bends (curvature), this new theory looks at how space "stretches" or fails to measure perfectly (called non-metricity, or Q).

Here is what they did, explained simply:

1. The Test Subjects: Cosmic Heavyweights

The authors didn't just build a theoretical model; they tested it against four real, heavy-duty stars in our galaxy: LMC X-4, SMC X-4, Cen X-3, and Vela X-1.
Think of these stars as cosmic anvils. They are incredibly dense, small, and heavy—so heavy that a teaspoon of their material would weigh billions of tons on Earth. These are neutron stars, the collapsed cores of dead stars.

2. The New Rulebook: f(Q) Gravity

In standard physics, gravity is like a rubber sheet that bends when you put a bowling ball on it. In this paper's "f(Q)" version, gravity is more like a stretchy fabric that also changes its own measuring tape.

• The authors assumed the stars inside are "anisotropic," which is a fancy way of saying the pressure pushing out isn't the same in every direction (like squeezing a stress ball that squishes differently depending on which way you push).
• They used a mathematical "shape" for the star called the Krori-Barua metric. Think of this as a specific mold they poured the star's physics into to see if it holds its shape.

3. The Balancing Act: Forces at War

Inside a neutron star, there is a massive tug-of-war:

• Gravity is trying to crush the star into a tiny dot.
• Nuclear Force (the pressure from the star's matter) is trying to push back and keep it from collapsing.

The authors found that in their new "f(Q)" model, the anisotropic factor (the difference in pressure directions) acts like a repulsive force. It's like having a team of internal springs pushing outward. They concluded that this outward push is strong enough to fight against gravity, keeping the star stable.

4. The Stress Test: Is the Star Real?

To make sure their model wasn't just math nonsense, they ran a series of "stress tests" on these four stars:

• Density Check: They checked if the star gets denser toward the center (like an onion) and less dense at the edge. Result: Yes, it behaves like a real star.
• Energy Check: They made sure the star isn't made of "exotic" or impossible matter. Result: The energy conditions were met; the star is made of "normal" (albeit very dense) stuff.
• Speed Limit Check: They checked if sound waves traveling inside the star move faster than light (which is impossible). Result: The speed of sound stayed safely below the speed of light.
• Stability Check: They calculated the "stiffness" of the star. If it's too squishy, it collapses. Result: The star is stiff enough to stay stable.

5. The "Chi-Square" Coin Flip

This is the most exciting part. The authors took the actual, observed mass of these four stars (what astronomers have measured with telescopes) and compared it to the mass their new f(Q) model predicted.

• They ran a statistical test called a Chi-Square test. Imagine flipping a coin 30 times to see if it's a fair coin.
• The Result: The test showed no significant difference between the real stars and their model. The model predicted the mass almost perfectly.
• The Conclusion: These four stars are indeed neutron stars, and they fit perfectly into this new "f(Q)" gravity framework.

6. The Final Verdict

The paper concludes that these four pulsars are neutron stars that exist comfortably within the limits of this new gravity theory.

• They are compact enough to be neutron stars (but not black holes).
• They are redshifted (light stretching out as it escapes) within safe limits.
• Most importantly, the "f(Q)" theory, which treats gravity as a mix of curvature and "stretching," successfully describes how these heavy stars hold themselves together without collapsing.

In a nutshell: The authors built a new mathematical model of gravity, used it to simulate four real, heavy neutron stars, and found that the stars behave exactly as they should. The model passed every test, suggesting that this new way of looking at gravity is a valid and accurate way to describe the universe's most extreme objects.

Technical Summary

Technical Summary: Evolution of Realistic Neutron Stars in the Framework of $f(Q)$ Gravity

Problem Statement
This study investigates the physical properties and evolutionary behavior of realistic compact objects, specifically neutron stars, within the framework of $f(Q)$ gravity. Unlike General Relativity (GR), where gravity is described by curvature, $f(Q)$ gravity posits that the gravitational interaction arises from non-metricity ($Q$). The authors aim to determine if this modified gravity theory can provide a viable description of neutron star interiors, specifically addressing the equation of state, stability, and mass-radius relations for observed pulsars (LMC X-4, SMC X-4, Cen X-3, and Vela X-1) without invoking singularities or exotic matter.

