We obtain an explicit formula for the full expansion of the spectral action on Robertson–Walker spacetimes, expressed in terms of Bell polynomials, using Brownian bridge integrals and the Feynman–Kac formula. We then apply this result to the case of multifractal Packed Swiss Cheese Cosmology models obtained from an arrangement of Robertson–Walker spacetimes along an Apollonian sphere packing. Using Mellin transforms, we show that the asymptotic expansion of the spectral action contains the same terms as in the case of a single Robertson–Walker spacetime, but with zeta-regularized coefficients, given by values at integers of the zeta function of the fractal string of the radii of the sphere packing, as well as additional log-periodic correction terms arising from the poles (off the real line) of this zeta function.

中文翻译：

---

多重分形罗伯逊-沃克宇宙论的光谱引力模型中的贝尔多项式和布朗桥

我们使用布朗桥积分和费曼-卡克公式获得了一个明确的公式，用于在罗伯逊-沃克时空上的频谱作用的完全展开，以贝尔多项式表示。然后，我们将此结果应用于从阿波罗尼球堆积中的罗伯逊-沃克时空排列中获得的多重分形包装瑞士奶酪宇宙学模型的情况。使用Mellin变换，我们表明频谱作用的渐近展开包含与单个Robertson-Walker时空相同的项，但是具有zeta正规系数，由分形字符串zeta函数整数的值给出球体填充半径的平方，以及此zeta函数的极点（偏离实线）产生的其他对数周期校正项。