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If a scalar field (eg. inflaton field) starts with a high potential.
Does the potential $V(φ)$ of the scalar field decrease with the expansion of space?
If it doesn’t decrease, would it mean that extra energy is created to fill in the additional space so that its potential $V(φ)$ remains the same throughout the space?
I’m a layman so a non-mathematical answer would be appreciated.
You're probably used to a Minkowski-space Lagrangian density such as $\frac{1}{2}\eta^{\mu\nu}\partial_\mu\phi\partial_\nu\phi -V(\phi)$. In curved spacetime, this generalises to $\sqrt{|g|}(\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi -V(\phi))$ with $g:=\det g_{\mu\nu}$.
We model the expansion of 3-dimensional space with $ds^2=g_{\mu\nu}d^\mu dx^\nu=dt^2-a^2(t)d\mathbf{x}^2$ (I won't go into the form of $d\mathbf{x}$ for now), so $\sqrt{|g|}\propto a^3$.
