The time-dependent cosmological term arises from the energy-momentum tensor calculated in a state different from the ground state. We discuss the expectation value of the energy-momentum tensor on the right hand side of Einstein equations in various (approximate) quantum pure as well as mixed states. We apply the classical slow-roll field evolution as well as the Starobinsky and warm inflation stochastic equations in order to calculate the expectation value. We show that, in the state concentrated at the local maximum of the double-well potential, the expectation value is decreasing exponentially. We confirm the descent of the expectation value in the stochastic inflation model. We calculate the cosmological constant Λ at large time as the expectation value of the energy density with respect to the stationary probability distribution. We show that $\Lambda \simeq {\gamma }^{\frac{4}{3}}$ where γ is the thermal dissipation rate.

中文翻译：

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半经典爱因斯坦方程：下降到基态

时间相关的宇宙学术语来自于在不同于基态的状态下计算出的能量动量张量。我们讨论了各种（近似）量子纯态和混合态中爱因斯坦方程右侧的能量动量张量的期望值。为了计算期望值，我们应用经典的慢滚场演化以及Starobinsky和暖通货膨胀随机方程。我们表明，在集中在双阱势的局部最大值的状态下，期望值呈指数下降。我们确认了随机通胀模型中期望值的下降。我们将宇宙常数Λ长时间作为能量密度相对于平稳概率分布的期望值进行计算。我们证明 $\Lambda \simeq {\gamma }^{\frac{4}{3}}$ 其中，γ是散热率。