The Big Bang — Question 9

Question 9

The particle horizon is a history integral in a radiation universe

Assume $a(t)=A t^{1/2}$ from t=0 and define proper particle horizon $D_p(t)=a(t)\int_0^t c\,dt^\prime/a(t^\prime)$.

1. Evaluate the integral and compare with c times the cosmic age.

2. At t=100 s find Dp using c=$3.00\times10^8$ m/s, and compare with the Hubble radius in this model.

Question 9 - Solution

The particle horizon is a history integral in a radiation universe

(a) Method and calculation

$$D_p=At^{1/2}(c/A)\int_0^t(t^\prime)^{-1/2}dt^\prime.$$$$\boxed{D_p=2ct\ne ct}.$$

(b) Interpretation and further calculation

$$\boxed{D_p(100\ {\mathrm{s}})=6.00\times10^{10}\ {\mathrm{m}}}.$$$$H=1/(2t)\Rightarrow\boxed{c/H=2ct=D_p\ \text{for this particular history}}.$$

Physical check

The equality of the Hubble radius and particle horizon here is model-specific. The integral assumes the power law extends back to zero.