SQUARE45

Inflation is modeled by the action

S = \int d^4 x \sqrt{-g} \left( \frac{1}{2} M_{Pl}^2 R - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi) \right)

. Assuming a slow-roll regime, the equation of motion for the inflaton field

\phi

is approximated by the Klein-Gordon equation in a curved background:\n

\ddot{\phi} + 3H \dot{\phi} + V'(\phi) = 0

The slow-roll conditions require

\frac{1}{2} M_{Pl}^2 \left(\frac{V'(\phi)}{V(\phi)}\right)^2 \ll 1

and

M_{Pl}^2 \left|\frac{V''(\phi)}{V(\phi)}\right| \ll 1

, ensuring

\ddot{\phi} \ll 3H \dot{\phi}

Inflationary Epoch