Difference between revisions of "Peculiarity of dynamics of scalar field coupled to dark matter"

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[[Category:Interactions in the Dark Sector|6]]
 
[[Category:Interactions in the Dark Sector|6]]
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__NOTOC__
 
__NOTOC__
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= Interacting quintessence model =
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Given that the quintessence field and the dark matter have unknown physical natures, there seem to be no a priori reasons to exclude a coupling between the two components. Let us consider a two-component system (scalar field $\varphi$ + dark matter) with the energy density and pressure
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\[\rho=\rho_\varphi+\rho_{dm},\quad p=p_\varphi+p_{dm}\]
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(we do not exclude the possibility of warm DM ($p_{dm}\ne0$).)
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If some interaction exists between the scalar field and DM, then
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\[\dot\rho_{dm}+3H(\rho_{dm}+p_{dm})=Q\]
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\[\dot\rho_\varphi+3H(\rho_\varphi+p_\varphi)=-Q.\]
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Using the effective pressures $\Pi_\varphi$ and $\Pi_{dm}$,
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\[Q=-3H\Pi_{dm}=3H\Pi_\varphi\]
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one can transit to the system
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\begin{align}
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\nonumber
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\dot\rho_{dm}+3H(\rho_{dm}+p_{dm}+\Pi_{dm}) & =0,\\
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\nonumber
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\dot\rho_\varphi+3H(\rho_\varphi+p_\varphi+\Pi_\varphi) & =0.
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\end{align}
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 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
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Revision as of 11:38, 8 November 2013





Interacting quintessence model

Given that the quintessence field and the dark matter have unknown physical natures, there seem to be no a priori reasons to exclude a coupling between the two components. Let us consider a two-component system (scalar field $\varphi$ + dark matter) with the energy density and pressure \[\rho=\rho_\varphi+\rho_{dm},\quad p=p_\varphi+p_{dm}\] (we do not exclude the possibility of warm DM ($p_{dm}\ne0$).) If some interaction exists between the scalar field and DM, then \[\dot\rho_{dm}+3H(\rho_{dm}+p_{dm})=Q\] \[\dot\rho_\varphi+3H(\rho_\varphi+p_\varphi)=-Q.\] Using the effective pressures $\Pi_\varphi$ and $\Pi_{dm}$, \[Q=-3H\Pi_{dm}=3H\Pi_\varphi\] one can transit to the system \begin{align} \nonumber \dot\rho_{dm}+3H(\rho_{dm}+p_{dm}+\Pi_{dm}) & =0,\\ \nonumber \dot\rho_\varphi+3H(\rho_\varphi+p_\varphi+\Pi_\varphi) & =0. \end{align}


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