Polytropic equation of state - Revision history

Cosmo All at 01:39, 3 December 2012

2012-12-03T01:39:24Z

Igor at 19:51, 19 November 2012

2012-11-19T19:51:15Z

Igor: /* Problem 2 */

2012-11-19T19:48:29Z

‎Problem 2

Igor at 19:47, 19 November 2012

2012-11-19T19:47:19Z

Cosmo All at 22:02, 12 November 2012

2012-11-12T22:02:55Z

Cosmo All: Created page with "3 NOTOC

=== Problem 1 === Consider a general..."

2012-11-12T22:01:23Z

Created page with "3 __NOTOC__ <div id="polytr1"></div> <div style="border: 1px solid #AAA; padding:5px;"> === Problem 1 === Consider a general..."

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[[Category:Dynamics of the Expanding Universe|3]]

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=== Problem 1 ===
Consider a generalized polytropic equation of state of the form \[
p = w\rho + k\rho ^{1 + 1/n}
\]
This equation of state represent the sum of a standard linear equation of state $p=w\rho $ and a polytropic equation of state $p=k\rho ^{\gamma } $, where $k$ is the polytropic constant and $\gamma \equiv 1+1/n$ is the polytropic index. We assume $-1\le w\le 1$. Проанализируйте характер космологических решений для различных значений параметров $w,k,n$. [http://arxiv.org/abs/1208.0797 P-H. Chavanis, arXiv:1208.0797]
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<p style="text-align: left;">For an equation of state of the form $p=w\rho +k\rho ^{1+1/n} $ with positive index $n>0$, the polytropic component dominates the linear component when the density is high. These models describe the early Universe. Conversely, when$n<0$, the polytropic component dominates the linear component when the density is low. These models will be studied in '''Section 9.5.1???''', describe the late Universe. Case $w+k\rho ^{1/n} \ge -1$corresponds to the "normal" case where the density decreases with the increase of scale factor. The opposite case $w+k\rho ^{1/n} <-1$, leading to a "phantom universe" where the density increases with the increase of scale factor '''(see 9.5.4)???'''</p>
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=== Problem 2 ===
For the equation of state }$p=w\rho +k\rho ^{1+1/n} $ (\textbf{$1+w+k\rho ^{1/n} >0$) find dependence $\rho (a)$and analyze limits $a\to 0$и $a\to \infty $. [http://arxiv.org/abs/1208.0797 P-H. Chavanis, arXiv:1208.0797]
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<p style="text-align: left;">For the polytropic equation of state the conservation equation becomes
\[\dot{\rho }+3H\left(1+w+k\rho ^{1/n} \right)=0\]
Assuming $1+w+k\rho ^{1/n} >0$, this equation can be integrated into
\[\rho =\frac{\rho _{*} }{\left[\left(\frac{a}{a_{*} } \right)^{3(1+w)/n} \mp 1\right]^{n} } \]
where $\rho _{*} $=$\left[\left(1+w\right)/\left|k\right|\right]^{n} $ and $a_{*} $ is a constant of integration. The upper sign corresponds to$k>0$, and the lower sign corresponds to $k<0$.
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For $k>0$the density is defined only for $a>a_{*} $ When $a\to a_{*} $
\[\frac{\rho }{\rho _{*} } \sim \left[\frac{n}{3\left(1+w\right)} \right]^{n} \frac{1}{\left(a/a_{*} -1\right)^{n} } \to \infty \]
When $a\to \infty $
\[\frac{\rho }{\rho _{*} } \sim \left(\frac{a_{*} }{a} \right)^{3(1+w} \to 0\]
Last result corresponds to the linear equation of state $\left(k=0\right)$: for$n>0$linear component dominates polytropic component when the density is low. In the same limits, $p\to \infty $ and $p\to 0$, respectively.
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For $k<0$, the density is defined for all $a$. When $a\to 0$ the density $\rho \to \rho _{*} $. When $a\to \infty $ the density $\rho \to 0$ corresponding to the linear equation of state. In the same limits, $p\to -\rho _{*} $ and $p\to 0$, respectively.
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=== Problem 3 ===
For the equation of state }$p=w\rho +k\rho ^{1+1/n} $ (\textbf{$1+w+k\rho ^{1/n} >0$) find a possible inflexion point$(q=\ddot{a}=0)$ in the curve $a(t)$. Consider the case only flat Universe.
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<p style="text-align: left;">We can rewrite generalized polytropic equation of state as
\[\begin{array}{l} {p=w(t)\rho ,} \\ {w(t)=w\pm \left(w+1\right)\left(\frac{\rho }{\rho _{*} } \right)^{1/n} } \end{array}\]
For a flat Universe
\[q(t)=\frac{1+3w(t)}{2} =\frac{1+3w}{2} \pm \frac{3}{2} \left(w+1\right)\left(\frac{\rho }{\rho _{*} } \right)^{1/n} \]
It follows that critical density$\rho _{c} $corresponding to inflexion point $q=0$ is
\[\rho _{c} =\rho _{*} \left[\mp \frac{1+3w}{3(1+w)} \right]^{n} \]
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=== Problem 1 ===

