I am trying to understand step-by-step Guth's famous paper "Inflationary universe: A possible solution to the horizon and flatness problems"

I am stuck at his discussion of the flatness problem. On eq. (2.13), p. 349 (from the link above), he estimates the present-day entropy contribution from photons as

$$S_{\gamma} > 3 \times 10^{85}. \tag{2.13}$$

I am not able to obtain the number on the right-hand side. Following Guth's notation, $S = R^3 s$, where $R$ is the scale factor and $s$ the entropy density. Again following Guth's computations,

$R > \frac13 H^{-1} = 10^9$ years $\approx 10^{17}$ seconds.

Putting back the $c$ to get the correct dimensions gives

$R > \frac13 \frac{c}{H}$ $\approx 3 \times 10^{25}$ meters.

The entropy density is given by eq. (2.6), p. 348

$$s = \frac{2 \pi}{45} \mathcal N T_{\gamma}^3 \approx 0.88 \times (2.7)^3 \approx 18\ ,\tag{2.6}$$

where $T_{\gamma}$ is the CMB temperature and $\mathcal N = 2$ are the photon's degrees of freedom. Again, the dimensions need to be adjusted as

$s = \frac{2 \pi}{45} \mathcal N T_{\gamma}^3 \frac{k_B^4}{\hbar^3 c^3} \approx 7 \times 10^{-15} \frac{\mathrm{joules}}{\mathrm{kelvin}}$

Putting the two together gives me

$S_{\gamma} = R^3 s > 27 \times 10^{75} \times 7 \times 10^{-15} = 1.9 \times 10^{62}$.

This estimate is so far off from Guth's estimate that I am convinced I am overlooking something very simple, but I cannot see it.


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