Initial perturbations in the Universe - Revision history

Cosmo All: /* Problem 18 */

2014-02-25T02:10:51Z

‎Problem 18

Cosmo All: /* Problem 18 */

2014-02-25T02:09:33Z

‎Problem 18

Cosmo All: /* Problem 18 */

2014-02-25T02:08:53Z

‎Problem 18

Cosmo All: /* Problem 10 */

2014-02-25T02:04:49Z

‎Problem 10

Cosmo All: /* Problem 5 */

2014-02-25T02:02:53Z

‎Problem 5

Cosmo All: /* Problem 5 */

2014-02-25T02:00:47Z

‎Problem 5

Cosmo All at 01:32, 25 February 2014

2014-02-25T01:32:32Z

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Cosmo All at 01:31, 25 February 2014

2014-02-25T01:31:01Z

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Cosmo All at 11:36, 18 February 2014

2014-02-18T11:36:13Z

Cosmo All: Created page with "4"

2014-02-11T11:29:48Z

Created page with "4"

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[[Category:Perturbation theory in cosmology|4]]

@@ Line 497: / Line 497: @@
    \mathcal{P}_{\varphi}(k)=\frac{ H_k^2}{(2\pi)^{2}}.
 \end{equation}
-Relations $$u=-z \mathcal{R},~~z=\frac{a\dot{\varphi}}{H}$$ from problems \ref{inf-z-u} and \ref{inf-z-u1}  give the connection between$\widetilde{\psi}$ and $\widetilde{R}$:
+Relations $$u=-z \mathcal{R},~~z=\frac{a\dot{\varphi}}{H}$$ from problems [[#inf-z-u]] and [[#inf-z-u1]]  give the connection between$\widetilde{\psi}$ and $\widetilde{R}$:
 \begin{equation}\label{mathcal_R_psi}
      \mathcal{R}=-\frac{H}{\dot{\varphi}} \widetilde{\psi}.

@@ Line 493: / Line 493: @@
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">In the slow-roll approximation, the expression for the
-$ u $ is exactly the same as for $ \chi $ in problem \ref{infl_vac_fluc_hor-p}, while expression $\widetilde{\psi} $ coincides with the expression for $\chi/a$, which plays as perturbations operator of inflanton field in simplified analysis in problem [[#infl_vac_fluc_hor-p]]. Beyond the horizon operator $\widetilde{\psi} $ is constant and equals the right--hand side of \eqref{phi_qun_chi-hor3} and corresponding Gaussian random field has a power spectrum \eqref{power_spectrum-hor}:
+$ u $ is exactly the same as for $ \chi $ in problem [[#infl_vac_fluc_hor-p]], while expression $\widetilde{\psi} $ coincides with the expression for $\chi/a$, which plays as perturbations operator of inflanton field in simplified analysis in problem [[#infl_vac_fluc_hor-p]]. Beyond the horizon operator $\widetilde{\psi} $ is constant and equals the right--hand side of \eqref{phi_qun_chi-hor3} and corresponding Gaussian random field has a power spectrum \eqref{power_spectrum-hor}:
 \begin{equation}\nonumber
    \mathcal{P}_{\varphi}(k)=\frac{ H_k^2}{(2\pi)^{2}}.

@@ Line 492: / Line 492: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">In the slow--roll approximation, the expression for the
+     <p style="text-align: left;">In the slow-roll approximation, the expression for the
-$ u $ is exactly the same as for $ \chi $ in problem \ref{infl_vac_fluc_hor-p}, while expression $\widetilde{\psi} $ coincides with the expression for $\chi/a$, which plays as perturbations operator of inflanton field in simplified analysis in problem \ref{infl_vac_fluc_hor-p}. Beyond the horizon operator $\widetilde{\psi} $ is constant and equals the right--hand side of \eqref{phi_qun_chi-hor3} and corresponding Gaussian random field has a power spectrum \eqref{power_spectrum-hor}:
+$ u $ is exactly the same as for $ \chi $ in problem \ref{infl_vac_fluc_hor-p}, while expression $\widetilde{\psi} $ coincides with the expression for $\chi/a$, which plays as perturbations operator of inflanton field in simplified analysis in problem [[#infl_vac_fluc_hor-p]]. Beyond the horizon operator $\widetilde{\psi} $ is constant and equals the right--hand side of \eqref{phi_qun_chi-hor3} and corresponding Gaussian random field has a power spectrum \eqref{power_spectrum-hor}:
 \begin{equation}\nonumber
    \mathcal{P}_{\varphi}(k)=\frac{ H_k^2}{(2\pi)^{2}}.
@@ Line 501: / Line 501: @@
      \mathcal{R}=-\frac{H}{\dot{\varphi}} \widetilde{\psi}.
 \end{equation}
-Using the results of the problem \ref {inf-z-u1}, one can deduce that  field $ \widetilde {\psi} $ is the same as the inflaton field perturbations in Minkowski space, and hence is Gaussian, and its power spectrum is calculated in the same way as in the problem \ref {infl_vac_fluc_hor-p}. Thus the power spectrum of the spatial curvature of hypersurfaces generated by fluctuations of the inflaton field in the comoving reference frame is given by
+Using the results of the problem [[#inf-z-u1]], one can deduce that  field $ \widetilde {\psi} $ is the same as the inflaton field perturbations in Minkowski space, and hence is Gaussian, and its power spectrum is calculated in the same way as in the problem \ref {infl_vac_fluc_hor-p}. Thus the power spectrum of the spatial curvature of hypersurfaces generated by fluctuations of the inflaton field in the comoving reference frame is given by
 \begin{equation}\label{mathcal_P_varphi}
    \mathcal{P}_{\varphi}(k)=\left.\left(\frac{ H^2}{2\pi\dot{\varphi}}\right)^2\right|_{\eta_k},
@@ Line 513: / Line 513: @@
 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 19 ===
 <p style= "color: #999;font-size: 11px">problem id: </p>

