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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.