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 the main idea about possible features of the evolution of scale factor in the slow-roll regime of inflation.

Chaotic Inflation (Inflation with Power Law Potential)

The chaotic inflation, or the inflation with high field, is considered as a rule with the power law potentials of the form $$V=g\varphi^n,$$ where $g$ is a dimensional constant of interaction: $$[g]=(\mbox{mass})^{4-n}$$ It should be noted that the slow-roll conditions for the given potential are always satisfied for sufficiently high values of the inflaton field $$\varphi\gg\frac{nM_{Pl}}{4\sqrt{3\pi}},$$ therefore the slow-roll takes place at field values which are great compared to Planck units.}

Problem 1

Consider inflation with simple power law potential

     $$V=g\phi^n,$$

and show that there is wide range of scalar field values where classical Einstein equations are applicable and the slow-roll regime is realized too. Assume that the interaction constant $g$ is sufficiently small in Planck units.

Problem 2

Estimate the total duration of chaotic inflation in the case of power law potentials of second and forth order.

Problem 3

Express the slow-roll parameters for power law potentials in terms of $e$-folding number $N_e$ till the end of inflation.

Problem 4

Obtain the number $N$ of $e$-fold increase of the scale factor in the model \[V\left( \varphi \right) = \lambda \varphi ^4 \quad \left( {\lambda = 10^{ - 10} } \right).\]

solution

Use the result of the previous problem to obtain for the power-law potential $V\left( \varphi \right) = \lambda \varphi ^n $ and $$ \varphi _i \gg \varphi _e $$ $$ N \sim \int_{\varphi _e }^{\varphi _i } \frac{\varphi d\varphi }{M_{Pl}^2 } \sim \frac{\varphi _i^2 }{M_{Pl}^2 }. $$ As $V(\varphi _i ) \sim M_{Pl}^4 $, in the considered model $\varphi _i \sim \lambda ^{ - 1/4} M_{Pl} $ and therefore $$ N \sim 10^5. $$

Problem 5

Express the slow-roll parameters for power law potentials in terms of $e$-folding number $N_e$ till the end of inflation.