Evolution of vacuum fluctuations generated during and before inflation

Abstract

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 $\rho \sim 3(H_{I}H{)}^2/32{\pi }^2$ in the matter-dominated (MD) period, where $H_I$ 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 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 parameter $H_P(>H_I)$. 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 $\rho \sim (a/a_0)(H_{P}H{)}^2/32$ in the MD period, where $a$ and $a_0$ are the scale factors at each time and at present. Hence, if $H_P$ is of the order of the Planck scale $M_P$, $\rho$ becomes comparable to the critical density $3(M_{P}H{)}^2$ 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.

54 pages, 12 figures; v2: references added; v3:version publised in PRD

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• Observational signatures of dark energy produced in an ancestor vacuum: Forecast for galaxy surveys