I analyze inflation in this following scenario: Suppose that at some very early epoch, $t_1 ≤ t ≤ t_2$ (where $t_1 ≪ t_2 ≪ t_r$ and $t_r$ is the time at the recombination epoch), the universe resides in a “false vacuum” state: A scalar field $\phi$ fills spacetime and provides an effective vacuum energy $ρ_{vac} ≃ V (\phi)$. This vacuum energy acts like a cosmological constant $Λ = 8πρ_{vac}$. As the universe expands, the potential slowly evolves. Around time t2, the scalar field decays into standard model particles, and the stress-energy tensor is no longer dominated by $V (\phi)$. These particles provide the matter and radiation content for our universe; it is then radiation dominated until recombination, and matter dominated thereafter.
We want to estimate $N=\sqrt{\Lambda/3}(t_2-t_1)$ such that the coordinate distance that a photon travels since the big bang till the recombination $r_H$ equals the distance it traveled from the CMB to an observer on earth $r_{CMB}$. Take $z$ at the recombination to be $1100$, and assume $\Lambda$ is known and is $\sim10^{71} s^{-2}$.
My attempt: the scale factor takes the form $t^{2/3}$ in the matter dominated epoch, $\sqrt{t}$ in the radiation dominated epoch, and $\exp(\sqrt{\Lambda/3}t)$ during the inflationary period. Also, $a(t_0)=1$ (today). Thus, we find the exact form of $a$ by the conditions and imposing its continuity. (For example $a(t)=(t/t_0)^{2/3}$ during matter epoch, and thus $a(t)=(t_r/t_0)^{2/3}(t/t_r)^{1/2}$ during the radiation epoch after inflation). Now, we have $r_H=\int dt/a$. Similarly we find $r_{CMB}$. When I equate them, however, I can't isolate $N$. Any suggestions?
