Single Scalar Cosmology

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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_00

Show that if the Universe is filled by a substance which satisfies the null energy condition then the Hubble parameter is a semi-monotonically decreasing function of time.

Problem 3

problem id: SSC_0_1

For single-field scalar models express the scalar field potential in terms of the Hubble parameter and its derivative with respect to the scalar field.

Problem 4

problem id: SSC_1

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

Problem 5

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 6

problem id: SSC_3

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

Problem 7

problem id: SSC_4

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

Problem 8

problem id: SSC_5

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

Problem 9

problem id: SSC_6_00

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

Problem 10

problem id: SSC_6_0

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

Problem 11

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 12

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 13

problem id: SSC_8

Estimate main contribution to total expansion factor of the Universe.

Problem 14

problem id: SSC_9_0

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

Problem 15

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 16

problem id: SSC_10

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

Problem 17

problem id: SSC_11

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

Problem 18

problem id: SSC_12

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

Problem 19

problem id: SSC_13

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

Problem 20

problem id: SSC_14_

When considering dynamics of scalar field $\varphi$ in flat Universe, let us define a function $f(\varphi)$ so that $\dot\varphi=\sqrt{f(\varphi)}$. Obtain the equation describing evolution of the function $f(\varphi)$. (T. Harko, F. Lobo and M. K. Mak, Arbitrary scalar field and quintessence cosmological models, arXiv: 1310.7167)

Exact Solutions for the Single Scalar Cosmology

after Harko (arXiv:1310.7167v4)

Problem 21

problem id: ES_0

Rewrite the equations of the single scalar cosmology \begin{equation} 3H^{2} =\rho _{\phi }=\frac{\dot{\phi}^{2}}{2}+V\left( \phi \right) , \label{H} \end{equation} \begin{equation} 2\dot{H}+3H^{2}=-p_{\phi }=-\frac{\dot{\phi}^{2}}{2}+V\left( \phi \right), \label{H1} \end{equation} \begin{equation} \ddot{\phi}+3H\dot{\phi}+V^{\prime }\left( \phi \right) = 0, \label{phi} \end{equation} in terms of the parameter $G(\phi)$ introduced as \[\dot\phi^2=2V(\phi)\sinh^2 G(\phi).\]

Problem 22

problem id: ES_1

Obtain equation to determine the parameter $G$ as function of time.

Problem 23

problem id: ES_2

Obtain equation to determine the parameter $G$ as function of scale factor.

Problem 24

problem id: ES_3

Obtain the deceleration parameter $q$ in terms of the parameter $G$.

Problem 25

problem id: ES_4

Obtain solution of the equation (\ref{fin}) with \begin{equation} V=V_{0}\exp \left( \sqrt{6}\alpha _{0}\phi \right). \label{pp} \end{equation} in the case $\alpha _0\neq \pm 1$.

Problem 26

problem id: ES_5

Obtain explicit solution of the problem #ES_4 in the case $\alpha _{0}=\pm \sqrt{2}$.

Problem 27

problem id: ES_6

Obtain explicit solution of the problem #ES_4 in the case $\alpha _{0}=\pm \sqrt{3/2}$.

Problem 28

problem id: ES_7

Obtain explicit solution of the problem #ES_4 in the case $\alpha _{0}=\pm 2/\sqrt{3}$.

Problem 29

problem id: ES_8

Obtain a particular solution of the problem #ES_4 in the case $G(\phi)=G_{0}=\mathrm{constant}$.

Problem 30

problem id: ES_9

Obtain solution of the problem #ES_4 in the case $\alpha _0= \pm 1$.

Problem 31

problem id: ES_10

Obtain solution of the equation (\ref{fin}) with \begin{equation} \frac{1}{2V}\frac{dV}{d\phi }=\sqrt{\frac{3}{2}}\;\alpha _{1}\,\tanh G, \end{equation} where $\alpha _{1}$ is an arbitrary constant.

Problem 32

problem id: ES_11

Obtain solution of the equation (\ref{fin}) for the case \begin{equation} G=\mathrm{arccoth}\left( \sqrt{\frac{3}{2}}\frac{\phi }{\alpha _{2}}\right) ,\qquad \alpha _{2}=\mathrm{constant}. \end{equation}

Problem 33

problem id: ES_12

Rewrite the equation (\ref{fin}) in form of the two linear differential equations for the variable $w=e^{-G}$.

Problem 34

problem id: ES_13

Obtain a consistency integral relation between the separation function $M(\phi )$ and the self-interaction potential $V(\phi )$, corresponding to the equations for the variable $w$, obtained in the previous problem.

Problem 35

problem id: ES_14

Obtain exact solution of the equation (\ref{fin}) in the case $M\left( \phi \right) =\sqrt{V}$.

Problem 36

problem id: ES_15

Obtain exact solution of the equation (\ref{fin}) in the case $M\left( \phi \right) =V^{-3/2}$.