Time-dependent Cosmological Constant - Revision history

Cosmo All at 18:20, 2 December 2012

2012-12-02T18:20:16Z

Cosmo All: /* Problem 1 */

2012-12-02T18:19:20Z

‎Problem 1

Cosmo All at 18:14, 2 December 2012

2012-12-02T18:14:39Z

Cosmo All: Created page with "3"

2012-12-01T13:19:55Z

Created page with "3"

New page

[[Category:Dark Energy|3]]

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 <div id="DE20"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 2 ===
 Show that when $G$ is constant, $\Lambda$ is also a constant if and only if the ordinary energy-momentum tensor $T_{\mu\nu}$ is also conserved.
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@@ Line 56: / Line 56: @@
 <div id="DE21"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 3 ===
 Show that in case the cosmological constant depends on time, the energy density related to the latter can be converted into matter.
 <div class="NavFrame collapsed">
@@ Line 73: / Line 73: @@
 <div id="DE22"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 4 ===
 Derive the time dependence of the scale factor for a flat Universe with the $\Lambda$--dynamical constant $\Lambda=\Lambda_0(1+\alpha t)$.
 <div class="NavFrame collapsed">
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 <div id="DE23"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 5 ===
 Construct the dynamics of the Universe in the cosmological model with $\Lambda=\sigma H ,\; \sigma>0.$
 <div class="NavFrame collapsed">
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 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 6 ===
 Construct the dynamics of the Universe in the cosmological model with $\Lambda(H)=\sigma H +3\beta H^2$ and $\rho=\rho_\Lambda+\rho_m$.
 <!--<div class="NavFrame collapsed">

@@ Line 96: / Line 96: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">Friedman equations for the considered model take the following form (see problem \ref{DE21}):
+     <p style="text-align: left;">Friedman equations for the considered model take the following form (see problem [[#DE21]]):
 \[\dot \rho _m  + 3H\rho _m  =  - \dot \rho_\Lambda k^2;\]
 \[\frac{3H^2 }  {k^2 } = \rho _m  + \rho_\Lambda k^2,\]
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 <div id="DE24_0"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 1 ===
 Construct the dynamics of the Universe in the cosmological model with $\Lambda(H)=\sigma H +3\beta H^2$ and $\rho=\rho_\Lambda+\rho_m$.

