Non-linear interaction in the dark sector - Revision history

Cosmo All at 11:18, 8 November 2013

2013-11-08T11:18:51Z

Cosmo All: /* Problem 1 */

2013-11-08T11:09:42Z

‎Problem 1

Cosmo All: /* Problem 1 */

2013-11-08T11:07:07Z

‎Problem 1

Cosmo All at 11:05, 8 November 2013

2013-11-08T11:05:20Z

Show changes

Cosmo All at 12:25, 5 November 2013

2013-11-05T12:25:54Z

Cosmo All: Created page with "5 NOTOC"

2013-11-05T11:20:18Z

Created page with "5 __NOTOC__"

New page

[[Category:Interactions in the Dark Sector|5]]

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@@ Line 43: / Line 43: @@
 <div id="IDE_33"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 2 ===
 Consider a Universe with more than two CDM components interacting with each other. What is the asymptotic behavior of the individual densities of the components in the limit $t\to\infty$?
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@@ Line 58: / Line 58: @@
 <div id="IDE_34"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 3 ===
 Consider a Universe containing a cold dark matter  and a dark energy, in which the dark energy behaves like a cosmological constant.  Show that in such model dark energy is a perpetual component of the Universe.
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@@ Line 81: / Line 81: @@
 <div id="IDE_35"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 4 ===
 Consider a two-component Universe with the interaction $Q=\gamma\rho_1\rho_2$. Let one component is CDM ($\rho_1=\rho_{dm}$, $w_1=0$), and the second is the dark energy with arbitrary state equation ($\rho_2=\rho_{de}$, $w_2=\gamma_{de}-1$). (The case considered in the previous problem corresponds to $\gamma_{de}=0$.) Find the relation between the dark energy and dark matter densities.
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@@ Line 117: / Line 117: @@
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 5 ===
 Let interaction term $Q$ be a non-linear function of the energy densities of the components and/or the total energy density. Motivated by the structure
 \[\rho_{dm} = \frac r{1+r}\rho, \quad \rho_{de} = \frac r{1+r}\rho,\]
@@ Line 136: / Line 136: @@
 <div id="IDE_37"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 6 ===
 Find analytical solution of non-linear interaction model covered by the ansatz of previous problem for $(m,n,s)=(1,1,-2)$, $Q=3H\gamma\rho_{de}\rho_{dm}/\rho$.
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@@ Line 160: / Line 160: @@
 <div id="IDE_38"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 7 ===
 Find analytical solution of non-linear interaction model for $(m,n,s)=(1,2,-2)$, $Q=3H\gamma\rho_{dm}^2/\rho$.
 <div class="NavFrame collapsed">
@@ Line 175: / Line 175: @@
 <div id="IDE_39"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 8 ===
 Find analytical solution of non-linear interaction model for $(m,n,s)=(1,0,-2)$, $Q=3H\gamma\rho_{de}^2/\rho$.
 <div class="NavFrame collapsed">
@@ Line 190: / Line 190: @@
 <div id="IDE_40"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 9 ===
 Consider a flat Universe filled by CDM and DE with a polytropic equation of state
 \[p_{de}=K\rho_{de}^{1\frac1n}\]
@@ Line 209: / Line 209: @@
 <div id="IDE_41"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 10 ===
 Show that under certain conditions the interacting polytropic dark energy with $Q=3\alpha H\rho_{de}$ behaves as the phantom energy.
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@@ Line 228: / Line 228: @@
 <div id="IDE_42"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 11 ===
 Find deceleration parameter for the system considered in the [[#IDE_40|problem]].
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@@ Line 87: / Line 87: @@
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">The general case for the considered model is described by the following equations:
-\begin{eqnarray}\label{IDE_35_1}
+\begin{equation}\label{IDE_35_1}
-\dot\rho_{dm} + 3H\rho_{dm}=\gamma\rho_{dm}\rho_{de}, \\
+\dot\rho_{dm} + 3H\rho_{dm}=\gamma\rho_{dm}\rho_{de},
-\dot\rho_{de} = - \gamma_{de} H \rho_{dm} - \gamma\rho_{dm}\rho_{de}, \label{IDE_35_2} \\
+\end{equation}
-H^2=\rho_{dm}+\rho_{de}, \label{IDE_35_3}
+\begin{equation}\label{IDE_35_2}
-\end{eqnarray}
+\dot\rho_{de} = - \gamma_{de} H \rho_{dm} - \gamma\rho_{dm}\rho_{de},
 Addition (\ref{IDE_35_1}) and (\ref{IDE_35_2}) yields
 \begin{equation}

@@ Line 87: / Line 87: @@
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">The general case for the considered model is described by the following equations:
-\begin{eqnarray}
+\begin{eqnarray}\label{IDE_35_1}
-\dot\rho_{dm} + 3H\rho_{dm}=\gamma\rho_{dm}\rho_{de},\label{IDE_35_1} \\
+\dot\rho_{dm} + 3H\rho_{dm}=\gamma\rho_{dm}\rho_{de}, \\
 \dot\rho_{de} = - \gamma_{de} H \rho_{dm} - \gamma\rho_{dm}\rho_{de}, \label{IDE_35_2} \\
 H^2=\rho_{dm}+\rho_{de}, \label{IDE_35_3}
@@ Line 112: / Line 112: @@
 <div id="IDE_36"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 1 ===
 Let interaction term $Q$ be a non-linear function of the energy densities of the components and/or the total energy density. Motivated by the structure

← Older revision		Revision as of 12:25, 5 November 2013
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−	[[Category:Interactions in the Dark Sector\|5]]	+	[[Category:Interactions in the Dark Sector\|4]]

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Non-linear interaction in the dark sector - Revision history

Cosmo All at 11:18, 8 November 2013

Cosmo All: /* Problem 1 */

Cosmo All: /* Problem 1 */

Cosmo All at 11:05, 8 November 2013

Cosmo All at 12:25, 5 November 2013

Cosmo All: Created page with "5 __NOTOC__"

Cosmo All: Created page with "5 NOTOC"