Newtonian cosmology - Revision history

Igor at 16:38, 13 October 2012

2012-10-13T16:38:50Z

Igor at 10:11, 10 August 2012

2012-08-10T10:11:30Z

Igor at 10:25, 4 August 2012

2012-08-04T10:25:17Z

Cosmo All at 16:30, 23 July 2012

2012-07-23T16:30:34Z

Cosmo All at 16:24, 23 July 2012

2012-07-23T16:24:18Z

Cosmo All: /* Problem 5. */

2012-06-18T21:27:37Z

‎Problem 5.

Cosmo All: /* Problem 5. */

2012-06-18T21:26:31Z

‎Problem 5.

Cosmo All: /* Problem 5. */

2012-06-18T21:24:27Z

‎Problem 5.

Cosmo All at 21:04, 18 June 2012

2012-06-18T21:04:20Z

Cosmo All: /* Problem 1. */

2012-06-18T21:03:15Z

‎Problem 1.

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 [[Category:Dynamics of the Expanding Universe|6]]
 __TOC__
 <div id="equ48"></div>
 === Problem 1: the first equation ===
 Obtain the first Friedman equation basing only on Newtonian mechanics.
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 where $k =  2E/m$. Therefore the first Friedman equation corresponds to the energy conservation law in the Newtonian interpretation.</p>
    </div>
-</div>
+</div></div>
-<div id="equ49"></div>
+<div id="equ49"></div><div style="border: 1px solid #AAA; padding:5px;">
 === Problem 2: the second equation ===
 Derive the analogue of the second Friedman equation in the Newtonian mechanics.
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 The difference with the second Friedman equation is in the absence of pressure $p$, which is a purely relativistic effect. We stress that Newtonian gravity can only lead to decelerated expansion.</p>
    </div>
-</div>
+</div></div>
-<div id="equ51"></div>
+<div id="equ51"></div><div style="border: 1px solid #AAA; padding:5px;">
 === Problem 3: conservation law ===
 Obtain the conservation equation for non-relativistic matter from the continuity equation for the ideal fluid.
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 and using the Hubble's law $V = HR$ to obtain the conservation law $\dot \rho  + 3H\rho  = 0$.</p>
    </div>
-</div>
+</div></div>
-<div id="equ55"></div>
+<div id="equ55"></div><div style="border: 1px solid #AAA; padding:5px;">
 === Problem 4: no static Universe in nonrelativistic theory ===
 Show that equations of Newtonian hydrodynamics and gravity prohibit the existence of a  uniform, isotropic and static cosmological model, i.e. a Universe constant in time, uniformly filled by ideal fluid.
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 represents a solution, where $\vec{C}$ and  $K$ are arbitrary integration constants. The continuity equation also reduces to identity. But the equations of motion (reduced to the condition $\nabla\Phi=0$) can be satisfied by no choice of $\vec{C}$ and $K$. Therefore there are no solutions of the system with required properties.</p>
    </div>
-</div>
+</div></div>
-<div id="equ_peeb1"></div>
+<div id="equ_peeb1"></div><div style="border: 1px solid #AAA; padding:5px;">
 === Problem 5: cosmic energy and the Layzer-Irvine equation ===
 Find the generalization of the Newtonian energy conservation equation to an expanding cosmological background$^*$.
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 $^*$It was derived independently in W.~M.~Irvine, Ph.D. thesis, Harvard University, (1961) and D.~Layzer, Astrophys. J. 138, 174 (1963).</p>
    </div>
-</div>
+</div></div>
-−
 === Problem 6: the virial theorem ===
 Using the Layzer-Irvene equation, discussed in the previous problem, recover the Newtonian virial theorem.
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 which is the Newtonian virial theorem.</p>
    </div>
-</div>
+</div></div>

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 <div id="equ51"></div>
-=== Problem 3: the conservation equation ===
+=== Problem 3: conservation law ===
 Obtain the conservation equation for non-relativistic matter from the continuity equation for the ideal fluid.
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 <div id="equ55"></div>
-=== Problem 4: no solutions in nonrelativistic theory ===
+=== Problem 4: no static Universe in nonrelativistic theory ===
 Show that equations of Newtonian hydrodynamics and gravity prohibit the existence of a  uniform, isotropic and static cosmological model, i.e. a Universe constant in time, uniformly filled by ideal fluid.
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-=== Problem 6: virial theorem ===
+=== Problem 6: the virial theorem ===
 Using the Layzer-Irvene equation, discussed in the previous problem, recover the Newtonian virial theorem.
 <div class="NavFrame collapsed">

