Fastrot-spec: Neutron Star Oscillation Code

• Fastrot-spec is a specialized framework that computes axisymmetric and non-axisymmetric oscillation frequencies in fast, rigidly rotating isolated neutron stars using pseudospectral methods.
• It couples primitive-variable hydrodynamics with the xCFC approximation to efficiently solve cold barotropic equilibrium problems on modest computational setups.
• The dual-grid architecture with a moving-surface boundary accurately captures stellar deformations and supports gravitational-wave signal extraction in the kHz band.

“fastrot-spec” is, as an Editor’s term, a concise designation for the ROXAS framework in its intended regime of fast, rigidly rotating isolated neutron stars: a new pseudospectral, nonlinear, general-relativistic code for computing stellar oscillations and extracting axisymmetric and non-axisymmetric mode frequencies relevant to the kHz gravitational-wave band expected from binary-neutron-star post-merger remnants (Servignat et al., 2024). The code is formulated for cold barotropic equations of state, uses primitive-variable hydrodynamics coupled to the extended conformal-flatness condition (xCFC), and has the notable capability to follow the stellar surface at every step, which is central for non-spherical rotating equilibria (Servignat et al., 2024).

1. Scientific setting and target regime

ROXAS, short for Relativistic Oscillations of non-aXisymmetric neutron stArS, is aimed at the oscillation problem for isolated rotating neutron stars, with explicit focus on the regime most relevant to hypermassive neutron stars formed after binary neutron star mergers (Servignat et al., 2024). The underlying motivation is observational: next-generation gravitational-wave detectors are expected to improve sensitivity in the kHz band, where such remnants are expected to emit, so robust mode-frequency predictions are needed for gravitational-wave interpretation and equation-of-state constraints (Servignat et al., 2024).

Within its current implementation, the framework supports fast, rigidly rotating isolated neutron stars, cold barotropic equations of state, and both axisymmetric and non-axisymmetric oscillations (Servignat et al., 2024). The paper also emphasizes that the code is very lightweight and can run on office computers (Servignat et al., 2024). This combination of target physics and modest computational footprint places the method between perturbative oscillation calculations and full dynamical spacetime evolutions: it is constructed for precision frequency extraction in a constrained but astrophysically important sector of parameter space.

A common misunderstanding is to read the “fastrot” label as implying differential rotation or merger-scale microphysics. The present implementation does not do so. The supported rotation law is rigid rotation, and the matter model is cold, barotropic, and $\beta$-equilibrated (Servignat et al., 2024). Thermal effects and out-of-$\beta$-equilibrium physics are identified instead as future extensions.

2. Relativistic formulation and equilibrium-preserving evolution

The hydrodynamics is written in terms of primitive variables rather than conserved variables, following the formalism cited in the paper, with the perfect-fluid stress-energy tensor

$T^{\mu\nu} = (e+p)u^\mu u^\nu + pg^{\mu\nu}.$

The matter model is zero-temperature, $\beta$-equilibrated baryonic matter with a barotropic EoS. The evolved variables are the log-enthalpy,

$H = \ln\left(\frac{e+p}{m_B n_B}\right),$

and the Eulerian velocity $U_i$, with sound speed

$c_s^2 = \frac{dp}{de}.$

The evolution equations for $H$ and $U_i$ are given explicitly in the paper as Eqs. $(\ref{eq:evolH})$ and $\beta$0 (Servignat et al., 2024).

For the spacetime sector, ROXAS adopts the xCFC reduction of Einstein’s equations in the fully constrained formalism. The gauge and geometric conditions are

$\beta$1

corresponding respectively to maximal slicing, Dirac gauge, and conformal flatness (Servignat et al., 2024). Under these assumptions, the metric problem reduces to a hierarchical system of two linear vector and two nonlinear scalar Poisson-like PDEs (Servignat et al., 2024). In practice, this is the key approximation that makes the method substantially cheaper than full GR evolution while remaining accurate enough for the frequency studies reported.

A central numerical feature is the well-balanced formulation. Every evolved quantity is decomposed into equilibrium plus perturbation, for example

$\beta$2

For stationary axisymmetric equilibrium, the first integral is written as

$\beta$3

The rewritten perturbative evolution equations are designed to remove cancellation errors, preserve equilibrium exactly, and exploit double precision more effectively (Servignat et al., 2024). This is significant because oscillation calculations are particularly sensitive to spurious drift away from equilibrium; the formalism is therefore tuned not merely for stability, but for long-time fidelity of small-amplitude mode content.

