We calculate the time evolution of the expectation value of the energy-momentum tensor for a minimally coupled massless scalar field in cosmological spacetimes, with an application to dark energy in mind. We first study the evolution from inflation until the present, fixing the Bunch-Davies initial condition. The energy density of a quantum field evolves as rho similar to 3(HIH)(2)/32 pi(2) in the matter-dominated (MD) period, where HI and H are the Hubble parameters during inflation and at each moment. Its equation of state, w = rho/p, changes from a negative value to w = 1/3 in the radiation-dominated (RD) period, and from 1= 3 to w = 0 in the MD period. We then consider possible effects of a Planckian universe, which may have existed before inflation, by assuming there was another inflation with the Hubble parameterH(P)(> H-I). In this case, modes with wavelengths longer than the current horizon radius are mainly amplified, and the energy density of a quantum field grows with time as rho similar to (a/a(0))(HPH)(2)/32 in the MD period, where a and a(0) are the scale factors at each time and at present. Hence, ifHP is of the order of the Planck scale M-P,M- rho becomes comparable to the critical density 3(MPH)(2) at the present time. The contribution to rho from the long wavelength fluctuations generated before the ordinary inflation has w = -1= 3 in the free field approximation. We mention a possibility that interactions further amplify the energy density and change the equation of state.