Numerical simulations of oscillating and differentially rotating neutron stars

This paper extends the ROXAS pseudospectral code to simulate oscillating, differentially rotating neutron stars in the conformal flatness approximation, successfully validating axisymmetric results against existing literature while providing the first frequency values for non-axisymmetric modes in such configurations to improve post-merger gravitational wave modeling.

The Gist

Imagine two massive, city-sized balls of neutron-rich matter (neutron stars) crashing into each other in a cosmic dance. When they merge, they don't just disappear; they often form a new, super-dense, spinning object called a Hypermassive Neutron Star (HMNS). This new star is like a spinning top made of the densest stuff in the universe, wobbling and vibrating as it tries to settle down.

These vibrations are crucial. As the star wobbles, it ripples the fabric of space and time itself, sending out gravitational waves—invisible sound waves that we can try to hear with detectors like LIGO.

However, there's a catch. The new star doesn't spin like a rigid basketball (where every part moves at the same speed). Instead, it spins differentially, meaning the core might be spinning like a race car while the outer layers are moving more like a slow jogger. This "differential rotation" makes the star's wobbles incredibly complex and hard to predict.

The Problem: The "Rigid" Blind Spot

For a long time, scientists trying to simulate these stars had to make a simplifying assumption: they treated the star as if it were rigid or ignored how the star's own gravity changed as it wobbled. It's like trying to predict the sound of a drum by assuming the drum skin never stretches or changes shape. While this is easier to calculate, it can lead to "ghost" sounds—frequencies that appear in the math but don't actually exist in reality.

The Solution: ROXAS Gets an Upgrade

The authors of this paper, Santiago, Jérôme, and Micaela, have upgraded a computer program called ROXAS. Think of ROXAS as a high-tech "virtual wind tunnel" for stars.

• Before: ROXAS could only simulate stars that spun like a solid top (rigid rotation).
• Now: They've taught ROXAS to handle differential rotation. It can now simulate a star where the center spins fast and the edges spin slow, just like a real post-merger remnant.

They also improved the math to ensure that when the star wobbles, the code doesn't get confused by the star's own gravity changing in real-time.

What They Discovered

Using their upgraded "virtual wind tunnel," they ran thousands of simulations and found some fascinating things:

1. The "Ghost" Frequency: In the old, simplified simulations (called the "Cowling approximation"), scientists saw a second type of wobble (a "secondary fundamental mode") that looked real. The authors proved that this is a fake. It's an artifact of the simplified math. When you run the full, realistic simulation (where gravity moves and changes), this ghost frequency disappears. It's like realizing a shadow you saw on the wall was just a trick of the light, not a real object.
2. New Sounds: They are the first to map out the specific "notes" (frequencies) that these differentially rotating stars sing when they wobble in a non-symmetrical way. This is like finding a new instrument in an orchestra that no one knew how to play before.
3. Speed and Efficiency: Their code is incredibly fast. While other super-computer simulations might take days to run one scenario, ROXAS can do it in minutes on a standard office computer. It's the difference between waiting for a slow train and hopping on a high-speed bullet train.

Why This Matters

Right now, our current gravitational wave detectors aren't sensitive enough to "hear" these specific high-pitched wobbles from merging stars. But the next generation of detectors (like the Einstein Telescope) will be able to hear them.

When those detectors come online, they will need a "dictionary" to translate the sounds they hear into physical facts about the star. This paper provides a huge chunk of that dictionary. By understanding how differential rotation changes the star's song, scientists will be able to:

• Figure out the Equation of State (the recipe for how neutron star matter behaves under extreme pressure).
• Understand the internal structure of these cosmic giants.
• Predict exactly what signal to look for when the next big merger happens.

In short: The authors built a faster, more realistic simulator for spinning neutron stars, proved that some previous "sounds" were fake, and gave us the first real map of the unique vibrations these cosmic giants make. This is a major step toward decoding the secrets of the universe's most extreme objects.

Technical Summary

1. Problem Statement

The remnants of binary neutron star (NS) mergers are expected to be hypermassive neutron stars (HMNS) supported temporarily by differential rotation before collapsing into black holes. These remnants emit gravitational waves (GWs) in the kilohertz (kHz) band, which encode information about the neutron star's internal structure and Equation of State (EoS).

Current numerical codes face a trade-off:

• Full General Relativity (GR) codes are computationally expensive, often requiring supercomputers, making large-scale parametric studies difficult.
• Simplified codes often assume rigid rotation or use the Cowling approximation (static metric), which may introduce artifacts or fail to capture the complex dynamics of differentially rotating post-merger remnants.

There is a need for a lightweight, efficient code capable of dynamically evolving differentially rotating neutron stars in a relativistic framework to generate accurate GW templates for next-generation detectors (e.g., Einstein Telescope, Cosmic Explorer).

2. Methodology

The authors extended the existing pseudospectral code ROXAS (Relativistic Oscillations of non-aXisymmetric neutron stArS) to handle differential rotation.

