Discuss the below:
(1) Describe the concept of our past and future light-cone. Explain the meaning of the terms particle horizon distance, event horizon distance and world-line and discuss the difference between time-like and space-like locations.
(2) Show that in an Einstein-de Sitter Universe in which the scale factor α(t) at time t follows α(t) ∝t^(2/3), the particle horizon is at 3ct and the event horizon is at infinity.
(3) Suppose that the scale-factor were given by α(t) ∝ exp(mt) where m is a positive constant. Show that the event horizon is finite and that the particle horizon grows exponentially when t >> 1/m.
(4) Explain how such behaviour of the particle horizon might be useful in explaining observations of the cosmic microwave background.