Universal Relations and Correlation Analysis of Proto-Neutron Star Properties in Energy-Momentum Squared Gravity

This study investigates the macroscopic properties and universal relations of proto-neutron stars within Energy-Momentum Squared Gravity, revealing that while thermodynamic variables and the modified gravity parameter significantly alter individual stellar characteristics, the underlying correlations between these properties remain robust and largely unaffected.

The Gist

Imagine the universe as a giant cosmic kitchen. In this kitchen, when a massive star runs out of fuel and collapses, it doesn't just vanish; it gets squeezed into a tiny, incredibly dense ball called a Proto-Neutron Star (PNS). Think of a PNS as a "freshly baked" neutron star. It is still incredibly hot, full of trapped particles (like lemons in a cake), and hasn't had time to cool down yet. Eventually, it cools off and becomes a standard, cold Neutron Star (NS).

This paper is a recipe book and a physics experiment combined. The author, Sayantan Ghosh, wants to understand how these "freshly baked" stars behave, not just under our current understanding of gravity (General Relativity), but under a new, slightly tweaked theory called Energy-Momentum Squared Gravity (EMSG).

Here is a breakdown of the study using simple analogies:

1. The Ingredients: The "Equation of State"

To bake a star, you need a recipe. In physics, this recipe is called the Equation of State (EOS). It tells us how the star's matter reacts to pressure and heat.

• The author used four different recipes (called NITR, IOPB-I, MODEL I, and IUFSU).
• They adjusted the "temperature" of the recipe by changing two main ingredients:
  • Entropy (S): How "hot" and chaotic the star is. $S=0$ is a cold, finished star. $S=1$ or $2$ is a hot, fresh PNS.
  • Lepton Fraction (Yl): The amount of "trapped particles" (like neutrinos) inside. More trapped particles make the star puffier.

2. The New Oven: EMSG vs. General Relativity

For decades, we've used Einstein's General Relativity (GR) to describe gravity. It works perfectly for things like planets and apples. But in the extreme heat and density of a neutron star, maybe GR needs a tweak.

• The Analogy: Imagine GR is a standard oven that bakes bread perfectly. EMSG is a new oven with a special dial (called $\alpha$) that adds a little extra "squared energy" to the mix.
• The Result: In weak gravity (like on Earth), this new oven looks exactly like the old one. But in the extreme gravity of a neutron star, the dial changes things.
  • If you turn the dial positive, the star's "crust" gets stiffer, making the star slightly larger and heavier.
  • If you turn it negative, the star gets squeezed tighter, becoming smaller and lighter.

3. What Happens to the Star? (The Results)

The author ran simulations to see how changing the heat (Entropy), the trapped particles (Leptons), and the gravity dial ($\alpha$) changed the star's properties:

• Size and Mass: Hotter stars (higher entropy) are bigger because the heat pushes outward, like steam in a pressure cooker. However, the new gravity dial ($\alpha$) can make them even bigger or smaller depending on the setting.
• Oscillations (The "Hum"): Neutron stars aren't static; they vibrate like a struck bell. This is called the $f$-mode. The study found that if the star is "puffier" (larger radius), it vibrates at a lower pitch (frequency). The new gravity dial changes this pitch, but the relationship holds true.
• Binding Energy: This is how tightly the star is held together. The study found that while the new gravity dial changes the numbers, the stars remain "bound" (they don't fall apart), though they are slightly less tightly held when the star is hotter.

4. The Big Surprise: Universal Relations

This is the most important part of the paper. Usually, if you change the recipe (EOS), the cake looks different. If you change the oven (Gravity), the cake looks different.

• The Analogy: Imagine you have four different types of flour (the four recipes) and you are baking in three different ovens (different gravity settings). You would expect the cakes to look totally different from each other.
• The Discovery: The author found that despite changing the flour, the heat, and the oven dial, the relationship between the cake's height and its weight remained almost exactly the same.
• In physics terms, these are called Universal Relations (URs). Even though the specific numbers for mass, radius, and vibration changed, the mathematical link between them stayed strong and consistent.
  • For example, no matter which recipe or gravity dial was used, if you knew the star's "compactness" (how dense it is), you could accurately predict its vibration frequency.

5. The Correlation: The "Glue" of the Universe

Finally, the author measured how "connected" these relationships were using a correlation score (a number between 0 and 1, where 1 is perfect).

• The Finding: Even with all the changes in temperature, particle count, and gravity theory, the connection score stayed incredibly high (between 0.92 and 1.0).
• The Metaphor: It's like having a group of friends who all change their clothes, hair, and jobs. You might think they are unrecognizable. But if you ask them to stand in a line by height, they still line up in the exact same order. The "order" (the correlation) is unbreakable, even if the individuals (the specific star properties) change.

Summary

The paper concludes that while the specific details of a Proto-Neutron Star (how big it is, how heavy it is, how it vibrates) are very sensitive to its temperature, its trapped particles, and the specific theory of gravity used, the underlying rules connecting these properties are incredibly robust.

Even if we are wrong about the exact theory of gravity (GR vs. EMSG) or the exact temperature of the star, the "Universal Relations" act as a reliable map. They tell us that the universe has a consistent structure that doesn't break, even when we tweak the fundamental laws or the conditions of the star. This gives scientists a powerful tool to understand these mysterious objects without needing to know every single tiny detail about their internal composition.

