Expansion of cosmological perturbations in helicities - Revision history

Cosmo All at 22:56, 24 February 2014

2014-02-24T22:56:55Z

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Cosmo All: /* Problem 22 */

2014-02-24T22:53:19Z

‎Problem 22

Cosmo All: /* Problem 22 */

2014-02-24T22:52:07Z

‎Problem 22

Cosmo All: /* Problem 18 */

2014-02-24T22:50:31Z

‎Problem 18

Cosmo All: /* Problem 15 */

2014-02-24T22:49:17Z

‎Problem 15

Cosmo All: /* Problem 7 */

2014-02-24T22:47:24Z

‎Problem 7

Cosmo All: /* Problem 7 */

2014-02-24T22:44:29Z

‎Problem 7

Cosmo All: /* Problem 6 */

2014-02-24T22:43:22Z

‎Problem 6

Cosmo All: /* Problem 2 */

2014-02-24T22:41:39Z

‎Problem 2

Cosmo All: /* Problem 2 */

2014-02-24T22:40:22Z

‎Problem 2

@@ Line 600: / Line 600: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">Substituting the expansion of the tensor perturbation \eqref{hij_gen} into the expression, defining the spatial part of the Einstein tensor (it is the only non-trivial under such expansion) from problem \eqref{Ein_gen_prob} and keeping only the terms linear in $ h_ {ij} ^ {TT} $ (since they are responsible for the tensor perturbations), one obtain
+     <p style="text-align: left;">Substituting the expansion of the tensor perturbation \eqref{hij_gen} into the expression, defining the spatial part of the Einstein tensor (it is the only non-trivial under such expansion) from problem [[#per20G]] and keeping only the terms linear in $ h_ {ij} ^ {TT} $ (since they are responsible for the tensor perturbations), one obtain
 \begin{equation}\label{GijTT}
 a^2\delta G^i_j  = \widetilde{h}''_{ij} + 2\frac{a'}{a} \widetilde{h}'_{ij}- \Delta \widetilde{h}_{ij}=0,

@@ Line 600: / Line 600: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">Substituting the expansion of the tensor perturbation \eqref {hij_gen} into the expression, defining the spatial part of the Einstein tensor (it is the only non--trivial under such expansion) from problem \eqref{Ein_gen_prob} and keeping only the terms linear in $ h_ {ij} ^ {TT} $ (since they are responsible for the tensor perturbations), one obtain
+     <p style="text-align: left;">Substituting the expansion of the tensor perturbation \eqref{hij_gen} into the expression, defining the spatial part of the Einstein tensor (it is the only non-trivial under such expansion) from problem \eqref{Ein_gen_prob} and keeping only the terms linear in $ h_ {ij} ^ {TT} $ (since they are responsible for the tensor perturbations), one obtain
 \begin{equation}\label{GijTT}
 a^2\delta G^i_j  = \widetilde{h}''_{ij} + 2\frac{a'}{a} \widetilde{h}'_{ij}- \Delta \widetilde{h}_{ij}=0,
 \end{equation}
-Since all terms in this equation have double helicity (Freud is crying?:))), we use the simplified notation: $h_{ij}^{TT} = \widetilde{h}_{ij}$. As it was shown in problem \ref{per22ntenzor}, energy-momentum tensor has components with a double helicity, hence the equation for gravitational waves is
+Since all terms in this equation have double helicity (Freud is crying?:))), we use the simplified notation: $h_{ij}^{TT} = \widetilde{h}_{ij}$. As it was shown in problem [[per22ntenzor]], energy-momentum tensor has components with a double helicity, hence the equation for gravitational waves is
      \begin{equation}\label{widetildehij}
          \widetilde{h}''_{ij} + 2\frac{a'}{a} \widetilde{h}'_{ij}- \Delta \widetilde{h}_{ij}=0,
@@ Line 618: / Line 618: @@
 <div id="gr_wave_eq_action"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 23 ===
 <p style= "color: #999;font-size: 11px">problem id: gr_wave_eq_action</p>

@@ Line 475: / Line 475: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">The components of perturbed Ricci tensor were obtained in problem \ref{per23}. Its spatial component is
+     <p style="text-align: left;">The components of perturbed Ricci tensor were obtained in problem [[#per23]]. Its spatial component is
 $$
 \widetilde{R}_{ ij}=\delta_{ij}\left(\mathcal{H} '+\Delta \Phi-4  \Phi \mathcal{H} ' - \Phi '' +2\mathcal{H}^2 -8\mathcal{H}^2\Phi-6\mathcal{H}\Phi '\right).
@@ Line 489: / Line 489: @@
 Using the definition of scalar Ricci curvature, one can express its spatial part as
 $$\delta R^{(3)}= g^{ij}  \delta R_{ ij}. $$
-Using the linearized Einstein equations from problem \ref{per28nnn}:
+Using the linearized Einstein equations from problem [[#per28nnn]]:
 \begin{equation}\label{R3phi}
     R^{(3)}= -\frac{4}{a^2}\Delta\widetilde{\Phi},
@@ Line 509: / Line 509: @@
    </div>
 </div></div>
 == Evolution of vector and tensor perturbations ==

