Difference between revisions of "Different Models of Inflation"

From Universe in Problems
Jump to: navigation, search
(→‎Problem 1)
Line 4: Line 4:
  
 
=== Problem 1 ===
 
=== Problem 1 ===
−
A scalar field $\varphi(\vec r,t)$ in a potential $V(\varphi)$ on flat background is described by Lagrangian
+
 
−
\[L=\frac12\left(\dot\varphi^2-\nabla\varphi\cdot\nabla\varphi\right)-V(\varphi)\]
+
Problem Text
−
Obtain the equation of motion (evolution) for this field from the least action principle.
+
 
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
−
     <p style="text-align: left;">Variation of the action with respect to $\varphi $ reads
+
     <p style="text-align: left;">
−
$$\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.
+
 
−
$$
+
Solution text
−
Integrate the second term by parts and use the boundary condition $\delta \varphi = 0$ фе infinity to obtain
+
 
−
$$
+
</p>
−
\frac{\partial L}{\begin{array}{l}
+
  </div>
−
\partial \varphi \\
+
</div>
−
\\
+
 
−
\end{array}}
+
 
−
- \frac{\partial }{\partial x^\mu }\left( {\frac{\partial L}{\partial \left( {\partial _\mu \varphi } \right)}} \right) = 0.
+
<div id="inf2"></div>
−
$$
+
<div style="border: 1px solid #AAA; padding:5px;"></div>
−
For the considered Lagrangian one obtains
+
 
−
$$
+
=== Problem 2 ===
−
\ddot \varphi - \nabla ^2 \varphi + V'(\varphi ) = 0.
+
 
−
$$
+
Problem Text
−
In the case of spatially homogeneous field
+
 
−
$$
+
<div class="NavFrame collapsed">
−
\ddot \varphi + V'(\varphi ) = 0.
+
  <div class="NavHead">solution</div>
−
$$</p>
+
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 3 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 4 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 5 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 6 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 7 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 8 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 9 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 10 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 11 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 12 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 13 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 14 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 15 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 16 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 17 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 18 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 19 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 20 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 21 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 22 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 23 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 24 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 25 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 26 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 27 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 28 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 29 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 30 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 31 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 32 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 33 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 34 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 35 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 36 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 37 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 38 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 +
  </div>
 +
</div>
 +
 
 +
 
 +
<div id="inf2"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;"></div>
 +
 
 +
=== Problem 39 ===
 +
 
 +
Problem Text
 +
 
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
 
 +
Solution text
 +
 
 +
</p>
 
   </div>
 
   </div>
−
</div></div>
+
</div>
  
  
 
<div id="inf2"></div>
 
<div id="inf2"></div>
−
<div style="border: 1px solid #AAA; padding:5px;">
+
<div style="border: 1px solid #AAA; padding:5px;"></div>

Revision as of 11:30, 27 August 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 main idea about possible features of the evolution of inflaton and scale factor in the slow-roll regime.

Problem 1

Problem Text


Problem 2

Problem Text


Problem 3

Problem Text


Problem 4

Problem Text


Problem 5

Problem Text


Problem 6

Problem Text


Problem 7

Problem Text


Problem 8

Problem Text


Problem 9

Problem Text


Problem 10

Problem Text


Problem 11

Problem Text


Problem 12

Problem Text


Problem 13

Problem Text


Problem 14

Problem Text


Problem 15

Problem Text


Problem 16

Problem Text


Problem 17

Problem Text


Problem 18

Problem Text


Problem 19

Problem Text


Problem 20

Problem Text


Problem 21

Problem Text


Problem 22

Problem Text


Problem 23

Problem Text


Problem 24

Problem Text


Problem 25

Problem Text


Problem 26

Problem Text


Problem 27

Problem Text


Problem 28

Problem Text


Problem 29

Problem Text


Problem 30

Problem Text


Problem 31

Problem Text


Problem 32

Problem Text


Problem 33

Problem Text


Problem 34

Problem Text


Problem 35

Problem Text


Problem 36

Problem Text


Problem 37

Problem Text


Problem 38

Problem Text


Problem 39

Problem Text


Retrieved from "http://universeinproblems.com/index.php?title=Different_Models_of_Inflation&oldid=1561"