# Inflation Theory. Horizon problem. Couldn't understand

I'm reading a text from MIT, Inflationary Cosmology and the Horizon and Flatness Problems: The Mutual Constitution of Explanation and Questions that is available online.

However I failed to understand this sentence,

For the numbers discussed above, this yields $$d_H(t_{\rm end}) = \frac{1}{H}ce^{100} \approx 10^{19}~{\rm cm}$$ At present, the radius of the observable universe is of order $3ct\approx 10^{28}$ cm. At the end of the inflationary era, $R(t)$ was smaller by about a factor of $10^{-27}$. Thus, at that time, the presently observed observable universe had a physical size of about $10$ cm.

Please enlighten me, as to how the value of $R(t)$ was calculated to be smaller by a factor of $10^{-27}$. Thank you.

P.s. - I understand how horizon distance dh was calculated at the end of inflation. The only trouble I'm having is with the evaluation of $R(t)$ at the end of inflation.

After inflation the universe goes through a radiation dominated epoch, followed by a matter dominated stage, the transition occurs at

$$1 + z_{\rm eq} \approx 2.4\times10^{4}\Omega_{m,0}h^2 \tag{1}$$

More recently ($z\lesssim 0.6$) the density content of the universe is dominated by the cosmological constant, but I will neglected it. During the first stage the scale factor $a$ (or $R$) follows the expression

$$a\sim t^{1/2} ~~~\mbox{for}~~~ z > z_{\rm eq}$$

therefore, if $a_{\rm end}$ is the scale factor after inflation we have

$$\frac{a_{\rm eq}}{a_{\rm end}} = \left(\frac{t_{\rm eq}}{t_{\rm end}} \right)^{1/2} \tag{2}$$

Similarly, during the matter-dominated epoch of the universe

$$a\sim t^{2/3}~~~~\mbox{for}~~~ z < z_{\rm eq}$$

So, if $a_0$ represents the scale factor today we have

$$\frac{a_0}{a_{\rm eq}} = \left(\frac{t_{0}}{t_{\rm eq}} \right)^{2/3} \tag{3}$$

Putting together (2) and (3) we arrive to

$$\frac{a_0}{a_{\rm end}} = \left(\frac{a_0}{a_{\rm eq}}\right)\left(\frac{a_{\rm eq}}{a_{\rm end}}\right) = \left(\frac{t_{0}}{t_{\rm eq}} \right)^{2/3} \left(\frac{t_{\rm eq}}{t_{\rm end}} \right)^{1/2} \approx 10^{26}$$

So the size of the universe has increased in a factor of around $10^{26}$ after inflation