Inflation of small true vacuum bubble by quantization of Einstein-Hilbert action

We study the quantization of the Einstein-Hilbert action for a small true vacuum bubble without matter or scalar field. The quantization of action induces an extra term of potential called quantum potential in Hamilton-Jacobi equation, which gives expanding solutions including the exponential expansion solutions of the scalar factor $a$ for the bubble. We show that exponential expansion of the bubble continues with a short period (about a Planck time $t_p$), no matter whether the bubble is closed, flat or open. The exponential expansion ends spontaneously when the bubble becomes large, i.e., the scalar factor $a$ of the bubble approaches a Planck length $l_p$. We show that it is quantum potential of the small true vacuum bubble that plays the role of the scalar field potential suggested in the slow-roll inflation model. With the picture of quantum tunneling, we calculate particle creation rate during inflation, which shows that particles created by inflation have the capability of reheating the universe.