Difference between revisions of "New problems"

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== NEW Problems in Dark Energy Category ==
 
== NEW Problems in Dark Energy Category ==
[[Single Scalar Cosmology|Single Scalar Cosmology]]
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[[Single Scalar Cosmology|'''Single Scalar Cosmology''']]
  
 
The discovery of the Higgs particle has confirmed that scalar fields play a fundamental
 
The discovery of the Higgs particle has confirmed that scalar fields play a fundamental

Revision as of 03:06, 20 December 2013


NEW Problems in Dark Energy Category

Single Scalar Cosmology

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 1

problem id: SSC_1

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


Problem 1

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 1

problem id: SSC_3

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


Problem 1

problem id: SSC_4

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


Problem 1

problem id: SSC_5

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