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3.1. Friedmann's Equation rightarrow Exponential Expansion

One way to describe inflation is that during inflation, a Lambdainf term dominates Eq. 11. Thus, during inflation we have,

H^{2}    &=&     \frac{\Lambda_{\rm inf}}{3}\\
\frac{dR}{dt}  &=&     R\sqrt{\frac{\Lambda_{\rm inf}}{3}}\\
%\frac{dR}{R}  &=&     dt\sqrt{\frac{\Lambda_{inf}}{3}}\\
\int^{R}_{R_{i}}\frac{dR}{R}   &=&     \int^{t}_{t_{i}}dt\sqrt{\frac{\Lambda_{\rm inf}}{3}}\\
ln \frac{R}{R_{i}}  &=&  \sqrt{\frac{\Lambda_{\rm inf}}{3}}\;(t - t_{i})\\
R             &\approx & R_{i} e^{Ht}
\end{eqnarray} (22)





where ti and Ri are the time and scale factor at the beginning of inflation. To get Eq. 26 we have assumed 0 approx ti << t < te (where te is the end of inflation) and we have used Eq. 22. Equation 26 is the exponential expansion of the Universe during inflation. The e-folding time is 1/H. The doubling time is (ln 2) / H. That is, during every interval Delta t = 1/H, the size of the Universe increases by a factor of e = 2.718281828... and during every interval Deltat = (ln 2) / H the size of the Universe doubles.

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