Friedman equations - Revision history

Cosmo All at 23:33, 8 January 2013

2013-01-08T23:33:36Z

Cosmo All at 23:33, 8 January 2013

2013-01-08T23:33:12Z

Igor: /* Problem 26: condition $\dot X = 0$ */

2012-11-10T17:50:27Z

‎Problem 26: condition $\dot X = 0$

Cosmo All: /* Problem 23: evolution of the state parameter */

2012-11-05T20:58:10Z

‎Problem 23: evolution of the state parameter

Cosmo All at 20:56, 5 November 2012

2012-11-05T20:56:34Z

Cosmo All: /* Problem 26: condition $\dot X = 0$ */

2012-11-05T20:21:18Z

‎Problem 26: condition $\dot X = 0$

Cosmo All at 20:19, 5 November 2012

2012-11-05T20:19:56Z

Igor: /* Problem 1: derivation of Friedman equations */

2012-10-13T16:35:52Z

‎Problem 1: derivation of Friedman equations

Igor: /* Problem 1: derivation of Friedman equations */

2012-10-13T16:35:16Z

‎Problem 1: derivation of Friedman equations

Igor: /* Problem 26: non-relativistic state parameter */

2012-10-13T16:27:58Z

‎Problem 26: non-relativistic state parameter

@@ Line 460: / Line 460: @@
 <div id="equ68n"></div><div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 23: evolution of the state parameter ===
+=== Problem 24: evolution of the state parameter ===
 Evaluate the derivatives of the state parameter $w$ with respect to cosmological and conformal times.
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-=== Problem 24: EoS via the expansion dynamics ===
+=== Problem 25: EoS via the expansion dynamics ===
 Consider an FLRW Universe dominated by a substance, such that Hubble parameter depends on time as $H=f(t)$, where $f(t)$ is an arbitrary differentiable function. Find the state equation for the considered substance.
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-=== Problem 25: upper bound on w ===
+=== Problem 26: upper bound on w ===
 Find the upper bound for the state parameter $w$.
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@@ Line 525: / Line 525: @@
 <div id="equ52"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 26: non-relativistic state parameter ===
+=== Problem 27: non-relativistic state parameter ===
 Show that for non-relativistic particles the state parameter $w$ is much less than unity.
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-=== Problem 26: condition $\dot X = 0$ ===
+=== Problem 28: condition $\dot X = 0$ ===
 The Friedman equation can be regarded as a constraint of the form
 \[

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 === Problem 26: condition $\dot X = 0$ ===
-The Friedman equation can be regarded as a constrain of the form
+The Friedman equation can be regarded as a constraint of the form
 \[
 X \equiv H^2  + \frac{k}{a^2} - \frac{8\pi G}{3}\rho  = 0
 \]
-Show that the conservation equation $\dot \rho  + 3H(\rho  + p)$ represent condition $\dot X = 0$.
+Show that the conservation equation $\dot \rho  + 3H(\rho  + p)$ represents the condition $\dot X = 0$.
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    <div class="NavHead">solution</div>
@@ Line 557: / Line 557: @@
 \dot X = 2H\left( {\dot H - \frac{k}{a^2}} \right) - \frac{8\pi G}{3}\dot \rho
 \]
-Using \[
+Using
-\dot H = \frac{k}{a^2} - 4\pi G\left( {\rho  + p} \right)
+ \[\dot H = \frac{k}{a^2} - 4\pi G\left( {\rho  + p} \right),\]
-\]
+one obtains
-obtain \[
+\[\dot X = \dot \rho  + 3H\left( {\rho  + p} \right) = 0.\]
 \dot X = \dot \rho  + 3H\left( {\rho  + p} \right) = 0.
 \]
 </p>
    </div>
 </div></div>

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-=== Problem 23: evolution of the state parameter ===
+=== Problem 23: relation between the total pressure and the deceleration parameter ===
 Find the relation between the total pressure and the deceleration parameter for a flat Universe.
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 <div id="equ68n"></div><div style="border: 1px solid #AAA; padding:5px;">
 === Problem 23: evolution of the state parameter ===
 Evaluate the derivatives of the state parameter $w$ with respect to cosmological and conformal times.

