We prove existence and uniqueness of solutions of the semiclassical Einstein equation in flat cosmological spacetimes driven by a quantum massive scalar field with arbitrary coupling to the scalar curvature. In the semiclassical approximation, the backreaction of matter to curvature is taken into account by equating the Einstein tensor to the expectation values of the stress-energy tensor in a suitable state. We impose initial conditions for the scale factor at finite time, and we show that a regular state for the quantum matter compatible with these initial conditions can be chosen. Contributions with derivative of the coefficient of the metric higher than the second are present in the expectation values of the stress-energy tensor and the term with the highest derivative appears in a non-local form. This fact forbids a direct analysis of the semiclassical equation, and in particular, standard recursive approaches to approximate the solution fail to converge. In this paper, we show that, after partial integration of the semiclassical Einstein equation in cosmology, the non-local highest derivative appears in the expectation values of the stress-energy tensor through the application of a linear unbounded operator which does not depend on the details of the chosen state. We prove that an inversion formula for this operator can be found, furthermore, the inverse happens to be more regular than the direct operator and it has the form of a retarded product, hence, causality is respected. The found inversion formula applied to the traced Einstein equation has thus the form of a fixed point equation. The proof of local existence and uniqueness of the solution of the semiclassical Einstein equation is then obtained applying the Banach fixed point theorem.

我们证明了半经典爱因斯坦方程在平坦宇宙时空中的解的存在性和唯一性，该方程由与标量曲率任意耦合的量子质量标量场驱动。在半经典近似中，通过将爱因斯坦张量等同于合适状态下应力-能量张量的期望值来考虑物质对曲率的逆反应。我们在有限时间内为比例因子施加初始​​条件，并且我们表明可以选择与这些初始条件兼容的量子物质的规则状态。度量系数的导数高于第二个的贡献存在于应力-能量张量的期望值中，并且具有最高导数的项以非局部形式出现。这一事实禁止对半经典方程进行直接分析，特别是，逼近解的标准递归方法无法收敛。在本文中，我们表明，在对宇宙学中的半经典爱因斯坦方程进行偏积分后，非局部最高导数出现在应力-能量张量的期望值中，通过应用线性无界算子，该算子不依赖于所选状态的详细信息。我们证明可以找到该算子的反演公式，而且，逆恰好比直接算子更规则，并且具有延迟乘积的形式，因此尊重因果关系。发现的反演公式应用于跟踪的爱因斯坦方程，因此具有不动点方程的形式。