We will now use the FRW equation to solve for \(a(t)\) when \(w=-1\). The \(a(t)\) we obtain will describe how \(a\) changes with \(t\) in a Universe where space is flat and where the only form of energy present is dark/vacuum energy. If you imagine removing all the stars, planets, atoms, radiation, dark matter, etc. from the Universe so that the Universe was totally empty, dark energy would essentially be the energy of empty space! The energy density of dark energy is represented by \(ρ_0\). The FRW equation in a Universe where space is flat and where the only form of energy density is \(ρ_0\) simplifies to
\begin{equation}
\label{6:12}
H^2(t)=\biggl(\frac{da/dt}{a}\biggl)^2=\frac{8}{3}ρ_0πG.
\end{equation} The energy density \(ρ_0\) due to dark energy is a constant of nature and, as far as we can tell, never changes. Its value is given roughly by \begin{equation}
\label{Eq:I:6:13}
ρ_0=(10^{-29}/8π)g/cm^3.
\end{equation} Since \(ρ_0\) is a constant, it follows that
\begin{equation}
\label{Eq:I:6:14}
H(t)=\frac{da/dt}{a}=\sqrt{\frac{8}{3}πGρ_0}=constant.
\end{equation}
Therefore, in this Universe with flat space where there is only dark energy, the Hubble parameter \(H(t)=constant\) for all values of \(t\). In such a Universe (using equation (3)) \(V∝D\) not just for all values of \(x^i\) (due to the distribution of mass being homogeneous and isotropic) but for all values of \(t\) as well. Multiplies equation (14) by \(a\) on both sides, we get \(\frac{da}{dt}=H(t)a\) where \(H(t)=\sqrt{\frac{8}{3}πGρ_0}=constant\). The solution to a differential equation where \(\frac{da}{dt}∝a\) is an exponential, so we get
\begin{equation}
\label{Eq:I:6:15}
a(t)=Ce^{Ht}=Ce^{\sqrt{\frac{8}{3}πGρ_0}t}.
\end{equation}
This is the relationship between \(D\) and \(t\) in a Universe with flat space where the only form of energy present is dark energy. Using equation (1) we find that the distance between any two galaxies (or, more specifically, point in space) in this particular Universe is \begin{equation}
\label{Eq:II:12:13}
D=a(t)∆r=Ce^{Ht}∆r.
\end{equation}This means that the distance \(D\) between any two points \(x^i_0\) and \(x^i\) is exponentially increasing with time and are therefore accelerating apart. The galaxies which, at all times, are overlapping these points are also accelerating away from one another at an exponential and accelerating rate.
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References
1. Leonard Susskind. "Matter and radiation dominated universes". theoreticalminimum.com.
2. Wikipedia contributors. "Dark energy." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 9 May. 2017. Web. 18 May. 2017.