Difference between revisions of "Dynamics of the Universe in terms of redshift and conformal time"

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(Problem 11.)
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[[Category:Dynamics of the Universe in the Big Bang Model|8]]
 
[[Category:Dynamics of the Universe in the Big Bang Model|8]]
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=== Problem 1. ===
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=== Problem 1: the first Friedman equation ===
 
Express the first Friedman equation in terms of redshift and analyze the contribution of different terms in different epochs.
 
Express the first Friedman equation in terms of redshift and analyze the contribution of different terms in different epochs.
 
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\Omega _{r0}(1+z)^4 + \Omega_{m0}(1+z)^3+
 
\Omega _{r0}(1+z)^4 + \Omega_{m0}(1+z)^3+
 
\Omega_{k0}(1+z)^2\right].\]
 
\Omega_{k0}(1+z)^2\right].\]
At large $z$ (early Universe) the term with the highest power becomes dominating -- that is, the one with radiation. As the term with curvature is very small at present (see [[The role of curvature in the dynamics of the Universe#dyn3|problem]]), it was all the more negligible before. On the other hand, it should become dominating in the future ($1+z\to 0$), but in the standard cosmological model, which will be discussed in the corresponding chapter later, the cosmological constant enters the play before that, and the curvature term does not ever have the chance to shine.</p>
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At large $z$ (early Universe) the term with the highest power becomes dominating -- that is, the one with radiation. As the term with curvature [[The role of curvature in the dynamics of the Universe#dyn3|is very small at present]], it was all the more negligible before. On the other hand, it should become dominating in the future ($1+z\to 0$), but in the standard cosmological model, which will be discussed in the corresponding chapter later, the cosmological constant enters the play before that, and the curvature term does not ever have the chance to shine.</p>
 
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=== Problem 2. ===
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=== Problem 2: $\eta(a)$ in one-component Universe ===
 
Find the conformal time as function of the scale factor for a Universe with domination of '''a)''' radiation and '''b)''' non-relativistic matter.
 
Find the conformal time as function of the scale factor for a Universe with domination of '''a)''' radiation and '''b)''' non-relativistic matter.
 
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$a(t) \sim t^{2/3}$, so $\eta\sim a^{1/2}.$</p>
 
$a(t) \sim t^{2/3}$, so $\eta\sim a^{1/2}.$</p>
 
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=== Problem 3. ===
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=== Problem 3: $t(z)$ for matter domination ===
 
Find the relation between time and redshift in the Universe with dominating matter.
 
Find the relation between time and redshift in the Universe with dominating matter.
 
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\left(1-\frac{1}{(1 + z)^{3/2}}\right).\]</p>
 
\left(1-\frac{1}{(1 + z)^{3/2}}\right).\]</p>
 
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=== Problem 4. ===
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=== Problem 4: $a(\eta)$ for radiation domination ===
 
Derive $a(\eta)$ for a spatially flat Universe with dominating radiation.
 
Derive $a(\eta)$ for a spatially flat Universe with dominating radiation.
 
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\[a - a_0 = H_0\sqrt{\Omega _{r0}}(\eta-\eta_0 ).\]</p>
 
\[a - a_0 = H_0\sqrt{\Omega _{r0}}(\eta-\eta_0 ).\]</p>
 
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=== Problem 5. ===
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=== Problem 5: $t(\eta)$ for dominating radiation ===
 
Express the cosmic time through the conformal time in a Universe with dominating radiation.
 
Express the cosmic time through the conformal time in a Universe with dominating radiation.
 
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=\frac{1}{2}H_0\sqrt{\Omega _{r0}}\;\eta^2.\]</p>
 
=\frac{1}{2}H_0\sqrt{\Omega _{r0}}\;\eta^2.\]</p>
 
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=== Problem 6. ===
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=== Problem 6: $a(\eta)$ for dominating dust ===
 
Derive $a(\eta)$ for a spatially flat Universe with dominating matter.
 
Derive $a(\eta)$ for a spatially flat Universe with dominating matter.
 
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= \frac{H_0^2\Omega _{m0}}{4}(\eta-\eta_0)^2.\]</p>
 
= \frac{H_0^2\Omega _{m0}}{4}(\eta-\eta_0)^2.\]</p>
 
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=== Problem 7. ===
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=== Problem 7: $a(\eta)$ for radiation and dust ===
 
Find $a(\eta)$ for a spatially flat Universe filled with a mixture of radiation and matter.
 
Find $a(\eta)$ for a spatially flat Universe filled with a mixture of radiation and matter.
 
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-H_{0}\sqrt{\Omega_{m0}}\;\eta\]</p>
 
-H_{0}\sqrt{\Omega_{m0}}\;\eta\]</p>
 
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=== Problem 8. ===
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=== Problem 8: variable EoS parameter ===
 
Suppose a component's state parameter  $w_i=p_i/\rho_i$ is a function of time. Find its density as function of redshift.
 
Suppose a component's state parameter  $w_i=p_i/\rho_i$ is a function of time. Find its density as function of redshift.
 
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\frac{dz'}{(1 + z')}\right\}.\]</p>
 
\frac{dz'}{(1 + z')}\right\}.\]</p>
 
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=== Problem 9. ===
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=== Problem 9: $H(z)$ for dominating dust ===
 
Derive the Hubble parameter as a function of redshift in a Universe filled with non-relativistic matter.
 
Derive the Hubble parameter as a function of redshift in a Universe filled with non-relativistic matter.
 
