Difference between revisions of "Cosmological horizons"
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− | [[Category:Dynamics of the Universe in the Big Bang Model]] | + | [[Category:Dynamics of the Universe in the Big Bang Model|6]] |
− | + | __TOC__ | |
+ | |||
+ | == 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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\[a(t)\sim t^{\frac{2}{3}(w+1)^{-1}}.\] | \[a(t)\sim t^{\frac{2}{3}(w+1)^{-1}}.\] | ||
Then | Then | ||
− | \[L_{p}=t\;\Big(1+\frac{2}{1+3w}\Big).\] | + | \[L_{p}=t\;\Big(1+\frac{2}{1+3w}\Big).\]</p> |
− | + | ||
</div> | </div> | ||
− | </div> | + | </div></div> |
− | </div> | + | |
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− | === 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. | ||
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= \int_0^\eta d\eta '= \eta.\]</p> | = \int_0^\eta d\eta '= \eta.\]</p> | ||
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− | </div> | + | </div></div> |
− | </div> | + | |
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− | === 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. | ||
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<div class="NavHead">solution</div> | <div class="NavHead">solution</div> | ||
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− | <p style="text-align: left;">Using the | + | <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> | + | |
− | + | ||
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− | === 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 $p=w\rho$. | ||
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= H_0^{-1}a^{\frac{1}{2}(1 +3w)}.\]</p> | = H_0^{-1}a^{\frac{1}{2}(1 +3w)}.\]</p> | ||
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− | </div> | + | </div></div> |
− | </div> | + | |
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− | === 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 $w$. | ||
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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> | ||
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− | === 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. | ||
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<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> | <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> | + | |
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− | === 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. | ||
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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> | ||
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− | </div> | + | </div></div> |
− | </div> | + | |
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− | === 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. | ||
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<div class="NavHead">solution</div> | <div class="NavHead">solution</div> | ||
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− | <p style="text-align: left;">We use again the expression for the radius of the $3$-sphere (\ | + | <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} | \[\frac{V_H}{V_U} | ||
\approx\frac{(4\pi/3)R_{H}^3}{2\pi^2 a_{0}^{3}} | \approx\frac{(4\pi/3)R_{H}^3}{2\pi^2 a_{0}^{3}} | ||
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Taking into account that observational data provide $|\Omega_{curv}|<0.02$, we get $V_H/V_U<0.06\%$, so the considered Universe contains more than 1 600 Hubbe spheres. More realistic estimate, with particle horizon instead of the Hubble sphere in the Standard cosmological model, is made in chapter 11.</p> | Taking into account that observational data provide $|\Omega_{curv}|<0.02$, we get $V_H/V_U<0.06\%$, so the considered Universe contains more than 1 600 Hubbe spheres. More realistic estimate, with particle horizon instead of the Hubble sphere in the Standard cosmological model, is made in chapter 11.</p> | ||
</div> | </div> | ||
− | </div> | + | </div></div> |
− | </div> | + | |
− | + | ||
+ | == FLRW in terms of proper distances == | ||
The following five problems are based on work by F. Melia ([http://arxiv.org/abs/0711.4181 arXiv:0711.4181], [http://arxiv.org/abs/0907.5394 arXiv:0907.5394]). | The following five problems are based on work by F. Melia ([http://arxiv.org/abs/0711.4181 arXiv:0711.4181], [http://arxiv.org/abs/0907.5394 arXiv:0907.5394]). | ||
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<div id="dyn-Melia1"></div> | <div id="dyn-Melia1"></div> | ||
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− | === Problem 9 | + | |
+ | === Problem 9: cosmic horizon in a flat Universe === | ||
Show that in a flat Universe $R_{h}=H^{-1}(t)$. | Show that in a flat Universe $R_{h}=H^{-1}(t)$. | ||
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=\frac{1}{H(t)}.\]</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 $R=a(t)r$. | ||
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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 $r\to R_h$. | ||
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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> | ||
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− | </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 $w>-1$. | ||
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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;"> | + | <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)''' | + | '''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}}dR} \Big]^2} - \Phi^{-1}d{R^2} |
− | - | + | - R^{2}d{\Omega ^2},\\ |
− | \Phi& = 1 - | + | \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} | + | \quad \dot f = \frac{1}{2t}\\ |
− | d{s^2} &= \Phi {\Big[ {dt + \left( {\frac{R} | + | 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} | + | \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} | + | 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} | + | 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} | + | \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
Contents
- 1 Particle horizon and the Hubble sphere
- 1.1 Problem 1: particle horizon in one-component Universe
- 1.2 Problem 2: particle horizon and the age of the Universe
- 1.3 Problem 3: radiation domination
- 1.4 Problem 4: comoving Hubble
- 1.5 Problem 5: Hubble radius and particle horizon
- 1.6 Problem 6: observed Universe
- 1.7 Problem 7: particle horizon and curvature scale
- 1.8 Problem 8: the observable portion of the Universe
- 2 FLRW in terms of proper distances
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 $p=w\rho$.
