# Time evolution of scale factor

A de Sitter universe is a cosmological solution to the Einstein field equations of general relativity, named after Willem de Sitter. It models the universe as spatially flat and neglects ordinary matter, so the dynamics of the universe are dominated by the cosmological constant, thought to correspond to dark energy in our universe or the inflaton field in the early universe. My question is: Considering only a positive cosmological constant in Friedmann Robertson-Walker spacetime, $$ds^{2}=-dt^{2}+a(t)^{2}\left(dx^{2}+dy^{2}+dz^{2}\right),$$ how can I solve the Einstein field equations to find the time evolution of the scale factor $$a(t)$$?

The first and most obvious step is write down the EFE (with a cosmological constant) $$R_{\mu \nu} - \frac{1}{2}R g_{\mu \nu} + \Lambda g_{\mu \nu} = \kappa T_{\mu \nu} \ .$$ Plugging in the FLRW metric $$g$$ and doing the calculation leads to two independent field equations, the Friedmann and acceleration equations. As you're interested in the de-Sitter solution, a vacuum solution, we can ignore $$T_{\mu \nu}$$. All we're left with is $$3 H(t)^2 - \Lambda = 0 \ , \\ -3H(t)^2 - 2 \dot{H}(t) + \Lambda = 0 \ ,$$ where $$H(t)= \dot{a}(t)/a(t)$$ is the Hubble parameter. Solving for $$H(t)$$ gives $$H(t) = \pm \frac{\sqrt{\Lambda}}{\sqrt{3}}$$. (The positive and negative solutions describe expanding and contracting patches respectively - we're interested in the positive one.) Similarly, solving the differential equation for $$a(t)$$ is then simple, which is of the form $$a(t) \propto \exp (t H)$$, with $$H$$ defined previously.