宇宙多胞形与宇宙光态波函数
Cosmological Polytopes and the Wavefuncton of the Universe for Light States
摘要：
我们扩大调查的宇宙的后期波函数的结构，一类玩具模型的标量与时间相关的质量和多项式耦合，其中包含一般大规模标量FRW宇宙学。我们关联一个通用的被积函数到每个费曼图有助于宇宙的波函数。对于一定的（轻）质量，这样的被积函数满足递归关系，涉及微分算子，连接不同质量的状态，并有，作为种子，无质量标量（它描述了一个保形耦合标量作为一个特殊情况）。我们表明，它是一个退化的限制的规范形式的一个概括的宇宙多面体描述的波函数的无质量标量。有趣的是，平坦空间的散射振幅表现为更高的余维面：当总能量取为零时，它编码洛朗展开中的前导项，面的余维提供了总能量极点的阶数。宇宙波函数的其他奇点也有同样的面与罗朗展开系数之间的联系，它们都与平空间过程有关。当退化极限被采用时，多面体的标准形的一些奇点相互坍缩，产生高阶极点。最后，我们把质量看作微扰耦合，证明了在共形耦合态附近进行微扰展开时，质量两点耦合图对波函数的贡献可以用宇宙学多面体标准形的退化极限来表示;或作为本文所介绍的宇宙学多面体扩张的标准型的双重退化极限，如果微扰展开是围绕最小耦合态进行的。
英文摘要：
We extend the investigation of the structure of the late-time wavefunction of the universe to a class of toy models of scalars with time-dependent masses and polynomial couplings, which contains general massive scalars in FRW cosmologies. We associate a universal integrand to each Feynman diagram contributing to the wavefunction of the universe. For certain (light) masses, such an integrand satisfies recursion relations involving differential operators, connecting states with different masses and having, as a seed, the massless scalar (which describes a conformally coupled scalar as a special case). We show that it is a degenerate limit of the canonical form of a generalisation of the cosmological polytopes describing the wavefunction for massless scalars. Intriguingly, the flat-space scattering amplitude appears as a higher codimension face: it is encoding the leading term in the Laurent expansion as the total energy is taken to zero, with the codimension of the face providing the order of the total energy pole. The same connection between the other faces and the Laurent expansion coefficients holds for the other singularities of the wavefunction of the universe, all of them connectable to flat-space processes. As the degenerate limit is taken, some of the singularities of the canonical form of the polytope collapse onto each other generating higher order poles. Finally, we consider the mass as a perturbative coupling, showing that the contribution to the wavefunction coming from graphs with mass two-point couplings can be identified with a degenerate limit of the canonical form of the cosmological polytope, if the perturbative expansion is done around the conformally coupled state; or as double degenerate limit of the canonical form of the extension of the cosmological polytopes introduced in the present paper, if the perturbative expansion is done around minimally coupled states.
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