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Exponentially expanding space-times play a central role in contemporary cosmology, most importantly in the theory of inflation and in the Dark Energy driven expansion in the late universe. In this work, we give a complete list of de Sitter solutions of the semiclassical Einstein equation (SCE), where classical gravity is coupled to the expected value of a renormalized stress-energy tensor of a free quantum field in the Bunch-Davies state. To achieve this, we explicitly determine the stress-energy tensor associated with the Bunch-Davies state using the recently proposed `moment approach' on the cosmological coordinate patch of de Sitter space. From the energy component of the SCE we thus obtain an analytic consistency equation for the model's parameters which has to be fulfilled by solutions to the SCE. Using this equation we then investigate the number of solutions and the structure of the solution set in dependency on the coupling parameter of the quantum field to the scalar curvature and renormalization constants using analytic arguments in combination with numerical evidence. We also identify parameter sets where multiple expansion rates separated by several orders of magnitude are possible. Potentially for such parameter settings, a fast (semi-stable) expansion in the early universe could be compatible with a late time `Dark Energy-like' behavior of the universe.