Difference between revisions of "Single Scalar Cosmology"

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from which the solutions $H[\varphi]$ can be reconstructed (see figure below).
 
from which the solutions $H[\varphi]$ can be reconstructed (see figure below).
  
[[File:Epsgt0v2.png|center|thumb|400px|Solutions $H[\varphi]$ for quadratic potentials (problem [[#SSC_6_1]]) with $v_0 > 0$ (left), and $v_0 < 0$ (right).]]
+
[[File:Epsgt0v2.png|center|thumb|400px|Solutions $H[\varphi]$ for quadratic potentials (problem [[#SSC_6_1]]) with $v_0 > 0$.]]
 +
[[File:Epslt0v2.png|center|thumb|400px|Solutions $H[\varphi]$ for quadratic potentials (problem [[#SSC_6_1]]) with $v_0 < 0$.]]
 
</p>
 
</p>
 
   </div>
 
   </div>
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<div id="SSC_8"></div>
 
<div id="SSC_8"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
 +
 
=== Problem 11 ===
 
=== Problem 11 ===
 
<p style= "color: #999;font-size: 11px">problem id: SSC_8</p>
 
<p style= "color: #999;font-size: 11px">problem id: SSC_8</p>
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<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 13 ===
 
=== Problem 13 ===
<p style= "color: #999;font-size: 11px">problem id: </p>
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<p style= "color: #999;font-size: 11px">problem id: SSC_9</p>
 
Show that the exponentially decaying scalar field
 
Show that the exponentially decaying scalar field
 
\[
 
\[
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   <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;">The Hubble parameter and potential giving rise to this solution can be constructed following the same procedure as for the eternally oscillating field (see problem \ref{SSC_2}), with the result
+
     <p style="text-align: left;">The Hubble parameter and potential giving rise to this solution can be constructed following the same procedure as for the eternally oscillating field (see problem [[#SSC_2]]), with the result
 
\[
 
\[
 
H = h + \frac{1}{4}\, \omega \varphi^2, \quad V[\varphi] = v_0 - \frac{\mu^2}{2}\, \varphi^2 + \frac{\lambda}{4}\, \varphi^4,
 
H = h + \frac{1}{4}\, \omega \varphi^2, \quad V[\varphi] = v_0 - \frac{\mu^2}{2}\, \varphi^2 + \frac{\lambda}{4}\, \varphi^4,
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solutions $H[\varphi]$ coming close to the maximum of the potential (see figure below).
 
solutions $H[\varphi]$ coming close to the maximum of the potential (see figure below).
  
[[File:Epsgt0v2.png|center|thumb|400px|Critical curves of stationary points and solutions $H[\varphi]$ for a quartic potential with spontaneous symmetry breaking.]]
+
[[File:Quarticv2.png|center|thumb|400px|Critical curves of stationary points and solutions $H[\varphi]$ for a quartic potential with spontaneous symmetry breaking.]]
 
</p>
 
</p>
 
   </div>
 
   </div>
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<div id="SSC_10"></div>
 
<div id="SSC_10"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
 +
 
=== Problem 14 ===
 
=== Problem 14 ===
 
<p style= "color: #999;font-size: 11px">problem id: SSC_10</p>
 
<p style= "color: #999;font-size: 11px">problem id: SSC_10</p>

Revision as of 00:03, 27 December 2013



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 $\varphi$.


Problem 1

problem id: SSC_0

Show that the Hubble parameter cannot increase with time in the single scalar cosmology.


Problem 2

problem id: SSC_1

Obtain first-order differential equation for the Hubble parameter $H$ as function of $\varphi$ and find its stationary points.


Problem 3

problem id: SSC_2

Consider eternally oscillating scalar field of the form $\varphi(t) = \varphi_0 \cos \omega t$ and analyze stationary points in such a model.


Problem 4

problem id: SSC_3

Obtain explicit solution for the Hubble parameter in the model considered in the previous problem.


Problem 5

problem id: SSC_4

Obtain explicit time dependence for the scale factor in the model of problem #SSC_2.


Problem 6

problem id: SSC_5

Reconstruct the scalar field potential $V(\varphi)$ needed to generate the model of problem #SSC_2.


Problem 7

problem id: SSC_6_00

Describe possible final states for the Universe governed by a single scalar field at large times.


Problem 8

problem id: SSC_6_0

Formulate conditions for existence of end points of evolution in terms of the potential $V(\varphi)$.


Problem 9

problem id: SSC_6_1

Consider a single scalar cosmology described by the quadratic potential \[ V = v_0 + \frac{m^2}{2}\, \varphi^2. \] Describe all possible stationary points and final states of the Universe in this model.


Problem 10

problem id: SSC_7

Obtain actual solutions for the model of previous problem using the power series expansion \[ H[\varphi] = h_0 + h_1 \varphi + h_2 \varphi^2 + h_3 \varphi^3 + ... \] Consider the cases of $v_0 > 0$ and $v_0 < 0$.


Problem 11

problem id: SSC_8

Estimate main contribution to total expansion factor of the Universe.


Problem 12

problem id: SSC_9_0

Explain difference between end points and turning points of the scalar field evolution.


Problem 13

problem id: SSC_9

Show that the exponentially decaying scalar field \[ \varphi(t) = \varphi_0 e^{-\omega t} \] can give rise to unstable end points of the evolution.


Problem 14

problem id: SSC_10

Analyze all possible final states in the model of previous problem.


Problem 15

problem id: SSC_11

Express initial energy density of the model of problem #SSC_9 in terms of the $e$-folding number $N$.


Problem 16

problem id: SSC_12

Estimate mass of the particles corresponding to the exponential scalar field considered in problem #SSC_9.


Problem 17

problem id: SSC_13

Calculate the deceleration parameter for flat Universe filled with the scalar field in form of quintessence.