Using the definition
\[L_{p}\left( t \right)
= a\left( t \right)\int_0^t
\frac{dt'}{a\left(t' \right)},\]
so if $a(t)\sim t^{n}$, we get
\[L_{p}=\frac{t}{1-n}.\]
Then in each specific case
a) $a\sim t^{1/2}\;\Rightarrow\;L_{p}( t) = 2t$,
b) $a\sim t^{2/3}\;\Rightarrow\;L_{p}( t) = 3t$,
c) From the energy conservation law in the form $\rho a^{3(1+w)}=const$ and the first Friedman equation we get:
\[a(t)\sim t^{\frac{2}{3}(w+1)^{-1}}.\]
Then
\[L_{p}=t\;\Big(1+\frac{2}{1+3w}\Big).\]
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.
The comoving horizon is \[L_{p}^{(comov)}(t)\equiv\frac{L_{p}(t)}{a(t)} = \int_0^t \frac{dt'}{a(t')} = \int_0^\eta d\eta '= \eta.\]
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.
Using the 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 \[H = \frac{\dot a}{a} = \frac{1}{2t},\quad \text{and}\quad R_H = H^{-1}=2t = {L_{p}}.\]
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 $p=w\rho$.
Again using $a(t)=A_{0} t^{\frac{2}{3(1+w)}}$, we get \[\frac{R_H}{a} = (aH)^{-1} = H_0^{-1}a^{\frac{1}{2}(1 +3w)}.\]
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 $w$.
Starting from the definition, \[L_{p}^{(comov)}\equiv\frac{L_{p}}{a} = \int\limits_{0}^{t} \frac{dt'}{a\left( t' \right)} =\int\limits_{0}^{a}\frac{da'}{H(a')^2} = \int\limits_{0}^{a} R_Hd(\ln a').\]
Problem 6: observed Universe
Show that in an open Universe filled with dust the number of observed galaxies increases with time.
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.
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.
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 \[ a_{0}=\frac{H_{0}^{-1}}{\sqrt{|\Omega_{curv}|}},\] we get that \[\frac{L_{p}}{R_{curv}}\bigg|_{present} \sim \sqrt{|\Omega_{curv}|}.\] Since the observational data dictate $|\Omega_{curv}|\leq 0.02$, at present we have $L_{p}/R_{curv}\ll 1$. Taking into account that for a power-law evolution of $a(t)$ we have $L_{p}\sim t\sim H^{-1}$, for the time evolution of this ratio we obtain \[\frac{L_{p}}{R_{curv}}\sim \frac{t}{a(t)}.\] Both for a Universe dominated by matter and by radiation the power exponent in $a(t)\sim t^{n}$ is less then unity, so $L_{p}/R_{curv}$ increases with time. In the present epoch $n=2/3$ so $L_{p}/R_{curv}\sim t^{1/3}$. In the early Universe, filled with radiation, $n=1/2$ and \[\frac{L_{p}}{R_{curv}}\sim t^{1/2}.\] Thus if curvature is negligible in some sence now, then it was all the more insignificant at any moment in the past.