@@ Line 249: / Line 249: @@
      <p style="text-align: left;">The following conditions hold in the slow--roll approzimation, as is well known:
      \begin{equation}\label{slow-row}
-     \varepsilon = \frac{1}{2}M_{Pl}^{2} \left( {\frac{{V'}}{V}} \right)^2\ll 0,~~\eta = \frac{1}{2}M_{Pl}^{*2} \frac{{V''}}{V}\ll 0.
+     \varepsilon = \frac{1}{2}M_{Pl}^{2} \left( {\frac{V'}{V}} \right)^2\ll 0,~~\eta = \frac{1}{2}M_{Pl}^{*2} \frac{V''}{V}\ll 0.
      \end{equation}
      Condition on $\eta$ can be rewritten as
@@ Line 286: / Line 286: @@
 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 11 ===
 <p style= "color: #999;font-size: 11px">problem id: </p>

@@ Line 148: / Line 148: @@
 and equation of motion reads
 $$\nabla _\mu \nabla ^\mu  \psi  + V''(\varphi_b)\psi.$$
-Proceeding as in problem \ref{inf5}:
+Proceeding as in problem [[#inf5]]:
 \begin{equation}\label{inf_eq_fp}
 \ddot\psi+ 3H\dot\psi-\frac{1}{a^2}\partial_i\partial_i\psi + V''(\varphi_b)\psi = 0

@@ Line 172: / Line 172: @@
 \end{equation}
-Recalling the features of the inflationary regime, this equation can be simplified for qualitative analysis. one of the main features of inflationary phase is the slow evolution of Hubble parameter $H$ and fast growth of scale factor. Physical momentum $q(t) = k/ a(t)$ is large compared to Hubble parameter and, moreover, $q^2 \gg V''(\varphi_b)$. The largest terms in \eqref{ddot_psi_per}, thus, are first and third terms; field $\psi_{\vec{k}}(t)$ oscillates rapidly and behaves mostly like in Minkowski space--time. In other words, mode $\psi_{\vec{k}}(t)$ is below the horizon, space--time curvature is effectively small and has little effect on its evolution. The results of problem \ref{infl_vac_fluc1} can be used in this regime and amplitudes of vacuum fluctuations decrease in correspondence with cosmological ''reddening'' of the momentum. At later times the relation $q(t)\ll H$ holds and term $3H\dot\psi_{\vec{k}}$ dominates the equation \eqref{ddot_psi_per}. Effective wavelength of the mode in this regime is much greater than cosmological horizon and doesn't depend on time, so that its amplitude  ''freezes'', although its wavelength increases and physical momentum decreases.</p>
+Recalling the features of the inflationary regime, this equation can be simplified for qualitative analysis. one of the main features of inflationary phase is the slow evolution of Hubble parameter $H$ and fast growth of scale factor. Physical momentum $q(t) = k/ a(t)$ is large compared to Hubble parameter and, moreover, $q^2 \gg V''(\varphi_b)$. The largest terms in \eqref{ddot_psi_per}, thus, are first and third terms; field $\psi_{\vec{k}}(t)$ oscillates rapidly and behaves mostly like in Minkowski space-time. In other words, mode $\psi_{\vec{k}}(t)$ is below the horizon, space--time curvature is effectively small and has little effect on its evolution. The results of problem [[#infl_vac_fluc1]] can be used in this regime and amplitudes of vacuum fluctuations decrease in correspondence with cosmological "reddening" of the momentum. At later times the relation $q(t)\ll H$ holds and term $3H\dot\psi_{\vec{k}}$ dominates the equation \eqref{ddot_psi_per}. Effective wavelength of the mode in this regime is much greater than cosmological horizon and doesn't depend on time, so that its amplitude  "freezes", although its wavelength increases and physical momentum decreases.</p>
    </div>
 </div></div>
@@ Line 179: / Line 179: @@
 <div id="infl_vac_per_app"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 7 ===
 <p style= "color: #999;font-size: 11px">problem id: infl_vac_per_app</p>

← Older revision		Revision as of 11:36, 18 February 2014
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−	[[Category:Perturbation theory in cosmology\|4]]	+	[[Category:Perturbation theory in cosmology\|5]]