|
|
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| </div> | | </div> |
| </div> --></div> | | </div> --></div> |
− |
| |
− | === Problem 2 ===
| |
− |
| |
− | 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 id="inf2"></div> |
− | <div style="border: 1px solid #AAA; padding:5px;"></div> | + | <div style="border: 1px solid #AAA; padding:5px;"> |
| + | === Problem 1 === |
| | | |
− | === Problem 3 === | + | 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 Text
| + | <!-- <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"> |
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| </p> | | </p> |
| </div> | | </div> |
− | </div> | + | </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">
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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.}