• Formalism:

  • The code uses the 3+1 decomposition of spacetime with the Extended Conformal Flatness Condition (xCFC). This approximates the full Einstein equations by assuming the spatial metric is conformally flat, significantly reducing computational cost while maintaining high accuracy.
  • The hydrodynamics are solved using primitive variables (enthalpy, velocity) rather than conserved variables, utilizing a well-balanced formulation. This involves decomposing variables into an equilibrium state ($eq$) plus perturbations ($\bar{}$), ensuring numerical stability for small perturbations on top of large equilibrium values.
  • Differential Rotation: The equilibrium state is defined by a rotation law relating the angular velocity $\Omega$ to the specific angular momentum $F$. The authors implemented the Komatsu-Eriguchi-Hachisu (KEH) profile, $F(\Omega) = A^2(\Omega_c - \Omega)$, which allows for a smooth transition from rigid rotation to differential rotation. The formalism was updated to include integral terms and additional source terms in the evolution equations arising from the radial dependence of $\Omega$.

• Numerical Implementation:

  • Grids: The code uses a multi-domain spherical grid. The hydrodynamic grid covers the star (nucleus + shell), while the metric grid extends to infinity using a compactification variable ($u=1/r$).
  • Perturbations: Simulations are initiated by perturbing the equilibrium star with specific modes in log-enthalpy ($H$) or Eulerian velocity ($U^i$). Perturbations can be axisymmetric ($l=m=0, 2$) or non-axisymmetric ($l=m=2$).
  • Validation: The code was tested against the "B sequence" of differentially rotating stars (Dimmelmeier et al. 2006) using a polytropic EoS ($p = \kappa n_B^\gamma$ with $\gamma=2$).

3. Key Contributions

1. Extension of ROXAS: The code was successfully updated to evolve differentially rotating neutron stars dynamically, moving beyond the previous rigid-rotation-only capability.
2. Well-Balanced Scheme for Differential Rotation: The authors derived and implemented the specific source terms required in the hydrodynamic evolution equations to maintain equilibrium stability when $\Omega$ varies spatially.
3. First Non-Axisymmetric CFC Results: The paper reports, for the first time, oscillation frequencies for non-axisymmetric modes in differentially rotating configurations evolved under the Conformal Flatness Condition (CFC).
4. Identification of Cowling Artifacts: The study explicitly demonstrates that the "secondary fundamental mode" ($F_{II}$) observed in Cowling approximation simulations is a numerical artifact that disappears in full dynamical spacetime simulations.

4. Results

The authors performed simulations on the B sequence (models B0–B9) with varying degrees of flattening and rotation rates.

• Stability: Unperturbed differentially rotating stars remained stable over 25 ms (many rotation periods), confirming the consistency of the equilibrium solver and the dynamical evolution code.
• Cowling Approximation vs. Dynamical Spacetime:
  • Cowling: Simulations showed two fundamental peaks: the primary mode ($F$) and a secondary mode ($F_{II}$) around 2.6 kHz.
  • Dynamical (CFC): Simulations showed only a single fundamental mode ($F$). The $F_{II}$ peak vanished, confirming it is an artifact of the static metric assumption in the presence of differential rotation.
• Frequency Agreement:
  • Axisymmetric Modes: Frequencies for $F$, $H_1$, $2f_0$, and $2p_1$ agreed with literature (Stergioulas et al. 2004; Dimmelmeier et al. 2006) within ~1-2% for most models.
  • Non-Axisymmetric Modes: The $2f_2$ and $2f_{-2}$ modes were extracted. For highly rotating models (B7–B9), the $2f_2$ frequency became negative, indicating a transition to retrograde rotation in the co-rotating frame (CFS instability regime).
  • Discrepancies: Some tension was observed in the fundamental frequency ($F$) for the fastest rotating models (B7, B9) compared to Dimmelmeier et al. (2006), likely due to differences in resolution and simulation time, though the CFC approximation itself was validated as reliable in previous rigid-rotation tests.
• Computational Efficiency: The code is lightweight, capable of running on standard office computers. It is estimated to be ~5 times faster than CoCoNuT for radial simulations, with potential speedups of 25x (axisymmetric) and 125x (non-axisymmetric) due to the pseudospectral method's efficiency.

5. Significance

• Realistic Modeling: This work bridges the gap between simplified analytic models and computationally prohibitive full-GR simulations, enabling the study of realistic post-merger remnants with differential rotation.
• GW Data Analysis: By providing accurate frequency templates for differentially rotating stars, the code aids in the interpretation of future kHz-band GW signals from next-generation detectors.
• Artifact Correction: The finding that the secondary fundamental mode is an artifact of the Cowling approximation is crucial for preventing misinterpretation of GW spectra.
• Future Outlook: The framework is designed to be extensible. Future work will incorporate more complex rotation laws (e.g., from merger simulations) and realistic nuclear EoS, making ROXAS a versatile tool for multi-messenger astrophysics.