Difference between revisions of "Different Models of Inflation"

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(The Novel Inflation (the Inflation Near Minimum of the Potntial))
(The Novel Inflation (the Inflation Near Minimum of the Potntial))
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where $n\geq 3$. The most often considered potential is the forth order one
 
where $n\geq 3$. The most often considered potential is the forth order one
 
$$V(\phi)=V_0-\frac{\lambda}{4}\phi^4.$$
 
$$V(\phi)=V_0-\frac{\lambda}{4}\phi^4.$$
The inflation model with such potential is called the {\bf novel inflation}.
+
The inflation model with such potential is called novel inflation.
  
 
<div id="inf35"></div>
 
<div id="inf35"></div>

Revision as of 09:13, 3 September 2013


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

Check whether the chaotic inflation model agrees with the experimental data, which give the value $r=\mathcal{P}_\mathcal{T}/\mathcal{P}_\mathcal{R}<0.2$ for the tensor perturbation amplitude and $n_s=0.94\div 0.99$ for the spectrum slope. For the inflaton potential take $V(\varphi)=m^2\varphi^2/2$.

Problem 13

Consider chaotic inflation with potential $V(\varphi)=m^2\varphi^2/2$ and obtain the difference between the spectrum slopes for the waves corresponding to cosmological perturbations of sizes $100\ kpc$ and $10\ Gpc$.

Problem 14

What is the difference between the chaotic inflation model by Linde and its original version by Starobinsky--Guth?

The Novel Inflation (the Inflation Near Minimum of the Potntial)

As the reader might notice in the previous subsection, the chaotic inflation requires to include the super-Planck field values, however it is worth noting that there is an inflation model free of such requirement. Conditions of possible inflation start in this model considerably differ from the chaotic initial data. At the same time the flatness requirement for the inflaton potential is present in this model too.

Consider an inflaton potential shown on figure and assume that for small $\varphi$ it takes the form $$V(\phi)=V_0-g\phi^n,$$ where $n\geq 3$. The most often considered potential is the forth order one $$V(\phi)=V_0-\frac{\lambda}{4}\phi^4.$$ The inflation model with such potential is called novel inflation.

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.