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← Older revision		Revision as of 22:02, 12 November 2012
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-=== Problem 1 ===
+=== Problem 4 ===
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+__TOC__
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-=== Problem 1 ===
+=== Problem 1: generalized polytropic EoS ===
 Consider a generalized polytropic equation of state of the form
 \[p = w\rho  + k\rho ^{1 + 1/n} \]
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-=== Problem 2 ===
+=== Problem 2: $\rho(a)$ ===
 Find the dependence $\rho (a)$ and analyze the limits $a\to 0$ and $a\to \infty$ for the equation of state $p=w\rho +k\rho ^{1+1/n}$ ($1+w+k\rho ^{1/n} >0$).
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-=== Problem 3 ===
+=== Problem 3: inflection points ===
 Find the possible inflection point $q=\ddot{a}=0$ of the curve $a(t)$ for the equation of state $p=w\rho +k\rho ^{1+1/n} $ ($1+w+k\rho ^{1/n} >0$) in a flat Universe.
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 === Problem 1 ===
-Consider a generalized polytropic equation of state of the form \[
+Consider a generalized polytropic equation of state of the form
-p = w\rho  + k\rho ^{1 + 1/n}
+\[p = w\rho  + k\rho ^{1 + 1/n} \]
-\]
+This equation of state represents the sum of the standard linear term $w\rho $ and the polytropic term $k\rho ^{\gamma } $, where $k$ is the polytropic constant and $\gamma \equiv 1+1/n$ is the polytropic index. We assume $-1\le w\le 1$. Analyze the cosmological solutions for different values of parameters $w,k,n$.
-This equation of state represent the sum of a standard linear equation of state $p=w\rho $ and a polytropic equation of state $p=k\rho ^{\gamma } $, where $k$ is the polytropic constant and $\gamma \equiv 1+1/n$ is the polytropic index. We assume $-1\le w\le 1$. Проанализируйте характер космологических решений для различных значений параметров $w,k,n$. [http://arxiv.org/abs/1208.0797 P-H. Chavanis, arXiv:1208.0797]
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    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">For an equation of state of the form $p=w\rho +k\rho ^{1+1/n} $ with positive index $n>0$, the polytropic component dominates the linear component when the density is high. These models describe the early Universe. Conversely, when$n<0$, the polytropic component dominates the linear component when the density is low. These models will be studied in '''Section 9.5.1???''', describe the late Universe. Case $w+k\rho ^{1/n} \ge -1$corresponds to the "normal" case where the density decreases with the increase  of scale factor.  The opposite case $w+k\rho ^{1/n} <-1$, leading to a "phantom universe" where the density increases with the increase  of scale factor '''(see 9.5.4)???'''</p>
+     <p style="text-align: left;">For the given equation of state with positive index $n>0$, the polytropic term is  dominating when the density is high. Such models describe the early Universe. Conversely, when $n<0$, the polytropic term dominates the linear term when the density is low. These models will be studied in '''Section 9.5.1???''', and describe the late Universe. The case $w+k\rho ^{1/n} \ge -1$ corresponds to the "normal" case when  density decreases with the increase  of scale factor.  The opposite case $w+k\rho ^{1/n} <-1$ corresponds to the "phantom universe", in which density increases with the increase  of scale factor '''(see 9.5.4)???'''</p>
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 === Problem 2 ===
-For the equation of state }$p=w\rho +k\rho ^{1+1/n} $ (\textbf{$1+w+k\rho ^{1/n} >0$) find dependence $\rho (a)$and analyze limits $a\to 0$и $a\to \infty $. [http://arxiv.org/abs/1208.0797 P-H. Chavanis, arXiv:1208.0797]
+For the equation of state $p=w\rho +k\rho ^{1+1/n} $ (\textbf{$1+w+k\rho ^{1/n} >0$) find dependence $\rho (a)$and analyze limits $a\to 0$и $a\to \infty $. [http://arxiv.org/abs/1208.0797 P-H. Chavanis, arXiv:1208.0797]
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    <div class="NavHead">solution</div>
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 === Problem 3 ===
-For the equation of state }$p=w\rho +k\rho ^{1+1/n} $ (\textbf{$1+w+k\rho ^{1/n} >0$) find a possible inflexion point$(q=\ddot{a}=0)$ in the curve $a(t)$. Consider the case only flat Universe.
+Find the possible inflection point $q=\ddot{a}=0$ of the curve $a(t)$ for the equation of state $p=w\rho +k\rho ^{1+1/n} $ ($1+w+k\rho ^{1/n} >0$) in a flat Universe.
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    <div class="NavHead">solution</div>
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-     <p style="text-align: left;">We can rewrite generalized polytropic equation of state as
+     <p style="text-align: left;">We can rewrite the generalized polytropic equation of state as
 \[\begin{array}{l} {p=w(t)\rho ,} \\ {w(t)=w\pm \left(w+1\right)\left(\frac{\rho }{\rho _{*} } \right)^{1/n} } \end{array}\]
 For a flat Universe
 \[q(t)=\frac{1+3w(t)}{2} =\frac{1+3w}{2} \pm \frac{3}{2} \left(w+1\right)\left(\frac{\rho }{\rho _{*} } \right)^{1/n} \]
-It follows that critical density$\rho _{c} $corresponding to inflexion point $q=0$ is
+Then the critical density $\rho _{c}$, corresponding to the inflection point $q=0$, is
-\[\rho _{c} =\rho _{*} \left[\mp \frac{1+3w}{3(1+w)} \right]^{n} \]
+\[\rho _{c} =\rho _{*} \left[\mp \frac{1+3w}{3(1+w)} \right]^{n}. \]
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