Interacting holographic dark energy - Revision history

Cosmo All: /* Problem 8 */

2013-11-11T04:32:06Z

‎Problem 8

2013-11-11T04:30:37Z

‎Problem 8

2013-11-11T04:29:08Z

‎Problem 8

2013-11-11T04:28:46Z

‎Problem 8

2013-11-11T04:28:04Z

‎Problem 7

2013-11-11T04:18:50Z

‎Problem 6

2013-11-11T04:17:50Z

‎Problem 6

2013-11-11T04:14:58Z

‎Problem 6

2013-11-11T04:14:19Z

2013-11-11T04:13:45Z

@@ Line 474: / Line 474: @@
 \gamma & {\rm ~for~~} Q=3\gamma H\rho_{tot}
   \end{array}
-  \right}
+  \right..
 \end{equation}
   From
@@ Line 482: / Line 482: @@
   w_q=-1+\frac{2}{3n}\sqrt{\Omega_q}-\frac{Q}{3H\rho_q},
 \end{equation}
- where
+where
 \begin{equation}\label{eq18}
   \frac{Q}{3H\rho_q}=\left\{
   \begin{array}{ll}
-  \alpha & {\rm ~for~~} Q=3\alpha H\rho_q \\ \vspace{0.75mm}
+  \alpha & {\rm ~for~~} Q=3\alpha H\rho_q \\
   \beta\left(\Omega_q^{-1}-1\right) & {\rm ~for~~} Q=3\beta H\rho_m\\
   \gamma\,\Omega_q^{-1} & {\rm ~for~~} Q=3\gamma H\rho_{tot}
@@ Line 492: / Line 492: @@
   \right..
 \end{equation}
- Using Eq.~(\ref{eq14}), the deceleration parameter is given by
+Using Eq.~(\ref{eq14}), the deceleration parameter is given by
 \begin{equation}\label{eq19}
   q\equiv -\frac{\ddot{a}a}{\dot{a}^2}=-1-\frac{\dot{H}}{H^2}

@@ Line 466: / Line 466: @@
   -\frac{Q}{3M_p^2 H^3}\right],
 \end{equation}
- where
+where
 \begin{equation}\label{eq16}
   \frac{Q}{3M_p^2 H^3}=\left\{
@@ Line 474: / Line 474: @@
 \gamma & {\rm ~for~~} Q=3\gamma H\rho_{tot}
   \end{array}
-  \right..
+  \right}
 \end{equation}
   From

@@ Line 445: / Line 445: @@
 T=\int_{0}^{a}{\frac{d{a}'}{H{a}'}}.
 \end{equation}
- of interacting agegraphic dark energy and the deceleration parameter, for sets of   interaction term $Q=3\alpha H\rho_q;\; 3\beta H\rho_m;\; 3\gamma H\rho_{tot}.$
+of interacting agegraphic dark energy and the deceleration parameter, for sets of   interaction term $Q=3\alpha H\rho_q;\; 3\beta H\rho_m;\; 3\gamma H\rho_{tot}.$
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>

@@ Line 438: / Line 438: @@
 Find  the equation of motion for the relative density
-   \begin{equation}\label{eq7}
+\begin{equation}\label{eq7}
- \Omega_q=\frac{n^2}{H^2T^2},
+\Omega_q=\frac{n^2}{H^2T^2},
 \end{equation}
 were
@@ Line 469: / Line 469: @@
 \begin{equation}\label{eq16}
   \frac{Q}{3M_p^2 H^3}=\left\{
- \begin{array}{ll}
+\begin{array}{ll}
 \alpha\Omega_q & {\rm ~for~~} Q=3\alpha H\rho_q \\
 \beta\left(1-\Omega_q\right) & {\rm ~for~~} Q=3\beta H\rho_m \\