Methodology
The authors employ a theoretical approach based on the following steps:

1. Theoretical Framework: The study utilizes the $f(Q)$ gravity theory proposed by Jiménez et al., where the action is defined by a function of the non-metricity scalar $Q$. The authors deduce that for the Schwarzschild (anti-) de Sitter solution to be an exact exterior solution, the second derivative of the function with respect to $Q$ must vanish ($f_{QQ} = 0$). This constraint leads to a linear form for the function: $f(Q) = aQ + b$, where $a$ and $b$ are integration constants.
2. Metric and Field Equations: To solve the field equations for the interior of the star, the authors adopt the Krori-Barua (KB) metric potential, characterized by $\mu(r) = Ar^2$ and $\phi(r) = Br^2 + C$. This choice ensures that gravitational potentials and their derivatives remain finite at the center, avoiding singularities.
3. Anisotropic Fluid: The matter distribution is modeled as an anisotropic fluid, distinguishing between radial pressure ($p_r$) and tangential pressure ($p_t$). The anisotropic factor is defined as $\Delta = p_t - p_r$.
4. Boundary Conditions: The interior solution is matched to the exterior Schwarzschild vacuum solution at the stellar surface ($r=R$) to determine the constants $A$, $B$, and $C$ in terms of the observed mass ($M$) and radius ($R$).
5. Physical Analysis: The authors derive expressions for energy density ($\rho$), pressures, and the equation of state (EoS) parameters. They validate the models against:
   • Energy Conditions: Null (NEC) and Strong (SEC) energy conditions.
   • Causality and Stability: The speed of sound ($v_s^2$) is checked to ensure it lies within $0 < v_s^2 < 1$. The adiabatic index ($\Gamma$) is analyzed to ensure $\Gamma > 4/3$ for dynamical stability. The Tolman-Oppenheimer-Volkoff (TOV) equation is used to verify hydrostatic equilibrium among gravitational, hydrostatic, and anisotropic forces.
   • Statistical Validation: A Chi-Square ($\chi^2$) test is performed using 30 distinct values of the parameter $a$ to compare model-generated masses against observed masses.

Key Contributions and Results

• Linear $f(Q)$ Model: The paper establishes that a linear function of non-metricity ($f(Q) = aQ + b$) is sufficient to model these compact objects, with the ratio $b/a$ corresponding to a cosmological constant-like term.
• Physical Viability:
  • Density and Pressure: The derived density and pressure profiles are positive, finite at the center, and monotonically decreasing toward the surface. The gradients of these quantities are negative, satisfying physical requirements for stable stellar configurations.
  • Anisotropy: The anisotropic factor $\Delta$ is found to be positive and monotonically increasing. This implies $p_t > p_r$, generating a repulsive anisotropic force that counteracts gravitational collapse.
  • Energy Conditions: The models satisfy NEC and SEC throughout the stellar interior, indicating the absence of exotic matter.
  • Equation of State: The EoS parameters ($\omega_r$ and $\omega_t$) are positive and lie within the range $0 < \omega < 1$, consistent with non-exotic matter.
• Stability Analysis:
  • Causality: The squared speed of sound in both radial and tangential directions remains within the permissible limit ($0 < v_s^2 < 1$), satisfying the cracking criterion for stability.
  • Adiabatic Index: The adiabatic index $\Gamma$ is consistently greater than $4/3$, confirming dynamical stability against radial perturbations.
  • Equilibrium: The TOV equation analysis shows that the gravitational force is balanced by the combined action of hydrostatic and anisotropic forces.
• Mass-Radius and Compactness:
  • The mass-radius relation shows that mass increases with radius.
  • The compactness parameter ($u = M/R$) for all four pulsars falls within the range $0.1 < u < 0.25$, identifying them as neutron stars. Crucially, all values remain below the Buchdahl limit ($u < 4/9$).
  • Surface redshift ($z_s$) values are calculated to be well within the permissible limit ($z_s \leq 5.211$).
• Statistical Consistency: The Chi-Square test results ($\chi^2_{cal} < \chi^2_{tab}$) for all four pulsars indicate that there is no statistically significant difference between the observed masses and the masses generated by the model. The null hypothesis is accepted.

Significance and Claims
The paper claims that the $f(Q)$ gravity framework, utilizing a linear function of non-metricity and the Krori-Barua metric, provides a robust and physically consistent description of realistic neutron stars. The study demonstrates that:

1. The modified gravity background successfully reproduces the observed properties of LMC X-4, SMC X-4, Cen X-3, and Vela X-1 without requiring singularities or exotic matter.
2. The repulsive nature of the anisotropic force in this model is sufficient to oppose gravitational attraction, preventing immediate collapse.
3. The model passes rigorous stability tests (causality, adiabatic index, TOV equilibrium) and statistical validation against observational data.

The authors conclude that these pulsars can be interpreted as neutron stars existing in a modified gravity background of $f(Q)$, offering a viable alternative to standard GR descriptions for compact stellar objects. The work suggests that future studies could extend this consistency check to other modified gravity frameworks and LIGO observational constraints.