Difference between revisions of "Cosmological horizons"

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[[Category:Dynamics of the Universe in the Big Bang Model]]
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[[Category:Dynamics of the Universe in the Big Bang Model|6]]
__NOTOC__
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__TOC__
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 +
== Particle horizon and the Hubble sphere ==
 
<div id="dyn47"></div>
 
<div id="dyn47"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1. ===
+
=== Problem 1: particle horizon in one-component Universe ===
 
Calculate the particle horizon for a Universe with dominating
 
Calculate the particle horizon for a Universe with dominating
  
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'''a)''' $a\sim t^{1/2}\;\Rightarrow\;L_{p}( t) = 2t$,<br/>
 
'''a)''' $a\sim t^{1/2}\;\Rightarrow\;L_{p}( t) = 2t$,<br/>
 
'''b)''' $a\sim t^{2/3}\;\Rightarrow\;L_{p}( t) = 3t$,<br/>
 
'''b)''' $a\sim t^{2/3}\;\Rightarrow\;L_{p}( t) = 3t$,<br/>
'''c)''' From the energy conservation law in the form $\rho a^{3(1+w)}=const$ and the first Friedman equation we get (\ref{dyn-a(t-w)}):
+
'''c)''' From the energy conservation law in the form $\rho a^{3(1+w)}=const$ and the first Friedman equation we get:
 
\[a(t)\sim t^{\frac{2}{3}(w+1)^{-1}}.\]
 
\[a(t)\sim t^{\frac{2}{3}(w+1)^{-1}}.\]
 
Then
 
Then
\[L_{p}=t\;\Big(1+\frac{2}{1+3w}\Big).\]
+
\[L_{p}=t\;\Big(1+\frac{2}{1+3w}\Big).\]</p>
\end{description}</p>
+
 
   </div>
 
   </div>
</div>
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</div></div>
</div>
+
  
  
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<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
  
=== Problem 2 ===
+
=== Problem 2: particle horizon and the age of the Universe ===
 
Show the the comoving particle horizon equals to the age of the Universe in conformal time.
 
Show the the comoving particle horizon equals to the age of the Universe in conformal time.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
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= \int_0^\eta  d\eta '= \eta.\]</p>
 
= \int_0^\eta  d\eta '= \eta.\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
<div id="dyn49"></div>
 
<div id="dyn49"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 3. ===
+
=== Problem 3: radiation domination ===
 
Show that, if ultrarelativistic matter is dominating in the matter content of a spatially flat Universe ($k=0$), its particle horizon coincides with the Hubble radius.
 
Show that, if ultrarelativistic matter is dominating in the matter content of a spatially flat Universe ($k=0$), its particle horizon coincides with the Hubble radius.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">Using the result of problem \ref{dyn47}, for radiation $L_{p}=2t$. On the other hand, from $a\sim t^{1/2}$ we have
+
     <p style="text-align: left;">Using the [[#dyn47|expression for particle horizon in one-component Universe]], we see that for radiation $L_{p}=2t$. On the other hand, from $a\sim t^{1/2}$ we have
 
\[H = \frac{\dot a}{a} = \frac{1}{2t},\quad
 
\[H = \frac{\dot a}{a} = \frac{1}{2t},\quad
 
\text{and}\quad R_H = H^{-1}=2t = {L_{p}}.\]</p>
 
\text{and}\quad R_H = H^{-1}=2t = {L_{p}}.\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
<div id="dyn35"></div>
 
<div id="dyn35"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 4. ===
+
 
 +
=== Problem 4: comoving Hubble ===
 
Find the comoving Hubble radius $R_{H}/a$ as function of the scale factor for a spatially flat Universe that consists of one component with equation of state $p=w\rho$.
 
Find the comoving Hubble radius $R_{H}/a$ as function of the scale factor for a spatially flat Universe that consists of one component with equation of state $p=w\rho$.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">Again using (\ref{dyn-a(t-w)}), we get
+
     <p style="text-align: left;">Again using $a(t)=A_{0} t^{\frac{2}{3(1+w)}}$, we get
 
\[\frac{R_H}{a} = (aH)^{-1}
 
\[\frac{R_H}{a} = (aH)^{-1}
 
= H_0^{-1}a^{\frac{1}{2}(1 +3w)}.\]</p>
 
= H_0^{-1}a^{\frac{1}{2}(1 +3w)}.\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
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+
  
  
 
<div id="dyn50"></div>
 
<div id="dyn50"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 5. ===
+
 
 +
=== Problem 5: Hubble radius and particle horizon ===
 
Express the comoving particle horizon $L_{p}/a$ through the comoving Hubble radius $R_{H}/a$ for the case of domination of a substance with state parameter $w$.
 
