# S15 Pressure and Energy Density Evolution SOLUTIONS

Box \(\PageIndex{1}\)

**Exercise 15.1.1:** Use the above equation to find \(P(\rho)\) for non-relativistic matter, given that \(\rho \propto a^{-3}\).

**Exercise 15.1.2:** Use the above equation to find \(P(\rho)\) for relativistic matter, given that \(\rho \propto a^{-4}\).

**Exercise 15.1.3**: Use the above equation to find \(P(\rho)\) for a cosmological constant, given that \(\rho \propto a^0\).

Solutions: The "above equation" is

\[a\frac{d\rho}{da} = -3(P/c^2+\rho)\]

Plugging in \(\rho \propto a^n\) we get \(n\rho = -3(P/c^2 + \rho)\). Solving for \(P\) for \(n=-3, -4, 0\) we find \(P=0\), \(P=\rho c^2/3\), and \(P = -\rho c^2\) respectively.