← Older revision		Revision as of 18:14, 2 December 2012
Line 1:		Line 1:
	[[Category:Dark Energy\|3]]		[[Category:Dark Energy\|3]]
		+
		+	__NOTOC__
		+
		+
		+	'''''(Inspired by I.Shapiro,J.Sola, H.Stefancic, [http://arxiv.org/abs/hep-ph/0410095 hep-ph/0410095])'''''
		+
		+
		+
		+
		+	<div id="DE19"></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 1 ===
		+	Obtain the analogue of the conservation equation $\dot\rho+3H(\rho+p)=0$ for the case when the gravitation constant $G$ and the cosmological "constant" $\Lambda$ depend on time.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">\[R_{\mu\nu}-\frac12Rg_{\mu\nu}
		+	=8\pi G\tilde{T}_{\mu\nu}, \quad
		+	\tilde{T}_{\mu\nu}\equiv T_{\mu\nu}
		+	+\rho_\Lambda g_{\mu\nu},\quad
		+	\rho_\Lambda=\frac{\Lambda}{8\pi G}.\]
		+	Here $T_{\mu\nu}$ is the ordinary energy-momentum tensor associated with isotropic matter and radiation, and $\rho_\Lambda$ represents the vacuum energy density. Then the Bianchi identities imply that $\nabla^\mu(G\tilde{T}_{\mu\nu})=0$ and with the help of the FLRW metric a straightforward calculation yields
		+	\[\frac{d}{dt}\left[G(\rho_\Lambda+\rho)\right]
		+	+3GH(\rho+p)=0.\]
		+	The simplest possible case is when both $G$ and $\Lambda$ are constants. In this case we revert to the conservation equation its original form
		+	\[\dot\rho+3H(\rho+p)=0.\]</p>
		+	</div>
		+	</div></div>
		+
		+
		+
		+	<div id="DE20"></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 1 ===
		+	Show that when $G$ is constant, $\Lambda$ is also a constant if and only if the ordinary energy-momentum tensor $T_{\mu\nu}$ is also conserved.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">If $G=const$,
		+	\[\frac{d}{dt}\left[G(\rho_\Lambda+\rho)\right]
		+	+3GH(\rho+p)=\dot\rho_\Lambda+\dot\rho
		+	+3H(\rho+p)=0.\]
		+	If the ordinary energy-momentum tensor is separately conserved
		+	\[\nabla^\mu T_{\mu\nu}=0\quad
		+	\Rightarrow\quad\dot\rho+3H(\rho+p)=0.\]
		+	Consequently,
		+	\[\dot\rho_\Lambda=0\quad
		+	\Rightarrow\quad \Lambda=const.\]
		+	The first non-trivial situation appears when $\Lambda=\Lambda(t)$ but $G$ is still constant. This possibility will be discussed bellow.</p>
		+	</div>
		+	</div></div>
		+
		+
		+
		+	<div id="DE21"></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 1 ===
		+	Show that in case the cosmological constant depends on time, the energy density related to the latter can be converted into matter.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">Conservation equation for the case of Universe composed by matter and dark energy takes the form
		+	\[\dot \rho _{tot} + 3H(\rho _{tot} + p_{tot} ) = 0; \] where $p_m =0$ and $p_\Lambda = - \rho _\Lambda$. Assuming that $\dot\rho _\Lambda \ne 0$, one obtains
		+	\[\dot \rho _m + 3H\rho _m
		+	= - \dot \rho _\Lambda.\]
		+	Under assumption that the cosmological constant decays, i.e. $\dot \rho _\Lambda < 0$, one can see from this equation that the cosmological constant can be converted into matter.</p>
		+	</div>
		+	</div></div>
		+
		+
		+
		+	<div id="DE22"></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 1 ===
		+	Derive the time dependence of the scale factor for a flat Universe with the $\Lambda$--dynamical constant $\Lambda=\Lambda_0(1+\alpha t)$.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">\[\left(\frac{\dot a}{a}\right)^2=\frac\Lambda3=\frac{\Lambda_0}{3}(1+\alpha t);\]
		+	\[\ln a=\frac{2}{3\alpha}\sqrt{\frac{\Lambda_0}{3}}(1+\alpha t)^{3/2}+C;\]
		+	$$
		+	a = a(t=0)\exp \left\{ {{2 \over {3\alpha }}\sqrt {{{\Lambda _0 } \over
		+	3}} \left[ {\left( {1 + \alpha t} \right)^{3/2} - 1} \right]}
		+	\right\}.$$</p>
		+	</div>
		+	</div></div>
		+
		+
		+
		+	<div id="DE23"></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 1 ===
		+	Construct the dynamics of the Universe in the cosmological model with $\Lambda=\sigma H ,\; \sigma>0.$
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">Friedman equations for the considered model take the following form (see problem \ref{DE21}):
		+	\[\dot \rho _m + 3H\rho _m = - \dot \rho_\Lambda k^2;\]
		+	\[\frac{3H^2 } {k^2 } = \rho _m + \rho_\Lambda k^2,\]
		+	where $k^2=8\pi G$.
		+	<br/>
		+	From these equations one finds that
		+	\[\rho _m = -{2\dot H}/{k^2},\]
		+	and substitution into
		+	\[{3H^2 } / {k^2 } = \rho _m + \rho_\Lambda k^2\] yields the equation governing the evolution of the Universe
		+	\[{3H^2 } + {2\dot H} - \sigma H = 0.\]
		+	Its solution reads
		+	\[a = C\left[
		+	{\exp \left( {\sigma t/2} \right) - 1} \right]^{2/3}.\]
		+	Scale factor dependencies for the energy densities are
		+	\[\rho _m = {\frac{\sigma ^2 C^3 }{3a^3 }}
		+	+ \frac{\sigma ^2 C^{3/2} }{3a^{3/2} };\]
		+	\[\rho_\Lambda
		+	= \frac{1}{k^2}\left(\frac{\sigma ^2 } 3
		+	+ \frac{\sigma ^2 C^{3/2} } {3a^{3/2} }\right).\]
		+	Time dependence of the Hubble parameter is
		+	\[H =\frac {\sigma /3} {1
		+	- \exp \left( { - \sigma t/2} \right)}.\]
		+	Finally,
		+	\[H(z) = H_0 \left[ {1 - \Omega _{m0}
		+	+ \Omega _{m0} \left( {z + 1} \right)^{3/2} } \right],\]
		+	where
		+	$\Omega _{m0} = \rho _{m0} 8 \pi G /\left( {3H_0^2 } \right)$ is current value of relative density of matter.</p>
		+	</div>
		+	</div></div>
		+
		+
		+
		+	<div id="DE24_0"></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 1 ===
		+	Construct the dynamics of the Universe in the cosmological model with $\Lambda(H)=\sigma H +3\beta H^2$ and $\rho=\rho_\Lambda+\rho_m$.
		+	<!--<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;"></p>
		+	</div>
		+	</div>--></div>