Igor: pr1: link to previous section

2013-10-09T13:40:08Z

pr1: link to previous section

Igor: /* Problem 3 */ no solution

2013-10-09T13:32:37Z

‎Problem 3: no solution

Cosmo All at 13:52, 29 September 2013

2013-09-29T13:52:36Z

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=== Problem 1 === Consider a flat universe with several components $\rho=\sum_i \rho_i$, each with density $\rho_i$ and partial pressure $p_i$ being..."

2013-09-29T13:51:37Z

Created page with "<div id="Horizon34"></div> === Problem 1 === Consider a flat universe with several components $\rho=\sum_i \rho_i$, each with density $\rho_i$ and partial pressure $p_i$ being..."

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<div id="Horizon34"></div>
=== Problem 1 ===
Consider a flat universe with several components $\rho=\sum_i \rho_i$, each with density $\rho_i$ and partial pressure $p_i$ being related by the linear state equation $p_i =w_i \rho_i$. Find the particle horizon at present time $t_0$.
<div class="NavFrame collapsed">
<div class="NavHead">solution</div>
<div style="width:100%;" class="NavContent">
<p style="text-align: left;">
The particle horizon $L_p$ at the current moment $t_0$ is (see \ref{LpDp})
\[L_p =\lim\limits_{t_e \to 0} D_p (t_e ,t_0)\]
The proper distance can be written as
\[D_p = a_0\int_t^{t_0} \frac{dt'}{a(t')}
=- \int_{z(t)}^{z(t_0)} \frac{dz'}{H(z')}
=\int_0^z \frac{dz'}{H(z')}. \]
The $i$-th energy density in terms of redshift $z$ is
\[\rho_i =\rho_{0i}(1+z)^{3(1+w_i)},\]
and
\[H(z)=H_0 \sqrt{\sum \Omega_{0i}(1+z)^{3(1+w_i)}},\]
so
\begin{equation}
L_p (t_0)=H_0^{-1}\int \limits_{0}^{\infty}
\frac{dz}{\sqrt{\sum \Omega_{0i}(1+z)^{3(1+w_i)}}}
\end{equation}
</p>
</div>
</div>

<div id="Horizon35"></div>
=== Problem 2 ===
Suppose we know the current material composition of the Universe $\Omega_{i0}$, $w_i$ and its expansion rate as function of redshift $H(z)$. Find the particle horizon $L_p (z)$ and the event horizon $L_p (z)$ (i.e the distances to the respective surfaces along the surface $t=const$) at the time that corresponds to current observations with redshift $z$.
<div class="NavFrame collapsed">
<div class="NavHead">solution</div>
<div style="width:100%;" class="NavContent">
<p style="text-align: left;">
For the particle horizon we rewrite the previous result in terms of redshifts
\begin{equation}
L_p (z)=H^{-1} (z)\int \limits_{0}^{\infty}
\frac{dz'}{\sqrt{\sum \Omega_{i}(z)(1+z')^{3(1+w_i)}}},
\end{equation}
and take into account how the partial densities $\Omega_i$ depend on time: they are defined to satisfy
\begin{equation}
H^2 (z)=H_0^2 \sum \Omega_{i0} (1+z)^{3(1+w_i)}.
\end{equation}
at any $z$ (or, equivalently, $t$), thus $\Omega_i (z)$ by definition is the ratio of the $i$th term of the sum to the whole sum at any moment of time:
\begin{equation}
\Omega_i (z) = \Omega _{i0} \frac{H_0^2 }{H^2 (z)}(1 + z)^{3(1+w_i)}.
\end{equation}
Then for the particle horizon (and for event horizon in the same way) we obtain
\begin{align}
L_p (z) &= \frac{1}{H(z)}\int_0^\infty
\frac{dz'}{\sqrt {\sum\limits_i \Omega_i (z)(1 + z')^{3(1 +w_i )}}}; \label{LpZ}\\
L_e (z) &= \frac{1}{H(z)}\int_{-1}^{0}
\frac{dz'}{\sqrt {\sum\limits_i \Omega_i (z)(1 + z')^{3(1 +w_i )}}} \label{LeZ}.
\end{align}
</p>
</div>
</div>

