AI 中文总结
研究κ变形非对易时空中的暴胀原初微扰，利用κ变形星积形式构建作用量、推导方程并求解，计算功率谱和谱指数，通过贝叶斯 MCMC 分析约束κ变形长度尺度，为量子引力现象学提供潜在窗口。
AI 中文摘要
我们研究κ-Minkowski 非对易时空中的暴胀原初微扰，它是由量子引力情景激发的规范时空的李代数型变形。利用κ变形星积形式，我们构建曲率微扰的双线性作用量，推导κ变形的 Mukhanov-Sasaki 方程并得到微扰解。进而计算原初功率谱和谱指数，表明功率谱的主导阶修正诱导出与(ln k)^2 成比例的尺度依赖项。谱指数也呈现明确的 ln k 依赖性。我们还用 ACT DR6 数据进行贝叶斯 MCMC 分析，在 1σ置信水平将κ变形长度尺度约束为λ=6.32^{+6.00}_{-4.30}×10^{-30}m，比普朗克尺度大约大四阶，证明κ变形时空通过精确宇宙学为量子引力现象学提供了一个潜在窗口。
英文摘要
We study the inflationary primordial perturbations in $κ$-Minkowski non-commutative space-time, a Lie-algebraic type deformation of canonical space-time motivated by quantum gravity scenarios. Employing the $κ$-deformed star product formalism, we construct the bilinear action for curvature perturbations and derive the $κ$-deformed Mukhanov-Sasaki equation and obtain the perturbative solutions. Further, we compute the primordial power spectrum and spectral index, showing that the leading order corrections to the power spectrum induces scale-dependent term proportional to $(\ln k)^2$. The spectral index also exhibits an explicit $\ln k$ dependence, which persists even when the slow-roll parameters are constant. We also perform a Bayesian MCMC analysis using ACT DR6 data and constrain the $κ$-deformation length scale to $λ=6.32^{+6.00}_{-4.30}\times10^{-30}m$ at $1σ$ CL, approximately four orders of magnitude larger than the Planck scale, demonstrating that the $κ$-deformed space-time offers a potential window into quantum gravity phenomenology through precision cosmology.
