SQUARE45

Let

d_{obs}

be the comoving distance between two points separated by the observed CMB temperature

T_{CMB}

. The particle horizon

d_H(t)

at the time of decoupling

t_{dec}

is defined by the integral:

d_H(t_{dec}) = a(t_{dec}) \times \frac{1}{a(t_{dec})} \times \text{integral}_{t_{initial}}^{t_{dec}} dt'

The Horizon Problem arises because the observed distance

d_{obs}

for regions separated by angular scale

\theta

(corresponding to

d_{obs} \neq 0

) exceeds the causal horizon

d_H(t_{dec})

calculated using the standard Big Bang model, implying that these regions could not have been in causal contact prior to

t_{dec}

Horizon Problem