Different Models of Inflation

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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 $$V\left( \varphi \right) = \lambda \varphi ^4 \quad \left( {\lambda = 10^{ - 10} } \right).$$

Problem 5

Estimate the range of scalar field values corresponding to the inflation epoch in the model $$V(\varphi ) = \lambda \varphi ^4 \left( {\lambda \ll 1} \right).$$

Problem 6

Show that the classical analysis of the evolution of the Universe is applicable for the scalar field value $\varphi\gg M_{Pl}$, which allows the inflation to start.

Problem 7

Find the time dependence for the scale factor in the inflation regime for potential $(1/n)\lambda\varphi^n$, assuming $\varphi\gg M_{Pl}$.

Problem 8

The inflation conditions definitely break down near the minimum of the inflaton potential and the Universe leaves the inflation regime. The scalar field starts to oscillate near the minimum. Assuming that the oscillations' period is much smaller than the cosmological time scale, determine the effective state equation near the minimum of the inflaton potential.

Problem 9

Show that effective state equation for the scalar field, obtained in the previous problem for potential $V\propto\varphi^n$, in the case $n=2$ corresponds to non-relativistic matter and for $n=4$ - to the ultra-relativistic component (radiation).

Problem 10

Obtain the time dependence of scalar field near the minimum of the potential.

Problem 11

Find the energy-momentum tensor of a homogeneous scalar field in the regime of fast oscillations near the potential's minimum.

Problem 12

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

Problem 13

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

Problem 14

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

Problem 15

Estimate the range of scalar field values corresponding to the inflation epoch in the model $$V(\varphi ) = \lambda \varphi ^4 \left( {\lambda \ll 1} \right).$$

Problem 16

Show that the classical analysis of the evolution of the Universe is applicable for the scalar field value $\varphi\gg M_{Pl}$, which allows the inflation to start.

Problem 17

Find the time dependence for the scale factor in the inflation regime for potential $(1/n)\lambda\varphi^n$, assuming $\varphi\gg M_{Pl}$.

Problem 18

The inflation conditions definitely break down near the minimum of the inflaton potential and the Universe leaves the inflation regime. The scalar field starts to oscillate near the minimum. Assuming that the oscillations' period is much smaller than the cosmological time scale, determine the effective state equation near the minimum of the inflaton potential.

Problem 19

Show that effective state equation for the scalar field, obtained in the previous problem for potential $V\propto\varphi^n$, in the case $n=2$ corresponds to non-relativistic matter and for $n=4$ - to the ultra-relativistic component (radiation).

Problem 20

Obtain the time dependence of scalar field near the minimum of the potential.