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# S15 Pressure and Energy Density Evolution SOLUTIONS

• Page ID
7882
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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.