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    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">\textbf{Solution.}\begin{equation}
+     <p style="text-align: left;">\begin{equation}
 \ddot{\delta \phi} + 3\frac{\dot{a}}{a}\dot{\delta \phi} + \frac{k^2
    \delta \phi}{a^2} + 2\Phi V'(\phi)  -4\dot{\Phi}\dot{\phi} +
@@ Line 442: / Line 442: @@
 <div id="Tmunu_inf"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 16 ===
 <p style= "color: #999;font-size: 11px">problem id: Tmunu_inf</p>

@@ Line 209: / Line 209: @@
 The similar representation is used for $\delta\rho,\,\, \delta p,\,\, v_i$ and other quantities. Einstein equations for scalar perturbations in momentum representation are obtained by substitution  $\partial_i\longleftrightarrow ik_i$:
 \begin{equation}
-  k^2\Phi +3\frac{a'}{a}\Phi '+3\frac{a'^2}{a^2}\Phi &=&-4\pi Ga^2\delta\rho \label{kPhi}
+  k^2\Phi +3\frac{a'}{a}\Phi '+3\frac{a'^2}{a^2}\Phi =-4\pi Ga^2\delta\rho \label{kPhi}
 \end{equation}
 \begin{equation}
-    \Phi ''+3\frac{a'}{a}\Phi+\left(2\frac{a''}{a}-\frac{a'^2}{a^2}\right) \Phi &=&4\pi Ga^2\delta p \label{Phi_}
+    \Phi ''+3\frac{a'}{a}\Phi+\left(2\frac{a''}{a}-\frac{a'^2}{a^2}\right) \Phi =4\pi Ga^2\delta p \label{Phi_}
 \end{equation}</p>
    </div>

@@ Line 208: / Line 208: @@
 $$
 The similar representation is used for $\delta\rho,\,\, \delta p,\,\, v_i$ and other quantities. Einstein equations for scalar perturbations in momentum representation are obtained by substitution  $\partial_i\longleftrightarrow ik_i$:
-\begin{eqnarray}
+\begin{equation}
-  k^2\Phi +3\frac{a'}{a}\Phi '+3\frac{a'^2}{a^2}\Phi &=&-4\pi Ga^2\delta\rho \label{kPhi}  \\
+  k^2\Phi +3\frac{a'}{a}\Phi '+3\frac{a'^2}{a^2}\Phi &=&-4\pi Ga^2\delta\rho \label{kPhi}
     \Phi ''+3\frac{a'}{a}\Phi+\left(2\frac{a''}{a}-\frac{a'^2}{a^2}\right) \Phi &=&4\pi Ga^2\delta p \label{Phi_}
-\end{eqnarray}</p>
+\end{equation}</p>
    </div>
 </div></div>
@@ Line 218: / Line 220: @@
 <div id="per30nnn"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 8 ===
 <p style= "color: #999;font-size: 11px">problem id: per30nnn</p>

@@ Line 181: / Line 181: @@
 \end{equation}
 Using the linearized Einstein tensor $\delta G_\mu^\nu$ from previous problem and linearized energy--momentum tensor $\delta T_\mu^\nu = \widetilde{T}_\mu^\nu -T_\mu^\nu$:
-\begin{eqnarray}
+\begin{equation}
-\Delta \Phi - 3\mathcal{H}\Phi ' - 3\mathcal{H}^2\Phi  =4\pi G a^2  \sum_i \delta\rho_{i};\label{Ein_1}\\
+\Delta \Phi - 3\mathcal{H}\Phi ' - 3\mathcal{H}^2\Phi  =4\pi G a^2  \sum_i \delta\rho_{i};\label{Ein_1}
-\Phi ' +  3\mathcal{H}\Phi  =  4\pi G a^2 \sum_i  \left(\rho_i - p_i\right) v_i;\label{Ein_2}\\
+\end{equation}
 \Phi ''+ 3\mathcal{H}\Phi '+ 2\mathcal{H}'+\mathcal{H}^2=4\pi G a^2  \sum_i \delta p_{i}.\label{Ein_3}
-\end{eqnarray}</p>
+\end{equation}</p>
    </div>
 </div></div>
@@ Line 192: / Line 196: @@
 <div id="per29nnn"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 7 ===
 <p style= "color: #999;font-size: 11px">problem id: per29nnn</p>

@@ Line 76: / Line 76: @@
 \delta p+4\mathcal{H} \left( \rho + p\right)v +(p+\rho)\Phi+\left[(\rho + p)v\right ]'=0
 \end{eqnarray}
-and can be obtained from equation in solution of [[Introduction_to_relativistic_theory_of_small_perturbations#per23ntenzor problem ]]  by substitution $h_{00}=2\Phi,~h_{ij}=2\Phi\delta_{ij},v_i=\partial _iv$.</p>
+and can be obtained from equation in solution of [[Introduction_to_relativistic_theory_of_small_perturbations#per23ntenzor| problem ]]  by substitution $h_{00}=2\Phi,~h_{ij}=2\Phi\delta_{ij},v_i=\partial _iv$.</p>
    </div>
 </div></div>