Dark energy and dark matter are two of the biggest mysteries of modern cosmology, and our understanding of their fundamental nature is incomplete. Many parametrizations of couplings between the two in the continuity equation have been studied in the literature, and observational data from the growth of perturbations can constrain these parametrizations. Assuming standard general relativity with a simple Yukawa-type coupling between dark energy and dark matter fields in the Lagrangian, we use the Boltzmann equation to analytically express and calculate the interaction kernel Q in the continuity equation and compare it to that of a typical parametrization. We arrive at a comparably very small result, as expected. Since the interaction is a function of the dark matter mass, other observational data sets can be used to constrain the mass. This calculation can be modified to account for other couplings of the dark energy and dark matter fields. This calculation required obtaining a distribution function for dark energy that leads to an equation of state parameter that is negative, which neither Bose–Einstein nor Fermi–Dirac statistics can supply, and this is the main result of this paper. Treating dark energy as a quantum scalar field, we use adiabatic subtraction to obtain a finite analytic approximation for its distribution function that assumes the FLRW metric and nothing more.

中文翻译：

---

从玻尔兹曼方程推导出连续性方程中的暗物质-暗能量相互作用项

暗能量和暗物质是现代宇宙学的两大谜团，我们对其基本性质的理解并不完整。文献中已经研究了连续性方程中两者之间耦合的许多参数化，并且来自扰动增长的观测数据可以约束这些参数化。假设标准广义相对论在拉格朗日中暗能量和暗物质场之间存在简单的 Yukawa 型耦合，我们使用玻尔兹曼方程来解析表达和计算相互作用核问在连续性方程中，并将其与典型的参数化进行比较。正如预期的那样，我们得到了一个相对非常小的结果。由于相互作用是暗物质质量的函数，因此可以使用其他观测数据集来约束质量。可以修改此计算以考虑暗能量和暗物质场的其他耦合。该计算需要获得一个暗能量分布函数，该函数导致状态方程参数为负，而玻色-爱因斯坦和费米-狄拉克统计都无法提供，这是本文的主要结果。将暗能量视为量子标量场，我们使用绝热减法来获得其分布函数的有限解析近似，该分布函数假设 FLRW 度量，仅此而已。