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GPU-Accelerated X-ray Pulse Profile Modeling

Tianzhe Zhou, Chun Huang

TL;DR

This work presents a public GPU-accelerated X-ray pulse-profile modeling framework that overcomes the long-standing accuracy-speed bottleneck in neutron-star forward modeling. By combining an oblate--Schwarzschild spacetime treatment, atmosphere-based beaming from NSX-H tables, full-surface temperature maps, and NICER-style instrument response, the authors achieve $\sim 2$--$5~\mathrm{ms}$ per evaluation with $10^{3}$--$10^{4}\times$ speedups on a modern GPU while maintaining $\sim 10^{-3}$ relative waveform accuracy against benchmarks. A key contribution is the identification and mitigation of a bias from atmosphere-table interpolation near grid boundaries, via a mixed-order interpolation that enforces linearity at grazing angles. The GPU framework enables high-resolution, comprehensive hotspot modeling and robust Bayesian inference on accessible hardware, opening the door to reanalyzing NICER targets with physically motivated temperature maps and broader parameter exploration. Overall, this work significantly enhances the practicality and reliability of PPM for current and future X-ray missions.

Abstract

Pulse-profile modeling (PPM) of thermal X-ray emission from rotation-powered millisecond pulsars enables simultaneous constraints on the mass $M$, radius $R$, and hence the equation of state of cold, dense matter. However, Bayesian PPM has faced a hard accuracy-speed bottleneck: current production resolutions used to keep inference tractable can under-resolve extreme hotspot geometries and bias the waveform computation, whereas the higher resolutions that remove this bias push forward models to minutes per evaluation, making inference impractical. We break this trade-off with, to our knowledge, the first public GPU-accelerated X-ray PPM framework that matches established benchmarks to within $\sim10^{-3}$ relative accuracy even for extreme geometries, while collapsing minutes-long high-fidelity computations to $2$--$5$ ms on an RTX 4080 ($10^{3}$--$10^{4}\times$ speedups), enabling posterior exploration at resolutions and complexities previously out of reach. We further uncover a bias near the interpolation boundaries of atmosphere lookup tables, demonstrate it with two diagnostic tests, and counter it with a mixed-order interpolator. Together, these advances enlarge the feasible hotspot model space and reduce key systematics in PPM, strengthening inferences for current and future X-ray missions.

GPU-Accelerated X-ray Pulse Profile Modeling

TL;DR

This work presents a public GPU-accelerated X-ray pulse-profile modeling framework that overcomes the long-standing accuracy-speed bottleneck in neutron-star forward modeling. By combining an oblate--Schwarzschild spacetime treatment, atmosphere-based beaming from NSX-H tables, full-surface temperature maps, and NICER-style instrument response, the authors achieve -- per evaluation with -- speedups on a modern GPU while maintaining relative waveform accuracy against benchmarks. A key contribution is the identification and mitigation of a bias from atmosphere-table interpolation near grid boundaries, via a mixed-order interpolation that enforces linearity at grazing angles. The GPU framework enables high-resolution, comprehensive hotspot modeling and robust Bayesian inference on accessible hardware, opening the door to reanalyzing NICER targets with physically motivated temperature maps and broader parameter exploration. Overall, this work significantly enhances the practicality and reliability of PPM for current and future X-ray missions.

Abstract

Pulse-profile modeling (PPM) of thermal X-ray emission from rotation-powered millisecond pulsars enables simultaneous constraints on the mass , radius , and hence the equation of state of cold, dense matter. However, Bayesian PPM has faced a hard accuracy-speed bottleneck: current production resolutions used to keep inference tractable can under-resolve extreme hotspot geometries and bias the waveform computation, whereas the higher resolutions that remove this bias push forward models to minutes per evaluation, making inference impractical. We break this trade-off with, to our knowledge, the first public GPU-accelerated X-ray PPM framework that matches established benchmarks to within relative accuracy even for extreme geometries, while collapsing minutes-long high-fidelity computations to -- ms on an RTX 4080 (-- speedups), enabling posterior exploration at resolutions and complexities previously out of reach. We further uncover a bias near the interpolation boundaries of atmosphere lookup tables, demonstrate it with two diagnostic tests, and counter it with a mixed-order interpolator. Together, these advances enlarge the feasible hotspot model space and reduce key systematics in PPM, strengthening inferences for current and future X-ray missions.

Paper Structure

This paper contains 27 sections, 98 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Model framework and computational recipe
  3. Geometrical Setup
  4. Spacetime and Oblate--Schwarzschild (OS)
  5. Photon Propagation: Light Bending and Time Delays
  6. Rotation: Relativistic Doppler Effect
  7. Surface Emission and Atmospheric Beaming
  8. Surface Temperature Maps and Surface Discretizations
  9. Interstellar Absorption and Instrument Response
  10. Computation Recipe
  11. Surface discretization
  12. Lensing table generation
  13. Number flux calculation
  14. Interstellar absorption and instrument response
  15. GPU Implementation
  16. ...and 12 more sections

Figures (9)

  • Figure 1: Demonstration of the geometric setup used in our X-ray PPM forward computation, shown in the lab-frame coordinate system $\hat{x}-\hat{y}-\hat{z}$. A surface patch (orange square) on the stellar surface at colatitude $\theta$ and rotational phase $\phi$, viewed along the observer’s line-of-sight unit vector $\hat{k}$. Definitions of all unit vectors are given in Table \ref{['tab:vectors']}, and angle conventions are summarized in Table \ref{['tab:angles']}.
  • Figure 2: Monochromatic ($E_{\rm obs}=1~\mathrm{keV}$) waveforms under the oblate-Schwarzschild (OS) approximation for the OS1 test suite (subset shown), computed with our CPU (blue) and GPU (orange) implementations. The theoretical benchmarks from Bogdanov_19 are shown in black. Residuals are taken with respect to the benchmark; red dashed lines indicate $\pm 0.1\%$. Test-case definitions follow Bogdanov_19.
  • Figure 3: Demonstration of two geometric test cases for atmosphere interpolation. Test 1: A point-like hotspot on the southern hemisphere; the line of sight is aligned with the spin axis and lies above the rotational north pole. Test 2: A point-like hotspot at the south pole; the observer is located in the northern hemisphere.
  • Figure 4: Simulated phase--energy--resolved X-ray pulse profiles for Interpolation Test 1, computed using four interpolation schemes applied to the NSX tables: (a) cubic interpolation in all dimensions; (b) as in (a), with negative intensities clipped to zero; (c) cubic interpolation in all dimensions with linear interpolation enforced at table boundaries; (d) the baseline configuration used in this work, cubic along two axes and linear along the remaining two, with linear interpolation enforced at table boundaries.
  • Figure 5: Simulated phase--energy--resolved X-ray pulse profiles for Interpolation Test 2, computed using four interpolation schemes applied to the NSX tables: (a) cubic interpolation in all dimensions; (b) as in (a), with negative intensities clipped to zero; (c) cubic interpolation in all dimensions with linear interpolation enforced at table boundaries; (d) the baseline configuration used in this work, cubic along two axes and linear along the remaining two, with linear interpolation enforced at table boundaries.
  • ...and 4 more figures