Dynamical Forms of Dark Energy - Revision history

Cosmo All: /* Problem 61 */

2012-12-03T00:51:21Z

‎Problem 61

Cosmo All: /* Problem 105 */

2012-12-03T00:44:57Z

‎Problem 105

Cosmo All: /* Problem 104 */

2012-12-03T00:44:41Z

‎Problem 104

Cosmo All: /* Problem 104 */

2012-12-03T00:44:11Z

‎Problem 104

Cosmo All: /* Problem 93 */

2012-12-03T00:43:13Z

‎Problem 93

Cosmo All: /* Problem 86 */

2012-12-03T00:41:48Z

‎Problem 86

Cosmo All: /* Problem 85 */

2012-12-03T00:41:18Z

‎Problem 85

Cosmo All: /* Problem 83 */

2012-12-03T00:40:31Z

‎Problem 83

Cosmo All: /* Phantom Energy */

2012-12-03T00:28:52Z

‎Phantom Energy

Cosmo All: /* Problem 50 */

2012-12-03T00:27:56Z

‎Problem 50

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-     <p style="text-align: left;">Proceeding like in problem \ref{inf5} of Chapter \ref{inf}, one obtains
+     <p style="text-align: left;">Proceeding like in problem \ref{inf5} of Chapter Inflation, one obtains
 $$
 \delta S = \int {d^4 x\sqrt { - g} } \left[ {  \left( {\nabla ^\mu  \nabla _\mu  \varphi } \right) - V_{,\varphi } (\varphi )} \right]\delta \varphi  - \int {d^4 x\sqrt { - g} } \nabla _\mu  \left( {\delta \varphi \nabla ^\mu  \varphi } \right)
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 <div id="DE98"></div>
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 === Problem 62 ===
 Find the energy density and pressure of the phantom field.

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-     <p style="text-align: left;">For the Universe model considered  the following holds (see problem [#DE55_2]):
+     <p style="text-align: left;">For the Universe model considered  the following holds (see problem [[#DE55_2]]):
 \[w(z)=\frac{\frac23(1+z)\frac{d\ln H}{dz}-1}{1-\frac{H_0^2}{H^2}\Omega_{m0}(1+z)^3}.\]
 Then the condition $w(z)=-1$ can be rewritten as
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 <div id="DE118"></div>
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 === Problem 106 ===
 Consider a model with the scale factor of the form

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    <div class="NavHead">solution</div>
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-     <p style="text-align: left;">The time derivative of the scalar field can be expressed as (see problem \ref{DE117_11})
+     <p style="text-align: left;">The time derivative of the scalar field can be expressed as (see problem [[#DE117_11]])
 \[\dot\varphi=\frac{d\varphi}{dw}\frac{dw}{dt}=-3H(c_s^2-w)(1+w)\frac{d\varphi}{dw}.\]
 On the other hand,

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 === Problem 104 ===
-Realize the procedure described in the problem [#DE117_11] for the case of a minimally coupled scalar field $\varphi$ with potential $V(\varphi)$ in a spatially flat Universe.
+Realize the procedure described in the problem [[#DE117_11]] for the case of a minimally coupled scalar field $\varphi$ with potential $V(\varphi)$ in a spatially flat Universe.
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 <div id="DE117_2"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 105 ===
 Consider the case of Universe filled with non-relativistic matter and quintessence and show that the condition to cross the phantom divide line $w=-1$ is equivalent to sign change in the following expression

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 === Problem 93 ===
 Reconstruct the potential of the scalar field model, producing the cosmological evolution
-\begin{equation}H(t)=\frac{S}{t^\alpha},\label{RIPS_68}\end{equation}
+\begin{equation}\label{RIPS_68}
 using the technique described in the previous problem.
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    <div class="NavHead">solution</div>
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-     <p style="text-align: left;">begin{equation}\dot\varphi = \pm \sqrt{-\frac23\dot H}=\pm\sqrt{\frac23\alpha S}t^{-\frac{\alpha+1}{2}}.\label{RIPS_68_1}\end{equation}
+     <p style="text-align: left;">\begin{equation}\dot\varphi = \pm \sqrt{-\frac23\dot H}=\pm\sqrt{\frac23\alpha S}t^{-\frac{\alpha+1}{2}}.\label{RIPS_68_1}\end{equation}
 We shall choose the positive sign, without loosing generality. Integration gives
 \begin{equation}\varphi(t) = \sqrt{\frac23\alpha S}\frac{2t^{-\frac{\alpha+1}{2}}}{1-\alpha}.\label{RIPS_68_2}\end{equation}
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 == The Statefinder ==

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 === Problem 86 ===
-Consider the previous problem with $\alpha=-1$ and $k<0$. This equation of state was introduced by Nojiri and Odintsov (see problem \ref{RIPS_7_3}). Chavanis re-derives their results in a more transparent form.
+Consider the previous problem with $\alpha=-1$ and $k<0$. This equation of state was introduced by Nojiri and Odintsov (see problem [[#RIPS_7_3]]). Chavanis re-derives their results in a more transparent form.
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 <div id="RIPS_8"></div>
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 === Problem 87 ===
 Show that for any bound system the rip always occurs when either $H$ diverges or $\dot H$ diverges (assuming $\dot H>0$ ( expansion of Universe is accelerating)).