Express the comoving particle horizon $L_{p}/a$ through the comoving Hubble radius $R_{H}/a$ for the case of domination of a substance with state parameter $w$.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
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= \int\limits_{0}^{a} R_Hd(\ln a').\]</p>
 
= \int\limits_{0}^{a} R_Hd(\ln a').\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
<div id="dyn52"></div>
 
<div id="dyn52"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 6. ===
+
=== Problem 6: observed Universe ===
 
Show that in an open Universe filled with dust the number of observed galaxies increases with time.
 
Show that in an open Universe filled with dust the number of observed galaxies increases with time.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">The velocity of Hubble sphere's recession is $V = c(1 + q)$ (see problem \ref{V=c(1+q)}). In an open Universe filled with dust $q>0$ so the Hubble sphere's velocity exceeds $c$ by $cq$ and overtakes the galaxies that are situated on it at the moment. Thus the galaxies originally outside of the Hubble sphere enter within and the number of observed galaxies increases.</p>
+
     <p style="text-align: left;">The velocity of Hubble sphere's recession is $V = c(1 + q)$. In an open Universe filled with dust $q>0$ so the Hubble sphere's velocity exceeds $c$ by $cq$ and overtakes the galaxies that are situated on it at the moment. Thus the galaxies originally outside of the Hubble sphere enter within and the number of observed galaxies increases.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
<div id="dyn56"></div>
 
<div id="dyn56"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 7. ===
+
 
 +
=== Problem 7: particle horizon and curvature scale ===
 
Show that even in early Universe the scale of particle horizon is much less than the curvature radius, and thus curvature does not play significant role within the horizon.
 
Show that even in early Universe the scale of particle horizon is much less than the curvature radius, and thus curvature does not play significant role within the horizon.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">At present $L_{p}\sim t_{0}\sim H_{0}^{-1}$ and using the expression (\ref{a-H-Om}) for $a$ through the observables
+
     <p style="text-align: left;">At present $L_{p}\sim t_{0}\sim H_{0}^{-1}$ and using the expression $a_{0}=\frac{H_{0}^{-1}}{\sqrt{|\Omega_0 -1|}}$ for $a$ through the observables
 
\[ a_{0}=\frac{H_{0}^{-1}}{\sqrt{|\Omega_{curv}|}},\]
 
\[ a_{0}=\frac{H_{0}^{-1}}{\sqrt{|\Omega_{curv}|}},\]
 
we get that
 
we get that
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Thus if curvature is negligible in some sence now, then it was all the more insignificant at any moment in the past.</p>
 
Thus if curvature is negligible in some sence now, then it was all the more insignificant at any moment in the past.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
<div id="dyn53"></div>
 
<div id="dyn53"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 8. ===
+
 
 +
=== Problem 8: the observable portion of the Universe ===
 
Estimate the ratio of the volume enclosed by the Hubble sphere to the total volume of the closed Universe.
 
Estimate the ratio of the volume enclosed by the Hubble sphere to the total volume of the closed Universe.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">We use again the expression for the radius of the $3$-sphere (\ref{a-H-Om}) through observables in form $a_{0}=(H_{0}\sqrt{|\Omega_{curv}|})^{-1}$, and the one for its volume $V=2\pi^{2}a^{3}$ (problem \ref{equ22} of chapter 2). Then neglecting curvature on the scales of the Hubble sphere (see problem (\ref{dyn56})), we obtain
+
     <p style="text-align: left;">We use again the expression for the radius of the $3$-sphere ($ a_{0}=\frac{H_{0}^{-1}}{\sqrt{|\Omega_0 -1|}}$) through observables in form $a_{0}=(H_{0}\sqrt{|\Omega_{curv}|})^{-1}$, and the one for its volume $V=2\pi^{2}a^{3}$ ([[Friedman-Lemaitre-Robertson-Walker_(FLRW)_metric#equ22|problem of chapter 2]]). Then neglecting curvature on the scales of the Hubble sphere (see [[#dyn56|problem]]), we obtain
 
\[\frac{V_H}{V_U}
 
\[\frac{V_H}{V_U}
 
\approx\frac{(4\pi/3)R_{H}^3}{2\pi^2 a_{0}^{3}}
 
\approx\frac{(4\pi/3)R_{H}^3}{2\pi^2 a_{0}^{3}}
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Taking into account that observational data provide $|\Omega_{curv}|<0.02$, we get $V_H/V_U<0.06\%$, so the considered Universe contains more than 1 600 Hubbe spheres. More realistic estimate, with particle horizon instead of the Hubble sphere in the Standard cosmological model, is made in chapter 11.</p>
 
Taking into account that observational data provide $|\Omega_{curv}|<0.02$, we get $V_H/V_U<0.06\%$, so the considered Universe contains more than 1 600 Hubbe spheres. More realistic estimate, with particle horizon instead of the Hubble sphere in the Standard cosmological model, is made in chapter 11.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
 +
== FLRW in terms of proper distances ==
 
The following five problems are based on work by F. Melia ([http://arxiv.org/abs/0711.4181 arXiv:0711.4181], [http://arxiv.org/abs/0907.5394 arXiv:0907.5394]).
 