3. Pseudospectral architecture and moving-surface hydrodynamics

ROXAS is built on LORENE and uses Chebyshev pseudospectral methods in spherical coordinates for both metric and hydrodynamic sectors (Servignat et al., 2024). The code employs separate grids for the metric solve and the fluid evolution because the two subsystems impose different geometric requirements.

Subsystem	Domain structure	Spectral representation
Metric grid	spherical nucleus, multiple spherical shells, compactified external domain	Chebyshev in radius; spherical harmonics / Fourier in angle
Hydro grid	spherical nucleus, one or more shells, deformed outer boundary following the stellar surface	Chebyshev in radius; spherical harmonics / Fourier in angle

The metric grid is used to solve the xCFC equations for $\beta$4, $\beta$5, $\beta$6, and $\beta$7, and its use of spherical domains is aligned with the requirements of the elliptic spacetime solver (Servignat et al., 2024). The hydrodynamic grid, by contrast, must conform to the actual stellar surface, which is non-spherical for rotating stars. The split-grid architecture is therefore not incidental; it is a direct response to the incompatibility between a metric solver that prefers spherical domains and a fluid domain that must adapt to stellar oblateness and oscillatory deformation.

The code further includes origin regularization, axis regularization, and exponential filters to suppress aliasing (Servignat et al., 2024). For axis regularization, the paper notes that scalar fields are transformed from Fourier representation to spherical-harmonic representation and back. These are standard but essential ingredients in spectral computations of rotating configurations, where coordinate singularities and modal contamination can otherwise dominate the error budget.

The most distinctive hydrodynamic feature is the moving outer boundary. The surface radius $\beta$8 is evolved from the impermeable-boundary condition written as a time-evolution equation for the boundary itself (Servignat et al., 2024). The paper stresses why this matters: a fixed spherical outer boundary would poorly represent the matter-vacuum interface, whereas a surface-following boundary maintains a clean hydrodynamic domain and improves the representation of rotating equilibria and their perturbations. The appendix validates this boundary-evolution law with a rotating ellipsoid test (Servignat et al., 2024).

4. Oscillation excitation, spectral diagnostics, and gravitational-wave extraction

Mode identification in ROXAS is performed from the stellar surface deformation, specifically through the spherical-harmonic decomposition of the radius (Servignat et al., 2024). The paper uses the following surface coefficients as diagnostics: $\beta$9 for quasi-radial modes, $T^{\mu\nu} = (e+p)u^\mu u^\nu + pg^{\mu\nu}.$0 for axisymmetric quadrupolar modes, and $T^{\mu\nu} = (e+p)u^\mu u^\nu + pg^{\mu\nu}.$1 for non-axisymmetric quadrupolar modes (Servignat et al., 2024).

The perturbation used to excite oscillations is

$T^{\mu\nu} = (e+p)u^\mu u^\nu + pg^{\mu\nu}.$2

The plus sign excites a mixture of $T^{\mu\nu} = (e+p)u^\mu u^\nu + pg^{\mu\nu}.$3 and $T^{\mu\nu} = (e+p)u^\mu u^\nu + pg^{\mu\nu}.$4 content, whereas the minus sign excites a mixture of $T^{\mu\nu} = (e+p)u^\mu u^\nu + pg^{\mu\nu}.$5 and $T^{\mu\nu} = (e+p)u^\mu u^\nu + pg^{\mu\nu}.$6 content (Servignat et al., 2024). Frequency extraction is then performed by Fourier transforming the time series of the selected surface-mode coefficient, with Fourier-space resolution

$T^{\mu\nu} = (e+p)u^\mu u^\nu + pg^{\mu\nu}.$7

The mode labels used in the paper are $T^{\mu\nu} = (e+p)u^\mu u^\nu + pg^{\mu\nu}.$8 for the fundamental quasi-radial mode, $T^{\mu\nu} = (e+p)u^\mu u^\nu + pg^{\mu\nu}.$9 for its first overtone, $\beta$0 for the fundamental $\beta$1 mode, and $\beta$2 and $\beta$3 for the co- and counterrotating $\beta$4 modes (Servignat et al., 2024).