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H(z)=H_0\sqrt {\Omega _{m0}} (1 + z)^{3/2}.\]</p>
 
H(z)=H_0\sqrt {\Omega _{m0}} (1 + z)^{3/2}.\]</p>
 
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=== Problem 10. ===
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=== Problem 10: $\dot z$ for dominating dust ===
 
The redshift of any object slowly changes with time due to acceleration (or deceleration) of the Universe's expansion. Find the rate of change of redshift $\dot{z}$ for a Universe with dominating non-relativistic matter.
 
The redshift of any object slowly changes with time due to acceleration (or deceleration) of the Universe's expansion. Find the rate of change of redshift $\dot{z}$ for a Universe with dominating non-relativistic matter.
 
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\left(1 - \sqrt{\Omega _{m0}}(1 +z)^{1/2}\right).\]</p>
 
\left(1 - \sqrt{\Omega _{m0}}(1 +z)^{1/2}\right).\]</p>
 
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=== Problem 11. ===
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=== Problem 11: optical horizon ===
 
The Universe is known to have become transparent for electromagnetic waves at $z\approx 1100$ (in the process of formation of neutral hydrogen,  recombination), i.e when it was $1100$ times smaller than at present. Thus in practice the possibility of optical observation of the Universe is limited by the so-called optical horizon: the maximal distance that light travels since the moment of recombination. Find the ratio of the optical horizon to the particle one for a Universe dominated by matter.
 
The Universe is known to have become transparent for electromagnetic waves at $z\approx 1100$ (in the process of formation of neutral hydrogen,  recombination), i.e when it was $1100$ times smaller than at present. Thus in practice the possibility of optical observation of the Universe is limited by the so-called optical horizon: the maximal distance that light travels since the moment of recombination. Find the ratio of the optical horizon to the particle one for a Universe dominated by matter.
 
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\approx 1.031.\]</p>
 
\approx 1.031.\]</p>
 
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=== Problem 12. ===
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=== Problem 12: particle horizon ===
 
Derive the particle horizon as function of redshift for a Universe filled with matter and radiation with relative densities $\Omega_{m0}$ and $\Omega_{r0}$.
 
Derive the particle horizon as function of redshift for a Universe filled with matter and radiation with relative densities $\Omega_{m0}$ and $\Omega_{r0}$.
 
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     <p style="text-align: left;">Analogously to the [[#dyn51|problem]], for the model with dominating of radiation and matter we obtain
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     <p style="text-align: left;">In the same way as for the [[#dyn51|optical horizon]], for the model with domination of radiation and matter we obtain
 
\[L_{p} =\int\limits_{0}^{z}
 
\[L_{p} =\int\limits_{0}^{z}
 
\frac{dz'}{H_0\sqrt{
 
\frac{dz'}{H_0\sqrt{
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We have taken into account here that $\Omega _{m0} + \Omega _{r0} = 1$.</p>
 
We have taken into account here that $\Omega _{m0} + \Omega _{r0} = 1$.</p>
 
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=== Problem 13. ===
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=== Problem 13: time dilation at cosmological horizon ===
 
Show that any signal emitted from the cosmological horizon will arrive to the observer with infinite redshift.
 
Show that any signal emitted from the cosmological horizon will arrive to the observer with infinite redshift.
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Revision as of 13:44, 14 October 2012

Problem 1: the first Friedman equation

Express the first Friedman equation in terms of redshift and analyze the contribution of different terms in different epochs.


Problem 2: $\eta(a)$ in one-component Universe

Find the conformal time as function of the scale factor for a Universe with domination of a) radiation and b) non-relativistic matter.


Problem 3: $t(z)$ for matter domination

Find the relation between time and redshift in the Universe with dominating matter.


Problem 4: $a(\eta)$ for radiation domination

Derive $a(\eta)$ for a spatially flat Universe with dominating radiation.


Problem 5: $t(\eta)$ for dominating radiation

Express the cosmic time through the conformal time in a Universe with dominating radiation.


Problem 6: $a(\eta)$ for dominating dust

Derive $a(\eta)$ for a spatially flat Universe with dominating matter.


Problem 7: $a(\eta)$ for radiation and dust

Find $a(\eta)$ for a spatially flat Universe filled with a mixture of radiation and matter.


Problem 8: variable EoS parameter

Suppose a component's state parameter $w_i=p_i/\rho_i$ is a function of time. Find its density as function of redshift.


Problem 9: $H(z)$ for dominating dust

Derive the Hubble parameter as a function of redshift in a Universe filled with non-relativistic matter.


Problem 10: $\dot z$ for dominating dust

The redshift of any object slowly changes with time due to acceleration (or deceleration) of the Universe's expansion. Find the rate of change of redshift $\dot{z}$ for a Universe with dominating non-relativistic matter.


Problem 11: optical horizon

The Universe is known to have become transparent for electromagnetic waves at $z\approx 1100$ (in the process of formation of neutral hydrogen, recombination), i.e when it was $1100$ times smaller than at present. Thus in practice the possibility of optical observation of the Universe is limited by the so-called optical horizon: the maximal distance that light travels since the moment of recombination. Find the ratio of the optical horizon to the particle one for a Universe dominated by matter.


Problem 12: particle horizon

Derive the particle horizon as function of redshift for a Universe filled with matter and radiation with relative densities $\Omega_{m0}$ and $\Omega_{r0}$.


Problem 13: time dilation at cosmological horizon

Show that any signal emitted from the cosmological horizon will arrive to the observer with infinite redshift.