For models with the equation of state p=w ρc^2 -1<w ≤1 show that R ∝t^(2)/(3(1+w)) (see proble

Step 1: Start with the equation of state for the models, which is given by $pu003Dw u005Crho c^2$, where

For models with the equation of state p=w ρc^2 -1<w ≤1...

For models with the equation of state $$ p=w \rho c^2 \quad-1<w \leq 1 $$ show that $R \propto t^{\frac{2}{3(1+w)}}$ (see problem 39) and hence that $$ q=\frac{4 \pi G \rho}{3 H^2}(1+3 w) . $$ Hence show that accelerating models do not have a particle horizon while decelerating models do. What happens if $q=0$ ? Without making assumptions about the equation of state we can show that positive pressure is a sufficient condition for the existence of a horizon. Show this by considering the integral $$ \int \frac{\mathrm{d} t^{\prime}}{R\left(t^{\prime}\right)}=\int \frac{\mathrm{d} R}{R \dot{R}} $$ and using the Friedmann equation and energy equation to estimate $\dot{R}$ as a function of $R$. (Recall that at early times the spatial curvature is small compared to the mass density in the Friedmann equation.)

For models with the equation of state
$$
p=w \rho c^2 \quad-1<w \leq 1
$$
show that $R \propto t^{\frac{2}{3(1+w)}}$ (see problem 39) and hence that
$$
q=\frac{4 \pi G \rho}{3 H^2}(1+3 w) .
$$
Hence show that accelerating models do not have a particle horizon while decelerating models do. What happens if $q=0$ ?

Without making assumptions about the equation of state we can show that positive pressure is a sufficient condition for the existence of a horizon. Show this by considering the integral
$$
\int \frac{\mathrm{d} t^{\prime}}{R\left(t^{\prime}\right)}=\int \frac{\mathrm{d} R}{R \dot{R}}
$$
and using the Friedmann equation and energy equation to estimate $\dot{R}$ as a function of $R$. (Recall that at early times the spatial curvature is small compared to the mass density in the Friedmann equation.)

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