The following five problems are based on work by F. Melia ([http://arxiv.org/abs/0711.4181 arXiv:0711.4181], [http://arxiv.org/abs/0907.5394 arXiv:0907.5394]).
  
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<div id="dyn-Melia1"></div>
 
<div id="dyn-Melia1"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 9. ===
+
 
 +
=== Problem 9: cosmic horizon in a flat Universe ===
 
Show that in a flat Universe $R_{h}=H^{-1}(t)$.
 
Show that in a flat Universe $R_{h}=H^{-1}(t)$.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
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=\frac{1}{H(t)}.\]</p>
 
=\frac{1}{H(t)}.\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
<div id="dyn-Melia2"></div>
 
<div id="dyn-Melia2"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 10. ===
+
=== Problem 10: FLRW in terms of proper distances ===
 
Represent the FLRW metric in terms of the observer-dependent coordinate $R=a(t)r$.
 
Represent the FLRW metric in terms of the observer-dependent coordinate $R=a(t)r$.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
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\end{align*}</p>
 
\end{align*}</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
<div id="dyn-Melia3"></div>
 
<div id="dyn-Melia3"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 11. ===
+
=== Problem 11: divergences and the horizon ===
 
Show, that if we were to make a measurement at a fixed distance $R$ away from us, the time interval $dt$ corresponding to any measurable (non-zero) value of $ds$ must go to infinity as $r\to R_h$.
 
Show, that if we were to make a measurement at a fixed distance $R$ away from us, the time interval $dt$ corresponding to any measurable (non-zero) value of $ds$ must go to infinity as $r\to R_h$.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
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Function  $\Phi\to 0$ as $R\to R_h$, thus for any measurable (non-zero) value of $ds$ the interval $dt$ must go to infinity as $R\to R_h$. In the context of black-hole physics (see Chapter 4), we recognize this effect as the divergent gravitational redshift measured by a static observer outside of the event horizon.</p>
 
Function  $\Phi\to 0$ as $R\to R_h$, thus for any measurable (non-zero) value of $ds$ the interval $dt$ must go to infinity as $R\to R_h$. In the context of black-hole physics (see Chapter 4), we recognize this effect as the divergent gravitational redshift measured by a static observer outside of the event horizon.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
<div id="dyn-Melia4"></div>
 
<div id="dyn-Melia4"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 12. ===
+
=== Problem 12: horizon only increases ===
 
Show that $R_h$ is an increasing function of cosmic time $t$ for any cosmology with $w>-1$.
 
Show that $R_h$ is an increasing function of cosmic time $t$ for any cosmology with $w>-1$.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
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"$R_h$ is fixed only for de Sitter, in which $\rho$ is a cosmological constant and $w=-1$. In addition, there is clearly a demarcation at $w=-1/3$. When $w<-1/3$, $R_h$ increases more slowly than lightspeed ($c=1$ here), and therefore our universe would be delimited by this horizon because light would have traveled a distance $t_0$ greater than $R_{h}(t_0)$ since the big bang. On the other hand, $R_h$ is always greater than $t$ when $w>-1/3$, and our observational limit would then simply be set by the light travel distance $t_0$".</p>
 
"$R_h$ is fixed only for de Sitter, in which $\rho$ is a cosmological constant and $w=-1$. In addition, there is clearly a demarcation at $w=-1/3$. When $w<-1/3$, $R_h$ increases more slowly than lightspeed ($c=1$ here), and therefore our universe would be delimited by this horizon because light would have traveled a distance $t_0$ greater than $R_{h}(t_0)$ since the big bang. On the other hand, $R_h$ is always greater than $t$ when $w>-1/3$, and our observational limit would then simply be set by the light travel distance $t_0$".</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
<div id="dyn-Melia5"></div>
 
<div id="dyn-Melia5"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 13. ===
+
=== Problem 13: simple examples ===
 
Using FLRW metric in terms of the observer-dependent coordinate $R=a(t)r$, consider specific cosmologies:
 
