Polytropic equation of state

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Problem 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

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 $p=w\rho +k\rho ^{1+1/n}$ ($1+w+k\rho ^{1/n} >0$).

P-H. Chavanis, arXiv:1208.0797

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.

Problem 4