@@ Line 417: / Line 417: @@
   where the  roots of the characteristic polynomial associated with the second order linear differential equation (\ref{9}) are  $\gamma_{s}$ and $\gamma^{+}=(\beta\alpha -u)/{\gamma_s}$.
 In what follows, we adopt $\gamma^{+}=1$ in order to mimic the dust-like behavior of the Universe at early times. In that case, the general solution of (\ref{9}) is   $\rho=b_1a^{-3\gamma_s}+b_2a^{-3}$,  from which we obtain
 \begin{equation}
 \label{10a}
@@ Line 427: / Line 425: @@
 \rho_x=\frac{(\alpha-\gamma_s)b_1a^{-3\gamma_s}+(\alpha-1)b_2a^{-3}}{\Delta}.
 \end{equation}
 Interestingly, Eqs. (\ref{10}) tell us that  the interaction (\ref{8}) seems to be a good candidate for alleviating the cosmic coincidence problem, since the ratio $\Omega_{c}/\Omega_{x}$ becomes bounded for all times.
 </p>
@@ Line 437: / Line 434: @@
 <div id="IDE_103"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 8 ===
 Find  the equation of motion for the relative density

@@ Line 272: / Line 272: @@
   where $x=\ln{a}$.
 The solution to eq. (\ref{EOH}) is obtained as
- \begin{eqnarray}
+\begin{eqnarray}\label{solution}
- \label{hubbleparameter}
+\frac{H^{2}}{H_{0}^{2}}    &=& A_{+} e^{\sigma_{+} x} + A_{-} e^{\sigma_{-} x}
-  \frac{H^{2}}{H_{0}^{2}}    &=& A_{+} e^{\sigma_{+} x} + A_{-} e^{\sigma_{-} x}
              + A_{\gamma} e^{-4 x} + A_{k} e^{-2 x} ,
   \end{eqnarray}
 where

@@ Line 228: / Line 228: @@
     Q=\gamma H \rho_{_{\cal R}},
    \end{eqnarray}
-  where $\gamma$ is a dimensionless parameter,
+where $\gamma$ is a dimensionless parameter,
 find the dependence of the density of dark energy and dark matter on the scale factor.
 <div class="NavFrame collapsed">
@@ Line 234: / Line 234: @@
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">The energy density of dark energy in the IRDE model is defined as
-\begin{eqnarray}
+\begin{equation}\label{RDE}
- \label{rde}
+   \rho_{_{\cal R}} = 3\alpha M_{p}^{2} \left( \dot{H}+2H^{2} + \frac{k}{a^{2}}\right),
-   \rho_{_{\cal R}}
+\end{equation}
-       &=& 3\alpha M_{p}^{2}
-       \left (
-       \dot{H}+2H^{2} + \frac{k}{a^{2}}
-       \right),
-                \label{RDE}
- \end{eqnarray}
   where $\alpha$ is a dimensionless parameter.
   Note that $\rho_{_{\cal R}}$ is proportional to the Ricci scalar curvature
- \begin{eqnarray}
+\begin{equation}
-  {\cal{R}}= -6
+  {\cal{R}} = -6\left(\dot{H}+2H^{2}+\frac{k}{a^{2}}\right).
-                      \left(\dot{H}+2H^{2}+\frac{k}{a^{2}}
+\end{equation}
-                      \right).
- \end{eqnarray}
 The time evolution of the scale factor $a(t)$ is described by the Friedmann equation

@@ Line 17: / Line 17: @@
 three-dimensional object. The holographic principle consists of two
 main statements:
-\begin{enumerate}
-\item All information contained in some region of space can be
+) All information contained in some region of space can be
 ''recorded'' (represented) on the boundary of that region.
-\item The theory, formulated on the boundaries of the considered
 region of space, must have no more than one degree of freedom per
 Planck area:
@@ Line 27: / Line 28: @@
      N\le \frac{A}{A_{pl}},\quad A_{pl}=\frac{G\hbar}{c^3}.
 \end{equation}
-\end{enumerate}
 Thus, the key piece in the holographic principle is
 the assumption that all the information about the Universe can be