Constraining the $f$-mode oscillations frequency in Neutron Stars through Universal Relations in the realm of Energy-Momentum Squared Gravity

The Gist

Imagine the universe as a giant cosmic laboratory. Inside this lab, there are some of the most extreme objects imaginable: Neutron Stars. These are the dead, super-dense cores of massive stars that have collapsed. They are so heavy that a single teaspoon of their material would weigh a billion tons on Earth. Because they are so dense, their gravity is incredibly strong, making them perfect places to test the rules of how gravity works.

For a long time, scientists have used Einstein's theory of General Relativity (GR) to describe gravity. It works great, but it has some gaps, especially when dealing with the very beginning of the universe or the centers of black holes. So, scientists are looking for "backup theories" to see if they fit the data better.

This paper explores one such backup theory called Energy-Momentum Squared Gravity (EMSG).

The "Recipe" for a Star

To understand a neutron star, you need a recipe. In physics, this recipe is called an Equation of State (EOS). It tells us how the star's material behaves under extreme pressure. The authors tested three different "flavors" of recipes:

• Stiff: Like a very hard, unyielding rock.
• Soft: Like a squishy sponge.
• Intermediate: Somewhere in between.

The New Ingredient: The "Alpha" Parameter

The main twist in this study is a new ingredient added to the gravity recipe called $\alpha$ (alpha).

• In standard Einstein gravity, this ingredient is zero.
• In this new theory (EMSG), $\alpha$ can be a tiny positive number or a tiny negative number.

Think of $\alpha$ as a volume knob for gravity.

• If you turn the knob to positive, the gravity gets "stiffer" (the star resists being squished more).
• If you turn it to negative, the gravity gets "softer" (the star squishes down more easily).

The "Singing" of the Stars

Neutron stars aren't just sitting still; they vibrate. Imagine hitting a bell. It rings with a specific pitch. Neutron stars ring too, but instead of sound, they send out ripples in space-time called Gravitational Waves.

The specific "note" or frequency the star sings is called the f-mode.

• The Goal: The authors wanted to figure out what note a neutron star would sing if the universe followed the rules of this new EMSG theory instead of standard Einstein rules.
• The Discovery: They found that the "pitch" of the star changes depending on the $\alpha$ knob.
  • Positive $\alpha$: The star becomes harder to deform, so it sings a lower note (lower frequency).
  • Negative $\alpha$: The star becomes easier to deform, so it sings a higher note (higher frequency).

The Universal "Cheat Codes"

One of the coolest things about neutron stars is that they follow "Universal Relations." These are like cheat codes or shortcuts.

• Even though we don't know the exact recipe (EOS) of a specific star, we know that its size, its weight, and its singing pitch are mathematically linked.
• The authors used these links to create a map. If we know how much a star wobbles (tidal deformability) from a collision, we can predict exactly what note it should sing.

Putting It to the Test

The authors took real data from two famous cosmic collisions detected by gravitational wave observatories: GW170817 and GW190814.

1. They used the "Universal Relations" to calculate what the "singing pitch" (f-mode frequency) should be for these events.
2. They checked how this pitch changed when they tweaked the $\alpha$ knob (the new gravity theory).
3. The Result: They found that the new theory (EMSG) changes the predicted pitch. For example, with the standard theory (Einstein), a 1.4-solar-mass star might sing at about 2.66 kHz. With the new theory, that pitch could shift up or down depending on whether $\alpha$ is positive or negative.

The Phase Transition Surprise

The study also looked at what happens inside the star when the pressure gets incredibly high.

• They found that for the "Stiff" recipe, the star undergoes a phase transition (like water turning to ice, but for star stuff) at very high densities.
• This happens at different depths depending on the recipe and the $\alpha$ knob. It's like finding a hidden layer of chocolate inside a cake that only appears when you bake it at a specific temperature.

The Bottom Line

This paper doesn't claim to have found a new star or changed how we build bridges. Instead, it's a theoretical exercise. It says:
"If gravity works slightly differently than Einstein thought (specifically with this EMSG theory), then neutron stars would vibrate at slightly different frequencies than we currently predict."

By listening to the "songs" of neutron stars in the future, astronomers might be able to tell if the $\alpha$ knob is turned up, down, or if it's actually zero (meaning Einstein was right all along). The paper provides the mathematical map to help us listen for those differences.

Technical Summary

Technical Summary: Constraining f-mode Oscillations in Neutron Stars via Universal Relations in Energy-Momentum Squared Gravity

Problem Statement
Neutron stars (NSs) serve as extreme astrophysical laboratories for testing gravitational theories in high-curvature regimes. While General Relativity (GR) has been successful, unresolved issues regarding dark matter, dark energy, and singularities motivate the exploration of modified gravity theories. Specifically, Energy-Momentum Squared Gravity (EMSG), which extends GR by including nonlinear terms involving the energy-momentum tensor ($T_{\mu\nu}T^{\mu\nu}$), offers a framework to probe deviations from GR in dense matter environments. A critical challenge in this domain is constraining the fundamental-mode (f-mode) oscillation frequencies of NSs within the EMSG framework. While Universal Relations (URs) linking oscillation frequencies to macroscopic properties (like compactness and tidal deformability) have been established in GR, their behavior and applicability in EMSG, particularly regarding the free parameter $\alpha$, remain unexplored.