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.
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 $V=2\pi^{2}a^{3}$ (problem of chapter 2). Then neglecting curvature on the scales of the Hubble sphere (see problem), we obtain \[\frac{V_H}{V_U} \approx\frac{(4\pi/3)R_{H}^3}{2\pi^2 a_{0}^{3}} =\frac{2}{3\pi}|\Omega_{curv}|^{3/2}.\] 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.
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 $R_{h}=H^{-1}(t)$.
By definition \[R_{h}=\Big(\frac{3}{8\pi G \rho}\Big)^{1/2} =\frac{1}{H(t)}.\]
Problem 10: FLRW in terms of proper distances
Represent the FLRW metric in terms of the observer-dependent coordinate $R=a(t)r$.
It is convenient to recast FLRW metric using a new function $f(t)$ \[a(t)=e^{f(t)}.\] In that case \[d{s^2} = {c^2}d{t^2} - {e^{2f(t)}}\left( {d{r^2} + {r^2}d{\Omega ^2}} \right).\] Making the coordinate transformation to the radial coordinate $r=R\,e^{-t}$, we obtain \[ds^2 = \Phi \Big[dt +\big(R\dot f\big){\Phi^{ - 1}}dR\Big]^2 -\Phi^{- 1} dR^2 - R^2 d\Omega^2,\] where for convenience we have defined the function \[\Phi=1-\big(R\dot f\big)^{2}.\] It is easy to see, that the radius of the cosmic horizon for the observer at the origin is \[R_{h}=1/\dot{f}=\frac{a}{\dot{a}}=\frac{1}{H}\] and \[\Phi=1-\frac{R}{R_h}.\] Finally, we obtain \begin{align*} ds^{2}&=\Phi\Big[dt+\frac{R}{R_h}\Phi^{-1}dR\Big]^{2} -\Phi^{-1}dR^2 -R^{2}d\Omega^2 =\\ &=\Big(1-\frac{R}{R_h}\Big) \Big[dt+\frac{R/R_h}{1-R/R_h}dR\Big]^{2} -\frac{dR^2}{1-R/R_h}-R^2 d\Omega^2. \end{align*}
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 $r\to R_h$.
Let us examine the behavior of the interval $ds$ connecting any arbitrary pair of spacetime events at $R$. For an interval produced at $R$ by the advancement of time only $dR=d\Omega=0$, metric obtained in previous problem gives \[ds^{2}=\Phi dt^{2}.\] 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.
Problem 12: horizon only increases
Show that $R_h$ is an increasing function of cosmic time $t$ for any cosmology with $w>-1$.
It is straightforward to demonstrate from Friedman equations that \[\dot{R}_{h}=\frac{3}{2}(1+w).\] Consequently, $\dot{R}_{h}>0$ for $w>-1$. "$R_h$ 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 $t_0$".
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 ($w=0$).
a) De Sitter
\[H=H_{0}=const,\quad a(t)=e^{H_{0}t},\quad
f=\ln a(t)=H_{0}t.\]
In this case $\dot{R}_{h}=0$ and therefore $R_h$ is fixed
\[R_{h}=\frac{1}{H_0}.\]
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}
d{s^2}& = \Phi {\Big[ {cdt + \big(\frac{R}{t}\big)
{\Phi ^{ - 1}}dR} \Big]^2} - \Phi^{-1}d{R^2}
- R^{2}d{\Omega ^2},\\
\Phi& = 1 - \left(\frac{R}{ct}\right)^2
\end{align}
The cosmic time $dt$ diverges for a measurable line element as $R\to R_h =t$.
c) In the case of radiation domination
\begin{align*}
a(t)& = {\left( {2{H_0}t} \right)^{1/2}},
\quad f(t) = \frac{1}{2}\ln \left( {2{H_0}t} \right),
\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},\\
\Phi & = 1 - \left( {\frac{R}{2t}} \right)^2.
\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.
d) Matter domination:
\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}\\
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}
\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.