标题： 参数不稳定性的解析研究：膨胀宇宙中振荡标量场的不稳定特性 显示英文标题

标题： Analytical study of the parametric instability of an oscillating scalar field in an expanding universe

摘要： 我们研究了膨胀宇宙中围绕其有效势最小值振荡的惯性标量场扰动的动力学。假设哈勃参数与振荡频率之比很小，我们应用了快速运动和慢速运动分离的技术。将振荡相位和能量密度视为快慢变量，我们推导出了描述涨落模的Hill方程，在该方程中能量密度被视为缓慢变化的参数。我们发展了一种求解这类方程的一般摄动方法，该方法基于Floquet理论以及在“冻结”参数解附近的渐近展开。作为一个例子，我们考虑了 $\phi ^{2}-\phi ^{4}$ 势，并构造了相应Lamé方程的近似解。所得的解与直接数值积分的结果非常吻合。 显示英文摘要

摘要： We investigate the dynamics of the perturbations of the inflaton scalar field oscillating around a minimum of its effective potential in an expanding universe. With the assumption of smallness of the ratio of the Hubble parameter to the oscillation frequency we apply the technique of separation of fast and slow motions. Considering the oscillation phase and the energy density as fast and slow variables we derive the Hill equation for the fluctuation modes in which the energy density is treated as a slowly varying parameter. We develop a general perturbative approach to solving the equations of this type, which is based on the Floquet theory and asymptotic expansions in the vicinity of the solutions with the "frozen" parameters. As an example, we consider the $\phi ^{2}-\phi ^{4}$ potential and construct the approximate solutions of the corresponding Lam\'{e} equation. The obtained solutions are found to be in a good agreement with the results of the direct numerical integration.