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| <div style="border: 1px solid #AAA; padding:5px;"> | | <div style="border: 1px solid #AAA; padding:5px;"> |
| === Problem 1 === | | === Problem 1 === |
| + | <p style= "color: #999;font-size: 11px">problem id: SSC_0</p> |
| Show that the Hubble parameter cannot increase with time in the single scalar cosmology. | | Show that the Hubble parameter cannot increase with time in the single scalar cosmology. |
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| </div> | | </div> |
| </div></div> | | </div></div> |
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| <div id="SSC_1"></div> | | <div id="SSC_1"></div> |
| <div style="border: 1px solid #AAA; padding:5px;"> | | <div style="border: 1px solid #AAA; padding:5px;"> |
| === Problem 1 === | | === Problem 1 === |
| + | <p style= "color: #999;font-size: 11px">problem id: SSC_1</p> |
| Obtain first-order differential equation for the Hubble parameter $H$ as function of $\varphi$ and find its stationary points. | | Obtain first-order differential equation for the Hubble parameter $H$ as function of $\varphi$ and find its stationary points. |
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| </div> | | </div> |
| </div></div> | | </div></div> |
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| <div id="SSC_2"></div> | | <div id="SSC_2"></div> |
| <div style="border: 1px solid #AAA; padding:5px;"> | | <div style="border: 1px solid #AAA; padding:5px;"> |
| === Problem 1 === | | === Problem 1 === |
| + | <p style= "color: #999;font-size: 11px">problem id: SSC_2</p> |
| Consider eternally oscillating scalar field of the form $\varphi(t) = \varphi_0 \cos \omega t$ and analyze stationary points in such a model. | | Consider eternally oscillating scalar field of the form $\varphi(t) = \varphi_0 \cos \omega t$ and analyze stationary points in such a model. |
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| </div> | | </div> |
| </div></div> | | </div></div> |
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| <div id="SSC_3"></div> | | <div id="SSC_3"></div> |
| <div style="border: 1px solid #AAA; padding:5px;"> | | <div style="border: 1px solid #AAA; padding:5px;"> |
| === Problem 1 === | | === Problem 1 === |
| + | <p style= "color: #999;font-size: 11px">problem id: SSC_3</p> |
| Obtain explicit solution for the Hubble parameter in the model considered in the previous problem. | | Obtain explicit solution for the Hubble parameter in the model considered in the previous problem. |
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| </div> | | </div> |
| </div></div> | | </div></div> |
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| <div id="SSC_4"></div> | | <div id="SSC_4"></div> |
| <div style="border: 1px solid #AAA; padding:5px;"> | | <div style="border: 1px solid #AAA; padding:5px;"> |
| === Problem 1 === | | === Problem 1 === |
| + | <p style= "color: #999;font-size: 11px">problem id: SSC_4</p> |
| Obtain explicit time dependence for the scale factor in the model of problem [[#SSC_2]]. | | Obtain explicit time dependence for the scale factor in the model of problem [[#SSC_2]]. |
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| </div> | | </div> |
| </div></div> | | </div></div> |
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| <div id="SSC_5"></div> | | <div id="SSC_5"></div> |
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| <p style= "color: #999;font-size: 11px">problem id: SSC_5</p> | | <p style= "color: #999;font-size: 11px">problem id: SSC_5</p> |
| Reconstruct the scalar field potential $V(\varphi)$ needed to generate the model of problem [[#SSC_2]]. | | Reconstruct the scalar field potential $V(\varphi)$ needed to generate the model of problem [[#SSC_2]]. |
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| <div class="NavHead">solution</div> | | <div class="NavHead">solution</div> |
The discovery of the Higgs particle has confirmed that scalar fields play a fundamental
role in subatomic physics. Therefore they must also have been present in the early Universe and played a part
in its development. About scalar fields on present cosmological scales nothing is known, but in view of the observational evidence for accelerated expansion it is quite well possible that they take part
in shaping our Universe now and in the future. In this section we consider the evolution of a flat, isotropic and homogeneous Universe in the presence of a single cosmic
scalar field. Neglecting ordinary matter and radiation, the evolution of such a Universe is described by two
degrees of freedom, the homogeneous scalar field $\varphi(t)$ and the scale factor of the Universe $a(t)$. The
relevant evolution equations are the Friedmann and Klein-Gordon equations,
reading (in the units in which $c = \hbar = 8 \pi G = 1$)
\[
\frac{1}{2}\, \dot{\varphi}^2 + V = 3 H^2, \quad \ddot{\varphi} + 3 H \dot{\varphi} + V' = 0,
\]
where $V[\varphi]$ is the potential of the scalar fields, and $H = \dot{a}/a$ is the Hubble parameter.
Furthermore, an overdot denotes a derivative w.r.t.\ time, whilst a prime denotes a derivative w.r.t.\ the
scalar field $\varphi$.