Using FLRW metric in terms of the observer-dependent coordinate $R=a(t)r$, consider specific cosmologies:
  
'''a)''' the De Sitter Universe ;
+
'''a)''' the De Sitter Universe;
  
 
'''b)''' a cosmology with  $R_h =t$, ($w=-1/3$);
 
'''b)''' a cosmology with  $R_h =t$, ($w=-1/3$);
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   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">''''a)''' De Sitter
+
     <p style="text-align: left;">
 +
'''a)''' De Sitter
 
\[H=H_{0}=const,\quad a(t)=e^{H_{0}t},\quad
 
\[H=H_{0}=const,\quad a(t)=e^{H_{0}t},\quad
 
f=\ln a(t)=H_{0}t.\]
 
f=\ln a(t)=H_{0}t.\]
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\[R_{h}=\frac{1}{H_0}.\]
 
\[R_{h}=\frac{1}{H_0}.\]
  
'''b)''' an equation of state $w=-1/3$ is the only one for which the current age, $t_0$, of the Universe can equal the light-crossing time, $t_{h}=R_h$. In this case
+
'''b)''' the equation of state $w=-1/3$ is the only one for which the current age, $t_0$, of the Universe can equal the light-crossing time, $t_{h}=R_h$. In this case
 
\begin{align}
 
\begin{align}
 
d{s^2}& = \Phi {\Big[ {cdt + \big(\frac{R}{t}\big)
 
d{s^2}& = \Phi {\Big[ {cdt + \big(\frac{R}{t}\big)
{\Phi ^{ - 1}}dR} \Big]^2} - {\Phi ^{ - 1}}d{R^2}
+
{\Phi ^{ - 1}}dR} \Big]^2} - \Phi^{-1}d{R^2}
- {R^2}d{\Omega ^2},\\
+
- R^{2}d{\Omega ^2},\\
\Phi& = 1 - {\left( {\frac{R}{{ct}}} \right)^2}
+
\Phi& = 1 - \left(\frac{R}{ct}\right)^2
 
\end{align}
 
\end{align}
 
The cosmic time $dt$ diverges for a measurable line element as $R\to R_h =t$.<br/>
 
The cosmic time $dt$ diverges for a measurable line element as $R\to R_h =t$.<br/>
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a(t)& = {\left( {2{H_0}t} \right)^{1/2}},
 
a(t)& = {\left( {2{H_0}t} \right)^{1/2}},
 
\quad f(t) = \frac{1}{2}\ln \left( {2{H_0}t} \right),
 
\quad f(t) = \frac{1}{2}\ln \left( {2{H_0}t} \right),
\quad \dot f = \frac{1}{{2t}}\\
+
\quad \dot f = \frac{1}{2t}\\
d{s^2} &= \Phi {\Big[ {dt + \left( {\frac{R}{{2t}}} \right){\Phi ^{ - 1}}dR} \Big]^2} - {\Phi ^{ - 1}}d{R^2} - {R^2}d{\Omega ^2},\\
+
d{s^2} &= \Phi {\Big[ {dt + \left( {\frac{R}{2t}} \right){\Phi ^{ - 1}}dR} \Big]^2} - {\Phi ^{ - 1}}d{R^2} - {R^2}d{\Omega ^2},\\
\Phi & = 1 - \left( {\frac{R}{{2t}}} \right)^2.
+
\Phi & = 1 - \left( {\frac{R}{2t}} \right)^2.
 
\end{align*}
 
\end{align*}
 
Thus, measurements made at a fixed $R$ and $t$ still produce a gravitationally-induced dilation of $dt$ as $R$ increases, but this effect never becomes divergent within that portion of the Universe (i.e., within $t_0$) that remains observable since the Big Bang.<br/>
 
Thus, measurements made at a fixed $R$ and $t$ still produce a gravitationally-induced dilation of $dt$ as $R$ increases, but this effect never becomes divergent within that portion of the Universe (i.e., within $t_0$) that remains observable since the Big Bang.<br/>
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'''d)''' Matter domination:
 