Although xCFC suppresses true radiative degrees of freedom, the code also computes approximate gravitational-wave signals via the quadrupole formula and its relativistic corrections. The paper writes

$\beta$5

together with the stress formula

$\beta$6

The practical amplitudes $\beta$7 and $\beta$8 are built using the weighted potential $\beta$9 and the relativistic density

$H = \ln\left(\frac{e+p}{m_B n_B}\right),$0

These diagnostics do not convert ROXAS into a full waveform generator; rather, they provide a consistent approximate radiative readout of the fluid oscillations represented within the xCFC approximation (Servignat et al., 2024).

5. Validation, recovered spectra, and accuracy claims

The principal validation strategy is comparison against semi-analytic perturbative methods, including axisymmetric perturbative/Cowling studies, CFC results, and full GR results for non-axisymmetric modes (Servignat et al., 2024). The general outcome reported is that most extracted frequencies agree to about 1% or better, while some low-frequency modes show larger relative discrepancies, up to 3–4%, and in a few cases more than 10% (Servignat et al., 2024). The paper associates the larger errors with the lowest-frequency modes and with short simulation lengths (~25 ms), which limit Fourier resolution.

For the realistic SLy4-like rotating star used as an example, the paper quotes

• $H = \ln\left(\frac{e+p}{m_B n_B}\right),$1: ROXAS $H = \ln\left(\frac{e+p}{m_B n_B}\right),$2 kHz versus literature $H = \ln\left(\frac{e+p}{m_B n_B}\right),$3 kHz, a 1.2% difference;
• $H = \ln\left(\frac{e+p}{m_B n_B}\right),$4: ROXAS $H = \ln\left(\frac{e+p}{m_B n_B}\right),$5 kHz versus literature $H = \ln\left(\frac{e+p}{m_B n_B}\right),$6 kHz, a 3.7% difference (Servignat et al., 2024).

The code also recovers spherically symmetric modes in the non-rotating BU0 model. The extracted $H = \ln\left(\frac{e+p}{m_B n_B}\right),$7 and $H = \ln\left(\frac{e+p}{m_B n_B}\right),$8 frequencies are stated to be consistent with the earlier spherically symmetric version of the code (Servignat et al., 2024). This is methodologically important: it shows that the 2D/3D extension preserves continuity with the simpler radial implementation rather than introducing an incompatible discretization artifact.

The broader interpretive point is that these comparisons are used to support the adequacy of CFC/xCFC for oscillation-frequency extraction in this setting. The data do not claim exact equivalence to full GR, but they do indicate that the approximation is sufficiently accurate for the targeted mode studies while being computationally much cheaper (Servignat et al., 2024).

6. Limitations, planned extensions, and terminological ambiguity

The current formulation is explicitly limited to isolated stars, fast rigid rotation, cold barotropic EoS, $H = \ln\left(\frac{e+p}{m_B n_B}\right),$9-equilibrium, no shocks, and the xCFC approximation rather than full GR (Servignat et al., 2024). The paper also notes the use of spatial symmetries that reduce angular ranges, identifies interpolation as the main computational cost, and remarks that longer runs are needed to improve low-frequency precision (Servignat et al., 2024). These restrictions are not peripheral: they define the exact domain in which the reported accuracy statements should be interpreted.

Planned future extensions include out-of-$U_i$0-equilibrium matter, finite-temperature effects, relaxed spatial symmetries, beyond-CFC spacetime evolution, and differential rotation profiles relevant to HMNSs (Servignat et al., 2024). This suggests a staged development trajectory in which the present code serves as a controlled high-accuracy baseline for fast-rotator oscillation calculations before moving toward more merger-realistic physics.

A plausible source of confusion is the suffix “spec.” In plasma physics, SPEC ordinarily denotes the Stepped-Pressure Equilibrium Code, extended to field-aligned flow and rigid rotation within MRxMHD (Qu et al., 2020). In programming-language research, a spec is a behavior-descriptive entity combining interface contracts, test rules, and test cases (Wang, 2011). Neither of those usages denotes ROXAS. This suggests that “fastrot-spec” is best treated not as a standardized acronym, but as an informal shorthand for the ROXAS fast-rotator oscillation setting (Servignat et al., 2024).