Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 10D / c

Table of contents
- Solution c

Solution

For the radiation-era law beginning at

t_{r}

, the present particle-horizon estimate is

d_{H} (t_{0}) = 2 c t_{0} (1 - t_{r} / t_{0}) ≃ 2 c t_{0} ≃ 6 \times 1 0^{25} m .

Rescale this present physical distance back to

t_{r}

using

a (t_{r}) / a (t_{0}) = t_{r} / t_{0}

. Its physical size was

2 c t_{r} t_{0} ≃ 0.6 m

, of order one metre as required. The matched inflationary Hubble rate is

H = 1/ (2 t_{r}) ≃ 5 \times 1 0^{34} s^{- 1}

, with Hubble radius

c / H ≃ 6 \times 1 0^{- 27} m

.

If inflation produces

N

e-folds before

t_{r}

, this patch began with size

0.6 e^{- N}

metres. It fits within one Hubble radius if

e^{N} ≳ 1 0^{26}, N ≳ 26 lo g 10 ≃ 60.

This estimate uses the idealized radiation-only history stated in the question; substituting the actual matter-dominated history would change the numerical estimate.

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