Different Models of Inflation
There is a number of inflationary models. All of them deal with potentials of scalar fields which realize the slow-roll regime during sufficiently long period of evolution, then the inflation terminates and Universe enters the hot stage. It is worth noting that the models considered below are among the simplest ones and they do not exhaust all the possibilities, however they give main idea about possible features of the evolution of inflaton and scale factor in the slow-roll regime.
Problem 1
A scalar field $\varphi(\vec r,t)$ in a potential $V(\varphi)$ on flat background is described by Lagrangian \[L=\frac12\left(\dot\varphi^2-\nabla\varphi\cdot\nabla\varphi\right)-V(\varphi)\] Obtain the equation of motion (evolution) for this field from the least action principle.
Variation of the action with respect to $\varphi $ reads $$\delta S = \int {d^4 x\left[ {\frac{\partial L}{\partial \varphi }\delta \varphi + \frac{\partial L}{\partial \left( {\partial _\mu \varphi } \right)}\delta \left( {\partial _\mu \varphi } \right)} \right]} = 0. $$ Integrate the second term by parts and use the boundary condition $\delta \varphi = 0$ фе infinity to obtain $$ \frac{\partial L}{\begin{array}{l} \partial \varphi \\ \\ \end{array}} - \frac{\partial }{\partial x^\mu }\left( {\frac{\partial L}{\partial \left( {\partial _\mu \varphi } \right)}} \right) = 0. $$ For the considered Lagrangian one obtains $$ \ddot \varphi - \nabla ^2 \varphi + V'(\varphi ) = 0. $$ In the case of spatially homogeneous field $$ \ddot \varphi + V'(\varphi ) = 0. $$