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    </div>
 </div></div>
 <div id="equ68n"></div><div style="border: 1px solid #AAA; padding:5px;">
 === Problem 23: evolution of the state parameter ===
 Evaluate the derivatives of the state parameter $w$ with respect to cosmological and conformal times.

@@ Line 519: / Line 519: @@
 \[w=\frac{p}{\rho}
 	=\frac{\left<v^2\right>}{3 c^2}\ll 1.\]</p>
    </div>
 </div></div>

@@ Line 19: / Line 19: @@
 		- \frac{1}{2}\delta _\nu ^\mu T \right).\]
 Using the [[Friedman-Lemaitre-Robertson-Walker_(FLRW)_metric#equ28|explicit expressions for the Ricci tensor components of of the FLRW metric]], one obtains from the component $\binom{0}{0}$ the following equation:
-\[ - 3\frac{{\ddot a}}{a}
+\[ - 3\frac{\ddot a}{a}
 	= 8\pi G \cdot \frac{1}{2}(\rho  + 3p),\]
 and from the $\binom{1}{1}$ component

@@ Line 10: / Line 10: @@
      \frac{\ddot a}{a} &=- \frac{4\pi G}3\big(\rho+3p).
 \end{align}
-Consider matter an ideal fluid with $T^{\mu}_{\nu}=diag(\rho,-p,-p,-p)$ ([[Friedman-Lemaitre-Robertson-Walker_(FLRW)_metric#equ29n|see problem]]).
+Consider matter an ideal fluid with $T^{\mu}_{\nu}=diag(\rho,-p,-p,-p)$.
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    <div class="NavHead">solution</div>
@@ Line 18: / Line 18: @@
 	= 8\pi G\left( T^\mu_\nu
 		- \frac{1}{2}\delta _\nu ^\mu T \right).\]
-Using the results of problems [[Friedman-Lemaitre-Robertson-Walker_(FLRW)_metric#equ28|this]], [[Friedman-Lemaitre-Robertson-Walker_(FLRW)_metric#equ28n|this]] and [[Friedman-Lemaitre-Robertson-Walker_(FLRW)_metric#equ29n|this]], one obtains from the component $\binom{0}{0}$ the following equation:
+Using the [[Friedman-Lemaitre-Robertson-Walker_(FLRW)_metric#equ28|explicit expressions for the Ricci tensor components of of the FLRW metric]], one obtains from the component $\binom{0}{0}$ the following equation:
 \[ - 3\frac{{\ddot a}}{a}
 	= 8\pi G \cdot \frac{1}{2}(\rho  + 3p),\]
@@ Line 28: / Line 28: @@
 Excluding the second derivative from the two independent equations one obtains the first Friedman equation, and on excluding the first derivative --- the second Friedman equation.
-$^*$This is why the stress-energy tensor was taken in the form of stress-energy tensor of ideal fluid; otherwise we would have come to contradiction. More on this see in [[Friedman-Lemaitre-Robertson-Walker_(FLRW)_metric#equ29n|problem ]].
+$^*$This is why the stress-energy tensor was taken in the form of stress-energy tensor of ideal fluid; otherwise we would have come to contradiction. More on this see in [[Friedman-Lemaitre-Robertson-Walker_(FLRW)_metric#equ29n|the problem on cosmological energy-momentum tensor]].
 </p>
    </div>
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 <div id="equ66"></div><div style="border: 1px solid #AAA; padding:5px;">
 === Problem 2: reformulation in terms of conformal time ===
 Derive the Friedman equations in terms of conformal time.

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    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">For a non-relativistic gas
+     <p style="text-align: left;">For non-relativistic gas
 \[p=nkT=\frac{p}{mc^2}\cdot
 	\frac{m\left<v^2\right>}{3}