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This note by Alan Guth says that
The false vacuum, however, cannot rapidly lower its energy density, so the energy density remains constant and the total energy increases. Since energy is conserved, the extra energy must be supplied by the agent that pulled on the piston.
Why is it that the false vacuum cannot lower its energy rapidly?
Do we know of a substance which behaves in this unusual fashion? In short, why is the energy density doesn't dilute with expansion?
From continuity equation ($\varepsilon$ - energy density, $p$ - pressure, $H$ - Hubble parametre) \begin{equation} \dot{\varepsilon}=-3H(p+\varepsilon) \end{equation} you can see that for a substance with negative pressure $p=-\varepsilon$, above equation gives $\varepsilon=const$, with or without expansion. Such equation of state is exactly what we need from the inflaton field during the inflation. Since pressure and energy density are defined as $$ \varepsilon=\frac{1}{2}\dot{\phi}^2+V(\phi)~,~~p=\frac{1}{2}\dot{\phi}^2-V(\phi)~, $$ when potential energy dominates, $V(\phi)\gg\frac{1}{2}\dot{\phi}^2$, we have $p\approx-\varepsilon$.
For further details see "Physical foundations of Cosmology" (inspire-hep link) by Mukhanov, or Baumann's lecture notes (pdf link, on pages 19-20 derivation of the continuity equation is given).