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 [[Category:Dynamics of the Expanding Universe|6]]
-__NOTOC__
+__TOC__
 <div id="equ48"></div>
-=== Problem 1. ===
+=== Problem 1: the first equation ===
 Obtain the first Friedman equation basing only on Newtonian mechanics.
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 <div id="equ49"></div>
-=== Problem 2. ===
+=== Problem 2: the second equation ===
 Derive the analogue of the second Friedman equation in the Newtonian mechanics.
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 <div id="equ51"></div>
-=== Problem 3. ===
+=== Problem 3: the conservation equation ===
 Obtain the conservation equation for non-relativistic matter from the continuity equation for the ideal fluid.
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 <div id="equ55"></div>
-=== Problem 4. ===
+=== Problem 4: no solutions in nonrelativistic theory ===
 Show that equations of Newtonian hydrodynamics and gravity prohibit the existence of a  uniform, isotropic and static cosmological model, i.e. a Universe constant in time, uniformly filled by ideal fluid.
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-=== Problem 5. ===
+=== Problem 5: cosmic energy and the Layzer-Irvine equation ===
 Find the generalization of the Newtonian energy conservation equation to an expanding cosmological background$^*$.
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 <div id="equ_peeb2"></div>
-=== Problem 6. ===
+=== Problem 6: virial theorem ===
 Using the Layzer-Irvene equation, discussed in the previous problem, recover the Newtonian virial theorem.
 <div class="NavFrame collapsed">

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 [[Category:Dynamics of the Expanding Universe|6]]
+__NOTOC__
-= Newtonian cosmology =
-−
 <div id="equ48"></div>
 === Problem 1. ===

← Older revision		Revision as of 16:24, 23 July 2012
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−	[[Category:Dynamics of the Expanding Universe]]	+	[[Category:Dynamics of the Expanding Universe\|6]]

	= Newtonian cosmology =		= Newtonian cosmology =

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 and the Hamiltonian is
 \[H = \sum\limits_i
-	{\frac{{\vec p_i^2}}{{2{m_i}{a^2}}}}  + MV
+	{\frac{\vec p_i^2}{2{m_i}{a^2}}}  + MV
 		= M(K + V).\]
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 Then from
-\[\frac{{dH}}{{dt}} = \frac{{\partial H}}{{\partial t}}\]
+\[\frac{dH}{dt} = \frac{\partial H}{\partial t}\]
 where the partial derivative is computed at fixed particle coordinates and momenta. At fixed $\vec{x}_i$ and $\vec{p}_i$ we have $K\sim a^{-1}$ and $V \sim {\rho _b}{a^2} \propto {a^{ - 1}}$. So the last equation is reduced to
 \[\frac{dE}{dt}= \frac{d}{dt}(K + V) =  - \frac{\dot a}{a}(2K + V).\]

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 \begin{align*}
 V &=  - \frac{1}{2}G{\rho _b}{a^2}
-	\int {{d^3}x\frac{{\xi (x)}}{x}}
+	\int {{d^3}x\frac{\xi (x)}{x}}
 	=  - 2\pi G{\rho _b}{J_2},\\
 J_2&\equiv \int\limits_0^\infty  {\xi (r)dr}
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 \[\frac{{dH}}{{dt}} = \frac{{\partial H}}{{\partial t}}\]
 where the partial derivative is computed at fixed particle coordinates and momenta. At fixed $\vec{x}_i$ and $\vec{p}_i$ we have $K\sim a^{-1}$ and $V \sim {\rho _b}{a^2} \propto {a^{ - 1}}$. So the last equation is reduced to
-\[\frac{{dE}}{{dt}}
+\[\frac{dE}{dt}= \frac{d}{dt}(K + V) =  - \frac{\dot a}{a}(2K + V).\]
-	= \frac{d}{{dt}}(K + V) =  - \frac{{\dot a}}{a}(2K + V).\]
 This equation is known as the cosmic energy or the Layzer-Irvine equation$^*$.

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    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">Let us consider the mass distribution of a collection of particles of masses $m_i$ at coordinate positions $\vec{x}_i$. We neglet nongravitational interaction, which assumes the fluctuations from homogeneity are appreciable only on scales small compared to the Hubble length, and peculiar motion is nonrelativistic. In this limit the Lagrangian for the system is
-\[L = \frac{1}{2}\sum\limits_i {{m_i}{a^2}
+\[L = \frac{1}{2}\sum\limits_i {{m_i}{a^2}\dot{\vec{x}_i^2} - MV}. \]
-	\dot \vec{x}_i^2 - MV}. \]
 The sum is over the kinetic energy of the particles in some large comoving volume. The total mass in this volume is $M=\sum_{i} m_{i}$. The gravitational potential energy per unit mass is
 \[V =  - \frac{1}{2}\frac{G{a^5}}{M}
-	\int {{d^3}{x_1}} {d^3}{x_2}\frac{{\left[ {\rho \left( {x_1} \right)-{\rho _b}} \right]
+	\int {{d^3}{x_1}} {d^3}{x_2}\frac{\left[ {\rho \left( {x_1} \right)-{\rho _b}} \right]
-\left[ {\rho \left( {{x_2}} \right) - {\rho _b}} \right]}}
+\left[ {\rho \left( {x_2} \right) - {\rho _b}} \right]}
-	{{{x_{12}}}}.\]
+	{x_{12}}.\]
 It is determined by the difference between the local mass density $\rho(\vec{x})$ and its mean value $\rho_b$.