A Summing Up — Question 6

Question 6

Matter and vacuum energy affect acceleration with opposite signs

For a flat universe with pressureless matter and a cosmological constant, use $q_0=\Omega_m/2-\Omega_\Lambda$. Let $\Omega_m=.30$ and $\Omega_\Lambda=.70$, with densities scaling as a to the power -3 and a to the power 0.

1. Find the present deceleration parameter and identify acceleration or deceleration.

2. Find the scale factor and redshift when acceleration begins, using $\rho_m=2\rho_\Lambda$.

Question 6 - Solution

Matter and vacuum energy affect acceleration with opposite signs

(a) Method and calculation

$$\boxed{q_0=.30/2-.70=-.55}.$$$$q=-a\ddot a/\dot a^2<0\Rightarrow\boxed{\ddot a>0}.$$

(b) Interpretation and further calculation

$$.30a^{-3}=2(.70)\Rightarrow\boxed{a_{\mathrm{acc}}=(.30/1.40)^{1/3}\simeq.5984}.$$$$\boxed{z_{\mathrm{acc}}=a_{\mathrm{acc}}^{-1}-1\simeq.6711}.$$

Physical check

The transition to acceleration differs from matter-vacuum equality because pressure contributes to the acceleration equation.