Technical Summary

Technical Summary: Universal Relations and Correlation Analysis of Proto-Neutron Star Properties in Energy-Momentum Squared Gravity

Problem Statement
Proto-neutron stars (PNSs) represent the hot, lepton-rich remnants of core-collapse supernovae, evolving into cold neutron stars (NSs) through neutrino emission and cooling. While NSs serve as laboratories for testing General Relativity (GR) in strong-field regimes, PNSs offer a unique environment where finite-temperature effects and high lepton fractions significantly influence the equation of state (EOS). Standard GR tests in the weak-field regime (e.g., Solar System experiments) cannot distinguish between GR and modified gravity theories like Energy-Momentum Squared Gravity (EMSG), as both yield identical parametrized post-Newtonian parameters. Consequently, there is a need to probe EMSG effects in the strong-field regimes of compact objects. Furthermore, while universal relations (URs)—EOS-independent correlations between macroscopic properties—have been extensively studied for cold NSs in EMSG, their applicability and stability for PNSs within the EMSG framework, particularly under varying entropy ($S$) and lepton fractions ($Y_l$), remain unexplored.

Methodology
The study employs a relativistic mean-field (RMF) formalism to generate finite-temperature EOSs for four distinct models: NITR, IOPB-I, MODEL I, and IUFSU. These models are constrained by current observational limits on NS maximum masses ($\approx 2 M_\odot$) and nuclear matter properties.

• Thermodynamics: The EOSs are constructed by fixing the entropy per baryon ($S=0$ for cold NSs; $S=1, 2$ for PNSs) and varying the lepton fraction ($Y_l$). The crust is modeled using the HS(DD2) finite-temperature data from COMPOSE.
• Gravity Framework: The study extends beyond GR by incorporating EMSG, which modifies the field equations by adding a term proportional to the square of the energy-momentum tensor ($T_{\mu\nu}T^{\mu\nu}$), scaled by a free parameter $\alpha$.
• Numerical Solution: The modified Tolman-Oppenheimer-Volkoff (TOV) equations are solved numerically to obtain mass-radius ($M-R$) profiles. Causality is verified by ensuring the squared sound speed ($c_s^2$) remains within physical limits ($0 \le c_s^2 \le 1$).
• Oscillations and Binding Energy: Non-radial $f$-mode oscillations are calculated using the Cowling approximation (ignoring metric perturbations), and gravitational binding energy ($B$) is computed as the difference between gravitational and baryonic masses.
• Universal Relations: The study establishes URs linking the $f$-mode frequency scaled by canonical mass ($f_f M_{1.4}$) and binding energy per mass ($B/M$) to compactness ($C$) and dimensionless tidal deformability ($\Lambda$). These relations are fitted using polynomial expansions, and their EOS-independence is quantified via reduced chi-squared ($\chi^2_r$) values and linear correlation coefficients ($r$).

Key Results

1. Macroscopic Properties:

   • Entropy and Lepton Fraction: Increasing $S$ and $Y_l$ leads to larger radii and slightly higher maximum masses due to increased thermal pressure. However, higher $Y_l$ at fixed $S$ tends to decrease the maximum mass.
   • EMSG Parameter ($\alpha$): Positive $\alpha$ values stiffen the EOS at low to intermediate densities, resulting in moderately higher masses and larger radii, while negative $\alpha$ softens the EOS, leading to smaller radii. The effect of $\alpha$ on the radius is more pronounced for PNSs with higher $S$ and $Y_l$ compared to cold NSs.
   • Oscillations and Binding Energy: The $f$-mode frequency decreases with higher $S$ and $Y_l$. The variation in frequency due to $\alpha$ is smaller for high-entropy PNSs ($S=2, Y_l=0.4$) compared to cold NSs. Gravitational binding energy remains negative (indicating a bound system) across all variations, though the system becomes less tightly bound as $S$ and $Y_l$ increase.

2. Universal Relations (URs):

   • The study successfully establishes URs for $f_f M_{1.4}$ vs. $C$, $f_f M_{1.4}$ vs. $\Lambda$, $|B|/M$ vs. $C$, and $|B|/M$ vs. $\Lambda$ within the EMSG framework.
   • Despite significant variations in macroscopic properties induced by changes in $S$, $Y_l$, and $\alpha$, the fitting coefficients for these relations remain robust. The reduced chi-squared values indicate that the relations maintain their EOS-insensitive nature.

3. Correlation Analysis:

   • Linear correlation analysis reveals that $f_f M_{1.4}$ and $|B|/M$ are positively correlated with compactness ($C$) and negatively correlated with tidal deformability ($\Lambda$).
   • Crucially, the absolute correlation coefficients ($|r|$) for all four URs remain in the range of 0.92 to 1.0 across all variations of $S$, $Y_l$, and $\alpha$. This indicates that the strength of the correlations is largely unaffected by the modifications to the gravity theory or the thermodynamic state of the star.

Significance and Claims
The paper claims to be the first study to investigate the macroscopic properties of PNSs and the correlations of their universal relations specifically within the context of Energy-Momentum Squared Gravity. The primary significance lies in demonstrating that while the specific values of macroscopic observables (mass, radius, frequency, etc.) are sensitive to the thermodynamic state ($S, Y_l$) and the modified gravity parameter ($\alpha$), the underlying universal relations and their strong correlations persist. This suggests that URs remain a powerful, EOS-independent tool for constraining PNS structure and testing gravity theories, even when detailed internal composition data is inaccessible. The findings reinforce the utility of correlation analysis in interpreting astrophysical observations, such as gravitational waves, in the strong-field regime.