Methodology
The study employs a multi-step theoretical and numerical approach:

1. Theoretical Framework: The authors derive the field equations for EMSG by modifying the Einstein-Hilbert action with a term $\alpha T_{\mu\nu}T^{\mu\nu}$. This leads to modified Einstein field equations where the standard energy-momentum tensor is replaced by an effective tensor ($T^{\mu\nu}_{eff}$) containing effective energy density ($E_{eff}$) and effective pressure ($P_{eff}$).
2. Hydrostatic Equilibrium: The modified Tolman-Oppenheimer-Volkoff (TOV) equations are derived and solved numerically. The study utilizes three distinct Equations of State (EOS): Stiff, Intermediate, and Soft. The free parameter $\alpha$ is varied across a range of values ($-5.01 \times 10^{-38}$ to $+5.01 \times 10^{-38}$ cm$^3$/erg) to generate mass-radius relations.
3. Non-Radial Oscillations: Within the Cowling approximation (neglecting metric perturbations), the authors solve the linearized perturbation equations for non-radial oscillations. They focus specifically on the quadrupolar ($l=2$) f-mode frequency, utilizing the effective energy and pressure derived from the EMSG framework.
4. Universal Relations (URs): The study establishes two key URs:
   • f-Love Relation: A correlation between the normalized f-mode frequency ($\bar{\omega}$) and the tidal Love number ($\Lambda$).
   • C-f Relation: A correlation between the stellar compactness ($C = M/R$) and the f-mode frequency.
     These relations are fitted using least-squares polynomial approximations for various $\alpha$ values.
5. Constraints: Using observational data from the gravitational wave events GW170817 and GW190814, specifically their tidal deformability constraints, the authors apply the derived URs to constrain the canonical f-mode frequencies ($f_{1.4}$) for NSs with a mass of $1.4 M_{\odot}$.

Key Contributions

• Derivation of EMSG TOV Equations: The paper provides the explicit modified TOV equations incorporating the $\alpha$ parameter and solves them for multiple EOS models.
• Establishment of EMSG Universal Relations: This work is the first to investigate and quantify the f-Love and C-f universal relations specifically within the EMSG framework. It demonstrates how these relations depend on the sign and magnitude of $\alpha$.
• Frequency Constraints: The study successfully constrains the f-mode oscillation frequencies for canonical NSs under EMSG, comparing them against GR predictions and previous literature.
• Phase Transition Analysis: The analysis of pressure-density profiles reveals distinct first-order phase transitions for the Stiff, Intermediate, and Soft EOSs, with the Stiff EOS undergoing transition at the highest energy densities.

Results

• Impact of $\alpha$ on Oscillations: The f-mode frequency exhibits a clear dependence on $\alpha$. For a fixed mass, the frequency decreases as $\alpha$ becomes more positive and increases as $\alpha$ becomes more negative.
• Universal Relation Robustness: The f-Love and C-f relations remain valid in EMSG but show EOS sensitivity that varies with $\alpha$. The study finds that negative $\alpha$ values enhance the EOS-insensitivity of the f-Love relation, while positive $\alpha$ values increase the reduced chi-squared ($\chi^2_r$) errors, suggesting a slight degradation in universality compared to the GR ($\alpha=0$) case.
• Canonical Frequencies: For the GR case ($\alpha=0$), the study infers a canonical f-mode frequency of $f_{1.4} = 2.659^{+0.458}_{-0.491}$ kHz for GW170817 and $2.143^{+0.127}_{-0.153}$ kHz for GW190814. These values are approximately 30-35% higher than some previous GR studies, a difference the authors attribute to the use of the Cowling approximation in this work versus full-GR formalisms in others.
• Mass-Radius Constraints: Using the C-f UR, the authors map the allowed Mass-Radius ($M-R$) regions. They find that for a fixed frequency, positive $\alpha$ values generally lead to smaller radii (higher compactness), while negative $\alpha$ values lead to larger radii.
• Sound Speed: The sound speed profiles ($c_s^2$) across the stellar radius satisfy causality ($c_s^2 \leq 1$) and conformal limits for all tested $\alpha$ values, confirming the physical validity of the models.

Significance and Claims
The paper claims that EMSG introduces significant modifications to the macroscopic properties of neutron stars compared to GR, altering the stiffness of the EOS and consequently affecting mass-radius relations, tidal deformability, and oscillation frequencies. The study posits that the derived Universal Relations in EMSG provide a viable tool for testing alternative gravity theories. By constraining the f-mode frequencies using observational data from GW170817 and GW190814, the work suggests that future gravitational wave observations could potentially distinguish between GR and EMSG by detecting deviations in oscillation frequencies and tidal deformabilities. The authors conclude that EMSG offers a compelling approach for studying strong-field gravity and dense matter, with the potential for future GW observations to validate or constrain the theory.