Cosmo All at 10:36, 8 November 2013

2013-11-08T10:36:56Z

Cosmo All: /* Problem 1 */

2013-11-08T10:28:56Z

‎Problem 1

Cosmo All at 10:28, 8 November 2013

2013-11-08T10:28:14Z

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Cosmo All at 12:25, 5 November 2013

2013-11-05T12:25:30Z

Cosmo All at 12:24, 5 November 2013

2013-11-05T12:24:30Z

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Cosmo All at 11:18, 5 November 2013

2013-11-05T11:18:22Z

Cosmo All: Created page with "1 NOTOC"

2013-11-05T11:17:56Z

Created page with "1 __NOTOC__"

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[[Category:Interactions in the Dark Sector|1]]

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-=== Problem 1 ===
+=== Problem 2 ===
 Obtain the equation for the evolution of the DE  energy density for   $Q=\delta(a) H\rho_{dm}$.
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+=== Problem 3 ===
-=== Problem 1 ===
 Find $\rho_{dm}$ and $\rho_{de}$ in the case  $Q=\delta H\rho_{dm}$, $\delta=const$, $w_{de}=const$.
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-=== Problem 1 ===
+=== Problem 4 ===
 As was shown above, interaction between dark matter and dark energy leads to non--conservation of matter, or equivalently, to scale dependence for the mass of particles that constitute the dark matter. Show that, within the framework of the model of previous problem ($Q=\delta H\rho_{dm}$, $\delta=const$, $w_{de}=const$) the relative change of particles mass per Hubble time equals to the interaction constant.
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-=== Problem 1 ===
+=== Problem 5 ===
 Find $\rho_{dm}$ and $\rho_{de}$ in the case  $Q=\delta H\rho_{de}$, $\delta=const$, $w_{de}=const$.
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-=== Problem 1 ===
+=== Problem 6 ===
 Find $\rho_{dm}$ and $\rho_{de}$ in the case  $Q=\delta(a) H\rho_{de}$, $\delta(a)=\beta_0a^\xi$, $w_{de}=const$.
 <br/>
-(after  \cite{1111.6743}.)
+(after  [http://arxiv.org/abs/1111.6743])
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-=== Problem 1 ===
+=== Problem 7 ===
 Let's  look at a more general linear model for the expansion of a Universe that contains two interacting fluids with the equations of state
 \[p_1 = (\gamma_1-1)\rho_1,\]
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 	\[\dot\rho_1+3H\gamma_1\rho_1 = -\beta H\rho_1 + \alpha H\rho_2,\]
 	\[\dot\rho_2+3H\gamma_2\rho_2 = \beta H\rho_1 - \alpha H\rho_2.\]
-Here $\alpha$ and $\beta$ are constants describing the energy exchanges between the two fluids. Obtain the equation for $H(t)$ and find its solutions (After \cite{9702029,0604063}).
+Here $\alpha$ and $\beta$ are constants describing the energy exchanges between the two fluids. Obtain the equation for $H(t)$ and find its solutions (After [http://arxiv.org/abs/physics/9702029], [http://arxiv.org/abs/gr-qc/0604063]).
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-=== Problem 1 ===
+=== Problem 8 ===
 Show that the energy balance equations (modified conservation equations) for $Q\propto H$ do not depend on H.
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-=== Problem 1 ===
+=== Problem 9 ===
 The Hubble parameter is present in the first Friedmann equation quadratically. This gives rise to a useful symmetry within a class of FLRW models. Because of this quadratic dependence, Friedmann's equation remains invariant under a transformation $H\to-H$ for the spatially flat case. This means it describes both expanding and contracting solutions. The transformation $H\to-H$ can be seen as a consequence of the change $a\to1/a$  of the scale factor of the FLRW metric. If, instead of just the first Friedmann equation, we want to make the whole system of Universe-describing equations invariant relative to this transformation, we must expand the set of values that undergo symmetry transformations. Then, when we refer to a duality transformation, we have in mind the following set of transformations
 \[H\to\bar H=-H,\quad \rho\to\bar\rho=\rho,\quad p\to\bar p=-2\rho-p,\quad \gamma\equiv\frac{\rho+p}{\rho}\to\bar\gamma\equiv\frac{\bar\rho+\bar p}{\bar\rho}=-\gamma.\]
-Generalize the duality transformation to the case of interacting components.(after \cite{0505096}.)
+Generalize the duality transformation to the case of interacting components.(after [http://arxiv.org/abs/gr-qc/0505096].)
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      <p style="text-align: left;">The conservation equation for DE density is
-\[\dot\rho_{de}+3H(1+w_{de})\rho_{de} + \delta(a) H\rho_{dm}=0/\]
+\[\dot\rho_{de}+3H(1+w_{de})\rho_{de} + \delta(a) H\rho_{dm}=0\]
 It is more useful to work with the variable $u=\ln a$. Substitute the solution for $\rho_{dm}$, obtained in previous problem
 	\[\rho_{dm}(u)=\rho_{dm0}e^{-3u}\exp\left(-\int\limits_u^0\delta(u')du'\right)\]
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 === Problem 1 ===
 Find $\rho_{dm}$ and $\rho_{de}$ in the case  $Q=\delta H\rho_{dm}$, $\delta=const$, $w_{de}=const$.

← Older revision		Revision as of 12:25, 5 November 2013
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← Older revision		Revision as of 11:18, 5 November 2013
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−	[[Category:Interactions in the Dark Sector\|1]]	+	[[Category:Interactions in the Dark Sector\|3]]

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Simple linear models - Revision history