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 Consider the polytropic equation of state
 \[p=\alpha\rho+k\rho^{1+1/n}\equiv-\rho+\rho\left(1+\alpha+k\rho^{1/n}\right)\]
-under assumption $-1<\alpha\le1$. The case $\alpha=-1$ is treated separately in Problem \ref{RIPS_7_4}. The additional assumption $1+\alpha+k\rho^{1/n}\le0$ (and necessary condition $k<0$) guarantees that the density increases with the scale factor. This corresponds to phantom Universe. Find explicit dependence $\rho(a)$ and analyze limits $a\to0$ and $a\to\infty$.
+under assumption $-1<\alpha\le1$. The case $\alpha=-1$ is treated separately in Problem [[#RIPS_7_4]]. The additional assumption $1+\alpha+k\rho^{1/n}\le0$ (and necessary condition $k<0$) guarantees that the density increases with the scale factor. This corresponds to phantom Universe. Find explicit dependence $\rho(a)$ and analyze limits $a\to0$ and $a\to\infty$.
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 <div id="RIPS_7_4"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 86 ===
 Consider the previous problem with $\alpha=-1$ and $k<0$. This equation of state was introduced by Nojiri and Odintsov (see problem \ref{RIPS_7_3}). Chavanis re-derives their results in a more transparent form.

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 === Problem 83 ===
-Solve the previous problem using the condition for absence of future singularities obtained in Problem [#RIPS_1].
+Solve the previous problem using the condition for absence of future singularities obtained in Problem [[#RIPS_1]].
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 <div id="RIPS_7_2"></div>
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 === Problem 84 ===
 Formulate the condition for the absence of a finite-time future (Big Rip) singularity in terms of function $\rho(a)$ .

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 == Phantom Energy ==
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-''The full amount of available cosmological observational data shows that the state equation parameter $w$ for dark energy lies in a narrow range near the value $w=-1$. In the previous subsections we considered the region $-1\le w\le-1/3$. The lower bound $w=-1$ of the interval corresponds to the cosmological constant, and all the remainder can be covered by the scalar fields with canonic Lagrangians. Recall that the upper bound $w=-1/3$ appears due to the necessity to provide the observed accelerated expansion of Universe. What other values of parameter $w$ can be used? The question is very hard to answer for the energy component we know so little about. General Relativity restricts possible values of the energy - momentum tensor by the so-called "energy conditions" (see Chapter 2). One of the simplest among them is the so-called Null Dominant Energy Condition (NDEC) $\rho+p\ge0$. The physical motivation of the latter is to avoid the vacuum instability. Applied to the dynamics of Universe, the NDEC requires that density of any allowed energy component cannot grow with the expansion of the Universe. The cosmological constant with $\dot\rho_\Lambda=0$, $\rho_\Lambda=const$ represents the limiting case. Because of our ignorance concerning the nature of dark energy it is reasonable to question whether this mysterious substance can differ from the already known "good" sources of energy and if it can violate the NDEC. Taking into account that dark energy must have positive density (it is necessary to make the Universe flat) and negative pressure (to provide the accelerated expansion of Universe), the violation of the NDEC must lead to $w<-1$. Such substance is called the phantom energy. The phantom field $\varphi$ minimally coupled to gravity has the following action:
+''The full amount of available cosmological observational data shows that the state equation parameter $w$ for dark energy lies in a narrow range near the value $w=-1$. In the previous subsections we considered the region $-1\le w\le-1/3$. The lower bound $w=-1$ of the interval corresponds to the cosmological constant, and all the remainder can be covered by the scalar fields with canonic Lagrangians. Recall that the upper bound $w=-1/3$ appears due to the necessity to provide the observed accelerated expansion of Universe. What other values of parameter $w$ can be used? The question is very hard to answer for the energy component we know so little about. General Relativity restricts possible values of the energy - momentum tensor by the so-called "energy conditions" (see Chapter 2). One of the simplest among them is the so-called Null Dominant Energy Condition (NDEC) $\rho+p\ge0$. The physical motivation of the latter is to avoid the vacuum instability. Applied to the dynamics of Universe, the NDEC requires that density of any allowed energy component cannot grow with the expansion of the Universe. The cosmological constant with $\dot\rho_\Lambda=0$, $\rho_\Lambda=const$ represents the limiting case. Because of our ignorance concerning the nature of dark energy it is reasonable to question whether this mysterious substance can differ from the already known "good" sources of energy and if it can violate the NDEC. Taking into account that dark energy must have positive density (it is necessary to make the Universe flat) and negative pressure (to provide the accelerated expansion of Universe), the violation of the NDEC must lead to $w<-1$. Such substance is called the phantom energy. The phantom field $\varphi$ minimally coupled to gravity has the following action:''
 \[S=\int d^4x \sqrt{-g}L=-\int d^4x \sqrt{-g}\left[\frac12g^{\mu\nu}\frac{\partial\varphi}{\partial x_\mu} \frac{\partial\varphi}{\partial x_\nu}+V(\varphi)\right],\]
-which differs from the canonic action for the scalar field only by the sign of the kinetic term.''
+''which differs from the canonic action for the scalar field only by the sign of the kinetic term.''
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 == Disintegration of Bound Structures ==

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      <p style="text-align: left;">\begin{align*}
 V & \propto \varphi ^{ - n} ;\\ \varphi & \propto t^\alpha   = t^{\frac{2}{n + 2}} ;\\
-\dot \varphi & \propto t^{\frac{2}{n + 2} - 1}  = t^{ - \left( {\frac{n}{{n + 2}}} \right)} ;\\
+\dot \varphi & \propto t^{\frac{2}{n + 2} - 1}  = t^{ - \left( {\frac{n}{n + 2}} \right)} ;\\
 \dot \varphi ^2 & \propto t^{ - \frac{2n}{n + 2}}  = \varphi ^{ - n};
 \end{align*}
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 <div id="DE75"></div>
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 === Problem 51 ===
 Find the equation of state parameter $w_\varphi\equiv p_\varphi/\rho_\varphi$ for the scalar field of problem [[#DE73]].