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Slow-roll approximation

{title2=$\phi=\phi_i e^{-M_P\sqrt{\lambda/(6\pi)}(t-t_i)}$}

For $V(\phi)=\lambda\phi^4/4$ with $\lambda>0$, use <natural units> and the unreduced <Planck mass> $M_P=G^{-1/2}$. The <slow-roll equations> give $H\simeq\sqrt{2\pi\lambda/3}\,\phi^2/M_P$ and $\dot\phi\simeq-M_P\sqrt{\lambda/(6\pi)}\,\phi$. Integrating gives the displayed exponential field decay, valid on either sign branch, and
$$
\frac{a(t)}{a_i}\simeq\exp\left[\frac{\pi}{M_P^2}(\phi_i^2-\phi(t)^2)\right].
$$
The <potential slow-roll parameters> require $\phi^2\gg M_P^2$. The expression is valid only during <slow-roll inflation>; its formal late-time saturation does not describe the subsequent <reheating> or radiation era.

Quartic-potential slow-roll solution

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 61 / 3 / Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 15B / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 14A / e / Solution

Quartic-potential slow-roll solution

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Quartic-potential slow-roll solution

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