Time evolution of the bubble radius for a constant (magenta curves) and exponentially growing (green curves) source in the energy-driven limit. The solid (dashed) lines show the radius evolution considering (neglecting) the effects of gravity. Solid curves have been obtained by numerically solving Eq. (29). The assumed parameters for the exponential input energy source Lw(t) = Lw,0et/tSal are the same as those in Fig. 2. The constant source has Lw(t) = Lw,0. The vertical dotted lines are as in Fig. 2. The density of the ambient medium is assumed to be ρgas = 10-25 g cm-3, i.e. about 1000 times the average of the Universe at z = 6. The power-law asymptotic behaviour at early (R ∝ t3 / 5, magenta dashed line) and at late (R ∝ t1 / 5, magenta dotted line) times can be appreciated for the constant energy source. For the exponentially growing case, such a density field reduces the size of the bubble radius by ~ 25% at late times.
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