Polytropic equation of state

3.3.1

Problem 3.3.1: Generalized polytropic EoS

Consider a generalized polytropic equation of state of the form \[p = w\rho + k\rho ^{1 + 1/n} \] 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 $w,k,n$.

P-H. Chavanis, arXiv:1208.0797

solution

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)???

3.3.2

Problem 3.3.2: $\rho(a)$

Find the dependence $\rho (a)$ and analyze the limits $a\to 0$ and $a\to \infty$ for the equation of state $p=w\rho +k\rho ^{1+1/n}$ ($1+w+k\rho ^{1/n} >0$).

P-H. Chavanis, arXiv:1208.0797

solution

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 $k<0$.
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.
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.

3.3.3

Problem 3.3.3: 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.

solution

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 \[q(t)=\frac{1+3w(t)}{2} =\frac{1+3w}{2} \pm \frac{3}{2} \left(w+1\right)\left(\frac{\rho }{\rho _{*} } \right)^{1/n} \] Then the critical density $\rho _{c}$, corresponding to the inflection point $q=0$, is \[\rho _{c} =\rho _{*} \left[\mp \frac{1+3w}{3(1+w)} \right]^{n}. \]

3.3.4

Problem 3.3.4

solution