'''d)''' Matter domination:
 
\begin{align*}
 
\begin{align*}
a(t) &= {\left( {3/2{H_0}t} \right)^{2/3}},\quad f(t) = \frac{2}{3}\ln \left( {3/2{H_0}t} \right),\quad \dot f = \frac{2}{{3t}}\\
+
a(t) &= {\left( {3/2{H_0}t} \right)^{2/3}},\quad f(t) = \frac{2}{3}\ln \left( {3/2{H_0}t} \right),\quad \dot f = \frac{2}{3t}\\
d{s^2} &= \Phi {\Big[ {dt + \left( {\frac{R}{{3t/2}}} \right){\Phi ^{ - 1}}dR} \Big]^2} - {\Phi ^{ - 1}}d{R^2} - {R^2}d{\Omega ^2},\\
+
d{s^2} &= \Phi {\Big[ {dt + \left( {\frac{R}{3t/2}} \right){\Phi ^{ - 1}}dR} \Big]^2} - {\Phi ^{ - 1}}d{R^2} - {R^2}d{\Omega ^2},\\
\Phi  &= 1 - {\left( {\frac{R}{{3t/2}}} \right)^2}
+
\Phi  &= 1 - {\left( {\frac{R}{3t/2}} \right)^2}
 
\end{align*}
 
\end{align*}
 
The situation is similar to that for a radiation dominated universe, in that $R_h$ always recedes from us faster than lightspeed. Although dilation is evident with increasing $R$, curvature alone does not produce a divergent redshift.</p>
 
The situation is similar to that for a radiation dominated universe, in that $R_h$ always recedes from us faster than lightspeed. Although dilation is evident with increasing $R$, curvature alone does not produce a divergent redshift.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+

Latest revision as of 22:03, 12 November 2012

Particle horizon and the Hubble sphere

Problem 1: particle horizon in one-component Universe

Calculate the particle horizon for a Universe with dominating

a) radiation,

b) dust,

c) matter with state equation $p=w\rho$.


Problem 2: particle horizon and the age of the Universe

Show the the comoving particle horizon equals to the age of the Universe in conformal time.


Problem 3: radiation domination

Show that, if ultrarelativistic matter is dominating in the matter content of a spatially flat Universe ($k=0$), its particle horizon coincides with the Hubble radius.


Problem 4: comoving Hubble

Find the comoving Hubble radius $R_{H}/a$ as function of the scale factor for a spatially flat Universe that consists of one component with equation of state $p=w\rho$.


Problem 5: Hubble radius and particle horizon

Express the comoving particle horizon $L_{p}/a$ through the comoving Hubble radius $R_{H}/a$ for the case of domination of a substance with state parameter $w$.


Problem 6: observed Universe

Show that in an open Universe filled with dust the number of observed galaxies increases with time.


Problem 7: particle horizon and curvature scale

Show that even in early Universe the scale of particle horizon is much less than the curvature radius, and thus curvature does not play significant role within the horizon.


Problem 8: the observable portion of the Universe

Estimate the ratio of the volume enclosed by the Hubble sphere to the total volume of the closed Universe.

FLRW in terms of proper distances

The following five problems are based on work by F. Melia (arXiv:0711.4181, arXiv:0907.5394).

Standard cosmology is based on the FLRW metric for a spatially homogeneous and isotropic three-dimensional space, expanding or contracting with time. In the coordinates used for this metric, $t$ is the cosmic time, measured by a comoving observer (and is the same everywhere), $a(t)$ is the expansion factor, and $r$ is an appropriately scaled radial coordinate in the comoving frame.

F.Melia demonstrated the usefulness of expressing the FRLW metric in terms of an observer-dependent coordinate $R=a(t)r$, which explicitly reveals the dependence of the observed intervals of distance, $dR$, and time on the curvature induced by the mass-energy content between the observer and $R$; in the metric, this effect is represented by the proximity of the physical radius $R$ to the cosmic horizon $R_{h}$, defined by the relation \[R_{h}=2G\,M(R_h).\] In this expression, $M(R_h)$ is the mass enclosed within $R_h$ (which terns out to be the Hubble sphere). This is the radius at which a sphere encloses sufficient mass-energy to create divergent time dilation for an observer at the surface relative to the origin of the coordinates.


Problem 9: cosmic horizon in a flat Universe

Show that in a flat Universe $R_{h}=H^{-1}(t)$.


Problem 10: FLRW in terms of proper distances

Represent the FLRW metric in terms of the observer-dependent coordinate $R=a(t)r$.


Problem 11: divergences and the horizon

Show, that if we were to make a measurement at a fixed distance $R$ away from us, the time interval $dt$ corresponding to any measurable (non-zero) value of $ds$ must go to infinity as $r\to R_h$.


Problem 12: horizon only increases

Show that $R_h$ is an increasing function of cosmic time $t$ for any cosmology with $w>-1$.


Problem 13: simple examples

Using FLRW metric in terms of the observer-dependent coordinate $R=a(t)r$, consider specific cosmologies:

a) the De Sitter Universe;

b) a cosmology with $R_h =t$, ($w=-1/3$);

c) radiation dominated Universe ($w=1/3$);

d) matter dominated Universe ($w=0$).