Minimally coupled massless scalar field (source code)

= Minimally coupled massless scalar field
{title2=$S=-\tfrac12\int d^dx\sqrt{-g}\,g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$}

The displayed <action functional> contains no <mass> term and no $R\phi^2$ curvature coupling. Its <Euler-Lagrange field equation> is $\partial_\mu(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\phi)=0$. In a flat <FLRW metric>, a <Fourier mode> obeys $\phi_k''+2(a'/a)\phi_k'+k^2\phi_k=0$ in four dimensions. The expansion can therefore freeze these modes; the curvature term of a <conformally coupled scalar field> changes that conclusion.