AI 中文总结
本文研究将切比雪夫插值应用于爱因斯坦-玻尔兹曼代码，其在更少节点数下精度更高、速度更快，已在SymBoltz中实现并可避免非整数ℓ插值的问题。
AI 中文摘要
爱因斯坦-玻尔兹曼代码用于计算宇宙学模型的理论预测，且在很大程度上依赖于对其自变量的插值：时间τ、波数k和多极矩ℓ。本文对切比雪夫多项式插值给出实用总结，该插值对光滑函数收敛迅速，因此与无近似的爱因斯坦-玻尔兹曼代码天然适配。通过在k和ℓ的切比雪夫节点处求解扰动和视线积分，我们表明切比雪夫多项式用更少的显式解实现了比传统三次样条更高的精度。在物质和宇宙微波背景（CMB）的一组示例谱上，我们发现使用相同数量的点时，插值误差降低了多达四个数量级。对于同时在k和ℓ上插值的典型CMB温度谱，切比雪夫多项式仅用每个变量50-80个点就收敛到10⁻⁴-10⁻⁵的相对误差，而三次样条用200个点才达到10⁻⁴的误差，这意味着速度提升了2.5倍至4倍。确切的改进程度取决于目标函数，通常在高精度水平下更为显著。标准切比雪夫ℓ插值需要推广到非整数ℓ的视线积分，但我们展示了一种通过将节点取整为整数来避免此问题的方法。切比雪夫插值已在SymBoltz中实现，可在此https URL获取。
英文摘要
Einstein-Boltzmann codes compute theoretical predictions of cosmological models and rely heavily on interpolation in their independent variables: time $τ$, wavenumber $k$ and multipole $\ell$. We give a practical summary of interpolation with Chebyshev polynomials, which converges rapidly for smooth functions and thus pairs naturally with approximation-free Einstein-Boltzmann codes. By solving the perturbations and line-of-sight integrals at Chebyshev nodes in $k$ and $\ell$, we show that Chebyshev polynomials interpolate to higher precision than traditional cubic splines from fewer explicit solutions. On a set of example spectra for matter and the cosmic microwave background (CMB), we find up to four orders of magnitude lower interpolation error using the same number of points. For a typical CMB temperature spectrum computed with interpolation in both $k$ and $\ell$, Chebyshev polynomials converge to $10^{-4}$-$10^{-5}$ relative error with only 50-80 points per variable, while cubic splines approach $10^{-4}$ error with 200 points, translating to a $2.5\times$-$4\times$ speedup. The exact improvement depends on the target function and is generally more dramatic at high precision levels. Standard Chebyshev $\ell$-interpolation needs line-of-sight integrals generalized to non-integer $\ell$, but we show a way to avoid this by rounding the nodes to integers. Chebyshev interpolation is implemented in SymBoltz, which is available at https://github.com/hersle/SymBoltz.jl.
Comments8 pages, 7 figures, submitted to A&A, SymBoltz is available at https://github.com/hersle/SymBoltz.jl
