标题： 一个（半）精确的哈密顿量用于曲率扰动$ζ$ 显示英文标题

标题： A (semi)-exact Hamiltonian for the curvature perturbation $ζ$

摘要： 广义相对论中的总哈密顿量包含了第一类哈密顿约束和动量约束，它弱场下消失。然而，当作用量在经典解附近展开时（如单标量场膨胀模型的情况），会出现非零的哈密顿量以及额外的第一类约束；但此时理论在涨落场的数量上变为微扰论。我们证明可以重新组织这个展开并精确求解哈密顿约束，从而得到一个显式的全阶作用量。另一方面，在张量模（$\gamma_{ij}$）的情况下，可以通过仍保持曲率扰动（$\zeta$）依赖关系精确地以微扰方式求解动量约束。这样，在规范固定后，可以获得一个半精确的哈密顿量$\zeta$，它只受到与张量模（h）相互作用的修正（因此当张量扰动设为零时，哈密顿量变为精确）。运动方程清楚地展示了$\zeta$的演化何时涉及对数时间依赖性，这是文献中争论的一个微妙点。我们讨论了长波长和晚期时间极限，并得到了一些简单但非平凡的经典零模解$\zeta$。 显示英文摘要

摘要： The total Hamiltonian in general relativity, which involves the first class Hamiltonian and momentum constraints, weakly vanishes. However, when the action is expanded around a classical solution as in the case of a single scalar field inflationary model, there appears a non-vanishing Hamiltonian and additional first class constraints; but this time the theory becomes perturbative in the number of fluctuation fields. We show that one can reorganize this expansion and solve the Hamiltonian constraint exactly, which yield an explicit all order action. On the other hand, the momentum constraint can be solved perturbatively in the tensor modes $\gamma_{ij}$ by still keeping the curvature perturbation $\zeta$ dependence exact. In this way, after gauge fixing, one can obtain a semi-exact Hamiltonian for $\zeta$ which only gets corrections from the interactions with the tensor modes (hence the Hamiltonian becomes exact when the tensor perturbations set to zero). The equations of motion clearly exhibit when the evolution of $\zeta$ involves a logarithmic time dependence, which is a subtle point that has been debated in the literature. We discuss the long wavelength and late time limits, and obtain some simple but non-trivial classical solutions of the $\zeta$ zero-mode.