<div id="Horizon36"></div>
=== Problem 3 ===
When are $L_p$ and $L_e$ equal? It is interesting to know whether both horizons might have or not the same values, and if so, how often this could happen.
\paragraph{No solution.}

<div id="Horizon37"></div>
=== Problem 4 ===
Find the particle and event horizons for any redshift $z$ in the standard cosmological model -- $\Lambda$CDM.
<div class="NavFrame collapsed">
<div class="NavHead">solution</div>
<div style="width:100%;" class="NavContent">
<p style="text-align: left;">
Using the general formulae (\ref{LpZ}--\ref{LeZ}), one obtains
\begin{align}
L_p(z) &= \frac{1}{H(z)}\int_0^\infty
\frac{dz'}{\sqrt {\Omega_m (z) (1 + z')^3 + \Omega_\Lambda (z)}},\\
L_e(z) &= \frac{1}{H(z)}\int_{-1}^{0}
\frac{dz'}{\sqrt {\Omega_m (z) (1 + z')^3 + \Omega_\Lambda (z)}},\\
&\Omega_m (z)=\Omega _{m0}\frac{H_0^2}{H(z)^2} (1 + z)^3 ,\\
&\Omega_\Lambda (z) = \Omega_{\Lambda 0} \frac{H_0^2}{H(z)^2}.
\end{align}
</p>
</div>
</div>

<div id="Horizon38"></div>
=== Problem 5 ===
Express the particle $L_p (z)$ and event $L_e (z)$ horizons in $\Lambda$CDM through the hyper-geometric function.
<div class="NavFrame collapsed">
<div class="NavHead">solution</div>
<div style="width:100%;" class="NavContent">
<p style="text-align: left;">

\begin{align}
L_p (z) &= \frac{2\sqrt {A(z)}}{H_0 \sqrt {\Omega _{\Lambda 0}}} F\left(\frac{1}{2},\frac{1}{6},\frac{7}{6}; - A(z) \right),\\
L_e (z) &= \frac{1}{H_0 \sqrt {\Omega _{\Lambda 0}}} F\left(\frac{1}{2},\frac{1}{3},\frac{4}{3}; - \frac{1}{A(z)} \right),\\
&A(z) = \frac{\Omega_{\Lambda 0}}{\Omega _{m0}} (1 + z)^{-3}.
\end{align}

← Older revision		Revision as of 13:52, 29 September 2013
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		+	[[Category:Horizons\|3]]
		+	__TOC__
	<div id="Horizon34"></div>		<div id="Horizon34"></div>
	=== Problem 1 ===		=== Problem 1 ===

@@ Line 8: / Line 8: @@
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">
-The particle horizon $L_p$ at the current moment $t_0$ is (see \ref{LpDp})
+The particle horizon $L_p$ at the current moment $t_0$ is (see [[Simple_Math#LpDp|this relation]])
 \[L_p =\lim\limits_{t_e \to 0} D_p (t_e ,t_0)\]
 The proper distance can be written as

@@ Line 61: / Line 61: @@
 === Problem 3 ===
 When are $L_p$ and $L_e$ equal? It is interesting to know whether both horizons might have or not the same values, and if so, how often this could happen.
 <div id="Horizon37"></div>
 === Problem 4 ===
 Find the particle and event horizons for any redshift $z$ in the standard cosmological model -- $\Lambda$CDM.

Composite Models - Revision history

Igor: pr1: link to previous section

Igor: /* Problem 3 */ no solution

Cosmo All at 13:52, 29 September 2013

Cosmo All: Created page with " === Problem 1 === Consider a flat universe with several components $\rho=\sum_i \rho_i$, each with density $\rho_i$ and partial pressure $p_i$ being..."

Cosmo All: Created page with "
=== Problem 1 === Consider a flat universe with several components $\rho=\sum_i \rho_i$, each with density $\rho_i$ and partial pressure $p_i$ being..."